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University of Southern California Dissertations and Theses
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Integration of truck scheduling and routing with parking availability
(USC Thesis Other)
Integration of truck scheduling and routing with parking availability
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In tegration of T ruc k Sc heduling and Routing with P arking A v ailabilit y b y Filip e de Almeida Araujo Vital A Dissertation Presen ted to the F A CUL TY OF THE GRADUA TE SCHOOL UNIVERSITY OF SOUTHERN CALIF ORNIA In P artial F ulfillmen t of the Requiremen ts for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) Ma y 2022 Cop yrigh t 2022 Filip e de Almeida Araujo Vital Dedication I dedicate this thesis to my family and friends for their c onstant supp ort and unc onditional love. ii A c kno wledgmen ts There are man y who help ed me along the w a y on this journey . I w an t to tak e a momen t to thank them. First, I wish to thank m y advisor, Prof. P etros Ioannou, for his con tin uous supp ort and patience during m y PhD study . I w ould also lik e to thank the mem b ers of m y qualification and dissertation committees, Prof. Maged Dessouky , Prof. Ash utosh Na yy ar, Prof. Rah ul Jain, and Prof. Ketan Sa vla, for their helpful feedbac k and guidance. I w ould also lik e to thank Shane Go o doff and Diane Demetras for their supp ort in na vigating life at USC o v er the past sev eral y ears. Y our presence alw a ys made the departmen t a more pleasan t, w elcoming and fun place. I also thank all m y labmates and the w onderful staff at the Electrical Engineering Departmen t for alw a ys b eing so helpful and friendly . T o m y friends, m y paren ts, and m y brother: y ou put up with me b eing far a w a y , distracted, and missing man y ev en ts. I am forev er grateful for y our patience and understanding. I w ould not b e here without y ou. iii T able of Con ten ts Dedication ii A c kno wledgmen ts iii List of T ables viii List of Figures x Abstract xv Chapter 1: In tro duction 1 1.1 Motiv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Con tribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: In telligen t T ruc k P arking Surv ey 8 2.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Bac kground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Sensing Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.1 Coun ting Metho ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.3 Comm unication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.4 Pilot Pro jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Resource Allo cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 Mo deling & Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Mo del Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Routing & Sc heduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Chapter 3: Assumptions 35 3.1 USA’s Hours of Service Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Information a v ailabilit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Lo w turno v er at nigh t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Occupancy Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 Driv ers ma y not b e allo w ed to rest or w ait at clien t lo cations . . . . . . . . . . . . . 38 iv Chapter 4: T ruc k Driv er Sc heduling Problem 39 4.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.1 P arking A v ailabilit y Constrain ts . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.2 W eekly Constrain ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Exp erimen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4.1 P arking A v ailabilit y Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4.2 Long T rips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Chapter 5: Shortest P ath and T ruc k Driv er Sc heduling Problem 55 5.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1.1 Scien tific con tributions and structure . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.1 P arking restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.2 P ath planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.4 Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.1 Extended Net w ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.2 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.4.3 Constrain ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.4.4 SPTDSP-P A F orm ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5 Lab el-Correcting Metho d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5.1 Ov erview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5.2 P artial Solution Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.5.3 Dominance R ules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.6.1 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.6.2 Estimating the cost of disregarding parking information . . . . . . . . . . . . 83 5.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.7 Randomized Net w orks Exp erimen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.7.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.7.2 Exp erimen t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.7.3 Exp erimen t 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Chapter 6: Long Haul Battery Electric T ruc k Planning 97 6.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3 Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3.1 Consumption Mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3.2 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4 Dynamic Programming F orm ulation and Rollout Algorithm . . . . . . . . . . . . . . 107 6.4.1 Constrain t Propagation and F easible Decision Space . . . . . . . . . . . . . . 108 6.4.2 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.3 Cost Lo w er Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4.4 Graph Prepro cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 v 6.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.6 Exp erimen ts on Random Net w orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.6.1 Without fast c hargers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.6.2 F ast c hargers with fixed w ait time . . . . . . . . . . . . . . . . . . . . . . . . 134 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Chapter 7: Optimizing Long Haul T ruc ks’ P ollutan t Emissions under Sto c hastic P arking A v ailabilit y 139 7.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.1.1 Related W ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.2 Mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3 P arking A v ailabilit y Uncertain t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.3.1 Recourse A ctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3.2 P olicy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3.3 Imp erfect Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4 Exp erimen ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.4.1 Static Net w ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.4.2 Time-Dep enden t Net w ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4.3 Uncertain P arking A v ailabilit y . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Chapter 8: Balancing P arking Demand 165 8.1 In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.2 Related W ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.2.1 Demand-Side Managemen t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.2.2 Direct Allo cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.2.3 Indirect Allo cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.2.4 Significan t differences to truc k parking . . . . . . . . . . . . . . . . . . . . . . 171 8.3 Ov erview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.4.1 Non-co op erativ e Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.5 F orm ulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.5.1 Agen t in teraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.5.2 Individual Beha vior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 8.5.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8.6 Demand Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.6.1 Sensitivit y to HOS conditions and uniform time slot prices . . . . . . . . . . 186 8.6.2 Resp onse to price c hanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Chapter 9: Concluding Remarks and Prop osed Researc h Directions 196 9.1 Prop osed Researc h Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Bibliograph y 200 vi App endix A TDSP Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 A.1 Time-dep enden t T ruc k Driv er Sc heduling Problem . . . . . . . . . . . . . . . . . . . 216 A.2 Multiple Clien ts with Service Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 A.2.1 W eekly Constrain ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 A.3 Electric V ehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 App endix B T rip Duration Lo w er-Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 App endix C Dominance R ules Deriv ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 C.1 Equal time, no slac k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 C.2 Equal time, non-zero slac k, after impro v emen t . . . . . . . . . . . . . . . . . . . . . . 231 C.3 Differen t time, non-zero slac k, b efore impro v emen t . . . . . . . . . . . . . . . . . . . 232 C.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 App endix D Optimalit y Pro of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 App endix E Lab el Impro v emen t Pseudo co de . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 App endix F P ath Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 App endix G Chapter 6 Exp erimen t Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 App endix H Chapter 7 Exp erimen t Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 H.1 Static Net w orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 H.1.1 Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 H.1.2 Lo w er Bound Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 H.2 Time-dep enden t Net w orks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 H.2.1 Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 H.2.2 Lo w er b ound impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 H.3 Sto c hastic Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 vii List of T ables 2.1 Pilot Pro jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 T ruc k P arking related w ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1 V ariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Resource Extension F unctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Exp erimen t P arameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Exp erimen t Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 P arking Shortage Lev el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5 Random Net w orks Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.6 A v erage T rip Duration Increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1 Mo del P arameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 Resource Extension F unctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.3 Solution Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Exp erimen t P arameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.1 V ariables and P arameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 8.2 P opulation P arameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.3 Exp erimen t Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 C.1 Effect of decision with duration u at rest no des . . . . . . . . . . . . . . . . . . . . . 233 H.1 Static Net w ork Scenarios: CO2 Emissions . . . . . . . . . . . . . . . . . . . . . . . . 255 H.2 Static Net w ork Scenarios: Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 H.3 Static Net w ork Scenarios: Nonp enalized Cost . . . . . . . . . . . . . . . . . . . . . . 258 H.4 Static Net w ork Scenarios: R unning Time . . . . . . . . . . . . . . . . . . . . . . . . 259 H.5 Time-dep enden t Scenarios: CO2 Emissions . . . . . . . . . . . . . . . . . . . . . . . 262 H.6 Time-dep enden t Scenarios: Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 viii H.7 Time-dep enden t Scenarios: Nonp enalized Cost . . . . . . . . . . . . . . . . . . . . . 265 H.8 Time-dep enden t Scenarios: R unning time . . . . . . . . . . . . . . . . . . . . . . . . 266 H.9 Sto c hastic Scenarios 1-3: A v erage Duration/Emissions/Cost . . . . . . . . . . . . . . 269 H.10 Sto c hastic Scenarios 1-3: P arking-related P erformance . . . . . . . . . . . . . . . . . 270 H.11 Sto c hastic Scenarios 4-6: A v erage Duration/Emissions/Cost . . . . . . . . . . . . . . 271 H.12 Sto c hastic Scenarios 4-6: P arking-related P erformance . . . . . . . . . . . . . . . . . 273 ix List of Figures 2.1 MSE of the o ccupancy estimate for target time 20:45 with v arying prediction horizons 30 2.2 Classification p erformance (Sp ecificit y/ Sensitivit y) for all parking lots for target time 20:45 with v arying prediction horizons . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Classification p erformance (Y ouden Index) for all parking lots for target time 20:45 with v arying prediction horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Time of Da y When Driv ers Exp erienced Most Difficult y in Finding Safe P arking. A T A = American T ruc king Asso ciations; OOID A = Owner Op erator Indep enden t Driv ers Asso ciation. Source: [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 T w en t y-four Hours P arking A ccum ulation Profile. Source: [1] . . . . . . . . . . . . . 38 4.1 Simple route with 5 lo cations (origin, 3 rest areas and destination) with 3 time- windo ws eac h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Route used on short trip exp erimen t. San Diego to Seattle through the I-5 freew a y . The triangles (base mo del) and+ s (new mo del) represen t truc k stops c hosen for daily rests, and the square (base mo del) and (new mo del) represen t the ones c hosen for short breaks. The gra y circles represen t the truc k stops near the c hosen route. . . . . 49 4.3 F easibilit y rate of the sc hedules generated without considering the parking con- strain ts. The feasibilit y rate of sc hedules that consider parking constrain ts is alw a ys 100%, so it w as omitted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.4 A v erage trip duration of sc hedules generated with and without parking constrain ts. 51 4.5 A v erage trip duration of sc hedules calculated b y the 3 metho ds with v arying total driving time, and the lo w er b ound of solutions that use the rolling time-windo w. The v ertical dotted line marks the on-dut y time w eekly limit (60h) and the horizon tal dotted line represen ts a trip duration of 1 w eek (168h). . . . . . . . . . . . . . . . . 52 4.6 Solv e time of the 3 presen ted metho ds, with v arying n um b er of lo cations a nd total tra v el distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.1 Example of simplified road net w ork. The no des with a n um b er index are road no des (in tersections, branc hing or merging sp ots) and the ones with a letter index are TPLs. 63 5.2 Sub-net w orks used to mo del non-driving activities. . . . . . . . . . . . . . . . . . . . 65 5.3 Lab el-correcting algorithm w orkflo w diagram. . . . . . . . . . . . . . . . . . . . . . . 71 x 5.4 Net w ork used for exp erimen ts. Based on a main route going from San Diego to Seattle via the I-5 freew a y indicated in red with double arro ws, together some p ossible detours indicated with blac k arro ws. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 A v erage trip cost (when disregarding parking information) for differen t irregular parking p enalties according to the t yp e of parking a v ailabilit y time-windo ws con- sidered for the main route. In this exp erimen t, the results did not v ary with the alternativ e routes’ tra v el sp eed so the plots for other sp eeds w ere omitted. . . . . . . 86 5.6 A v erage trip cost/duration (when using parking information for planning) accord- ing to parking a v ailabilit y time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.7 Num b er of instances that used alternativ e routes for the single clien t case, according to the time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. . . . . 87 5.8 A v erage driving time of solutions for the single clien t case, according to the time- windo ws (main route) and tra v el sp eed (alternativ e routes) used. . . . . . . . . . . . 88 5.9 Num b er of instances that used alternativ e routes for the t w o clien ts case, according to the time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. . . . . 89 5.10 A v erage driving time of solutions for the t w o clien ts case, according to the time- windo ws (main route) and tra v el sp eed (alternativ e routes) used. . . . . . . . . . . . 89 5.11 A v erage running time o v er randomized net w orks. . . . . . . . . . . . . . . . . . . . . 92 5.12 A v erage running time for instances with v arying n um b er of clien ts and cost (trip duration in hours). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.13 A v erage running time for instances with v arying driving and off-dut y time. . . . . . . 95 5.14 A v erage running time for instances with v arying n um b er of clien ts and off-dut y time ratio (off-dut y time/trip duration). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 The green and bro wn regions are examples of p ossible feasible regions in a 2D space. The figures sho w ho w the exact (6.1a) and appro ximate (6.1b) feasible spaces are calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Example graph fo cusing on the road net w ork. F o cuses on rest area (no des with letter indexes) placemen t along main roads. Easy to visualize but has a large n um b er of in termediate no des (no des with n um b er indexes). . . . . . . . . . . . . . . . . . . . . 122 6.3 Stop-based graph generated from Figure 6.2 to fo cus on the connection b et w een p ossible stops (rest areas, clien ts, origin, destination). Eac h p ossible stop is directly connected to do wnstream stops satisfying predetermined conditions. Dashed arro ws exemplify edges that could b e remo v ed for b eing to o short or to o long. . . . . . . . . 122 6.4 Net w ork used for exp erimen ts. Arc lengths are giv en in kilometers. . . . . . . . . . 123 6.5 T rip duration under differen t parking a v ailabilit y and c harging infrastructure condi- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 xi 6.6 CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure con- ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.7 T rip feasibilit y under differen t parking a v ailabilit y and c harging infrastructure con- ditions for scenarios with 600k Wh battery capacit y . . . . . . . . . . . . . . . . . . . . 127 6.8 T rip duration under differen t parking a v ailabilit y and c harging infrastructure condi- tions for scenarios with 600k Wh battery capacit y . . . . . . . . . . . . . . . . . . . . . 127 6.9 CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure con- ditions for scenarios with 600k Wh battery capacit y . . . . . . . . . . . . . . . . . . . . 127 6.10 A v erage trip duration under differen t parking a v ailabilit y and c harging infrastructure conditions. Includes only scenarios that allo w sp eed reduction. . . . . . . . . . . . . 130 6.11 CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure con- ditions. Includes only scenarios that allo w sp eed reduction. . . . . . . . . . . . . . . 131 6.12 T rip cost in US dollars under differen t parking a v ailabilit y and c harging infrastruc- ture conditions. Includes only scenarios that allo w sp eed reduction. . . . . . . . . . . 132 6.13 P ercen tage of instances that w ere feasible under differen t parking a v ailabilit y and c harging infrastructure conditions. All diesel truc k scenarios w ere feasible. . . . . . . 133 6.14 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, and without fast c hargers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.15 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 50k W fast c hargers, and 1h w ait. . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.16 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 100k W fast c hargers, and 1h w ait. . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.1 Subgraph represen ting the actions that can b e tak en at rest areas after inclusion of recourse actions and alternativ e parking lo cations. . . . . . . . . . . . . . . . . . . . 148 7.2 CO2 emissions as a fraction of the baseline emission. The baseline emission for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 152 7.3 T rip duration as a fraction of the baseline duration. The baseline duration for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 152 7.4 Nonp enalized cost as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 153 7.5 R unning time as a fraction of the baseline cost. The baseline running time for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 153 7.6 CO2 emissions as a fraction of the baseline emission. The baseline emission for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 155 7.7 T rip duration as a fraction of the baseline duration. The baseline duration for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 155 xii 7.8 Nonp enalized cost as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . 156 7.9 R unning time as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. . . . . . . . . . . . . . . . . . . . . 156 7.10 F unction used to define the probabilit y of finding parking at a giv en rest area and time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.11 Probabilit y distribution of CO2 emissions, trip duration and trip cost (with CO2 p enalt y parameter set to 1) of the decision p olicies obtained for net w ork 5. . . . . . 160 7.12 Probabilit y distribution p erformance measures related to illegal parking for net w orks 0,2,4 and 5, with CO2 p enalt y set to 1. . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.13 Probabilit y distribution of CO2 emissions, trip duration and trip cost (with CO2 p enalt y parameter set to 1) of the decision p olicies obtained for net w ork 5. . . . . . 162 7.14 Probabilit y distribution p erformance measures related to illegal parking for net w orks 0,2,4 and 5, with CO2 p enalt y set to 1. . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.1 Simplified represen tation of ho w parking demand is generated. Green b o xes represen t information that is unique to the system, regardless of whether it can b e measured. Red b o xes represen t information that is sp ecific to eac h compan y . The red horizon tal lines on the parking demand plots represen t eac h facilit y’s maxim um capacit y . . . . . 173 8.2 System Diagram. The blue b o xes represen t our system’s comp onen ts and output. The red b o x represen ts the distributed system encompassing all truc k companies and their decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.3 Distribution of the parking demand generated b y 23h trips without parking c harges and with HOS initial condition restricted to 0,1. . . . . . . . . . . . . . . . . . . . . 187 8.4 Distribution of the parking demand generated b y 23h trips with hourly op erational cost in the in terv al [60;80] and with eac h HOS initial condition v arying b et w een 0 and its regulation limit min us 1h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 8.5 Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. . . . . . . . 192 8.5 Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. . . . . . . . 193 8.5 Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. . . . . . . . 194 F.1 T rip duration, in hours, when only the edges of the k shortest paths are included in the net w orks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 F.2 R unning time, in seconds, when only the edges of the k shortest paths are included in the net w orks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 F.3 Solution’s driving time, in hours, when only the edges of the k shortest paths are included in the net w orks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 G.1 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 50k W fast c hargers, and 0.5h w ait. . . . . . . . . . . . . . . . . . . . . . . . . . 251 xiii G.2 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 100k W fast c hargers, and 0.5h w ait. . . . . . . . . . . . . . . . . . . . . . . . . 252 G.3 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 50k W fast c hargers, and 2h w ait. . . . . . . . . . . . . . . . . . . . . . . . . . . 253 G.4 A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging sta- tions, 100k W fast c hargers, and 2h w ait. . . . . . . . . . . . . . . . . . . . . . . . . . 254 H.1 Histogram of the running time of sim ulations p erformed with b ounds 1 and 2 sepa- rated b y the p enalt y m ultiplier used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 H.2 Cum ulativ e distribution of the ratio b et w een results (left: running time, righ t: ob- jectiv e function) obtained with b ounds 2 and 1. . . . . . . . . . . . . . . . . . . . . . 262 H.3 Histogram of the running time of sim ulations p erformed with b ounds 1 and 2 sepa- rated b y the p enalt y m ultiplier used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 H.4 Cum ulativ e distribution of the ratio b et w een results (left: running time, righ t: ob- jectiv e function) obtained with b ounds 2 and 1. . . . . . . . . . . . . . . . . . . . . . 269 xiv Abstract A ccording to the American T ransp ortation Researc h Asso ciation surv eys, truc k parking is cur- ren tly one of the truc king industry’s main issues. Due to hours-of-service (HOS) regulations, commercial driv ers are required to stop and rest regularly , th us reducing fatigue-related crashes. Ho w ev er, as the parking infrastructure cannot cop e with the demand generated b y these required stops, new issues arise. First, not finding appropriate rest lo cations ma y cause driv ers to w ork b e- y ond the allo w ed time limit, increasing the c hances of fatigue-related crashes. Second, driv ers ma y c ho ose to park illegally (road shoulders, ramps, abandoned lots, etc.), p osing safet y risks. Third, truc k parking shortage ma y adv ersely affect industry costs in m ultiple w a ys, suc h as increased fuel consumption due to idling or lo oking for parking and higher acciden t-related costs and insurance premiums. F ourth, the increase in fuel consumption will also negativ ely affect the en vironmen t. T ruc ks are resp onsible for a significan t p ortion of the U.S.A. ’s greenhouse gas emissions, and emis- sions generated b y truc k idling can cause substan tial deterioration of the surrounding region’s air qualit y . This dissertation prop oses incorp orating truc k parking a v ailabilit y information early in the planning pro cess to b etter utilize the installed parking infrastructure. By promoting a b etter dis- tribution of truc k parking demand o v er time and space, w e aim to mitigate truc k parking shortages and ensure that driv ers do not find themselv es in a situation where they are required to stop at a region without appropriate parking lo cations. First, as the usage of parking a v ailabilit y infor- mation is the fo cus of this w ork, w e p erformed a surv ey on the curren t status of in telligen t truc k parking systems. Although truc k parking information is not ubiquitous, progress is b eing made in dev eloping the tec hnology necessary for the implemen tation of in telligen t truc k parking systems, and some pilot sites are already in place. Second, w e prop ose a sc heduling mo del that accoun ts for truc k parking a v ailabilit y information b y using time-windo w constrain ts to mo del when parking xv is exp ected to b e a v ailable. In addition, w e prop ose a form ulation for the U.S.A. ’s HOS rules for trips longer than one w eek, whic h are usually neglected in the literature. W e study ho w parking a v ailabilit y time-windo ws affect problem feasibilit y and trip duration. Third, w e incorp orate path planning in to the previous sc heduling problem and prop ose a resource-constrained shortest path problem (R CSPP) form ulation along with a tailored lab el-correcting algorithm to find an optimal solution. W e estimate the cost of disregarding parking information b y sim ulating driv ers’ b eha vior when parking is una v ailable and applying time and cost p enalties. The effect of parking a v ailabilit y and alternativ e routes on trip costs is also studied, aiming to illustrate that disregarding these factors can lead to significan t errors in cost estimates. Although imp osing parking constrain ts can significan tly increase costs, prev en ting a cciden t-related costs can mak e it adv an tageous in the long term. W e then extended the R CSPP form ulation to study other problem v arian ts fo cusing on battery electric truc ks (BET), emissions reduction, and sto c hastic parking a v ailabilit y . W e studied the impact of co ordinating rest and rec harge needs on BET s’ p erformance compared to diesel truc ks. Computational exp erimen ts w ere used to estimate the effects of differen t lev els of c harging and parking infrastructure. When studying the trade-offs b et w een prioritizing emissions reduction or trip duration, w e found that, although fo cusing on emissions reduction can increase trip duration significan tly , this impact is greatly reduced when considering scenarios with limited parking a v ail- abilit y . When considering sto c hastic parking a v ailabilit y , w e studied ho w solutions are affected b y the lev el of information pro vided to driv ers. W e found that ignoring uncertain t y in parking a v ailabilit y results in inconsisten t p erformance ev en when restricting parking to p erio ds when the probabilit y of finding parking is high. F urthermore, results migh t not reflect the in ten t of the cost function used, e.g., minimizing illegal parking ev en ts or the priorit y assigned to emissions reduction. Finally , w e presen t preliminary w ork on the issue of ho w T ruc k Driv er Sc heduling Problem (TDSP) mo dels can b e used to aid in redistributing truc k parking demand. W e presen t a form ula- tion for the problem that uses parking prices to influence driv er b eha vior, using a mo dified TDSP mo del to sim ulate ho w driv ers w ould react to price c hanges. The TDSP with parking costs is v ery sensitiv e to parking rates, but price up dates can cause the system to oscillate, aggra v ating parking issues b efore sho wing an y impro v emen ts. The dev elopmen t of pricing mec hanisms able to a v oid xvi suc h oscillations and concerns regarding system optima and desired prop erties will b e left for future w ork. This dissertation exp osed the imp ortance of using truc k parking a v ailabilit y information during planning and prop osed metho ds to do so. Besides helping individual truc k driv ers with trip plan- ning, the metho ds dev elop ed in this pro ject can sim ulate differen t scenarios and aid p olicymak ers in estimating the impacts of p olicy and infrastructure in v estmen t decisions. xvii Chapter 1 In tro duction It w as estimated that, in the y ear of 2013, truc ks w ere resp onsible for carrying around 70% (in w eigh t) of USA’s total freigh t shipmen ts, without considering m ultimo dal shipmen ts that use truc ks at some p oin t [2]. It is exp ected that this v alue will still b e as high as 66% b y the y ear of 2040, despite substan tial increases in m ultimo dal and rail shipmen ts [2]. This sho ws just ho w imp ortan t truc ks are to the USA econom y . Ho w ev er, the increasing demand for truc ks comes with a need for supp orting infrastructure and legislation, in particular, appropriate parking infrastructure. In 2015, a surv ey b y the F ederal High w a y A dministration iden tified truc k parking shortages in 36 US states, with more pronounced issues along ma jor trade corridors [1]. The lac k of truc k parking can ha v e significan t impact on road safet y , industry costs, and the en vironmen t [3, 4], and rank ed among the truc king industry’s top concerns in recen t surv eys b y the American T ransp ortation Researc h Institute (A TRI) [5]. As expanding the infrastructure w ould require significan t capital in v estmen t, this issue p oin ts to the need for b etter utilization of the existing truc k parking capacit y . One of the requiremen ts for efficien t utilization of the truc k parking capacit y is ha ving enough information ab out the truc k parking system. Recen tly , the topic of smart parking systems has b ecome increasingly p opular, with researc h on parking o ccupancy sensing and information distri- bution systems for b oth passenger v ehicles and truc ks [6, 7]. Nev ertheless, driv ers’ direct usage of o ccupancy information can only help with last min ute adjustmen ts to the sc hedule and migh t not b e as helpful in high-demand areas where all rest areas migh t b e full during p eak hours. P arking a v ailabilit y information could b e used more effectiv ely if included in the earlier stages of planning, when the route and sc hedule are decided. 1 Ideally , an in tegrated planning system with access to data on the lo cation and predicted a v ail- abilit y of all rest areas w ould b e able to c ho ose an optimal route and sc hedule, suc h that: w orking hours regulations are satisfied; and off-dut y time is sc heduled only at lo cations that are exp ected to ha v e a v ailable parking at the time of arriv al. This t yp e of system has y et to b e dev elop ed and shall b e the sub ject of this researc h. 1.1 Motiv ation Due to the Hours-of-Service (HOS) regulations, truc k driv ers are required to tak e regular breaks. These rules aim to ensure that driv ers tak e adequate rest and a v oid fatigue-related acciden ts. Ho w ev er, when appropriate parking is scarce, driv ers ma y find themselv es ha ving to c ho ose b et w een driving b ey ond the legal limits or parking in unauthorized and often unsafe lo cations, suc h as high w a y shoulders and freew a y ramps. In recen t surv eys, most driv ers rep orted using unauthorized parking lo cations at least once a w eek [8, 1, 3, 9]. As truc k crashes can b e v ery costly [10, 11], suc h practices ma y lead to significan t losses for the truc king industry due to p oten tial acciden ts. A ccording to a study b y the Virginia Departmen t of T ransp ortation, 25% of all truc k-related crashes along Virginia’s ma jor corridors o ccurred on en trance and exit ramps [12]. Although the data is not sp ecific to park ed truc ks, it sho ws that parking on ramps p oses a significan t safet y risk. T ruc k insurance premiums ha v e increased in recen t y ears and curren tly represen t 5% of the a v erage marginal op erational cost of truc king (not including w ork ers comp ensation costs/insurance, ph ysical damage, jury a w ards, and out-of-court settlemen ts) [13]. This is in part due to recen t increases in w ork ers comp ensation claims, settlemen ts, and jury a w ards, whic h at times can surpass $10 million [14, 15]. These gro wing financial risks push b oth truc king and insurance companies to reev aluate safet y and ho w m uc h risk they are willing to tak e. A ccording to A TRI [15], the increase in insurance costs hits small carriers particularly hard, and is considered a primary reason for closing. T ruc k parking shortage costs go b ey ond safet y-related ones [4, 8]. Surv eys rep ort that driv ers often sp end more than 30 min utes lo oking for parking [16, 17]. Although a surv ey b y A TRI [8] rep orted that most surv ey ed driv ers sp en t less than 15 min utes lo oking for parking, the driv ers had, on a v erage, park ed one hour earlier than required b y the regulation, whic h also con tributes to decreasing their daily rev en ue-earning miles. Stopping for rest early or sp ending a long time 2 lo oking for parking is inefficien t use of driv er’s time. In particular, A TRI estimated that stopping on a v erage one hour earlier than necessary could represen t a reduction of 9300 rev en ue-earning miles p er y ear [8]. As most truc k driv ers are not paid b y the hour, this can ha v e a significan t impact on their comp ensation [8]. Sp ending long times lo oking for parking also means higher fuel consumption, increasing b oth costs and emissions. Besides, as driv ers often idle the truc k’s engine to p o w er their appliances, when truc ks end up parking near residen tial areas, the emissions generated can significan tly affect the region’s air qualit y [18]. The inclusion of HOS rules in sc heduling algorithms, the truc k driv er sc heduling problem (TDSP), w as approac hed in man y studies in recen t y ears, either as the main fo cus of the study [19, 20, 21, 22, 23, 24] or as part of a v ehicle routing and truc k driv er sc heduling problem (VR TDSP) [25, 26, 27, 28, 29, 30, 31], whic h is a v arian t of the v ehicle routing problem (VRP). In [19], Arc hetti et al. considered the problem of determining whether a sequence of n full truc kload transp orta- tion requests is feasible giv en a set of HOS regulations and pic k-up time-windo ws. The prop osed metho d allo ws driv ers to park an ywhere and finds a feasible sc hedule in O(n 3 ) time. In [20], Go el considered a similar problem using the Europ ean regulations, and in [22], presen ted an algorithm to find feasible sc hedules to visit n lo cations using the US regulations in O(n 2 ) time. Ho w ev er, these metho ds assumed that driv ers could park an ywhere, whic h is not v alid in practice. This assump- tion is also presen t in [32, 27, 28, 29, 21]. In [23], Go el presen ted a mixed in teger programming form ulation for the TDSP that restricts parking to clien t lo cations and calculates a sc hedule with minim um trip duration. Rest areas w ere mo deled as clien ts with zero service time and un b ounded time-windo ws. Similar mo dels w ere used in [26, 24], fo cusing on differen t asp ects of the problem. In [26], K ok et al. addressed the issue of traffic congestion b y considering time-dep enden t tra v el times and prop osed a heuristic approac h to in tegrate the TSDP mo del in to a VRP metho d. In [24], K o ç et al. approac hed the en vironmen tal impact caused b y truc k idling and ho w it is affected b y the truc k’s equipmen t and rest areas’ infrastructure. The driv ers can only stop at rest areas, whic h ha v e differen t t yp es of infrastructure a v ailable. The cost function accoun ts for the t yp e of idling used in eac h stop giv en the equipmen t installed on the truc k and the infrastructure a v ailable at eac h rest area. This metho d w as later used as a base for a VR TDSP algorithm with the same fo cus [31]. In [30], Gaddy et al. prop osed a VRP algorithm where only a subset of the clien ts allo w ed parking, but the ones that did w ere alw a ys a v ailable. 3 Although some progress has b een ac hiev ed in in tegrating HOS rules in to trip planning and taking in to accoun t existen t rest areas, up to no w only the static information of the rest areas’ existence w as considered. The fact that parking facilities ma y b e una v ailable at certain times is ignored in the literature. W e b eliev e that, giv en the curren t shortage in truc k parking, this line of researc h has p oten tial to impro v e the planning efficiency of the truc king industry and hop efully the w orking conditions of truc k driv ers. 1.2 Problem Description An imp ortan t asp ect of truc k parking a v ailabilit y is its time dep endence. Although driv ers often rep ort difficulties finding parking and truc k stops rep ort op erating o v ercapacit y , they usually refer to the p erio d b et w een 7 PM and 5 AM when driv ers are lo oking for o v ernigh t parking [1, 17, 8, 9]. This suggests that it ma y b e p ossible to mitigate the truc k parking shortage b y encouraging driv ers to plan their stops on off-p eak p erio ds, th us balancing the demand. Balancing the demand with resp ect to space ma y also b e an option. Sev eral factors influence driv ers’ c hoice of parking lo cation; ho w ev er, lac k of information can lead driv ers to park illegally ev en when there are facilities with a v ailable parking nearb y [3]. In [1], less than 50% of truc k stops rep orted w orking o v ercapacit y . Man y of the facilities op erating under capacit y ma y b e in regions with lo w parking demand, y et, some ma y b e in high-demand areas and are underutilized b ecause driv ers do not kno w they are viable options. With this in mind, it is w orth considering to include parking a v ailabilit y earlier in the planning pro cess. Instead of only pro viding driv ers information on facilities along their path to allo w for on-trip decisions, if parking information is considered when c ho osing the path itself, the existing parking capacit y could b e b etter utilized. Dep ending on the situation, it ma y b e adv an tageous for the driv er to tak e a longer route if that can guaran tee a v ailable parking. This suggestion of addressing parking issues at the supply c hain lev el has b een brough t up b efore in [8, 33]; ho w ev er, literature on the topic is still scarce. Previous studies fo cused on dev eloping metho ds to estimate truc k parking demand, predict parking o ccupancy , and measure parking o ccupancy for parking managemen t systems. Some pilot pro jects distribute real-time a v ailabilit y information through w ebsites or v ariable-message signs on nearb y high w a ys. Ho w ev er, judging ho w to b est use the 4 information, if at all, is up to the driv ers. These systems can only assist driv ers with small on- trip sc hedule adjustmen ts as using all this information to generate regulation-complian t itineraries is not trivial, and most systems do not include an o ccupancy prediction feature either. On the other hand, the literature on regulation-complian t routing and sc heduling do es not tak e parking a v ailabilit y in to accoun t, generating the main researc h gap addressed in this dissertation. Another in teresting topic to b e considered is ho w HOS regulations and parking shortages affect battery electric truc ks (BET). Among the concerns regarding hea vy-dut y v ehicles electrification are the reduced range, longer rec harge time and the reduction in maxim um pa yload due to added battery w eigh t. Ho w ev er, it is often o v erlo ok ed that commercial driv ers are required to stop and rest regularly regardless of the v ehicle’s range. Ev en if long-haul truc ks driv e on a v erage 600 miles p er da y [34], they are required to stop for at least 30 min utes ev ery 8 hours and for at least 10 hours after 11 hours of driving time. A dequate infrastructure and efficien t trip planning can mitigate the range issue b y rec harging the v ehicle during mandatory stops. Nev ertheless, rec harging do es tak e longer than refueling and c harging stations are not abundan t, making c harging station a v ailabilit y a p oten tial issue for BET s. T ruc k parking is already considered a critical issue for the truc king industry , and the similar issue of rec harge station a v ailabilit y and electric v ehicles’ range are obstacles to the adoption of hea vy-dut y BET s. Therefore, this dissertation studies the follo wing questions: 1. Can parking a v ailabilit y information b e obtained? What is the curren t state of truc k parking managemen t systems? 2. Ho w to in tegrate parking a v ailabilit y information in to the planning of truc k driv ers’ itineraries? 3. Ho w are trip cost and duration affected b y stricter parking restrictions? Is it financially in teresting to the industry? 4. Do es the com bination of w orking hour regulations and parking restrictions mitigate the dis- adv an tages of battery electric truc ks relativ e to diesel truc ks? 5. Ho w can w e use parking-a w are planning systems to balance truc k parking demand within a region? T o answ er the ab o v e questions, this dissertation in tro duces the ‘T ruc k Driv er Sc heduling Prob- lem under P arking A v ailabilit y Constrain ts’ (TDSP-P A) and studies m ultiple of its v arian ts. 5 1.3 Con tribution Motiv ated b y the questions in section 1.2, this dissertation fo cuses on the design and ev aluation of planning metho ds that use parking a v ailabilit y information to pro vide safer and more realistic itineraries for truc k driv ers. The main con tributions of this w ork are the follo wing: • W e in tro duce and prop ose an MIP form ulation for the truc k driv er sc heduling problem under parking a v ailabilit y constrain ts (TDSP-P A). W e also presen t an extension to consider the USA’s regulation for long trips, whic h is usually left out in the TDSP literature. T o the exten t of our kno wledge, this w as the first w ork to consider rest areas with time-dep enden t parking a v ailabilit y and the rolling time windo w nature of the USA HOS regulation for long trips in the TDSP; • W e prop ose a v ersatile resource-constrained shortest path problem (R CSPP) form ulation to the shortest path and truc k driv er sc heduling problem under parking a v ailabilit y constrain ts (SPTDSP-P A) along with a tailored lab el-correcting algorithm that solv es the problem to optimalit y . W e used the prop osed mo del to study the impact of parking conditions on route c hoice, cost and feasibilit y . The cost/duration increase caused b y imp osing parking a v ailabilit y constrain ts can b e seen as an estimate of ho w m uc h driv ers and companies w ould need to sp end in order to ensure safe itineraries for the driv ers, and prev en t acciden ts and other costs related to the difficult y in finding appropriate rest lo cations. • W e analyze the impact that parking constrain ts ha v e on trip duration and compare it to estimated p oten tial costs of disregarding parking during planning. W e estimate the cost of disregarding parking information under differen t parking shortage sev erit y lev els b y sim ulating driv ers’ b eha vior when parking is una v ailable. Dep ending on the illegal parking p enalties considered, the cost sa vings from prev en ting said p enalties can exceed, or at least offset, the cost increase caused b y parking constrain ts. Due to the high direct and indirect costs of truc k-related acciden ts, w e b eliev e that these p enalties are lik ely high, and that safer parking practices w ould b enefit the truc king industry not only from a safet y standp oin t, but also from an economic standp oin t. • W e extend the R CSPP form ulation to v arian ts requiring energy/fuel consumption manage- men t, i.e. battery electric truc k (BET) planning and emissions reduction. The extended mo dels w ere used to study the follo wing topics: – the impact of co ordinating rest and rec harge needs on BET s’ p erformance and its com- parison to diesel truc ks; – the effects of parking and c harging infrastructure conditions on BET s’ p erformance; – the trade-offs b et w een prioritizing emissions reduction or trip duration during planning, and ho w they are affected b y parking conditions; • W e extend the R CSPP form ulation to consider sto c hastic parking a v ailabilit y at eac h rest area and mo del driv ers p ossible recourse actions when unable to find parking. W e study ho w solutions are affected b y the lev el of information pro vided to driv ers/planners. W e found that ignoring uncertain t y in parking a v ailabilit y results in inconsisten t p erformance ev en when restricting parking to p erio ds when probabilit y of finding parking is high. F urthermore, results migh t not reflect the used cost function’s in ten t, e.g., minimizing illegal parking ev en ts and/or the priorit y assigned to emissions reduction. 6 1.4 Thesis Structure The thesis is organized as follo ws: Chapter 2 presen ts a surv ey on the curren t state of in telligen t truc k parking infrastructure. Chapter 3 presen ts assumptions that are used in the subsequen t c hapters. Chapters 4 and 5 in tro duce the truc k driv er sc heduling problem under parking a v ailabilit y constrain ts without and with path planning capabilities. Chapter 6 extends the problem to the case of battery electric truc ks and studies the effect of parking and c harging infrastructure on trip duration. Chapter 7 studies the trade-offs b et w een prioritizing emissions reduction or trip costs, extends the form ulation to accoun t for uncertain parking a v ailabilit y , and studies ho w the accuracy of the information pro vided to planners/driv ers affects p erformance. Chapter 8 presen ts preliminary w ork on balancing truc k parking demand b y utilizing TDSP mo dels to estimate parking prices’ impact on driv er b eha vior. Chapter 9 presen ts the conclusion and future researc h directions. 7 Chapter 2 In telligen t T ruc k P arking Surv ey Chapter based on the publication: • F. de Almeida Araujo Vital, P . Ioannou, and A. Gupta, “Surv ey on In telligen t T ruc k P arking: Issues and Approac hes,” IEEE In telligen t T ransp ortation Systems Magazine, 2020. 2.1 In tro duction 2.1.1 Bac kground It w as estimated that, in the y ear of 2013, truc ks w ere resp onsible for carrying around 70% (in w eigh t) of USA’s total freigh t shipmen ts, without considering m ultimo dal shipmen ts that use truc ks at some p oin t [2]. It is exp ected that this v alue will still b e as high as 66% b y the y ear of 2040, despite substan tial increases in m ultimo dal and rail shipmen ts [2]. This sho ws just ho w imp ortan t truc ks are to the USA econom y . Ho w ev er, the increasing demand for truc ks comes with a need for supp orting infrastructure and legislation, in particular, appropriate parking infrastructure. A surv ey b y the American T ransp ort Researc h Institute ( A TRI ) determined the top issues in the truc king industry , among whic h are the Hours-of-Service ( HOS ) rules, Compliance, Safet y and A ccoun tabilit y scores and T ruc k P arking [35]. Ov er the past y ears some states ev aluated their truc k parking a v ailabilit y and the impact of shortages in parking lo cations. The state of California is one of the states with the largest n um b er of parking spaces. Ho w ev er, due to the large high w a y net w ork and hea vy truc k traffic, these parking spaces are to o sparse compared to the real necessities of the state. As of 2000, California had estimated the state’s total n um b er of parking spaces as 8600, whic h is 38% of the estimated 8 demand of 22700 parking spaces [36]. Similarly , a 2015 rep ort b y the Virginia Departmen t of T ransp ortation calculated a statewide deficit of nearly 5000 parking spaces, whic h means that the state only satisfies around 60% of the calculated demand (12500 spaces) [12]. A ccording to the U.S. Departmen t of T ransp ortation ( USDOT), 36 states are exp eriencing shortages in rest areas, either public or priv ate, whic h negativ ely affect truc k parking [1]. During a surv ey , a large p ercen tage of driv ers rep orted difficult y in finding safe parking from 4PM to 5AM, while less than 10% rep orted difficult y from 5AM to 4PM [1]. Ho w ev er, another surv ey sho w ed that less than 50% of truc k stops op erate o v ercapacit y from 7PM to 5AM [1]. These results suggest that the existing parking capacit y is not b eing fully utilized, p ossibly due to driv ers not ha ving enough information on where parking is a v ailable, and redistributing the parking demand in time and space can mitigate the truc k parking shortage. The shortage of truc k parking leads to the follo wing consequences: 2.1.1.1 Illegal P arking Surv eys carried out b y some states ha v e iden tified sev eral h undred illegal or unofficial parking lo cations suc h as freew a y shoulders, freew a y en trance and exit ramps, roadw a ys accessing freew a y ramps, lo cal streets and commercial areas [1]. The use of these lo cations p oses serious safet y hazards to other motorists and truc k driv ers themselv es, and exp ose driv ers to b ecome targets of ill-in ten tioned p eople. 2.1.1.2 Unsafe Driving With the driving time limits imp osed b y the HOS rules, a driv er unable to find a suitable parking lo cation ma y c ho ose to either park illegally or driv e illegally and tired. A study b y the AAA F oundation for T raffic Safet y found that 21% of all acciden ts in whic h a p erson w as killed in v olv ed a dro wsy driv er [37]. Although the data used w as not sp ecific to truc ks, it sho ws ho w dangerous dro wsy driving can b e. 9 2.1.1.3 En vironmen tal Impact The shortage of parking spaces forces driv ers to driv e around lo oking for parking and/or park at inappropriate lo cations. Both actions result in an increase in fuel consumption and emissions. While in some truc k stops the driv ers are able to plug in their v ehicles to the grid and a v oid idling, no illegal parking lo cation will ha v e this kind of service a v ailable, forcing the truc k to idle for sev eral hours. Idling is a large source of emissions, fuel exp enditure and engine w ear, so man y states already ha v e la ws and incen tiv es for idling reduction [38]. If the driv ers often need to find parking in the lo cal streets, they migh t impact the air qualit y and health of the nearb y comm unities [18]. 2.1.1.4 Cost increase As men tioned b efore, the shortage of parking can ha v e a substan tial impact on fuel consumption, b e it b ecause of the time sp en t lo oking for parking or the time sp en t idling for lac k of prop er infrastructure. A study b y the Univ ersit y of California, Da vis has estimated idling time to b e resp onsible for 8.7% of the total fuel consumption of long-haul truc ks [39]. F uel is resp onsible for a large share of the op erational costs in the truc king industry , making the o v erall cost highly dep enden t on fuel costs [40]. Other than the fuel consumption there is still the cost related to v ehicle main tenance (10% of total cost in 2015) [40], whic h can b e increased b y almost $2,000 a y ear due to idling [41]. Insurance premiums are another p ossibilit y of impacted costs as they can b e affected b y the n um b er of acciden ts and robb eries in v olving this kind of v ehicle. Recen tly , there ha v e b een man y adv ances in the field of autonomous v ehicles. A TRI published a rep ort on the p ossible impact this tec hnology w ould ha v e in the truc k industry [42], whic h includes a drop in the truc k parking demand. The use of high-lev el automation tec hnologies can p oten tially cause c hanges to the HOS rules and eliminate the need to find parking for the mandatory rest stops. A t this stage these b enefits are theoretical in nature as safet y issues asso ciated with high- lev el automation tec hnologies are y et to b e resolv ed to the lev el of b eing attractiv e enough for widespread adoption. Un til then, the demand for parking will b e high and the scarcit y of truc k parking will con tin ue to p ose problems. 10 While the topic of smart parking has b ecome more common o v er the past decade and reviews ha v e b een done on differen t asp ects of the problem [6, 43, 44], the problem of truc k parking did not receiv e as m uc h atten tion. Although the topics of urban smart parking and truc k parking ha v e m uc h in common, they exist in v ery differen t con texts. Hence metho ds used in urban smart parking ma y not b e suitable for implemen tation in rest areas and truc k stops, or ma y not fully utilize the particularities of truc k parking. Urban en vironmen ts ha v e surrounding infrastructure whic h can b e used to gather or rela y data, lik e traffic ligh t sensors, parking meters, street cameras, and ev en the dense street traffic. T ruc k stops and rest areas tend to a differen t set of users, and ha v e a differen t set of a v ailable infrastructure and data. This mak es it incon v enien t to use certain metho ds, but also allo ws the exploitation of the particular asp ects of truc k parking, e.g. all truc k driv ers are sub ject to HOS rules, and driv ers usually ha v e a set sc hedule to follo w and kno w exactly when they in tend to lea v e. Th us the need for researc h fo cused on the truc king industry . The main con tribution of this pap er is to presen t, in a organized w a y , the curren t state of the differen t researc h areas related to in telligen t truc k parking ( ITP ), starting from ho w data is acquired, ho w it can b e enhanced b y mathematical mo dels and ending with an unexplored w a y to use it in routing & sc heduling. Curren tly , there is no thorough review on the topic. The surv ey ed rep orts and pap ers usually review ed only the topics directly related to their sp ecific applications. The broadest review so far w as prepared b y the Univ ersit y of Wisconsin [45], whic h fo cuses mainly on pilot pro jects and on the studies realized in states mem b ers of the Mid America Asso ciation of T ransp ortation Officials ( MAASTO ), only briefly co v ering sensing tec hnology . This surv ey pap er fo cuses on the in telligen t truc k parking researc h efforts, on ho w to alleviate the curren t truc k parking shortage b y b etter utilizing the existing capacit y . It giv es an o v erview of the curren t state of researc h on in telligen t truc k parking, as w ell as discusses some studies on urban passenger v ehicle parking under the premise that they migh t b e directly or indirectly useful when considering truc k parking problems. Ev en though truc k parking is also an issue in urban en vironmen ts, the w ork surv ey ed in this pap er fo cuses on parking in truc k stops/rest areas a w a y from cities. The motiv ation to address the truc k parking problem is presen ted in section 2.1.1, the problem description and ensuing researc h needs are presen ted in section 2.1.2. This surv ey separates the review ed w ork in 4 categories: Sensing Infrastructure in section 2.2, Resource 11 Allo cation in section 2.3, Mo deling & Prediction treated in section 2.4 and Routing & Sc heduling in section 2.5. Section 2.6 presen ts the conclusion. 2.1.2 Problem Description There are 2 general approac hes to deal with the shortage in parking spaces: Increase parking capacit y An increase in parking capacit y w ould require either the construc- tion of new parking lots or the expansion/reorganization of existing parking lots, and b oth are costly prop ositions. Therefore, researc h on the b est w a y to mak e existing infrastructure more ef- ficien t in managing parking a v ailabilit y b y using adv anced tec hnologies is needed. Understanding demand and driv er b eha vior helps understand the actual needs in terms of additional n um b er of spaces needed when expansion of the infrastructure is inevitable. Decrease p eak demand By reorganizing the parking demand o v er time and space, the existing parking capacit y can b e used more efficien tly as demand is distributed o v er all a v ailable parking capacit y . F or example, a system that com bines the information regarding driv ers origin, destina- tion, HOS R ules and a v ailable parking at rest stops will allo w driv ers and shipmen t companies to optimally plan their shipping sc hedules, routes and rest stops. With these approac hes in mind w e can think of the follo wing researc h topic needs: • Sensing infrastructure that can gather real-time data accurately under differen t en vironmen ts of the rest stops with lo w main tenance costs, whether or not the lo cation has easy access to p o w er and/or ground comm unication; • Comm unication infrastructure that can gather data from the rest stops indep enden t of their lo cation and efficien tly distribute information to the driv ers; • Resource allo cation algorithms that can redistribute driv ers in a certain region according to their driving time restrictions and the parking lots exp ected demand; • Regional/lo cal demand mo dels that can b e used to supp ort decision making when planning future in v estmen ts; • Prediction mo dels that can use a v ailable data to predict the future parking a v ailabilit y of a certain truc k stop; • Routing/sc heduling algorithms that accoun t for HOS rules and parking a v ailabilit y . 12 2.2 Sensing Infrastructure There is little w ork on sensing infrastructure that fo cuses on commercial truc k parking mainly b ecause the sensing infrastructure for passenger and commercial v ehicles is v ery similar. This section presen ts the c haracteristics and issues of the o ccupancy detection systems that ha v e b een tested. 2.2.1 Coun ting Metho ds The most common w a ys to access the o ccupancy of a parking lot are individual space o ccupancy detection and in/out coun ting. This section fo cuses on these metho ds, but other w a ys to obtain o ccupancy information are men tioned in section 2.2.2. 2.2.1.1 Individual Space Occupancy Detection This metho d consists of c hec king the o ccupancy of ev ery single parking space and then coun ting the n um b er of o ccupied sp ots. It has the adv an tage that sensing errors in a single parking space do not ha v e a big impact on the total o ccupancy and do not accum ulate o v er time. It also has an easier time dealing with detac hable parts, lik e truc ks dropping/pic king up trailers, and with passenger v ehicles parking in truc k parking spaces. Its main dra wbac k is that the n um b er of sensors needed is prop ortional to the n um b er of parking spaces, whic h increases the installation and main tenance costs for large facilities. Naturally , some tec hnologies are more scalable than others, e.g. a single camera could co v er sev eral parking spaces while more than one magnetic sensor migh t b e needed for eac h parking space. Another issue is the need for individual spaces to exist, and that driv ers park within the spaces. 2.2.1.2 In/Out Coun ting This metho d consists of, giv en the parking lot’s initial o ccupancy , coun ting the n um b er of v ehicles that en ter and exit the lot, then calculating ho w man y v ehicles are curren tly park ed. Its adv an tage is that it is v ery scalable. Ev en if the parking area is v ery large, the infrastructure needed is limited b y the n um b er of en trances/exits. This system can b e as simple as only ha ving 13 a gated en try with a terminal, or it can ha v e m ultiple sensors to detect and classify eac h v ehicle without h uman in terv en tion. The absence of parking space markings and the misuse of parking spaces do not affect the o ccupancy coun ting in this case, but they still affect the total capacit y of the parking lot, compromising a v ailabilit y estimates. F o cusing all resources on the access p oin ts also creates some disadv an tages for this metho d. One of them is that its error is cum ulativ e and can only b e corrected with ground truth information. A miscoun t will b e carried on un til a ground truth correction is made or a miscoun t in the opp osite direction is made. Ev en if the sensors at the en try/exit p oin ts do not fail, errors can still app ear in certain situations. F or exam ple, a truc k en ters carrying a trailer and drops it in a parking space b efore lea ving, the system w ould not consider the trailer in its calculation and will undercoun t the o ccupancy . These detac hable parts are in visible to the system. So this t yp e of system needs frequen t ground truth corrections. Another problem, that is usually solv able through extra sensors and/or soft w are up dates, is the b eha vior of driv ers at the en trance/exit p oin ts. Beha viors lik e tailgating and using the wrong lanes to en ter/exit can affect detection, so the system’s design has to tak e these situations in to accoun t to a v oid errors. 2.2.2 Data acquisition The previous section presen ted the most common metho ds of o ccupancy detection, but only describ ed their in trinsic adv an tages and disadv an tages. This section presen ts the sensors that are used to implemen t these metho ds and also other t yp es of systems that can b e used to measure or estimate parking o ccupancy . 2.2.2.1 Magnetic These sensors can detect v ehicles b y measuring the disturbance they cause on the magnetic field. As sho wn on T able 2.1, this kind of sensor has b een used in sev eral pilot pro jects for ITP , b oth for in/out coun ting and individual space detection. It w as also used in urban smart parking systems, suc h as [46, 47, 48, 49]. A ccording to these studies, the sensors are v ery energy efficien t and can last for sev eral y ears b efore running out of battery . Other strengths include easy installation, robustness to w eather conditions, lo w-bandwidth and requiring little main tenance. Ho w ev er, they 14 are sensitiv e to the p ositioning of the v ehicle and m ultiple sensors are needed p er parking space when used for individual space detection. F urthermore, they are vulnerable to cross-in terference caused b y adjacen t v ehicles. This issue w as treated either through comm unication b et w een nearb y sensors [50] or usage of more than one t yp e of sensor [51]. In [52], F allon installed magnetic sensors at a rest area to test in/out coun ting. The system reac hed a v ehicle detection accuracy of 96%, but had issues with the cum ulativ e o ccupancy error. Some sensors use more than one t yp e of sensing at the same time to impro v e p erformance. In [50], Haghani et al. dev elop ed a parking sensor with magnetic sensors, using a temp erature sensor and soft w are to comp ensate for the temp erature drift and magnetic cross-in terference. The system used individual space o ccupancy detection and reac hed an a v erage detection accuracy of 96%, with man y of the errors b eing caused b y signal blo c kage. The ones in [51, 53] used magnetic and infrared sensors to detect v ehicles on individual spaces, whereas Siemens Wimag sensors [54] com bine magnetic field sensors with MicroRadar. In [47], Sifuen tes used an optically triggered magnetic sensor to lo w er energy consumption. 2.2.2.2 Laser This t yp e of sensor uses lasers to detect ob jects and measure its distance to the sensor. A Danish pilot pro ject [55] and a German pilot pro ject [56] organized the truc ks with similar or non-conflicting departure times in columns and used this kind of sensor to measure the remaining space in eac h column of the parking lot. The Siemens Sitraffic Conduct+ system [57] uses laser scanners at the en trance and exit of the parking lot to iden tify the t yp e of v ehicle that is en tering. [58] tested side scanners, o v erhead scanners and ligh t curtains com bined with a Doppler r adar for in/out coun ting, reac hing detection accuracy rates of 99%. Ho w ev er, it w as noticed that ice buildup on the detector can imp ede prop er w orking of the system, and that the required p ositioning of the radar mak es it vulnerable to tamp ering. 2.2.2.3 Video Image pro cessing This metho d consists of using images from video surv eillance to detect the v ehicles. While this approac h is more computationally exp ensiv e and needs more bandwidth than others, it is attractiv e 15 due to the fact that surv eillance cameras are already installed on parking lots for other reasons. Another adv an tage of cameras when used in individual sp ot sensing metho ds is that a small quan tit y of cameras can co v er a wide area. Reference [59] prop osed a v ehicle detection algorithm based on gra y-lev el segmen tation, accum ulator agen ts and threshold that uses edge detection as a w a y to a v oid the false detection caused b y nearb y cars’ shado ws. In [60], the author used con v olutional neural net w orks to detect the parking spaces o ccupancy on regular color camera images. As far as the w ork dev elop ed sp ecially for truc ks is concerned, [61, 62, 63] used sev eral images of the same area from differen t p oin ts of view to reconstruct a 3D represen tation of that scene. The 3D represen tation is then matc hed with real-w orld co ordinates and used to c hec k if there is a v ehicle o ccup ying the p osition of eac h parking space. This metho d a v oids the problems that 2D approac hes ha v e with shado ws and o cclusion, and ac hiev ed 95% detection accuracy in [63]. The detection accuracy w as affected b y sno wfall and rain, but the effect w as small and the system still p erformed w ell under these conditions. [64, 65] used cameras at the en trance and exit of the parking lot to detect v ehicles en tering and lea ving the area. In [66], Mo di et al. used mixture of Gaussians com bined with a shado w remo v al tec hnique for foreground detection, using the area co v ered b y the v ehicle to determine the o ccupancy of a parking sp ot. 2.2.2.4 GPS data analysis GPS data has b een used for freigh t p erformance in [67] and parking utilization analysis in [68, 69]. In [69], the author compared data from the w eigh-in-motion traffic coun ting station with GPS data to estimate, for eac h time p erio d, the p ercen tage of truc ks captured b y the GPS data. Then used this factor to correct the o ccupancy coun t generated b y the GPS data. In [68], the authors compare 4 differen t mo dels for truc k parking utilization: a P oisson, a negativ e binomial, a P oisson with prop ensit y , and a P oisson with prop ensit y and threshold sp ecific constan t. V ariables suc h as time of da y , n um b er of v ehicles detected on/off-ramp, a v erage sp eed close to the parking lot and truc k v olume w ere used to estimate the parameters for these mo dels. GPS data can pro vide the n um b er of v ehicles in the general area of the parking lots and also information ab out the curren t truc k traffic flo w in high w a ys that lead to that parking lot. Ho w ev er, applications that need to use GPS data in real-time migh t ha v e issues with data acquisition. Some p eople migh t b e concerned ab out the driv ers’ priv acy . Considering that the participation of a certain 16 p ercen tage of the truc ks in op eration is required in order to get reliable data, driv ers/companies engagemen t is crucial and migh t b e a problem. In the con text of urban parking, [70] utilized GPS, along with the accelerometer and blueto oth sensors of phones in order to detect when and where the device’s o wner park ed his/her car and when the parking space w as released. 2.2.2.5 Cro wdsourcing Cro wdsourcing consists of enlisting a large n um b er of p eople, usually the service users them- selv es, to obtain the information needed instead of setting up sp ecific infrastructure to acquire the information directly . This approac h has b een studied mostly for urban parking. In [71] users and parking op erators can publish information, and the system uses driv er b eha vior to impro v e the parking a v ailabilit y estimation. In [72], Hoh et al. suggest a game-theoretic incen tiv e platform to address the issues of user participation and information qualit y . On the commercial v ehicles’ side, apps lik e T ruc k erpath [73], P arc kr [74] and P arkMyT ruc k [75] are already op erational, pro viding information to truc k driv ers. T ruc k erP ath and P arc kr get information from b oth driv ers and parking pro viders, while P arkMyT ruc k only allo ws registered pro viders to up date their a v ailabilit y . 2.2.3 Comm unication W e can sub divide this topic in to system connectivit y , and information dissemination. 2.2.3.1 System Connectivit y The surv ey ed w ork did not fo cus on the differen t tec hnologies existen t for sensor connectivit y and data transmission. Most pro jects tested off-the-shelf equipmen t, th us b eing restricted to what w as offered b y the man ufacturer, man y using wired connections. A summary of the wireless sensor net w ork tec hnologies that are usually considered for smart parking applications can b e found in [6]. The connection b et w een the test site and the cen tral serv er usually used a cellular net w ork or cable. 17 2.2.3.2 Information Dissemination The usual w a ys o v er whic h the parking information can b e distributed to the users are v ariable message signs ( VMS ), w ebsite, smartphone apps, radio and in-cab devices. Differen t com binations of these metho ds ha v e b een adopted in the surv ey ed pro jects, as can b e seen on T able 2.1. An imp ortan t consideration is the lev el of acceptance of eac h metho d. User participation can b e an issue, as v erified on a surv ey realized b y W o o dro offe et al. [65] in Mic higan in 2015. 75% of the 60 surv ey ed driv ers rep orted that information ab out parking is v aluable and sa v es time. Nev ertheless, when ask ed ho w they determined parking a v ailabilit y at the curren t site, only 30% used the VMS information, whereas 55% said they knew parking w as a v ailable based on past exp erience and 17% did not kno w parking w as a v ailable. This lo w participation lev el happ ened despite VMS b eing rep orted as the preferred information source b y most of the surv ey ed driv ers. Ev en though truc k driv ers ac kno wledge the usefulness of parking a v ailabilit y information, most of them are not using it. Ho w to close this gap b et w een the rep orted v alue of the information pro vided b y ITP systems and actual user participation is still an op en problem. 2.2.4 Pilot Pro jects There ha v e b een sev eral initiativ es to impro v e the curren t situation of the truc k parking infras- tructure. These pro jects usually aim to either pro vide b etter information to truc k driv ers or to increase the quan tit y and qualit y of parking lo cations. Efforts b y the Europ ean Union ([76, 77, 55]), USDOT ([52, 64, 58]), MAASTO ([78, 79]) and Florida Departmen t of T ransp ortation ([80, 81]) generated pilot sites that aimed to test the p erformance of differen t t yp es of ITP systems. T able 2.1 lists the pilot pro jects men tioned in the surv ey ed w ork with their lo cation, t yp e of sensor used and their p ositioning at the site, the n um b er of sites using the system and the in tended metho d of comm unication with the public. As it includes pro jects not men tioned in section 2.2.2 due to limited information, this table giv es a broader view of the t yp es of sensors b eing tested and the tests’ scale. The priv ate sector is also starting to mak e their o wn impro v emen ts. Some priv ate truc k stops are implemen ting their o wn reserv ation systems or pro viding their parking information to third parties as T ruc k erpath, P arc kr and P arkMyT ruc k. Pilot Flying J and High w a y P ark w ere included in the table as examples of suc h cases. They ha v e a reserv ation system for part of the 18 parking spaces in their lo cations, and users can c hec k the a v ailabilit y and mak e reserv ations o v er the in ternet or the phone. Although man y pilot pro jects w ere tested, sev eral fo cused only on ev aluating a certain t yp e of sensor’s accuracy or the system’s uptime, and most pro jects did not ha v e a large enough scale to cause a significan t impact on the net w ork. A few Europ ean pro jects had a larger scale and already included the information distribution asp ect, but w e w ere unable to find their p erformance ev aluations. MAASTO’s T ruc k P arking Information Managemen t System [78] started op erating in 2019 and pro vides real-time truc k parking information for o v er a 100 parking facilities in 8 states. A ccording to a surv ey b y A TRI, 34% of surv ey ed driv ers rep orted using unauthorized parking lo cations less often after the system’s implemen tation [82]. 2.2.5 Remarks There are sev eral factors that influence the viabilit y of eac h sensor t yp e, suc h as the a v ailable budget, the coun ting metho d and the c haracteristics of eac h parking lot. F rom the pilot pro ject rep orts w e concluded that small impro v emen ts to the organization of the rest area b efore the installation of an y system w ould b e v ery b eneficial. Whenev er p ossible, clear markings for the parking sp ots and separation b et w een passenger and commercial v ehicles should b e used to simplify system implemen tation and impro v e the parking lot’s efficiency . F urthermore, w e understand that in/out coun ting is more con v enien t for large truc k stops, whereas individual space detection is a b etter option for small ones. As men tioned b efore, in/out coun ting systems ha v e the adv an tage of scalabilit y , but need to b e monitored and corrected often due to cum ulativ e errors. A large n um b er of parking spaces guaran tees that the error build up w ould not reac h a significan t p ercen tage of the total capacit y v ery fast. Large parking lots are also more lik ely to ha v e manned facilities, with p eople able to help with ground-truth correction. On the other hand, individual space detection systems’ costs increase prop ortionally to the n um b er of parking spaces, but ha v e no error build-up issues. Small parking lots are v ery sensitiv e to error build up and are not as affected b y the cost difference, so individual detection is a con v enien t c hoice. T est sites w ere prepared using magnetic sensors, laser sensors and cameras with image pro cessing soft w are, and man y systems ac hiev ed an a v erage detection accuracy of more than 95% [52, 50, 58, 19 T able 2.1: Pilot Pro jects Coun try Lo cation Sensor T yp e Sensor P osition 1 Sites Comm. 2 German y A8[77] magnetic EE 1 V A5[77] magnetic EE 5 V, W A3[55, 83] gate EE 1 W High w a y P ark sys- tem[55] man ual - 20 W A3[55, 56] laser C 1 V, W A9[84, 55] laser, radar EE 21 R, W, A A2[77] magnetic, cameras P 1 V P ort of Ham burg [53] magnetic/ IR P 1 V A ustria Vienna[77] man ual - 10 V, W F rance A10[77] magnetic P 2 V A7[55] - P or EE 61 V A13[55, 85] - - 12 V Hungary M1-M0[55] cameras - 1 V Denmark Ustrup East[55] gate, laser EE, C 1 V USA I-10, FL[51] magnetic/ IR P 2 W I-75, FL[81] magnetic/ IR, radar P 2 V MN[61, 62] cameras - 1 V, W, I US1, MA[52] magnetic EE 1 W I-95, MA[52] magnetic EE, P 1 W I-90, MA[64] cameras EE 1 W I-75, TN[58] laser, radar EE 1 W I-94, MI[65, 86, 79] cameras, magnetic EE 15 W I-95, MD[50] magnetic P 1 W V A[87, 88] magnetic/ radar P 6 V, W 1 EE = en trance & exit, P = parking sp ot, C = parking column 2 V = VMS, W = w ebsite, R = radio, A = smartphone app, I = in-cab device 61, 63] sho wing that curren t tec hnology is already capable of pro ducing reasonable results when effort is put in to adjusting the system to the sp ecific conditions of eac h rest area. Ho w ev er, the tests sho w ed that most systems are not v ery robust short after installation, sp ecially in the case of irregular driv er b eha vior. The researc hers needed to monitor the system closely and mak e c hanges to the system often in order to adapt the tec hnology to this application and ac hiev e a go o d p erformance. This is exp ected during the dev elopmen t of a new application, but the need for p ost-installation mo difications to the system should b e minimized in a pro duct exp ected to 20 b e broadly adopted. Limited testing still hinders our abilit y to study the adv an tages of eac h system in differen t en vironmen ts, adjustmen ts that eac h one migh t need and problems that migh t o ccur during regular op eration. This lac k of tried-and-true systems is a barrier to the adoption of in telligen t truc k parking systems. 2.3 Resource Allo cation A system capable of measuring the real-time o ccupancy of rest areas op ens other w a ys to manage those facilities. Lik e using a reserv ation system or a resource allo cation algorithm to b etter distribute the spaces from a certain region among incoming v ehicles. As men tioned b efore, parking shortage during p eak hours is a big problem. Ho w ev er, not all parking lots suffer from that shortage. A resource allo cation algorithm that tak es in to accoun t the driv ers’ constrain ts and the parking lots’ historical o ccupancy profile can redistribute driv ers among a set of parking facilities in a more efficien t w a y compared to the driv ers’ o wn decisions. In [89, 90], the authors used a m ulti-agen t system for the managemen t of parking reserv ations among requesting truc ks. When the v ehicle en ters the road net w ork it sends its origin, destination and preferred parking to the system manager. If the rest area has a v ailable sp ots a temp orary reserv ation will b e made. If the rest area do es not ha v e a v ailable sp ots the negotiation proto col is initiated. Eac h driv er receiv es a list of p ossible rest areas to b e graded according to his/her preferences. Eac h driv er’s v ote is w eigh ted according to their remaining allo w ed driving time, giving priorit y to driv ers that are closer to reac hing their legal driving limit. The scores for eac h driv er are summed for eac h feasible solution and the solution with the largest score is selected. Note that this algorithm assumes that there is at least 1 feasible solution for a giv en problem. F ollo wing the selection, all driv ers and rest areas are notified of the new allo cation. The system’s robustness to c hanges in the a v ailable parking areas and the system’s scalabilit y w ere tested through sim ulation and the results w ere promising, sho wing a substan tial reduction of the necessit y for driv ers to park in illegal areas. This resource allo cation problem w as also treated b efore in the con text of urban parking. In [91] the resource allo cation problem w as defined as a sequence of Mixed In teger Linear Programming 21 problems solv ed o v er time sub jected to a set of fairness constrain ts. [92] uses in terv al sc heduling algorithms to try to optimally allo cate the parking spaces. 2.4 Mo deling & Prediction Although m uc h w ork has b een done on the mo deling and prediction of passenger v ehicle urban parking, the w ork on commercial v ehicle’s parking is v ery limited. In [93], Fleger et al. dev elop ed a parametric mo del for the demand of truc k parking on high w a y segmen ts during p eak hours and used it to estimate the ann ual increase in demand. The simplified mo del estimates the demand along a high w a y segmen t based on the truc k-hours of tra v el p er da y on that segmen t and the a v erage parking time p er truc k-hour of tra v el. The final mo del considers factors to accoun t for p eak traffic, the ratio b et w een long-haul and short-haul truc ks, loading/unloading time and other v ariables. In [94], instead of fo cusing only on the p eak hour demand, W ang generated regression mo dels to estimate demand throughout the da y , and used them to estimate future parking shortages. The mo dels used information from in terstate high w a ys in Virginia to estimate the demand of truc k stops dep ending on factors suc h as distance to the next truc k stop/rest area and to the high w a y , p ercen tage and v olume of truc k traffic, etc. In [95], Heinitz estimated the parking demand for rest areas in a high w a y section based on the upstream traffic. The upstream region of a high w a y section w as divided in iso c hronal 60-min rings. The historic traffic flo w on these iso c hronals w as used to estimate ho w man y truc ks will pass b y that section and their elapsed driving time. Then a c hoice mo del w as used to estimate where eac h truc k is going to park. The c hoice mo del used utilit y functions based on lo cation, service and fees to estimate the probabilit y of the driv er stopping at eac h parking lot. This mo del is more flexible than the previous ones, and could b e used to sim ulate not only ho w the demand will increase on follo wing y ears, but also ho w the parking b eha vior can b e affected b y c hanges in the infrastructure and regulations. Although 2 of the presen ted demand mo dels also generate time-dep enden t demand/o ccupancy estimates, all 3 fo cus on the in v estmen t decision supp ort asp ect of the problem, not on pro viding information to driv ers. These t yp es of mo dels can help with in v estmen t planning b y estimating presen t and future parking shortages, and ho w eac h region w ould b enefit from in v estmen ts. The mo dels presen ted next fo cus on predicting future a v ailabilit y to help driv ers plan their trips. In 22 [51], Ba yraktar installed a smart parking-managemen t system in a Florida parking lot and used the acquired data to feed a Kalman-filter based o ccupancy prediction mo del. In [63], Morris et al. used the data from their tests on computer vision parking detection to test 2 prediction mo dels. The first w as a Kalman Filter based on [51]. The second w as a harmonic forecasting mo del, whic h describ ed the parking space o ccupancy coun t as the sum of a series of phase-shifted sin usoidal basis functions, capturing the p erio dic nature of the system. Both mo dels w ere tested for prediction horizons of 15 min utes and 1 hour. The harmonic mo del p erformed b etter on b oth cases, but the impro v emen t w as greater for the 1-hour predictions, with errors around 10-15% of the total capacit y . In [96], Sadek et al. also to ok adv an tage of the cyclical b eha vior of parking o ccupancy and used F ourier Mo dels to predict o ccupancy at an y time during the presen t da y . The authors observ ed m ultiple trends in the data, and, for eac h trend and w eekda y , fitted F ourier mo dels to the a v erage o ccupancy curv e of past data with the same w eekda y and trend. The mo del w as tested for prediction horizons significan tly longer than the previously cited mo del, but its a v erage error w as b elo w 5% of the total capacit y . In [97], Brab ec et al. used toll transactions data to create a pro xy v ariable to the parking a v ailabilit y , and used this pro xy v ariable to estimate the future parking a v ailabilit y . A generalized additiv e P oisson mo del w as used to describ e eac h parking lot, and a Mon te Carlo approac h w as used to calculate predictions with an horizon of up to 2.5 hours ahead. Considering ho w scarce real-time parking a v ailabilit y data is, b eing able to use indirect data for the predictions is v ery adv an tageous. In [98], T a v afoghi et al. decomp ose v ehicle arriv als according to their duration and use a queuing mo del to prop ose prediction metho ds that pro vide a real-time probabilistic forecast of parking o ccupancy . The P arc kr app [74] cites on their w ebsite that they ”fuse floating car data, real-time sensor data and comm unit y feedbac k to estimate and forecast the o ccupancy of truc k parkings”, and that this service is already a v ailable in 8 EU coun tries. Due to the larger n um b er of mo dels a v ailable for urban parking prediction there is also a large arra y of opp ortunities to lo ok for w a ys to adapt some of these metho ds or at least the general idea b ehind them for use in truc k parking. [99] prop osed a m ultiv ariate spatiotemp oral approac h whic h detrends the v ariation in parking a v ailabilit y at eac h time in terv al, and then uses an autoregressiv e mo del with information from all parking lots in the area to estimate the future detrended v ariation. In [100] the author used genetically optimized neural net w orks for short term parking o ccupancy 23 prediction. Genetic algorithms are used to optimize the learning rate and momen tum of bac k- propagation training algorithm, as w ell as the structure of the hidden la y er and lo ok-bac k time windo w. [101] used a com bination of neural net w orks and classification metho ds in order to predict if the parking lots will b e full or not, according to time, ev en ts happ ening in the vicinit y and w eather data. The first part of the mo del tested b oth generalized regression neural net w ork ( GRNN ) and m ultila y er feed-forw ard net w ork ( MLFN ) to predict the parking a v ailabilit y , ac hieving b etter results with the GRNN. The second part of the mo del is the classification metho d that is used to reduce the n um b er of false negativ es generated when the parking lot has high o ccupancy . The metho ds tested w ere a naiv e Ba y es classifier and a classification and regression tree ( CAR T). [102] uses parking reserv ations to predict the v ehicles’ arriv al and departure times and therefore the future parking a v ailabilit y in real-time. This system used a discrete c hoice mo del with on-line and historical information to sim ulate the arriv als, c ho ose their allo cation and their departure times. Other w orks use P oisson random pro cesses to mo del parking a v ailabilit y . [103] mo dels the lot’s a v ailabilit y as a P oisson random pro cess, estimating its parameter using historical data. [104] mo deled the arriv al rates of the parking lots as P oisson random pro cesses, the duration of sta y as an exp onen tial v ariable, and mo deled the parking lot using a con tin uous-time Mark o v mo del. [105] also used a con tin uous-time Mark o v mo del, but presen ted a w a y to estimate b oth arriv al and departure rates based only on o ccupancy data, whereas [104] required additional assumptions. [106] used sim ulation metho ds in its predictions. An agen t-based mo del ( ABM ) w as com bined with Mark o v Chain Mon te Carlo ( MCMC ) in order to ac hiev e more accurate results than either metho d applied separately . The sim ulated data generated through ABM w as used to create the prop osal distribution for the MCMC. 2.4.1 Mo del Comparison In order to c hec k ho w some of the published metho ds compare with the predictions and rec- ommendations/classifications generated b y simple historical o ccupancy a v erages, w e tested a few algorithms on a dataset obtained through collecting data a v ailable at the w ebsite of a priv ate truc k stop c hain with an op erational reserv ation service [107]. P arking reserv ations start at 4pm on the da y of reserv ation un til 3pm the next calendar da y . The w ebsite w as c hec k ed ev ery 15 min utes 24 from 00:00 un til 20:45 from late April 2017 up to mid Octob er 2017. Data collection stopp ed at 20:45 b ecause 21:00 the w ebsite stopp ed taking reserv ations for the curren t da y and data b ecame una v ailable. When the data collection restarts at 00:00, the data concerns reserv ations for the fol- lo wing da y , not the real-time o ccupancy of the facilit y . The truc ks’ departure time is not captured b y the data, an increase in the n um b er of a v ailable sp ots while data w as b eing collected w ould mean a canceled reserv ation. Ho w ev er, this case w as v ery rare. The data collected p ertains to 14 priv ate stops lo cated in the state of California. The p erio d from July 8th to A ugust 16th w as used for training and the p erio d from Septem b er 5th to Octob er 14th w as used for testing. As this dataset has discon tin uities at the end of eac h da y , w e c hose mo dels that could b e adapted to a v oid these discon tin uities more easily . The metho ds used for comparison are: a non-homogeneous P oisson mo del (NH P oisson); the Multiv ariate Spatiotemp oral mo del (MSM) from [99]; the curv es similarit y approac h from [108]; Historical a v erage; Historical a v erage b y w eekda y . 2.4.1.1 Prediction Mo dels Non-homogeneous P oisson mo del This mo del w as defined in a similar w a y to the m ultiv ari- ate spatiotemp oral mo del [99] men tioned b efore. The normalized parking o ccupancy v ariation of parking lot l at time t , v l;t , is defined as: v l;t = (a l;t a l;t1 ) c l (2.1) where c l is the total capacit y of pa rking lot l and a l;t is the o ccupancy of parking lot l at time t . a l;t and c l are measured in n um b er of v ehicles (or reserv ations), v l;t is dimensionless. The a v erage normalized parking o ccupancy v ariation v l;t , is estimated from the historical data as a w eigh ted a v erage, o v er the lastW w eeks (curren t one not included), of windo ws of widthH starting at time t of the same w eekda y as the one for whic h the estimate is needed, i.e. v l;t = 1 H W X j=0 ! i H1 X i=0 v l;tijTw (2.2) whereH is the length of the mo ving windo w,T w is the length of a w eek,W is the n um b er of w eeks and ! i are p ositiv e w eigh ts to emphasize the data of most recen t w eeks with P W i=0 ! i = 1 . 25 In this sp ecific dataset, reserv ation cancellations w ere v ery rare and can b e ignored, allo wing us to in terpret the n um b er of reserv ations made as a non-homogeneous P oisson random v ariable with parameter c l v l;t . Although not used in the presen ted exp erimen ts, w e could use this assumption to estimate the probabilities of ha ving a sp ecific n um b er of spaces a v ailable at a certain time. Giv en the o ccupancy at time t , a l;t , the estimated o ccupancy at a future time t+k , ^ a l;t+k , can b e calculated b y : ^ a l;t+k = a l;t + k X i=1 E[Poisson(c l v l;t+i )] = a l;t +c l k X i=1 v l;t+i (2.3) Note that it is p ossible to predict an o ccupancy ab o v e the parking lot capacit y . In the con text of truc k parking it is not uncommon for parking lo ts to op erate o v er-capacit y , so it can b e in terpreted as suc h. It should also b e noted that k is not un b ounded; t+k should represen t a time for whic h the reserv ations are still v alid, preferably b efore 9PM on the same calendar da y as t . After that, data is either una v ailable or it concerns reserv ations for a differen t da y . Multiv ariate Spatiotemp oral Mo del [99] The normalized parking o ccupancy v ariationv l;t , is defined as in (2.1). v l;t is mo deled as the sum of a trend comp onen t v l;t and a sto c hastic comp onen t ~ v l;t . v l;t = v l;t + ~ v l;t (2.4) where v l;t is defined as in (2.2). Let ~ V t b e theL1 v ector of the sto c hastic comp onen ts (~ v l;t ) of all parking lots at time t and let E t b e a zero-mean random v ector of i.i.d. white random pro cesses. The follo wing autoregressiv e mo del is used to estimate the sto c hastic comp onen t: ~ V t , 2 6 6 6 6 4 ~ v 1;t . . . ~ v L;t 3 7 7 7 7 5 ;E t , 2 6 6 6 6 4 e 1;t . . . e L;t 3 7 7 7 7 5 (2.5) ~ V t = M X m=1 A m ~ V tm +E t (2.6) 26 whereL is the total n um b er of parking lots and theA m co efficien ts are LL matrices that need to b e estimated. The order of the mo del M can b e found using Akaik e’s information criterion [109]. Let ^ V t b e an un biased estimator of ~ V t then: ^ V t =E[ ~ V t j ~ V ti ;i = 1;:::M] = M X m=1 ^ A m ~ V tm (2.7) where ^ A m are the estimated co efficien t matrices, whic h are estimated using historical data [99]. The predicted parking o ccupancy is calculated b y: ^ a i;t+1 =E[a i;t jI t ] =a i;t +~ a i;t+1 (2.8) ~ a i;t+1 = ( v i;t+1 + ^ v i;t+1 )c i (2.9) where I t denotes all a v ailable parking information related to time t , ^ a i;t+1 denotes the predicted parking o ccupancy , ~ a i;t+1 is the exp ected c hange in o ccupancy b et w een time stepst andt+1 , and ^ v i;t is the i th elemen t of ^ V t . Curv es Similarit y This forecasting mo del, presen ted in [108], w as dev elop ed with hotel reser- v ations in mind. As the gathered data come from a reserv ation system, this metho d w as selected to b e tested and compared with the other metho ds that w ere dev elop ed for real-time o ccupancy data. It used a curv es similarit y approac h that compares the curren t/incomplete reserv ation curv e for a target da y , with curv es from other da ys obtained from historical data. This metho d assumes that the curv e for the target da y will b eha v e as an a v erage of all the historical curv es that had a similarit y measure b elo w a certain threshold defined b y the user. The predicted o ccupancy for an y future da y is tak en as the v alue that this a v eraged curv e w ould tak e at that sp ecific da y . A detailed description of the metho d can b e found in [108]. The similarit y measure w as defined as follo ws: S i = v u u t T X k=0 (a l;d;tk a l;i;tk ) 2 (2.10) where d is the forecasted da y , i is the da y to whic h it is b eing compared, T is the time of the last a v ailable data p oin t for the forecasted da y . a l;d;t is the parking o ccupancy at lo cation l, da y d 27 and time t;t = 0 represen ts the first data p oin t of the da y . Let Z b e the set of da ys for whic hS i is smaller than a threshold . The estimated parking o ccupancy ^ a l;d;t of a l;d;t can b e calculated b y: Z =fijS i < g (2.11) ^ a l;d;t = 1 jZj X i2Z a l;i;t (2.12) Historical a v erage Let a l;d;t b e the parking o ccupancy at lo cation l , da y d and time t , where t2 N and t = 0 means the first data p oin t of the da y . The estimate ^ a l;d;t when considering only the time of the da y is defined as: ^ a l;d;t+n = 1 jSj X i2S a l;i;t (2.13) where S is the set of da ys for whic h data for time t is a v ailable. Historical a v erage b y w eekda y The estimate ^ a l;d;t when considering the time and the w eekda y is defined as: ^ a l;d;t+n = 1 jS d j X i2S d a l;i;t (2.14) whereS d is the set of da ys that are the same w eekda y asd and for whic h data for timet is a v ailable. 2.4.1.2 Classification Mo del The classification (F ull/Not-full) w as done based on the predicted relativ e o ccupancy for eac h time. A threshold w as defined for eac h metho d and whenev er the predicted o ccupancy is ab o v e that threshold, the parking lot is classified as full. In order to c ho ose the threshold, the exp erimen ts w ere run for a differen t set of 40 target da ys. The receiv er op erating c haracteristic (R OC) curv e w as generated for eac h prediction mo del and the threshold w as selected for eac h curv e b y maximizing the Y ouden index [110], whic h is defined as: YoudenIndex =sensitivity +specificity1 (2.15) sensitivity = TP TP +FN (2.16) 28 specificity = TN TN +FP (2.17) where TP is the n um b er of true p ositiv es (parking lot is full and w as classified as full) in the exp erimen t, FN is the n um b er of false negativ es (parking lot is full and w as classified as a v ailable), TN is the n um b er of true negativ es (parking lot is a v ailable and is classified as a v ailable) and FP is the n um b er of false p ositiv es (parking lot is a v ailable and is classified as full). 2.4.1.3 Exp erimen tal Results The comparison w as made b y c ho osing the p eak demand time of the a v ailable data, 20:45 in this case, as the target time for the prediction and v arying the prediction horizon considered b y the prediction mo dels. Figure 2.1 sho ws the mean squared error (MSE) calculated using the relativ e o ccupancy v alues (o ccupancy/capacit y) of eac h parking lot for eac h prediction horizon. Figure 2.2 sho ws the sensitivit y and sp ecificit y v alues, defined in (2.16) and (2.17), calculated based on the n um b er of true/false p ositiv es and true/false negativ es for all 14 parking lots for eac h prediction horizon. The sensitivit y v alue measures ho w lik ely the system is to correctly classify the p ositiv es, and the sp ecificit y measures ho w lik ely the system is to correctly classify the negativ es. Figure 2.3 sho ws ho w the Y ouden index of eac h metho d v aries with the prediction horizon. This index, defined on eq. 2.15, is a p erformance measure that giv es equal w eigh t to the sensitivit y and sp ecificit y of the tested metho ds. So, according to the exp erimen ts, the tested metho ds’ abilit y to iden tify when the parking lot w on’t b e full (sp ecificit y) is not significan tly affected b y the prediction horizon. Ho w ev er, their abilit y to iden tify when the parking lot is going to b e full (sensitivit y) is highly dep enden t on the prediction horizon. It is p ossible to see that, for prediction horizons smaller than 3 hours, if prioritizing sensitivit y , and 5 hours, if considering the Y ouden index, the 3 tested metho ds p erform significan tly b etter than the simple forecast metho ds b oth in estimation error and in classification p erformance. A 3-5 hour prediction horizon is still not at the necessary standard to b e used in a full off-line trip planning, but it is already enough to pro vide the driv ers with useful information to plan their stops and/or up date their routes during the trip, with or without an optimization soft w are’s assistance. F urthermore, it is also noted that the classification p erformance ’ranking’ of the metho ds is not necessarily the same for all circumstances. Differen t prediction horizons or priorities on the sensitivit y and sp ecificit y measures can c hange whic h algorithm is 29 Figure 2.1: MSE of the o ccupancy estimate for target time 20:45 with v arying prediction horizons more appropriate to use. T esting sev eral metho ds in differen t con texts and v erifying whic h one w orks b etter, when and for what is also a p ossible approac h to impro v e prediction capabilities. It can b e exp ected that, when compared to using simple forecast metho ds, further dev elopmen t of truc k parking prediction mo dels can pro vide significan t impro v emen ts to the driv ers and companies’ planning efficiency . 30 Figure 2.2: Classification p erformance (Sp ecificit y/ Sensitivit y) for all parking lots for target time 20:45 with v arying prediction horizons Figure 2.3: Classification p erformance (Y ouden Index) for all parking lots for target time 20:45 with v arying prediction horizons 31 2.5 Routing & Sc heduling Another w a y to address the parking shortage is to plan the routes and sc hedules already taking in to accoun t the lik eliho o d of finding parking. This implies solving a com bined optimization problem of c ho osing b oth route, sc hedule and parking in a w a y that minimizes the o v erall cost sub ject to practical constrain ts. The HOS regulations men tioned previously pla y an imp ortan t part in the truc k driv er sc heduling problem, so this section fo cus on metho ds that tak e them in to accoun t. As this is not a concern for passenger v ehicles, the problems of parking guidance and trip planning for passenger v ehicles are v ery differen t from routing & sc heduling of commercial truc ks. Therefore, literature on passenger v ehicles w as not included in this section. The v ehicle routing problem ( VRP ) and its v arian ts ha v e b een studied for sev eral decades and man y metho ds to solv e them ha v e b een dev elop ed. Surv eys on the differen t metho ds dev elop ed to solv e the VRP and its extensions can b e found in [111, 112]. Curren t researc h has already started taking in to consideration the HOS rules for a few v ariations of the VRP . Some authors treat the com bined routing and sc heduling problem, whereas others fo cus only on the sc heduling for a giv en route. Due to the problem complexit y , most researc hers use heuristic metho ds based on dynamic programming or searc h algorithms to solv e the com bined routing and sc heduling problem. The largest differences b et w een the approac hes are whether they are lo oking for feasible or optimal sc hedules, if they consider sp ecific parking lo cations for the driv er, and whic h HOS rules are tak en in to accoun t. Xu et al. [113] considered a pic kup and deliv ery problem and presen ted a dynamic programming approac h to find a feasible HOS-complian t sc hedule. Ho w ev er, they did not consider sp ecific parking lo cations. So the problem is treated as if the truc ks could park at an y p oin t along the route, whic h is not true in practice. This assumption is also used in [19, 20, 22, 32, 27, 28, 29]. In some cases the authors consider that driv ers can tak e breaks only at clien t lo cations [30, 25, 23]. In [19, 20], the authors treat a simplified v ersion of the pic k-up and deliv ery problem, and presen t searc h algorithms that can find a feasible sc hedule. In [23], the sc heduling problem w as mo deled as a mixed in teger linear programming problem that minimizes the total trip duration and solv ed using dynamic programming. In [25], the author addresses the lac k of time-dep endency in other approac hes and presen ts a dynamic programming heuristic for the time-dep enden t VRP , whic h 32 accoun ts for the traffic congestion in b et w een clien t lo cations. [26] analyzed the v ehicle departure time optimization as a p ost-pro cessing problem to the VRP with time windo ws and then prop osed an in tegrated solution to heuristically solv e the VRP with time-dep enden t tra v el times and HOS rules. In [30], Gaddy applied a mo dified Clark e-W righ t Sa vings Heuristic to a VRP in order to generate HOS complian t routes with certain restrictions to parking lo cations. The clien t lo cations are also used as parking, but only part of the clien t lo cations are made a v ailable for this purp ose. In [24, 31], the author fo cus on the en vironmen tal impact asso ciated with the differen t idling options. The driv ers can only stop in rest areas, whic h ha v e differen t t yp es of infrastructure a v ailable. The cost function accoun ts for the t yp e of idling used in eac h stop giv en the equipmen t installed in the truc k and the infrastructure a v ailable at eac h rest area. [29] presen ted the only exact metho d for the routing and sc heduling problem. The problem w as mo deled as a shortest path problem with resource constrain ts o v er an extended auxiliary net w ork and w as solv ed using a branc h and price algorithm. These approac hes usually limit the planning horizon to less than a w eek, so some HOS rules are still not fully considered. Rest areas are considered, or can b e included, in some mo dels, but they w ould ha v e to b e lo cated along a route already defined. Although some progress has b een ac hiev ed in in tegrating the HOS rules in the trip planning and taking in to accoun t existen t rest areas, up to no w only the static information of the rest areas’ existence w as considered. No w ork that used an estimated parking a v ailabilit y when calculating the routes and sc hedules of commercial truc ks w as found during this surv ey . 2.6 Conclusion This pap er presen ted a surv ey of recen t efforts on truc k parking, whic h include studies and tests on system mo deling, resource allo cation, routing in tegration and sensing infrastructure, together with some relev an t w ork in urban parking. T able 2.2 con tains a list of the men tioned pap ers and rep orts whic h fo cused sp ecifically on the truc k industry . W e used real data from a truc k parking reserv ation system to compare the p erformance of differen t prediction mo dels. The results sho w ed that prediction mo dels p erform m uc h b etter than simple historical a v erages. Nev ertheless, efforts on dev eloping suc h mo dels are rather limited. In 33 order to effectiv ely use these mo dels in ITP applications, suc h as routing & sc heduling, more w ork is needed for their tuning, v alidation and testing based on real data. In addition to truc k parking issues suc h as sensing, prediction, etc., in tegrating parking a v ail- abilit y in to truc k routing is an imp ortan t area of researc h whic h can pro vide optimized decisions. T able 2.2: T ruc k P arking related w ork T opic Sub-topic Reference Sensing Infrastructure Video Detection [64, 61, 62, 66, 63] Magnetic Sensor [52, 51, 50] GPS Data [69, 68] Laser sensors [58, 56] Resource Allo cation Negotiation [90, 89] Mo del & Prediction Demand Mo del [95, 93, 94] Occupancy Prediction [51, 96, 97, 63, 98] Routing & Sc heduling Unrestricted P arking Lo cations [113, 32, 27, 28, 29] Restricted P arking Lo cations [25, 26, 30, 31] Sc heduling F easibilit y [19, 20, 22] Optimalit y [23, 24] 34 Chapter 3 Assumptions Although truc k parking information is not ubiquitous, progress is b eing made in dev eloping the tec hnology necessary for the implemen tation of in telligen t truc k parking systems and some pilot sites are already in place. Unless otherwise stated, the remaining c hapters consider the follo wing assumptions: 1. USA HOS regulations; 2. P arking a v ailabilit y information will b e a v ailable for a sufficien t n um b er of truc k parking lo cations in the near future; 3. T ruc k parking o ccupancy lev els do not oscillate significan tly at nigh t; 4. Not all clien ts allo w driv ers to rest or w ait at their facilit y . 3.1 USA’s Hours of Service Regulations The USA HOS regulation restricts for ho w long driv ers can driv e/w ork, and ho w long they should rest b efore b eing allo w ed to driv e again. It differen tiates b et w een driving time, on-dut y time and off-dut y time. In summary , driving time is the time sp en t op erating the truc k, on-dut y time is the time from when the driv er is required to b e ready for w ork un til he/she is reliev ed from w ork, and off-dut y is the time when the driv er is not on-dut y . W e refer to the off-dut y p erio ds required b y the regulation based on their minim um duration: br e aks (0.5h), daily r ests (10h) and we ekly r ests (34h). Note that the longer off-dut y p erio ds can b e used to reset the restrictions related to the shorter ones. The USA HOS regulation can b e summarized as follo ws [114]: • 11-hour Driving Time Limit: A driv er ma y driv e at most 11 hours b et w een 2 consecutiv e daily r ests . • 14-Hour Elapsed Time Limit: A driv er cannot driv e after 14 hours ha v e elapsed since the last daily r est ended. • Rest Breaks: A driv er m ust tak e a br e ak after 8 cum ul ativ e hours of driving time. Recen t c hanges in the regulation allo w this constrain t to b e satisfied b y an y non-driving p erio d of 30 35 consecutiv e min utes. The studies in c hapters 4 and 5 w ere p erformed b efore the c hanges in the regulation and th us consider elapsed time instead of driving time, and the constrain t can only b e reset b y off-dut y p erio ds [115]. • 60-Hour Limit: A driv er cannot driv e after ha ving b een on dut y for 60 hours in an y p erio d of 7 consecutiv e da ys. The 7 da ys p erio d can b e reset b y taking a we ekly r est . W e do not consider the sleep er b erth pro vision, whic h allo ws daily r ests to b e split. 3.2 Information a v ailabilit y This assumption is made based on the fact that man y lo cations already ha v e op erational truc k parking managemen t systems, e.g. the priv ate truc k stops that pro vided the data in [98], and the Midw est facilities participating in MAASTO’s system [78]. It is reasonable to assume that as the tec hnology matures it will b ecome more efficien t and accessible. Therefore, information a v ailabilit y should increase in the follo wing y ears. 3.3 Lo w turno v er at nigh t This assumption is used to justify the c hoice of mo deling parking a v ailabilit y as time-windo w constrain ts. Surv ey ed driv ers and parking o ccupancy data rep ort a prev alence of truc k parking issues at nigh t, and data on driv ers’ duration of sta y suggest that nigh t stops are predominately long. Giv en this information, w e h yp othesize that truc k stops turno v er is lo w during the nigh t p erio d, and once they are full or o v ercapacit y , they are lik ely to sta y at a similar o ccupancy lev el for most of the nigh t. Although some v ariation in parking o ccupancy migh t o ccur due to driv ers that arriv ed v ery early or that are taking short stops, these sp ots are lik ely to b e quic kly filled. Therefore, w e think that, for the purp ose of small scale parking recommendation systems, truc k parking a v ailabilit y can b e mo deled as time-windo ws. 3.3.1 Occupancy Profile A ccording to a surv ey b y the USDOT [1], difficult y in finding parking is fo cused on the o v ernigh t p erio ds, as can b e seen in Figure 3.1, and o v ernigh t parking demand is seen as the main problem b y most studies. In [93], Fleger et al. fo cus on the estimating parking demand b et w een 10pm and 6am, whic h is said to b e the p erio d of p eak demand. The results and data presen ted b y Heinitz 36 Figure 3.1: Time of Da y When Driv ers Exp erienced Most Difficult y in Finding Safe P arking. A T A = American T ruc king Asso ciations; OOID A = Owner Op erator Indep enden t Driv ers Asso ciation. Source: [1] and Hesse in [95] also fo cused on the o v ernigh t p erio d and sho w ed that the studied facilit y w as w orking o v er capacit y for most of the nigh t. The data presen ted b y W ang and Garb er in [94] suggest that truc ks’ a v erage duration of sta y is longer at nigh t, and that o ccupancy at most truc k stops increased throughout the da y . Ho w ev er, the authors note that truc k stops on one of the high w a ys studied b eha v ed differen tly , ha ving t w o p eak p erio ds (afterno on and ev ening) and ha ving lo w o ccupancy early morning. The authors h yp othesized that driv ers prefer not to sta y o v ernigh t in that high w a y . Data presen ted b y Sadek et al. in [96] sho ws a cyclic o ccupancy profile with p eaks at nigh t and v alleys in the middle of the da y , and a similar pattern is sho wn in the example data from [98]. In [98], T a v afoghi et al. analyzed data from 29 truc k parking lo cations. It w as v erified that the arriv al rate of truc ks requiring short stops p eaks around no on, whereas the arriv al rate of truc ks requiring long stops p eaks around 6pm. The parking spaces o ccupied b y long stop arriv als w ould tak e significan tly longer to b e v acated, causing parking o ccupancy to remain high throughout most of the nigh t. This b eha vior supp orts the o ccupancy profiles v erified in other studies, and is consisten t with the data rep orted b y the USDOT in [1] (see Figure 3.2). 37 Figure 3.2: T w en t y-four Hours P arking A ccum ulation Profile. Source: [1] 3.4 Driv ers ma y not b e allo w ed to rest or w ait at clien t lo cations This assumption motiv ates our c hoice to restrict parking to only rest areas and to blo c k early arriv als at clien t lo cations and rest areas. A ccording to [1, 116], short-term staging due to w arehouse or terminal hours is a source of truc k parking demand. In fact, Jason’s La w, a bill to implemen t a pilot program to establish truc k parking facilities, w as named after a truc k driv er that w as m urdered while resting at an abandoned gas station just 12 miles from the deliv ery lo cation, whic h w as not y et op en to receiv e deliv eries [1]. Therefore, w e b eliev e that the abilit y to accommo date driv ers whenev er they arriv e should not b e tak en for gran ted. 38 Chapter 4 T ruc k Driv er Sc heduling Problem Section based on the publication: • F. Vital and P . Ioannou, “Long-Haul T ruc k Sc heduling with Driving Hours and P arking A v ailabilit y Constrain ts,” in 2019 IEEE In telligen t V ehicles Symp osium (IV), v ol. June, jun 2019, pp. 620–625. 4.1 In tro duction A ccording to the U.S. Departmen t of T ransp ortation (USDOT), 36 states are exp eriencing shortages in rest areas [1]. T ruc k parking shortages can lead to illegal parking, dro wsy driving and prolonged time lo oking for parking, affecting negativ ely driv ers’ safet y , op erating costs and the en vironmen t. As expanding the infrastructure w ould require significan t capital in v estmen t, this issue p oin ts to the need for b etter utilization of the existing truc k parking capacit y . One of the requiremen ts for efficien t utilization of the truc k parking capacit y is ha ving enough information ab out the truc k parking system. Recen tly , the topic of smart parking systems has b ecome increasingly p opular, with researc h on parking o ccupancy sensing and information distri- bution systems for b oth passenger v ehicles and truc ks [6, 51, 63]. Nev ertheless, the direct usage of the o ccupancy information b y the driv ers can only help with last min ute adjustmen ts to the sc hedule and migh t not b e as helpful in high-demand areas where all rest areas migh t b e full dur- ing p eak hours. P arking a v ailabilit y information could b e used more effectiv ely if included in the earlier stages of planning, when the route and sc hedule are decided. Ideally , an in tegrated planning system with access to data on the lo cation and predicted a v ailabilit y of all rest areas w ould b e 39 able to c ho ose an optimal route and sc hedule, suc h that the sc hedule satisfies the driving hours regulations and only sc hedules stops at lo cations that are exp ected to ha v e a v ailable parking at the time of arriv al. The truc k driv er sc heduling problem (TDSP) under HOS regulations has b een studied b oth as a part of a v ehicle routing problem [29, 25] and b y itself[19, 20, 22, 23, 24, 26]. Differen t v ersions of the TDSP w ere considered, the main differences b eing: parking restrictions, the HOS regulation considered, and if the solution is optimal or only feasible. In [19, 20, 22, 29], the authors do not consider sp ecific parking lo cations, treating the problem as if truc ks could park at an y p oin t along the route, whic h is not true in practice. In some cases, the authors restrict parking to clien t lo cations [25, 26, 23], and prop ose to mo del rest areas as customer lo cations with zero service time and an un b ounded time-windo w. In [23], a mixed in teger programming (MIP) mo del to the TDSP w as presen ted. This mo del restricts parking to clien t lo cations and considers the common t yp es of restrictions presen t in HOS regulations, allo wing it to mo del or appro ximate differen t regulations. Similar MIP mo dels w ere used in [24, 26]. In [24], K o ç includes an en vironmen tal impact factor dep enden t on the t yp es of idling used in eac h stop. P arking is restricted to rest areas, but they are assumed to b e alw a ys a v ailable. In [26], K ok studied the problem of optimizing the departure time and sc hedule with time-dep enden t tra v el times. Although man y metho ds already restrict parking to suitable areas suc h as customer sites or rest areas, the p ossibilit y of the rest areas b eing full at certain times is not considered. F urthermore, the existing w ork usually limits the planning horizon and/or the total on-dut y time to a v oid the rules that regulate longer trips. In this pap er w e fo cus on the issues of parking a v ailabilit y and long trips HOS regulations. Our first con tribution is in tro ducing the TDSP with parking a v ailabilit y , whic h is a v arian t of the TDSP that assumes that parking lo cations are sub ject to a v ailabilit y constrain ts. The second con tribution is to include the USA HOS rule for long trips, i.e. trips with more than 60 hours of on-dut y time, in the MIP mo del for the TDSP . This pap er is organized as follo ws: 4.2 presen ts the problem description; section 4.3 describ es the MIP mo del; section 4.4 describ es the exp erimen ts and results; and section 4.5 presen ts the conclusion and future w ork. 40 v 0 v 1 v 2 v 3 v 4 d 0;1 d 1;2 d 2;3 d 3;4 [t min 1;1 ;t max 1;1 ] [t min 1;2 ;t max 1;2 ] [t min 1;3 ;t max 1;3 ] [t min 2;1 ;t max 2;1 ] [t min 2;2 ;t max 2;2 ] [t min 2;3 ;t max 2;3 ] [t min 3;1 ;t max 3;1 ] [t min 3;2 ;t max 3;2 ] [t min 3;3 ;t max 3;3 ] [t min 4;1 ;t max 4;1 ] [t min 4;2 ;t max 4;2 ] [t min 4;3 ;t max 4;3 ] Figure 4.1: Simple route with 5 lo cations (origin, 3 rest areas and destination) with 3 time-windo ws eac h. 4.2 Problem Description W e consider the problem of sc heduling the rest stops for a long-haul truc k trip with a kno wn route and a single clien t while taking in to accoun t the USA HOS regulations and estimated parking a v ailabilit y windo ws for all rest areas along the route. It is assumed that the rest areas are lo cated on the route and require no detours to b e accessed. The parking a v ailabilit y time-windo ws are assumed kno wn. The route has n + 1 no des, 2 of whic h are the origin, no de 0, and destination of the truc k, no de n . The other n 1 are rest areas lo cated along the route. F or eac h no de i2f0;1;:::;ng the v ariable x i = (x i;a ;x i;d ) represen ts the arriv al and departure times of the truc k at that no de. Eac h rest area i has T i parking a v ailabilit y time-windo ws [t min i; ;t max i; ] , where 2f1;2;:::;T i g indicates the time-windo w’s index. The time-windo ws restrict the arriv al time at that no de and are only in effect when the truc k has to stop at that sp ecific no de, driving b y it is not constrained b y the time windo ws. F or eac h lo cation and time-windo w, a binary v ariable y i; represen ts if that sp ecific time windo w is b eing used (y es:1, no:0). Driving b y without stopping is represen ted b y the v ariable y i;0 (driv e b y:1, stop:0). The tra v el time d i;i+1 in b et w een no des is considered kno wn and indep enden t of time. The planning horizon is denoted b y t hor . The driv er m ust reac h its destination b efore the sp ecified planning horizon. Figure 4.1 sho ws an example of a route with originv 0 , 3 rest areasv 1 ;v 2 andv 3 with 3 time-windo ws eac h, and a destinationv 4 also with 3 time-windo ws. The sc hedule m ust comply with the HOS regulations describ ed in section 3.1. R is defined as the set of differen t t yp es of rest p erio d describ ed in the regulation. F or eac h r2R , t r defines the minim um duration of that t yp e of rest p erio d. C is the set of constrain ts imp osed b y the regulation. 41 C 1 C is the set of constrain ts con trolling the maxim um elapsed time b et w een off-dut y p erio ds. C 2 C is the set of constrain ts con trolling the maxim um accum ulated driving time b et w een off- dut y p erio ds. C 3 C is the set of constrain ts con trolling the maxim um accum ulated on-dut y time during a rolling time-windo w; the width of the time-windo w for a constrain t c2C 3 is represen ted b y c . In the USA regulation c is 7 da ys, so these rolling time-windo w constrain ts will b e referred to as w eekly constrain ts. F or eac h constrain t c2C , t c is the time limit imp osed b y the regulation and R c R is the set of rest t yp es that can reset this coun ter. The binary v ariable z i;r indicates whether a rest of t yp e r is tak en at lo cation i (y es:1, no:0). The driv er cannot tak e more than 1 t yp e of rest at the same lo cation. If no t yp e of rest is sc hedule for a rest area, the driv er cannot stop there. The departure time from the origin m ust b e within the in terv al [t 0 ;t dep ] . It is assumed that the driv er has b een off-dut y for long enough b efore the departure time, so that all constrain ts’ coun ters are reset b efore departure. T able 4.1 lists all the v ariables and parameters used in the mo del, some of whic h are defined in the follo wing section. 4.3 Mo del A MIP mo del for the TDSP under HOS regulations has b een prop osed b y Go el in [23]. This mo del considers that the prediction horizon is limited to 1 w eek, that driv ers ma y rest at clien t lo cations, and that parking lo cations are alw a ys a v ailable. Our mo del considers similar HOS reg- ulations, but restricts parking to rest areas with a v ailable parking spaces, and includes the USA regulations for longer trips. Section 4.3.1 presen ts the mo del used for short trips whic h includes the parking a v ailabilit y . The w eekly constrain ts are in tro duced on section 4.3.2. 4.3.1 P arking A v ailabilit y Constrain ts Other form ulations mo del all no des in the net w ork as required stops with clien t lo cations ha ving one or more time-windo ws constrain ts, and parking lo cations ha ving a single un b ounded time- windo w. In this mo del w e aim to address the issue of parking a v ailabilit y , so these past approac hes are not directly applicable. W e mo del parking a v ailabilit y as time-windo ws constrain ts for eac h parking lo cation. Ho w ev er, unlik e clien t lo cations, the rest areas are not required stops, and if the driv er is not stopping at a certain lo cation, there is no need to restrict the sc hedule with its parking 42 a v ailabilit y . Therefore, the time-windo ws m ust b e conditioned to the sc heduling of off-dut y p erio ds at the rest areas. The form ulation is as follo ws: Minimize T otal tra v el time =x n;a x 0;d (4.1) sub ject to x i;d +d i;i+1 =x i+1;a 80in1 (4.2) x i;a + X r2R t r z i;r x i;d 81in (4.3) x i;d x i;a +(1y 0; )t hor 81in (4.4) y i;0 + T i X =1 y i; = 1 81in (4.5) T i X =1 y i; = X r2R z i;r 81in1 (4.6) T i X =1 y i; t min i; x i;a 81in (4.7) x i;a t hor T i X =1 [y i; (t hor t max i; )] 81in (4.8) x k;a x i;d t c +t hor k1 X j=i+1 X r2Rc z j;r 80i<kn;c2C 1 (4.9) k1 X j=i d j;j+1 t c +t hor k1 X j=i+1 X r2Rc z j;r 80ikn;c2C 2 (4.10) x i 2 [0;t hor ] 2 ;y i 2f0;1g T i +1 ;z i 2f0;1g jRj 81in (4.11) x 0;d 2 [0;t dep ];y n;0 = 0 (4.12) The ob jectiv e function (4.1) is set to minimize the total trip duration. Co nstrain t (4.2) guaran tees that the arriv al time equals the departure time of the previous lo cation plus the driving time. Constrain t (4.3) states that the v ehicle m ust not depart b efore the arriv al time plus the minim um rest time decided for that lo cation. Constrain t (4.4) con trols what happ ens when the driv er do es not stop at a certain lo cation. If the v ehicle do es not stop at lo cation i , the arriv al time equals the departure time. This constrain t w orks with constrain ts (4.3,4.5,4.6) to assure this. Equalit y will hold when y i;0 = 1 . If y i;0 = 0 , then constrain t (4.4) is alw a ys true as t hor is large. Constrain t 43 (4.5) states that at an y lo cation, either exactly 1 time-windo w is used or the v ehicle do es not stop. Constrain t (4.6) states that the driv er only stops if an off-dut y p erio d is sc heduled. Constrain ts (4.7) and (4.8) c hec k the time-windo ws. Arriv al m ust happ en after the b eginning and b efore the end of the c hosen time windo w. Constrain t (4.9) c hec ks that the time elapsed since the last rest in R c ;c2 C 1 is less than t c . Constrain t (4.10) c hec ks if the accum ulated driving time b et w een rest p erio ds in R c ;c2C 2 is less than t c . Constrain t (4.11) sets the v ariables’ domains, and (4.12) guaran tees that the departure time from the origin is within the required p erio d and that the v ehicle will stop at the destination. 4.3.2 W eekly Constrain ts Some authors limit the planning horizon to less than a w eek and mo del the w eekly constrain t as an accum ulated driving time constrain t o v er the whole trip [24] or b et w een 2 consecutiv e w eekly rests [23]. The form ulation in [24] is unable to deal with trips requiring on-dut y time ab o v e the w eekly limit. Without the planning horizon assumption, the mo del in [23] loses its guaran tee of optimalit y due to not considering the differen t structure of the USA regulation, but still guaran tees a v alid sc hedule. In this mo del, the w eekly constrain t c w can b e defined as a constrain t of the set C 2 as follo ws: c w 2 C 2 ; t cw = 60; R cw =f we ekly r estg . W e will refer to this form ulation as Simplifie d mo del. In the USA, this w eekly constrain t is defined as a 7-da y rolling time-windo w in whic h the driv er cannot driv e after w orking for 60 hours. This restriction can b e mo deled as: i;c (t) =R(tx i;d )R(tx i+1;a ) R(tx i;d c )+R(tx i+1;a c ) 80in1; c2C 3 (4.13) c (t) = n1 X i=0 i;c (t); 8c2C 3 (4.14) c (t)t c 8t2fx 1;a ;x 2;a ;:::;x n;a g; c2C 3 (4.15) where R(t) is the unit ramp function. i;c (t) represen ts the accum ulated driving time generated b y the displacemen t b et w een lo cations i and i+1 at time t , and c (t) represen ts the accum ulated driving time o v er the last c hours at time t , b oth relativ e to constrain t c2C 3 . It is sufficien t to 44 c hec k these constrain ts at the arriv al times x i ;a . If the constrain ts are brok en an ywhere they will also b e brok en at the arriv al time that follo ws. 4.3.2.1 MIP form ulation The accum ulated driving time o v er the last c hours, c (t) , needs to b e ev aluated at all ar- riv al times x j;a , so eac h of its comp onen t functions i;c (t) m ust also b e ev aluated at these times. Constrain ts w ere defined for eac h ev aluated time using the metho d for writing piecewise linear functions in MIP mo dels describ ed in [117]. The domains of the functions i;c (t) are divided in sections according to when the slop e of the function c hanges and auxiliary v ariables are used to write t according to where it is lo cated relativ e to the sections’ b oundaries. F or eac h function and ev aluation time i (x j;a ) , the sets of v ariablesf i;j;p g ,f i;j;q g , andf i;j g are defined as follo ws: i;j;p;c 2f0;1g; i;j;q;c 2 [0;1] 80 i < j n; 0 p 4; 1 q 5; c2 C 3 (4.16) 1 i;j;0;c i;j;1;c i;j;1;c i;j;4;c i;j;5;c 80i<jn;c2C 3 (4.17) i;j;p;c < i;j;p+1;c +1 80i<jn;c2C 3 (4.18) x j;a =x i;d i;j;1;c +d i;i+1 i;j;2;c +( c d i;i+1 ) i;j;3;c +d i;i+1 i;j;4;c +t hor i;j;5;c 80i<jn; c2C 3 (4.19) i;j =d i;i+1 i;j;2 d i;i+1 i;j;4 80i<jn (4.20) j1 X i=0 i;j t c 81jn; c2C 3 (4.21) where i;j = i (x j;a ) . The ’s and ’s are auxiliary v ariables used to mo del the piecewise definition of i;c (t) . The ’s determine in whic h section of the function domaint is, and the ’s define its exact p osition within the section, the indexesp andq represen t the sections. Constrain ts (4.16, 4.17, 4.18) imply that, for a section q , whenev er 0 < ;q < 1 , then ;q = 1; ;q = 1; 8q < q , and ;q + = 0; ;p = 0; 8q + >q;pq . Constrain t (4.19) writes the time instan t to b e ev aluated,x j;a , as a function of the ’s and ’s. Constrain t (4.20) uses the ’s to calculate i (x j;a ) , and constrain t (4.21) calculates, and limits, the accum ulated driving time o v er the mo ving time-windo w relativ e 45 to regulation c2C 3 . This set of constrain ts substitutes constrain ts (4.13), (4.14) and (4.15), and guaran tees that the accum ulated driving time in an y p erio d of c consecutiv e hours is k ept b elo w t c . Due to (4.19) this problem w ould b e a quadratically constrained problem. Ho w ev er, as (4.19) only considersj >i , the v ariables i;j;p;c forp< 2 and i;j;q;c forq< 3 will b e alw a ys 1 and can b e defined as constan ts. This mo del still do es not include the p ossibilit y of using we ekly r ests to reset the constrain t, so it will b e referred to as No R eset mo del. 4.3.2.2 Reset for w eekly constrain t A ccording to USA’s regulation, a driv er ma y restart the 168 consecutiv e hours (7 da ys) p erio d, b y taking an we ekly r est . When this we ekly r est is tak en the system should b e able to set the w eekly accum ulated driving time at the end of that rest to zero and start coun ting again from there. This w as implemen ted using indicator constrain ts con trolled b y the v ariables z i;r . A set of v ariables f i;j g w as created to represen t the accum ulated driving time generated b y all trips starting at lo cationsf0;:::;ig measured at time x j ;a . The form ulation is as follo ws: i;j = 8 > > > < > > > : i1;j + i;j if P r2Rc z i;r = 0 i;j if P r2Rc z i;r = 1 81i<jn; c2C 3 (4.22) 0;j = 0;j ; 81i<jn (4.23) j1;j t c ; 81jn; c2C 3 (4.24) where constrain t (4.22) defines i;j and sets to zero all con tributions from no des b efore lo cation i when an appropriate rest is tak en at lo cation i . This mo del assumes that all constrain ts’ coun ters are reset b efore departure, so (4.23) sets the initial accum ulated driving time to zero. These 3 constrain ts replace constrain t (4.21). In the regulation considered, only we ekly r ests can b e used to reset this constrain t, sojR c j = 1 for c2 C 3 . In this case the conditions turn in to z i;r = 1 and z i;r = 0 . This mo del will b e referred to as R eset mo del. 46 T able 4.1: V ariables V ariables Sym b ol Description x i;a ;x i;d Arriv al/Departure times from lo cation i y i; Used time-windo w at lo cation i ? y i;0 Dro v e b y lo cation i ? z i;r Rest of t yp e r w as tak en at lo cation i ? i;j;c A ccum ulated driving time from a trip departing lo cation i at time x j;a , relativ e to constrain t c2C 3 i;j;c A ccum ulated driving time from trips departing lo cations 0 toi at timex j;a , relativ e to constrain t c2C 3 ; A uxiliary v ariables for ramp constrain ts P arameters Sym b ol Description T i Num b er of time-windo ws at lo cation i t min i; ;t max i; Lo w er/Upp er limit of -th time-windo w at lo cation i R Set of rest t yp es defined in the regulation C Set of constrain ts defined in the regulation t c Time limit related to constrain t c2C c Rolling time-windo w’s width for constrain t c2C 3 R c Set of rest t yp es that can reset constrain t c2C t r Minim um duration for rest of t yp e r2R d i;i+1 T ra v el time b et w een lo cation i and i+1 t hor Planning time horizon t dep Maxim um departure time from the origin 47 4.4 Exp erimen ts 4.4.1 P arking A v ailabilit y Impact This section describ es the exp erimen t used to test the impact of considering a v ailabilit y windo ws for ev ery parking lot along a truc k route. A mo del without the parking a v ailabilit y constrain ts w as used as baseline for comparison. A route, appro ximately 1960Km long, going from San Diego to Seattle using the I-5 high w a y w as c hosen. Data from the FHW A [118] w as used to find rest areas and truc k stops lo cated close to the route and p osition them along the route; 94 truc k stops and rest areas w ere considered. Figure 4.2 sho ws the parking lots along the route (gra y circles), as w ell as the c hosen parking lo cations for the base case (triangles and squares) and for one of the tested scenarios (crosses). This trip requires less than 60 hours of on-dut y time, so the rolling time-windo w constrain ts are not needed. In order to sim ulate parking a v ailabilit y , time windo ws with start and end times normally distributed w ere considered for eac h rest area/truc k stop. The distribution used for the start times had mean 5 hours (5am) and standard deviation of 0.5 hours, and the one for the end time had mean 20 hours (8pm) and standard deviatio n of 1 hour. F or this exp erimen t, a 100 scenarios with differen t parking a v ailabilit y time-windo ws w ere generated. It w as considered that the final destination has daily time-windo ws from 8am to 6pm. The da y w as divided in 8 3-hours long in terv als, and these in terv als w ere used as departure constrain ts. Both mo dels w ere solv ed for eac h pair of departure constrain ts and scenario using the solv er CPLEX. This exp erimen t compared the feasibilit y and the a v erage trip duration of solutions generated b y the t w o mo dels. As the scenarios do not affect the baseline mo del, only one solution w as generated for eac h departure in terv al, the solution’s feasibilit y w as then ev aluated in eac h of the 100 scenarios. Our mo del generates differen t solutions for eac h scenario, so the a v erage cost w as used for comparison. Our mo del includes additional constrain ts in the problem, and do es not relax the existing ones. Consequen tly , it is imp ossible for our mo del to reduce the optimal cost (trip duration). The adv an tage of our mo del lies in the practical feasibilit y of its solutions. As can b e seen on Figure 4.3, the sc hedules generated without considering parking are often infeasible, meaning that they sc heduled rest stops at lo cations without a v ailable parking. Ev en if rules of 48 Figure 4.2: Route used on short trip exp erimen t. San Diego to Seattle through the I-5 freew a y . The triangles (base mo del) and + s (new mo del) represen t truc k stops c hosen for daily rests, and the square (base mo del) and (new mo del) represen t the ones c hosen for short breaks. The gra y circles represen t the truc k stops near the c hosen route. 49 (0,3] (3,6] (6,9] (9,12] (12,15] (15,18] (18,21] (21,24] Departure Time Range (hours) 0 20 40 60 80 100 Feasibility Rate (%) Without Parking Constraints Figure 4.3: F easibilit y rate of the sc hedules generated without considering the parking constrain ts. The feasibilit y rate of sc hedules that consider parking constrain ts is alw a ys 100%, so it w as omitted. th um b w ere used to c ho ose departure constrain ts that impro v e feasibilit y , suc h measures can ha v e a significan t impact on the total duration/cost of the trip, as seen on Figure 4.4. When the departure w as restricted to earlier times the feasibilit y impro v ed, ho w ev er the cost deteriorated. Figure 4.4 also sho ws that the a v erage cost of our mo del’s solution is not significan tly higher than the baseline mo del. Therefore, our mo del is able to guaran tee the feasibilit y of its sc hedules with only a small impact to the cost. In fact, in this exp erimen t, if w e do not restrict the departure time, the a v erage cost of our solutions is practically equal to the optim um cost of the baseline mo del. It is also imp ortan t to note that these costs for the baseline mo del w ere generated as if the driv er did not incur an y p enalt y for not finding parking. In practice the driv er w ould ha v e to either k eep driving lo oking for parking or park illegally somewhere nearb y . In the first case, the searc h time itself w ould already increase the trip duration, and it migh t cause other c hanges in the rest of the sc hedule, further increasing the cost. In the second case, the driv er is sub ject to the p ossibilit y of b eing fined and to higher safet y risks. The estimation of the extra costs incurred b y infeasible sc hedules, in particular the second case, is non-trivial and w as not treated in this pap er. 4.4.2 Long T rips In order to test the long trip mo dels, a route w as generated with equally spaced truc k stops, the tra v el time b et w een t w o adjacen t truc k stops w as set to 1 hour. Lik e in the previous exp erimen t, 50 (0,3] (3,6] (6,9] (9,12] (12,15] (15,18] (18,21] (21,24] Departure Time Range (hours) 46 48 50 52 54 56 58 60 Average Trip Duration (hours) Without Parking Constraints With Parking Constraints Figure 4.4: A v erage trip duration of sc hedules generated with and without parking constrain ts. normal distributions w ere used to generate time-windo ws for eac h truc k stop. The distribution used for the start times had mean 4 hours (4am) and standard deviation of 1 hour, and the one for the end time had mean 21 hours (9pm) and standard deviation of 2 hours. It w as considered that the truc k m ust depart from the origin during the first 24 hours and that the final destination has daily time-windo ws from 8am to 6pm. This exp erimen t tested the p erformance of the 3 mo dels, R eset , No R eset , and Simplifie d , whic h w ere presen ted on section 4.3. 4.4.2.1 P erformance It can b e seen on Figure 4.5 that when the reset option is not implemen ted the total trip duration automatically increases to more than 1 w eek when the needed driving time is larger than the w eekly limit. When the reset is implemen ted the trip duration only increases b y the duration of the w eekly rest needed to reset the coun ter. This is the reason wh y the Simplifie d mo del is lik ely to find an optimal sc hedule for the USA regulations. In general, it is more efficien t for the driv ers to tak e the 34 hours rest and reset the coun ter than to reduce their a v erage daily driving hours to matc h the rolling time-windo w. Ho w ev er, this is not necessarily true for ev ery scenario. In a scenario with more restrictiv e time constrain ts, the need to extend off-dut y p erio ds b ey ond the minim um required in order to meet said constrain ts w ould lo w er the a v erage daily driving hours, making we ekly r ests not as adv an tageous. Figure 4.5 also sho ws, as ’TW Bound’, the lo w er b ound 51 0 20 40 60 80 Total Driving Time (h) 0 50 100 150 200 Total Trip Duration (h) No Reset Reset Simplified TW Bound Figure 4.5: A v erage trip duration of sc hedules calculated b y the 3 metho ds with v arying total driving time, and the lo w er b ound of solutions that use the rolling time-windo w. The v ertical dotted line marks the on-dut y time w eekly limit (60h) and the horizon tal dotted line represen ts a trip duration of 1 w eek (168h). for solutions that need the rolling time-windo w to b e found, i.e. solutions that assign at least 1 p erio d longer than 168 hours without a we ekly r est and with more than 60 hours of driving time. This lo w er b ound w as obtained b y solving this problem without restricting the parking lo cations, but forcing the solution to ha v e at least 1 in terv al with more than 60 hours of driving without a we ekly r est . As this b ound is dep enden t only on the HOS regulation and the total driving time, it can b e calculated off-line to b e used for comparison when needed. The solution generated b y the Simplifie d mo del has the minim um cost among the subset of solutions that do not use the rolling time-windo w constrain t. Therefore, if its cost is smaller than the lo w er b ound of the cost of solutions that use the rolling time-windo w constrain t, w e can guaran tee that this solution is optimal. F or the tested scenarios, the results for the Simplifie d mo del w ere optimal and significan tly lo w er than the b ound. F or sc hedules that exceed this b ound w e can only sho w what is the maxim um p ossible impro v emen t to the solution if the R eset mo del w ere used, and use this information to decide whether to accept the curren t solution or try to impro v e it b y using the R eset mo del. 4.4.2.2 Complexit y Figure 4.6 sho ws that the a v erage solv e times for all 3 mo dels are almost the same when the n um b er of lo cations used is smaller than 61. Due to fixed spacing b et w een lo cations used in the 52 40 50 60 70 80 Number of Locations 0 500 1000 1500 2000 2500 Solve time (s) No Reset Reset Simplified Figure 4.6: Solv e time of the 3 presen ted metho ds, with v arying n um b er of lo cations and total tra v el distance. exp erimen t, at 61 lo cations the total driving time reac hes the w eekly driving limit (60h) and the w eekly constrain ts start b eing needed. The solv e times for the R eset mo del rises sharply after that threshold. The solv e times for the other mo dels also increase, but at a slo w er rate. Although the R eset and No R eset mo dels ha v e a similar n um b er of constrain ts and v ariables, the indicator constrain ts mak e the R eset mo del’s solv e time increase significan tly faster. Unexp ectedly , ev en though the Simplifie d mo del has a notably smaller n um b er of v ariables and constrain ts, the v ariation of its solv e time w as v ery inconsisten t and did not sho w a significan t impro v emen t compared to the No R eset mo del. Nev ertheless, its solv e time is still shorter than the R eset mo del while finding solutions of same or similar costs, whic h are significan tly b etter than the No R eset solutions. The oscillations seen in the solv e time plot of Figure 4.6 sho w that the problem is significan tly harder to solv e in certain scenarios. This is most lik ely caused b y ho w the rest areas spacing and total driving time matc h with the regulations. W e b eliev e that the solv e time is v ery sensitiv e to the configuration of the rest areas, and that those oscillations will c hange for other configurations and regulations. 4.5 Conclusion In this study , a MIP mo del for the TDSP with parking a v ailabilit y w as presen ted and extended to include the w eekly constrain ts of the USA regulation. Moreo v er, the effects of the inclusion of parking a v ailabilit y and w eekly w orking hours constrain ts to the p erformance and complexit y of the mo del w ere studied. 53 The exp erimen ts sho w ed that b y including parking a v ailabilit y constrain ts, our mo del guaran tees the practical feasibilit y of the sc hedules with minimal impact on the cost for the driv er/compan y , whereas the practical feasibilit y of other mo dels’ solutions is compromised b y the risk of not finding parking. Ho w ev er, using only the simplified w eekly constrain t w as found to b e more adv an tageous compared to using the complete constrain t due to scalabilit y issues, except when the driv er is exp ected to consisten tly driv e less than the allo w ed daily limit. T o the exten t of our kno wledge, this w as the first w ork to consider the time-dep enden t parking a v ailabilit y of the rest areas and the rolling time windo w nature of the USA HOS regulation for long trips in the TDSP . As long as data is a v ailable, this mo del could b e used b y driv ers to b etter plan their sc hedules, b oth off-line and during the trip whenev er new data came in. In future w ork, it w ould b e in teresting to consider the impact of eac h truc k’s sc hedule on the parking a v ailabilit y and optimize sc hedules for large fleets. Another topic whic h w as not treated and can also b e pursued as future w ork is accoun ting for the uncertain t y in parking a v ailabilit y . 54 Chapter 5 Shortest P ath and T ruc k Driv er Sc heduling Problem Chapter based on the submitted pap er: • F. Vital and P . Ioannou, “Sc heduling and Shortest P ath for T ruc ks with W orking Hours and P arking A v ailabilit y Constrain ts,” T ransp ortation Researc h P art B: Metho dological 148 (2021): 1-37, doi:10.1016/j.trb.2021.04.002 5.1 In tro duction In 2015, a surv ey b y the F ederal High w a y A dministration iden tified truc k parking shortages in 36 US states, with more pronounced issues along ma jor trade corridors [1]. The lac k of truc k parking can ha v e significan t impact on road safet y , industry costs, and the en vironmen t [3, 4], and rank ed among the truc king industry’s top concerns in recen t surv eys b y the American T ransp ortation Researc h Institute (A TRI) [5]. Due to the Hours-of-Service (HOS) regulations, truc k driv ers are required to tak e regular breaks. These rules aim to ensure that driv ers tak e adequate rest and a v oid fatigue-related acciden ts. Ho w ev er, when appropriate parking is scarce, driv ers ma y find themselv es ha ving to c ho ose b et w een driving b ey ond the legal limits or parking in unauthorized and often unsafe lo cations, suc h as high w a y shoulders and freew a y ramps. In recen t surv eys, most driv ers rep orted using unauthorized parking lo cations at least once a w eek [8, 1, 3, 9]. As truc k crashes can b e v ery costly [10, 11], suc h practices ma y lead to significan t losses for the truc king industry due to p oten tial acciden ts. A ccording to a study b y the Virginia Departmen t of T ransp ortation, 25% of all truc k-related crashes along Virginia’s ma jor corridors o ccurred on en trance and exit ramps [12]. Although the data is not sp ecific to park ed truc ks, it sho ws that parking on ramps p oses a significan t safet y risk. T ruc k 55 insurance premiums ha v e increased in recen t y ears and represen ted 5% of the a v erage marginal op erational cost of truc king (not including w ork ers comp ensation costs/insurance, ph ysical damage, jury a w ards, and out-of-court settlemen ts) in 2019 [13]. This is in part due to recen t increases in w ork ers comp ensation claims, settlemen ts, and jury a w ards, whic h at times can surpass $10 million [14]. These gro wing financial risks push truc king companies to reev aluate safet y and ho w m uc h risk they are willing to tak e. T ruc k parking shortage costs go b ey ond safet y-related ones [4, 8]. Surv eys rep ort that driv ers often sp end more than 30 min utes lo oking for parking [16, 17]. Although a surv ey b y A TRI [8] rep orted that most surv ey ed driv ers sp en t less than 15 min utes lo oking for parking, the driv ers had, on a v erage, park ed one hour earlier than required b y the regulation, whic h also con tributes to decreasing their daily rev en ue-earning miles. Stopping for rest early or sp ending a long time lo oking for parking is inefficien t use of driv er’s time. As most truc k driv ers are not paid b y the hour, this can ha v e a significan t impact on their comp ensation [8]. Sp ending long times lo oking for parking also means higher fuel consumption, increasing b oth costs and emissions. Besides, as driv ers often idle the truc k’s engine to p o w er their appliances, when truc ks end up parking near residen tial areas, the emissions generated can significan tly affect the region’s air qualit y [18]. An imp ortan t asp ect of truc k parking a v ailabilit y is its time dep endence. Although driv ers often rep ort difficulties finding parking and truc k stops rep ort op erating o v ercapacit y , they usually refer to the p erio d b et w een 7 PM and 5 AM when driv ers are lo oking for o v ernigh t parking [1, 17, 8, 9]. This suggests that it ma y b e p ossible to mitigate the truc k parking shortage b y encouraging driv ers to plan their stops on off-p eak p erio ds, th us balancing the demand. Balancing the demand with resp ect to space ma y also b e an option. Sev eral factors influence driv ers’ c hoice of parking lo cation; ho w ev er, lac k of information can lead driv ers to park illegally ev en when there are facilities with a v ailable parking nearb y [3]. In [1], less than 50% of truc k stops rep orted w orking o v ercapacit y . Man y of the facilities op erating under capacit y ma y b e in regions with lo w parking demand, y et, some ma y b e in high-demand areas and are underutilized b ecause driv ers do not kno w they are viable options. With this in mind, it is w orth considering to include parking a v ailabilit y earlier in the planning pro cess. Instead of only pro viding driv ers information on facilities along their path to allo w for on-trip decisions, if parking information is considered when c ho osing the path itself, the existing parking capacit y could b e b etter utilized. Dep ending on the situation, it ma y b e 56 adv an tageous for the driv er to tak e a longer route if that can guaran tee a v ailable parking. This suggestion of addressing parking issues at the supply c hain lev el has b een brough t up b efore in [8, 33, 7]; ho w ev er, literature on the topic is still scarce. Although planning metho ds that accoun t for HOS rules, the truc k driv er sc heduling problem (TDSP), ha v e b een extensiv ely studied in the truc k sc heduling literature [19, 20, 21, 22, 23, 24, 119], the issue of truc k parking a v ailabilit y b eing time-dep enden t w as only considered b y Vital and Ioannou in [119]. The usual assumptions are that an y v alid parking lo cation is a v ailable 24/7, and that driv ers can arriv e at clien t lo cations as early as needed and w ait un til their deliv ery time- windo w. Giv en the truc k parking shortage in man y States, w e b eliev e that suc h assumptions can lead driv ers to lo cations that are unable to accommo date them at their arriv al time. Therefore, as in [119], w e consider that parking is restricted in b oth time (only within certain time-windo ws) and space (only at rest areas), and that early arriv als at clien t lo cations are not allo w ed. Ho w ev er, [119] studied a sc heduling problem where the path to b e tak en is giv en, whereas, in this pro ject, w e address a more general v ersion of the problem where the path tak en m ust also b e optimized. While HOS rules ha v e b een studied in the con text of shortest path problems in [29, 21, 120], these w orks do not accoun t for parking a v ailabilit y information. The metho ds presen ted b y Go el and Irnic h [29], and b y Drexl and Prescott-Gagnon [21], consider that driv ers ma y rest an ywhere along the path, whereas a recen t study b y Ma y erle et al. [120] restricts parking to appropriate facilities but do es not accoun t for the parking shortage. HOS-complian t sc hedules, efficien t path c hoices and safe parking conditions (time and lo cation) are all imp ortan t asp ects of planning for long-haul truc king. Although these topics ha v e b een stud- ied separately , w ork com bining t w o of them is v ery limited (i.e. HOS rules and parking a v ailabilit y in [119], and HOS rules and path planning in [120]), and the in tersection of all three topics has y et to b e in v estigated. Our purp ose is to study the in teraction b et w een HOS-complian t sc heduling, path planning and time-dep enden t parking a v ailabilit y , and the imp ortance of considering all three when facing truc k parking shortages. 57 5.1.1 Scien tific con tributions and structure In this study , w e fo cus on the issue of in tegrating time-dep enden t parking a v ailabilit y informa- tion in to long-haul truc k planning (path and sc hedule). The main con tributions of the study are the follo wing: First, w e in tro duce the shortest path and truc k driv er sc heduling problem with park- ing a v ailabilit y constrain ts (SPTDSP-P A). The SPTDSP-P A extends the T ruc k Driv er Sc heduling Problem (TDSP) b y adding a parking a v ailabilit y and a path optimization comp onen t. Eac h park- ing facilit y in the road net w ork has a set of parking a v ailabilit y time-windo ws that restrict when driv ers can park. Multiple paths ma y exist b et w een t w o consecutiv e clien ts, eac h one with its o wn set of parking facilities. Second, w e prop ose a resource-constrained shortest path form ulation for the SPTDSP-P A along with a tailored lab el-correcting algorithm used to find an optimal solution. Third, w e analyze the impact that parking constrain ts ha v e on trip duration and compare it to estimated p oten tial costs of disregarding parking during planning. The truc k parking shortage has b een recognized as a safet y concern b y the USA [1] and the Europ ean Union [121]. In addition, estimates p oin t to a substan tial impact in the econom y [121, 4, 122]. Nev ertheless, researc h on freigh t planning accoun ting for parking a v ailabilit y remains v ery limited. The same is true for w a ys to estimate ho w parking a v ailabilit y relates to trip duration, costs, and illegal parking. Our study pro vides a to ol for p olicymak ers to estimate ho w the curren t truc k parking infrastructure affects truc k driv ers’ abilit y to safely comply with the HOS regulations. The abilit y to estimate the a v erage cost of follo wing a safe sc hedule at differen t parking a v ailabilit y lev els can b e used to supp ort decisions on parking infrastructure sp ending. F rom the industry’s standp oin t, our mo del allo ws driv ers and companies to impro v e their op erations’ safet y standards and estimate their costs more accurately . This study is organized as follo ws: Section 5.2 reviews the relev an t literature. Section 5.3 presen ts the problem addressed. Section 5.4 describ es the prop osed form ulation. Section 5.5 de- scrib es the lab el-correcting algorithm used. Section 5.6 presen ts a case-study used to ev aluate the impact of the prop osed metho d. Section 5.7 presen ts the exp erimen ts used to measure the algorithms p erformance. Section 5.8 presen ts the conclusion. 58 5.2 Literature Review The inclusion of HOS rules in sc heduling algorithms, the truc k driv er sc heduling problem (TDSP), w as approac hed in man y studies in recen t y ears [19, 20, 22, 23, 24, 119]. Multiple regu- lations ha v e b een considered, including ones from the United States [22], Europ e [20] and Canada [123]. F urthermore, it is often studied as part of a v ehicle routing and truc k driv er sc heduling prob- lem (VR TDSP) [25, 26, 27, 28, 30, 31], whic h is a v arian t of the v ehicle routing problem (VRP) that accoun ts for HOS rules, and, less commonly , it is studied in the con text of shortest path problems (SPP) [21, 29, 120]. Besides the particular metho ds used, the differences b et w een problems treated in the literature usually relate to the follo wing asp ects: regulation considered, optimalit y of the solutions, parking restrictions, cost function, and main problem (TDSP , VRP or SPP). W e are most in terested in ho w they approac hed parking restrictions and path planning. 5.2.1 P arking restrictions Although truc k parking is curren tly a critical issue, it is often o v erlo ok ed in the literature, with man y metho ds not ev en restricting parking to appropriate facilities. In [19], Arc hetti et al. considered the problem of determining whether a sequence of n full truc kload transp ortation requests is feasible giv en a set of HOS regulations and pic k-up time-windo ws. The prop osed metho d allo ws driv ers to park an ywhere and finds a feasible sc hedule in O(n 3 ) time. In [20], Go el considered a similar problem using the Europ ean regulations, and in [22], presen ted an algorithm to find feasible sc hedules to visit n lo cations using the US regulations in O(n 2 ) time. Ho w ev er, these metho ds assumed that driv ers could park an ywhere, whic h is not v alid in practice. This assumption is also presen t in [32, 21, 27, 28, 29]. In [23], Go el presen ted a mixed in teger programming (MIP) form ulation and a dynamic programming algorithm for the TDSP that restricts parking to clien t lo cations and calculates a sc hedule with minim um tr ip duration. Rest areas w ere mo deled as clien ts with zero service time and un b ounded time-windo ws. Similar MIP mo dels w ere used in [26, 24, 119], fo cusing on differen t asp ects of the problem but k eeping parking restricted to appropriate facilities. In [26], K ok et al. addressed the issue of traffic congestion b y considering time-dep enden t tra v el times and prop osed a heuristic approac h to in tegrate the TDSP mo del in to a VRP metho d. In [24], K o ç et al. approac hed the en vironmen tal impact caused b y truc k idling and ho w it is affected b y 59 the truc k’s equipmen t and rest areas’ infrastructure. The driv ers can only park at rest areas, whic h ha v e differen t t yp es of infrastructure a v ailable. Early arriv al is allo w ed at clien t lo cations, but it do es not coun t as off-dut y time. The cost function accoun ts for the t yp e of idling used in eac h stop giv en the equipmen t installed on the truc k and the infrastructure a v ailable at eac h rest area. This metho d w as later used as a base for a VR TDSP algorithm with the same fo cus [31]. In [119], Vital and Ioannou approac hed the issue of truc k parking a v ailabilit y and US HOS rules for long trips. Their mo del considered a single clien t trip, whic h hinders driv ers’ abilit y to plan consecutiv e trips. P arking w as restricted to rest areas, and parking a v ailabilit y w as mo deled as time-windo w constrain ts for eac h rest area. Eac h rest area’s a v ailabilit y time-windo ws tak e effect only if a stop is sc heduled for that particular lo cation. Due to the fo cus on parking a v ailabilit y issues, the mo del assumed that parking is una v ailable outside of the deliv ery time-windo w and did not allo w early arriv al at the clien t or rest areas. As short-term staging due to w arehouse or terminal hours is a source of truc k parking demand [1, 116], w e see the restriction on early arriv als (also included in our mo del) as an imp ortan t distinction when considering truc k parking shortages. This study is the only one that considered time-dep enden t parking a v ailabilit y in the TDSP . Nev ertheless, as [119] addresses only the sc heduling problem, it do es not accoun t for alternativ e paths or parking lo cations that require a detour to b e reac hed. This limitation motiv ates the other asp ect of our w ork: path planning. 5.2.2 P ath planning The inclusion of parking constrain ts and HOS regulations when determining the shortest path b et w een lo cations is relev an t not only to individual driv ers that need to plan their itineraries, but also to carriers and other stak eholders that need to estimate op erational costs and allo cate resources. Hence wh y w e are in terested in the shortest path problem with resource constrain ts (SPPR C) that lies b et w een the TDSP and the VR TDSP . VR TDSP metho ds assume that the shortest path b et w een an y t w o clien ts is kno wn (and indep enden t of the curren t status of the HOS constrain ts), and use TSDP algorithms to calculate the cost of eac h route generated. The rest areas considered in these problems are lo cated along these kno wn shortest paths. If the driv er is allo w ed to rest an ywhere or only at clien t lo cations, this assumption do es not affect the route cost. Ho w ev er, when parking 60 is restricted and rest areas are considered, the minim um cost path b et w een t w o clien ts will dep end on the lo cation of ev ery reac hable rest area and the HOS constrain ts’ status at the departure time from the clien t. The inclusion of parking a v ailabilit y constrain ts mak es it ev en more imp ortan t to consider alternativ e paths and rest areas. When parking is scarce at the usual routes, it ma y b e cost-effectiv e to tak e a sligh tly longer path if it has b etter parking conditions. F ailing to consider ho w parking a v ailabilit y and HOS constrain ts affect the shortest path b et w een clien ts ma y cause planners to underestimate the trip’s duration and cost. This inaccuracy can upset op erations planning as w ell as fair driv er rem uneration (dep ending on ho w w ages are determined). The issue is aggra v ated when driv ers lac k the flexibilit y to adjust their route, as some of the driv ers surv ey ed in [124]. In this case, the driv er is limited to taking a sub-optimal route, further increasing the difference b et w een estimated and actual trip cost and duration. The shortest path problem with resource constrain ts (SPPR C) often app ears in column gen- eration solutions to the VRP [125] and sev eral approac hes ha v e b een prop osed for its v arian ts [29, 126, 127, 128, 129]. The SPPR C is often solv ed through dynamic programming-based lab eling algorithms, applying tailored dominance rules and b ound estimates to iden tify and discard inferior paths. SPPR C form ulations and algorithms are tailored to their o wn problem v arian ts and ma y not b e directly applicable to other problems. Hence the need to dev elop tailored metho ds for the SPPR C in the con text of HOS regulations and parking a v ailabilit y constrain ts. Ho w ev er, the n um- b er of studies using SPPR C form ulations in the con text of HOS-complian t planning is v ery limited. In [21], Drexl and Prescott-Gagnon presen t a SPPR C form ulation to the problem of finding HOS- complian t routes and sc hedules, and prop ose exact and heuristic lab eling algorithms. In [29], Go el and Irnic h prop ose an exact metho d for the VR TDSP using a branc h and price algorithm where a SPPR C is used to generate HOS-complian t routes and their costs. An auxiliary net w ork is used to mo del driv ers’ p ossible activities, but parking lo cations are not considered. Ev en though they consider HOS regulations, b oth [29] and [21] assume that driv ers ma y stop and rest an ywhere on a route. This limitation is partially addressed in [120], where Ma y erle et al. study the impact of Brazilian regulations in the planning of long-haul full truc kload shipmen ts. Differen tly from [21, 29], this study is not aimed at deciding whic h clien ts to visit and in what order for a VR TDSP , but at ho w c hanges to HOS rules affect the b est path to reac h a clien t. They use a lab eling algorithm and pruning heuristics to optimize the path a truc k tak es to reac h a single clien t, while sc heduling stops 61 at allo w ed lo cations to satisfy regulations. Their mo del includes some time-restrictions to all rest stops b y restricting departure times at the b eginning of eac h w ork da y , as w ell as the start time of lunc h breaks. Ho w ev er, they also o v erlo ok the question of whether those parking lo cations will b e a v ailable at the desired times. In addition, it shares the same single clien t limitation as [119]. 5.3 Problem Description The problem consists of planning a single truc k’s minim um cost path and sc hedule from the origin to an ordered set of clien t lo cations, while complying with the USA HOS regulations, clien t service time-windo ws and only sc heduling off-dut y time at truc k parking lo cations (TPL) whic h are exp ected to ha v e a v ailable parking at the time of arriv al. The road net w ork’s tra v el times are fixed and kno wn. P arking a v ailabilit y is mo deled as time-windo ws for eac h TPL, within whic h the driv er is guaran teed to find parking. The parking a v ailabilit y time-windo ws are assumed kno wn. The collection of data and statistical analysis required to generate reasonable time-windo ws is b ey ond the scop e of this study and is not addressed. Some recen t w ork on parking prediction can b e seen in [99, 51, 96, 130, 63, 98]. An imp ortan t difference b et w een this study and some of the sc heduling pap ers men tioned earlier, suc h as [23, 24], is that w e do not allo w early arriv al at no des with time-windo w constrain ts. In [23], the driv er w as allo w ed to arriv e early at a clien t lo cation and w ait un til the start of the service time-windo w. This w aiting time w as treated as an off-dut y p erio d and, when long enough, could b e used to satisfy HOS rest requiremen ts. In [24], the driv er w as also allo w ed to arriv e early at a clien t, ho w ev er, the w aiting time is treated as on-dut y time and could not b e used to satisfy the HOS rest requiremen ts. Both studies set un b ounded time-windo ws for TPLs, so early arriv als at these facilities w ere not a concern. In this study , w e assume that TPLs and clien ts can only accommo date new v ehicle arriv als within the time-windo ws. Therefore, allo wing early arriv al w ould b e equiv alen t to telling the driv er to stop outside the parking or customer facilit y and w ait un til it b ecomes a v ailable, whic h go es against the ob jectiv e of a v oiding stops at inappropriate lo cations. A ccording to [1, 116], short-term staging due to w arehouse or terminal hours is a source of truc k parking demand. Therefore, w e b eliev e that the abilit y to accommo date driv ers whenev er they arriv e should not b e tak en for gran ted. If a clien t allo ws driv ers to arriv e early , this should b e 62 explicitly mo deled as a TPL righ t b efore the customer lo cation. This TPL should ha v e its o wn time-windo ws to limit ho w early the driv er ma y arriv e, and p ossibly include restrictions on the duration of sta y . The problem is solv ed o v er a road net w ork that includes only the main routes the truc k can tak e for that sp ecific trip, and the TPLs around them. T ruc ks ha v e road restrictions due to size and w eigh t, so using the full road net w ork w ould include a lot of sup erfluous information. In addition, the order in whic h clien ts m ust b e visited can also b e used to narro w do wn the set of relev an t paths. The truc k route net w ork is generated from the full net w ork b y calculating a set of paths b et w een ev ery t w o consecutiv e clien ts, then connecting eac h path to the set of nearb y TPLs to b e considered. The n um b er of paths considered, as w ell as the relev an t TPLs for eac h path, is at the user’s discretion. App endix F presen ts a simple heuristic that could b e used to c ho ose the n um b er of paths and its impact on solution qualit y and running time. The net w ork is defined as an acyclic directed graph G = (V;A) , where V is the set of no des of the graph and A is the set of arcs. The no des represen t lo cations of in terest in the road net w ork, as TPLs, clien t lo cations, in tersections, and road branc hing sp ots. The arcs represen t road sections, and eac h (i;j)2A is assigned a fixed tra v el time d ij , and length l ij . Figure 5.1 sho ws an example net w ork, the no des with a n um b er index are road no des and the ones with a letter index are TPLs. The edges connected to TPLs are represen ted as dashed arro ws and the main paths as con tin uous arro ws. v 0 v 7 v 1 v 2 v 3 v 4 v 5 v 6 v 8 v a v b v c 1 Figure 5.1: Example of simplified road net w ork. The no des with a n um b er index are road no des (in tersections, branc hing or merging sp ots) and the ones with a letter index are TPLs. 5.4 Mo del The SPTDSP-P A is mo deled as a resource-constrained shortest path problem [129], where the time, cost, and coun ters for the differen t HOS regulations are treated as resources. This section 63 presen ts the extended net w ork and system dynamics used to mo del the problem presen ted in section 5.3. 5.4.1 Extended Net w ork The net w ork in tro duced in section 5.3 describ es the road sections and lo cations considered in the problem, but it do es not p ortra y the activities that tak e place at eac h lo cation. In order to represen t the differen t activities and decisions in v olv ed in the problem, w e define an extended net w ork G 0 = (V 0 ;A 0 ) . No des represen ting lo cations where non-driving activities tak e place are expanded to explicitly include these activities as edges in the graph. Figure 5.2 sho ws the subnet w orks that will replace the expanded no des. Eac h edge has its activit y indicated b elo w the arro w. Non-driv e edges that ha v e a fixed duration ha v e their duration indicated ab o v e the arro w. The incoming/outgoing edges of the subnet w ork are the incoming/outgoing edges of the expanded no de. Rest Areas Off-dut y p erio ds of differen t durations satisfy differen t HOS constrain ts, so rest area no des are expanded to include a differen t path for eac h t yp e of off-dut y p erio d. In this case, only 3 t yp es are considered. They are br e ak, daily r est and we ekly r est , whic h ha v e minim um durations of t b , t r and t w , resp ectiv ely . Figure 5.2a sho ws the subnet w ork represen ting the no de after expansion. The duration of the second half of eac h path is a non-negativ e decision v ariable used to mo del rest time b ey ond the minim um required. W e only restrict arriv al time, so time-windo ws that w ere assigned to the original rest area no de are assigned only to the en trance no dev in i . Ho w ev er, if a parking facilit y closes at certain times and requires v ehicles to lea v e b efore then, this restriction can b e mo deled b y a time-windo w at the exit no dev out i . Rest areas with restrictions on the duration of sta y can b e mo deled b y restricting the v alues that the second half of eac h path can tak e and b y remo ving paths with minim um duration exceeding the limit (e.g. if a rest area do es not allo w stops longer than 4 hours, the paths for daily and we ekly r ests can b e remo v ed, and the second half of the br e ak path restricted to at most 3.5 hours ). In the remainder of the c hapter, the cen tral no des of eac h path, v b i ,v r i and v w i will b e referred to as br e ak, daily r est and we ekly r est no des, resp ectiv ely . Let V b , V r and V w represen t the set of all br e ak, daily r est and we ekly r est no des, resp ectiv ely . Origin The c hoice of departure time from the Origin no de do es not ha v e the same effect as the c hoice of departure time from TPLs. W e assume it do es not affect HOS resources, but that it migh t affect cost. Therefore, it m ust b e treated differen tly . The Origin no de is replaced b y the subnet w ork in Figure 5.2b to mo del the v ehicle’s departure time. Departure time constrain ts can b e mo deled either b y the set of allo w ed v alues for the duration of that edge, or as a time-windo w constrain t on the departure no de v dep . Clien ts Clien t no des are expanded as sho wn in Figure 5.2c. The service edge in the subnet w ork is used to mo del the service time at eac h clien t. As the regulation differen tiates b et w een driving and on-dut y time, service time m ust b e treated differen tly from the driving time. The time-windo ws that w ere assigned to the original clien t no de will b e assigned only to the en trance no de v in i . 64 v in i v r i v b i v w i v out i t r daily rest daily rest t b break break t w weekly rest weekly rest drive drive (a) Sub-net w ork used to expand rest area no des. v 0 v dep departure drive (b) Sub-net w ork used to expand the origin no de. v in i v out i w i service drive drive (c) Sub-net w ork used to expand clien t no des. w i is the service time. Figure 5.2: Sub-net w orks used to mo del non-driving activities. 65 5.4.2 System Equations As the driv er’s sc hedule m ust comply with the HOS regulations, eac h regulation constrain t is mo deled b y a differen t resource that m ust b e k ept b elo w the limits describ ed in the regulation throughout the whole path. The resources considered are: • curren t time ( 0 ) • cost (c ) • elapsed time since last br e ak ( b ) • elapsed time since last daily r est ( r ) • accum ulated driving time since last daily r est ( r ) • accum ulated on-dut y time since last we ekly r est ( w ) Eac h state of the system can b e describ ed b y the tuple x = (i;) , where i2V 0 is the curren t no de and = ( 0 ;c; b ; r ; r ; w ) lists the curren t resource v alues. Eac h arc(i;j)2A 0 is assigned a set of allo w ed durations ij and a lengthl ij . When(i;j) represen ts a road section, then ij =fd ij g and l ij is the length of the asso ciated arc from A . F or the other t yp es of arcs ( br e ak, daily r est , we ekly r est , dep artur e , servic e ), presen t only inA 0 ,l ij is zero. If the arc has a fixed duration, ij is a single elemen t set con taining that duration. If the arc has a v ariable duration, lik e the departure arc and the arcs used to define off-dut y p erio d extensions, ij = [0;1) . This in terv al can b e further restricted to a v oid un w an ted decisions b eing c hec k ed, e.g. br e aks should not b e extended to the p oin t where they are longer than a daily r est ’s minim um duration. Let x k = (v k ; k ) , where k = ( 0 k ;c k ; b k ; r k ; r k ; w k ) , represen t the system’s state after k deci- sions. Let U(x k ) represen t the set of feasible decisions at state x k . Eac h decision is describ ed b y a tuple u k = (v k+1 ; k ) , where v k+1 is the next no de to b e visited, and k is the time required to reac h it. The ev olution of the system is describ ed b y: x k+1 =f(x k ;u k ); 8x k 2X;u k 2U(x k ); (5.1) U(x k ) =fu k = (v k+1 ; k )jf(x k ;u k )2X; x k = (v k ; k );(v k ;v k+1 )2A 0 ; k 2 v k v k+1 g; (5.2) where X is the set of feasible states. As the resources are affected differen tly b y the activities in v olv ed, the function f() is defined separately for eac h activit y . F urthermore, as the definition 66 T able 5.1: Resource Extension F unctions f d f s f b f r f w f 0 0 k+1 = 0 k + k c k+1 c k = d k + d k s k b k r k w k 0 k b k+1 = b k + k 0 b k r k+1 = r k + k 0 r k r k+1 = r k + k r k 0 r k w k+1 = w k + k w k 0 w k of f() is trivial for the up date of the next no de, i.e. it is alw a ys equal to the first elemen t of the decision tuple, this part will b e omitted. The resource up date rules are mo deled b y a set (one for eac h activit y) of resource extension functions (REF), whic h are describ ed in T able 5.1. These functions tak e the resource v ector k , the duration k and length k = l v k v k+1 of the c hosen arc as argumen ts, and return the new resource v ector k+1 . The functions f d , f s , f b , f r , f w and f 0 are used for the activities drive , servic e , br e ak, daily r est , we ekly r est and dep artur e , resp ectiv ely . F or example, if edge (v k ;v k+1 ) ’s activit y is drive , then b k+1 = b k + k , but if (v k ;v k+1 ) ’s activit y is br e ak , then b k+1 = 0 . The assignmen t of activities to the arcs of the extended net w ork is describ ed in section 5.4.1. The cost is mo deled as a linear com bination of the distance tra v eled and the time sp en t in eac h activit y throughout the trip. d , s , b , r , w and 0 are the hourly costs applied to activities drive , servic e , br e ak, daily r est , we ekly r est and dep artur e , resp ectiv ely . W e assume that 0 < b = r = w . d is the cost p er kilometer tra v eled, it is applicable only to the activit y drive . Related w orks usually use total trip duration, on-dut y time, tra v el distance or a com bination of those factors as the cost, so our form ulation is flexible enough to mo del most cost functions found in the literature. 5.4.3 Constrain ts There are t w o t yp es of constrain ts, time-windo w constrain ts, used to mo del arriv al time re- strictions at clien t lo cations and parking a v ailabilit y at TPLs, and HOS constrain ts to mo del the compliance to HOS regulations. A state x k = (v k ; k ) is only feasible if it satisfies all constrain ts. Belo w w e describ e eac h constrain t: 67 5.4.3.1 Time-Windo w Constrain ts Eac h no de i represen ting a clien t lo cation or TPL has a set of T i disjoin t time-windo ws. Eac h time-windo w is defined b y a tuple (t min i; ;t max i; ) represen ting the minim um and maxim um arriv al times allo w ed b y t hat time-windo w, where 2f1; ;T i g is the index of the windo w. A state x k with v k =i satisfies the time-windo ws constrain ts if and only if 0 k 2 T i S =1 [t min i ;t max i ] . 5.4.3.2 HOS Constrain ts Unless the driv er exp ects to b e on-dut y for less than 8.6 hours p er da y (60/7) on a v erage, taking we ekly r ests is more efficien t than not. Therefore, a simplified v ersion of the 60-Hour Limit w as considered in this pap er. Instead of restricting the on-dut y time o v er an y p erio d of 7 consecutiv e da ys, the on-dut y time b et w een t w o consecutiv e we ekly r ests w as restricted to 60 hours. This greatly simplifies the implemen tation of the 60-Hour limit, while still guaran teeing regulation compliance. Eac h HOS related resource has a maxim um allo w ed v alue defined b y the regulation in tro duced in section 3.1. Let t eb b e the limit for elapsed time b et w een br e aks , t er the limit for elapsed time b et w een daily r ests , t ar the limit for accum ulated driving time b et w een daily r ests , and t aw the limit for accum ulated on-dut y time b et w een we ekly r ests . Then a feasible state x k m ust satisfy: b k t eb ; r k t er ; r k t ar ; w k t aw (5.3) Note that the regulation describ ed in Section 3.1 restricts driving, but not other w orking activities. So, if a clien t has a parking facilit y that do es not require driving to b e reac hed, that parking facilit y can b e mo deled as a TPL righ t after the clien t exit no de and the HOS restrictions could b e relaxed for those particular clien t exit and TPL en trance no des. The time limits ma y b e exceeded during service, but the driv er w ould b e able to rest b efore driving, so the sc hedule w ould still satisfy the HOS regulations. 5.4.4 SPTDSP-P A F orm ulation The ob jectiv e is to find a minim um cost route and sc hedule that satisfies the ab o v e constrain ts. The cost is mo deled b y the resourcec . LetX d represen t the set of terminal states, i.e. the feasible 68 states at the destination no de. W e refer to the no de with no outgoing edges inG 0 as the destination no de. It is assumed that all terminal states are absorbing, i.e. f(x;u) =x;8x2X d . Let n b e the maxim um n um b er of decisions required to reac h the destination from the Origin no dev 0 . AsG 0 is an acyclic directed graph, n is finite and b ounded b y the n um b er of no des jV 0 j . Giv en an initial state x 0 , the SPTDSP-P A is form ulated as: min u 0 ;;u n1 c n (5.4) s.t. x k+1 =f(x k ;u k ); k = 0;1; ;n1 (5.5) x k 2X; u k = (v k ; k )2U(x k ) (5.6) where X is the set of feasible states. The ob jectiv e function (5.4) minimizes the total trip cost c n , defined as the cost resource at the last state, x n . Constrain t (5.5) con trols the ev olution of the system. Constrain t (5.6) defines the domains of the v ariables used. U(x k ) is defined in (5.2), and feasibilit y is defined in Section 5.4.3. In the follo wing section w e presen t a metho d for solving the SPTDSP-P A problem (5.4)–(5.6). 5.5 Lab el-Correcting Metho d This section describ es the dynamic programming-based lab el-correcting metho d used to solv e the SPTDSP-P A. An o v erview of resource constrained shortest path problems, including lab eling algorithm approac hes, can b e found in [129, 126]. The general idea b ehind lab el-correcting metho ds is to progressiv ely calculate the shortest path from the origin no de to all other no des un til the shortest path to the destination is found. Due to the large n um b er of paths generated, it is necessary to iden tify and discard inefficien t paths as so on as p ossible. 5.5.1 Ov erview This subsection explains the general w orking of the algorithm. Figure 5.3 sho ws the w ork-flo w of the algorithm used. T o ev ery partial solution (u 0 ;u 1 ; ;u k1 ) going from the origin no de v 0 to a no de v k 2V 0 w e assign a lab el consisting of the resource v ector k at no de v k , the slac ks that 69 will b e describ ed in Section 5.5.2.2, and information necessary to reconstruct the partial solution. F or simplicit y , w e refer to a lab el b y its asso ciated resource v ector k . Lab el Choice Let OPEN b e the list of un treated lab els. OPEN is initialized with the initial state’s lab el and w e use the small lab els first metho d [131] to c ho ose whic h lab el to treat first. New lab els are inserted at OPEN ’s start if smaller than the first lab el, and at the end otherwise. A t eac h iteration OPEN ’s first lab el is pic k ed and expanded to generate new lab els. Lab els are compared lexicographically b y their resources, follo wing the order ( 0 ;c; w ; r ; r ; b ) . W e giv e priorit y to resources that are reset less often. Giving priorit y to the arriv al time ( 0 ) also helps with the dominance c hec k as a lab el can only b e dominated b y lab els with lesser or equal arriv al time. The metho d used to manage OPEN is represen ted in Figure 5.3 b y the ‘Searc h Metho d’ blo c k. Expansion Eac h lab el describ es a certain state of the system (and also con tains information on the partial solution that generated it). The ‘expansion’ step consists of heuristically c ho osing a set of feasible decisions and applying them to that lab el, th us generating new lab els/partial solutions. This step is represen ted in Figure 5.3 b y the ‘Expand P artial Solution’ blo c k, and, as sho wn in the diagram, dep ends on sev eral factors. The set of feasible decisions at an y giv en state dep ends on the net w ork top ology and on the regulations, whic h include b oth the HOS and time-windo w/parking constrain ts, b eing considered. Ideally , all feasible decisions should b e tested during expansion. Ho w ev er, our mo del also has a n um b er of con tin uous decision v ariables, making it imp ossible to test all of them. The ‘Expansion Criteria’ blo c k refers to the heuristics defining ho w the set of feasible decisions is sampled. In addition, the A algorithm and dominance rules are also used to sp eed up computation b y discarding inefficien t partial solutions b efore their lab els are inserted in to OPEN . Lab el Impro v emen t Due to the usage of heuristics during no de expansion, the partial solutions generated ma y con tain inefficiencies. Our algorithm k eeps trac k of these inefficiencies and, at no des where the driv er can c ho ose to extend an off-dut y p erio d, it c hec ks whether future lab els can b e impro v ed b y up dating upstream decisions. This approac h is based on ho w Go el trac k ed unnecessary off-dut y time to adjust infeasible partial solutions in [23]. Under certain conditions, b y c hanging the departure time from the origin or the duration of some upstream rest stops, it is p ossible to reac h a particular no de at the same time but sp ending few er resources or at a lo w er cost. This step is represen ted b y the ‘Up date Upstream Lab els’ blo c k. Dominance Chec k W e sa y that a lab el is dominated when it cannot generate solutions that are b etter than the ones generated b y another lab el. A lab el can only dominate or b e dominated b y a lab el of the same no de, so the algorithm also k eeps a list of un treated lab els separated b y no de to sp eed up the dominance c hec k. When new lab els for rest no des are created, the algorithm p erforms dominance c hec ks b et w een the new lab el and the existing ones. If the new lab el is dominated b y an existing one, it is not inserted in to OPEN . An y lab els dominated b y the new one are discarded. In order to reduce the computation load, dominance is c hec k ed only on rest no des, other no des’ lab els are alw a ys accepted. T ermination Conditions The algorithm stops if it finds a solution that is close enough to the estimated lo w er-b ound, or if OPEN is empt y . 70 Open Set Get Next Label update upstream labels (if decision has controllable duration) Create Label Discard Dominance Check (for rest node labels) LB check Expand Partial Solution Destination? Search Method Regulations Initial Condition Empty? Dominance Criteria Expansion Criteria Network UB Estimator LB Estimator Add Label Stop Within tolerance Exceeds tolerance Stop Yes No Yes No Label Correcting Algorithm Problem and Method Parameters Check A* Condition Pass Fail Fail Pass Figure 5.3: Lab el-correcting algorithm w orkflo w diagram. 71 5.5.2 P artial Solution Expansion 5.5.2.1 Expansion Criteria If the no de b eing treated has no time duration decision, then there is only a finite n um b er of p ossible decisions and all feasible decisions should b e tested. If a time duration decision is required, i.e. at rest no des and at the origin, then it is necessary to c ho ose whic h decisions to test. LetA d A 0 represen t the set of all arcs with driving as their assigned activit y . F or ev ery no de pair (p;q) suc h that there is a directed path from p to q , letD(p;q) andD d (p;q) b e, resp ectiv ely , the minim um tra v el time (including service time) and minim um driving time b et w een no desp and q with all resource, time-windo w and HOS constrain ts relaxed: D(p;q) = 8 > > < > > : min( pq ); if (p;q)2A 0 min (p;k)2A 0 (min( pk )+D(k;q)); o/w (5.7) D d (p;q) = 8 > > > > > > < > > > > > > : 0; if (p;q)2A 0 nA d min( pq ); if (p;q)2A d min (p;k)2A 0 (D d (p;k)+D d (k;q)); o/w (5.8) If there is no directed path from p to q , thenD(p;q) =D d (p;q) =1 . Consider the lab el i = ( 0 i ;c i ; b i ; r i ; r i ; w i ) relativ e to a partial solution ending at no de v i . LetN(x i ) represen t the set of no des that can b e reac hed from state x i without resting, giv en b y: N(x i ) = v2V 0 jD(v i ;v)( i )^D d (v i ;v)t ar r i (5.9) ( i ) = min(t eb b i ;t er r i ;t aw w i ) (5.10) where ( i ) represen ts ho w m uc h time is left un til one of the HOS constrain ts related to elapsed time ( b i ; r i ) or on-dut y ( w i ) time is brok en. As the constrain ts considered in ( i ) are affected b y service time, ( i ) is compared toD(v i ;v) . The accum ulated driving time since last daily r est ( r ) is not affected b y service time, so t ar r i is considered separately and compared to 72 D d (p;q) . Let ~ x i = (v i ; ~ i ) represen t a mo dified x i where the resources already accoun t for p oten tial lab el impro v emen ts. This is used to a v oid ignoring decisions that are only feasible after lab el impro v emen t. Lab el impro v emen t is describ ed in section 5.5.2.2, and ~ i is defined in (5.17). When expanding i , for eac h time-windo w of eac h no de inN(~ x i ) , the shortest decision that can generate a path that will satisfy the time-windo w is tested. The set b U(x i ) of feasible decisions to b e tested is describ ed b y: b U(x i ) = n (v;)2U(~ x i )j9l2N(~ x i );2f1; ;T l g;s:t: =min B(x i ;v;l;) o (5.11) B(x i ;v;l;) = n y2 v i v 0 i +y +D(v;l)2 t min l ;t max l o (5.12) where B(x i ;v;l;) represen ts the duration of decisions passing through no de v that, giv en the curren t state x i , can generate paths that reac h no de l within the time-windo w t min l ;t max l . 5.5.2.2 Lab el Impro v emen t This step targets sp ecifically no des that ha v e a time duration decision, i.e. rest no des and the origin. These no des ha v e an outgoing arc with con trollable length, whic h is used to define if and ho w m uc h the driv er will w ait at the rest stop after completing the minim um required off- dut y p erio d, and when to depart from the origin. While driv ers ma y need to extend their rest duration to accommo date time-windo w constrain ts, it is imp ortan t to notice that the w aiting times at differen t no des are not equiv alen t. One example is that extending a br e ak is more ‘exp ensiv e’ than a we ekly/daily r est , b ecause br e aks affect more resources. In terms of w ait time ‘cost’ w e can rank the no de t yp es as follo ws: origin < we ekly/daily r ests< br e aks . So, when a w aiting time is needed, it is more efficien t to try to extend the w aiting time at the last upstream no de of a ‘c heap er’ t yp e. In order to do so w e k eep trac k of ho w m uc h w e can extend the w ait times of the origin no de and of the last upstream we ekly/daily r est no de without affecting the feasibilit y of solutions generated. Let N d denote the set of no des that ha v e a time duration decision, and N t the set of no des that ha v e time-windo w constrain ts. Consider a partial solution with path (v 0 ;v 1 ; ;v j ) , let i b e the duration of the decision tak en at no de v i , 0 i the arriv al time at no de v i , and [a i ;b i ] the time-windo w b eing used at no de v i , suc h that a i 0 i b i . The sets of slac k v ariablesf i;j g and 73 f i;j g are used to trac k b y ho w m uc h i can b e up dated. The v ariable i;j stores the surplus of off-dut y time b et w een no des v i andv j , whic h can b e in terpreted as b y ho w m uc h i can b e increased without affecting 0 j . i;j stores b y ho w m uc h i can b e extended without pushing 0 k outside of the time-windo w [a k ;b k ] for all i<kj . They are defined as follo ws: i;j = 8 > > > > > > < > > > > > > : 0; if j =i+1 i;j1 + j1 ; if j >i+1 and v j1 2N d i;j1 ; if j >i+1 and v j1 = 2N d (5.13) i;j = 8 > > > > > > > > > > < > > > > > > > > > > : 1; if j =i+1 and v j = 2N t i;j1 ; if j >i+1 and v j = 2N t b j 0 j + i;j ; if j =i+1 and v j 2N t min( i;j1 ;b j 0 j + i;j ); o/w (5.14) where v i 2 N d and i < j . W e define i;i = i;i = 0 . F or eac h generated lab el, the ’s and ’s stored are the ones relativ e to the origin and the last we ekly or daily r est . As can b e seen in the REF s on T able 5.1, w aiting at these t yp es of no des sp ends few er resources than w aiting at br e ak no des. In terms of resource exp enditure when extending an off-dut y p erio d, we ekly and daily r ests are equiv alen t, so only the information regarding the most recen t one is stored. Let (v l ;H)2 b U(x j ) b e one of the decisions c hosen to expand state x j = (v j ; j );v j 2 N d , and let l = ( 0 l ;c l ; b l ; r l ; r l ; w l ) b e the lab el generated b y this decision. If min( 0;j ; 0;j ) > 0 , then the w aiting time 0 at no de v 0 can b e increased to 0 as follo ws: 0 = min( 0;j ; 0;j +H)+ 0 (5.15) A new lab el 0 1 is created for no dev 1 using the new decision (v 1 ; 0 ) . Instead of putting this lab el in OPEN , it is treated immediately and separate from the others. This lab el will b e expanded passing b y the same no des as the partial solution that is b eing impro v ed, but taking the shortest decisions that will not decrease the arriv al time at an y in termediate no de. Note that the in termediate lab els generated are not stored in OPEN , and only one decision is used in their expansion. W e 74 w an t to generate an up dated v ersion of the partial solution b eing impro v ed, not a new tree of partial solutions. This up dated path will generate a lab el 0 l with cost resource c 0 l = c l + ( 0 r )min( 0;j ; 0;j +H) that dominates l . Before storing the lab el in OPEN , w e c hec k if other slac ks can b e remo v ed. Let v r b e the last we ekly or daily r est no de visited, with v r = v 0 if none w as visited. If min( 0;j ; 0;j ) = 0 , but r 6= j and min( r;j ; r;j ) > 0 , then the same pro cedure can b e used to up date r using r;j and r;j . Up dating r reduces r j (elapsed time since last daily r est ) b y min( r;j ; r;j +H) . The impro v ed lab el 0 l is giv en b y: 0 l = ( 0 l ; c l +( 0 r )min( 0;j ; 0;j +H); b l ; r l min( r;j ; r;j +H); r l ; w l ) (5.16) After b oth slac ks ha v e b een remo v ed, 0 l is stored in OPEN . A pseudo co de represen tation of this path up date pro cess is presen ted in App endix E. The ~ i used in Section 5.5.2.1 considers the maxim um impro v emen ts that could b e obtained (i.e. large H ) and is defined as: ~ i = ( 0 i ; c i +( 0 r ) 0;i ; b i ; r i r;i ; r i ; w i ) (5.17) In summary , the algorithm will, at first, only explore the minim um w aiting times necessary to satisfy the time-windo ws of no des reac hable without resting, not considering ho w that affects the stops that follo w. When the algorithm reac hes a state that indicates that earlier decisions can b e impro v ed, the curren t partial solution is up dated according to the decisions that will b e tested at the curren t state. This metho d reduces the n um b er of unnecessary lab els generated and also handles the con tin uous v ariables without discretizing and testing the whole decision space, whic h w ould b e computationally exp ensiv e. App endix D presen ts an optimalit y pro of for the algorithm prop osed. 5.5.2.3 A The A algorithm is used during expansion to discard lab els that cannot generate solutions with cost lo w er than the curren t upp er-b ound. Let v t b e the destination no de, UP b e an upp er- b ound for the optim um solution’s cost, and LOW(v i ;v t ; i ) b e a lo w er-b ound for the cost of a 75 trip from v i to v t with initial resources i . If c i +LOW(v i ;v t ; i ) > UP , then i is discarded. F aster termination can b e ac hiev ed b y setting a tolerance > 0 , and replacing the condition b y c i +LOW(v i ;v t ; i )>UP . By accepting only lab els that ca n impro v e the curren t upp er b ound b y at least , running sp eed is reduced while the solution obtained is k ept within of optimalit y . In order to find a lo w er b ound for the cost of solutions that can b e generated from a giv en state x i = (v i ; i ) , a relaxed v ersion of the problem is solv ed. The minim um driving time from the curren t no de to the destination is used to calculate the minim um trip duration un til the destination, relaxing the time-windo w constrain ts and allo wing the driv er to rest an ywhere. Let D HOS (d;) represen t the minim um duration of a HOS-complian t trip withd driving hours and initial resource v ector , assuming the driv er can rest an ywhere, and without considering service time and time- windo w constrain ts, i.e. if a driv er w ere at the b eginning of an empt y straigh t road of length d km where he/she can rest an ywhere, giv en an initial resource v ector , ho w long w ould he/she tak e to reac h the end of the road without breaking the HOS regulations. Whenev er a new statex i is generated, w e use the minim um driving time from v i to v t ,D d (v i ;v t ) , and the lab el’s resource v ector i to calculate a lo w er b ound D HOS (D d (v i ;v t ); i ) for the time sp en t driving or resting un til v t considering only HOS restrictions. App endix B describ es the metho d used to calculate D HOS (d;) with and without considering the curren t HOS resource v alues. Let D s (v i ;v t ) and D l (v i ;v t ) represen t the service time and minim um distance b et w een no des v i and v t , giv en b y: D s (p;q) = 8 > > > > > > < > > > > > > : 0; if (p;q)2A 0 nA s min( pq ); if (p;q)2A s min (p;k)2A 0 D s (p;k)+D s (k;q) ; o/w (5.18) D l (p;q) = 8 > > > > > > < > > > > > > : 0; if (p;q)2A 0 nA d l pq ; if (p;q)2A d min (p;k)2A 0 D l (p;k)+D l (k;q) ; o/w (5.19) 76 where A s A 0 is the set of servic e arcs. The lo w er b ound LOW(v i ;v t ; i ) for the cost of a trip from v i to v t with initial resource i is giv en b y: LOW(p;q;) = r D r (p;q;)+ s D s (p;q)+ d D d (p;q)+ d D l (p;q) (5.20) D r (p;q;) =D HOS (D d (p;q);)D d (p;q) (5.21) where D r () represen ts ho w m uc h time from D HOS () w ould b e sp en t resting. As D HOS (d;) is indep enden t of the net w ork top ology and time-windo ws, it can b e calculated b eforehand for differen t v alues of d and , and used in an y problem instance. Ho w ev er, discretizing the p ossible v alues of d and and storing the results for ev ery com bination w ould require a lot of storage space. Setting all HOS related resources to zero and discretizing only d when calculating and storing D HOS () greatly reduces storage space, but also generates a lo oser b ound. Another option is to store the results for a limited n um b er of (d;) com binations, and use those results to appro ximate others or accelerate their computation during run time. The form ulation presen ted in app endix B uses D HOS (d;(0;0;:::;0)) to calculate D HOS (d;) for a general . W e tested storing only the results ofD HOS (d;(0;0;:::;0)) and calculating the others during run time, but the lo w er- b ound impro v emen t w as not enough to comp ensate for the extra computations. Therefore, in our exp erimen ts, the lo w er-b ound is calculated without considering curren t HOS resources. The initial upp er b ound can b e tak en from a kno wn sub-optimal solution, or calculated based on the planning horizon and max distance to destination, and up dated as the algorithm finds b etter solutions. If the initial upp er b ound is set to o high, the solv e time can increase significan tly . Ho w ev er, w e noticed that, as the algorithm can quic kly determine that the problem is infeasible when the upp er b ound is to o lo w, it is efficien t to set a lo w upp er b ound and increase it gradually un til the problem b ecomes feasible. 5.5.3 Dominance R ules Let i = ( 0 i ;c i ; b i ; r i ; r i ; w i ) and 0 i = ( 00 i ;c 0 i ; b0 i ; r0 i ; r0 i ; w0 i ) denote lab els for t w o differen t partial solutions ending at the same no dev i . If i dominates 0 i , then for ev ery solution that can b e generated b y expanding 0 i , there is a b etter solution that can b e generated b y expanding i . F or 77 example, assume i = (10;9;7;10;10;10) and 0 i = (10;10;7;10;10;10) , and that neither lab el can b e impro v ed. As the arriv al times are equal, i.e. 0 i = 00 i , an y time-windo w that can b e satisfied starting from 0 i can also b e satisfied starting from i . As all HOS related resources are also equal, no lab el has an adv an tage regarding when a rest stop will b e required. Therefore, as i has a smaller cost (c i = 9; c 0 i = 10 ), this cost adv an tage will b e carried to all paths generated from i , making them c heap er than paths generated from 0 i . The example p ortra ys the base case for dominance c hec k, when 0 i = 00 i and neither lab el can b e impro v ed. In this case, if all resources in i are smaller or equal to the r esources in 0 i , with at least one b eing strictly smaller, i dominates 0 i . Including the effects of p ossible lab el impro v emen ts mak es the conditions a little more complicated. Similar dominance rules w ere used in [23]. Ho w ev er, as they consider that early arriv als are allo w ed and that lab el impro v emen t is alw a ys p erformed b efore dominance c hec ks, these rules cannot b e used in our mo del. As w e do not allo w early arriv als, when 0 i < 00 i , paths generated from i ma y b e unable to satisfy time-windo w constrain ts of do wnstream no des, otherwise satisfied b y paths generated from 0 i , due to arriving to o early . Ho w ev er, b y using the particular structure of the problem, dominance rules for when 0 i 00 i w ere established for we ekly r est , daily r est and br e ak no des. These are the only t yp es of no des with con trollable outgoing arc duration, whic h can b e used to equalize the arriv al times at the next no de. Th us, the dominance conditions w ere deriv ed b y using their REF s to define when ev ery lab el generated from 0 i through a decision (j;) is dominated (after lab el impro v emen ts) b y the lab el generated from i through the decision (j; + ) , where = 00 i 0 i 0 . In order to reduce the n um b er of dominance c hec ks p erformed, only lab els assigned to rest no des (v i 2 V b [V r [V w ) are c hec k ed. As rest no des’ dominance rules are not restricted to lab els with matc hing arriv al times, they ha v e a greater p oten tial for iden tifying and discarding inferior solutions. The dominance rules are giv en b y: v i 2V b [V r [V w (5.22) 0 i + = 00 i (5.23) 0; (5.24) b i b0 i (5.25) 78 r i r0 i (5.26) w i w0 i (5.27) c i c 0 i g c (5.28) r i r0 i g r ; if v i 2V b (5.29) c i c 0 i 0 w 0 r ( +w 0 ) (5.30) r i r0 i w r ; if v i 2V b (5.31) where g c , g r , w 0 and w r are defined as: g c = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 0 w 0 r ( +w 0 ); if 0;i 0;i + 0 (y 0 ) r y 0 ; if ( 0;i > 0;i + )^( 0 0;i 0 0;i ) min 0 (z 0 ) r z 0 ; 0 w 0 r ( +w 0 ) ; o/w (5.32) g r = 8 > > > > > > < > > > > > > : w r ; if r;i r;i + y r ; if ( r;i > r;i + )^( 0 r;i 0 r;i ) min(z r ; w r ); o/w (5.33) w 0 = 0 0;i 0;i ; y 0 = 0 0;i 0;i ; z 0 = 0 0;i 0;i (5.34) w r = 0 r;i r;i ; y r = 0 r;i r;i ; z r = 0 r;i r;i (5.35) where (5.22) defines for whic h no des these rules are applicable. Conditions (5.23)–(5.24) define and restrict its allo w ed v alues, so that lab els can only b e dominated b y lab els with a lo w er or equal arriv al time. Conditions (5.25)–(5.31) describ e the conditions on the cost and HOS resources. (5.25) is alw a ys v alid as this is c hec k ed only at rest no des, where the elapsed time since last br e ak resource is reset, i.e. b = 0 . As the resources represen ting cost (c ) and elapsed time since last daily r est ( r ) are affected b y lab el impro v emen ts, the dominance rule m ust c hec k if the solutions 79 generated from 0 i are inferior to ones generated from i ev en if lab el impro v emen t is p erformed at the curren t no de, or at a do wnstream no de. (5.28)–(5.29) include the effects of the curren tly a v ailable lab el impro v emen ts. (5.30)–(5.31) consider the impact of p oten tial lab el impro v emen ts if the slac ks cannot b e completely used at the next decision. A partial solution migh t app ear b etter at the curren t no de, but b e less flexible to adapt to do wnstream constrain ts, e.g. if i w as generated b y a path with v ery lo ose time-windo ws, it will ha v e more flexibilit y to adjust upstream arriv al times to reduce the amoun t of unnecessary off-dut y time at do wnstream no des. (5.32)–(5.35) define the auxiliary v ariables used to define (5.28)–(5.31). W e can sa y that i dominates 0 i if conditions (5.22)–(5.31) are satisfied, with at least one among (5.25)–(5.31) b eing a strict inequalit y . App endix C describ es in detail ho w the dominance rules w ere obtained. 5.6 Case Study T o ev aluate the prop osed algorithm a set of test scenarios is created using the net w ork sho wn in Figure 5.4. The net w ork is based on a route going from San Diego to Seattle via the I-5 freew a y , and includes some p ossible detours. F or easier visualization, the parking lots (p i no des) are displa y ed along the routes. On the actual graph, they are outside the routes, as in Figure 5.1, to allo w for them to b e b ypassed. F or simplicit y , the distances b et w een the routes and the parking lots are set to 0. W e w an t to study the impact of HOS rules and parking constrain ts on the shortest path to the destination and compare the cost of taking parking a v ailabilit y information in to accoun t with the estimated costs of not doing so. F urthermore, to illustrate the imp ortance of including parking information early in the planning pro cess as opp osed to only doing so at the sc heduling stage, w e 024h 93 80 41 64 68 96 129 64 295 193 290 161 206 161 224 161 81 150 129 79 161 203 80 140 O D v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 p 9 p 10 1 Figure 5.4: Net w ork used for exp erimen ts. Based on a main route going from San Diego to Seattle via the I-5 freew a y indicated in red with double arro ws, together some p ossible detours indicated with blac k arro ws. 80 also study scenarios where the driv er is restricted to the main route. As men tioned in [124], not all driv ers are allo w ed to c ho ose their routes freely . If the carrier determines the route and do es not consider parking a v ailabilit y information during route assignmen t, the trip’s duration and cost ma y b e significan tly larger than estimated b y the carrier, esp ecially if the driv er is not allo w ed to adjust the route. In this case study , w e tak e the trip duration as the cost, and w e assume that there is no extra cost for w aiting at the origin, i.e. = 0 = 0 and d = s = b = r = w = 1 . The follo wing asp ects are considered when ev aluating the algorithm: • What is the estimated cost of not using parking a v ailabilit y information during planning? • Ho w often the minim um duration path includes one or more alternativ e routes? • A v erage driving times under v arying alternativ e routes’ sp eed and in their absence. • A v erage trip duration under v arying alternativ e routes’ sp eed and in their absence. 5.6.1 Scenarios T aking longer routes (in terms of driving time) is b eneficial only if the time sa v ed due to b etter parking a v ailabilit y conditions is larger than the increase in driving time. The exp erimen ts fo cus on studying ho w the problem’s solution is affected b y the usage of parking a v ailabilit y information, b y the main route’s parking a v ailabilit y time-windo ws’ distribution and b y the difference in driv- ing time b et w een the alternativ e routes and the main one (b y v arying the a v erage sp eed on the alternativ e routes). T able 5.2 sho ws the distributions used to sample the start and end times of the three t yp es of parking a v ailabilit y time-windo ws used in the tests. It is assumed that eac h da y has a single time-windo w. The daily parking a v ailabilit y time-windo ws are defined b y sampling a start and an end time from normal distributions with the parameters listed in the table. The parameters giv en are in hours and follo w the 24h format. The route dra wn in red with double arro ws on Figure 5.4 is tak en as the main route. This route has the shortest driving time from origin to destination, so, in the absence of HOS and parking constrain ts, the optimal solution w ould go through this path. All road sections not included in the main route are considered alternativ e routes. F or simplicit y , w e set all main route links’ a v erage sp eed to 75km/h. The a v erage sp eed of alternativ e route links v ary with the scenario. T able 5.3 sho ws the parameters used for eac h set of scenarios, and ho w man y instances w ere tested. Scenarios 10 to 12 restrict the driv er to the main route, so alternativ e routes’ parameters are sho wn as blo cke d . The time-windo w configurations for departure and deliv ery 81 T able 5.2: Exp erimen t P arameters P arking Time-windo ws Distribution Start Time (h) End Time (h) Narro w N (9,1) N (16,1) Medium N (7,1) N (19,1) Wide N (5,1) N (22,1) Other Time-windo ws Start Time (h) End Time (h) Departure (unique) 0 24 Deliv ery at D (daily) 8 16 Deliv ery at v 4 (daily , 2 clien ts case) 12 16 T able 5.3: Exp erimen t Scenarios Time-windo ws A v erage Sp eed (km/h) Scenario Instances Main Route Alternativ e Routes Main Route Alternativ e Routes 1 100 Wide Wide 75 75 2 100 Medium Wide 75 75 3 100 Narro w Wide 75 75 4 100 Wide Wide 75 70 5 100 Medium Wide 75 70 6 100 Narro w Wide 75 70 7 100 Wide Wide 75 65 8 100 Medium Wide 75 65 9 100 Narro w Wide 75 65 10 100 Wide blo c k ed 75 blo c k ed 11 100 Medium blo c k ed 75 blo c k ed 12 100 Narro w blo c k ed 75 blo c k ed 82 times are sho wn in the lo w er part of T able 5.2. The departure time has a single time-windo w on the first da y (0h to 24h), whereas the deliv ery time at the destination ( D ) has daily time-windo ws (8h to 16h). Besides the single clien t case just describ ed, the t w elv e scenarios w ere also tested for the case when no dev 4 is a clien t with a daily time-windo w (12h to 16h) and zero service time. All other parameters are the same as for the t w elv e original scenarios. 5.6.2 Estimating the cost of disregarding parking information As previously stated, the truc k parking shortage has m ultiple negativ e consequences for driv ers, industry and so ciet y . While it is hard to accurately estimate the costs in v olv ed, in this section w e prop ose a recourse function used to sim ulate ho w the driv er w ould react if he/she w ere to arriv e at a parking facilit y and find it una v ailable, as w ell as the costs incurred. F or eac h one of the scenarios explained previously , a solution is generated without including parking a v ailabilit y constrain ts. Then the recourse function is applied to eac h scenario’s solution in order to sim ulate the driv er’s reactions. Three situations are considered: parking is a v ailable, parking is una v ailable and there is another facilit y within reac h, parking is una v ailable and no facilit y is within reac h. P arking is a v ailable: In this case the driv er follo ws the sc hedule. P arking is una v ailable and no other facilit y is within reac h: A new route and sc hedule are generated (also without considering parking a v ailabilit y) starting from the exit of the parking facilit y where the driv er is curren tly at. No cost or time p enalties are applied. P arking is una v ailable and no other facilit y is within reac h: It is assumed that the driv er sp ends 0.5 hours searc hing for parking, so a 0.5h time p enalt y is imp osed. It is also considered that the driv er is lik ely to park at an unofficial parking lo cation, so a cost p enalt y is applied. A new route and sc hedule are generated (also without considering parking a v ailabilit y) starting from the en trance of the parking facilit y where the driv er is curren tly at. The required rest is tak en at the curren t lo cation as if it w as an unofficial parking. In these exp erimen ts, the cost w as measured in hours, so the cost p enalt y is also con v erted to hours. As it is hard to estimate the cost of the financial risk a driv er is undertaking ev ery time he/she parks illegally , a sensitivit y analysis is p erformed with m ultiple p enalt y v alues. The v alues tested are: 2h, 4h, 6h, 8h, and 10h. F or comparison, considering an hourly marginal cost of op eration of $71.78 [13], the 4h ( $287) p enalt y is comparable to the fine for non-emergency stops on freew a ys in California ( $238) [132], and factoring in acciden t risks w ould increase p enalties further. The risk of getting in v olv ed in an acciden t or b eing fined v aries with the region, route, time of da y , and t yp e of v ehicle, as do acciden ts’ a v erage sev erit y and cost. Nev ertheless, when acciden ts do happ en, costs can b e substan tial. In 2005, the a v erage cost of truc k crashes w as $91 thousand o v erall, $195 thousand for injury crashes, and $3.6 million for crashes in v olving fatalities 83 [10]; around 1.2 thousand, 2.7 thousand and 50 thousand hours of op eration, resp ectiv ely . The costs presen ted in [10] are in 2005 dollars, and w ere calculated using 2002 v alues for the V alue of Statistical Life; curren t v alues should b e significan tly higher. Therefore, a 1% acciden t probabilit y w ould result in p enalties higher than 12 hours, already exceeding the tested p enalties. Other indirect costs ma y include a drop in driv er satisfaction, higher insurance premiums, loss of clien ts, etc. Data on all these factors are required for a compan y to accurately estimate the financial risks of illegal parking. In our exp erimen ts, w e assigned the same p enalties to all lo cations, but, with enough data, appropriate v alues can b e set for eac h region/lo cation. 5.6.3 Results 5.6.3.1 Estimated Cost of Disregarding P arking Information Figure 5.5 sho ws the a v erage costs (after adjustmen ts b y the recourse function) of trips that are planned without taking in to accoun t parking a v ailabilit y information. Both the single clien t (Fig. 5.5a) and the t w o clien t (5.5b) cases are sho wn. In the scenarios tested, the a v erage costs did not v ary with the alternativ e routes’ a v erage sp eed, so only the plot s for 75km/h are sho wn, the others can b e assumed equal. This result is lik ely caused b y the net w ork used, and is discussed later in 5.6.3.1. Figure 5.6 sho ws the a v erage trip cost when parking a v ailabilit y information is considered during planning. In this case, as w e assumed that parking a v ailabilit y is guaran teed within the time-windo ws, the sc hedules generated are alw a ys feasible and no p enalt y or rerouting is required. Therefore, the trip cost equals the trip duration. When comparing the plots in Figures 5.5 and 5.6, it can b e seen that the adv an tage of considering parking a v ailabilit y information v aries substan tially with the scenario and p enalties considered. As previously stated, stopping early to guaran tee appropriate parking affects driv er pro ductivit y [8]. The same is true for extending rest p erio ds to guaran tee parking at the next facilit y . The follo wing question can b e raised. Whic h one is more exp ensiv e: w aiting longer to guaran tee appropriate parking, or the financial risks of allo wing irregular parking? When parking is abundan t (wide time-windo ws), accoun ting for parking a v ailabilit y information is alw a ys b eneficial, as the extra w aiting times are not significan t. Ho w ev er, as parking b ecomes 84 scarce, the effect of the w aiting times required to guaran tee parking can b e considerable, esp ecially if alternativ e routes are slo w or nonexisten t. Stricter constrain ts cause longer w aiting times and, consequen tly , higher costs. This effect can b e seen on the cost increase caused b y narro wing time- windo ws, and b y adding a second clien t. Nev ertheless, situations with strict constrain ts are when driv ers need the most help. In these cases, the financial risk of illegal parking is the defining factor. F o cusing on the more restrictiv e t w o-clien ts scenarios of Figures 5.5b and 5.6b, it can b e seen that whether using parking informa- tion is c heap er or not, dep ends on the p enalt y v alues considered. F or example, if driv ers are fined ev ery time they park illegally , but the region they w ork at is v ery safe and no acciden ts or rob- b eries happ en, it w ould put the p enalt y for illegal parking at around 4h of op eration (assumptions explained in section 5.6.2), i.e. the y ello w bar with small circles on Figure 5.5b. In this particular case, if parking is v ery scarce along the m ain route (narro w time-windo ws) it is c heap er to consider parking information only if the tra v el sp eed of the alternativ e routes is 75km/h. If the alternativ e routes are slo w, the driv er w ould most lik ely prefer to park illegally and pa y the fines. The difficult y in accurately gauging this risk ma y lead driv ers and companies to mistak enly assume that it is c heap er to disregard parking constrain ts. Although these p enalties are v ery hard to estimate, the litigious en vironmen t that the truc king industry has b een facing is pushing these costs up [14]. Some insurance companies are already pushing their clien ts for the adoption of safet y tec hnologies, suc h as collision a v oidance or camera systems, and legal parking planning practices ma y b e a go o d complemen t to the industry’s safet y standards. Rerouting b eha vior W e exp ected that in some cases the HOS constrain ts w ould force the driv er to tak e an alternativ e route when rerouting. Ho w ev er, in the tested scenarios, whenev er parking is una v ailable the driv er do es not ha v e enough remaining driving time to reroute and head for a differen t parking facilit y . This causes the driv er to use irregular parking instead of rerouting. After resting, the driv er is less lik ely to tak e an alternativ e route due to HOS constrain ts. As this happ ens for all alternativ e route sp eeds, the a v erage cost do es not v ary with the sp eed parameter. 5.6.3.2 Effect on Shortest P ath 85 Wide Medium Narrow Main Route Time-Windows Width 50 55 60 65 70 Average Trip Cost (h) Alt. Route Travel Speed = 75 Irregular Parking Cost Penalty (h) 2 4 6 8 10 (a) Single clien t Wide Medium Narrow Main Route Time-Windows Width 50 55 60 65 70 75 Average Trip Cost (h) Alt. Route Travel Speed = 75 Irregular Parking Cost Penalty (h) 2 4 6 8 10 (b) T w o clien ts Figure 5.5: A v erage trip cost (when disregarding parking information) for differen t irregular parking p enalties according to the t yp e of parking a v ailabilit y time-windo ws considered for the main route. In this exp erimen t, the results did not v ary with the alternativ e routes’ tra v el sp eed so the plots for other sp eeds w ere omitted. Wide Medium Narrow Main Route Time-Windows Width 50 55 60 65 70 Average Trip Cost (h) Alternative Routes Travel Speed (km/h) 75 70 65 Only Main Route (a) Single clien t Wide Medium Narrow Main Route Time-Windows Width 50 55 60 65 70 75 Average Trip Cost (h) Alternative Routes Travel Speed (km/h) 75 70 65 Only Main Route (b) T w o clien ts Figure 5.6: A v erage trip cost/duration (when using parking information for planning) according to parking a v ailabilit y time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. Single Clien t Figures 5.7, 5.8 and 5.6a sho w ho w often alternativ e routes are used and ho w the a v erage driving time and a v erage trip duration v ary with the parameters used for eac h scenario. It can b e seen that when the main route has wide or medium parking time-windo ws, there is little or no b enefit in taking alternativ e routes. In these cases the main route has enough parking a v ailabilit y and alternativ e routes are rarely used. Ho w ev er, this do es not hold an ymore when the parking a v ailabilit y time-windo ws are narro w. When parking is scarce on the main route, alternativ e routes can significan tly lo w er costs. The usage of alternativ e routes is more pronounced when their tra v el sp eed do es not differ m uc h from the main one’s, but it can still b e seen ev en when the sp eed is lo w er. 86 Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 1% 0% 0% Time-Windows = Wide Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 9% 2% 0% Time-Windows = Medium 75 70 65 Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 98% 60% 10% Time-Windows = Narrow Figure 5.7: Num b er of instances that used alternativ e routes for the single clien t case, according to the time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. It is also imp ortan t to note that the a v eraging dilutes the con tribution of the instances that used the alternativ e paths. F or example, Figure 5.7 sho ws that alternativ e routes are used only 60% of the time when the time-windo ws are narro w and the sp eed is 70km/h, so the 2h impro v emen t in cost seen in Figure 5.6a is actually caused b y only half of the instances. Similarly , the 0:5h increase in driving time seen in Figure 5.8 is also caused b y only half of the instances. So, in the end, there are 60 instances where paths are, on a v erage, 1h longer, but still generate costs 4h shorter on a v erage. When more restrictions are applied and the problem complexit y increases, the b enefits of con- sidering m ultiple paths b ecome more eviden t, as can b e seen on the case with t w o clien ts. T w o Clien ts Figures 5.9, 5.10 and 5.6b sho w ho w often alternativ e routes are used and ho w the a v erage driving time and a v erage trip duration v ary with the parameters used for eac h scenario. In this case, the adv an tage of considering alternativ e routes is clear and they are used significan tly more often when the time-windo ws are medium or narro w. Ev en in the case where the tra v el sp eed of the alternativ e routes is 65km/h and the a v erage driving time is increased b y more than 1 hour, there is an impro v emen t of almost 5 hours to the a v erage trip duration. As exp ected, the b enefits of considering alternativ e routes increase when they are not sig- nifican tly longer than the main route and when the main route has limited parking a v ailabilit y . Moreo v er, it is imp ortan t to note that alternativ e routes can also impact problem feasibilit y . In 87 Wide Medium Narrow Main Route Time-Windows Width 26:0 26:2 26:4 26:6 26:8 27:0 Average Driving Time (h) Alternative Routes Travel Speed (km/h) 75 70 65 Only Main Route Figure 5.8: A v erage driving time of solutions for the single clien t case, according to the time- windo ws (main route) and tra v el sp eed (alternativ e routes) used. these exp erimen ts, the planning horizon is set high enough so that all instances w ould ha v e a fea- sible solution. Ho w ev er, man y of the solutions with larger costs could b ecome infeasible with a shorter planning horizon. When solving the VRP , the shortest paths are usually assumed kno wn, so no alternativ e paths are considered when running the sc heduling subroutines. As parking a v ailabilit y is usually not considered, these metho ds w ould b e exp ecting the trip cost/duration to b e similar to the v alues for wide parking a v ailabilit y time-windo ws sho wn in Figure 5.6. Ho w ev er, dep ending on the sev erit y of the region’s parking shortage, a driv er that attempted to plan the trip accoun ting for parking a v ail- abilit y could face considerably larger costs lik e the ones sho wn for narro w parking time-windo ws. With the w orst cost happ ening if the driv er tried to stic k to the exact path considered b y the compan y (bars for ‘Only Main Route’ case in Figure 5.6). Driv ers that do not consider parking when planning w ould b e sub jecting themselv es to safet y risks as the ones estimated in Figure 5.5. In either case, the trip costs used b y VRP algorithms that do not consider parking a v ailabilit y and that assume a fixed path b et w een an y t w o clien ts can b e a complete misrepresen tation of the costs the driv er or compan y will actually exp erience. 88 Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 28% 0% 0% Time-Windows = Wide Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 99% 65% 62% Time-Windows = Medium 75 70 65 Travel Speed on Alternative Routes (km/h) 0 100 Alt. Route Usage 100% 100% 100% Time-Windows = Narrow Figure 5.9: Num b er of instances that used alternativ e routes for the t w o clien ts case, according to the time-windo ws (main route) and tra v el sp eed (alternativ e routes) used. Wide Medium Narrow Main Route Time-Windows Width 26:00 26:25 26:50 26:75 27:00 27:25 27:50 27:75 28:00 Average Driving Time (h) Alternative Routes Travel Speed (km/h) 75 70 65 Only Main Route Figure 5.10: A v erage driving time of solutions for the t w o clien ts case, according to the time- windo ws (main route) and tra v el sp eed (alternativ e routes) used. 89 5.7 Randomized Net w orks Exp erimen ts In this section w e presen t exp erimen ts measuring the p erformance of the algorithm on randomly generated net w orks. The first exp erimen t fo cuses on net w orks with a small n um b er of clien ts and a large v ariation on the driving time required to reac h the last clien t. The second exp erimen t fo cuses on v arying the n um b er of clien ts with narro w er range of driving times. 5.7.1 Setup The graphs w ere created with the follo wing c haracteristics: • Net w ork organized in la y ers. • Random n um b er of la y ers b et w een consecutiv e clien ts. • Random n um b er of no des p er la y er. • Ev ery no de has a fixed probabilit y of ha ving an edge linking it to eac h no de of the follo wing la y er. • Ev ery no de that has an incoming edge, has at least one outgoing edge. • Branc hes leading to TPLs are inserted randomly along eac h edge according to a P oisson pro cess. • Multiple v alues for the a v erage spacing b et w een TPLs w ere tested. • Multiple v alues for the probabilities of a TPL ha ving a narr ow , me dium or wide a v ailabilit y windo w w ere tested. F or eac h graph, fiv e p ossible parking shortage lev els, with increasingly strict parking conditions, w ere tested. The parking shortage lev els in T able 5.4 define the probabilit y of eac h TPL ha ving a Narr ow , Me dium or Wide time-windo w. These three t yp es of time-windo ws are defined in T able 5.3. F or eac h graph and parking shortage lev el, 20 problem instances w ere generated. All clien ts ha v e daily time-windo ws from 9:00 to 17:00. The algorithm’s tolerance is set to find solutions within 0.25 hours of optimalit y . T able 5.4: P arking Shortage Lev el Probabilit y of eac h t yp e of time-windo w P arking Shortage Lev el Narro w Medium Wide 1 0.1 0.2 0.7 2 0.2 0.3 0.5 3 0.33 0.33 0.34 4 0.5 0.3 0.2 5 0.7 0.2 0.1 90 T able 5.5: Random Net w orks Configurations A vg. Spacing (km) Graphs T otal Instances 50 84 8400 100 86 8600 150 67 6700 200 44 4400 (9 infeasible) 5.7.2 Exp erimen t 1 In this exp erimen t, w e studied ho w the parking shortage affects total trip duration, and our algorithm’s p erformance in net w orks of v arying sizes. Note that our fo cus is long-haul truc king; driv ers tra v el long distances to visit a relativ ely small n um b er of clien ts. When studying larger net w orks, our in terest is in v arying the trip duration, the n um b er of p ossible paths and the n um b er of TPLs along eac h path. The n um b er of TPLs v aries from 0 to 288, the total driving time of the solutions found for eac h instance v aries from 3.3 to 60.7 hours, and the total n um b er of clien ts v aries from 1 to 5. T able 5.5 presen ts the a v erage spacing b et w een TPLs used to generate the test net w orks, as w ell as ho w man y net w orks w ere created with eac h spacing v alue. 5.7.2.1 P erformance W e implemen ted our algorithm in Python 3.8, and all exp erimen ts w ere run on a In tel Core i5, 3.1GHz CPU with 8Gb of RAM. W e w ould lik e to note that the obtained running times could b e reduced b y implemen ting the algorithm in faster languages, suc h as C, C++ or Ja v a, ho w ev er, this is not the fo cus of this w ork. Figure 5.11 sho ws ho w the a v erage running time v aries with the solutions’ total driving time and the total n um b er of TPLs presen t in eac h instance. Both the n um b er of TPL c hoices and the n um b er of rests that need to b e sc heduled increase the problem’s complexit y , so w e concluded that this w as the most meaningful w a y of presen ting the results. The metho d still presen ts some scalabilit y issues, with running times increasing sharply when trips approac h the w eekly driving limit and a large n um b er of TPLs is considered. Nev ertheless, most scenarios tested ha v e an a v erage running time b elo w 200 seconds, whic h is reasonable for a truc k driv er planning his itinerary for the follo wing w eek. Also, although the curren t p erformance do es 91 0 100 200 300 TPLs 10 20 30 40 50 60 Solution driving time (h) 10 2 10 1 10 0 10 1 10 2 10 3 Running time (s) Figure 5.11: A v erage running time o v er randomized net w orks. not allo w for this metho d to b e used at ev ery iteration of a VR TDSP algorithm, it can still b e used as a metho d to refine the final routes c hosen. W e also noticed that the time windo ws can significan tly affect running time. Dep ending on the parking shortage lev el and instance considered, the same net w ork ma y ha v e v astly differen t running times. 5.7.2.2 P arking Shortage’s Impact T able 5.6 presen ts the p ercen tual increase in trip duration of instances with parking shortage lev el 2 to 5 relativ e to lev el 1 instances. Lev el 5 instances ha v e 5.1% higher trip duration o v erall, and ev en considering only instances with an a v erage spacing of 50km b et w een TPLs the increase is 2.3%, whic h is not negligible giv en the truc king industry’s size. The case study presen ted in Section 5.6 illustrated ho w parking a v ailabilit y conditions can ha v e a significan t impact on trip durations. This exp erimen t o v er randomized net w orks further supp orts this claim, b y sho wing that significan t impacts are seen ev en on more complex net w orks. 92 T able 5.6: A v erage T rip Duration Increase P arking Shortage Lev el A vg. Spacing (km) 1 (baseline) 2 3 4 5 50 0% +0.2% +0.8% +1.3% 2.3% 100 0% +0.7% +1.4% +2.5% 4.2% 150 0% +0.6% +2.7% +5.9% 8.3% 200 0% +0.9% +3.5% +5.6% 7.1% Ov erall 0% +0.6% +1.9% +3.4% +5.1% 5.7.3 Exp erimen t 2 This exp erimen t fo cuses on studying ho w the algorithm b eha v es when the n um b er of clien ts is increased. As stated previously , our fo cus is on long-haul, whic h implies a small n um b er of clien ts visited o v er a long trip, so the scenarios tested in this exp erimen t div erge from our in tended use case. In order to a v oid the effects of driving time (and n um b er of TPLs) on the running time, sho wn in Figure 5.11, the net w orks used ha v e driving times around 30-40 hours, and the n um b er of clien ts v aries from 1 to 46 with zero service time. Therefore, the a v erage distance b et w een clien ts decreases as the n um b er of clien ts increases. These net w orks ha v e only 1 in termediate la y er b et w een clien ts, so there are not man y p ossible routes b efore accoun ting for the detours necessary to reac h TPLs. The a v erage spacing b et w een TPLs is set to 100 km and most net w orks tested ha v e b et w een 20 and 40 TPLs. Exp erimen t 2 w as run in the same computer as exp erimen t 1. Figures 5.12, 5.13, and 5.14 sho w ho w the running time v aries with the n um b er of clien ts, trip duration, off-dut y time, and driving time. In Figure 5.12 w e see that ha ving to accommo date constrain ts of a larger n um b er of clien ts in a relativ ely short trip causes large increases in trip duration despite similar driving times. W e b eliev e that the n um b er of clien ts’ effect on running time is caused mostly b y this increase in trip duration. Figures 5.12 and 5.13 sho w that the scenarios with larger trip duration and off-dut y times are also the ones with higher running times, p ossibly due to needing to sc hedule a larger n um b er of rest stops or testing longer rest duration v alues for eac h stop. Similarly , Figure 5.14 sho ws that running time increases with the off-dut y time ratio, and also that the off-dut y time ratio increases with the n um b er of clien ts. A p oin t w orth noting is that this t yp e of scenario with a large n um b er of clien ts close to eac h other do es not fit long-haul truc king usual jobs, it migh t b e closer to what happ ens in lo cal truc king, 93 0 10 20 30 40 Clients 50 100 150 200 250 Cost (h) 10 2 10 1 10 0 10 1 10 2 10 3 Running time's (s) Figure 5.12: A v erage running time for instances with v arying n um b er of clien ts and cost (trip duration in hours). where driv ers sta y within a smaller region. Ho w ev er, in the U.S., short-haul driv ers that op erate within a 150 air-mile radius of their normal w ork rep orting lo cation are sub ject to differen t, less restrictiv e, regulations [115]. As clien ts are exp ected to b e widely spaced and require non-negligible service time in the con text of long-haul truc king, w e judge that the limiting factor for our problem is not the n um b er of clien ts, but the trips’ exp ected driving/on-dut y time, duration and the n um b er of TPLs and paths considered during planning (as seen in Figure 5.11). 94 30 35 40 Driving Time (h) 50 100 150 200 O-Duty Time (h) 10 2 10 1 10 0 10 1 10 2 10 3 Running Time (s) Figure 5.13: A v erage running time for instances with v arying driving and off-dut y time. 0 10 20 30 40 Clients 0:4 0:5 0:6 0:7 0:8 O-Duty Time / Trip Duration 10 2 10 1 10 0 10 1 10 2 10 3 Running time (s) Figure 5.14: A v erage running time for instances with v arying n um b er of clien ts and off-dut y time ratio (off-dut y time/trip duration). 95 5.8 Conclusion In this study , w e in tro duced the shortest path and truc k driv er sc heduling problem with park- ing a v ailabilit y constrain ts (SPTDSP-P A), whic h in tegrates parking a v ailabilit y information in to the path planning and sc heduling of long-haul truc k shipmen ts. A resource-constrained shortest path form ulation is prop osed, and a tailored lab el-correcting a lgorithm is dev elop ed to solv e it b y efficien tly exploiting the problem’s particularities. W e p erformed a case study on a net w ork based on the US I-5 freew a y , under v arious parking a v ailabilit y conditions and alternativ e routes’ sp eed. The results sho w that, as exp ected, the cost of imp osing parking constrain ts increases as parking a v ailabilit y decreases. Therefore, illegal park- ing costs are the deciding factor when ev aluating the profitabilit y of imp osing parking constrain ts. Dep ending on the illegal parking p enalties considered, the cost sa vings from prev en ting said p enal- ties can exceed, or at least offset, the cost increase caused b y parking constrain ts. Due to the high direct and indirect costs of truc k-related acciden ts, w e b eliev e that these p enalties are lik ely high, and that safer parking practices w ould b enefit the truc king industry not only from a safet y standp oin t, but also from an economic standp oin t. T o illustrate the imp ortance of considering parking information when defining the paths b et w een clien ts, w e studied ho w parking a v ailabilit y and alternativ e paths affect trip costs. Considering alternativ e paths b y using the SPTDSP-P A form ulation instead of only solving a truc k driv er sc heduling problem can mitigate cost increases due to parking constrain ts and significan tly c hange estimated trip costs. Exp erimen ts on randomly generated net w orks sho w ed that, ev en on net w orks with more alter- nativ e paths and parking options, limited parking a v ailabilit y can ha v e a non-negligible impact on trip duration. In addition, this impact increases substan tially in net w orks where parking facilities are scarcer. The cost/duration increase caused b y imp osing parking a v ailabilit y constrain ts can b e seen as an estimate of ho w m uc h driv ers and companies w ould need to sp end in order to ensure safe itineraries for the driv ers, and prev en t acciden ts and other costs related to the difficult y of finding appropriate rest lo cations. By sim ulating ho w parking a v ailabilit y can affect trip duration, costs, and illegal parking, our mo del can aid in infrastructure and p olicy decisions. In future w ork, w e in tend to extend the mo del to include time-dep enden t tra v el times in order to accoun t for the effects of traffic congestion. 96 Chapter 6 Long Haul Battery Electric T ruc k Planning Section includes results published in: • F. Vital, and P . Ioannou, ”Effects of W orking Hour Regulations and P arking Shortages on T ruc k Electrification,” 2021 IEEE 24th In ternational Conference on In telligen t T ransp ortation Systems (ITSC). 6.1 In tro duction T ransp ortation electrification is seen as one of the main paths for emissions reduction. Electric v ehicles (EV) ha v e b een receiving increasing atten tion in recen t y ears. With the increase in p er- formance and v ariet y of EV mo dels in the mark et, it b ecomes essen tial to study ho w to efficien tly incorp orate them in to our so ciet y . BET s are exp ected to b e more efficien t than diesel truc ks and ha v e lo w er op erational costs [133]. Studies regarding the viabilit y of battery electric truc ks (BET s) sho w promising results; ho w ev er, concerns regarding reduced pa yload, limited range, long rec harge time, and the required supp orting infrastructure w ere raised [134, 135, 136, 137, 133, 34]. Curren t p o w ertrain efficiency and battery densit y mak e it so large batteries are required for long-range trips. As truc ks are sub ject to w eigh t constrain ts, the battery w eigh t reduces max pa yload. Moreo v er, large capacit y batteries tak e longer to c harge. Therefore, planning rec harging stops is the fo cal p oin t of man y EV-related problems. Another imp ortan t asp ect is estimating energy consumption, as it directly affects v ehicle range. Studies v ary significan tly in their treatmen t of these topics. Some mo dels consider that an EV will alw a ys b e fully c harged when stopping at a c harging station [138], whereas others allo w partial rec harging [139]. In [140], the authors treated the problem of finding an efficien t rec harging p olicy 97 for a fixed route while accoun ting for the cost of o v erc harging. The prop osed metho d assumed fixed tra v el costs and homogeneous c harging stations, but adapted v ersions w ere tested under sto c hastic tra v el costs and non-homogeneous c harging stations. In [141], the authors also considered o v erc harging costs, but fo cused on accoun ting for c harging infrastructure a v ailabilit y in the routing pro cess. F or eac h c harging station, the probabilit y of b eing a v ailable and the exp ected w aiting time when o ccupied w ere giv en. Heuristics w ere used to obtain adaptiv e routing and rec harging p olicies. The problem of trying to minimize the fuel/energy consumption and p ollutan ts emissions in the transp ortation sector is not new. Sev eral studies ha v e approac hed this topic b oth for passenger v ehicles and truc ks. A surv ey on ‘green’ v ehicle routing can b e found in [142], whic h giv es an o v erview of the t yp es of mo dels used for fuel consumption and emissions, and of the differen t v arian ts of the v ehicle routing problem whic h in v olv e en vironmen tal factors. Multiple mo dels ha v e b een dev elop ed to estimate the fuel consumption and p ollutan ts emissions based on differen t factors and targeting the usage on problems of differen t scales. Some mo dels consider only the a v erage sp eed of the car, as the one used in [143], but more precise mo dels ma y consider the v ehicle load [144], road incline, and if the v ehicle is accelerating, decelerating or cruising [145]. Reference [145] presen ts a comparison of differen t fuel consumption mo dels. These mo dels are then used to giv e an en vironmen tal asp ect to transp ortation problems. These problems can b e divided based on their time-dep endency (time-dep enden t traffic conditions or not), c hoice of decision v ariables (route, n um b er of v ehicles, tra v el sp eed, departure time, etc) and c hoice of cost function (only en vironmen tal factors or m ulti-ob jectiv e). The problem presen ted in [144] considers the impact of load and sp eed on carb on emissions and fuel consumption. Sim ulation results sho w ed fuel consumption reductions of up to 18%. The traffic conditions are not considered, so the prop osed mo del is time-indep enden t. Also, the sp eed is tak en as a parameter of the road, not as a decision v ariable for the mo del, so the mo del can c ho ose to use a faster or slo w er road when con v enien t, but it cannot tell the driv er to driv e slo w er than the road ‘regular’ sp eed. In [143], the sp eed is used as a decision v ariable to optimize fuel consumption during the trip and to facilitate the organization of truc k plato ons. Although the treated problem is a routing problem, the fuel consumption optimization is only p erformed after a shortest path has b een c hosen for eac h v ehicle. A t this p oin t it turns in to a sc heduling problem, as the route is giv en. Other v arian ts of the problem consider time-dep enden t traffic conditions on the road net w orks [146] or the impact of differen t 98 idling options used when resting [24]. Sev eral studies targeted the b enefits of truc k plato ons [143], [147, 148, 149], sho wing that it is p ossible to ac hiev e fuel consumption reductions in the range of 5%-15% dep ending on plato on sp eed, gap b et w een truc ks and p osition of the truc k inside the plato on. The problem is that most of these fuel/emissions-efficien t mo dels do not address imp ortan t practical constrain ts of the truc king industry , i.e. w orking hours or Hours-of-Service (HOS) regula- tions and parking a v ailabilit y . These factors are particularly imp ortan t to long-haul truc k driv ers, whic h are usually not the fo cus of these studies. In [24], the author considered the w orking hour regulations and included an en vironmen tal cost based on the emissions generated b y truc k idling dep ending on the equipmen t installed in the truc k and the one a v ailable at the parking lo cation. Ho w ev er, this mo del did not consider the p ossibilit y to optimize the emissions/fuel b y con trolling the sp eed of the v ehicle, and also did not consider traffic conditions. In most cases, the studies fo cused on the truc king industry whic h consider w orking hours regulations and, to a certain ex- ten t, parking fo cus on the monetary costs directly accrued b y the truc king compan y . Th us the ob jectiv e function usually considers only the total trip duration [23, 26], or a function of the total trip duration and the w orking time or tra v el distance [27]. There are studies on the com bined routing and sc heduling problem, and studies fo cused only on sc heduling, but it is the sc heduling problem the one resp onsible for the practical feasibilit y of the solutions when sub ject to regulations and parking a v ailabilit y . Some metho ds allo w the driv er to rest at an y p oin t during the route [27, 29, 28, 32, 113], not considering the need for an appropriate rest lo cation, others only allo w the driv er to rest at truc k stops and/or clien t lo cations [24, 23, 26]. Most studies do not consider time-dep enden t tra v el times, ho w ev er there are still some that do [26, 25, 150]. When accoun ting for the fact that driv ers need to rest due to regulations, the rec harge time issue migh t not b e as pronounced. Driv ers are required to rest regularly due to hours-of-service (HOS) regulations. If driv ers can use the mandatory rest stops to rec harge, the increase in trip duration due to rec harge times can b e reduced or eliminated. The sync hronization of rest and rec harge times w as studied for single-da y trips b y Sc hiffer et al. [151], sho wing impro v emen ts for BET s. A study b y Mareev et al. [134] used EU regulations to help estimate costs for long-haul BET s. Mareev used the EU regulation to generate a regulation-complian t baseline driving cycle. Ho w ev er, in practice, the trip’s 99 sc hedule dep ends on b oth clien ts and c harging stations; th us, it can differ significan tly from the regulation’s minim um requiremen ts. A part of this problem whic h is still o v erlo ok ed most of the time is the issue of parking a v ail- abilit y . Most mo dels assume that an y v alid parking lo cation will alw a ys b e free, whic h is unrealistic as appropriate truc k parking is an issue b oth in the U.S.A. [1] and in Europ e [152]. Due to limited infrastructure, BET s ma y face difficulties finding a v ailable rec harging stations. Ho w ev er, diesel truc ks are not en tirely free of suc h w orries. In [119, 153], Vital and Ioannou studied the problem of including b oth parking a v ailabilit y information and HOS regulations in the planning of long-haul transp ortation, but those studies do not co v er electric v ehicles. Sev eral v arian ts of the t w o sides of this problem, fuel/emissions optimization and sc heduling with w orking hours regulations, ha v e b een studied. Curren tly what w e miss is a mo del that can in tegrate b oth sides. A mo del able to generate solutions with reduced en vironmen tal cost, but that are still feasible in practice. In this pro ject w e plan to fill this gap b y extending a mo del dev elop ed in a previous pro ject, whic h fo cused on the w orking hours regulations and parking a v ailabilit y constrain ts, to consider the impact of traffic conditions and differen t tra v el sp eeds in the fuel consumption and p ollutan ts emissions of the truc ks, as w ell as ho w uncertain ties in the parking a v ailabilit y affect the problem’s solutions. Planning for freigh t transp ort electrification requires a go o d understanding of BET op erations’ p erformance and cost under realistic scenarios and ho w eac h scenario affects BET usage’s minim um requiremen ts. These scenarios include the need for driv ers to adapt their sc hedules to fulfill clien t restrictions, HOS regulations, parking a v ailabilit y , and c harging needs. In this pro ject, w e study ho w the p erformance gap b et w een BET s and diesel truc ks is affected b y practical constrain ts suc h as HOS regulations and limited parking a v ailabilit y . The pap er is organized as follo ws: Section 6.3.1 describ es the mo dels used to estimate energy/fuel consumption and emissions. Section 6.2 describ es the mathematical mo del used for BET trip planning under HOS and parking constrain ts. Sections 6.5 and 6.6 describ e the exp erimen ts p erformed and their results. Section 6.7 presen ts the conclusion. 100 6.2 Problem Description The problem consists of planning the path, tra v el sp eed and duration of rest and rec harge stops of a single battery-electric truc k from origin to destination with required stops at an ordered set of clien t lo cations. The solution m ust comply with the USA HOS regulations, and satisfy battery lev el, deliv ery time and parking a v ailabilit y constrain ts. This problem is a v arian t of the SPTDSP- P A (shortest path and truc k driv er sc heduling problem with parking a v ailabilit y constrain ts), whic h w as in tro duced in Chapter 5 [153]. This v arian t differs mainly in the inclusion of factors relev an t to EV planning, i.e., finite battery capacit y , partial rec harges, sp eed-dep enden t energy consumption, and sp eed con trol. The problem is solv ed o v er a simplified road net w ork that includes only the main routes the truc k can tak e b et w een t w o consecutiv e clien t lo cations, and the rest areas and c harging stations around them. The simplified road net w ork is defined as an acyclic directed graph G = (V;A) , where V is the set of no des of the graph and A is the set of edges. Eac h road section (i;j)2 A has a fixed length d ij and an allo w ed sp eed range [s ij ;s + ij ] , th us setting the allo w ed tra v el time to [ d ij s + ij ; d ij s ij ] . The sp eed limit is considered constan t within eac h road section, but the a v erage tra v el sp eed can b e adjusted within the allo w ed range to con trol the tra v el time and energy consumption. During long trips, HOS regulations require driv ers to rest along the w a y . Rest stops are restricted to rest areas and their minim um durations are defined b y the regulation. W e do not allo w for rests to b e tak en at clien t lo cations. Ho w ev er, note that service times longer than 30min can reset the 8h driving limit constrain t despite coun ting as on-dut y time for other constrain ts. Eac h parking lo cation has a set of time-windo ws represen ting the in terv als when parking spaces are exp ected to b e a v ailable. These time-windo ws restrict the v ehicle’s arriv al time. The v ehicle is not allo w ed to arriv e early and w ait. The regulation sets a minim um duration for the rest stops, but it do es not set a maxim um duration, so the driv er is allo w ed to extend the sta y when con v enien t. Similarly , eac h clien t has a set of time-windo ws constrain ts and a service time, whic h define when the truc k can arriv e at the clien t and the duration of sta y . Ho w ev er, driv ers cannot extend the service time at the clien t. As rest areas are not required stops, the graph G is built so that rest areas can b e b ypassed. Clien ts are mandatory stops, so all considered routes go through the clien t no des. 101 The v ehicle has a finite battery capacit y , and the energy mo dels used to calculate the energy consumption and rec harge rates are describ ed in section 6.3.1. The battery can only b e rec harged at c harging stations, whic h can b e either rest areas or clien t facilities with c harging infrastructure. In addition, w e consider the p ossibilit y of ha ving fast c hargers in the net w ork. As our use of parking a v ailabilit y time-windo ws is based on the assumption that v ehicle turno v er is lo w during o v ernigh t p erio ds as driv ers need to rest for long p erio ds, w e treat fast c hargers differen tly . W e limit the maxim um stop duration as w e consider that fast c hargers’ main purp ose is using short stops to mitigate range issues, not serving as rest lo cations. Instead of using time-windo ws to mo del a v ailabilit y , w e consider that fast c harging stations ha v e a kno wn w aiting time and an appropriate w aiting area. W e consider that driv ers migh t b e required to mo v e the v ehicle during the w ait, so they are not completely reliev ed of w ork resp onsibilities. Therefore, the w aiting time is treated as service time, but it do es not coun t to w ards the stop duration limit. The exp erimen ts in this c hapter consider time-indep enden t w aiting times and sp eed limits, but the rollout algorithm used also accepts time-dep enden t v arian ts. 6.3 Mo del W e use a mo dified v ersion of the R CSPP form ulation describ ed in section 5.4. Driving edges no w ha v e an in terv al of allo w ed tra v el times instead of a fixed one, c harging stations are treated as rest areas when generating the extended net w ork describ ed in section 5.4.1, and eac h edge also stores information regarding c harging capabilities, i.e., whether it is a c harging station or not, and the c harging rate. A priori w e consider that v ehicles cannot rec harge at clien t lo cations and that the idling consumption rate is the same for all clien t lo cations, but c harging and energy consumption information can also b e stored in the service edges to customize b eha vior at eac h clien t. In the case of fast c hargers, the daily/we ekly r est activities are not considered, and the w aiting time needs to b e included, so the no de is expanded in to arrival , char ge start , and out no des. The first t w o no des are connected b y a waiting/servic e edge, and the last t w o no des are connected b y a br e ak edge. Section 6.3.2 describ es the mo difications made to the resources and resource extension functions to include battery lev el as a resource and accoun t for the regulation c hanges. 102 6.3.1 Consumption Mo dels The energy/fuel consumption dep ends on the activit y b eing considered, so w e separate the mo del in three cases: driving, idling and c harging (BET only). The mo del parameters considered are listed in T able 6.1. 6.3.1.1 Driving The consumption mo dels used are based on mo dels found in the literature ([154, 155, 135, 156, 157] for BET s, and [158] for diesel truc ks). Both mo dels first estimate the v ehicle’s p o w er demand due to resistance forces acting on the v ehicle, then estimate the consumption rate based on the p o w er demand. W e consider the a v erage tra v el sp eed o v er eac h road section, and terms relativ e to acceleration and road grade w ere omitted. BET s LetP B (v) b e the p o w er demand (inkW ) to the battery due to the forces acting against the truc k’s mo v emen t, accoun ting for the battery to wheel efficiency , and B (v) b e the rate of energy consumption p er unit of distance tra v eled (in kWh=km ) defined as follo ws: P B (v) = AC D 25:92 v 2 +mgC R v 3600 bw (6.1) B (v) = P B (v)+P a v (6.2) where C D and C R are the co efficien ts of drag and rolling resistance, resp ectiv ely . The air densit y (kg=m 3 ) is giv en b y , and the acceleration due to gra vit y is giv en b y g . The terms v , m and A represen t the truc k’s sp eed(km=h) , mass(kg) and fron tal area (m 2 ), resp ectiv ely . bw represen ts the battery-to-wheels efficiency , and P a is the p o w er demand from the v ehicle’s accessories and supp ort systems, e.g. A/C, ligh ting, electric steering system. The battery lev el cannot b e negativ e, so an y displacemen t requiring more energy than curren tly stored in the battery is considered infeasible. Diesel T ruc ks F or diesel truc ks, w e used the mo del presen ted b y W ang and Rakha in [158]. More sp ecifically , the parameters used are the ones for a con v ex mo del of a F reigh tliner/FLD 120, 103 y ear 2001, lab eled as “HDDT8” in their pap er. This mo del c haracterizes fuel consumption as a second-order p olynomial function of the p o w er demand, as follo ws: P D (v) = AC D 25:92 v 2 +mgC R (c 1 v +c 2 ) v 3600 d (6.3) D (v) = 0 + 1 P D (v)+ 2 P D (v) 2 3600 v (6.4) where, similarly to (6.1)-(6.2), P D (v) represen ts the p o w er demand (kW) , and D (v) represen ts the fuel consumption p er distance (L=km ). C R , c 1 and c 2 are the rolling resistance parameters (unitless), d is the driv eline efficiency (unitless), 0 , 1 and 2 are v ehicle-sp ecific mo del co efficien ts calibrated in [158] using empirical data. The remaining parameters are defined as in (6.1). 6.3.1.2 Idling When ‘idling’, w e consider a fixed consumption rate P I (kW) for BET s andF I (L=h) for diesel truc ks. W e assume that P I is smaller than P a as some systems, suc h as electric steering, migh t b e inactiv e when the v ehicle is stopp ed. F or BET s, w e assume that c hargers can pro vide p o w er to these supp ort systems on top of c harging needs and idling consumption is not subtracted from the c harging rate at c harging stations. Ho w ev er, when estimating emissions, idling consumption is included in the energy exp enditure. 6.3.1.3 Charging BET s can rec harge their batteries at c harging stations lo cated on the road net w ork (and p ossibly clien ts with c harging infrastructure). W e consider a finite battery capacit y B (in kWh ), and, for eac h c harging station` , a constan t c harging rate ` (inkW ). The battery cannot store more energy than its capacit y , so w e assume that the battery will stop c harging when full. 6.3.2 System Equations W e consider the system’s state as b eing a v ectorx k = (v k ; k ) , where k = ( 0 k ; b k ; r k ; r k ; w k ;b k ) con taining the curren t lo cation of the truc k (v k ) and the curren t resource v alues ( k ) . The resources 104 T able 6.1: Mo del P arameters Battery Electric T ruc k P arameter Description V alue C D [156] co efficien t of drag 0.63 C R [156] co efficien t of rolling resistance 6.3E-3 bw [156] battery-to-wheels efficiency 0.85 m (kg) [156] truc k’s total mass 3.6E4 P a (k W) [34] supp ort systems p o w er demand 10 P I (k W) idling p o w er demand 3 A (m 2 ) [156] truc k’s fron tal area 7.2 Diesel T ruc k C D [158] co efficien t of drag 0.78 C R [158] co efficien t of rolling resistance 1.25E-3 c 1 [158] co efficien t of rolling resistance 0.0328 c 2 [158] co efficien t of rolling resistance 4.575 d [158] driv eline efficiency 0.94 m (kg ) [158] truc k’s total mass 3.6E4 A (m 2 ) [158] truc k’s fron tal area 10 0 [158] v ehicle-sp ecific mo del co efficien t 2.16E-3 1 [158] v ehicle-sp ecific mo del co efficien t 7.98E-5 2 [158] v ehicle-sp ecific mo del co efficien t 1.0E-8 F I (L=h ) [159] idling fuel consumption 3 General g(m=s 2 ) [158] gra vit y 9.8066 (kg=m 3 ) [156] air densit y 1.2256 d (kg=L) [160, 161] CO 2 emission factor for diesel 3.13 e (kg=kWh) [161] CO 2 emission factor for electricit y in California 0.2 105 T able 6.2: Resource Extension F unctions f d f s f b f r f w f 0 0 k+1 = 0 k + k b k+1 = b k + k ( 0; if k >t b b k ; o.w. 0 b k r k+1 = r k + k 0 r k r k+1 = r k + k r k 0 r k w k+1 = w k + k w k 0 w k b k+1 = b k k B ( k k ) ( max(B;b k + k ` ); if c harging b k k P I ; o.w. b k are resp onsible for trac king the HOS restrictions, battery lev el and arriv al time at eac h no de. The resources used are: 0 : Time when no de w as visited b : A ccum ulated driving time since last br e ak r : Elapsed time since last daily r est r : A ccum ulated driving time since last daily r est w : A ccum ulated on-dut y time since last we ekly r est b : Battery lev el The ev olution of the system is describ ed b y x k+1 = f (x k ;u k ) , where x k is the curren t state, x k+1 is the next state, and u k is the decision tak en. The decision u k is comp osed b y an edge e k = (v k ;v k+1 )2 A 0 , with length k , and a duration k included in e k ’s allo w ed duration set. When dealing with edges related to driving, this set is defined b y the length and allo w ed sp eed v alues of e k . The function f (x k ;u k ) defines ho w eac h elemen t of x k is affected b y a decision u k . As differen t activities ha v e differen t impacts on eac h resource, eac h edge of the extended net w ork has an activit y assigned to it. T able 6.2 sho ws ho w the resources are up dated dep ending on the activit y . The functions f d , f s , f b , f r , f w and f 0 describ e the up date rules for activities drive , servic e , br e ak, daily r est , we ekly r est and dep artur e , resp ectiv ely . Section 5.4.1 sho ws ho w the activities are assigned to eac h edge. Note that b ’s and f b ’s definitions differ from [153] due to recen t c hanges in the regulation. No w the 8h limit is applied to driving time instead of elapsed time, and an y non-driving p erio d longer than 30 min utes can satisfy this constrain t. 106 6.4 Dynamic Programming F orm ulation and Rollout Algorithm LetJ (x k ) b e the minim um cost to go from statex k to the destination, andX d the set of feasible states at the destination no de. This cost-to-go function is defined as: J (x k ) = 8 > > < > > : 0; if x k 2X d min u2U(x k ) g(x k ;u)+J (f (x k ;u)); o.w. (6.5) where g(x k ;u) is the cost accrued b y decision u at state x k , and U (x k ) is the set of decisions u for whic h f (x k ;u) is a feasible state. A state is considered feasible if all resources are within their resp ectiv e feasible ranges. If U (x k ) is empt y , w e sa y that the destination cannot b e reac hed from x k andJ (x k ) is infinite. The c hoice of g() determines what is b eing minimized. In this pro ject, w e use the trip duration as the ob jectiv e to b e minimized, but more complex ob jectiv e functions can b e used. F or example, w e can include the cost of energy and p erform a m ulti-ob jectiv e optimization that minimizes a w eigh ted sum of the cost generated b y the trip duration (driv er’s hourly salary) and costs generated b y energy exp enditure. Although an y no de has only a finite n um b er of outgoing edges, the decision space U (x k ) can ha v e uncoun tably man y elemen ts if the allo w ed duration set of one or more of these edges is a con tin uous in terv al. In order to mitigate this issue, w e first propagate the constrain ts of eac h no de to all upstream no des. This reduces the feasible space at eac h no de and the decision space to b e considered for eac h decision. During execution, the algorithm uses the prepro cessed feasible ranges to generate a reduced decision space, whic h is then discretized, generating a finite set of decisions. Nev ertheless, due to the curse of dimensionalit y , this approac h do es not scale w ell for large instances. Using a coarse decision space discretization can bring significan t impro v emen ts to computation time, but will also cause the cost to deteriorate. Therefore, w e use a rollout algorithm [162] to find sub optimal solutions while k eeping the computational demand in c hec k. The general idea is to use the cost obtained from applying a base p olicy as an appro ximate cost function, then 107 use this appro ximation to generate a one-step lo okahead p olicy . One-step lo okahead p olicies c ho ose the decision that minimizes the follo wing expression: min u k 2U(x k ) g(x k ;u k )+ e J (f (x k ;u k )) (6.6) where e J (x k ) is the appro ximated cost-to-go of statex k . Let the p olicy b e a function that returns a feasible decision(x k )2U (x k ) for ev ery statex k . J (x k ) is the cost-to-go when the p olicy is used to tak e decisions at ev ery state, and it can b e describ ed as: J (x k ) = 8 > > < > > : 0; if x k 2X d g(x k ;(x k ))+J (f (x k ;(x k ))); o.w. (6.7) In this pro ject, w e used e J (x) = J (x) , where is the p olicy generated b y solving the problem with a coarser discretization of the decision space. The strategy used to propagate constrain ts is included in Section 6.4.1. Section 6.4.4 describ es ho w the graphs w ere prepro cessed to reduce issues with short links. Section 6.4.2 and Section 6.4.3 sho w, resp ectiv ely , analytical solutions and cost lo w er b ounds that can b e used to sp eed-up the algorithm. The cost lo w er b ounds presen ted in Section 6.4.3 include cost functions that are a w eigh ted sum of time and energy consumption, whic h are more general then the cost functions considered in sections 6.5 and 6.6. 6.4.1 Constrain t Propagation and F easible Decision Space Consider the follo wing expression describ es ho w the states are up dated: x i+1 =f (x i ;u i ); u i 2U i (x i )U i (6.8) Let F i represen t the set of feasible states at no de v i . W e define U i (x i ) as: U i (x i ) =fu2U i j f (x i ;u)2F i+1 g (6.9) When c ho osing the decisions to test, w e can either sample U i and c hec k the feasibilit y of eac h decision or calculate the feasible decision space with an in v erse functionf 1 (F i+1 ;x i ) that returns 108 the elemen ts of U i that can generate a next state in F i+1 . As most edges up date the resources b y adding its duration to the curren t resource, in general this op eration consists of shifting the in terv als represen ting the constrain t for eac h resource, then taking the in tersection b et w een all of them, e.g. if the next no de has a time-windo w [10,15] and the curren t time is 5, then the decision duration m ust b e in the in terv al [5,10] to b e feasible. Differen t resources will generate differen t in terv als, and feasible decisions m ust satisfy all of them. Originally ,F i represen ts only the feasibilit y regarding the lo cal constrain ts at no dev i , ho w ev er, if w e consider constrain ts from other no des, w e ma y b e able to reduceF i , and consequen tly reduce U i (x i ) . Eac h no de’s lo cal constrain ts can b e propagated do wnstream and upstream to reduce other no des’ feasible spaces. 6.4.1.1 F orw ard Propagation LetF (F i ; F j ;U i ()) represen t a function that returns whic h states in F j can b e reac hed from F i , i.e., F (F i ; F j ;U i ()) =fx j 2F j j9x i 2F i ; 9u2U i (x i ); f (x i ;u) =x j g (6.10) The setR j of states that can b e reac hed at no de v j is giv en b y: R j = [ i; (v i ;v j )2A F (F i ;F j ;U i ()) (6.11) R j can b e o v erly complex due to the coupling b et w een resources, so w e try to appro ximate it b y propagating the constrain ts for eac h resource separately . Let F (r) i b e the pro jection of F i on the axis represen ting resource r , and f (r) the comp onen t of f that defines the ev olution of resource r . LetF (r) (F i ; F j ;U i ) b e a function that returns whic h v alues of resource r can b e reac hed at no de v j , defined as follo ws: F (r) (F i ; F j ;U i ) = n x (r) j 2F (r) j j9x (r) i 2F (r) i ; 9u2U i ; f (r) x (r) i ;u =x (r) j o (6.12) Let ^ R j appro ximateR j as follo ws: R i;j = Y r F (r) ^ R i ; F j ;U i (6.13) 109 ^ R j = Y r [ j; (v i ;v j )2A R (r) i;j (6.14) WhereR i;j is appro ximation accoun ting only for the constrain ts of upstream no de v i , andR i;j (r) is its pro jection on the axis represen ting resource r . Note that, for a giv en v i , if9r suc h that F (r) ^ R i ;F j ;U i =; , thenR i;j =; . That is, if states from v i cannot satisfy the constrain ts for 1 or more resources, then v i will not b e coun ted when calculating the reac hable states at v j . F urthermore, the edge (v i ;v j ) can b e remo v ed from the problem. A t the origin no de w e ha v e that R 0 = ^ R 0 . If the initial state is kno wn, it is the only reac hable state at the origin, otherwiseR 0 is the set of p ossible initial states. 6.4.1.2 Bac kw ard Propagation Bac kw ard propagation follo ws the same general idea as forw ard propagation. Let F i b e the reduced feasible space. Lik e ho w w e calculatedU i (x i ) , w e need a functionB(F i ; F j ;U i ()) that can calculate the v alues of x i at no de v i that can lead to at least one feasible state x j at one of the successors v j , i.e, B(F i ;F j ;U i ()) =fx i 2F i j9u2U i (x i ); f (x i ;u)2F j g (6.15) Ho w ev er, this function is hard to compute and generates complex regions t hat will require more space to store, and more time to c hec k during execution. Therefore, w e calculate separate regions for eac h resource and use it to generate an appro ximate feasible state space ^ F i as follo ws: C (r) (F i ; F j ;U i ) = n x (r) i 2F (r) i j9u2U i ; f (r) x (r) i ;u 2F (r) j o (6.16) ^ F i;j = Y r C (r) F i ; ^ F j ;U i (6.17) ^ F i = Y r [ j; (v i ;v j )2A F (r) i;j (6.18) As inR i;j , ^ F i;j is the empt y set if an y resource constrain t cannot b e satisfied. In this case, the edge (v i ;v j ) can b e remo v ed from the graph as it cannot generate feasible states. W e do the forw ard propagation b efore the bac kw ard, so, at the destination no de v n , w e ha v e that F n = ^ F n = ^ R n . 110 (a) Blue dashed line: correct feasible space. (b) Blue dashed line: appro ximate feasible space. Blue region: infeasible. Figure 6.1: The green and bro wn regions are examples of p ossible feasible regions in a 2D space. The figures sho w ho w the exact (6.1a) and appro ximate (6.1b) feasible spaces are calculated. F or example, if a no dev j has a time-windo w [10;15] and the edge (i;j) can ha v e a duration in the in terv al [2;5] , then v i m ust b e visited in the time-windo w [105;152] = [5;13] . If a differen t edge (v i ;v k ) generated a propagated time-windo w of [7;17] on v i , w e w ould consider the union of b oth time-windo ws, i.e., [5;17] . Then w e w ould tak e the in tersection of v i ’s original time-windo w, sa y [0;15] , and the time-windo ws obtained from propagating do wnstream constrain ts to obtain an estimated feasible time-windo w of [5;17] . Note that the in terv al [5;15] can b e divided in to an in terv al feasible for paths passing through v i , [5;13] , and one feasible for paths through v k , [7;15] . The same can happ en to other resource constrain ts. Therefore, it is p ossible that a state in ^ F i satisfies the time-windo w for a certain path and the HOS resource constrain ts for a differen t path. As all constrain ts are satisfied b y some path, the state is included in ^ F i , but, in practice, that state cannot generate feasible successors. So, w e ha v e that ^ F i migh t con tain states that cannot satisfy do wnstream solutions, i.e., F i ^ F i F i . Figure 6.1a and Figure 6.1b sho w a 2D example of the difference b et w een reduced feasible state space F i and its appro ximation ^ F i . The blue region in Figure 6.1b b elongs to ^ F i , but not to F i . 111 6.4.1.3 Propagating the resources W e separate the resource extension functions according to ho w they affect the resource b eing up dated. The resource extension functions either add a v alue to the resource (ADD), main tain the curren t resource v alue (NoEff ), or set the resource v alue to 0 (RESET). Let e = (v i ;v j ) b e an edge, [ e ; + e ] b e edgee ’s p ossible durations defined in U i . Let i ; + i b e the feasible v alues for resource r at no de v i . The appro ximate propagation functions describ ed previously are defined as follo ws for the 3 t yp es of REF: F orw ard Propagation ADD:F (r) (F i ; F j ;U i ) = i + e ; + i + + e T h j ; + j i NoEff: F (r) (F i ; F j ;U i ) = i ; + i T h j ; + j i RESET: F (r) (F i ; F j ;U i ) =f0g T h j ; + j i Bac kw ard Propagation ADD:C (r) (F i ; F j ;U i ) = h j + e ; + j e i T i ; + i NoEff: C (r) (F i ; F j ;U i ) = i ; + i T h j ; + j i RESET: C (r) (F i ; F j ;U i ) =f0g T h j ; + j i When a resource’s feasible range is a set of disjoin t in terv als, the functions ab o v e can b e applied to eac h in terv al separately and w e tak e the union of the resulting sets. Note that, in these REF s, the decision’s duration is directly used to up date the resource v alues. When energy/fuel consumption is included as a resource, the up date v alue will b e a function of the duration, so the propagation function will dep end on the consumption mo del used. 6.4.1.4 Reduced Decision Space The reduced decision space is generated follo wing the same idea. U i;j x i ;U i (); ^ F j = n u2U i (x i )jf (x i ;u)2 ^ F j o = \ r n u2U i (x i )jf (r) (x i ;u)2 ^ F (r) j o (6.19) 112 U i x i ;U i (); n ^ F j o = [ j; (v i ;v j )2A 0 U i;j x i ;U i (); ^ F j (6.20) Let i b e the curren t v alue of resource r . The other sym b ols are defined as in the previous section. ADD: U (r) i;j x i ;U i (); ^ F j = n u2U i (x i )jf (r) (x i ;u)2 ^ F (r) j o = h j i ; + j i i T [ e ; + e ] NoEff: U (r) i;j x i ;U i (); ^ F j = ( ;;if i = 2 ^ F (r) j [ e ; + e ]; o:w: RESET: U (r) i;j x i ;U i (); ^ F j = ( ;;if 0 = 2 ^ F (r) j [ e ; + e ]; o:w: 6.4.2 Analytical Solutions A t no des where the only p ossible next stop is the destination it is p ossible to analytically define the b est decision so that the algorithm do es not need to searc h o v er the remainder of that searc h tree branc h. Naturally , the decision dep ends on the cost function and constrain ts b eing considered in the problem. Consider the follo wing cost function for a decision of duration : C() = 8 > > < > > : + e ( e =); ifdriving (+ +); o:w: (6.21) , where is the truc ks hourly op erational cost (excluding fuel/energy) and is the cost p er unit of fuel/energy . F or non-driving decisions, is the hourly idling fuel/energy consumption, represen ts hourly costs incurred while stopp ed from sources other than idle energy consumption and op erational costs. F or driving decisions, e is the length of the road segmen t considered, and (v) is the fuel/energy consumption p er unit of distance. This cost function considers b oth time and energy/fuel related costs, and their relativ e imp ortance can b e adjusted using the parameters , , and . In this section, w e study the optimal decisions for the last driving and rest extension decisions. Although w e fo cus on BET s, the solutions for diesel truc ks can b e obtained b y ignoring the battery constrain t. 113 6.4.2.1 Last driving decision dC d =+ e d(v) d =+ e d(v) dv dv d = e d(v) dv e 2 =v 2 d(v) dv = 0 (6.22) Cost is minim um for ~ = e ~ v , suc h that ~ v is the ro ot of v 2 d(v) dv = . Assuming that (v) is a con v ex function, and, consequen tly , d dv is monotonically non-decreasing, w e can sa y that v 2 d(v) dv is strictly increasing o v er (max(0;v 0 );1) , where v 0 satifies d(v 0 ) dv = 0 . As and are p ositiv e, ~ v is unique. The functionv 2 d(v) dv do es not dep end on the edge, so ~ v can b e calculated b eforehand. Let [;] b e ’s domain, the optimal decision is giv en b y: = 8 > > > > > > < > > > > > > : ; if ~ < ; if ~ > ~ ; o:w: (6.23) 6.4.2.2 Last rest extension Let b e the rec harge rate at the curren t lo cation, 0 2 0 ; 0 the rest extension to b e c hosen, and ` 2 ` ; ` the duration of the decision at the follo wing edge, whic h is the last driving edge. The cost from the rest no de to the destination can b e written as C( 0 ; ` ) = (+ +) 0 + ` + e ( e = ` ) . Assume that, due to the destination no de’s resource constrain ts and the curren t state’s resource v alues, 0 + ` 2 D;D . The optimization problem b eing solv ed at the last rest decision can b e describ ed as: min 0 ; ` C( 0 ; ` ) = (+ +) 0 + ` + e ( e = ` ) (6.24) s:t: e ( e = ` ) 0 B 0 0 (6.25) 114 D 0 + ` D (6.26) 0 0 0 (6.27) ` ` ` (6.28) , where (6.25) guaran tees that the battery c harge is non-negativ e when arriving at the destination. (6.26) restricts the time to reac h the destination, and can b e related to b oth HOS and time-windo w constrain ts. (6.27) and (6.28) restrict the domains of 0 and ` to the reduced decision space, whic h is affected b y all constrain ts and the curren t state. Consider the follo wing definitions: H(v) =v 2 d(v) dv (6.29) P(v) = e (v)B 0 (6.30) rC = (+ +) (H(v)) (6.31) rg 1 = () (H(v)) (6.32) ~ v;H(~ v) = (6.33) ^ v;H(^ v) = (6.34) v;H( v) = (6.35) v;H( v) = 0 (6.36) v ;H(v ) = +(+ )+ (6.37) , where g 1 represen ts constrain t (6.25). H(v) and P(v) are auxiliary functions defined to simplify the notation and represen t, resp ectiv ely , the deriv ativ e of the energy consumption with resp ect to ` and the minim um feasible 0 giv en ` . Thev ’s with differen t accen ts are v alues used in the solution that can b e calculated offline. ~ v , ^ v , and v represen t, resp ectiv ely , the sp eeds at whic h the cost gradien trC is p erp endicular to (6.27), (6.26), and (6.25). v and v are the sp eeds at whic h (6.25) is parallel to (6.26) and (6.27), resp ectiv ely . Note that, giv en a distance e , eac h v also defines a duration ` , e.g. ^ ` = e ^ v . The accen ts on the ’s indicate whic h v generate them. First, consider the case when (6.25) is not activ e (e.g., diesel truc ks). The optim um p oin t is giv en b y: 115 T able 6.3: Solution Candidates P oin t Condition x 1 = (P(v ); ` ) - x 2 = D 2 ; 2 ; P( e = 2 ) =D 2 ` < 2 ` x 3 = (D 3 ; 3 ); P( e = 3 ) =D 3 ` 3 ^ ` x 4 = 0 ;P 1 ( 0 ) ~ ` P 1 ( 0 ) ` x 5 = 0 ;P 1 ( 0 ) ` P 1 ( 0 ) ` x 6 = (P( e = ` ); ` ) ` ` x 7 = P( e = ` ); ` ` ` 0 = 8 > > < > > : 0 ; if 0 + ` D min 0 ; D ` ;max 0 ; D ^ l ; D ` ; o:w: (6.38) ` = 8 > > > > > > < > > > > > > : ` ; if 0 + ` D D 0 ; if ( 0 + ` <D)^( 0 6= 0 ) min ` ; max ` ; ~ l ; D 0 ; o:w: (6.39) If the p oin t ( 0 ; ` ) satisfies (6.25), then it is optimal. Otherwise, it means that (6.25) m ust b e activ e. In this case, w e can define 7 candidate p oin ts and the sufficien t conditions for them to b e the optim um. The candidate p oin ts are giv en b y the p oin t along g 1 with minim um cost and the p oin ts where (6.25) in tersects other constrain ts, and the conditions conditions are deriv ed from eac h p oin t’s KKT conditions. T able 6.3 presen ts the candidate solutions and their conditions. F easibilit y is a basic necessary condition for an y solution, and w as th us omitted from the table. P 1 (v) refers to the in v erse ofP(v) o v er the domainv2 [ v;1) . The p oin tsx 2 andx 3 , represen ting the candidates where (6.25) and one of the constrain ts forming (6.26) in tersect, migh t b e computationally exp ensiv e to calculate, so w e can lea v e testing them for last. W e can also use appro ximate solutions instead of solving it exactly . Note that the conditions are generated from sp eeds that can b e calculated b eforehand. Therefore, w e ma y b e able to directly eliminate some candidate solutions based on ` ’s domain. 116 6.4.3 Cost Lo w er Bound LetA d A 0 represen t the set of all arcs with driving as their assigned activit y . F or ev ery no de pair (p;q) suc h that there is a directed path from p to q , letD(p;q) ,D d (p;q) , andD ` (p;q) b e, resp ectiv ely , the minim um tra v el time (including service time), minim um driving time and minim um tra v el distance b et w een no desp andq with all resource, time-windo w and HOS constrain ts relaxed: D(p;q) = 8 > > < > > : min( pq ); if (p;q)2A 0 min (p;k)2A 0 (min( pk )+D(k;q)); o:w: (6.40) D d (p;q) = 8 > > > > > > < > > > > > > : 0; if (p;q)2A 0 nA d min( pq ); if (p;q)2A d min (p;k)2A 0 (D d (p;k)+D d (k;q)); o:w: (6.41) D ` (p;q) = 8 > > > > > > < > > > > > > : 0; if (p;q)2A 0 nA d pq ; if (p;q)2A d min (p;k)2A 0 (D ` (p;k)+D ` (k;q)); o:w: (6.42) If there is no directed path from p to q , thenD(p;q) =D d (p;q) =D ` (p;q) =1 . LetD HOS (d; ) represen t the minim um duration of a HOS-complian t trip withd driving hours and initial resource v ector , assuming the driv er can rest an ywhere, and without considering service time and time-windo w constrain ts, i.e. if a driv er w ere at the b eginning of an empt y straigh t road with length equiv alen t to d driving hours where he/she can rest an ywhere, giv en an initial resource v ector , ho w long w ould he/she tak e to reac h the end of the road without breaking the HOS regulations. A metho d to calculate D HOS (d; ) is describ ed in Section B. LetD s (p;q) b e the service time required b et w een no des p and q. If the ob jectiv e w ere simply to minimize trip duration, the lo w er b oundL dur can b e calculated as: L dur (p;q; ) =D HOS (D d (p;q); )+D s (p;q) (6.43) 117 Ho w ev er, when considering a com bination of trip duration and energy/fuel consumption or emissions as the ob jectiv e function, the lo w er b ound generated using only the duration term (L dur (p;q; ) ) is to o lo ose and not as useful. Therefore, w e need a lo w er b ound on the fuel consumption/emissions. 6.4.3.1 Bound 1 Idling cost Let the energy/fuel consumption rate when idle (resting or service). A lo w er b ound on the idling cost is giv en b y: L idl1 (p;q; ) = ( +)(D HOS (D d (p;q); )D d (p;q)+D s (p;q)) (6.44) D s is fixed as clien t visits are mandatory . D d considers the minim um driving time of eac h edge, and D HOS (d; )d is monotonically increasing in d (required rest time cannot decrease when driving time increases), soL idl1 is a lo w er b ound on idling cost. Note that if the cost/consump- tion parameters for rest and service time are differen t, the term D s (p;q) will app ear separately m ultiplying its o wn parameter. Driving consumption Let v min b e the minim um tra v el sp eed allo w ed in the net w ork. W e assume that the fuel consumption p er time FC(v) is monotonically increasing in the range of sp eeds used in the problem, as is the case for the mo del w e use. Therefore,FC(v min ) giv es a lo w er b ound o n the energy/fuel consumption rate when driving. A lo w er b ound on the consumption due to driving is giv en b y: L f _dr1 (p;q) = 3600FC(v min )D d (p;q) (6.45) An alternativ e is using the minim um tra v el distanceD ` (p;q) and the sp eed v ` that minimizes the fuel consumption p er distance, (v) , (or the nearest feasible sp eed) to generate a energy/fuel consumption lo w er b ound. Cost Consider the cost function defined in (6.21). A cost lo w er b ound is giv en b y: 118 L cost1 (p;q; ) = L dur (p;q; )+L f _dr1 (p;q)+ L idl1 (p;q; ) (6.46) Note that the driving energy/fuel consumption b ound is calculated using the minim um tra v el sp eed, whereas the idling cost and trip duration b ounds are calculated using the maxim um tra v el sp eed. Therefore, this b ound is not tigh t. 6.4.3.2 Bound 2 When calculating analytical solutions in 6.4.2.1, w e sho w ed ho w to calculate the optimal sp eed based on energy/fuel and duration costs, and consumption mo del. W e no w use this information to refine the lo w er b ound. Driving time 6.4.3.1 used a driving time considering the maxim um tra v el sp eed. Ho w ev er, dep ending on the cost function, the cost increase due to fuel consumption at higher sp eeds ma y exceed sa vings due to shorter trip duration. Optimal solutions are exp ected to tend to w ards using the optimal sp eed ~ v (limited b y p ossible increases in required rest time). With this in mind, w e scale the driving time so that it represen ts the tra v el time at the optimal sp eed (or the nearest feasible sp eed). v t =max(min(~ v;v max );v min ) (6.47) e D d (p;q) =D d (p;q) v max v t (6.48) This scaling assumes that all edges ha v e the same sp eed limits and optim um sp eed. An alterna- tiv e (but still assuming that all edges ha v e the same optim um sp eed) w ould b e to use the length of the minim um length path,D ` (p;q) , to estimate a lo w er b ound on the driving cost when tra v eling with sp eed v t . A more general approac h w ould b e to, when building the graph, calculate v t for eac h edge, and store in eac h edge the tra v el time and cost asso ciated withv t . The stored costs can b e used to calculate a minim um cost path and its driving time. In b oth alternativ es, the minim um cost (w e refer to it asL dr _cost (p;q)) can b e used as a lo w er b ound on the driving related costs (due 119 to b oth emissions and duration) and w e w ould require only to complemen t it with a lo w er b ound on the idling costs (due to b oth emissions and duration). It is imp ortan t to remem b er that, due to HOS regulations, increasing driving time ma y end up increasing required rests. The extra rest time caused b y driving time scaling is giv en b y: =D HOS e D d (p;q); e D d (p;q)(D HOS (D d (p;q); )D d (p;q)) (6.49) T rip duration and fuel consumption are calculated follo wing the same ideas as 6.4.3.1 but using the scaled driving time and correcting trip duration and idling time to remo v e the extra rest time. T rip Duration The trip duration is calculated as follo ws: L dur2 (p;q; ) =D HOS e D d (p;q); +D s (p;q) (6.50) Idling cost The idling cost lo w er b ound is giv en b y the same expression as (6.44) due to the rest time correction, i.e., L idl2 (p;q; ) =L idl1 (p;q; ): (6.51) Driving consumption Energy/fuel consumption due to driving is giv en b y: L f _dr2 (p;q) = 3600FC(v t ) e D d (p;q) (6.52) Cost A cost lo w er b ound is giv en b y: L cost2 (p;q; ) = L dur2 (p;q; )+L f _dr2 (p;q)+ L idl2 (p;q; ) (6.53) Note that while L dur2 (p;q; ) L dur (p;q; ) and L f _dr2 (p;q) L f _dr1 (p;q) , L dur2 and L f _dr2 are consisten t with resp ect to the tra v el sp eed used for their calculation, and use a sp eed that minimizes cost (not accoun ting for mandatory rests). As the rest (idling) time is k ept as the one from the minim um duration path, the rest time is the minim um feasible. DecreasingL dur2 w ould imply that one or more edges are using a sp eed greater than the optimal, causing an increase in fuel consumption costs that exceeds the sa vings in trip duration costs. Similarly , decreasing 120 L f _dr2 , w ould cause an increase in trip duration costs, and increase o v erall cost. Therefore,L cost2 is a lo w er b ound. Eac h term is not a lo w er b ound for the v alue it appro ximates, but they are calculated so that they generate a cost lo w er b ound. If the driving cost lo w er b oundL dr _cost (p;q) is calculated directly , then the cost lo w er b ound is giv en b y: L cost2 (p;q; ) =L dr _cost (p;q)+(+ +)(D HOS (D d (p;q); )D d (p;q)+D s (p;q)) | {z } idling time (6.54) 6.4.4 Graph Prepro cessing In the appro ximate dynamic programming algorithm used, w e store the decision and cost for sev eral states at eac h no de. Therefore, ha ving a large n um b er of in termediate no des b et w een rest areas increases b oth the n um b er of decisions needed to reac h the destination and the storage space required b y the algorithm. F urthermore, when optimizing tra v el sp eed to reduce fuel consumption, the precision with whic h sp eed can b e adjusted dep ends on the time resolution used in the decision space, but also on the length of an y giv en edge. If an edge is to o short, an y c hange in duration migh t generate a tra v el sp eed outside of the allo w ed range. In order to reduce the n um b er of no des in the graph, w e use a stop-based graph based on the road net w ork and remo v e short edges b et w een nearb y rest areas (e.g., only consider rest areas that are at least 2h a w a y from the curren t no de). By stop-based graph w e mean a graph that directly links p ossible stop lo cations (origin, rest areas, clien ts), analogous to customer-based graphs used for v ehicle routing problems. Ho w ev er, the graph is not complete as eac h lo cation is connected only to lo cations that w ere do wnstream in the original road net w ork. As clien ts are mandatory stops and ha v e a fixed order, no des are not directly connected to no des do wnstream of the next clien t. It can b e seen as generating the stop-based graph based on the subnet w orks connecting eac h pair of consecutiv e clien ts, as opp osed to using the whole net w ork directly . Figure 6.2 sho ws a graph represen ting a road net w ork, whereas Figure 6.3 sho ws the stop-based graph that w ould b e generated from that net w ork. As our exp erimen ts set the same sp eed profile for all edges, eac h edge (i;j) of the stop-based graph w as generated using the length of the minim um distance path b et w een no des i and j in the road net w ork and setting the same sp eed profile used in the road net w ork. W e assume that a stop-based graph is kno wn or can b e obtained 121 O v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 D v 9 v 10 v a v b v c v d 1 Figure 6.2: Example graph fo cusing on the road net w ork. F o cuses on rest area (no des with letter indexes) placemen t along main roads. Easy to visualize but has a large n um b er of in termediate no des (no des with n um b er indexes). O D v a v b v c v d 1 Figure 6.3: Stop-based graph generated from Figure 6.2 to fo cus on the connection b et w een p os- sible stops (rest areas, clien ts, origin, destination). Eac h p ossible stop is directly connected to do wnstream stops satisfying predetermined conditions. Dashed arro ws exemplify edges that could b e remo v ed for b eing to o short or to o long. b y the user, and do not co v er the sp ecifics of its construction for general net w orks. Algorithms to construct customer-based graphs for time-dep enden t road net w orks w ere prop osed in [163]. Giv en a stop-based graph, w e remo v e edges that ha v e distance or minim um tra v el tim e shorter than c hosen limits, except when one of the edge’s no des is a clien t, the origin, or the destination. In our exp erimen ts, the time and distance limits w ere set to 2 h and 100km, resp ectiv ely . In addition, as HOS regulations limit driving time, edges with minim um tra v el time greater than 8h w ere also remo v ed. Although it is p ossible for the fastest path b et w een lo cations to v ary with time in time- dep enden t net w orks, w e assume that edge lengths (distance) are fixed in the stop-based graph. 122 T able 6.4: Exp erimen t P arameters P arking Time-windo ws Distribution Start Time (h) End Time (h) Narro w N (9,1) N (16,1) Medium N (7,1) N (19,1) Wide N (5,1) N (22,1) 93 km 80 km 41 km 64 km 68 km 96 km 129 km 64 km 295 km 193 km 290 km 161 km 206 km 161 km 224 km 161 km 81 km 150 km 129 km 79 km 161 km 203 km 80 km 140 km O D v0 v1 v2 v3 v4 v5 v6 v7 v8 p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 1 Figure 6.4: Net w ork used for exp erimen ts. Arc lengths are giv en in kilometers. 6.5 Case Study Exp erimen ts w ere p erformed on the graph net w ork used in section 5.6 (sho wn in Figure 6.4). Ev ery no dep i is assumed to b e a c harging station that allo ws for long term parking. No desv 4 andD are clien ts with daily time-windo ws [12;16] and [8;16] , resp ectiv ely . All c harging stations ha v e the same c harging p o w er (100 k W or 50 k W, dep ending on the scenario), and are sub ject to a v ailabilit y time-windo ws. Within eac h scenario, probabilit y distributions are defined for the start and end times of the a v ailabilit y time-windo ws. The same pair of distributions is used for all lo cations and da ys, but they are sampled separately . T able 6.4 describ es the probabilit y distributions used to generate the 3 differen t t yp es of time-windo ws used. The battery capacit y v alues tested w ere 400, 600, 800, and 1000k Wh. F uel constrain ts for diesel truc ks w ere relaxed b y setting a large fuel capacit y . The a v erage tra v el sp eed is set to 75km/h at all edges. As the energy consumption is affected b y sp eed, sp eed can also affect v ehicle range and problem feasibilit y . Therefore, for scenarios with feasibilit y issues, w e p erformed exp erimen ts where the v ehicle is allo w ed to reduce its sp eed to 70% of the a v erage tra v el sp eed (around 52.5km/h). Our ob jectiv e is to study ho w battery capacit y , c harging p o w er and c hargers/parking’ a v ailabilit y affect BET s p erformance in terms of trip duration, CO2 emissions and route feasibilit y , and compare it with a diesel truc k’s p erformance. 123 Figure 6.5 sho ws ho w trip duration is affected b y parking a v ailabilit y , c harging p o w er and battery capacit y . All scenarios w ere infeasible for the 400k Wh capacit y , so no results are sho wn. It can b e seen that when wide time-windo ws are used, diesel truc ks hold a significan t adv an tage in terms of trip duration, esp ecially when lo w er c harging p o w er and battery capacities are used. This result is exp ected and lik ely caused b y sc hedule adjustmen ts needed to accommo date longer rec harging stops. F or example, due to the HOS regulations and parking time-windo ws, short dela ys migh t push driv ers close to their driving limits and require them to add another daily r est (10h minim um) to the trip. Ho w ev er, as w e consider more sev ere parking shortage scenarios b y narro wing the time-windo ws, this adv an tage is gradually reduced. When narro w time-windo ws are considered, the BET’s p erformance w as comparable to the baseline diesel truc k’s. If parking a v ailabilit y is limited, diesel truc ks w ould also need to adjust their sc hedules to guaran tee appropriate parking regardless of ha ving longer range and faster refueling times. Figure 6.6 sho ws results regarding CO2 emissions. In this asp ect, BET s presen t a clear adv an tage in all feasible scenarios. In scenarios where 600k Wh battery capacit y w as considered w e encoun tered some feasibilit y issues, sho wn in Figure 6.7. As energy consumption is sp eed-dep enden t, w e rerun these exp erimen ts allo wing the sp eed to b e adjusted b et w een 70%-100% of the road’s a v erage sp eed. Figure 6.7 sho ws ho w this allo w ed sp eed reduction impro v es feasibilit y . The effects of the allo w ed sp eed reduction on the trip duration and CO2 emissions are sho wn in Figure 6.8 and Figure 6.9. These results sho w that diesel truc ks ha v e an adv an tage (regarding trip duration) under ideal parking a v ailabilit y conditions, but that adv an tage is greatly reduced when w e accoun t for curren t parking shortage issues. Although the n um b er of c harging stations a v ailable is not comparable to the n um b er of regular truc k stops and gas stations, the n um b er of BET s in op eration will also b e limited at first. F urthermore, in tegrating sp eed optimization in to the planning mitigates range and feasibilit y problems. While larger battery capacit y and c harging p o w er help narro wing the p erformance gap b et w een BET s and diesel truc ks, the battery and c harger requiremen ts are less restrictiv e when practical constrain ts are considered. It is true that BET s presen t a series of limitations and that they cannot replace diesel truc ks in ev ery situation. Ho w ev er, it is imp ortan t to remem b er that diesel truc ks do not op erate in ideal scenarios. Ha ving a larger range do es not mean that this range is alw a ys gonna b e needed. Being able to refuel fast do es not mean that breaks are limited to short refueling stops. Similarly , it is unrealistic to ev aluate BET s as if they 124 Figure 6.5: T rip duration under differen t parking a v ailabilit y and c harging infrastructure conditions. w ere supp osed to b e drop-in replacemen ts for diesel truc ks. BET s will require differen t itineraries, but, as seen in Figure 6.5, the impact of needed adjustmen ts ma y b e small dep ending on eac h application’s constrain ts. Therefore, when comparing the p erformance of BET s and diesel truc ks, it is imp ortan t to accoun t for the impacts of practical constrain ts suc h as HOS regulations, deliv ery and parking constrain ts. 125 Figure 6.6: CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure condi- tions. 126 Figure 6.7: T rip feasibilit y under differen t parking a v ailabilit y and c harging infrastructure condi- tions for scenarios with 600k Wh battery capacit y . Figure 6.8: T rip duration under differen t parking a v ailabilit y and c harging infrastructure conditions for scenarios with 600k Wh battery capacit y . Figure 6.9: CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure condi- tions for scenarios with 600k Wh battery capacit y . 127 6.6 Exp erimen ts on Random Net w orks 6.6.1 Without fast c hargers Exp erimen ts w ere p erformed on 10 random net w orks with b et w een 15 and 100 c harging stations. The a v erage distance b et w een c harging stations is set to 200km. All c harging stations ha v e the same c harging p o w er (100 k W or 50 k W, dep ending on the scenario), and are sub ject to a v ailabilit y time- windo ws. Within eac h scenario, probabilit y distributions are defined for the start and end times of the parking a v ailabilit y time-windo ws. The same pair of distributions is used for all lo cations and da ys, but they are sampled separately . T able 6.4 describ es the probabilit y distributions used to generate the 3 differen t t yp es of time-windo ws used. W e consider a [0;24] departure time-windo w (first da y only), a daily [8;16] deliv ery time-windo w for the destination, and 100h planning horizon. The battery capacit y v alues tested w ere 500, 800, and 1000 k Wh. F uel constrain ts for diesel truc ks w ere relaxed b y setting a large fuel capacit y . The maxim um sp eed limit is set to 80km/h at all edges. As the energy consumption is affected b y sp eed, sp eed can also affect v ehicle range and problem feasibilit y . Therefore, in the exp erimen ts, w e also v ary whether the sp eed limit is tak en as the a v erage tra v el sp eed or if the v ehicle is allo w ed to reduce its sp eed to 70% of the sp eed limit (around 56km/h). Our ob jectiv e is to study ho w battery capacit y , c harging p o w er, sp eed con trol and c hargers/parking’ a v ailabilit y affect BET s p erformance in terms of trip duration, CO2 emissions and route feasibilit y , and compare it with a diesel truc k’s p erformance. Figure 6.10 sho ws ho w a v erage trip duration is affected b y parking a v ailabilit y , c harging p o w er and battery capacit y . In scenarios with wide time-windo ws and 50k W c hargers, diesel truc ks hold a significan t adv an tage in terms of trip duration. This result is exp ected as sc hedule adjustmen ts ma y b e needed to accommo date longer rec harging stops. HOS regulations and parking time- windo ws ma y also exacerbate this effect in some instances, as short dela ys can push driv ers close to their w orking hours limits and require them to add another daily r est (10h minim um) to the trip. Ho w ev er, as w e consider sev ere parking shortage scenarios (narro w time-windo ws), this adv an tage is reduced. If parking a v ailabilit y is limited, diesel truc ks also need to adjust their sc hedules to guaran tee appropriate parking regardless of ha ving longer range and faster refueling times. When 100k W c hargers are considered, the a v erage trip duration of BET s with 1MWh battery is 128 comparable to the baseline diesel truc ks’ in all scenarios. Sho wing that, in these instances, adequate sync hronization of rec harging and resting times completely negated the c harging time and range issues. In terms of CO2 emissions, BET s presen t a clear adv an tage in all feasible scenarios, as sho wn in Figure 6.11. In addition to this adv an tage in emissions, BET s’ energy efficiency , com bined with increasing diesel prices, can also create cost adv an tages, as sho wn in Figure 6.12. The trip cost presen ted in Figure 6.12 considers a diesel cost of $0:95=L (US a v erage Octob er 2021 [164]), electricit y cost of $0:142=kWh (US a v erage Octob er 2021 [165]), and marginal hourly op erational cost (excluding fuel) of $54.71 [13]. Figure 6.13 presen ts ho w route feasibilit y w as affected b y parking a v ailabilit y , c harging p o w er, battery capacit y and sp eed flexibilit y . Note that the results sho wn are the aggregate of all 10 net w orks used. Although feasibilit y issues w ere mitigated b y higher c harging p o w er and tra v el sp eed flexibilit y , they w ere not resolv ed, and some net w orks had zero feasibilit y rate for one or more scenarios. Nev ertheless, the effec t of parking a v ailabilit y and sp eed optimization sho w that these are also imp ortan t factors to consider when assessing the viabilit y of truc k electrification. 129 Figure 6.10: A v erage trip duration under differen t parking a v ailabilit y and c harging infrastructure conditions. Includes only scenarios that allo w sp eed reduction. 130 Figure 6.11: CO2 emissions under differen t parking a v ailabilit y and c harging infrastructure condi- tions. Includes only scenarios that allo w sp eed reduction. 131 0 2000 4000 Cost ($) 3174 3055 3349 3761 Time-windows: Wide 3182 3113 3329 3517 Time-windows: Medium Chargers Power: 50 kW 3507 3204 3328 3682 Time-windows: Narrow Diesel 1000 800 500 Battery Capacity (kWh) 0 2000 4000 Cost ($) 3174 2489 2783 3640 Diesel 1000 800 500 Battery Capacity (kWh) 3182 2495 2914 3684 Diesel 1000 800 500 Battery Capacity (kWh) Chargers Power: 100 kW 3507 2735 3121 3679 Figure 6.12: T rip cost in US dollars under differen t parking a v ailabilit y and c harging infrastructure conditions. Includes only scenarios that allo w sp eed reduction. 132 Figure 6.13: P ercen tage of instances that w ere feasible under differen t parking a v ailabilit y and c harging infrastructure conditions. All diesel truc k scenarios w ere feasible. 133 6.6.2 F ast c hargers with fixed w ait time In these exp erimen ts, w e use 10 randomly generated net w orks to study the effect of fast c hargers. The a v erage spacing b et w een c harging stations is set to 100km, the departure time constrain t w as narro w ed to the in terv al [6;12] , and w e fix the a v ailabilit y time-windo ws to the mean v alues of the distributions used previously . The scenarios considered v ary the p ercen tage of c harging stations replaced b y fast c harging stations (0, 10, 20 or 30%), the c harging p o w er of the 2 t yp es of c hargers (regular c hargers: 50 or 100k W, fast c hargers: 150, 300, 500k W). Results for diesel truc ks are also giv en as a baseline. The stop duration at fast c hargers is limited to 3h. Charging stations with fast-c hargers do not ha v e a v ailabilit y time-windo ws, but, instead, ha v e a fixed w aiting time (0.5, 1 or 2h) b efore rec harge starts. The w aiting time is treated as service time, but it do es not coun t to w ards the stop duration limit. Figure 6.14 sho ws the a v erage trip duration for scenarios without fast c hargers. The results obtained are similar to the ones from Figure 6.10. When 50k W c hargers are used, the difference in trip duration b et w een BET s and diesel truc ks decreases when narro w time-windo ws are considered. When 100k W c hargers are used, BET s with 1MWh batteries p erform on the same lev el as diesel truc ks. Figure 6.15 and Figure 6.16 presen t results when part of the c harging stations are replaced b y fast c hargers with 1h w ait. When regular c hargers ha v e only 50k W p o w er, the inclusion of 300 and 500k W fast c hargers sho w trip duration reductions of up to 20% compared to scenarios without fast c hargers. In scenarios with 100 k W c hargers, the impro v emen t generated b y fast c hargers is significan tly reduced, b eing completely eliminated for BET s with 1 MWh batteries. F urthermore, the impro v emen ts gained from increasing the p ercen tage of fast c hargers are also significan tly smaller than in the scenarios with 50k W c hargers. This sho ws that although fast c hargers are the in tuitiv e w a y to address one of BET’s main w eak-p oin ts, dep ending on the con text, it migh t b e more adv an tageous to increase the n um b er or p o w er of regular c hargers instead of increasing the n um b er of fast c hargers. Figures G.1, G.2, G.3, and G.4 sho w the results for 0.5 and 2h w aiting time and can b e found in App endix G. These results shed some ligh t on ho w the c harging and parking infrastructure can impact BET p erformance and viabilit y . Similar sim ulations can b e used to determine appropriate infrastructure 134 Figure 6.14: A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging stations, and without fast c hargers. 135 Figure 6.15: A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging stations, 50k W fast c hargers, and 1h w ait. 136 Figure 6.16: A v erage trip duration for scenarios with 100km a vg. spacing b et w een c harging stations, 100k W fast c hargers, and 1h w ait. 137 lev els for a region giv en its exp ected BET p opulation and a v ailable budget. F or example, p oli- cymak ers can study whether it is more b eneficial to in v est in increasing the n um b er of c hargers, or increasing the p o w er of the installed c hargers. It can also b e used t o advise truc k driv ers and truc king companies as to whic h t yp es of v ehicles will b etter fit the region’s future infrastructure. 6.7 Conclusion In this pro ject, w e studied ho w practical constrain ts, suc h as HOS (hours-of-service) regula- tions and limited parking a v ailabilit y , impact the p erformance gap b et w een battery electric truc ks (BET s) and diesel truc ks. Both BET s and diesel truc ks need to rest regularly due to HOS regula- tions, and, giv en the curren t truc k parking shortage in the US and the risks asso ciated with illegal truc k parking, it is imp ortan t to include parking information in the planning pro cess. W e include energy/fuel consumption constrain ts in to the resource constrained shortest path form ulation for the shortest path and truc k driv er sc heduling problem with parking a v ailabilit y constrain ts pro- p osed b y the same authors in [153]. Exp erimen ts w ere p erformed on random net w orks to estimate the trip duration and C02 emissions of a baseline diesel truc k and BET s with differen t battery capacities under differen t a v ailable c harging p o w er and parking shortage sev erit y lev els. In terms of emissions, BET s v astly outp erformed the diesel truc k in all feasible scenarios. F urthermore, when parking a v ailabilit y is limited, the p erformance gap (in terms of trip duration) b et w een BET s and diesel truc ks is greatly reduced in scenarios with 50k W c hargers, and further reduced when 100k W c hargers are considered. In addition, scenarios with fast c harging stations sho w ho w the b enefits of replacing some lo w-p o w er c harging stations b y fast-c hargers v ary with the p o w er and exp ected a v ailabilit y/w ait of c hargers from either t yp e. This illustrates the imp ortance of considering the v arious constrain ts to whic h driv ers are sub jected when ev aluating the viabilit y of BET s, and com- pare not if the solutions used b y diesel truc ks are feasible for BET s, but whether solutions tailored for BET s ha v e a go o d enough p erformance in a particular region. 138 Chapter 7 Optimizing Long Haul T ruc ks’ P ollutan t Emissions under Sto c hastic P arking A v ailabilit y 7.1 In tro duction A ccording to the U.S. En vironmen t Protection Agency [166], the U.S. T ransp ortation sector is resp onsible for 28% of the US’s greenhouse gas emissions, 23% of whic h are caused b y medium- and hea vy-dut y truc ks. This means that 6.4% of all greenhouse gas emissions in the U.S. are generated b y truc ks. F urthermore, this issue is not particular to the U.S.A. The Europ ean Union faces a similar problem, with almost 5% of their CO2 emissions originating from hea vy-dut y v ehicles [167]. The Europ ean Commission prop osed, in 2018, targets for the reduction of emissions in new hea vy- dut y v ehicles, sho wing that there is a gro wing concern with the topic [167]. Similar measures ha v e already b een adopted b y the state of California as an effort to impro v e its fleet’s efficiency and curb CO2 emissions. Although California w as able to reac h its total emissions reduction targets early b y pushing for the usage of renew able energy and greener tec hnologies, the emissions caused b y the transp ortation sector k eep rising, and hea vy-dut y v ehicles still coun t for around 8% of the state’s CO2 emissions [168]. Considering the con tin uous gro wth of the truc king industry , it is clear the imp ortance of dev eloping more efficien t w a ys of using the truc ks, trying to reduce their emissions as m uc h as p ossible. In Chapter 6, w e approac hed the problem of using parking a v ailabilit y information to impro v e planning for battery electric truc ks (BET s), and study ho w HOS regulations and parking/c harging infrastructure can affect BET s’ p erformance. Ho w ev er, unless w e are able to replace most or all in ternal com bustion engine truc ks b y zero-emission ones, emissions reduction will remain an imp ortan t ob jectiv e. 139 In this c hapter, w e start b y adapting the metho ds from Chapter 6 to optimizing b oth the tra v el time and emissions of diesel truc ks on time-dep enden t net w orks. Most of the w ork presen ted in sections 6.2 and 6.4 is general enough to b e used for emissions reduction b y simply relaxing battery capacit y restrictions and adding only rest areas without c harging infrastructure to the net w ork, so w e will fo cus on clarifying the differences and assumptions. W e study the trade-off b et w een emissions and trip duration, and ho w it is affected b y parking conditions. Then, in order to mak e the mo del more realistic, w e extend it to include uncertain t y in parking a v ailabilit y . In practice, it is imp ossible to b e certain ab out the future parking a v ailabilit y of an y lo cation during planning. Therefore, w e include this uncertain t y in the mo del and study its effect on the solutions dep ending on the information pro vided to driv ers/planners. 7.1.1 Related W ork The time-dep enden t shortest path problem w as first studied b y Co ok e and Halsey (1966) [169], who extended b ellman’s equations to time-dep enden t net w orks and presen ted an dynamic pro- gramming solution for the discrete time problem. Since then, p olynomial time solutions ha v e b een prop osed for net w orks with the FIF O (first in, first out) prop ert y , i.e., one cannot arriv e earlier at the end of an arc b y departing later. [170] pro v es that, in problems where the net w ork has the FIF O prop ert y the complexit y of lab eling algorithms for time-dep enden t net w orks is the same as for static net w orks. The FIF O assumption holds in practice for man y net w orks, including transp orta- tion net w orks. Most algorithms prop osed for the time-dep enden t shortest path problem are based on the Dijkstra and A* algorithms often studied for the static problem [171, 172, 173, 174, 175, 176]. [175] prop osed an algorithm to calculate sim ultaneously all the shortest paths from all no des to a giv en destination no de and for ev ery discrete time step in a net w ork with time-dep enden t arc costs. They presen ted a lab el correcting metho d that uses a b ottom-up dynamic programming approac h to calculate shortest paths. The problem do es not assume FIF O net w orks. In [177], Dean studies the theoretical prop erties of time-dep enden t shortest path problems, and presen ts serial and paral- lel algorithms for the problem of calculating the earliest arriv al time at one or more no des in FIF O net w orks. In [178], Zhao generalizes the A* algorithm for time dep enden t net w orks. The algorithm 140 correctness is guaran teed if the used time-dep enden t estimator functions satisfy the prop osed suf- ficien t conditions. The landmark based AL T algorithm is also extended to the time-dep enden t case. Landmarks with precalculated optimal tra v el times to ev ery no de (at ev ery time) are used to estimate lo w er b ounds for the tra v el times from ev ery no de to the destination. These lo w er b ounds satisfy the sufficien t conditions prop osed and are used in the A* algorithm to guide the exploration of the searc h space, impro ving p erformance. In [174], Nannicini et al. presen t a bidirectional searc h metho d that is also based on AL T algorithms. In [173], Delling presen ts a time-dep enden t v ersion of the SHAR C algorithm, whic h uses prepro cessing routines based on high w a ys hierarc hies [179] and arc-flags [180] to sp eed-up a Dijkstra-based algorithm. Another imp ortan t consideration is whether w aiting at no des is allo w ed. In [181], Orda & Rom sho w ed that if w aiting is allo w ed, then a shortest path can b e found in p olynomial time ev en without the FIF O assumption. Ho w ev er, if w aiting is not allo w ed and the net w ork do es not ha v e the FIF O prop ert y , the problem is NP-hard [181]. Later, F osc hini et al studied in detail the complexit y of the arriv al time function and of algorithms searc hing for a minim um dela y path [182]. Omer and P oss (2019) [183] prop osed a p olynomial time algorithm for the case when w ait times are allo w ed at all no des, but when those w ait times are not considered in the cost function. The algorithm calculates the shortest paths while iterativ ely increasing the maxim um allo w ed total w ait time. The authors used FIF O net w ork and piece-wise linear tra v el time assumptions to pro v e that an optim um solution can b e found while testing only a finite n um b er of total w ait times. The length of the sequence of w ait times that need to b e considered dep ends on the total n um b er of breakp oin ts of all tra v el time functions. The sequence of w ait times to b e tested is c hosen so that shortest paths passing through eac h no de can b e written as a concatenation of a shortest path found in the previous iteration, the w ait time increase for the curren t iteration, and a path to the destination without w aits. The problem addressed do es not consider time-windo ws, the only constrain t is on total w aiting time. [184] uses a time-dela y neural net w ork with the same top ology as the road net w ork to calcu- late the shortest path to a giv en destination no de when tra v el time b et w een no des is defined b y piecewise constan t functions. The time complexit y dep ends on the pro duct of the n um b er of time- windo ws needed to describ e the tra v el time functions, and on the shortest path’s arriv al time at the 141 destination no de (in time steps). This is an in teresting line of researc h, as, compared to metho ds based on Dijkstra and A*, the time complexit y is not as affected b y the net w ork size. Most studies fo cus on minimizing trip duration, arriv al time or driving time. Ho w ev er, in practice, those are not the only relev an t ob jectiv es. F or example, the transp ortation companies migh t w an t to minimize fuel consumption, emissions, safet y risks, or monetary costs. These t yp es of ob jectiv e functions ma y not satisfy FIF O assumptions and can b e more problematic to deal with, as studied in [185]. [185] sho w ed that in some time-dep enden t minim um w eigh t path problems there is no finite optimal path, and prop osed conditions for the existence of finite optimal paths. [186] study path planning under time-dep enden t cost functions mo deled as a spatiotemp oral scalar field. As an example, they men tion that minimizing a w eigh ted sum of tra v el duration and exp osure to traffic migh t b e useful in reducing emissions-related health risks to long-haul truc k driv ers. They study the effects of allo wing w aiting under scalar fields defined b y linear com binations of Gaussian functions and prop ose lo cal conditions to prune searc h trees used b y graph searc h algorithms. In [187], He et al. study problem v arian ts where a subset of no des has p enalties for w aiting or where there is a limit on the total w aiting time at a subset of no des. They pro v ed that some v arian ts are NP-Hard and prop osed p olynomial time algorithms for the ones that are not. In [188], Cai et al. consider no de-dep enden t upp er-b ounds on the w aiting time at eac h no de. The w ork on time-dep enden t shortest paths problems with constrain ts is limited. The most common constrain ts considered are applied to the w aiting time, as in [187, 188], or total trip duration [189]. [190] addresses time-dep enden t lab el-constrained shortest path problems whic h restrict the structure of acceptable paths. Besides the usual setup for shortest path problems, eac h arc is ascrib ed a lab el, and the acceptable lab el sequences are defined b y a ‘language’ . F or example, the lab els can represen t tra v el mo des, suc h as w alk, driv e, and bus, and the language can sp ecify that only paths using at most 2 lab els (tra v el mo des) are acceptable. Although v arian ts with time-windo w constrain ts are common for the static shortest path problem that is not the case for the time-dep enden t v ersion. In [191], Spliet et al. studied a problem v arian t with time-windo w constrain ts in the con text of a routing problem. Spliet et al. presen ted an exact lab eling algorithm and a heuristic tabu searc h algorithm for the shortest path problem with a capacit y constrain t, time-dep enden t tra v el times, time windo w constrain ts on b oth the no des and on the arcs, and linear no de costs. The lab eling algorithm w as based on the algorithm prop osed b y [192] for a 142 similar v arian t, where the tra v el time is not time-dep enden t, but no des ha v e time-windo ws and time-dep enden t costs. [120] studied the time-dep enden t shortest path problem under time-windo w and hours-of-service regulation constrain ts, where the solution represen ts not only a path, but also a sc hedule sp ecifying for ho w long the driv er m ust rest at eac h lo cation. They define a state-graph where eac h no de is a certain stoppage configuration and presen t a Dijkstra-based algorithm with a pruning heuristic to find go o d solutions. A related problem w as studied in [26], where K ok et al. presen ted an in teger linear programming form ulation for the time-dep enden t truc k driv er sc heduling problem. Although their form ulation considers a fixed path, it can b e used as a p ost-pro cessing step for shortest path or v ehicle routing problems under time-windo w and hours-of-service regulation constrain ts. 7.2 Mo del The problem consists of planning the path and sc hedule for a truc k starting at an origin lo cation and visiting an ordered list of clien ts, where the last clien t is referred to as the destination. Eac h clien t has a fixed non-negativ e service time, and time-windo w constrain ts restricting the v ehicle’s arriv al time. The sc hedule m ust comply with HOS regulations, whic h imp ose restrictions on ho w long the driv er can w ork or driv e without resting, and the minim um duration of rests (rests of differen t durations satisfy differen t restrictions). Driv ers can rest only at rest areas, but arriv al time at rest area no des is also sub ject to time-windo w constrain ts (represen ting parking a v ailabilit y). As in the BET planning v arian t (section 6.2), the v ehicle consumes fuel/energy when driving or idling. W e consider the consumption mo del for diesel truc ks defined in section 6.3.1. The driving consumption rate is describ ed b y a non-linear sp eed-dep enden t function, and the idling consumption rate is tak en as constan t. Eac h road section has an allo w ed sp eed range, ho w ev er, in this case the range is time-dep enden t [s ij (t);s + ij (t)] . W e assume that the sp eed profiles are defined suc h that all edges satisfy FIF O assumptions when considering only one of the sp eed limits. W e do not add an y new activities or c hange ho w no des are expanded, so, b esides adding information regarding time-dep enden t sp eed profiles, there is no need to mak e an y mo difications to the extended net w ork describ ed in 5.4.1. 143 As this c hapter addresses the issue of emissions reduction, w e w an t this to b e reflected in our ob jectiv e function. W e tak e as ob jectiv e function a linear com bination of trip duration and fuel consumption. The trip duration term accoun ts for driv er w ages and op erational costs (excluding fuel), whereas the fuel term accoun ts for fuel and emissions costs. The emissions costs can b e seen b oth as some kind of carb on pricing, or simply the lev el of imp ortance attac hed to reducing emissions as opp osed to reducing trip duration. Consider the follo wing cost function for a decision of duration : g() = 8 > > < > > : + e ( e =); ifdriving (+ +); o:w: (7.1) , where is the truc ks hourly op erational cost (excluding fuel/energy) and is the cost p er unit of fuel/energy . F or non-driving decisions, is the hourly idling fuel/energy consumption, represen ts hourly costs incurred while stopp ed from sources other than idle energy consumption and op erational costs. F or driving decisions, e is the length of the road segmen t considered, and (v) is the fuel/energy consumption p er unit of distance. This cost function considers b oth time and energy/fuel related costs, and their relativ e imp ortance can b e adjusted using the parameters , , and . The dynamic programming form ulation presen ted in section 6.4 already represen ts the decision space as a function of the state (U(x k ) ), so it is general enough to represen t the time-dep enden t case and do es not need an y mo dification. Ho w ev er, the analytical solutions, lo w er b ounds and the metho ds presen ted for constrain t propagation are describ ed for time-indep enden t net w orks and need some clarification. Constrain t Propagation Although tra v el time affects all resources, due to the FIF O assump- tion, w e fo cus the mo difications on the time resource. Let the function a ij ( i ) represen t the arriv al time at no dej when departing no dei at time i , and w e use an under-bar to indicate when the minim um sp eed is b eing considered, and an o v er-bar to indicate when the maxi- m um sp eed is b eing considered. During forw ard propagation, w e replace i + e ; + i + + e b y a ij ( i );a ij ( + i ) . During bac kw ard propagation, w e replace i + e ; + i e b y h a 1 ij ( i );a 1 ij ( + i ) i , the sup erscript ‘1 ’ refers to the in v erse function. As w e assume that the sp eed profiles satisfy FIF O assumptions, b oth a ij and a ij are strictly increasing and ha v e unique in v erses. F or the other (non time) resources, w e simply tak e e and + e as the 144 lo w er and upp er b ounds for the tra v el time at an y time instan t. Another p ossibilit y is to use i ; + i to calculate the range of p ossible tra v el times for these departure/arriv al times. Ho w ev er, this metho d can only pro vide a b etter range if the in terv al i ; + i is narro w and do es not span a wide range of p ossible tra v el times. Analytical Solutions The analytical solution for the last driving decision, describ ed in section 6.4.2.1, is not affected b y the time-dep enden t tra v el time as the departure time is fixed and kno wn. The last rest extension decision is affected b y the c hange in tra v el time, so the one describ ed in section 6.4.2.2 is not v alid an ymore. In this case, w e did not calculate a new solution, and opted to stop using analytical solutions for those cases. Cost Lo w er Bound Similar to the case of HOS constrain ts propagation, w e use the upp er and lo w er sp eed b ounds o v er the whole planning p erio d to calculate the b ounds describ ed in section 6.4.3. 7.3 P arking A v ailabilit y Uncertain t y The mo dels presen ted in c hapters 4, 5 and 6 considered that parking a v ailabilit y could b e predicted with certain t y . Ho w ev er, in practice, there is a certain lev el of uncertain t y in an y pre- diction, and the longer the prediction horizons the less certain w e can b e ab out an y prediction. Therefore, w e no w mo del parking a v ailabilit y in a probabilistic w a y . The previous form ulation represen ted parking a v ailabilit y as time-windo ws at eac h rest area. T w o p ossible w a ys of extending this form ulation are: Sto c hastic Time-Windo ws: w e assume that there is a con tin uous time in terv al within whic h parking is guaran teed, but w e are unsure of the exact start and end times. The time-windo w’s start and end times are giv en b y random v ariables with kno wn distribution. P arking a v ail- abilit y is defined indirectly , dep ending on whether the arriv al time falls within that in terv al or not. So, w e w ould need to consider the probabilit y of arriving after the start of the time- windo w, but b efore its end. The deterministic mo del could b e seen as an appro ximation using the exp ected v alues of the time-windo w’s limits, or v alues that satisfy some confidence lev el. This mo del w ould ignore the small o ccupancy v ariations that can o ccur. F or exam- ple, o v ernigh t, most of the parking spaces are tak en b y long-term parking. Ho w ev er, unless ALL parking spaces are used for long-term parking, there will still b e some truc ks lea ving on o ccasion. Sto c hastic P arking A v ailabilit y: directly mo del parking a v ailabilit y as a random v ariable with a time-dep enden t probabilit y distribution, i.e., at an y time t , there is a probabilit y p i (t) of rest area i ha ving an a v ailable parking space. In this approac h, the probabilit y of finding parking can b e calculated directly , without w orrying ab out ho w the distributions of time- windo ws’ limits in teract. The small v ariations that o ccur ev en at high o ccupancy p erio ds can b e mo deled b y a v ery small, but non-zero, probabilit y of finding parking during that p erio d. The deterministic mo del can b e seen as time-windo ws defined b y the in terv als at whic h p i (t) exceeds a giv en threshold. 145 W e tak e the second approac h, mo deling sto c hastic parking a v ailabilit y directly . A new binary comp onen t w , represen ting whether parking is a v ailable at the curren t lo cation (Y es:1, No:0), is added to the state definition. This comp onen t can b e used to con trol the actions a v ailable to driv ers at rest areas, e.g., if parking is a v ailable (w = 1 ), the driv er needs to c ho ose for ho w long to rest, if the rest area is full, the driv er needs to revise the trip plan and decide whether to searc h for nearb y alternativ e parking lo cations or to con tin ue driving. The new state con tains the follo wing information: • Curren t no de (v ) • Time when no de w as visited ( 0 ) • A ccum ulated driving time since last break ( b ) • Elapsed time since last daily rest ( r ) • A ccum ulated driving time since last daily rest ( r ) • A ccum ulated on-dut y time since last w eekly rest ( w ) • P arking a v ailabilit y (w ) W e define the up date rule for w as w i+1 = f (w) (x i ;u i ;!) = ! , where ! is a binary random v ariable c haracterized b y a probabilit y distribution P(jx i ;u i ) . As w aims to mo del the parking a v ailabilit y at rest areas, it is set to default v alues at other lo cations as needed. The dynamic programming form ulation presen ted b efore can b e up dated as follo ws: J (x) = min u2U(x) E ! fg(x;u;!)+J (f (x;u;!))g = min u2U(x) 1 X !=0 P(!jx;u)(g(x;u;!)+J (f (x;u;!))) (7.2) It is imp ortan t to note that as w e are considering parking a v ailabilit y to b e sto c hastic, it migh t b e imp ossible to guaran tee parking at all times. As long as the probabilit y of finding parking at the visited lo cations is not 1, it is p ossible for a driv er to try to park at ev ery single lo cation without success un til the next lo cation is to o far to b e reac hed without exceeding HOS constrain ts. Therefore, the mo del m ust include what happ ens in those cases. If the driv er exceeds the HOS limits, the truc k’s monitoring equipmen t migh t automatically sh ut do wn the truc k, and the driv er w ould b e stuc k somewhere incon v enien t for a while. The driv er migh t face legal p enalties, ma yb e a fine or license susp ension. If the driv er stops at a road shoulder or high w a y ramp, there is an 146 asso ciated risk of causing acciden ts or b eing fined. In an y case, the mo del m ust consider that suc h scenarios are p ossible, what actions can b e tak en and what are their consequences/costs. 7.3.1 Recourse A ctions W e consider t w o p ossible w a ys for a driv er to react when unable to find parking at the curren t lo cation: reroute and try to rest at a do wnstream facilit y , or lo ok for an alternativ e parking lo cation in the surrounding region. Essen tially , the driv er needs to decide whether it is feasible to stop later or if they need to stop righ t a w a y . The deterministic mo del included three t yp es of actions at rest areas, eac h one represen ting an off-dut y p erio d that resets the coun ter for a certain set of regulations. In the sto c hastic mo del, if parking is a v ailable, the same set of actions is used, but when parking is una v ailable, w e consider that all 3 rest actions are prohibited. Instead, w e include the actions se ar ch and exit : Exit: represen ts the action of lea ving the rest area and heading to the next lo cation without resting, and is connected to the exit no de of the rest area in the problem’s graph represen tation. Searc h: represen ts the action of lo oking for an alternativ e parking option nearb y , and it leads to the en trance no de of an alternativ e parking lo cation. The alternativ e parking lo cation will b eha v e the same w a y as a regular parking lo cation, except b y the fact that p enalt y costs will b e incurred for its usage. Figure 7.1 sho ws a diagram of the actions a v ailable at rest areas after including the recourse actions. The se ar ch action’s duration can b e used as a time p enalt y that forces driv ers to adjust the rest of the trip, and it can ha v e cost p enalties included in it. Essen tially , it is an y action that will lead to ha ving a lo cation to rest without driving to another one of the facilities included in the graph. F or the purp ose of calculating fuel consumption, searc hing is treated as driving at a user-defined sp eed. In our exp erimen ts, w e only apply p enalties for the se ar ch action if its duration exceeds the driv er’s remaining allo w ed driving time. In this case, w e assume that the driv er w ould need to park at a lo cation ev en w orse than usual in order to a v oid HOS violations, hence incurring some extra p enalties (b oth a fixed p enalt y and one prop ortional to the excess duration). In the alternative p arking lo c ations , w e do not include time p enalties as that w as already considered in the se ar ch edge, but they ma y ha v e b oth fixed and v ariable costs assigned to them (on top of the usual time and fuel consumption costs). Note that w e w an t to mo del the fact that driv ers can react to the lac k of parking and w e use these generic actions to do so. Whether driv ers will driv e around for 147 Figure 7.1: Subgraph represen ting the actions that can b e tak en at rest areas after inclusion of recourse actions and alternativ e parking lo cations. a while lo oking for parking and then park at a road shoulder, or will mak e use of some service to arrange for appropriate parking, dep ends on the options a v ailable in the region, driv ers/companies preferences and the risks/costs in v olv ed with eac h option. One could ev en include m ultiple sets of alternativ e parking lo cations and recourse actions, e.g., one for lo oking for a road shoulder to park at, and another for using an exp ensiv e service that offers guaran teed parking or driv er replacemen t. 7.3.2 P olicy Our ob jectiv e is to giv e driv ers and planners go o d recommendations ab out ho w to plan their trips. The p olicy obtained tak es the curren t state of the system and outputs the decision that minimizes a certain cost (or whatev er estimate w e ha v e of that cost). As can b e seen in (7.2), in the sto c hastic case w e optimize an exp ected v alue of the cost function, so the p olicy cost is a single v alue represen ting that exp ectation. Ho w ev er, it migh t b e in teresting for the user to visualize more information ab out this cost. As sho wn in (6.7), to calculate the p olicy cost w e need to 148 sim ulate its effect on the system b y recursiv ely applying the p olicy . When doing this, w e generate a decision tree describing ho w the system ev olv es under that p olicy giv en an initial state. The tree branc hes out at decisions leading to rest areas, as the state when arriving at the rest area dep ends on the random v ariable represen ting parking a v ailabilit y . As parking a v ailabilit y can only tak e binary v alues, this system is easier to sim ulate than systems where the random v ariables can tak e a large or infinite n um b er of v alues (e.g., a con tin uous in terv al). Although that decision tree is readable only for v ery small problems, w e can use it to calculate the probabilit y distribution of an y cost function, regardless of if it w as actually considered during the p olicy generation or not. The decision tree giv es us the p ossible paths and their probabilities, so if w e kno w ho w to calculate the target information giv en the path, w e can calculate the probabilit y distribution of that information. F or example, w e can generate a p olicy that minimizes the estimated trip duration, then sim ulate that p olicy to find out ho w it affects the lik eliho o d of using alternativ e parking, the probabilit y distribution of emissions, etc. The time it tak es to calculate that information dep ends on ho w fast the p olicy function can b e ev aluated and ho w man y states need to b e visited, so it will increase if the trip is longer or has man y rest stops (rest stops cause branc hing). 7.3.3 Imp erfect Information Studying this question of sim ulating the cost/effects of a giv en p olicy raised another concern. Similar to ho w lo okahead p olicies optimize an appro ximate cost, whic h is usually differen t from its o wn p olicy cost, p olicies ma y b e calculated based on information or assumptions that do not matc h realit y . F or example, companies that disregard parking difficulties migh t generate p olicies based on the assumption that all rest areas are a v ailable 24/7. Ev en if that p olicy is calculated without using appro ximations and is the optimal p olicy for a scenario satisfying the assumption, it can sho w significan tly differen t results when applied to the real w orld. Therefore, w e think it is in teresting to study ho w imp erfect information affects differen t p olicies. This can giv e us some insigh t on the v alue of information and sev erit y of cost misestimation. F or clarit y , w e redefine the p olicy used, differen tiating b et w een the mo del used b y the plan- ner and the ones used for sim ulation, where the mo dels define the probabilit y distributions, cost functions and state transition functions considered. Let p (x k ) b e a one-step lo okahead p olicy 149 calculated using a mo delp , andJ p;s (x k ) b e the cost of applying p olicy p when a mo dels is tak en as the “w orld mo del” for sim ulation. p (x k ) = argmin u2U(x k ) E !p n g p (x k ;u;! p )+ e J (f p (x k ;u;! p )) o = argmin u2U(x k ) 1 X !p=0 P p (! p jx k ;u) g p (x k ;u;! p )+ e J (f p (x k ;u;! p )) (7.3) J p;s (x k ) = 8 > > < > > : 0; if x k 2X d E !s g s (x k ; p (x k );! s )+J p;s (f s (x k ; p (x k );! s )) ; o.w. (7.4) As in the deterministic case, w e use a rollout algorithm, so the appro ximate cost function is the p olicy cost of a base p olicy . Our base p olicy is the optimal p olicy o v er a coarsely discretized decision space. Ho w ev er, no w it is imp ortan t to emphasize that, as the sim ulation mo del migh t b e unkno wn to the planner, the planning mo del is the one used to calculate the base p olicy’s cost, i.e.,: e J (x k ) = 8 > > > < > > > : 0; if x k 2X d min u2 e U(x k ) E !p n g p (x k ;u;! p )+ e J (f p (x k ;u;! p )) o ; o.w. (7.5) , where e U (x k ) is the coarsely discretized decision space. This c hapters’ exp erimen ts consider the follo wing cases: F ull Information (Scenario 0): driv ers ha v e full access to parking a v ailabilit y probabilit y dis- tributions, and arriv al time at rest areas is not restricted. Time-windo w + Deterministic (Scenarios 1-3): driv ers assume that parking is guaran teed within certain time-windo ws generated from exp erience or data. As in the deterministic case, driv ers can only arriv e at rest areas within the giv en time-windo ws. Time-windo w + Uncertain t y (Scenarios 4-6): driv ers ha v e access to the parking a v ailabilit y probabilit y distribution within giv en time-windo ws. Ho w ev er, arriv al at rest areas is still limited to within these time-windo ws. Scenarios 1-3 w ere implemen ted while trying to limit mo difications to the system’s basic rou- tines. They use a deterministic view of the w orld, so they w ere implemen ted b y b ypassing the 150 usage of probabilit y distributions when calculating the p olicies. Mo difications to the curren t im- plemen tation of cost functions and p olicies are required to facilitate the decoupling b et w een the information used during planning and the “w orld mo del” used for sim ulation/ev aluation. 7.4 Exp erimen ts The follo wing sections presen t the results of exp erimen ts p erformed on static and time- dep enden t net w orks. The marginal op erational cost p er hour do es not accoun t for fuel costs and is set to $54.77 (similar to $54.71 found in [13]). The emission co efficien t is set to 3.13 kg CO 2 /L (0.44 from pro duction [161] and 2.69 from com bustion [160]), diesel prices w ere set to 1$/L, and the emission costs are set to 18e 3 $/kgCO 2 [193]. The relativ e imp ortance of reducing emissions is con trolled b y applying a p enalt y m ultiplier to the emission cost. T ables with results detailed p er net w ork w ere included in App endix H. The deterministic scenarios w ere also used to com- pare the effectiv eness of the cost lo w er b ounds presen ted in 6.4.3, and the results w ere included in App endix H. 7.4.1 Static Net w ork The results presen ted w ere obtained using the lo w er b ound describ ed in subsubsection 6.4.3.2. The rollout p olicy used a sa mpled decision space with 0.5h sampling in terv al, and the base p olicy is the optimal solution when using a sampled decision space with 5h sampling in terv al and stopping if a solution is within 5% of the cost lo w er b ound. Figure 7.2 presen ts the impro v emen ts obtained in CO2 emissions when the p enalt y m ultiplier is increased under differen t parking a v ailabilit y conditions. W e see up to 5-7% decrease in CO2 emissions, with the smaller impro v emen ts happ ening in scenarios with narro w time-windo ws. Fig- ure 7.3 and Figure 7.4 sho w the effects of differen t p enalt y m ultipliers on the a v erage trip duration and nonp enalized trip cost, resp ectiv ely . Although there are some outliers with more than 20% increase in trip duration or cost, the a v erage increases in duration and cost are under 15% and 7%, resp ectiv ely , in all scenarios. Figure 7.5 presen ts the a v erage running time v aries with the p enalt y m ultiplier. The instances placing a higher priorit y on emissions reductions sho w significan tly longer running times. This is p ossibly due to the b ounds on trip duration b eing tigh ter than the b ounds on 151 1 10 50 100 500 1000 Penalty Multiplier 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 CO2 Emissions relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on emissions (all networks) Figure 7.2: CO2 emissions as a fraction of the baseline emission. The baseline emission for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 1 10 50 100 500 1000 Penalty Multiplier 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Trip duration relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on trip duration (all networks) Figure 7.3: T rip duration as a fraction of the baseline duration. The baseline duration for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. the m ulti-ob jectiv e cost accoun ting for b oth duration and emissions. Nev ertheless, as the p enalt y m ultiplier do es not affect feasibilit y , solving faster instances first and using their solutions to gener- ate upp er b ounds for slo w er instances migh t impro v e p erformance. This approac h is tested in the exp erimen ts o v er time-dep enden t net w orks (subsection 7.4.2). T ables H.1-H.4 presen t the results for eac h net w ork separately . 152 1 10 50 100 500 1000 Penalty Multiplier 1.0 1.1 1.2 1.3 1.4 Cost relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on nonpenalized cost (all networks) Figure 7.4: Nonp enalized cost as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 1 10 50 100 500 1000 Penalty Multiplier 0 25 50 75 100 125 150 Running time relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on running time (all networks) Figure 7.5: R unning time as a fraction of the baseline cost. The baseline running time for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 153 7.4.2 Time-Dep enden t Net w ork Including time-dep enden t tra v el times increases problem complexit y so an extra heuristic w as included to impro v e p erformance. As can b e seen in the static net w ork exp erimen t results, the problem is significan tly faster to solv e when lo w er p enalt y v alues are used, so w e solv ed the instances with small p enalt y first and used the solutions to generate upp er b ounds for instances with larger p enalt y . The solution for p enalt y 1 instances w ere used to generate upp er b ounds for the instances with p enalt y 10, whic h w ere used to impro v e instances with p enalt y 50, and so on. The results presen ted w ere obtained using the lo w er b ound describ ed in subsubsection 6.4.3.2 (Bound 2) and do not include the time sp en t solving smaller instances. The rollout p olicy used a sampled decision space with 0.2h sampling in terv al, and the base p olicy is the optimal solution when using a sampled decision space with 2h sampling in terv al and stopping if a solution is within 1% of the cost lo w er b ound. Figure 7.6 presen ts the impro v emen ts obtained in CO2 emissions when the p enalt y m ultiplier is increased under differen t parking a v ailabilit y conditions. As in the static case, w e see up to 5-7% decrease in CO2 emissions, with the smaller impro v emen ts happ ening in scenarios with narro w time-windo ws. On the other hand, the impact of higher p enalt y v alues on trip duration and cost is substan tially larger than in the static case. Although Figure 7.7 sho ws that emissions reduction comes at the cost of significan t increases in trip duration, w e can also notice that when parking a v ailabilit y is scarce the increase in trip duration is not as sev ere. This b eha vior is also reflected in the costs sho wn in Figure 7.8. Whereas scenarios with wide and medium time-windo ws sho w ed a v erage cost increases of up to 30%, scenarios with narro w time windo ws sho w ed an a v erage cost increase of around 12% when the p enalt y m ultiplier is set to 1000. These results illustrate the significance of the impact of parking a v ailabilit y conditions and HOS regulations in the cost-b enefit analysis of prioritizing emissions reduction during planning. Figure 7.9 sho ws the increases in running time relativ e to the instances with p enalt y equal to 1. Ev en though the solutions of instances with lo w p enalt y are b eing used as upp er b ounds for the instances with high p enalt y , there is still a significan t increase in the a v erage running time. P ossible reasons for this are: the lo w er b ounds are tigh ter when lo w er p enalties are used; the significan tly larger increases in the p enalt y m ultiplier from 100 to 500 and from 500 to 1000 (compared to 1-10, 154 1 10 50 100 500 1000 Penalty Multiplier 0.85 0.90 0.95 1.00 CO2 Emissions relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on emissions (all networks) Figure 7.6: CO2 emissions as a fraction of the baseline emission. The baseline emission for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 1 10 50 100 500 1000 Penalty Multiplier 1.00 1.25 1.50 1.75 Trip duration relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier Parking Time-windows: Narrow Penalty’s effect on trip duration (all networks) Figure 7.7: T rip duration as a fraction of the baseline duration. The baseline duration for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 10-50) mak e it so the solutions used as upp er b ounds are not as efficien t as for scenarios with closer p enalties. Nev ertheless, the faster running times for lo w p enalt y solutions mak e it con v enien t to solv e a lo w p enalt y instance first and use its solution as an upp er b ound for high p enalt y instances. T ables H.5-H.8 presen t the results for eac h net w ork separately . 155 1 10 50 100 500 1000 Penalty Multiplier) 1.0 1.2 1.4 Cost relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier) Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier) Parking Time-windows: Narrow Penalty’s effect on nonpenalized cost (all networks) Figure 7.8: Nonp enalized cost as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 1 10 50 100 500 1000 Penalty Multiplier) 0 5 10 Running time relative to baseline Parking Time-windows: Wide 1 10 50 100 500 1000 Penalty Multiplier) Parking Time-windows: Medium 1 10 50 100 500 1000 Penalty Multiplier) Parking Time-windows: Narrow Penalty’s effect on running time (all networks) Figure 7.9: R unning time as a fraction of the baseline cost. The baseline cost for eac h scenario is the v alue obtained with p enalt y m ultiplier of 1. 156 7.4.3 Uncertain P arking A v ailabilit y Our exp erimen ts used the same probabilit y distribution for all lo cations. The parking a v ailabil- it y probabilit y distribution w as set so that the lik eliho o d of finding parking is lo w at nigh t and high during the da y . This b eha vior reflects driv ers’ usual complain ts regarding difficult y to find parking for o v ernigh t rests. In practice, historical data should b e used to define a probabilit y distribution for eac h rest area. Due to the increased complexit y , the minim um tra v el time b et w een rest areas w as increased to 4h. In addition, exp erimen ts w ere run only in a subset of the net w orks used in the deterministic exp erimen ts. The ob jectiv e function includes an hourly p enalt y of $10/h when park ed illegally , and a fixed p enalt y of $100 p er illegal parking ev en t. W e set the searc h when parking is una v ailable to 0.5h. W e apply a fixed p enalt y of $100 if the driv er has less than 0.5h of remaining driving time, plus a v ariable p enalt y of $300/h applied on the difference b et w een the searc h time and the remaining driv e time. Ev en when parking at unauthorized lo cations, driv ers still need some time to searc h for a less risky one. Therefore, w e apply this p enalt y to represen t b oth the increased risk of parking at a w orse lo cation due to lac k of time, and the risk of not b eing able to find a lo cation to park b efore exceeding regulations’ limits. As describ ed in section 7.3.3, the scenarios used v ary according to the information a v ailable to driv ers/planners and the w eigh t giv en to CO2 emissions. In Scenario 0, driv ers kno w the probabilit y of finding parking at ev ery lo cation and time. F or scenarios 1-3, driv ers assume that parking is alw a ys a v ailable within giv en time-windo ws. These time-windo ws are defined as time in terv als when the probabilit y of finding parking is equal to or larger than a certain threshold. Scenarios 1, 2 and 3 use, resp ectiv ely , 0.95, 0.77, and 0.59 as thresholds. Scenarios 4,5 and 6 use the same time-windo ws as scenarios 1,2 and 3, but driv ers kno w that parking is not guaran teed and kno w the probabilit y of finding parking at eac h time within those windo ws. 7.4.3.1 Results Figure 7.11 presen ts the emissions, trip duration, and trip cost of one of the test net w orks. When full information is used (scenario 0), the results reflect w ell the in ten t of the cost functions used. Scenarios with larger CO2 p enalties result in lo w er emissions ev en at the cost of higher 157 Figure 7.10: F unction used to define the probabilit y of finding parking at a giv en rest area and time. trip cost and duration, and w orse results ha v e decreasing probabilities. Scenarios 1 to 3 generate somewhat disorderly probabilit y distributions with a v erage p erformance inferior to scenario 0 and larger v ariances. In these scenarios, the CO2 p enalt y is unable to con trol the CO2 emissions as effectiv ely . F or example, in scenario 2 of Net w ork 5, the a v erage CO2 emissions with p enalt y 50 is larger than with p enalt y 1. Figure 7.12 presen ts the probabilit y distributions for parking-related p erformance measures for eac h test net w ork for scenarios 0 to 3. As for Figure 7.11, scenario 0 has a significan tly b etter p erformance, with decreasing probabilities for w orse outcomes and similar b eha vior for all net w orks. On the other hand, scenarios 1 to 3’s p erformance is less consisten t and more sensitiv e to the net w ork. Figures 7.13 and 7.14 presen t the same p erformance measures, but comparing scenarios 4 to 6 with scenario 0. In all these scenarios the driv er has full information ab out the probabilit y of finding parking. Ho w ev er, in the scenarios 4 to 6, driv ers are restricted to the same time-windo ws considered in scenarios 1 to 3. W e can see that b y informing driv ers ab out the parking uncertain t y and including this information in the planning algorithm, the p erformance is significan tly less affected b y the time-windo ws used. The drop in p erformance is not as pronounced, and the results 158 are less sensitiv e to scenario parameters, k eeping the exp onen tial-lik e shap e of the distributions found for scenario 0. Although restricting driv ers’ decisions to these time-windo ws can negativ ely impact p erformance, it significan tly reduces problem complexit y , th us reducing running time. One p ossible approac h it to use p olicies generated from narro w time-windo w scenarios as base p olicies for the rollout algorithm. T ables H.9 to H.12 presen t the exp ected v alues for the p erformance measures used in Figures 7.11 to 7.14 for eac h net w ork, scenario and CO2 p enalt y v alue. 159 Figure 7.11: Probabilit y distribution of CO2 emissions, trip duration and trip cost (with CO2 p enalt y parameter set to 1) of the decision p olicies obtained for net w ork 5. 160 Figure 7.12: Probabilit y distribution p erformance measures related to illegal parking for net w orks 0,2,4 and 5, with CO2 p enalt y set to 1. 161 Figure 7.13: Probabilit y distribution of CO2 emissions, trip duration and trip cost (with CO2 p enalt y parameter set to 1) of the decision p olicies obtained for net w ork 5. 162 Figure 7.14: Probabilit y distribution p erformance measures related to illegal parking for net w orks 0,2,4 and 5, with CO2 p enalt y set to 1. 163 7.5 Conclusion In this study , w e considered the case of sto c hastic parking a v ailabilit y , as opp osed to the deter- ministic time-windo ws, as w ell as an application in emissions reduction. When studying the trade-offs b et w een prioritizing emissions reduction or trip duration, w e found that although fo cusing on emissions reduction can increase trip duration significan tly , this impact is greatly reduced when considering scenarios with limited parking a v ailabilit y . Although not the fo cus of this study , w e also presen t a cost lo w er b ound that com bines information on optimal sp eeds for particular cost functions with HOS requiremen ts, and can b e used to significan tly sp eed-up problem solution in deterministic scenarios, b oth static and time-dep enden t. The resource-constrained shortest path form ulation w as further extended to mo del driv ers p ossi- ble recourse actions when unable to find parking and the ensuing costs. W e used this form ulation to study ho w the solutions are affected b y the lev el of information pro vided to driv ers. W e found that ignoring uncertain t y in parking a v ailabilit y results in inconsisten t p erformance ev en when restrict- ing parking to p erio ds when probabilit y of finding parking is high. F urthermore, results migh t not reflect the in ten t of the cost function used, e.g., minimizing illegal parking ev en ts and/or the prior- it y assigned to emissions reduction. Giving driv ers full information ab out the probabilit y of finding parking at an y time/lo cation significan tly impro v es p erformance and reduces illegal parking-related risks, but also substan tially increase complexit y and solv e time. Using full information regarding parking a v ailabilit y but restricting the parking times to high a v ailabilit y time-windo ws can reduce complexit y while main taining consisten t, although reduced, p erformance. 164 Chapter 8 Balancing P arking Demand 8.1 In tro duction In previous c hapters, w e fo cused on efficien tly planning a single truc k’s trips while accoun ting for HOS regulations and parking a v ailabilit y information, without considering ho w our decisions w ould affect the en tire system. This solution is appropriate when used only b y a small n um b er of v ehicles that do not significan tly affect the system. Ho w ev er, if adopted b y a large enough n um b er of driv ers, the parking a v ailabilit y information used w ould b ecome in v alid, and parking demand ma y turn un balanced again. This can easily happ en if unco ordinated individual selections of the same parking rest area at the b eginning of the trip lead to reac hing o v ercapacit y at arriv al time, violating the initial assumption of parking a v ailabilit y at the time of arriv al. W e con template a cen tral parking co ordinator (CP AR C) system that will ha v e access to parking a v ailabilit y from all parking areas as w ell as the demand of differen t truc k ers. While suc h infor- mation flo w ma y not b e a v ailable to da y , the industry and infrastructure are exp ected to b ecome more and more connected o v er time. The dev elopmen t of metho dologies that tak e adv an tage of this exp ected connectivit y in order to iden tify and quan tify the b enefits to industry and en viron- men t is v ery crucial in accelerating information tec hnologies in the area. The CP AR C system, with access to the supply and demand for parking, will b e able to mak e accurate predictions of parking a v ailabilit y/demand at all parking lo cations. Suc h a system could b e used to implemen t pricing sc hemes to con trol parking demand and ac hiev e a b etter balance b et w een supply and demand. In this c hapter, w e consider the issue of co ordinating the decisions of a large n um b er of truc ks. More sp ecifically , w e study ho w to mo del the b eha vior of a region’s driv er p opulation and ho w 165 it could b e influenced. The mo dels dev elop ed in previous c hapters can b e used to sim ulate ho w companies w ould revise their trip plans in resp onse to c hanges in differen t factors, suc h as parking prices and parking a v ailabilit y estimates. This information can then b e used to study ho w to con trol parking demand. 8.2 Related W ork The truc k parking shortage is a serious issue in sev eral American states as their curren t parking infrastructure cannot accommo date p eak demand. This could b e solv ed b y increasing capacit y or decreasing p eak demand. W e are in terested in the latter. As the general ob jectiv e is to manage demand, the p ossible approac hes dep end on the lev el of con trol w e ha v e o v er the system. 8.2.1 Demand-Side Managemen t If a mo del of ho w the demand reacts to certain parameters is kno wn, or if demand can b e rejected, it is p ossible to directly con trol demand. In this case, simply con trolling demand is often not the ob jectiv e, so demand con trol is used along with some allo cation, sc heduling or routing system to maximize an ob jectiv e or satisfy particular constrain ts. F or example, mobilit y-on-demand systems with electric v ehicles where the distribution of clien ts’ willingness to pa y is kno wn, allo wing demand to b e directly affected b y the price. In this case, the n um b er of clien ts b eing serviced at eac h lo cation can con trolled directly b y price, and the demand at c harging stations is con trolled directly b y routing decisions. Ho w ev er, the range of feasible demands at c harging stations will b e indirectly affected b y the price and demand c hanges o ccurring throughout the net w ork. There is also w ork on dynamic pricing applied to smart grid in order to maximize profit or reduce p eak demand. In [194], Moradipari and Alizadeh address the problem of managing demand on public EV c harging stations. Their system consists of a cen tral op erator that allo cates resources according to users’ v alue of time, c harging demand and tra v el preferences. The system’s ob jectiv es are to pro vide fair service with short w ait times to customers while managing the effects of EV s on the grid. A set of service options is pro vided to users, eac h one with differen t prices and probabilities of b eing 166 assigned to particular c harging stations and exp ected w aiting time. Incen tiv e compatible pricing- routing p olicies that maximize either a measure of so cial w elfare or the cen tral op erator profits are presen ted. They assumed that the users do not observ e the exact w ait times, the exp ected w ait time is constan t at equilibrium and giv en b y a function of the arriv al rates and routing probabilities generated b y the c hosen p olicy . They suggest the use of queueing mo dels to define the exp ected w aiting time functions, but this is not co v ered in this study . The v alue of time (V oT) used to mo del users utilit y and c hoice is considered a random v ariable with kno wn distribution. In [195], T uran et al. use reinforcemen t learning to generate a dynamic p olicy con trolling ride prices and routing/c harging decisions for an autonomous-mobilit y-on-demand (AMoD) fleet of autonomous EV s. The problem is mo deled as a mark o v decision problem with states determined b y electricit y prices at eac h no de, customers queue lengths for eac h origin-destination pair, and the n um b er of v ehicles at eac h no de and their energy lev els. The decision p olicy is defined b y a deep neural net w ork trained using T rust Region P olicy Optimization. The system ob jectiv e is to maximize op erator profit. Although exp ected w aiting time is unkno wn to customers and do not affect demand, the op erator is p enalized for w aiting time and th us tries to reduce total w aiting time. In [196], T uc k er et al. design a pricing framew ork for online electric v ehicle (EV) parking assignmen t and c harge sc heduling. Eac h user is defined b y the requested time in terv al, acceptable lo cations, required energy and utilit y obtained from eac h lo cation. Energy prices at eac h lo cation v ary with time. The system allo cates ho w the energy receiv ed b y eac h user v aries with time. As long as users demands are met, eac h lo cation can con trol c harging b eha vior in order to optimize its o wn costs. Eac h lo cation can generate a certain amoun t of solar energy at no cost, and can buy a limited amoun t of energy from the grid. After users send their information, the system generates a set of options defined b y cable reserv ation and c harging sc hedule, along with a price for eac h option. If a request is rejected, the user’s utilit y is set to zero and it is assumed that the user park ed at an auxiliary parking lot without c harging capabilities. The offline metho d assumes that all requests for a certain time in terv al are kno wn, and maximizes the so cial w elfare, defined as the difference b et w een the total user utilit y and op erational costs. A mathematical form ulation of the offline problem and its F enc hel dual are describ ed. In the online metho d, heuristic pricing functions are used along with an auction mec hanism. The system decides whether to accept or not a request 167 dep ending on the curren t prices and the user’s p oten tial utilit y gain. Prices are up dated after eac h user request is treated. The prop osed mec hanism is sho wn to b e -comp etitiv e. In [197], Tian et al. address the problem of optimizing a parking op erator’s rev en ue using a parking reserv ation system. P arking requests are mo deled b y a P oisson pro cess with an arriv al rate dep enden t on the parking price. Ho w ev er, it is assumed that demand for a certain time in terv al dep ends only on the price for that same in terv al in that particular parking lo cation. Solutions are prop osed for exp onen tial and linear demand functions. The authors justify the assumptions on price/demand relations as that b eing the p oin t of view of the parking manager. The manager is not a w are of the decision pro cess of the users and cannot observ e it, he/she can only analyze ho w the c hanges and price affect the arriv al rate. 8.2.2 Direct Allo cation If there is some base demand that is uncon trollable, but w e can directly con trol ho w that demand is distributed, w e can lo ok at it as a resource allo cation problem or a routing with load balancing problem. Smart parking systems that ha v e some flexibilit y in ho w to allo cate parking reserv ation requests are examples of the resource allo cation p oin t of view. In the case of load balancing, w e can think of the problem of routing m ultiple v ehicles/pac k ets o v er a transp ortation/comm unications net w ork where the cost of an edge dep ends on the n um b er of routes using it. Some routing studies use dynamic pricing sc hemes to incen tivize user participation. In [90], Cap devila et al. used a m ulti-agen t system for the managemen t of parking reserv ations among requesting truc ks. When a v ehicle en ters the road net w ork it sends its origin, destination and preferred parking to the system manager. If the rest area has a v ailable sp ots a temp orary reserv ation will b e made. If the rest area do es not ha v e a v ailable sp ots the negotiation proto col is initiated. Eac h driv er receiv es a list of p ossible rest areas to b e graded according to his/her preferences. Eac h driv er’s v ote is w eigh ted according to their maxim um allo w ed driving time and curren t driving time. These w eigh ts giv e priorit y to driv ers that are closer to reac hing their legal driving limit. The scores for eac h driv er are summed for eac h feasible solution and the solution with the largest score is selected. Note that this algorithm assumes that there is at least 1 feasible solution for a giv en problem. F ollo wing the selection all driv ers and rest areas are notified of the 168 new allo cation. While the n um b er of truc ks requesting parking reserv ations for a giv en parking lot is smaller than the n um b er of a v ailable spaces all of them are gran ted sp ots, but when there are more reserv ations than a v ailable spaces a negotiation proto col is used to c ho ose the parking allo cation. The negotiation in v olv es a v oting pro cedure that tak es in to accoun t the preferences of eac h truc k and its allo w ed stops. The system’s robustness to c hanges in the a v ailable parking areas and the system’s scalabilit y w ere tested through sim ulation and the results w ere promising, sho wing a substan tial reduction of the necessit y for driv ers to park in illegal areas. Similar resource allo cation problems w ere also treated b efore in the con text of urban parking. In [91] the resource allo cation problem w as defined as a sequence of Mixed In teger Linear Programming problems solv ed o v er time sub jected to a set of fairness constrain ts. [92] uses in terv al sc heduling algorithms to try to optimally allo cate parking spaces. In [198], Xie et al. approac h the problem of assigning health care w ork ers to home visits within certain time-windo ws. Practitioners ha v e differen t skill sets, time constrain ts and clien t preferences. Similarly , visits ha v e time-windo w constrain ts and skill requiremen ts. The pa ymen t sough t b y eac h practitioner dep ends on their skill lev el and the costs incurred to pro vide the service. The system w an ts to minimize service costs while guaran teeing that all visits are co v ered b y qualified practi- tioners. An iterativ e bidding framew ork is prop osed where pro viders calculate feasible sc hedules’ costs and bid for the sc hedule with the largest pa y off. The health agency pro visionally c ho oses sc hedules that satisfy its constrain ts while minimizing costs. A uction follo ws certain bidding rules regarding when bids can b e c hanged and b y ho w m uc h, and terminates when no v alid bids are up dated. In [199], K ordonis et al. prop ose mec hanisms to co ordinate truc k driv ers routing decisions to balance the traffic load and impro v e the o v erall traffic conditions and time dela ys exp erienced b y b oth truc k and passenger v ehicle driv ers. The mec hanisms use monetary incen tiv es and fees to steer individual driv ers’ decisions to w ards a system optim um without p enalizing driv ers compared to the user equilibrium. They prop ose fairness measures and 2 sets of constrain ts that w ould encourage driv ers to participate either as a group (either ev ery one or no one) or individually (eac h driv er see it as b eneficial to participate regardless of other driv ers’ decisions). The effect of routing assignmen ts on cost/tra v el time is assumed kno wn. P assenger v ehicle assignmen ts are assumed kno wn and fixed. Demand for eac h OD is a random v ariable with kno wn distribution. 169 8.2.3 Indirect Allo cation If w e can con trol only parameters that affect demand distribution indirectly , then the problem resem bles w ork using dynamic pricing to indirectly influence agen ts’ decisions in congestion pricing, and w ork on an ticipatory route guidance, whic h uses traffic predictions information to influence driv ers’ routing c hoices. This t yp e of system can also b e view ed as a non-co op erativ e game where w e a lo oking for a pricing p olicy/mec hanism that leads to an equilibrium state with particular prop erties. In [200], Hollander and Prashk er presen t a surv ey of applications of non-co op erativ e game theory in transp ortation problems. The problems are categorized based on the pla y ers in v olv ed: tra v elers x demon, tra v elers x tra v elers, tra v elers x authorities, authorit y x authorit y . Problems suc h as congestion pricing w ould fall in to the categories of games b et w een tra v elers or b et w een tra v elers and authorities, dep ending on ho w the pricing sc heme is included in the problem. F or example, the pricing sc heme can b e giv en as an input, and a game b et w een tra v elers is then used to study the equilibrium results from differen t inputs. The game itself w ould not output a p olicy but could b e used to study a p olicy’s impacts. Another p ossibilit y is that the authorit y can b e explicitly mo deled as a pla y er with its o wn ob jectiv e and constrain ts. In this case, studies often use bi-lev el form ulations where the upp er lev el optimizes the authorit y’s ob jectiv e, whereas the lo w er lev el is the user-equilibrium problem that defines ho w tra v elers react to the authorit y’s decisions. This game b et w een an authorit y and a collectiv e of tra v elers w ould output a p olicy recommendation. In [201], Kaufman et al. study an ticipatory route guidance that accoun ts for the b eha vior of an ticipatory v ehicles’ impact on the system. If a prediction is giv en to driv ers, they will c hange their b eha vior and in v alidate the prediction. So, this pap er calculates what traffic prediction should b e giv en to driv ers so that they b eha v e as predicted, i.e., a self-fulfilling prediction. This is calculated iterativ ely b y switc hing b et w een solving dynamic assignmen t problems and time-dep enden t shortest path problems un til the routing p olicy con v erges. In [202], Chen et al. use a bi-lev el optimization approac h to c ho ose the lo cation and capacit y of EV c harging stations suc h that construction costs and driv ers’ tra v el time and w aiting time are minimized. The lo w er lev el calculates v ehicle routing and c harging b eha vior at equilibrium giv en a set of lo cations and capacities, whereas the upp er lev el optimizes the decisions regarding 170 lo cation and capacit y sub ject to service lev el constrain ts at eac h c harging station. The effect of routing c hoices on tra v el time is a non-linear function tak en from the literature. W aiting time at c harging stations is mo deled as a queueing system, but due it b eing computationally exp ensiv e to use, an appro ximate function is prop osed based on Mon te-Carlo sim ulation. The problem is then reform ulated as a single-lev el mathematical programming with complemen tary constrain ts, whic h solv ed b y using standard NLP solv ers to solv e a sequence of relaxed problems. 8.2.4 Significan t differences to truc k parking Dynamic pricing and incen tiv e sc hemes are often studied for issues in the energy [203, 204, 205, 206], transp ortation [207, 195, 208], and comm unications [209] sectors. Ho w ev er, some of the assumptions made are not reasonable for the truc k parking managemen t problem. In [209], F alo w o et al. applied dynamic pricing to solv e a load balancing problem in wireless net w orks. In this case, one of their ob jectiv es w as to ac hiev e uniform load distribution, whic h ma y not b e necessary or reasonable in our case. Our main ob jectiv e is to a v oid o v erloading an y parking facilit y at an y time, so the load should b e balanced enough to a v oid p eaks that exceed capacit y , but not necessarily uniform. In [195], T uran et al. used reinforcemen t learning to manage (ride prices, routing and c harging decisions) an electric autonomous mobilit y-on-demand system. Ride prices w ere used to con trol customers’ arriv al rate at eac h no de. Ho w ev er, the authors considered that the base arriv al rates are not time-dep enden t, and that the customers’ willingness-to-pa y distribution (ho w m uc h eac h clien t is willing to pa y for a ride) is the same at all lo cations and times. In the case of truc k parking managemen t, due to the sev eral factors that influence the planners’ costs, the willingness of eac h planner to pa y to use a certain parking slot and the arriv al rate ma y v ary greatly with time and lo cation. In addition, some dynamic pricing sc hemes treat demands at differen t lo cations and times as if they are indep enden t and can simply b e eliminated. When demands are reduced due to higher prices, it is assumed that customers ga v e up, it do es not affect the demand at other times/lo cations. In our case, demand is usually not simply reduced, it is shifted. Although rerouting ma y cause the total parking time of a truc k to b e reduced (p ossibly turned in to extra driving time), most of it is only mo v ed to a differen t lo cation or time slot. This issue w as also raised in [206] with regards to energy consumption, as some customers react to dynamic prices b y c hanging their consumption 171 time instead of only reducing consumption. Nev ertheless, estimating clien ts’ b eha vior in order to predict when and b y ho w m uc h demand will b e shifted or reduced is v ery hard. In the congestion pricing /co ordinated routing problem, all driv ers using a certain route/link are affected b y the high usage rate of that route/link. Ho w ev er, in truc k parking, only driv ers that arriv e after a parking lot is full w ould exp erience cost increases. The cost of driv ers arriving early is not affected b y the high o ccupancy , so they migh t b e harder to influence. In the co ordinated routing problem all routes are assumed to b e con tained within a single time in terv al, so ev en though routes are comp osed of m ultiple links, the time dimension is ignored and all costs and effects affect all links at the same time. As truc ks ha v e large limitations on a v ailable routes, it is also somewhat reasonable to assume that the n um b er of p ossible routes is small. Esp ecially if w e assume that these form ulations fo cus on short-haul as they consider that trips are completed within a single time in terv al. In truc k parking, the time dimension is imp ortan t as the time a driv er o ccupies a certain rest area has a large impact on the times he/she is lik ely to stop next. And due to the fo cus on long-haul and the p ossibilit y of a large n um b er of rest areas existing along a route, the n um b er of p ossible com binations of rest areas used, stopping times and rest durations that form a sc hedule is v ery large and not straigh tforw ard to reduce. 8.3 Ov erview Figure 8.1 sho ws a simplified diagram of ho w truc k parking demand is generated. Eac h truc king compan y plans its trips according to some public information, suc h as regulations, traffic data and parking data when a v ailable, and its o wn priv ate information regarding its o wn op erational constrain ts and parameters, suc h as clien ts’ requireme n ts and driv ers’ rem uneration. F or simplicit y , w e refer to eac h truc k driv er/compan y that needs to plan a trip as a planner. Eac h planner acts in its o wn b est in terest, planning a route and sc hedule that minimizes its op erational cost. The driv ers will then follo w their itinerary and try to park at the planned lo cation and time. Ho w ev er, the planners do not p ossess information on ho w the decisions of other planners will affect the future state of the system. So, assuming planners ha v e access to the same public information, if a certain parking lot is lo w cost compared to others and usually a v ailable at a certain time, all planners will assume that they can use it at that time. Unable to see the whole picture, all truc ks ma y b e routed 172 Figure 8.1: Simplified represen tation of ho w parking demand is generated. Green b o xes represen t information that is unique to the system, regardless of whether it can b e measured. Red b o xes represen t information that is sp ecific to eac h compan y . The red horizon tal lines on the parking demand plots represen t eac h facilit y’s maxim um capacit y . to the same parking lots, causing an un balanced parking capacit y usage. Certain facilities ma y b e w orking o v ercapacit y , while others ma y ha v e plen t y of parking spaces left. Not b eing able to find parking at the exp ected lo cation, some driv ers ma y b e forced to park illegally , whic h ma y p ose significan t safet y and financial risks. F or the driv ers that could not find parking, the cost increase of c ho osing a sub-optimal route/sc hedule from the b eginning ma y b e lo w er than the cost increase caused b y not finding parking at the planned route. Ho w ev er, in order to consciously decide against using the optimal route, the planner needs to kno w in adv ance whether others’ decisions will turn its o wn decision infeasible. Planners’ decisions ma y b e influenced b y factors suc h as traffic, fuel costs, parking a v ailabil- it y , parking rates, and illegal parking p enalties. Man y cost-defining factors ma y b e particular to a certain planner, suc h as compan y p olicy ab out illegal parking, driving sp eed limits, fuel con- sumption, and driv er rem uneration. The same is true for the set of feasible sc hedules, as it can dep end on restrictions particular to eac h planner, suc h as the details of eac h trip (clien t lo cation, time-windo ws, etc.) and driv ers’ remaining driving time. This heterogeneit y means that differen t planners measure the adv an tage of using a certain parking lo cation and time differen tly . Our ob jectiv e is to influence this cost calculation so that planners ha ving flexible sc hedules will opt to use lo w-demand parking slots instead of high-demand ones. Consider the effects of the follo wing factors: hourly parking price; and illegal parking p enalties. If a certain parking lot is 173 exp ected to w ork o v ercapacit y at a particular time, the parking rates for that time and place can b e increased, motiv ating driv ers to park at differen t times or lo cations. The price v alue w ould con v ey ho w go o d or bad (for the system) it is for the planner to use that resource at that time. As eac h planner is solving a differen t optimization problem (the jobs, v ehicles, clien ts, and constrain ts ma y all b e differen t), eac h truc k has a differen t cost for c hanging itineraries. Therefore, as the prices increase, companies that ha v e reasonable alternativ e route options w ould c hange their itineraries to minimize costs. Whereas companies that are in urgen t need of that resource due to less flexible conditions, w ould k eep their itinerary and accept the cost increase. Differen t from urban parking dynamic pricing sc hemes that often aim to maximize parking lot profits, in this case, w e aim for a b etter utilization of the a v ailable parking capacit y and a reduction of the cases of illegal parking. The illegal parking p enalt y will help mo del and con trol planners’ un willingness to switc h routes, as w ell as measure the qualit y of alternativ e routes. If a planner considers more cost-effectiv e to risk parking illegally than to switc h routes, then it means that all alternativ es are to o exp ensiv e. In the future, the p enalt y v alues could also b e used to measure the need and b enefits of infrastructure in v estmen ts in certain areas. Ho w ev er, as the p enalt y’s effect dep ends on the probabilit y of finding parking, whic h v aries according to other planners’ decisions, it is not our main con trol input. F urthermore, it w ould b e hard to con trol the “real” p enalt y v alues as they dep end on the fines imp osed b y the go v ernmen t, on ho w strictly high w a y officers are enforcing parking restrictions, and other factors completely out of our con trol, suc h as acciden t rates, and a v erage litigation costs. In Figure 8.2, w e sho w the basic diagram of the prop osed pricing sc heme. The CP AR C system w ould first pro vide the planners with initial parking rates, calculated according to historical data. Planners w ould then calculate their routes and comm unicate their desired parking slots to CP AR C. Assuming full participation, CP AR C’s Demand Estimator w ould b e able to p erfectly calculate the hourly demand for eac h parking slot, and v erify whic h ones are o v ercapacit y and need to ha v e their prices increased. The new prices w ould then b e sen t to the planners. In the case of partial participation, CP AR C could generate demand estimates b y using the participan ts’ demands along with historical parking a v ailabilit y data. Similarly , the prices w ould b e adjusted according to the demand estimates and sen t bac k to the planners. This cycle of planning, demand estimation, and price up date w ould con tin ue un til an acceptable solution is reac hed. 174 Figure 8.2: System Diagram. The blue b o xes represen t our system’s comp onen ts and output. The red b o x represen ts the distributed system encompassing all truc k companies and their decisions. P erformance measures to determine what is an acceptable solution and ho w distan t w e are from ac hieving it will b e dev elop ed during the pro ject, and the pricing strategy will dep end on the measures used. F or example, w e could measure the parking shortage’s sev erit y b y measuring the excess demand at eac h parking lot. LetD i (t) represen t facilit yi ’s parking demand at timet ,C i is facilit y i ’s parking capacit y . Then w e can define the follo wing p erformance measures: E i (t) =max(0;D i (t)C i ) (8.1) i = Z 1 0 E i (t)dt (8.2) = X i i (8.3) whereE i (t) measures the excess demand of lo cationi at timet in parking spaces, i is a measure of the parking shortage at lo cationi inparking spaceshours , and measures the parking shortage of the whole system inparking spaceshours . Ideally , an acceptable solution w ould ha v e equal to zero, so no parking facilit y has a demand larger greater than it can supp ort. Ho w ev er, in order to prop erly adjust the prices, w e also need to kno w whic h facilities need to ha v e their demand decreased and at what time. This information is pro vided b yE i (t) and i . It ma y also b e necessary to trac k ho w the price c hanges are affecting the costs of eac h planner and of the whole system. If the system cost decreases, it ma y b e p ossible to use the gains of some planners to comp ensate for the losses of 175 others. This ideal situation w ould allo w for the system to b e main tained without extra in v estmen ts b esides infrastructure and managemen t costs. One of the c hallenges is finding a metho d to sim ulate planners’ reactions to price c hanges. Therefore, w e study ho w to use the mo dels dev elop ed in previous c hapters to sim ulate planners’ b eha vior. In our analysis, w e assume that the ob jectiv e of eac h planner is to minimize its o v erall cost, part of whic h is asso ciated with the parking cost. By c hanging the parking cost at a particular lo cation, the o v erall cost ma y no longer b e optim um when compared with a lo w er cost parking, whic h ma y require the planner to mo dify the initial route. P arameters unique to eac h planner, suc h as clien t lo cations, deliv ery constrain ts and driv er hourly w age, can b e sampled from a giv en distri- bution, whereas parameters suc h as traffic, parking lo cations, parking price and HOS regulations will b e the same for all planners. 8.4 Preliminaries LetV b e the set of differen t v ehicles or trips usually found in the region of in terest. Eac h v2V con tains all the parameters necessary to describ e the trip (e.g., origin, destination, time-windo ws, HOS constrain ts initial condition, etc). LetTDSP(v) b e the set of feasible sc hedules for v ehicle/trip v , without considering parking a v ailabilit y constrain ts, w e will also refer toTDSP(v) as the set of pure strategies for v ehicle v . Although the TDSP mo dels describ ed in c hapters 4 and 5 consider time-related v ariables as con tin uous, here w e assume that the decision space is discretized. Let A v 2TDSP(v) represen t a pure strategy for v ehiclev , and a pure strategy profile A = (A v ) v2V b e the v ector of pure strategies assigned to eac h v ehicle. The set of all p ossible pure strategy profiles is defined as S = Q v2V TDSP(v) . The function v (s) : TDSP(v)7! R is a mixed strategy for v ehicle v , defined as the probabilit y of v ehicle v c ho osing to use sc hedule s2 TDSP(v) . The mixed strategy profile =f v ()g v2V is the v ector of functions v () indicating the mixed strategy adopted b y eac h v ehicle. Giv en a system-wide ob jectiv e function () , the system optim um can b e defined as the solution to the follo wing optimization problem: min () (8.4) 176 s:t: : X s2TDSP(v) v (s) = 1; 8v2V (8.5) v (s) 0; 8v2V;s2TDSP(v) (8.6) P arking capacit y constrain ts are not included as w e consider that () already accoun ts for all relev an t costs, suc h as costs due to o v ercapacit y parking facilities, trip dela ys, etc. If feasible, o v ercro wding w ould b e a v oided b y setting high asso ciated costs sufficien tly high. Nev ertheless, this form ulation assumes a co op erativ e relationship where all v ehicles are con trolled b y a single agen t, and it is acceptable to increase the costs of certain v ehicles without limits as long as the o v erall system cost is reduced. 8.4.1 Non-co op erativ e Game The truc king industry is a comp etitiv e mark et comp osed b y a large n um b er of agen ts, with large and small companies commanding v arying fractions of the v ehicles in op eration and a large n um b er of truc k-o wners op erating indep enden tly . Although it is reasonable to assume co op eration within large companies, that is not the case for the o v erall system. The w a y agen ts b eha v e in a comp etitiv e mark et is b etter mo deled as a non-co op erativ e game, where eac h agen t is trying to optimize its o wn ob jectiv e dep ending on ho w it exp ects comp etitors will b eha v e. This scenario is usually analyzed in the literature b y assuming that agen ts ha v e some information on eac h others’ in ten tions and can plan accordingly . If the agen ts are able to reac h a stable solution, i.e., no one could impro v e their ob jectiv e b y c hanging b eha vior giv en that all other agen ts’ b eha viors remain unc hanged, that solution is called a Nash Equilibrium (NE). A game ma y ha v e zero, one or m ultiple equilibria, whic h are usually less efficien t than the system optim um. Therefore, the system manager is in terested in either pushing the system to w ards the NE that b est suits the system’s ob jectiv e, or implemen ting p olicies that generate NEs with b etter system-wide p erformance. The ratio b et w een the w orst-case NE’s cost and the system optim um is usually referred to as the Price of Anarc h y [210], it is often used in the literature to study the qualit y of a game’s equilibria. LetF v (A) b e the cost of v ehiclev under a pure strategy profile A2S and let(A v ;s v ) represen t the strategy profile obtained when, starting from a pure strategy profile A , v ehicle v switc hes to 177 strategy s v . F or con v enience, F v () is also used to represen t the exp ected cost of v ehicle v under a mixed strategy profile , and, in this case, it is defined as: F v () = X A2S p(A)F v (A) (8.7) p(A) = Y v2V p v (A v ) (8.8) , where p(A) is the probabilit y that pure strategy profile A is used, and p v (A v ) is the probabilit y of v ehicle v using pure strategy A v . A mixed strategy profile constitutes a NE if and only if it satisfies the follo wing constrain ts: F v ( v ;s v )F v (); 8v2V;s v 2TDSP(v) (8.9) X s2TDSP(v) v (s) = 1; 8v2V (8.10) v (s) 0; 8v2V;s2TDSP(v) (8.11) If the system manager w ere able to recommend sc hedules to all users, but those users w ould only follo w the recommendations if it optimizes their ob jectiv e giv en their exp ected b eha vior of the other agen ts, the b est recommendation could b e calculated b y finding min () suc h that (8.13)-(8.15) are satisfied. F or example, a planning soft w are used b y a large n um b er of v ehicles w ould b e able to b oth recommend sc hedules and sho w planners the exp ected cost of other options. The cost estimate is generated assuming that the other users will follo w the recommended sc hedule. Although unable to directly con trol the v ehicles and force the adoption of a system optim um strategy , the system manager w ould b e able to at least steer the system to w ards the most b eneficial NE. An ticipatory routing systems [211, 201] in terfere with the information pro vided to driv ers, instead of directly c harging for certain routes or imp osing decisions. F v () dep ends on predicting the strategy v used b y other agen ts and ho w the differen t strategies from a strategy profile in teract. Therefore, it can b e affected b y the information pro vided to driv ers regarding the exp ected effect of others on the system. Ho w ev er, if the pro vided information do es not matc h what driv ers actually exp erience, they w ould lose trust in the system and stop using it. So, the manager is restricted to using p olicies that mak e driv ers b eha v e as predicted. If w e consider that the imp ortance driv ers giv e to parking 178 conditions is already adequate and they simply lac k the information to act on it, w e could consider a system similar to an ticipatory routing. In this case, the system w ould giv e driv ers a parking a v ailabilit y prediction suc h that they realize that prediction. No w, consider the case when the system manager is unable to recommend sc hedules, or all NE ha v e undesirable costs. In this case, w e w an t to c hange the system suc h that the new system’s NEs ha v e acceptable costs. As describ ed in (8.13), the NEs dep end on the agen ts’ p erception of cost. So in order to shift the NEs to w ards more desirable solutions, w e need to someho w influence the agen ts’ costs. Go o d examples are the pricing mec hanisms in [199, 194]. In pricing mec hanisms the manager can directly c harge or giv e monetary incen tiv e for the usage of eac h resource, so it w ould b e equiv alen t to adding an extra cost v () to the cost function F v () used b y agen ts to optimize their decisions. Studies using this t yp e of approac h often explore concerns regarding user participation, fairness of the prices/incen tiv es used, whether the manager or users are making or losing money , and whether users are truthful when pro viding information to the system. This approac h can b e describ ed as solving the follo wing problem: min (; ) (8.12) s:t: :F v ( v ;s v )+ v ( v ;s v )F v ()+ v (); 8v2V;s v 2TDSP(v) (8.13) X s2TDSP(v) v (s) = 1; 8v2V (8.14) v (s) 0; 8v2V;s2TDSP(v) (8.15) , where = ( v ) v2V , and (; ) is a mo dified ob jectiv e function to accoun t for an y impact the pricing p olicy migh t ha v e on the system cost, e.g., the ob jectiv e function migh t include the balance of incen tiv es giv en and fees collected b y the system. 179 8.5 F orm ulation 8.5.1 Agen t in teraction In the case of truc k parking, the in teraction of agen ts will b e based on the demand of truc k parking lo cations. This effect could b e mo deled b y the recourse function used in [212] to estimate the cost of not finding parking or ma yb e a p enalt y for sc heduling a stop at a full rest area. One issue is whether w e assume that driv ers accoun t for this cost when planning their trips. In theory , driv ers do not kno w other driv ers’ decisions and ho w they will affect parking a v ailabilit y . W e could argue that freigh t mo v emen t within a region follo ws certain patterns and that driv ers w ould adapt to these patterns o v er time and reac h a NE as if ev ery driv er knew other driv ers’ decisions. With this assumption w e can calculate the impact of parking shortage p enalties b oth on individual driv ers and on the whole system. A priori w e assume that the parking shortage has no impact on the driv ers that arriv e b efore rest areas reac hing capacit y . It migh t b e in teresting and reasonable to consider that park ed driv ers can b e affected to a smaller exten t to o as o v ercro wding migh t cause problems within rest areas and driv ers can b e affected b y acciden ts caused in the rest area’s surroundings. Let i (t arrival ;t dep ;o i ) represen t p enalties due to parking conditions p erceiv ed b y driv ers parking from t 0 to t 1 at a lo cation i with parking demand profile o i . This function defines ho w m uc h driv ers care ab out parking conditions, and whether ev ery driv er using an o v ercro wded parking facilit y incurs p enalties or only those that arriv ed after it reac hed capacit y . 8.5.2 Individual Beha vior W e assume that eac h driv er seeks to minimize his/her o wn costs, and that they are a w are of the p enalt y costs that will b e incurred after equilibrium is reac hed. F or simplicit y , w e consider that all driv ers at a parking facilit y are p enalized equally when the facilit y is o v ercapacit y and this p enalt y will b e considered as part of the parking fees imp osed b y the system. As in Chapter 4, eac h v ehicle is sub ject to HOS constrain ts and clien t deliv ery time-windo ws. Ho w ev er, w e are no w in terested in using this mo del to estimate parking demand, so w e do not include parking a v ailabilit y time-windo w constrain ts. F urthermore, w e need to accoun t for parking costs and initial conditions 180 and to trac k the parking demand generated b y eac h v ehicle. Therefore, w e consider the follo wing mo dified TDSP form ulation: min v x v e v ;a x v s v ;d + e v X i=s v p v i + e v X i=s v v i (fy i g;o i ) (8.16) s:t: : x v i;d +d i =x v i+1;a ; 8s v ie v 1 (8.17) x v i;a + X r2R t r z v i;r x v i;d ; 8s v ie v 1 (8.18) x v i;d x v i;a + 1y v i;0 t hor ; 8s v ie v 1 (8.19) y v i;0 + T v i X =1 y v i; = 1; 8s v ie v (8.20) y v d;i;0 + T v i X =1 y v d;i; = 1; 8s v ie v 1 (8.21) y v d;i;0 =y v i;0 ; 8s v ie v 1 (8.22) 1y v i;0 = X r2R z v i;r ; 8s v ie v 1 (8.23) T v i X =1 y v i; t min i; x v i;a ; 8s v ie v (8.24) x v i;a t hor T v i X =1 y v i; t hor t max i; ; 8s v ie v (8.25) T v i X =1 y v d;i; t min i; x v i;d ; 8s v ie v 1 (8.26) x v i;d t hor T v i X =1 y v d;i; t hor t max i; ; 8s v ie v 1 (8.27) x v k;a x v i;d t c + k1 X j=i+1 X r2Rc z v j;r (t c +t r ); 8s v i<ke v ; c2C 1 (8.28) k1 X j=i d j t c +t c k1 X j=i+1 X r2Rc z v j;r ; 8s v i<ke v ; c2C 2 (8.29) x v k;a x v s v ;a +h v c t c + k1 X j=s v X r2Rc z v j;r (t c +t r ); 8s v <ke v ; c2C 1 (8.30) h v c + k1 X j=s v d j t c +t c k1 X j=s v X r2Rc z v j;r ; 8s v <ke v ; c2C 2 (8.31) 181 p v i T v i X k=1 CP i;k y v d;i;k y v i;k+1 ; 8s v ie v 1 (8.32) x v i 2 [0;t hor ] 2 ;y v i 2f0;1g T v i +1 ; 8s v ie v (8.33) y v d; i 2f0;1g T i +1 ;z v i 2f0;1g jRj ; 8s v ie v 1 (8.34) x v s v ;d 2 t v dep ;t v dep (8.35) o i; = X v2V X k=1 y v i;k 1 X k=1 y v d;i;k ! 8i; (8.36) T able 8.1 presen ts a description of the v ariables and parameters used. Most v ariables are defined as in Chapter 4, with the sup erscriptv represen ting to whic h v ehicle that v ariable b elongs to. Time is divided in time slots forming a partition of the in terv al [0; t hor ] , and w e included an extra set of v ariablesfy d g that trac k v ehicles’ departure time slots at eac h lo cation. The time slots on rest areas are not used to restrict arriv al time, but to trac k parking demand and calculate parking costs. Eac h time slot at eac h parking lo cation is assigned a certain cost, whic h is reflected in the v ariablesfCP i;k g . A consequence of calculating parking costs lik e this is that w e automatically discretize the set of pure strategies a v ailable for eac h v ehicle. The v ariablesfx v i g represen ting the arriv al and departure times at eac h lo cation are con tin uous, so, originally , the TDSP can ha v e an uncoun table n um b er of feasible solutions. Ho w ev er, as the differen t v ehicles only in teract through parking demand and its effect on parking costs, v arying arriv al or departure times within a giv en time slot will not c hange ho w the sc hedule affects parking demand. Therefore, the pure strategies are defined only b e the time slots used, not b y the detailed sc hedule describ ed b y the arriv al and departure times. Ideally , (8.16) should b e affected b y o i through the term P e v i=s v v i (fyg;fog) as describ ed in section 8.5.1, and the problem should b e solv ed for all v ehicles sim ultaneously , including the equi- librium conditions describ ed in section 8.4.1, in order to find the generated NE. Ho w ev er, this w ould b e in tractable. Therefore, w e c hose to solv e the problem iterativ ely , considering that the driv ers and system manager w ould consider data from the previous iteration when estimating p enalties or deciding parking rates. In addition, w e assume that an y o v ercro wding p enalt y can b e included in the parking rates and th us stop using the term P e v i=s v v i (fyg;fog) . This w a y , w e can solv e (8.16)-(8.35) for eac h v ehicle in parallel, then use (8.36) to calculate the parking demands needed 182 to adjust parking rates. When necessary , w e will refer to the simplified mo del that remo v es the term and solv es the optimizes eac h v ehicle indep enden tly as the de c ouple d mo del . F or the purp ose of estimating parking demand, w e can see eac h car as a system that receiv es the v ehicle’s parameters (initial conditions, start/end lo cations, departure/deliv ery constrain ts and hourly op erational cost) and all rest areas’ parking rates, and returns the trip duration, parking cost and parking demand for eac h lo cation and time slot. If the parameters’ distribution is kno wn for v ehicles of a certain region, it can b e used to estimate the costs and parking demand those v ehicles generate within the region. 8.5.3 Equilibrium A game where eac h pla y er’s set of strategies is describ ed as the set of feasible time slot usage configurations sub ject to (8.17)-(8.35) constitutes a finite game, and th us it has a Nash equilibrium [213]. F urthermore, if w e consider the set of p enalt y functions for whic h no p enalt y is applied at parking lo cations op erating at or under capacit y , w e can sa y that an y solution to the de c ouple d mo del that k eeps all parking facilities w orking at or under capacit y is a NE. Therefore, if w e are able to find parking prices for whic h the resulting de c ouple d mo del satisfies parking capacit y constrain ts, this solution will b e a NE. This happ ens b ecause all driv ers will already b e follo wing their optim um route, and the resulting parking demand will not generate p enalties that migh t mak e driv ers switc h strategies. 183 T able 8.1: V ariables and P arameters V ariables Sym b ol Description x v i;a ;x v i;d Arriv al/Departure times from lo cation i (y v i; ;y v d;i; ); > 0 V ehiclev stops at/departs from lo cationi within time slot . (T rue:1, F alse:0) y v i;0 =y v d;i;0 V ehicle v do es not stop at lo cation i . (T rue:1, F alse:0) z v i;r V ehicle v tak es rest of t yp e r at lo cation i p v i P arking cost incurred b y v ehicle v at lo cation i o i; Lo cation i ’s parking demand at time slot . P arameters Sym b ol Description d i T ra v el time b et w een lo cation i and i+1 s v ;e v Start/end lo cations of v ehicle v . v V ehicle v ’s hourly op erational cost. h t v dep ;t v dep i V ehicle v ’s departure time-windo w. t hor Planning time horizon T v i Num b er of time-windo ws at lo cation i h t v;min i; ;t v;max i; i Time slot of index at lo cation i for v ehicle v . CP i;k Lo cation i ’s cum ulativ e parking cost at time slot k . (Sum of the cost of all time slots from 0 to k) R Set of rest t yp es defined in the regulation C Set of constrain ts defined in the regulation C 1 C Set of constrain ts restricting elapsed time. C 2 C Set of constrain ts restricting driving time. t c Time limit related to constrain t c2C R c R Set of rest t yp es that can reset constrain t c2C t r Minim um duration for rest of t yp e r2R h v c V ehicle v ’s initial condition relativ e to constrain t c. v i F unction describing the p enalt y incurred b y v ehiclev at lo cationi giv en the time slots used (fy i g) and the demand profile o i . 184 8.6 Demand Estimation Let v b e a random v ector with kno wn distribution represen ting the parameters of v ehicles within a region of in terest. W e estimate parking demand b y sampling v and solving a TDSP for eac h sample. The results indicate ho w this v ehicle p opulation tend to b eha v e under the curren t parking rates. The results can b e scaled to matc h the region’s actual traffic v olume. When sim ulating a region’s v ehicles w e need to define eac h v ehicle’s start and destination no des, the time they “en ter” de region, an y departure or deliv ery time constrain ts, and the initial status of driv ers’ HOS coun ters. W e assume that the origin-destination matrices represen ting the in ten t of driv ers starting trips at differen t times/lo cations is kno wn and can b e used to generate reasonable samples of the trips start/end no des and the time driv ers are a v ailable to start the trip. In addition, w e need to generate the initial state of the driv ers’ HOS constrain ts coun ters. As these coun ters are not indep enden t, w e need to consider their coupling when defining the probabilit y distributions for a region’s v ehicles. Consider the HOS-related resources and resource extension functions defined in section 6.3.2: b 2 [0;t eb ] : A ccum ulated driving time since last br e ak r 2 [0;t ar ] : Elapsed time since last daily r est r 2 [0;t er ] : A ccum ulated driving time since last daily r est w 2 [0;t aw ] : A ccum ulated on-dut y time since last we ekly r est After accoun ting for ho w differen t activities affect eac h resource, w e ha v e that the follo wing inequalities m ust hold: b r r ; r w (8.37) The con v ex p olyhedron defined b y these inequalities is the set of p ossible initial conditions for HOS coun ters and can b e used to generate v alid initial conditions. One p ossible approac h is to use rejection sampling to uniformly sample the p olyhedron. W e can also generate samples b y sampling one resource at a time, using the sampled v alue for the smaller v ariables to define the domain of the larger ones, i.e.: b 2 [0;t eb ]; r 2 [ b ;t ar ] (8.38) 185 r 2 [ r ;t er ]; w 2 [ r ;t aw ] (8.39) This metho d do es not sample the p olyhedron uniformly , but w e do not need to c hec k the v alidit y of generated samples. 8.6.1 Sensitivit y to HOS conditions and uniform time slot prices The follo wing figures w ere generated b y indep enden tly sim ulating 100 v ehicles with the same origin, destination, departure/deliv ery time constrain ts. The trip has 23h of driving time and ev enly spaced rest areas with 1h of tra v el time b et w een an y t w o adjacen t rest areas. Eac h set of initial conditions w as generated b y sampling random in tegers from the in terv als describ ed un til v alues satisfying (8.37) w ere generated. Figure 8.3 presen ts the parking demand generated without parking c harges and with HOS initial conditions restricted to {0,1}. Ev en though there is a small n um b er of v alid initial conditions and no influence from parking prices w e already see the impact that ev en small c hanges in the initial conditions ha v e on parking demand. Figure 8.4 presen ts the parking demand generated with hourly op erational costs (in $/h) are in tegers sampled from [60;80] , and eac h HOS initial condition is an in teger b et w een 0 and its regulation limit min us 1h, e.g., b 2 [0;t eb 1] . Figure 8.4a do es not c harge for parking and Figure 8.4b uses parking c harges of $5 p er time slot. Despite c harging the same v alue at all times and lo cations, as the parking costs are calculated p er time slot, setting a non-zero parking rate already serv es to discourage driv ers from taking short stops. Consequen tly , Figure 8.4b has less narro w demand p eaks compared to Figure 8.4a. 186 Figure 8.3: Distribution of the parking demand generated b y 23h trips without parking c harges and with HOS initial condition restricted to 0,1. 187 (a) P arking rate: $0 (b) P arking rate: $5 p er time slot Figure 8.4: Distribution of the parking demand generated b y 23h trips with hourly op erational cost in the in terv al [60;80] and with eac h HOS initial condition v arying b et w een 0 and its regulation limit min us 1h. 188 8.6.2 Resp onse to price c hanges 8.6.2.1 Price Up date W e consider k eeping truc k stops demand b elo w capacit y as the main ob jectiv e. Consider a simple pricing strategy of increasing parking prices at lo cations/times where demand is ab o v e an upp er threshold and decreasing it when demand is b elo w a lo w er threshold. T o a v oid parking rates div erging, prices will b e restricted to a certain range. Let O and T b e matrices represen ting, resp ectiv ely , the estimated demand and target demand for eac h lo cation and time slot and let [`;u] b e an in terv al used to define the range of demand v alues that do not required direct in terv en tion (deadband). W e define the error matrix E as follo ws: E i;j = 8 > > < > > : 0; if `O i;j =T i;j u O i;j T i;j ; o:w: (8.40) The error matrix is used to feed an in tegral con troller. As prices are restricted to a giv en range, a bac k calculation an ti-windup metho d is used to correct the in tegrator state. 8.6.2.2 Exp erimen t Consider a route with 23 lo cations where the tra v el time b et w een an y pair of adjacen t lo cations is 1h. W e define 3 v ehicle p opulations with the parameters listed in T able 8.2. T able 8.2: P opulation P arameters All P1 P2 P3 Marginal Hourly Cost ($/h) [50, 70] Arriv al Time (h) [12, 18] Max W ait (h) 12 Start Lo cation 0 7 5 End Lo cation [5, 15] [14, 20] [17, 23] Resource 1 (h) [3, 7] [0, 7] [0, 7] Resource 2 (h) [3, 13] [0, 13] [0, 13] Resource 3 (h) [3, 10] [0, 10] [0, 10] Resource 4 (h) [3, 30] [0, 30] [0, 30] Size 100 100 30 189 T able 8.3: Exp erimen t Results Step Time Cost P arking Cost T otal Cost Max Demand 0 1571.18 0 1571.18 0.182609 1 1571.18 -0.0340742 1571.15 0.23913 2 1571.18 -0.329091 1570.86 0.321739 3 1571.18 -0.744367 1570.44 0.334783 4 1571.18 -0.751305 1570.43 0.326087 5 1571.18 -0.391361 1570.79 0.265217 6 1571.18 -0.103988 1571.08 0.217391 7 1571.18 0.00304427 1571.19 0.191304 8 1571.18 0.130596 1571.32 0.191304 9 1571.18 0.198699 1571.38 0.195652 10 1571.18 0.240001 1571.42 0.16087 11 1571.18 0.281829 1571.47 0.134783 12 1571.18 0.291691 1571.48 0.134783 13 1571.18 0.302905 1571.49 0.126087 14 1571.18 0.302278 1571.49 0.117391 Resources 1, 2, 3 and 4 refer to, resp ectiv ely , the coun ters for the HOS rules’ 8h (30min break) driving limit, 14h limit, 11h driving limit and 60h on-dut y time limit. P arameters defined b y an in terv al are sampled uniformly from that in terv al. All v ehicles consider daily deliv ery time-windo ws of [8;18] at the end no de. Consider that eac h lo cation represen ts a region with a finite truc k parking capacit y equal to a certain p ercen tage of the total truc k p opulation b eing considered. In this example w e set the capacit y to 12% of the truc k p opulation, i.e., no lo cation can accommo date more than 27 truc ks at an y giv en time. P arking demand is not considered at the end no de, but it is coun ted at the start no de. The deadband w as set to 10-100% of the parking capacit y at eac h lo cation. P arking rates are restricted to the in terv al [20;20] . P arking demand con v erged to v alues b elo w capacit y at ev ery lo cation after 14 iterations. Plots of the demand and parking price at eac h lo cation and time slot for some time steps are presen ted in 8.5. The follo wing table presen ts the a v erage time cost, parking cost, total cost and the maxim um parking demand at eac h step of the exp erimen t. Driv er Sensitivit y In this example, the sc heduling mo del sho w ed v ery high sensitivit y to parking prices. The scenario considers marginal hourly costs in the range of $50-$70, but the highest parking rate used to balance demand w as less than $1/h. T able 8.3 sho ws that the a v erage parking cost is less than 0.02% of the tr ip’s a v erage total cost. Nev ertheless, it is imp ortan t to note that required 190 parking rates dep end on the parking capacit y of the differen t lo cations, the n um b er of v ehicles and their parameters. As this example considers only 3 differen t v ehicle p opulations, a large part of the parking spaces are not utilized, making it easier for demand to b e redistributed. T ransien t Beha vior The con troller succeeds in adjusting the demand to within the target lev el. Ho w ev er, parking demand oscillates significan tly in the b eginning, reac hing lev els higher than in the original (uncon trolled) setting. T able 8.3 sho ws that the maxim um demand increases from 18% to 33% of the truc k p opulation in steps 0 to 3, b efore starting to decrease. It is only at step 10 that w e see an impro v emen t relativ e to the original condition. If this strategy w ere to b e applied directly to a region, it could disrupt the system for a while b efore it starts sho wing b enefits. Therefore, if w e can use this kind of sim ulation to estimate the b eha vior of a region’s truc k p opulation, w e can mitigate these transien t effects. 191 (a) t=0 (b) t=1 Figure 8.5: Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. 192 (c) t=2 (d) t=7 Figure 8.5: Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. 193 (e) t=10 (f ) t=14 Figure 8.5: Example results: Demand and prices for time steps 0, 1, 2, 7, 10 and 14. 194 8.7 Conclusion In this c hapter, w e presen ted preliminary w ork on using TDSP mo dels to dev elop a cen tral parking co ordinator that stim ulates a more balanced distribution of truc k parking demand. W e presen t a form ulation that uses a mo dified TDSP MIP mo del whic h trac ks parking usage b y dividing time in to time-slots and c harging driv ers p er time slot used. The parking rates dep end on b oth the time and lo cation b eing considered. Due to the complexit y of solving the problem for a large n um b er of driv ers and mo deling ho w to manage parking a v ailabilit y and driv ers in teractions within a single optimization problem, w e optimize eac h driv ers’ sc hedule separately using a common price matrix. Assuming w e ha v e information on the usual conditions of truc k driv ers op erating in a region (or a large n um b er of driv ers/companies willing to k eep their planning system con tin uously connected to the pricing co ordinator), w e use a sample p opulation to sim ulate the effect of price c hanges b efore actually implemen ting the price c hange. Sim ulations w ere used to study ho w driv ers react to price c hanges. Results sho w that the sc heduling mo del is v ery sensitiv e to ev en small c hanges in parking prices, whic h is conduciv e to using parking prices as a means to influence demand. Ho w ev er, under the price up date rule tested, the system oscillates significan tly b efore reac hing a v alid solution and the initial iterations migh t see an increased maxim um demand instead of the in tended demand redistribution. More study is required in or der to b etter understand the system’s prop erties, and ho w to a v oid or damp en these oscillations. F urthermore, due to HOS rules and clien t constrain ts, it migh t b e imp ossible to div ert demand from certain time slots and lo cations. Nev ertheless, this mo del could still b e used to iden tify these sp ots and ev aluate the need for infrastructure in v estmen ts. It is imp ortan t to note that w e assume that driv ers w ould b e using a planning to ol to optimize their itineraries and that they w ould accept the routes generated. Driv ers migh t ha v e their o wn preferences regarding itineraries, and do not necessarily follo w optimal sc hedules. 195 Chapter 9 Concluding Remarks and Prop osed Researc h Directions In this w ork, w e dev elop ed metho ds to in tegrate parking a v ailabilit y information in to the plan- ning pro cess for long-haul truc king and studied truc k parking shortages’ p oten tial impacts on the industry . First, as the usage of parking a v ailabilit y information is the fo cus of our prop osal, w e p erformed a surv ey on the curren t status of in telligen t truc k parking systems. Although truc k parking infor- mation is not ubiquitous, progress is b eing made in dev eloping the tec hnology necessary for the implemen tation of in telligen t truc k parking systems, and some pilot sites are already in place. Second, w e studied the truc k driv er sc heduling problem (TDSP), whic h considers a fixed route and aims to determine a minim um duration regulation-complian t sc hedule. W e prop osed a mixed in teger programming form ulation that uses conditioned time-windo w constrain ts to mo del the park- ing a v ailabilit y at parking facilities and mo ving windo w constrain ts to mo del the hours-of-service (HOS) constrain ts for long trips. Sim ulation results illustrate that sc hedules calculated without accoun ting for parking a v ailabilit y are often infeasible. Although parking constrain ts increased trip duration in some scenarios, these scenarios also sho w ed lo w er feasibilit y rates when ignoring parking information. W e follo w ed b y extending the TDSP under parking a v ailabilit y constrain ts to include path planning. W e prop osed a resource-constrained shortest path form ulation that uses a resource v ector to k eep trac k of the HOS and time constrain ts. The problem is solv er o v er an auxiliary net w ork that explicitly mo dels the differen t activities a v ailable to driv ers and ho w they affect eac h regulation constrain t. W e prop osed a tailored lab el-correcting algorithm that solv es the problem to optimalit y . Computational exp erimen ts sho w ed that parking conditions could significan tly affect 196 the route c hoice, illustrating the imp ortance of accoun ting for parking a v ailabilit y information early in the planning pro cess. W e also sim ulated the p oten tial costs of disregarding parking information under differen t parking shortage sev erit y lev els and ho w they compare to the cost increase caused b y imp osing parking restrictions. The results v ary greatly dep ending on the a v ailable routes’ qualit y , the parking shortage sev erit y , and the exp ected cost of illegal parking. The results underline the imp ortance of including parking information as early as p ossible to increase the quan tit y and qualit y of a v ailable routes and sc hedules. In addition, it also elicits the imp ortance of further researc h on estimating the p oten tial costs and risks of illegal truc k parking. Next, w e extended the resource-constrained shortest path form ulation to the case of battery- electric truc ks (BET s). W e studied the impact of co ordinating rest and rec harge needs on BET s’ p erformance and its comparison to diesel truc ks. Computational exp erimen ts w ere used to estimate the effects of differen t lev els of c harging and parking infrastructure. Although BET s generate only a small fraction of diesel truc ks’ CO2 emissions, BET s require longer trip durations in most scenarios. Ho w ev er, this gap in trip duration dep ends on battery capacit y , c harging infrastructure (p o w er of regular and fast c hargers), and parking/c harging facilities’ a v ailabilit y (regular c hargers’ time- windo ws, and fast c hargers n um b er and w ait time). A common concern regarding the utilization of BET s for long-haul truc king is the infrastructure required to quic kly c harge large batteries, reducing the disparit y to diesel truc ks’ refueling time. Nev ertheless, our exp erimen ts sho w that, although fast-c hargers can significan tly reduce trip duration in man y scenarios, trip duration is ev en more sensitiv e to the p o w er and a v ailabilit y of the regular c hargers used for long (o v ernigh t) rests. It is imp ortan t to note that these results do not mean that fast c hargers are without b enefit. The adv an tages of particular infrastructure decisions will v ary for eac h case. Our exp erimen ts illustrate that, while trying to ha v e BET s op erating in similar itineraries to curren t diesel truc ks (e.g., using fast c hargers to reduce rec harging time) migh t b e the instinctiv e w a y to approac h truc k electrification, it is not the only one. It is lik ely not the b est approac h either. Afterw ards, w e considered the case of sto c hastic parking a v ailabilit y , as opp osed to the deter- ministic time-windo ws considered in previously , as w ell as an application in emissions reduction. The resource-constrained shortest path form ulation w as further extended to mo del driv ers p ossible recourse actions when unable to find parking and the ensuing costs. W e used this form ulation to study ho w the solutions are affected b y the lev el of information pro vided to driv ers. W e found that 197 ignoring uncertain t y in parking a v ailabilit y results in inconsisten t p erformance ev en when restrict- ing parking to p erio ds when probabilit y of finding parking is high. Giving driv ers full information ab out the probabilit y of finding parking at an y time/lo cation significan tly impro v es p erformance and reduces illegal parking-related risks, but also substan tially increase complexit y and solv e time. Using full information regarding parking a v ailabilit y but restricting the parking times to high a v ailabilit y time-windo ws can reduce complexit y while main taining consisten t, although reduced, p erformance. Finally , w e approac h the issue of ho w TDSP mo dels can b e used to aid in redistributing truc k parking demand. W e presen t an MIP mo del to the TDSP with m ultiple v ehicles under o ccupancy constrain ts. Ho w ev er, due to the complexit y of solving this problem for a large n um b er of v ehicles, w e prop ose using a mo dified TDSP mo del where parking costs are considered to optimize eac h v ehicle’s sc hedule separately using parking prices to influence their decisions. W e found that the TDSP with parking costs is indeed v ery sensitiv e to parking rates, but that small c hanges can cause the system to oscillate, aggra v ating parking issues b efore sho wing an y impro v emen ts. Concerns regarding design pricing mec hanisms with desirable prop erties w ere not addressed, and will b e left for future w ork. In this dissertation, w e exp osed the imp ortance of using truc k parking a v ailabilit y informa- tion during planning, and prop osed metho ds to do so. Besides helping individual truc k driv ers with trip planning, the metho ds dev elop ed in this pro ject can sim ulate differen t scenarios and aid p olicymak ers in estimating the impacts of p olicy and infrastructure in v estmen t decisions. 9.1 Prop osed Researc h Directions Existence of prop er p olicies One of the issues w e found in c hapters 6 and 7 w as that, due to the resource constrain ts, not all v alid states can generate a feasible solution. W e use constrain t propagation to discard some un w an ted states, but w e a only able to appro ximate the space of states able to generate feasible solutions. Due to the presence of these v alid but infeasible states, w e cannot guaran tee the existence of prop er p olicies, and generated p olicies ma y lead to infeasible states ev en if the initial state is feasible. Therefore, an in teresting researc h direction is to study ho w to guaran tee the existence of prop er p olicies b y either creating an efficien t metho d to calculate the set of feasible states, or to mo dify the form ulation so that all states are feasible 198 T ruc k P arking Demand Balancing In c hapter 8, w e started exploring using TDSP mo dels to dev elop a pricing system aiming to redistribute parking demand. Ho w ev er, w e fo cus on prop osing an initial form ulation and studying ho w sensitiv e the mo del is to price c hanges. The design of pricing mec hanisms that reac h a system optimal, or that guaran tee prop erties suc h as incen tiv e compatibilit y still need to b e addressed. 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The piecewise constan t definition of tra v el sp eed guaran tees that a driv er cannot reac h an edge’s end earlier b y departing later from its start no de, satisfying the “first-in-first-out” (FIF O) prop ert y . The mo del presen ted in c hapter 4 (with simplified w eekly constrain ts) can b e extended b y adding the follo wing constrain ts. m i X r=1 U i;r = 1; 8 0in (A.1) g i;r U i;r x i;d;r ; 8 0in; 0rm i (A.2) g i;r+1 U i;r x i;d;r ; 8 0in; 0rm i (A.3) m i X r=1 x i;d;r =x i;d ; 8 0in (A.4) d i a i;r +b i;r (x i;d g i;r )+M (U i;r 1); 80in; 0rm i (A.5) x i;d;r 2 [0;t hor ];U i;r 2f0;1g; 8 0in; 0rm i (A.6) The tra v el time from lo cationi toi+1 is defined b y m i functions of the forma i;r +b i;r (x i;d g i;r ) , where g i;r ; 0 r m i , are the times at whic h c hanges of slop e o ccur. The binary v ariables U i;r 216 are used to define in whic h in terv al [g i;r ;g i;r+1 ] the departure time x i;d is lo cated. The v ariables x i;d;r tak e the v alue of x i;d if U i;r = 1 and zero otherwise. (A.1) guaran tees that only one U i;r can tak e the v alue 1 at a time. (A.2)and (A.3) mak e it so all the v ariables x i;d;r , except the one relativ e to the U i;r with v alue 1, are set to 0. (A.4) forces the non-zero x i;d;r to equal to x i;d , and, consequen tly , U i;r can tak e v alue 1 only if x i;d 2 [g i;r ;g i;r+1 ] . (A.5) defines the tra v el time when U i;r = 1 , is alw a ys satisfied when U i;r = 0 . (A.6) defines the domain of the v ariables x i;d;r andU i;r . A.2 Multiple Clien ts with Service Time W e consider the problem of sc heduling the rest stops for a long-haul truc k trip with a kno wn route and m ultiple clien ts while taking in to accoun t the USA HOS regulations and estimated parking a v ailabilit y windo ws for all rest areas along the route. It is assumed that the rest areas are lo cated on the route and require no detours to b e accessed. The parking a v ailabilit y time-windo ws are assumed kno wn. The route has n+1 no des, 2 of whic h are the origin, no de 0, and destination of the truc k, no de n . The other no des are either clien ts, rest areas, or dumm y lo cations used to mo del sp ecial cases. Let N =f0;1;:::;ng b e the set of all lo cations, N r N the set of rest areas, N c N the set of clien t lo cations, and N cr N is the set of dumm y lo cations used to mo del resting at a clien t lo cation immediately after completing the required service. Ev ery no de in N cr is p ositioned righ t after a no de in N c , and the tra v el time b et w een them is zero. The dumm y lo cations used for resting at a clien t lo cation b efore starting the service are treated as regular rest areas. The tra v el time, d i , b et w een lo cations i and i +1 is assumed constan t, x i;a and x i;d represen t the arriv al and departure time of the truc k at lo cation i , resp ectiv ely . Eac h lo cation i has T i time-windo ws h t min i; ;t max i; i , where 2f1;2;:::;T i g indicates the time-windo w’s index. The time-windo ws restrict the arriv al time at that no de and are only in effect when the truc k has to stop at that sp ecific no de, driving b y it is not constrained b y the time windo ws. F or eac h lo cation and time-windo w, a binary v ariabley i; represen ts if that sp ecific time windo w is b eing used (y es:1, no:0). Driving b y without stopping is represen ted b y the v ariable y i;0 (driv e b y:1, stop:0). In the case of rest areas, the time-windo ws represen t the area’s parking a v ailabilit y , whereas for clien t lo cations the time-windo ws represen t deliv ery time constrain ts imp osed b y the clien t. The driv er 217 m ust stop at all clien ts, so y i;0 = 0 for all clien t lo cations. Eac h clien t has a service time s i whic h is assumed constan t. The sc hedule m ust comply with the USA HOS regulations. Some HOS rules regulate driving time or elapsed time, so they are not affected b y the inclusion of service time in the mo del. Ho w ev er, regulations that consider on-dut y time m ust accoun t for the service time, as the driv er is w orking during that time. R is defined as the set of differen t t yp es of rest p erio d describ ed in the regulation. F or eac h r2 R , t r defines the minim um duration of that t yp e of rest p erio d. C is the set of constrain ts imp osed b y the regulation. C 1 C is the set of constrain ts con trolling the maxim um elapsed time b et w een off-dut y p erio ds. C 2 C is the set of constrain ts con trolling the maxim um accum ulated driving time b et w een off-dut y p erio ds. C 3 C is the set of constrain ts con trolling the maxim um accum ulated on-dut y time during a rolling time-windo w; the width of the time-windo w for a constrain t c2 C 3 is represen ted b y c . In the USA regulation c is 7 da ys, so these rolling time-windo w constrain ts will b e referred to as w eekly constrain ts. F or eac h constrain t c 2 C , t c is the time limit imp osed b y the regulation and R c R is the set of rest t yp es that can reset this coun ter. The binary v ariable z i;r indicates whether a rest of t yp e r is tak en at lo cation i (y es:1, no:0). The driv er cannot tak e more than one t yp e of rest at the same lo cation. If no t yp e of rest is sc hedule for a rest area, the driv er cannot stop there. The departure time from the origin m ust b e within the in terv al [t 0 ;t dep ] . It is assumed that the driv er has b een off-dut y for long enough b efore the departure time, so that all constrain ts’ coun ters are reset b efore departure. It is imp ortan t to note that these rules state that the driv er cannot driv e after the time limit is reac hed, not that he/she cannot w ork. Therefore, the limit can b e exceeded during service time as long as the driv er can rest at the clien t lo cation immediately after, without ha ving to driv e. Because of this the dumm y no des used to mo del resting at clien t lo cations after w orking need to b e treated differen tly . The extended mo del without the w eekly constrain ts is the follo wing: minx n;a x 0;d (A.7) s.t.: x i;d +d i =x i+1;a ; 80in1 (A.8) 218 x i;a + X r2R t r z i;r x i;d ; i2NnN c (A.9) x i;a +s i =x i;d ; i2N c (A.10) x i;d x i;a +(1y i;0 )t hor ; 81in (A.11) y i;0 + T i X =1 y i; = 1; 81in (A.12) y i;0 = 0; i2N c (A.13) T i X =1 y i; = X r2R z i;r ; 8i62N c [f0g (A.14) X r2R z i;r = 0; 8i2N c [f0g (A.15) T i X =1 y i; t min i; x i;a ; 81in (A.16) x i;a t hor T i X =1 y i; t hor t max i; ; 81in (A.17) x k;a x i;d t c +t hor k1 X j=i+1 X r2Rc z j;r ; 80i<kn;k62N cr ;c2C 1 (A.18) x k;a x i;d t c +t hor k X j=i+1 X r2Rc z j;r ; 80i<kn;k2N cr ;c2C 1 (A.19) k1 X j=i d j t c +t hor k1 X j=i+1 X r2Rc z j;r ; 80i<kn;c2C 2 (A.20) x i 2 [0;t hor ] 2 ;y i 2f0;1g T i +1 ;z i 2f0;1g jRj ; 80in+1 (A.21) x 0;d 2 [0;t dep ] (A.22) The ob jectiv e function (A.7) is set to minimize the total trip duration. Constrain t (A.8) guar- an tees that the arriv al time equals the departure time of the previous lo cation plus the driving time. Constrain t (A.9) states that the v ehicle m ust sta y at a lo cation (clien ts excluded) at least for the minim um rest time decided for that lo cation. Constrain t (A.10) states that the v ehicle m ust depart a clien t lo cation as so on as the service time is o v er. Constrain t (A.11) con trols what happ ens when the driv er do es not stop at a certain lo cation. If the v ehicle do es not stop at lo cation i , the arriv al time equals the departure time. This constrain t w orks with constrain ts (A.9), (A.12) 219 and (A.14) to assure this. Equalit y will hold when y i;0 = 1 . If y i;0 = 0 , then constrain t (A.11) is alw a ys true as t hor is large. Constrain t (A.12) states that at an y lo cation, either exactly one time-windo w is used or the v ehicle do es not stop. Constrain t (A.13) states that the driv er m ust stop at all clien t lo cations. Constrain t (A.14) states that the driv er only stops if an off-dut y p erio d is sc heduled. Constrain ts (A.16) and (A.17) c hec k the time-windo ws. Arriv al m ust happ en after the b eginning and b efore the end of the c hosen time windo w. Constrain ts (A.18) and (A.19) c hec k that the time elapsed since the last re st inR c ;c2C 1 is less thant c . Constrain t (A.20) c hec ks if the accum ulated driving time b et w een rest p erio ds inR c ;c2C 2 is less thant c . Constrain t (A.21) sets the v ariables’ domains, and (A.22) guaran tees that the departure time from the origin is within the required p erio d and that the v ehicle will stop at the destination. A.2.1 W eekly Constrain ts The constrain ts describ ed here are from the setC 3 , for simplicit y w e will consider thatjC 3 j = 1 and will omit the index c , whic h indicates to whic h rule the v ariable refers to, from the v ariables. In the USA regulation only one rolling time-windo w constrain t is used at a time, so this assumption is reasonable. Nev ertheless, if more constrain ts are needed for a sp ecific coun try , they can b e added separately . The constrain ts C 3 can b e simplified to limit the on-dut y time b et w een t w o w eekly rests, instead of limiting the on-dut y time during a rolling time-windo ws. This form ulation will giv e sub optimal results in certain scenarios, but also guaran tees regulation complian t sc hedules. These t w o form ulations will b e presen ted separately to a v oid confusion. A.2.1.1 Simplified Constrain t As men tioned previously , this simplified form ulation limits the on-dut y time b et w een w eekly rests, accoun ting for the p ossibilit y of resting immediately after finishing w ork at a clien t. k1 X j=i d j + k1 X j=i+1 s j t c +t hor k1 X j=i+1 X r2Rc z j;r ; 80ikn; k2NnN cr ; c2C 3 (A.23) k1 X j=i d j + k1 X j=i+1 s j t c +t hor k X j=i+1 X r2Rc z j;r ; 80ikn;k2N cr ;c2C 3 (A.24) 220 Constrain t (A.23) c hec ks if the accum ulated driving and service time b et w een departure from lo cation i and arriv al at lo cation k is b elo w the limit; the constrain t is automatically satisfied if an appropriate rest is tak en at an y lo cation in b et w een, not including i and k . Constrain t (A.24) treats the particular case of resting after w orking at a clien t. Similarly , it c hec ks if the accum ulated driving and service time b et w een departure from lo cation i and arriv al at lo cation k is b elo w the limit, but no w, an appropriate rest at k can also satisfy the constrain t. As k2N cr , resting at k is equiv alen t to resting at the clien t lo cation righ t after the w ork is completed. The driv er do es not need to driv e b et w een k1 and k , so the regulation is not violated ev en if the on-dut y time limit is exceeded during the service at k . A.2.1.2 Rolling Time-Windo w Constrain t This constrain t mak es it so the driv er cannot driv e after ha ving accum ulated more than t c on- dut y hours o v er the last c hours, with c2C 3 . Let i (t) represen t the accum ulated driving time generated b y the service at lo cationi and the displacemen t b et w een lo cationsi andi+1 measured at time t . i (t) is describ ed b y: i (t) =R(tx i;d )R(tx i+1;a )R(tx i;d c )+R(tx i+1;a c ); 80in1; i2NnN c ; c2C 3 (A.25) i (t) =R(tx i;a )R(tx i+1;a )R(tx i;a c )+R(tx i+1;a c ); 80in1;i2N c ;c2C 3 (A.26) where R(t) is the unit ramp function. Equation (A.25) represen ts the accum ulated on-dut y time generated when i is not a clien t lo cation. As an y time sp en t at i is used for resting, the on-dut y time only starts b eing coun ted after departure (t =x i;d ), the coun t stops up on arriv al at the next lo cation (t =x i+1;a ), and sta ys constan t un til t =x i;d + c , when that on-dut y time starts lea ving the rolling time-windo w un til reac hing zero again att =x i+1;a + c . Equation (A.26) is used when i is a clien t lo cation, so the service time at i also needs to b e coun ted. Therefore, the on-dut y time starts b eing coun ted when the driv er arriv es at i , t =x i;a . The functions i (t) are included in the 221 mo del as piecewise linear functions. The domains of the functions are divided in sections according to when the slop e of the functions c hange, and auxiliary v ariables are used to write t according to where it is lo cated relativ e to the sections’ b oundaries. F or this problem, it suffices to ev aluate i (t) at the arriv al timesx j;a ;j >i , so w e define the v ariables i;j = i (x j;a ) . The sets of v ariables f i;j;p g ,f i;j;q g are used to define f i;j g as follo ws: i;j;p 2f0;1g; i;j;q 2 [0;1]; 80i<jn; 0p 4; 1q 5 (A.27) 1 i;j;0 i;j;1 i;j;1 ::: i;j;4 i;j;5 ; 80i<jn (A.28) i;j;p < i;j;p+1 +1; 80i<jn (A.29) x j;a =x i;d i;j;1 +d i i;j;2 +( c d i ) i;j;3 +d i i;j;4 +t hor i;j;5 ; 80i<jn; i2NnN c ; c2C 3 (A.30) x j;a =x i;a i;j;1 +(d i +s i ) i;j;2 +( c d i s i ) i;j;3 +(d i +s i ) i;j;4 +t hor i;j;5 ; 80i<jn; i2N c ; c2C 3 (A.31) i;j = (d i +s i )( i;j;2 i;j;4 ); 80i<jn (A.32) where the ’s and ’s are auxiliary v ariables used to mo del the piecewise definition of i;j . The ’s determine in whic h section of the function domain t is, and the ’s define its exact p osition within the section, the indexes p and q represen t the sections. Constrain ts (A.27), (A.28) and (A.29) imply that, for a section q , whenev er 0 < ;q < 1 , then ;q = 1; ;q = 1; 8q < q , and ;q + = 0; ;p = 0; 8q + > q;p q . Constrain ts (A.30) and (A.31) write the time instan t to b e ev aluated, x j;a , as a function of the ’s and ’s. Constrain t (A.32) uses the ’s to calculate i (x j;a ) . Due to (A.30) this problem w ould b e a quadratically constrained problem. Ho w ev er, as (A.30) only considers j > i , the v ariables i;j;p for p < 2 and i;j;q for q < 3 will b e alw a ys 1 222 and can b e defined as constan ts. The accum ulated driving time generated b y all trips or service starting at lo cationsf0;:::;ig measured at time x j;a is represen ted b y i;j . i;j = 8 > > < > > : i1;j + i;j ;if P r2Rc z i;r = 0 i;j ;if P r2Rc z i;r = 1 81i<jn; c2C 3 (A.33) 0;j = 0;j ; 81i<jn (A.34) j1;j t c ; 81jn;j = 2N cr ;c2C 3 (A.35) where constrain t (A.33) defines i;j and sets to zero all con tributions from no des b efore lo cation i when an appropriate rest is tak en at lo cation i . This mo del assumes that all constrain ts’ coun ters are reset b efore departure, so (A.34) sets the initial accum ulated driving time to zero. Constrain t (A.35) limits the accum ulated on-dut y time to the regulation limit t c , for all no des except if they are used for after-w ork rest at a clien t lo cation. Similarly to what w as explained for constrain t (A.24), the limit can b e exceed during a service time, as long as the driv er rests immediately after, without driving. A.3 Electric V ehicles Consider the previous problem, but no w the truc k used is electric, more sp ecifically , battery electric. The truc k has a battery with total capacit yB inkWh . When driving along a link (i;i+1) the battery is consumed at a rate of a i kWh=h , this rate is assumed constan t for eac h link. The route has a setN s NnN c of c harging stations that allo w the driv er to b oth rec harge the v ehicle’s battery and rest. It is assumed that all c harging stations can b e used for resting, but not all rest areas are c harging stations. During a sta y at a c harging station the battery will b e rec harged at a rate of v i kWh=h , whereas during a sta y at a regular rest area the battery c harge is consumed at a rate v i kWh=h due to the truc ks auxiliary systems, lik e A C. As rest areas/c harging stations ha v e v arying lev els of infrastructure a v ailable, the rates of consumption/rec harge can v ary b et w een lo cations. It is assumed that the battery cannot b e c harged during service at a clien t lo cation. The battery c harge cannot exceed the capacit y B and it cannot b e negativ e either. The time that a v ehicle sta ys at a c harging station is not limited, but it is assumed that the battery will 223 automatically stop c harging when full. The effectiv e c harging time at lo cation i2N s is represen ted b y i . A t lo cations without c harging stations the v ehicle will b e consuming energy during the whole sta y , so i will b e equal to the whole duration of sta y . F or simplicit y , the mo del assumes that all parking spaces of a c harging station are electrified, so the parking a v ailabilit y time-windo ws are also the c harging station a v ailabilit y time-windo ws. If necessary , electrified and regular parking spaces can b e separated b y including a dumm y lo cation sp ecifically for the c harging station with a differen t time-windo w adjacen t to the regular rest area. This can also b e used to separate spaces with differen t c harging rates at the same lo cation. Let (b i;a ;b i;d ) represen t the battery c harge at the time of arriv al and departure at lo cationi . The battery c h arge constrain ts can b e describ ed as follo ws: i x i;d x i;a ; 8i2N s (A.36) i =x i;d x i;a ; 8i2NnN s (A.37) b i;d =b i;a + i v i ; 81in (A.38) b i;a =b i1;d +d i a i ; 81in (A.39) i 0; (b i;a ;b i;d )2 [0;B] 2 ; 80in (A.40) Constrain t (A.36) defines the effectiv e c harging time at c harging stations, whereas constrain t (A.37) fixes the effectiv e consumption time to the whole sta y ev erywhere else. Constrain t (A.38) up dates the battery lev el after a sta y at an y lo cation. Constrain t (A.39) up dates the battery lev el after a displacemen t. Note that the ’s will b e negativ e when consuming energy and p ositiv e when c harging the battery . Constrain t (A.40) defines the domains of the b ’s and ’s. This mo del allo ws the optimization of the sc hedules of electric truc ks with heterogeneous c harg- ing stations and rest areas. This extension can b e directly added to the mo del presen ted previously , so the HOS and parking a v ailabilit y constrain ts are still considered, as w ell as the p ossibilit y of servicing m ultiple clien ts. Although this mo del allo ws for differen t consumption rates to b e used at eac h trip section, the rates are fixed. Energy consumption mo dels found in the literature represen t the energy consumption as a non-linear function of the v ehicle sp eed, but this kind of consumption 224 cannot b e directly included in the curren t mo del. This mo del generates energy-feasible sc hedules, but do es not optimize the energy consumption. 225 App endix B T rip Duration Lo w er-Bound The sc hedule is affected b y the truc k stops’ lo cations, b y their a v ailabilit y windo ws, and b y the HOS regulation. In order to calculate lo w er b ounds for this problem, w e can consider an ideal scenario where truc ks ha v e no restriction on parking lo cation and time, only considering the HOS regulation. As the b ounds are dep enden t only on the regulation b eing used, they can b e calculated b eforehand and used to ev aluate the partial solutions obtained during the optimization. Here w e consider the same structure of the USA regulation and assume that all resources are set to zero. An extension for the non-zero resources is presen ted later. Also, as the regulation treats service time and driving time differen tly , this form ulation calculates the minim um trip duration for a trip with zero service time. When this is not the case, the service time can b e added to the calculated lo w er-b ound afterw ard. The b ound will b e lo oser in this case, but it still is a v alid lo w er-b ound for the trip duration. W e calculated the lo w er b ounds b y optimizing the total trip duration for trip lengths (in terms of driving time) that do not use daily r ests , then used this as a building blo c k to optimize trips that do not need we ekly r ests , then for trips of an y length. The parameters used are defined as follo ws: t b minim um br e ak duration; t r minim um daily r est duration; t w minim um we ekly r est duration; t eb limit for elapsed time b et w een br e aks ; t er limit for elapsed time b et w een daily r ests ; t ar limit for accum ulated driving time b et w een daily r ests ; t aw limit for accum ulated on-dut y time b et w een we ekly r ests ; arbitrarily small p ositiv e constan t. 226 First w e calculate the minim um trip duration for a trip with less than a da y’s w orth of driving time,L d () . L d (x) =x+t b x t eb ; 0xt ar (B.1) where x represen ts the trip length in hours, i.e. the total driving time of the trip. As x w as limited to less than the daily driving limitt ar , w e just need to calculate the n um b er of br e aks that will b e necessary during the da y . The is used to a v oid including a br e ak when the driving time is exactly on the limit. ThenL d () is used to describ e the trip duration for trips with less than a w eek’s w orth of driving time,L w () , as follo ws: u(x) = 8 > > < > > : 1 if x> 0 0 if x = 0 (B.2) g w (y) =L d (y 0 )+ n X i=1 [t r u(y i )+L d (y i )]; y2R n+1 ;0kyk 1 t ar (B.3) A(x) = y2R n+1 0kyk 1 t ar ; kyk 1 =x;n = x t eb (B.4) L w (x) = 8 > > < > > : min y2A(x) g w (y); t ar <xt aw L d (x); 0xt ar (B.5) The ’shorter than a w eek’ trips describ ed here can b e divided in m ultiple ’less than a da y’ trips separated b y daily r ests . F unction g w () optimizes these smaller sections usingL d () and adds a daily r ests for eac h non-zero section, calculating the minim um trip duration giv en a v ector y = (y 0 ;y 1 ;:::;y n ) comp osed of the n + 1 section lengths y i . Equation (B.5) optimizes the trip duration o v er all v alid com binations of section lengths, with up to n daily r ests . Equation (B.4) defines the v alid section length v ectors, c ho osing n suc h that the optimization considers enough daily r est to accoun t for the case of taking a daily r est ev ery time a br e ak is needed. 227 The same approac h is used to calculate the trip duration for longer trips,L l () , as follo ws: g l (y) =L w (y 0 )+ n X 1 [t w u(y i )+L w (y i )]; y2R n+1 ;0kyk 1 t aw (B.6) B(x) = y2R n+1 0kyk 1 t aw ;kyk 1 =x;n = x L (B.7) L l (x) = ( min y2B(x) g l (y); t aw <xL w (x); 0xt aw (B.8) where t eb L t aw should b e c hosen in a w a y that creates a v ector long enough to test the differen t p ossibilities of mo ving driving hours b et w een w eeks to a v oid br e aks and daily r ests when p ossible. This v alue affects the maxim um n um b er of we ekly r ests that the problem will consider in the optimization. T aking L = t eb certainly w orks as it is the maxim um driving time allo w ed without an y kind of rest. In the case of USA regulations, the optimization w ould consider ev en the case of taking a we ekly r est ev ery 8h of driving. F or long trips this resolution migh t b e excessiv e and will unnecessarily slo w do wn the calculations due to the large n um b er of p ossible v ectors. An optimal sc hedule will ha v e a high driving time to trip duration ratio, so w e suggest takingL as the length of the trip when the in termediate w eekly cycles (the ones that are follo w ed b y a we ekly r est ) w ould b e the most efficien t, i.e. L = argmax xtaw x L w (x)+t w . In the case of the regulations considered in this pro ject, L is equal to 55 hours. This form ulation considers that all resources are set to zero, so it pro vides a v ery lo ose b ound in most cases. A t the cost of computational complexit y and/or memory space, this can b e impro v ed b y including the curren t resources in the calculation. Consider the v ariables b , r and w defined as follo ws: b =t eb b (B.9) e =t er r (B.10) r =min(t er r ;t ar r ) (B.11) w =t aw w (B.12) 228 where b , r , r and w are the curren t v alues of the relev an t resources. The resources w ere defined in section 5.3. The lo w er b ounds with non-zero resources are defined as follo ws: L 0 d (x; b ) =x+t b +t b x b t eb ;0xt ar (B.13) L 0 w (x; b ; r ; e ) = min zmin( r;x) L 0 d (z; b ) e L 0 d (z; b )+t r u(xz)+L w (xz) ; 0xt aw (B.14) L 0 l (x; b ; r ; e ; w ) = min zmin( w;x) L 0 w (z; b ; r ; e )+t w u(xz)+L l (xz) ; 0x (B.15) The D HOS (x;) used in 5.5.2.3 is giv en b yL 0 l (x; b ; r ; w ) , where equations (B.9)–(B.12) are used to calculate b ; r and w from . 229 App endix C Dominance R ules Deriv ation F or con v enience, the notation used in this app endix is sligh tly differen t from the rest of the pap er. Consider t w o lab els i = ( 0 i ;c i ; b i ; r i ; r i ; w i );i2f1;2g , with asso ciated slac k v ariables ( 0;i ; 0;i ; r;i ; r;i );i2f1;2g , assigned to the same no de v . F or the ’s and resources, the index (1 or 2) is used to differen tiate b et w een the t w o lab els b eing compared. F or the ’s and ’s, the first index indicates if the slac k is relativ e to the origin (0 ) or the last daily/we ekly r est (r ), the second index indicates if the slac k is part of lab el 1 or 2 . W e sa y that 2 is dominated b y 1 , if 2 cannot generate solutions b etter than the ones generated b y 1 . The in tuitiv e dominance condition is to ha v e eac h resource in 1 b e smaller or equal to the corresp onding resource in 2 , with at least one b eing strictly smaller. Ho w ev er, as the partial solu- tions can b e impro v ed b y remo ving the slac ks in the sc hedule, these slac ks m ust also b e considered. Besides that, as w e imp ose time-windo w constrain ts on the arriv al time instead of on the activit y starting time, the time resource has to b e treated differen tly from other pap ers that use similar dominance rules, suc h as [23]. Without allo wing early arriv al, if 0 1 < 0 2 , then the solutions gener- ated from 1 ma y b e unable to satisfy the same time-windo ws as solutions generated from 2 due to arriving to o early . C.1 Equal time, no slac k The base case for the dominance c hec k is when the complexities of time-windo w constrain ts and slac ks need not b e considered. Let 0;i = 0;i = r;i = r;i = 0;i2f1;2g and 0 1 = 0 2 . In this case, the lab els cannot b e impro v ed b y remo ving slac ks and there is no difference b et w een the 230 time-windo ws that can b e satisfied b y paths generated from either lab el. 2 is dominated b y 1 if all the follo wing conditions hold, with at least one inequalit y b eing strict: 0 1 = 0 2 (C.1) c 1 c 2 (C.2) b 1 b 2 (C.3) r 1 r 2 (C.4) r 1 r 2 (C.5) w 1 w 2 (C.6) C.2 Equal time, non-zero slac k, after impro v emen t First, consider ho w lab el impro v emen t affects the resources. The new cost resource c i after remo ving slac ks relativ e to the origin is giv en b y: c i =c i + 0 min( 0;i ; 0;i ) r min( 0;i ; 0;i ) (C.7) The slac k remo v al shifts time from activities with r hourly cost to one with 0 . When the slac k regarding daily/we ekly r ests , ( r;i ; r;i ) , is remo v ed, the cost do es not c hange as these activities ha v e the same hourly cost as br e aks . In this case, to define dominance rules w e pa y atten tion to ho w these slac ks can affect the resource r i . The impro v ed resource r i is giv en b y: r i = r i min( r;i ; r;i ) (C.8) After lab el impro v emen t w e ha v e that the new slac k v ariables ha v e the follo wing prop ert y min( 0;i ; 0;i ) = min( r;i ; r;i ) = 0;i2f1;2g . Assume that 2 and 1 are impro v ed lab els, so min( 0;i ; 0;i ) = min( r;i ; r;i ) = 0;i2f1;2g . If all ’s are equal to zero, it falls on the previous case where no lab el impro v emen t is p ossible b y up dating upstream decisions, no w or in the future. Ho w ev er, if that is not the case, it is still p ossible for the remaining slac ks to b e used in the future if do wnstream decisions mak e the ’s 231 increase, e.g. if 2 w as generated b y a path with v ery lo ose time-windo ws, it ma y ha v e large ’s; this slac k can b e used to reduce the amoun t of unnecessary off-dut y time at do wnstream no des. It is imp ortan t to note that the usage of this remaining slac k is not guaran teed. The slac ks ma y b e reset or the v ehicle ma y reac h the final destination b efore the ’s are increased. Therefore, the dominance criteria m ust consider b oth the case when the slac k remains un used and when it is fully used. The case when the slac ks remain un used is co v ered b y (C.1)–(C.6). T o accoun t for the slac k usage, the follo wing constrain ts are included: c 1 c 2 ( 0 r )[ 0;2 0;1 ] (C.9) r 1 r 2 r;1 r;2 (C.10) Note that, un til the no de of reference is c hanged, the ’s can only increase, and the ’s can only decrease. So the ’s giv e an upp er-b ound on ho w m uc h future lab els can b e impro v ed b y up dating the decision tak en at the curren t reference no de. (C.9) and (C.10) c hec k if a path generated from 1 will b e sup erior to one from 2 ev en if the slac ks are fully used. A giv en path will cause the same resource exp enditures regardless of the resource v ector of the lab el from whic h it w as generated, so the resource v ariations caused b y do wnstream decisions do not sho w up in (C.9) and (C.10). 1 dominates 2 if conditions (C.1)–(C.6), (C.9) and (C.10) hold, with at least one inequalit y b eing strict. These dominance rules are v alid for an y no de, but only for impro v ed lab els with matc hing 0 . As lab el impro v emen t affects path feasibilit y and it is only p erformed during decisions at rest no des, the dominance rules for lab els that ha v e not b een impro v ed y et will b e treated in the next section, along with lab els with differen t 0 . C.3 Differen t time, non-zero slac k, b efore impro v emen t Due to the existence of time-windo w constrain ts 1 cannot dominate 2 if 0 1 > 0 2 as some time-windo ws ma y b e imp ossible to satisfy due to late arriv al at do wnstream no des. Therefore, w e assume that = 0 2 0 1 >= 0 . Ho w ev er, as w e assume early arriv als are not allo w ed, ha ving 0 1 < 0 2 will also cause problems if not treated prop erly . Therefore, w e only allo w > 0 at no des where this arriv al time difference can b e corrected b y the next decision, i.e. rest no des. The option 232 T able C.1: Effect of decision with duration u at rest no des Next lab el info v2V b v2V r [V w 00 i = 0 i + u 0 i + u c 0 i = c i + r u c i + r u b0 i = b i b i r0 i = r i + u r i r0 i = r i r i w0 i = w i w i 00 i = 0;i 0;i 00 i = 0;i + u 0;i + u r0 i = r;i r;i r0 i = r;i + u r;i to extend the duration of rests mak es it p ossible to matc h the arriv al time of lab els generated b y 1 to the ones generated b y 2 . The dominance conditions are deriv ed b y studying when ev ery lab el generated from 2 through a decision u 2 = (j;) is dominated (after lab el impro v emen ts) b y the lab el generated from 1 through the decisionu 1 = (j;+ ) . Note that 1 and 2 are not impro v ed lab els, w e simply calculate the effects that lab el impro v emen t could ha v e on their descendan ts. Let V b ,V r andV w represen t the set of all br e ak, daily r est and we ekly r est no des, resp ectiv ely . T able C.1 summarizes ho w a decision u with duration u tak en at a rest no de affects the resources and slac k v ariables, the second column refers to decisions tak en at br e ak no des, whereas the third column refers to decisions tak en at daily r est or we ekly r est no des. This table differs sligh tly from T able 5.1 b ecause it refers sp ecifically to decisions tak en at rest no des, and it also includes the slac k v ariables. Note that decisions tak en at a rest no de alw a ys lead to the TPL’s exit (no de v out i in Figure 5.2a), whic h do es not ha v e time-windo w constrain ts, and that resources requiring resetting are already zero. Let the resources with sup erscript represen t the resources generated after lab el expansion and lab el impro v emen t, in this order. The lab el expansion uses the decisions listed previously ( u 1 =+ for 1 and u 2 = for 2 ). Lab el impro v emen t effects are describ ed in (C.7) and (C.8). If the follo wing conditions are satisfied (with at least one inequalit y b eing a strict inequalit y) for all v alues of , w e can sa y that 1 dominates 2 . 233 v2V b [V r [V w (C.11) 0 1 + + = 0 1 = 0 2 = 0 2 + (C.12) b 1 = b 1 b 2 = b 2 (C.13) r 1 = r 1 r 2 = r 2 (C.14) w 1 = w 1 w 2 = w 2 (C.15) c 1 =c 1 +( 0 r )min( 0;1 ; 0;1 + + )+ r ( + ) (C.16) c 2 =c 2 +( 0 r )min( 0;2 ; 0;2 +)+ r (C.17) c 1 c 2 (C.18) r 1 = 8 > > < > > : r 1 min( r;1 ; r;1 + + )+ + ; if v2V b 0; if v2V r [V w (C.19) r 2 = 8 > > < > > : r 2 min( r;2 ; r;2 +)+; if v2V b 0; if v2V r [V w (C.20) r 1 r 2 (C.21) c 1 c 2 ( 0 r )( 0;2 0;1 ) r (C.22) r 1 r;1 + r 2 r;2 ; if v2V b (C.23) (C.11) defines that these rules are applicable only to rest no des. (C.12) is alw a ys satisfied due to ho w is defined. (C.13) is alw a ys satisfied b ecause b is alw a ys zero at rest no des and do es not c hange when the rest is extended. (C.14) and (C.15) c hec k the conditions on the accum ulated driving time since the last daily r est and on-dut y time since the last we ekly r est , resp ectiv ely . These first four conditions are indep enden t of the v alue of , so they can b e easily c hec k ed from 1 and 2 . Equations (C.16)–(C.21) c hec k the conditions on elapsed time since the last daily r est ( r ) and accum ulated cost (c ) assuming lab el impro v emen t will b e p erformed when that lab el is expanded. Conditions (C.22)–(C.23) are lik e (C.9)–(C.10) in that they guaran tee that dominance will still b e v alid ev en if remaining slac ks are fully used do wnstream. Note that (C.22)–(C.23) are generated 234 from (C.16)–(C.21) b y using the ’s to b ound the effects of future lab el impro v emen ts, equiv alen t to taking a large . Conditions (C.16)–(C.21) dep end on , so w e determine sufficien t conditions to guaran tee that they hold for ev ery . Note that, for v 2 V b , the structure of the conditions on r is the same as the ones on c . Therefore, w e only describ e ho w to find sufficien t conditions to satisfy (C.16)–(C.18). Analogous conditions for (C.19)–(C.21) can b e found b y taking 0 = 0 and r = 1 , and b y replacing the resource and slac k v ariables b y the ones relativ e to r . Eqs. (C.16)–(C.18) can b e rewritten as: c 1 c 2 +( 0 r ) min( 0;2 ; 0;2 +) min( 0;1 ; 0;1 + + ) r (C.24) The subscripts ‘0 ’ of the ’s and ’s will b e omitted for con v enience. F urthermore, it is assumed that 0 < r . The sufficien t conditions are determined b y minimizing the righ t side of (C.24) o v er all v alues of . Case 1: 1 1 + ^ 2 2 c 1 c 2 +( 0 r )( 2 1 ) r =c 2 + 0 ( 2 1 )+ r ( 2 + 1 ) (C.25) Case 2: 1 1 + ^ 2 > 2 c 1 c 2 +( 0 r ) min( 2 ; 2 +) 1 r (C.26) W e determine the minim um of the righ t side of (C.26) and use it to determine a sufficien t condition indep enden t of . Note that ( 0 r )< 0 . max 0 min( 2 ; 2 +) 1 = 2 1 (C.27) ) c 1 c 2 +( 0 r )( 2 1 ) r =c 2 + 0 ( 2 1 )+ r ( 2 + 1 ) c 2 +( 0 r ) min( 2 ; 2 +) 1 r (C.28) Case 3: 1 > 1 + ^ 2 2 c 1 c 2 +( 0 r ) 2 min( 1 ; 1 + + ) r (C.29) 235 Determine the minim um of the righ t side of (C.29). max 0 2 min( 1 ; 1 + +) = 2 1 (C.30) ) c 1 c 2 +( 0 r )( 2 1 ) r =c 2 + 0 ( 2 1 )+ r ( 2 + 1 ) c 2 +( 0 r ) 2 min( 1 ; 1 + + ) r (C.31) Case 4: 1 > 1 + ^ 2 > 2 c 1 c 2 +( 0 r ) min( 2 ; 2 +) min( 1 ; 1 + + ) r (C.32) If 1 1 < 2 2 , the righ t side is minimized b y a large : max 0 min( 2 ; 2 +)min( 1 ; 1 + + ) = 2 1 (C.33) ) c 1 c 2 +( 0 r )( 2 1 ) r =c 2 + 0 ( 2 1 )+ r ( 2 + 1 ) c 2 +( 0 r ) min( 2 ; 2 +) min( 1 ; 1 + + ) r (C.34) If 1 1 2 2 , the righ t side is minimized b y = 0 : max 0 min( 2 ; 2 +)min( 1 ; 1 + + ) = 2 1 (C.35) ) c 1 c 2 +( 0 r )( 2 1 ) r =c 2 + 0 ( 2 1 )+ r ( 2 + 1 ) c 2 +( 0 r ) min( 2 ; 2 +) min( 1 ; 1 + + ) r (C.36) C.3.1 Summary Conditions (C.16)–(C.21) should b e replaced b y the follo wing conditions: Cost constrain t: Giv en 0 < r . If 0;1 0;1 + : c 1 c 2 + 0 ( 0;2 0;1 )+ r ( 0;2 + 0;1 ) (C.37) If 0;1 > 0;1 + ^ 0;2 0;2 : c 1 c 2 + 0 ( 0;2 0;1 )+ r ( 0;2 + 0;1 ) (C.38) 236 If 0;1 > 0;1 + ^ 0;2 > 0;2 : c 1 c 2 + 0 ( 0;2 0;1 )+ r ( 0;2 + 0;1 ) ^ c 1 c 2 + 0 ( 0;2 0;1 )+ r ( 0;2 + 0;1 ) (C.39) r constrain t, for v2V b : If r;1 r;1 + : r 1 r 2 r;2 + r;1 (C.40) If r;1 > r;1 + ^ r;2 r;2 : r 1 r 2 r;2 + r;1 (C.41) If r;1 > r;1 + ^ r;2 > r;2 : r 1 r 2 r;2 + r;1 ^ r 1 r 2 r;2 + r;1 (C.42) 237 App endix D Optimalit y Pro of Definition 1. W e use the term ‘R estricte d Pr oblem’ to r efer to a sub-pr oblem that c onsiders a single p ath over the extende d network G 0 = (V 0 ;A 0 ) and a single time-window at e ach lo c ation, without mo difying the HOS c onstr aints. L et v 0 and v n r epr esent, r esp e ctively, the origin and destination no des. L et Tv j S =1 [t min v j ;t max v j ] r epr esent the set of time-windows for no de v j , wher e T v j is the numb er of disjoint time-windows at no de v j . A n instanc e of the r estricte d pr oblem is define d by a p ath p = (v 0 ;v 1 ; ;v n ) such that(v i ;v i+1 )2 A 0 for al l i = 0; ;n1 , and, for e ach lo c ation v j in p , a time-window [a j ;b j ] = [t min v j ;t max v j ];2 f1; ;T v j g . F or no des without time-windows, c onsider [a j ;b j ] = (1;+1) . The HOS c onstr aints r emain the same as in the original pr oblem. Definition 2. L et k = ( 0 k ;c k ; b k ; r k ; r k ; w k ) b e the r esour c e ve ctor define d in Se ction 5.4.2. L et ( i;k ; i;k );i2f0;rg b e the slack ve ctors define d in Se ction 5.5.2.2, wher e r is the index of the last daily or we ekly r est no de visite d. The extende d state ve ctor is describ e d by the curr ent lo c ation, the r esour c e ve ctor and the slack ve ctor, i.e. x k = (v k ; 0 k ;c k ; b k ; r k ; r k ; w k ; 0;k ; 0;k ; r;k ; r;k ) . Definition 3. A solution to the original pr oblem is c omp ose d of a p ath and a sche dule. It c an b e uniquely describ e d by a ve ctor of extende d states x = (x 0 ;x 1 ; ;x i ) , or by a triplet of initial statex 0 , p ath and de cisions ve ctor (x 0 ;p i ;) , wher ep i = (v 0 ;v 1 ; ;v i ) and = ( 0 ; 1 ; ; i1 ) . When c onsidering a r estricte d pr oblem, as p i is given, (x 0 ;) is enough to r epr esent a sche dule, and, c onse quently, a solution. Definition 4. Consider a r estricte d pr oblem with p ath p = (v 0 ; ;v e ; ;v n ) and time-windows f[a i ;b i ]g;i = 1; ;n . L et D l (i;e) , D d (j;e) , D s (j;e) and D m (j;e) b e, r esp e ctively, the length, 238 driving time, servic e time and the mandatory off-duty time b etwe en no des v j and v e . L et V b , V r and V w r epr esent the set of al l br e ak, daily r est and we ekly r est no des, r esp e ctively. L et N d = V b [V r [V w [fv 0 g b e the set of lo c ations with c ontr ol lable dur ation de cisions, i.e. origin and r est no des. L etbr = maxfijie;i2N d g b e the index of the last r est no de,r = maxfijie;i2N d nV b g the index of the last daily or we ekly r est no de, and w = maxfijie;i2V w g the index of the last we ekly r est no de. A ssuming zer o initial c onditions, given the p artial de cision ve ctor ( 0 ; 1 ; ; e1 ) for a p artial sche dule ending at no de v e , the r esour c e ve ctor e and its c onstr aints c an b e written as: a e 0 e = 0 +D d (1;e)+D s (1;e)+D m (1;e) + X k2[1;e1] v k 2N d k b e (D.1) c e = 0 0 + d D l (1;e)+ d D d (1;e)+ s D s (1;e) + r D m (1;e)+ X k2[1;e1] v k 2N d k (D.2) r e =D d (r;e)+D s (r;e)+D m (r;e) + X k2[r+1;e1] v k 2N d k t er (D.3) b e =D d (br;e)+D s (br;e)t eb (D.4) r0 e =D d (r;e)t ar (D.5) w0 e =D d (w;e)+D s (w;e)t aw (D.6) L etC(i;j) andD(i;j) r epr esent, r esp e ctively, the fixe d p art of the c ost and trip dur ation b etwe en no des v i and v j . W e c an r ewrite (D.1) , (D.2) and (D.3) as: a e 0 e =D(0;e)+ X k2[0;e1] v k 2N d k b e (D.7) c e =C(0;e)+ 0 0 + r X k2[1;e1] v k 2N d k (D.8) 239 r e =D(r;e)+ X k2[0;e1] v k 2N d k t er (D.9) Lemma 1. Consider the two r epr esentations of a p artial solution: x = (x 0 ;x 1 ; ;x e ) , and (x 0 ;p e ;) , wher e p e = (v 0 ;v 1 ; ;v e ) and = ( 0 ; 1 ; ; e1 ) . W e have that, for every 0i je : i;j = 0 ! X k2[i+1;j1] v k 2N d k = 0 Pr o of. F ollo ws from the definition of i;j , (5.13) . Corollary 1. L et x = (x 0 ;x 1 ; ;x n ) b e a solution for an instanc e of a r estricte d pr oblem. If 0;n = 0 , then the solution x has minimum trip dur ation for that r estricte d pr oblem. Lemma 2. Consider a r estricte d pr oblem with p ath p and time-windowsf[a i ;b i ]g;i = 1; ;n . Consider the two r epr esentations of a p artial solution: x = (x 0 ;x 1 ; ;x e ) , and (x 0 ;) , wher e = ( 0 ; 1 ; ; e1 ) . W e have that, for every 0ijen : i;j = 0 !9k2 [i+1;j]; 0 k =b k and i;k = 0 (D.10) Pr o of. F ollo ws from the definition of i;j , (5.14). i;i+1 is set to1 or b i+1 0 i+1 + i;i+1 at no de v i+1 , and, at ev ery no dev k ,i<kj , i;k is set to min( i;k1 ;b k 0 k + i;k ) . So, if i;j = 0 , then b k 0 k + i;k m ust b e zero for at least one no de v k , i < k j . This can only happ en if 0 k = b k and i;k = 0 . Prop osition 1. Consider a r estricte d pr oblem with p athp and time-windowsf[a i ;b i ]g;i = 1; ;n . The state ve ctor (x 0 ;x 1 ; ;x e ) and de cision ve ctor ( 0 ; 1 ; ; e1 ) r epr esenting the p artial sche d- ule found at the end of e ach iter ation have the fol lowing pr op erties: (a) min 0;i ; 0;i = 0;81ie . (b) min( r;i ; r;i ) = 0;81ie; r = maxfkjki;v k 2V r [V w [fv 0 gg . (c) i > 0!9j >i; 0 j =a j and9l 0!9j > 0; 0 j =a j . [Note that in (c) and (d), ifj >e , 0 j is the e arliest p ossible arrival time atv j given the curr ent p artial sche dule, i.e., 0 j = 0 e +D(e;j) .] 240 Pr o of. Prop osition 1(a) follo ws from the lab el impro v emen t p erformed when lab els for TPLs’ exits are generated. Only rest extension decisions can increase the ’s, whereas the ’s can only decrease. So, if the slac k is remo v ed at a TPL’s exit no de, it cannot increase un til the next TPL exit is visited. As the lab el impro v emen t is p erformed at ev ery TPL exit, the slac k is alw a ys zero. The same reasoning applies to Prop osition 1(b). Prop osition 1(c) follo ws from the heuristic used to c ho ose the decisions to test and the effects of lab el impro v emen t. The c hosen rest extension is the smallest duration required to satisfy do wnstream time-windo ws, ho w ev er, if this v alue w ould generate a slac k in the sc hedule, the lab el impro v emen t pro cedure up dates the sc hedule to remo v e this slac k. Therefore, if the rest extension is greater than zero ev en after the lab el impro v emen t, it means that b oth ’s are zero, and, b y Lemma 2, there is an upstream no de v l with 0 l = b l . F urthermore, as i is the smallest v alue that satisfies do wnstream time-windo ws, then at least one do wnstream lo cation v j m ust ha v e 0 j = a j , otherwise i could b e reduced. The same reasoning applies to Prop osition 1(d). Prop osition 2. If a fe asible sche dule exists, the algorithm finds a sche dule with minimum trip dur ation for the r estricte d pr oblem (fixe d p ath, single time-windows). Pr o of. A t the last iteration w e ha v e that, b y Prop osition 1(a), min( 0;n ; 0;n ) = 0 . If 0;n = 0 then the sc hedule has no nonmandatory off-dut y time, and the sc hedule has minim um trip duration. If 0;n > 0 , then at least one nonmandatory off-dut y time is nonzero. Let i = maxfkj k > 0g . By Prop osition 1, w e ha v e that9j > i; 0 j = a j , and that min( 0;j ; 0;j ) = 0 . As i > 0 , w e ha v e that 0;j > 0 and 0;j = 0 . This implies that 0 and 0 1 cannot b e increased due to time-windo w constrain ts b et w een no des v 0 and v j . As 0 1 cannot b e increased and 0 j cannot b e decreased, the trip duration at v j , giv en b y 0 j 0 1 , is optimal. If j = n , then the trip duration is optimal. If j < n , w e ha v e that, b y the definition of i , k = 0 for all k2 [i+1;n1] . As no nonmandatory off-dut y time is included b et w een v j and v n , the trip duration at v n is also optimal. Prop osition 3. If a fe asible sche dule exists, the algorithm finds a minimum c ost sche dule for the r estricte d pr oblem (fixe d p ath, single time-windows). Pr o of. F rom prop osition Prop osition 2 w e ha v e that 0 n 0 1 is minim um, and, consequen tly , P k2[1;n1] k is also minim um. F rom Prop osition 1, w e ha v e that, if 0 > 0 then9j > 0; 0 j = a j , 241 and 0 cannot b e decreased without increasing P k2[1;j1] k . Ho w ev er, as 0 < r , decreasing 0 b y increasing the other ’s w ould only increase costs. Therefore, the solution has minim um cost. Prop osition 4. If a solution exists, the algorithm finds a minimum c ost solution (p ath and sche dule) for the unr estricte d pr oblem. Pr o of. As describ ed in Section 5.5.2.1, when expanding a lab el, the lab el correcting algorithm ex- plores ev ery outgoing edge that can generate a feasible solution. Therefore, all paths are considered b y the algorithm. In addition, at ev ery decision with a con trollable time duration, the algorithm tests the minim um v alues able to satisfy eac h time-windo w of eac h reac hable do wnstream lo ca- tion, suc h that all p ossible time-windo w usages are accoun ted for. Eac h com bination of path and time-windo ws used represen ts an instance of the restricted problem. The algorithm discards partial paths and sc hedules that are infeasible, dominated or unable to impro v e the curren t upp er-b ound (A*). Therefore, not all instances of the restricted problem are explicitly generated. Ho w ev er, if the dominance rules and the estimate used b y the A* metho d are correctly defined and do not dis- card go o d solutions, then if a solution exists the restricted problem instance con taining the optimal solution (or a solution within the c hosen tolerance) is guaran teed to b e generated. Prop osition 3 states that the algorithm finds a feasible solution for the restricted problem if one exists. As the algorithm explores all restricted problem instances that migh t con tain the optimal solution, then the optimal solution to the original problem is found b y c ho osing the b est solution among restricted problems’ optimal solutions. 242 App endix E Lab el Impro v emen t Pseudo co de F unction up dateP ath input : (x j ) 0jk ; ( j ) 0j<k ; h; i ; output: (^ x j ) 0jk ; ( ^ j ) 0j<k ; /* x j = (v j ;( 0 j ;c j ; b j ; r j ; r j ; w j )) , and ^ x j follows the same format. */ 1 for j 0 to i1 do 2 ^ x j x j ; 3 ^ j j ; 4 end 5 ^ x i x i ; 6 ^ i i +h ; 7 ^ x i+1 f(^ x i ;(v i+1 ; ^ i )) ; 8 for j i+1 to k1 do 9 if v j 2N d then 10 p 0 j+1 ^ 0 j ; 11 ^ j max(p;0) ; 12 else 13 ^ j j ; 14 end 15 ^ x j+1 f(^ x j ;(v j+1 ; ^ j )) ; 16 end F unction up dateP ath tak es a partial solution giv en b y ((x j ) 0jk ; ( j ) 0j<k ) , increases the decision at no de v i b y h , then up dates the rest of the partial solution, returning a new one. As describ ed in Section 5.5.2.2, the new partial solution follo ws the same path as the original one, and eac h decision with con trollable duration tak es the smallest v alue that will not decrease the arriv al time ( 0 ) at the follo wing no de. W e assume that v j 2N d can b e easily v erified. 243 F unction expand tak es a lab el’s information (partial solution, slac k v ariables and last daily/we ekly r est no de), along with the set of decisions to b e tested ( ^ U(~ x i ) as defined in Section 5.5.2.1) and generates new partial solutions/lab els, p erforming lab el impro v emen t when necessary . Note that, as describ ed in Section 5.5.2.1, if the curren t lab el do es not require a time duration decision, all decisions are tested, i.e. ^ U(~ x i ) =U(x i ) . Lines 1 - 6 expand lab els that do not require time duration decisions. W e assume that v i 62N d can b e easily v erified. Line 9 calculates b y ho w m uc h the departure time from the origin should b e dela y ed to arriv e at the next no de at the desired time without slac ks relativ e to the origin. Lines 10-16 generate an up dated partial solution with the new departure time. Lines 17 and 23 recalculate the duration of the decision required to reac h the next no de at the desired time accoun ting for the up dates p erformed. Lines 18-22 do the same as lines 9-16 but with regards to the last daily/we ekly r est no de instead of the origin no de. Lines 24-26 use the up dated path and decision to calculate the next state and generate the lab el for the expanded partial solution, whic h is then inserted in to OPEN . F or simplicit y , w e consider that an y necessary c hec ks, suc h as A* b ound conditions and dominance rules, are included in the pro cess of creating a lab el and inserting it in to OPEN . 244 F unction expand input : (x j ) 0ji ;( j ) 0j 0 then 11 (^ x j ) 0ji , ( ^ j ) 0j 0 then 21 (^ x j ) 0ji , ( ^ j ) 0j
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Creator
de Almeida Araujo Vital, Filipe
(author)
Core Title
Integration of truck scheduling and routing with parking availability
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2023-05
Publication Date
01/03/2022
Defense Date
12/03/2021
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
hours-of-service regulations,label-correcting algorithm,long-haul truck planning,OAI-PMH Harvest,parking constraints,resource constrained shortest path,truck driver scheduling
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Ioannou, Petros (
committee chair
), Dessouky, Maged (
committee member
), Nayyar, Ashutosh (
committee member
), Savla, Ketan (
committee member
)
Creator Email
2004vital@gmail.com,fvital@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC110455714
Unique identifier
UC110455714
Legacy Identifier
etd-deAlmeidaA-10328
Document Type
Dissertation
Format
application/pdf (imt)
Rights
de Almeida Araujo Vital, Filipe
Type
texts
Source
20220112-usctheses-batch-907
(batch),
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given.
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Repository Email
cisadmin@lib.usc.edu
Tags
hours-of-service regulations
label-correcting algorithm
long-haul truck planning
parking constraints
resource constrained shortest path
truck driver scheduling