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Hybrid scheduling methods for the general routing problem

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Hybrid scheduling methods for the general routing problem

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Content Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HYBRID SCHEDULING METHODS FOR THE GENERAL ROUTING PROBLEM by Majid Mohammad Aldaihani A Dissertation Presented to the FACULITY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment o f the Requirements for the Degree DOCTOR OF PHILOSOPHY (INDUSTRIAL AND SYSTEMS ENGINEERING) May 2002 Copyright 2002 M ajid M ohamm ad Aldaihani Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UM I Number: 3073738 Copyright 2002 by Aldaihani, Majid Mohammad All rights reserved. __ ___ __® UMI UMI Microform 3073738 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DEDICATION To my parents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS This research would not be at hand without the continuous encouragement, support, guidance and unlimited assistance o f my advisor, Professor Maged Dessouky. I would like to thank him for being more than an advisor for me during my 4-year study at USC. I have learned many valuable things from him, which are not less important than academic achievements. I like his way o f teaching students, managing people, organizing time and communicating with others. Professor Dessouky is an exceptional advisor, teacher, and friend. I would like also to thank the members o f the Guidance Committee, Professor Satish Bukkapatnam and Professor Richard McBride for their efforts and guidance. Their feedback was very helpful in enriching my research. Many thanks go to the members o f my qualifying committee, Professor Randolph Hall and Professor F. Stan Settles for their efforts. Special thanks go to Professor Hall for providing valuable comments and ideas during my qualifying exam, which have improved this research. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TA B L E O F CONTENTS D E D IC A T IO N .................................................................................................................................................. '' A C K N O W LE D G EM E N TS......................................................................................................................... iii LIST OF T A B L E S .......................................................................................................................................... viii LIST OF FIG URES......................................................................................................................................... ix A B ST R A C T ....................................................................................................................................................... xi Chapter 1. IN TR O D U C TIO N .................................................................................................................... I 1.1 Importance o f Routing...................................................................................... 1 1.2 Research M otivation ......................................................................................... 2 1.3 Hybrid D A R P ...................................................................................................... 4 1.4 Research Contribution....................................................................................... 6 1.5 Dissertation O rganization................................................................................. 9 2. LITERATURE R E V IE W ....................................................................................................... 10 2.1 Exact Algorithm s.............................................................. 12 2.2 Heuristic A lgorithm s....................................................... 15 2.2.1 Tabu Search H euristic................................................................... 21 2.3 Hybrid D ial-A -R ide Problem .......................................................................... 22 3. PROBLEM DESCRIPTION................................................................................................... 27 4 . A V T A D A TA A N A L Y SIS..................................................................................................... 31 4.1 Background......................... 31 4 .2 Statistical Data A n alysis.................................................................................... 34 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter Pil=c 4.2.1 M iles Traveled Per Request......................................................... 34 4 .2.2 Pick-up T im e ..................................................................................... 34 4.2.3 D rop-off T im e ........................................................................ 34 4.2.4 V ehicle Occupancy R a te.............................................................. 38 4.2.5 Num ber o f Passengers per R equest........................................... 38 4.2.6 Ridesharing......................................................................................... 40 4.2.7 M iles Traveled per D a y................ 41 4.2.8 Sum m ary............................................ 42 5. M ATHEM ATICAL F O R M U L A T IO N ................................................................................... 45 5.1 Notations and S e ts.............................................................................................. 45 5.2 Mathematical Form ulation............................................................................. 48 6. HEURISTIC A PPR O A C H ............................................................................................................ 52 6.1 N otations................................................................................................................ 55 6.2 Phase I o f the H euristic .............................................................. 56 6.3 Initial Results................................................................................................................. 58 6.4 Phase II o f the H euristic.................................................................................... 6 1 6.4.1 V eh icle S election ............................................................................... 64 6 .4 .1 .1 Illustration............................................................................... 66 6.5 Improvement Procedure............................................................................................ 66 7. COM PUTATIONAL EXPERIM ENTS..................................................................................... 70 7.1 Information A v a ila b le................................................................................................ 70 7 .2 Manipulating the D ata................................................................................................ 71 7.3 Performance M easures o f the H euristics............................................................. 72 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter Page 9.6 Computational Experiment............................................................................. 130 9.7 Sim ulation............................................................................................................ 133 9.7.1 Comparison Between Simulation and Analytical M o d e l..................................... 136 9 .7 .1.1 Analytical M odel Results...................................... 136 9.7.1.2 Simulation R esu lts................................................... 136 10. C O N C L U SIO N S ...................................................................................................................... 138 10.1 C ontribution....................... 138 10.2 Future Research ........................................................................................ 139 REFERENCES................................................................................................................................................. 142 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES T able Page 4.1 Summary o f the statistics............................................................................................................. 43 7.1 D aily FI and F2 valu es.............................................................................. 75 7.2 D aily vehicle distance and custom er time (capacity = 2 ) ......... 78 7.3 D aily vehicle distance and customer time (capacity = 3 ) ................................................ 79 7.4 Total vehicle distance and customer tim e.................................................... 80 7.5 Components o f custom er trip time for im provem ent h eu ristic..................................... 82 7.6 Components o f custom er trip time for improvement heuristic...................................... 83 7.7 Sensitivity analysis on number o f vehicles (capacity = 2 ) ............................................... 85 7.8 Sensitivity analysis on number o f vehicles (capacity = 3 )............................................... 87 8.1 Total vehicle distance and customer tim e............................................................................... 101 8.2 Daily vehicle distance and customer tim e .............................................................................. 102 8.3 C PU time (seconds) to implement the heuristics................................................................. 103 8.4 Components o f customer trip time for T A B U -S ........................... 104 8.5 Components o f customer trip time for T A B U -A .................................................. 105 8.6 D aily vehicle distance and customer tim e .............................................................................. 107 9.1 Num ber o f bus trips....................................................................................................................... 120 9.2 Length o f bus trips.......................................................................................................................... 121 9.3 Average bus trip .............................................................................................................................. 122 9.4 Parameter values.............................................................................................................................. 131 9.5 Total cost valu es.............................................................................................................................. 132 9.6 Summary o f the simulation results........................................................................................... 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure Page 2.1 Solution methods and problem types........................................................................................ 11 4.1 Snapshot o f A rcview for the A V TA d ata.............................................................................. 32 4 .2 Dial-A-Ride service area for A V T A ....................................................................................... 33 4.3 M iles traveled per request............................................................................................................ 35 4.4 Actual pick-up tim e..................................................................... 36 4 .5 Actual drop-off tim e ......................................................................................................... 37 4 .6 Occupancy rate-percentage tim e.............................................................................................. 39 4.7 Occupancy rate-percentage m ile s............................... ............................................................ 39 4.8 Number o f passengers per request ............................................................................ 40 4.9 Ridesharing........................................................................................................................................ 41 4.10 M iles traveled per day per driver............................................................................................... 42 6.1 Hybrid network................................................................................................................................. 54 6 .2 Number o f candidate requ ests.................................................................................................... 59 6.3 Number o f candidate p ath s.......................................................................................................... 60 6.4 Insertion heuristic flow chart....................................................................................................... 62 6.5 Vehicle selection flow chart........................................................................................................ 65 6.6 Improvement heuristic flow chart.............................................................................................. 68 7.1 Sensitivity analysis (vehicle distance)..................................................................................... 76 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 7.2 Sensitivity analysis (custom er tim e)........................................................................................ 76 8.1 Proposed m ethod ology................................................................................................................. 91 8.2 Illustration o f the Sn and En con cep t........................... 94 8.3 Illustration o f the saving concept...................... 98 9.1 Hybrid network (n = 3 ).................................................................................................................. 109 9.2 Expected vehicle travel distance................................................................................................ 115 9.3 Average distance dvi and dv2 .................................................................................................... 117 9.4 Hybrid network shapes according to n .................................................................... 118 9.5 db vs. n ............................................................................................................................................. 126 9.6 Total cost function vs. n ................................................................................................... 133 9.7 Sample o f the simulated data....................................................................................................... 134 X Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A BSTRA CT We study an extension to the general routing problem, which deals with integrating fixed route service with the general pick up and delivery problem to create a hybrid routing problem. The primary application for such a service is a dial-a-ride system used by transit agencies to transport disabled or elderly individuals. The main aim o f the integration is to have higher productivity and/or less cost while not significantly reducing the customer service level. Due to the combinatorial nature of the problem, we propose a heuristic approach that provides an approximate solution, which is computationally efficient for solving large sized problems. Our initial solution is first derived using an Insertion procedure. The Insertion procedure consists of two phases. In the first phase, all the candidate routes/paths that meet a certain criterion for each request are identified. In the second phase, a feasible path from the candidates' list that has the shortest on-demand vehicle distance is selected and inserted into the vehicle schedule. The solution of the Insertion procedure is fed into an Improvement procedure. In this procedure, we try to identify an alternative path for requests that have multiple hybrid paths that can satisfy the demand. The solution from the Improvement procedure is further improved by two system-wide strategies implemented by a Tabu Search technique, which are Re-sequencing (TABU-S) and Re-assigning (TABU-A). We tested our heuristics on actual data from a transit agency. The most effective heuristic in terms o f the trade-off between solution quality and xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. computation time was TABU-S. The results of the recommended methodology showed that shifting some o f the demand to a hybrid service route (18.6% o f the requests) reduces the on-demand vehicle distance by 16.6% and the overall custom er trip time by 8.7% over the manual schedule provided by the transit agency. However, for these customers who take the hybrid delivery method (18.6% o f the requests), their trip time will increase on average by 5.4%. In addition to developing algorithms to improve the scheduling o f a hybrid system, we also develop a model that aids decision-makers in designing a hybrid network. That is, we determine the optimal number o f zones in an area where each zone is served by a number o f on-demand vehicles, which transfer passengers to a fixed route line if the destination is in a different zone or to its final destination if it is within the same zone. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PTER 1 IN TRO D U C TIO N 1.1 Importance of Routing Routing problems have received an extensive amount o f research, from mathematicians, computer scientists and operations researchers, during the last five decades mainly because these problems are easy to describe, hard to solve and applicable to a wide range o f realistic systems. For example, airline scheduling, material handling, production scheduling, postal and delivery services, dial-a-ride services, and many more practical systems can be formulated as routing problems. Therefore, deciding an optimal methodology to solve routing problems can save a tremendous amount of money for many government, industrial and business sectors. The common objective o f routing problems is simply to determine the best sequence o f visiting points that minimizes either travel distance or time. However, the difficulty comes from the combinatorial nature of the problem and the various constraints that represent the different requirements o f the systems. The foundation o f most o f the routing problems is the traveling salesman problem (TSP). To describe the TSP, consider a network consisting o f a num ber of points (nodes). Each link (arc or edge) between any two nodes has a cost value. The objective is to find the least cost route that starts from the depot, visits each node exactly once and comes back to the same depot. I Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A generalization of the TSP is the vehicle routing problem (VRP). The objective of the VRP is to minimize the cost of constructing a set o f m routes that are done by a set of m vehicles with a limited capacity, to satisfy a demand of n customers (points) such that each customer (point) is visited exactly once and the capacity o f each vehicle is not violated. Therefore, the VRP is a TSP if there is only one vehicle, one depot, no vehicle capacity limit and no custom er demands. The pickup and delivery problem (PDP) is an extension of the VRP such that each customer has an origin and destination point and is restricted by the precedence and coupling constraints. The precedence constraint restricts the vehicle to visit the origin point before the destination point o f each custom er while the coupling constraint demands the two points of each custom er to be visited by the same vehicle. When the demand at the nodes o f the PDP represents people, the problem is referred to as the Dial-A-Ride problem (DARP). The focus o f this research is to develop effective approaches to solve the DARP by integrating two modes o f transportation, which are a curb-to-curb system and a fixed bus route system. 1.2 Research Motivation The growth o f personalized public transit and demand responsive transit (DRT) began in the late 1970’s and early 1980’s with large demonstration projects developed in Rochester, NY and Santa Clara County, C A among others. These 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. early systems failed to meet expectations due to low demand requests and deficiency in communication and computer technology to effectively manage such systems (Lave, Teal, and Piras, 1996). However, with the passage of the Americans with Disabilities Act (ADA), which requires that transit agencies provide paratransit or on demand service for the disabled, there has been renewed interest in demand responsive transit. The passage o f ADA has increased the obligations of the transit providers to adhere to strict service standards. In addition, the demand o f these types of transit services is likely to continue increasing rapidly (Levine, 1997). For example, as reported in the June 24, 1993 issue of the W all Street Journal, the paratransit market was a $500 million industry. Today, it is around a $ 1 billion industry. In Los Angeles County alone, more than 5,000 vans and 4,200 cabs provide service, generating 8 million trips per year. Many researchers emphasize the potential benefits o f deploying Intelligent Transportation Systems (ITS) in DRT systems. For example, Chira-Chavala (1999) and Chira-Chavala and V enter (1997) describe the benefits o f implementing ITS Technology at Santa Clara Valley Transportation Authority. They study the cost and productivity impact o f implementing an ITS based system. After implementing ITS, they show a 17% increase in ridesharing, 13% savings in transportation cost, and 28% reduction in total personnel salaries. Higgins, Laughlin and Turnbull (2000) evaluate the impact on the performance of implementing ITS at Houston METROLift paratransit service. By deploying 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Since most Dial-A-Ride programs tor the transport of elderly and disabled persons are heavily subsidized programs, the increased usage o f those curb-to-curb services has put significant budget pressures on most transit agencies. In fact this is such a major concern in Los Angeles County that the agency responsible for paratransit services, Access, is allowing all ADA eligible passengers to ride for free on their fixed route bus lines in order to shift some o f the passengers to this mode o f transit service. Although there is a significant body o f research on scheduling curb-to-curb systems, there has been limited work that attempts to integrate curb-to-curb services with the fixed route bus lines. We refer to a transit system that integrates these two modes of transportation as a “hybrid” service delivery method. The main aim of this research is to develop approaches to solve the DARP by integrating the curb-to-curb system and the fixed route system in order to have higher productivity and/or less cost while not significantly reducing the custom er service level. Clearly, a curb-to-curb system as opposed to a hybrid system minimizes the travel time for the passenger. However, shifting some o f the demand to fixed routes may alleviate some o f the demand pressure caused by ADA requirements. The fundamental question that will be addressed is whether transferring passengers to fixed routes will result in significant reduction in service level. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. heuristic algorithms is to provide near optimal or good solutions to large-sized problems in a reasonable amount o f computation time. Our initial solution is first derived using an Insertion procedure. The Insertion procedure consists o f two phases. In the first phase, all the candidate routes/paths that meet a certain criterion for each request are identified. In the second phase, a feasible path from the candidates’ list that has the shortest on- demand vehicle distance is selected and inserted into the vehicle schedule. The solution of the Insertion procedure is fed into an Improvement procedure. In this procedure, we try to identify an alternative path for requests that have multiple hybrid paths that can satisfy the demand. The solution from the Improvement procedure is further improved by two system-wide strategies implemented by a Tabu Search technique, which are Re-sequencing (TABU-S) and Re-assigning (TABU-A). The limited research in this area includes the work o f Liaw, White, and Bander (1996), and Hickman and Blume (2000). Both of the above approaches are based on developing insertion heuristics. Hickman and Blume (2000) use the insertion procedure developed by Jaw, Odoni, Psaraftis, and Wilson (1986) in their approach. Liaw, White, and Bander (1996) test their heuristic on a data set from Ann Arbor, Michigan while Hickman and Blum (2000) test their insertion heuristic on a data set from Houston, Texas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We build on this earlier work by expanding on the insertion heuristic approaches by adding an improvement as well as a Tabu Phase to the solution procedure. Tabu search is considered to be one of the most useful meta-heuristics for solving routing problems. For example, Rochat and Taillard (1995), Badeau, Buertin, Gendreau, Potvin, and Taillard (1997), and Taillard, Badeau, Gendreau, Guertin, and Potvin (1997) use a tabu search procedure for solving the vehicle routing problem, while Nanry and Barnes (2000) use it to solve the pickup and delivery problem. We test our methodology on data from an actual paratransit service provider. The selected agency is Antelope Valley Transit Authority (AVTA). This agency is selected since their service area is ideal for a hybrid system. For example, most ADA passengers travel to a central area where the hospitals and shopping malls are located. In addition, the distances traveled by many o f the passengers are large enough to justify a transfer to a fixed route bus line. Moreover, we present a network design analytical model that studies the hybrid problem from a strategic point o f view. The objective o f the model is to balance the trade o ff between the operating cost and the custom er service level by optimizing the number o f service zones and fixed bus lines in the system. We test our optimization procedure on randomly generated data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.5 Dissertation Organization The rest of this dissertation is organized as follows. An academic review of relevant literature is presented in chapter 2. A formal problem statement o f the research topic is given in chapter 3. The statistical analysis o f data provided by Antelope Valley Transit Authority is presented in chapter 4. The mathematical formulation is shown in chapter 5. The proposed heuristic approaches are described in chapter 6. Computational experiments are shown in chapter 7. The system-wide strategies and their related computational experiments are described in chapter 8. The network design analytical model is presented in chapter 9. We end the dissertation with the conclusions in chapter 10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PT E R 2 LIT ER A TU R E R EV IE W In this chapter, we review the literature on the pick up and delivery problem (PDP) and the Dial-A-Ride problem (DARP). Transporting objects from many origin points to many destination points is referred to as the pick up and delivery problem. When these objects represent people, the problem is referred to as the dial-a-ride problem. The algorithms in this review are divided into two main categories: exact algorithms and heuristic algorithms. Each category involves two types o f problems, which are the static problems and the dynamic ones. Also, in each o f these categories, we specify if the solution introduced for the problem is either single-vehicle or multi-vehicle and either with or without a time window. This classification is shown in Figure 2.1. Also, we pay more attention in our review to the tabu search technique when we discuss the heuristic algorithms because o f two reasons. First, it is one of the most useful meta-heuristics for solving routing problems and second it is implemented in this research. In addition to the above, we review the limited work on the hybrid delivery method and describe how this research is going to address the deficiencies o f the previous work and fill the gaps in this area. Exact algorithms guarantee to find the optimal solution, if it exists, but usually with a tremendous amount o f computational time; while heuristic algorithms find an effective or sometimes optimal solution with less amount o f 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. computational time. Another aspect of the DARP or PDP is the number of vehicles in service. Usually the multiple-vehicle problems are more difficult to deal with than the single-vehicle ones. Furthermore, DARP or PDP is referred to as a static problem when all the transportation requests are made in advance, e.g., one day before the service day. On the other hand, if the requests can be received while the service is going on, the problem is referred to as dynamic DARP. In some instances, when customers identify desired delivery times or pick up times, the problem is classified as DARP with time windows. Savelsberg and Sol (1995) provide an excellent review for the Pickup and Delivery Problem. Exact Heuristic f * ____ ~---------1 Static Dynam ic f — ----- --------- 1 f Single V ehicle M ultiple V ehicle 1 r * — ------ =rr, . With T im e W indow W ithout Tim e W indow Figure 2.1. Solution methods and problem types. There are three main broad categories o f the conventional DARP: many-to- many, many-to-few and many-to-one (Dial, 1995). M any-to-one (MTO) is the simplest type o f DARP in terms of finding a good solution. There is one common point in the system (e.g., medical center, shopping mall or any attraction center), 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which all passengers want to go to or come from. The passenger needs to identify only the other endpoint o f the trip. Although it is relatively easy to find an effective schedule, MTO restricts itself to less transportation demand. Many-to- few (MTF) extends the service for more than one common point. Therefore, more trips are made from origin points to those few attraction centers, which yields higher transportation demand and probably higher productivity. The most complicated category o f the DARP is the many-to-many (MTM). In this category, every passenger’s request has a different origin point and destination point. An example o f MTM service is taxis. The high fare o f taxis reflects the low productivity o f the many-to-many DARP. 2.1 Exact Algorithms Dynamic programming can be used to find an optimal solution o f the DARP. An exact algorithm based on dynamic programming is presented by Psaraftis (1980) to solve the static and dynamic versions of the single vehicle many-to-many DARP. The objective function is a linear combination o f the rider inconvenience and the total time traveled. The inconvenience cost is a linear function o f the waiting time and the riding time. This objective function yields to a limited search space. The algorithm checks the routes’ feasibility in terms o f a set o f four screening tests in a forward and backward operation. Then the request with the least objective function is selected. The computational effort o f the algorithm 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. grows exponentially in terms of the size of the problem. They were able to optimally solve problems with up to 10 requests. As an extension to his previous dynamic programming approach, Psaraftis (1983a) comes up with a modified version that considers pick up and delivery time windows. The computational effort o f the modified algorithm also grows exponentially with the size o f the problem. Kalantari, Hill, and Arora (1985) develop an algorithm to solve the single vehicle static PDP optimally. The algorithm is based on a branch and bound method and is capable o f solving the traveling salesman problem with pick up and delivery points (e.g., single PDP). The main idea of the algorithm is to remove in each branch all the arcs that violate the precedence constraint o f the pick up and delivery points. A problem consisting o f 18 requests has been solved using this exact algorithm. Desrosiers, Dumas, and Souman (1986) present an exact algorithm based on dynamic programming. The algorithm solves the static single-vehicle DARP with time windows. The main idea is to find the best routes based on the state (S,i). This state is basically for checking the availability of a feasible route that visits all the nodes in S and stops in node i, where S is a subset o f the total node set. This method can reduce the search space. A problem consisting o f 40 requests has been tested using this approach. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dumas. Desrosiers. and Soumis (1991) develop an approach that can solve optimally the multiple vehicle advanced request PDP with time windows and can handle problems with different objective functions, multiple depot, and heterogeneous fleet o f vehicles. They formulate the problem as follows: min ^ c rxr re R subject to ^ c i irx r = 1 re/? J \ v r = \ M \ re R X r € {0,1} Where, xr is a binary variable to check whether the rlh route is implemented or not. cr is the cost o f implementing the rth route. air is a binary variable to check whether request i is served using the rlh route or not. Enumerating all the possible routes is not computationally possible; therefore, they use a column generation method to handle this difficulty. A problem consisting of 55 requests and 22 vehicles has been solved using the proposed approach. Ruland and Rodin (1997) formulate the single-vehicle PDP as an integer programming model. The main idea o f the formulation is to identify all the feasible 14 V/e N Vie N Vre R Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rouies. They also develop a branch-and-cui algorithm 1 0 optimally solve the problem. 2.2 Heuristic Algorithms The real world size problems o f DARP and PDP cannot be solved by exact methods because o f their large size and their inherent computational difficulty to solve optimally. Therefore, most researchers develop heuristic algorithms and approximation methods to handle this problem and find good solutions within reasonable processing time. The following section describes some o f the important work that has been done for solving the DARP and PDP using heuristic algorithms. Stein (1978) presents a heuristic algorithm to solve the single-vehicle DARP without capacity constraints. He suggests dividing the service area into k sub-areas. The number o f vehicles in the system (m) is greater than the sub-areas (k). Each set o f these sub-areas is served by one vehicle. The arrival time o f the customers’ requests are assumed to be random variables following a Poisson distribution. Every time period, each vehicle performs two TSP tours in its assigned set o f sub-areas. The first one is to pick up the customers and the second is to drop them off. In these two TSP tours, customers may be transferred from one vehicle to another if his/her origin and destination are not located in the same set o f subareas. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another heuristic algorithm to solve the single-vehicle many-to-many DARP is proposed by Psaraftis (1983b). The algorithm recommends that one TSP tour can include both pick up and delivery stops while making sure that for each request pick-up is visited before delivery. The initial TSP is made by generating a Minimum Spanning Tree (MST) for these pick up and delivery points. The overall DARP is then improved by using one o f the fc-opt algorithms. The time complexity o f Psaraftis’s algorithm is quadratic and for problems o f size 50 customers or less, the solutions are less than 15% worse than the optimal solution. The algorithm worst case performance is three times above the optimum. Psaraftis showed that his algorithm generates a better solution than Stein’s approach for problems o f the above size. Haines and W olff (1982) present a dial-a-ride system that is used for real time vehicle scheduling. The objective function is a linear combination of the riders’ inconvenience and the vehicle occupancy. The rider’s inconvenience includes the waiting time, total time in system, service time, and the deviations from the requested pick up and drop off times. A schedule of every vehicle is given at each time period. The system evaluates the insertion o f any new custom er’s request in each o f the given temporary vehicle schedules and then chooses the one with the least objective function cost. Fiala and Pulleyblank (1990) introduce a heuristic algorithm to solve the static single-vehicle PDP without time windows. Although their application was 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. helicopter scheduling, it is basically a PDP. The heuristic, which is based on insertion improvement methods, finds a solution for a problem that has sets of customers with different priorities. Daganzo (1984) makes a comparison between three kinds o f transportation system configurations: curb-to-curb system, fixed bus route system and hybrid curb-to-curb and fixed bus route system. The comparison is based on an objective function o f the custom ers’ inconvenience and the actual operating cost. He found that the fixed bus route system is the best in high demand environments while the curb-to-curb system is the best in low demand environments. Sexton and Bodin (1986a; 1986b) introduce a heuristic to solve the static single-vehicle DARP with time windows. The objective is to minimize the deviation from the desired delivery time and the excess ride time. Their algorithm is based on a nonlinear integer programming model. A problem consisting of 20 requests was tested using this approach. The solution was better than the practical solutions. Bruggen, Lenstra and Schuur (1990) present a heuristic for the static single vehicle PDP with time windows. The algorithm has two phases: a construction and an improvement. In Phase I, an objective function that measures the infeasibility is used to find an initial feasible solution. In phase II, another objective function is used to find better solutions. Problems consisting of up to 50 requests were tested using this approach. The solutions fall within only 1% o f the optimal solution. 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Cullen. Jarvis, and Ratliff (1981) introduce a heuristic algorithm for the static multiple-vehicle DARP. The main idea o f this algorithm is to create and select clusters to meet the demand o f the entire requests and link these clusters to form a PDP. They formulate the clustering problem as follows: min X C/-vy jeJ Subject to = 1 Vi e N i*j y>e{0,l} y / 6 J W here J is the set o f all possible clusters, Cj is the cost o f cluster j, y ,- is a binary variable that denotes whether cluster j is selected or not, a^ is a binary variable that denotes whether request i is assigned to cluster j or not. Problems consisting of up to 100 requests show the effectiveness of the algorithm. Jaw and others (1986) propose a heuristic algorithm for the static multiple- vehicle many-to-many DARP with time windows. This algorithm, which treats one request at a time, is based on two main steps. First, the algorithm searches all the feasible insertions o f a request into the existing vehicle schedules. Second, all the feasible insertions found in the first phase are evaluated in terms o f the objective function, and the best one is chosen. The objective function is a linear combination o f the rider inconvenience and the consumption o f the vehicle resources. The proposed algorithm outperformed a clustering and routing algorithm that was also introduced by the same authors. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dumas. Desrosiers, and Sournis (1989) propose a heuristic algorithm to solve the multiple-vehicle advanced request DARP with time windows. The algorithm is based on the idea of mini-clustering. A mini-cluster is a small group o f requests that can be served by one vehicle and is treated as a one request. The vehicle is empty before and after serving the mini-cluster. Another mini-cluster approach is presented by Ioachim, Desrosiers, Dumas, Solomon, and Villeneuve (1995). After creating the mini-clusters, they solve them optimally using an optimization algorithm. A problem consisting of 190 requests and 31 vehicles was solved using this algorithm. An insertion heuristic algorithm for the DARP with multiple objectives is given by Madsen, Ravn, and Rygaad (1995). The algorithm is developed for a demand responsive transportation system for elderly and people with disabilities. They ranked the requests in terms o f their degree o f difficulty. The more special needs the customer requires the higher degree of difficulty the request is. The insertion priority is always for the most difficult request. The system evaluates the insertion of any new custom er request in each o f the existing vehicle schedules and then chooses the one with the least objective function cost. This method is time consuming; therefore they add two hard constraints per request. These constraints are for time window and m axim um traveling time. Adding these constraints makes the time o f insertion o f a new request as fast as only one second. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ioachim and others (1995) develop an approximation method to mini clustering for solving a multi-vehicle pick-up and delivery problem with time windows by column generation. They define the mini-cluster as “a feasible segment of an itinerary where the vehicle is never empty between the first-pick-up point and the last drop off point; feasibility is imposed with respect to the time window, capacity, priority and coupling constraints.” Moreover, they develop a parallel insertion heuristic algorithm for mini-clustering. In this algorithm, the mini-cluster is initiated with the longest processing time request and then other requests are added. They compared the two approaches and found the optimized- based approach is better. Also, for a problem o f more than 2500 requests, the first approach improved the total traveling time by 5.9% and reduced the fleet size by 9.7%. Toth and Vigo (1997) develop constructive and improving algorithms for the general PDP with time windows. A parallel insertion heuristic (constructive) algorithm is first implemented to find an initial solution. This is done by performing two main steps. First, the minimum number of routes needed to serve a number o f trips based only on the vehicle capacity is estimated. Second, the unrouted trips are inserted into the existing routes if they are feasible. In case o f unfeasibility, a new route needs to be created. Then, the solution found from the constructive algorithm is improved by using a Tabu Thresholding procedure. They 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. show ed that the schedule found by the proposed approach is better than the hand m ade schedules for a data set from the city o f Bologna. 2.2.1 Tabu Search Heuristic Tabu search is considered to be the most successful metaheuristic applied to solve the routing problem with time windows. The basic TS algorithm moves at each iteration from a solution to its neighborhood until an end criterion is fulfilled. To avoid cycling, short-term memory forbids to revisit solutions that are recently visited. Long-term memory is implemented to discourage frequently made moves and to encourage the search process to move to undiscovered regions o f the solution space and look for a better solution. A parallel TS to solve the multiple vehicle routing problem with hard time windows has been used by Rochat and Taillard (1995). They employed the principle o f probabilistic tabu search to direct he neighborhood selection in order to overcome the local optimum trap. They used a master-slave like approach to parallelize some of the most time consuming steps in their algorithm. Another parallel tabu search heuristic is introduced by Badeau and others (1997) to solve the multiple vehicle routing problem with hard time windows. Their implementation showed that parallelization o f the original sequential algorithm can substantially improve the solution quality when the time available for com puting a solution is limited in practice. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Taillard and others (1997) describes a tabu search heuristic for the multiple vehicle routing problem with soft time windows. The neighborhood o f the current solution is created through an exchange procedure, which is extended from the general edge exchange heuristic k-opt. The new starting solutions for the tabu search are produced through a combination o f routes taken from an adaptive memory, which is a pool containing the best solutions visited by the search. Nanry and Braines (2000) solved the multiple pickup and delivery problem with hard time windows using reactive tabu search. The main difference between the reactive tabu search and the classical tabu search is that the parameters are adjusted dynamically according to the assessment o f the exploration quality. Three distinct move neighborhoods are employed in their approach to capitalize on the dominance o f the pairing and priority constraints. These moves are re-sequencing requests within the routes, re-assigning by insertion method and re-assigning by swapping method. After the initial solution is constructed, these neighborhoods are examined under a hierarchical multi-neighborhood search method, and the best nontabu move is selected. 2.3 Hybrid Dial-A-Ride problem Liaw and others (1996) introduce an algorithm to solve the hybrid DARP with the two modes o f transportation: fixed route buses and the paratransit vehicles. The main idea o f their approach is to have two systems for scheduling the 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transportation requests: on-line and off-line systems. In the on-line, they insert the requests, if they are feasible, in the order they receive them in the existing vehicle schedules. If a request is not feasible, they put it in the stand by request list. Therefore, an initial feasible schedule is created from performing the on-line system. In the off-line when no more requests are arriving, iterative improvements are performed to insert the stand by requests. After finding the final schedule, the customers on the stand by list and the customers with corrected schedules are called and informed with the final schedule. The fixed bus route is chosen if the distance from the origin point to the entry bus stop plus the distance from the exit bus stop to the destination point is less than a certain percentage (e.g., 50% or 70%) o f the direct distance from the origin to destination. They tested the algorithm on simulated and real data. The results show that the off-line system is effective and is capable o f increasing the number o f requests served. Our work will contrast their work in several aspects: 1. Mathematical Formulation. Liaw and others (1996) come up with a nonlinear formulation to the problem that cannot be solved using a standard integer program solver. The objective o f the formulation is to give a mathematical description o f the problem. However, our formulation can be easily adjusted to a mixed integer linear formulation, which can be solved by an integer program solver on small-sized problems. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2. Selecting the Fixed Bus Route. The fixed bus route is selected based on minimizing the distance traveled by paratransit vehicles. The riders’ convenience is not considered in this criterion. In some instances, the passenger might be scheduled on a very long fixed bus route. We treat this issue in the second criterion o f phase I o f the heuristic. In our approach, in addition to minimizing the distance traveled by paratransit vehicles, we restrict the fixed bus route distance to be less than or equal to the distance traveled by the paratransit vehicle, which reduces the riders’ inconvenience. 3. Insertion Procedure. In their In their approach, if the request is a candidate for the hybrid system, the hybrid path (the entry and exit points of the selected fixed bus route) that minimizes the partransit vehicle distance is chosen and inserted in an existing schedule. If the insertion is not feasible, another hybrid path is selected. Using the above method may not be the most effective way to select the hybrid path. This is because the hybrid path that minimizes the vehicle’s distance does not consequently give the best final schedule o f the vehicle. Therefore, we build on this earlier work by expanding heuristic approaches by adding an improvement as well as a Tabu Phase to the solution procedure. Specifically, in the improvement phase, we first find all the possible fixed bus routes for every request and all the candidate paths on these fixed routes for every request. After that, each candidate path is evaluated separately and the one with the best insertion, which leads to the best schedule for the vehicle, is selected. Also, in 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the system-wide procedure (Tabu Phase), we consider the successor trips in creating the on-demand vehicle schedule. Hickman and Blume (2000) introduce an approach that integrates the demand-responsive service with the fixed route service. They consider both the operating cost and rider’s service level in their model. The main objective o f their approach is to schedule both the passenger trips and vehicle trips such that the total operating cost is minimized. A request is eligible for the integrated system if it passes the following screening procedures: 1. The direct distance between the origin and destination exceeds a minimum distance. 2. Common fixed route serves both origin and destination points. 3. The time on shortest path is acceptable. The paratransit vehicle routing and scheduling is done using the insertion heuristic proposed by Jaw and others (1986) and Wilson and W eissberg (1976). The proposed approach is implemented using the existing transit service in Houston, Texas. They found that the proposed approach reduced the daily operating cost by 15% by transferring 26% of the trips to the fixed route service. In considering the last two researchers that deal with integrating the fixed route with demand responsive service, what is still lacking is: I. A mathematical formulation of the problem that can be linearized to determine optimal solutions for small sized problems. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R 3 PR O B L E M D ESC R IPTIO N The problem under study in this research is the advanced request many-to- many multiple-vehicles Dial-A-Ride problem with hard time windows for hybrid systems. As the literature survey shows there has been very little work in scheduling o f hybrid systems beside that of Liaw and others (1996) and Hickman and Blume (2000). Our research will contrast with their work in several aspects including the mathematical formulation, selecting the fixed route, insertion procedure, improvement procedure and system-wide strategies as well as development a network design analytical model (see the literature review chapter for more details). We next present a formal statement o f the studied problem. In a certain workday, there are a number o f requests o f a set o f A f customers who need to be picked up from origin points and dropped o ff at destination points. Every request has a desired pick up time or drop off time. Clearly, for each rider, the pick up stop must occur before the drop off stop. In this problem, we have two modes o f transportation: (a) paratransit curb- to-curb system and (b) fixed bus route system. The main idea is to integrate these two modes in order to have higher productivity and/or less cost while not significantly reducing the custom er service level. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The paratransit curb-to-curb system includes a set o [ ' M paratransit vehicles, with a known capacity, that are used to pick up the passengers from their origins or from the bus stops and drop them off at their final destinations or at the bus stops. All the paratransit vehicles dispatch and return to their depot at the end of the workday. On the other hand, the fixed bus route system includes a set o f R fixed bus routes (lines). Every fixed bus route has a number o f buses that travel through it, a set of bus stops and a time schedule. The capacity o f the buses is relatively large and is not considered a constraint in this problem. We assume that at most two transfers can be made for any custom er in the system and no transfers can be made between the same mode. To clarify the transferring matter, here is an example o f performing a request of a custom er who wants to travel from an origin point to a final destination point. The following are five scenarios that can satisfy the request: Scenario 1. Paratransit vehicle V picks up the custom er from his/her origin and drops him/her off at his/her final destination. Scenario 2. Bus X picks up the customer from his/her origin point (Bus stop B I) and drops him/her off at his/her final destination (bus stop B2). Bus stops B i and B2 represent the custom er’s origin and destination points respectively. Scenario 3. Paratransit vehicle VI picks up the custom er from his/her origin and drops him/her o ff at bus stop B 1 to wait for bus X. Bus X picks up the 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. custom er and travels to bus stop B2 where paratransit vehicle V2 is waiting to pick up the customer from bus stop B2 and drop him/her off at his/her final destination. Scenario 4 . Paratransit vehicle VI picks up the customer from his/her origin and drops him/her off at bus stop BI to wait for bus X. Bus X picks up the customer and travels to bus stop B2. Bus stop B2 represents the custom er’s final destination. Scenario 5. Bus X picks up the customer from bus stop B 1 and travels to bus stop B2 where paratransit vehicle V2 is waiting to pick up the custom er from bus stop B2 and drop him/her off at his/her final destination. Bus stop B1 represents the customer’s origin point. Another important assumption in this problem is that all requests are made in advance. For example, they are made one day before the service day. This assumption makes the problem a static DARP. To evaluate the efficiency o f the hybrid delivery method, we will compare the strictly curb-to-curb system with the hybrid curb-to-curb and fixed route system. These two systems will be tested using analyzed data provided by Antelope Valley Transit Authority. The performance metrics that will be studied include: I. Vehicle productivity. This metric is often measured as the number o f riders per trip. By decreasing the on-demand vehicle miles (increasing productivity), transit service providers can decrease their direct costs (e.g., 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. gasoline, labor) and capital costs (e.g., fleet size). In Los Angeles County, the average number of riders per trip is around l.l. Ridesharing in demand responsive transit is low because schedulers mostly dispatch vehicles in a manner similar to taxis. Access Services Inc. (ASl), the agency responsible for coordinating paratransit service within Los Angles County, believes that through improved vehicle scheduling, there is a potential in Los Angeles County to increase this metric to at least 1.8. 2. Passenger travel tim e. Clearly, curb-to-curb systems minimize travel time over a hybrid system. However, it may be possible to shift some o f the demand to accessible fixed route lines without significantly increasing passenger travel times, thereby alleviating some o f the high demand pressure generated by the ADA requests. In our heuristic approach, we try to balance the trade off between the above two performance measures since they represent the operating cost and the customer service level. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PT E R 4 A VTA DATA ANALYSIS 4.1 Background Antelope Valley Transit Authority (AVTA) was selected based on suggestions from Access Services personnel, the agency responsible for coordinating paratransit service within the county, who felt that this region was where it might beneficial to use a hybrid fixed route and curb-to-curb system. The travel distances in Antelope Valley are large enough to justify a transfer point between the two different types of transit services. Furthermore, most o f the disabled and elderly passengers travel to a central location where most o f the hospitals are located. This is shown in Figure 4.1. The area around Palmdale Street (Lower circle) is mostly a residential area while the area around Avenue K (Upper circle) is mostly hospitals and a commercial district. Furthermore, there are fixed bus routes that connect these two areas, which are also shown in the figure by dark lines. Finally, AVTA is a small to mid-size agency so that there is opportunity for effective communication between their fixed route and paratransit services. AVTA provided data o f their operations including pickup time, travel distances, fleet size, etc. This data will be used to form statistical distributions of representative paratransit operations. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Sample data ArcView GIS 3.2 Antelope Valley Transit Authority Feb. 2000 AvenueJG - “'■ j Avenue i Avenue_K AvenueM -jr'nrrr T ii Balmdaile wHIIhs3 '' i n •^A % ejiu£oL i ' x r -t- v - • . -m - ~ "y- . ; /-• • • -: ^ _ « * * . w i .f r " ! E S r £ 3 T T § Figure 4.1. Snapshot o f Arcview for the AVTA data. Antelope Valley Transit Authority provides fixed route and paratransit services in Lancaster County. The total service area is divided in three parts, Urban Zone, Rural Zone 1 and Rural Zone 2 as shown in Figure 4.2. We received data in the hard format for the period of two weeks, February 14, 2000 to February 25, 2000. We considered weekdays. Thus, we received nine days o f data. AVTA operates from 6:00 a.m. to 7:30 p.m., during the weekdays. The total number o f requests served during these days was 1,242. The number o f drivers per day varied from 4 to 12 during this period (Average = 9.22). The average number o f requests served per day per driver was 15.79 with the maximum 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. num ber being 30.33 on any given day. The data provides the following details for each requested trip: 1. The unique identification number. 2. Number of attendants and companions. 3. Pick-up/drop-off destinations. 4. Requested pick-up/drop-off time window. 5. Actual pick-up/drop-off time. 6. Cumulative miles. Los Angeles County Line RURAL ZONE 2 RURAL ZONE 1 RURAL ZONE 2 Lancaster URBAN ZONE Palmdale Angeles National Forest Boundary Littlerock Pearblosso n Acton Angeles 1 National Forest Boundary Figure 4.2. Dial-A-Ride service area for AVTA. We entered this data in a spreadsheet and the distributions for vehicle travel time, pick-up time, travel distance and other important results were tabulated. 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Statistical D ata A nalysis The collected data is analyzed to determine the travel patterns of a representative DRT operation. Below we present some o f the important statistics. 4.2.1 Miles Traveled Per Request Figure 4.3 shows a histogram of the distance traveled in miles. As the figure shows, the most common request is within the range from 6 to 10 miles. However, there still is a significant amount o f passengers traveling greater than 10 miles which is significant enough o f a distance to justify a transfer to an accessible fixed route bus. 4.2.2 Pick-up Time Figure 4.4 shows the frequency o f the actual pick-up times. Most o f the requests were during late morning and early afternoon. The number o f requests after 7:00 p.m. is very low since the working hours for AVTA is from 6:00 a.m. to 7:30 p.m. Almost in all the cases, the demand was satisfied within the requested time window. W e note that the peak of the pick-up time distribution matches closely with the high frequency periods of the fixed routes. 4.2.3 Drop>off Time Figure 4.5 shows the frequency o f the actual drop-off times. There is no drop-off tim e in the first time bin from 6:00 a.m. to 6:30 a.m. because it is the start 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the clay. Although AVTA closes at 7:30 p.m., there are some drop-offs after this time since the request was accepted during the working day, and AVTA is responsible for dropping off these requests even after working hours. As can be seen, there is no significant difference between the shape o f the distribution o f drop-off and pick-up times. The drop-off times are simply shifted to the right o f the pick-up times. 300 250 200 150 50 • M ies Traveled Per Request Figure 4.3 Miles traveled per request. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pickup Time Figure 4.4. Actual pick-up time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Dropo tT Time Figure 4.5. Actual drop-off time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2.4. Vehicle O ccupancy Rale Figure 4.6 shows the fraction of the total travel time during which the vehicle was occupied by a request. When occupied, there may be more than one passenger in the vehicle. As is shown in the figure, more than half of the time the vehicle is empty implying possibly that the vehicle travels great distances to pickup passengers. Improved coordination between pickup locations in the schedule could improve this performance measure. Figure 4.7 shows the results based on miles instead o f time. Note there is not much difference in the figures. 4.2.5 Number of Passengers Per Request Figure 4.8 provides details on the number o f passengers picked up per request. In some cases, elderly, disabled and patients need one more person as their attendant to assist them. Also a companion can travel with the passenger. As shown in the graph, almost in 90% o f the cases, there is only one passenger associated with each request. In some cases, there are two passengers associated with each request and in rare cases there are three. These results are useful for determining vehicle capacity requirements. 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44.69% Empty Miles Miles traveled with at least one passenegr in car 55.31% Figure 4.6. Occupancy rate— percentage time. Tirrt traveled with at least one passenger in car 44.95% 55.05% Enpty time Figure 4.7. Occupancy rate— percentage miles. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.8. Number of passengers per request. 4.2.6 Ridesharing Figure 4.9 shows the fraction o f time in which there was more than one request in the vehicle. As shown in the figures, about one third o f the time there were more than one request in the vehicle. We differentiate between number of requests and num ber o f passengers since each request can be associated with more than one passenger. W e consider the case only when there are multiple requests currently being served by the vehicle as ridesharing. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tim e w ith no passenger or one request in vehicle □ 68% Tim e w ith more than one request in vehicle (Ride Sharing) Figure 4.9. Ridesharing. 4.2.7 Miles Traveled Per Day Figure 4.10 shows the distribution o f the number o f miles traveled by a driver per day. We can see from the graph that the maximum that was driven by a driver was over 180 miles. The average number o f miles driven by a driver is around 165 miles. Note that a hybrid system has the potential to reduce these averages. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Miles Per D.iy Per Driver Figure 4.10. Miles traveled per day per driver. 4.2.8 Summary After presenting and discussing each aspect o f the data separately, Table 4.1 summarizes the results o f the statistical analysis. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1. Summary of the statistics 1 Travel Distance Per Request in M iles Average 12.75 Standard Deviation 13.00 Median 10.00 Mode 4 .00 Maximum 163.00 Minimum 0 .00 2 Pickup Time Average 1 1.87 Standard Deviation 3.41 Median 11.58 Mode 9.55 Minimum 5.48 3 Drop-off Time Average 11.89 Standard Deviation 3 .4 1 Median 11.58 M ode 9.55 Maximum 20.07 Minimum 5.48 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tabic 4 . 1 (continued) 4 Total number o f requests served 1242.00 5 Total number o f passengers transported 1872.00 6 Number o f passengers per request (%) One passenger 9 1.03 T w o passenger 8.88 Three passenger 0.09 7 Percentage o f ridesharing 3 1.78 8 Percentage o f time traveled with passenger 55.05 9 Percentage o f time traveled em pty 44.95 10 Percentage o f distance traveled with passenger 5 5 .3 1 11 Percentage o f distance traveled em pty 44.69 12 Average number o f requests served per day per driver 15.79 13 M aximum number o f requests served per day per driver 30.33 14 Average distance traveled per day 1132.69 15 Distance traveled per driver per day Average 165.45 Standard Deviation 49.06 Median 174.00 M ode 166.00 Maximum 262.00 Minimum 53.00 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 MATHEMATICAL FORMULATION For the purpose of describing and understanding the hybrid problem, we present in this chapter a complete mathematical formulation, which can be used to solve small sized problems. 5.1 Notations and Sets Let set ne N represent the customers in the network. Let pn + be the index for the pickup point and pn' be the index for the delivery point for customer request n. Each request has a load of qn, which has a positive value at pn+ and a negative value at pn\ The set V+ represents the indices o f all the origin points and the set V' represents the indices o f all the delivery points. The set V (V+u V ‘) represents the pickup and delivery points o f all the requests. Associated with each request n is a time window [LTn,UTn], that specifies the range o f feasible pickup times. The set k e M represents the on-demand vehicles used in the network. Each vehicle has a fixed capacity of Q. Let the index for each vehicle’s start location be k+ and end location be k \ Let set W contain the value o f all the vehicle start and end location indices. The set re R represents the fixed bus routes in the network. Each route r has a set o f bus stops Br. The set B ( u re r Br) represents all the bus stops in the fixed 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bus routes. Each bus stop be B has a set of arriving buses Ob- Erich bus oeO b has a scheduled arrival time at bus stop b, TBSbo- Also, let S b,n be the travel time between any two bus stops on a given route r where (b,m )eB r. Each index in sets V and W is a separate node defined in our network. We also define a separate node in our network for each pair (n,b), n e N and b eB , to represent a vehicle movement to a fixed bus stop to either pickup or drop-off a hybrid request. Thus, a new set G is defined with each index in G representing a unique (n,b) combination. Let Gb be the indices in G that represent stop B and G„ be the indices in G that represent stop n. Finally, let G„b :=GnnG b. Note that IGnbl=L For all the nodes in the network (i,je V u W u G ), let tjj denote the travel time from i to j and cy denote the cost o f traveling from i to j. Note that the indices for the set V range from 1 IVI, the indices for the set W range from IVI+l,...,IVI+IWI, and the indices for the set G range from IVI+IWI+1,..., IVI+IWI+IGI. The following are the decision variables: Z„k A binary variable, which equals I if request n is a door-to- door request that is assigned to vehicle k and equals 0 otherwise. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SNnb" sxn b k BSn b o As W TCnb Yi ASXnb A binary variable, which equals 1 if request n is a hybrid request that is assigned to vehicle k with bus stop b as an entry bus stop and equals 0 otherwise. A binary variable, which equals I if request n is a hybrid request that is assigned to vehicle k with bus stop b as an exit bus stop and equals 0 otherwise. A binary variable, which equals I if the o‘ h bus at bus stop b is used by custom er n and equals 0 otherwise. A binary variable, which equals 1 if vehicle k travels from i to j and equals 0 otherwise. Vehicle arrival time at node i. The waiting time o f customer n at bus stop b Load o f the vehicle arriving at node i. The arrival time o f customer N at bus stop b. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 M athem atical Form ulation Below we show the mathematical formulation o f the problem: min / ( * ) = £ X X cuX ij keMieVUWUG jeVUWUG subject to 2I Z . ‘+ I I + I I = 2 v " € N ke M keM baB ke M he B I I = X I Vne N. r e R ATeA/ beBr KeM beB, X ^ « /> + S ^ < I V b e B ,n e N k e \ t keM X * i P = I X / ' / = Z « + V "6 ^ **€ " ieVU**UC ie VUG X*‘,= X V/ie N, p = p~.be B.ke M 16VUG I'eVUA'UG X *fs= X X k.=SNkb + SXkb V b e B ,n e N .g e G nh,k e M feVU**UG /el'LI* “UG I /el'*U G V k e M X ^ .* * -= l Vice M fe V U G Ar =0 Vice M A . > A . V n e N Pn P. (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. AjZlAi+t^Xfj LTi <A,. <UTi a k > a p Vi, j e V \JW \jG.ke M (5.11) (5.12) X ^ i , teW ASX,,, =(Ag +WTCn h + *„,,)( J^SN k nh £ S X ‘ m \k e A t keAl , Vbe B,ne N .p = p+ ,geG nh (5.13) Vne N,be B.me B ,g e G nh (5.14) Ag > ASX nb r \ £C V * keM n b a „ > a k WTC,lh = ,sxk .„ k e \t X TBSh l) BSn h ll -A , < *oh ,s n :> ! I B S n h o = £SAf‘, ueO h K r = 0 ^<(2 ieAf . JteAf Vfce B.ne N ,g e G t nh (5.15) Vbe B.ne N ,p = pn, g e G nh (5.16) Vfce B.ne N .g e G nb V b e B .n e N V keM Vie VUG (5.17) (5.18) (5.19) (5.20) V i ,je V \J W \jG ,k e M (5.21) Vne N.be B, p = p ~ ,g e G nh (5.22) Vne N,be B,p = p*,ge Gnb (5.23) 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Z* e {0,1} Vne N .k e M (5.24) S/V‘ „e{0,l} Vne N.be B .keM (5.25) SX*he{0,\\ Vne N.be B.ke M (5.26) BSnhne{ 0,1} Vne N.be B.oeOh (5.27) * * .e (0 ,l} V/, j e V \jW \jG .ke M (5.28) Aj >0 V /e V 'U 'f U C (5.29) WTC th >0 V ne N.be B (5.30) Yi >0 V ieV U W 'U C (5.31) ASX„b ^ 0 V ne N.be B (5.32) Constraint (5.1) determines whether the request is a door-to-door o r hybrid request and assigns only one vehicle if it is a door-to-door request and two vehicles (and two bus stops) if it is a hybrid request. Constraint (5.2) ensures that the entry bus stop and the exit bus stop of every hybrid request are located on the same fixed bus line (route). The above two constraints enforce the fact that every hybrid request has exactly one entry bus stop and one exit bus stop. Constraint (5.3) ensures that the same bus stop cannot be both an entry and an exit bus stop for a customer. Constraints (5.4) and (5.5) ensure that a vehicle visits an origin or destination point if and only if the request is a door-to-door, first leg o f a hybrid 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. request, or second leg of a hybrid request. Constraint (5.6) ensures that a vehicle visits a bus stop (to pickup or deliver a hybrid request) if and only if the request is assigned to this particular vehicle and bus stop. Constraints (5.7) and (5.8) make sure that every vehicle starts and ends its route from its depot. Constraints (5.9)-(5.18) form the precedence constraints. Specifically, constraint (5.9) sets the vehicle arrival time at the starting location ’’depot” to a zero value. Constraint (5.10) ensures that the vehicle arrival time at the delivery point of every request must be greater than or equal to the vehicle arrival time at the pickup point o f the same request. Constraint (5.11) determines the vehicle arrival time at each point. Constraint (5.12) ensures that vehicle arrival time at the pickup point o f every request falls within the time window. Constraint (5.13) makes sure that for every hybrid request, the vehicle arrival time at the entry bus stop must be greater than or equal to the vehicle arrival time at the pickup point. Constraint (5.14) determines the customer arrival time at the exit bus stop. Constraint (5.15) ensures that the on-demand vehicle arrival time at the exit bus stop must be greater than the custom er arrival time at the exit bus stop. Constraint (5.16) makes sure that for every hybrid request, the vehicle arrival time at the final destination must be greater than the vehicle arrival time at the exit bus stop. Constraint (5.17) determines the custom er waiting time at the entry bus stop. Constraint (5.18) ensures that the hybrid customer will use only one bus at the entry bus stop. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The constraints from (5.19) through (5.23) form the capacity constraints. Constraint (5.19) initiates the vehicle load at the start location with a zero value. Constraint (5.20) makes sure that the vehicle load at every node does not exceed the vehicle capacity. Constraint (5.21) determines the load o f the vehicle at every node the vehicle visits. Constraint (5.22) sets the load o f the entry bus stop to be same as the load o f the delivery point o f the request assigned to that bus stop. Constraint (5.23) sets the load o f the exit bus stop to be same as the load of the origin point of the request assigned to that bus stop. Constraints from (5.24) to (5.28) are for the binary variables while the constraints from (5.29) to (5.32) are the non-negativity constraints. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 HEURISTIC APPROACH Due to the combinatorial nature o f the problem, it is not possible to find optimal solutions to practical size problems in a reasonable amount o f computation time. Therefore, heuristics are necessary to find a near optimal or good solution to the problem. We developed two types o f heuristics to the problem. They are referred to as Insertion and Improvement. In the Insertion heuristic, the best route is selected based on one request at a time with no look-ahead capability. The Improvement heuristic will incorporate look-ahead capabilities in selecting the route. The Insertion procedure consists of two phases. In the first phase, all the candidate routes/paths that meet a certain criterion for each request are identified. In the second phase, a feasible path from the candidates’ list that has the shortest on-demand vehicle distance is selected and inserted into the vehicle schedule. The solution o f the Insertion procedure is fed into an Improvement procedure. In this procedure, we try to identify an alternative path for requests that have multiple hybrid paths that can satisfy the demand. To help motivate the heuristics and illustrate the combinatorial nature o f the problem, Figure 6.1 shows a simple network that has hybrid service. In this example, the triangle with a one is the origin while the triangle with a two is the 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. destination. There are two fixed route bus lines in this example. Obviously, one path is to take the direct distance using only the on-demand vehicle. Another path is to take a vehicle to stop 1C then get on a fixed bus line to either stops TC, IB, and IA and transfer to another vehicle. Clearly one of the functions of the heuristic is to select the drop-off bus stop. In this case, stop 1A is the closest to the final destination but requires the passenger to sit in the bus longer. Furthermore, the request that the vehicle serves next may be closer to stop TC. Hence, the heuristics need to consider all these factors in selecting the best route/path. 1A (m i 3 mi 5 mi .2m l PATH 3 m i i § 1-2 1-1C-2C-2 .72, 1-1C-1A-2 1-2D-2C-2 1-1C-TC-2 (l.S,4)=10 .80,1 1-1C-1B-2 (1,8,4)=13 .61,-625 Figure 6.1. Hybrid Network 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 N otations Before presenting the methodology of the proposed heuristic algorithms, we present some additional notations. BB (r, Bl, B2) Distance from bus stop B l to bus stop B2 on fixed route r DD (n) Door to door distance o f request n PB(r, Bl,n) Distance from the origin point o f request n to bus stop Bl on route r DB(r, B2, n) Distance from bus stop B2 on route r to the drop off (destination) point o f request n For each bus route, we now have the distance between bus stop B 1 to each pick-up point and the distance between bus stop B2 to each drop-off point. This is the total distance of request n that is traveled using the paratransit vehicle for a hybrid system. Let this variable be DBD (r, Bl, B2, n). DBD (r, Bl, B2, n) = PB (r, Bl, n) + DB (r, B2, n) (6.1) The hybrid total distance is the total distance that is traveled by the on- demand vehicle and fixed bus line. This distance is always greater than or equal to the direct distance of the same request. Let this variable be HYB (r, Bl, B2, n). 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. HYB (r, Bl, B2, n) = (DBD (r, Bl, B2, n) +BB (r. Bl, B 2))>D D (n) (6,2) 6.2 Phase I of the Heuristic Phase I o f the heuristic identifies candidate paths that meet the following three criteria: 1. The ratio of the direct distance over the hybrid distance must be greater than or equal to a threshold level FI. DD (n)/HYB (r, Bl, B2, n) > FI (6.3) 2. The ratio o f the distance traveled by the on-demand vehicle over the distance on the fixed bus route must be less than or equal to a threshold level F2. DBD (r, Bl, B2, i)/BB (r, Bl, B2)<F2 (6.4) 3. The door-to-door distance o f the request must be greater than or equal to a threshold level F3. DD (n) >F3 (6.5) W e can look at the first condition as the rider’s service level or convenience. The second condition ensures that the distance traveled by the paratransit vehicle is minimized or in other words, increases the productivity o f the paratransit vehicle. The third condition makes sure that no transfer to a fixed route 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is made if the distance o f the request is short. The above three conditions work together and a path needs to satisfy all o f them in order to be considered a candidate path. Although, phase I o f the algorithm treats the problem from the distance point o f view, the problem has another important aspect, which is the time or the scheduling aspect. However, we should mention here that implementing phase I before phase II is important for two reasons: First, it is less computationally intensive than phase II. That is, enumeration o f all the possible paths can be done quickly. Second, it will reduce the search space o f the vehicle scheduling part (Phase II). Figure 6.1 shows how the candidate paths for the hybrid system can be found using FI = 0.7, F2 = 1 and F3 = 8. By following phase I of the algorithm, one can conclude that path 1-IC-TC-2 is a candidate path for this particular transportation request. Also, path 1-1C-2C-2 is not a candidate because it violates one o f our assumptions, which is “no transfer between same transportation mode.” In the first phase, we enumerate all the possible paths from the origin point to the destination point o f each request. Therefore, the total number o f paths of each request depends on the number o f the fixed bus routes and the number o f bus stops o f each route. Specifically, it is: Af P A T H S = P * ' + P 2S - + ... + / > / ' ( 6 . 6 ) 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. considered a candidate if it has at least one candidate path. Thus many requests may have many alternative candidate paths. Fl=0.4 Fl=0.5 2 25 Fl=0.6 Z 15 Fl=0.8 FI=0.9 0.2 0.4 0.6 0.8 1.2 1.6 1.4 Figure 6.2. Num ber o f candidate requests. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 700 600 Fl=0.' 500 a . 400 Fl=0.. “ 300 Fl=0.6 200 Fl=0.7 100 Fl=0.8 ■ E L f O.9 1.6 0.8 0.2 0.4 0.6 1 1.2 1.4 0 F2 Figure 6.3. Number o f candidate paths. As it is shown in the above two charts, the number of candidate requests increases as the ratio FI decreases and ratio F2 increases. Also the number o f candidate paths increases as the ratio FI decreases and ratio F2 increases. Using the values o f FI = 0.7 and F2 = 1, 25 (53%) requests are found to be candidates for using the hybrid system. These requests have a total o f 63 candidate paths. If we let FI = 0.4 and maintain F2 = 1, 43 (91%) requests are found to be candidates for using the hybrid system with a total o f 321 candidate paths. The sensitivity of these ratios represents a trade off between the customer service level and overall efficiency o f the system. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Phase U of the H euristic As mentioned earlier, some candidate requests may have many candidate paths. A crucial question is; which candidate path is the best? In other words, which candidate path will lead to the best vehicle schedule as well as the best customer schedule ? In the Insertion heuristic algorithm, for each candidate request, the candidate paths are sorted based on the distance traveled by the on-demand vehicles (e.g., DBD (r, B S 1, BS2, n)) the distance from origin to entry bus stop plus the distance from exit bus stop to the final destination). After that, the shortest candidate path is selected for each candidate request. We note here that the shortest candidate path may not lead to finding the shortest total distance traveled by the on- demand vehicle. Figure 6.4 shows the steps of the Insertion heuristic. The following are the steps o f the Insertion heuristic algorithm; 1. In the first step o f the Insertion heuristic, all first legs o f hybrid requests (one first leg per hybrid request) and the door-to-door requests are inserted in set “N.” That is, the number o f requests in N is actually the number o f total requests o f the day. However, they are divided into two types; candidate (hybrid) requests and door-to-door requests. Then, the requests in set N are sorted based on their desired pickup time. 2. Next, the request with the earliest pickup time in N is selected for scheduling purposes. Again, this request might be a door-to-door request or just one leg o f a hybrid request. 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. No Yes No Yes Any more requests in N No "i \ t least one c a r ' ' which can satisfy the \ time window ^ .Yes Is this a hybrid request on the first Add vehicle Determine the pickup time and point o f the second leg and insert it in N Stop Remove the current request from N Select request with the earliest pickup time in N For each request, find shortest candidate path if any exist Put all first leg hybrid and Door-to- Door requests in set N. Son N based on pickup time Select the vehicle which results in the smallest increase in the total distance traveled and insert request in the schedule Figure 6.4. Insertion heuristic flow chart. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. A question is asked here, is there at least one vehicle that can satisfy the time window of the current request? If the answer is NO, another vehicle will be added to the system and the same question is asked again. If the answer is YES, then go to the next step. 4. Among all the vehicles that satisfy the current request time window, select the one that results in the smallest increase in the total distance traveled by an on-demand vehicle. This step is further described below (see section 6.4.1). 5. After finding the best vehicle to satisfy the request, another question is asked: Is the current request a hybrid request on the first leg? If the answer is NO, then the current request (door-to-door request or second leg o f a hybrid request) is removed from set N because it is satisfied. However, if the answer is YES, this means that we have just satisfied the first leg o f a hybrid request. Therefore, the second leg o f the hybrid request needs to be satisfied. This includes determining the pickup point and time of the second leg. Finally, the second leg (with all its information needed) is inserted in set N and the first leg is removed from set N. 6. The last question in the algorithm is: are there any more requests in set N? If the answer is YES, then select the request with the earliest pickup time and continue the algorithm. If the answer is NO, insert vehicle idle time if possible to reduce the customer waiting time at the entry bus stop and then stop the algorithm. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.1 Vehicle Selection In this part o f the algorithm, we are dealing just with the vehicles that can satisfy the time window o f the current request and at the same time not violating the vehicle capacity. The on-demand vehicle that results in the smallest increase in the total distance traveled by the on-demand vehicle will be selected. Figure 6.5 shows the steps o f this procedure. The following are the steps: 1. Select vehicle “k,” let the current schedule o f vehicle k be Sk- 2. Identify the last pickup request in Sk, call it “n.” 3. Insert the pickup and drop-off points o f the current request in all feasible combinations after request n. Feasible combinations are those in which the pickup point of a request precedes the delivery point o f the same request and the vehicle capacity is not violated. See the illustration below. 4. Select the feasible scenario with the total shortest distance DISk. 5. Are there any more vehicles? If the answer is YES, go back and select another vehicle “k” and continue the algorithm. If the answer is NO, select the vehicle “k*” with the shortest distance DISk. 6. Insert the current request in vehicle k* based on the best combination and update Sk* 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Yes Any more vehicles No Insert the current request in vehicle k* and update Sk* Select vehicle "k," Let current schedule o f vehicle k be Sk Identify the last pickup request in Sk. call it “n” Select the vehicle k* with the shortest distance DISk Select the feasible scenario with the shortest distance DISk traveled by the on-demand vehicle Insert the pickup and drop-off points of the current request in all feasible combinations after request n Figure 6.5. Vehicle selection flow chart. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.1.1 Illustration. The following are two examples to show how we can insert a request in an existing schedule of a vehicle. In the first case, the vehicle pickups the first passenger and then drops the same passenger off before picking up the second passenger (PID1P2D2). In the second case, the vehicle pickups both passengers first and then drops them off (P1P2D1D2). We now illustrate the possible combinations o f inserting a new passenger request. In the first case, three alternatives are evaluated while in the second case nine alternatives are evaluated. 6.5 Improvemeni Procedure In the previous algorithm, we select the candidate path that minimizes locally the total distance traveled by the on-demand vehicle for a particular request. However, this might not be the most efficient method to reduce the total distance traveled by the on-demand vehicles. Consider the following scenario: Consider a hybrid request that has two candidate paths for the hybrid system and both o f them are feasible to be inserted in an existing vehicle schedule. Suppose that there is a custom er waiting for a pickup at or near the next stop o f the f PI D l P 2P 3 D3 D2 PI D l P2 D2 J P 1 D IP 2 P 3 D 2 D 3 „ PI D l P 2 D 2 P 3 D 3 f PI P2 P3 D3 D l D2 PI P2P 3 D l D3 D2 PI P2 P3 D l D2 D3 PI P2 P3 D3 D 2D 1 PI P2 D l D2 J P 1 P 2 P 3 D 2 D 3 D 1 PI P2 P3 D 2D 1 D3 PI P2D 1 P3 D3 D2 PI P2 D l P3 D2 D3 ^ PI P 2 D I D 2 P 3 D 3 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Find an iniiial schedule using the Insertion algorithm. Let TVD* and TTT* be the resulting total distance traveled by the on-demand vehicle and total passenger trip time. 2. Select a candidate (hybrid) request “n” with the earliest pickup time in set N. No Yes Any more hybrid requests in N Yes No Any other candidate path yields to smaller - '~ J V D and TTT___ Stop Update TVD and T T Select the new path Find iniiial schedule using the Insertion heuristic Select hybrid request n with the earliest pickup time in N Evaluate alternative paths for request n by replacing the candidate path in the current solution with all other possible candidate paths Figure 6.6. Improvement heuristic flow chart. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Evaluate all the candidate paths o f request n by replacing the candidate path in the current solution with all other possible candidate paths. 4. Any other candidate path which leads to a feasible schedule with a smaller total vehicle distance TVD* and total passenger trip time TTT*? If yes, update TVD* and TTT* and select the new path. 5. Remove the current request from set N. 6. Are there more candidate requests that are not enumerated yet in set N? If the answer is YES, go to the second step. If the answer is not, then stop. Note that the above procedure does not enumerate all possible combinations in determining the solution. That is, once the best candidate path for a particular request is found it is held fixed. Suppose that there are 4 candidate (hybrid) requests that have 8, 7, 6 and 9 candidate paths, respectively. The total number o f solutions considered will be 8 + 7 + 6 + 9 = 30. This means that 30 different solutions are considered. There is no dispute that a better solution will be found if we enumerate all the possible combinations of the candidate paths. However, the small improvement that would be gained is going to be at the expense of a very long computation time (e.g., 3,024 solutions will be considered in this example, 8 * 7 * 6 * 9 = 3,024). Furthermore, the solution space will explode for realistic sized problem when there are more candidate requests and paths. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 COMPUTIONAL EXPERIMENTS This chapter presents the computational experiments that have been performed to test the two heuristics using the real data from A VTA. Next we show the information that we have for the experiments. Then we describe the manipulations that need to be done for the data in order to be compatible with our programming code. After that, we introduce the performance measures that are needed for comparison purposes. Finally, we show the results o f two hybrid delivery methods (Insertion and Improvement) and one Door-to-Door delivery method and compare them to the manual schedule from A VTA. 7.1 Information Available Each transportation request obtained from the A VTA data includes the following: Address o f the pickup point, address o f the delivery point, vehicle arrival time at the pickup point, vehicle arrival time at the delivery point, vehicle number, vehicle cumulative mileage at the pickup point, and vehicle cumulative mileage at the delivery point. On the other hand, the bus schedule o f the Antelope Valley area, which includes 10 fixed routes, is available. Each route has many bus stops. The location o f each bus stop and the arrival time of the buses at each bus stop are known. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7.2 M anipulating the Data The format o f the data received from A VTA and the bus schedule need to be modified to be in a format that can be compatible with the programming code. First, the locations o f the bus stops, pickup point, and delivery point need to be converted from the address form to the coordinates form (Longitude and Latitude). ArcView software is used for this task. Since the bus stops are relatively few, the coordinate conversion is done easily. However, this is not the case for the pickup and delivery points, which are around 2200 points. Some points that are unmatched in the software are approximated. Second, the manual schedule is found as follows: 1. Determining vehicle sequence o f visiting the points. This can be found from the arrival time of every vehicle at each point. 2. Finding points’ locations by geocoding the addresses o f the pickup and delivery points using ArcView software. 3. Measuring the direct (Euclid) distance between every two consecutive points visited by the same vehicle using the sphere equation: DISTANCE = (3963*ACOS (COS (RADIANS (90-LAT1))*COS (RADIANS (90- LAT2))+SIN (RADIANS (90-LATi))*SIN (RADIANS (90-LAT2))*COS (RADIANS (LONG 1-LONG2)))) Calculating the performance measures o f the manual schedule for each working day: 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1. Total custom er time. 2. Total on-demand vehicle distance. 3. Num ber of on-demand vehicles that are used. 7.3 Performance Measures of the Heuristics The performance measures o f our results can be mainly divided into three major categories; vehicle productivity, custom er service level and operating cost. Below we show these categories and their subcategories: 7.3.1. Vehicle Productivity 1. Total Serving Distance traveled by on-demand vehicle (TSD). One or more customers are in board. 2. Total Empty Distance traveled by on-demand vehicle (TED). No passengers on board, vehicle is going to pickup a customer. 3. Total Vehicle Distance (TVD). TVD = TSD + TED. 7.3.2 Customer Service Level 1. Total DD Passengers Time in on-demand Vehicle (TT_DD). 2. Total Hybrid Requests Time in on-demand Vehicle (TV_HYB). 3. Total Hybrid Requests Time in Bus (TB_HYB). 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. Total Hybrid Requests Time at entry bus stop waiting for the bus (TBS1). 5. Total Hybrid Requests Time at exit bus stop waiting foi the on- demand vehicle (TBS2). 6. Total Hybrid Requests Time (TT_HYB). TT__HYB = TV HYB + TB_HYB + TBS 1 + TBS2. 7. Operating Cost: number of required on-demand vehicles to service the requests. 8. CPU Times: number of seconds needed to implement a program code. 7.4 Results This section shows the results obtained from computation experiments executed for two hybrid delivery methods (Insertion and Improvement) and one Door-to-Door delivery method. The Door-to-Door delivery method is a special case o f the hybrid delivery method (Insertion) where the number o f requests using the fixed route system equals zero. Two sizes o f on-demand vehicle capacity (two and three) are investigated for every delivery method. The three delivery methods are compared using the AVTA data in terms of total on-demand vehicle distance and total custom er trip time. W e benchmark the results against their manual 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. distance results in a high custom er trip time. Thus, tor this day we use a combination o f (FI = 0.6, F2 = 1.1). Table 7.1. Daily FI and F2 values V ehicle Capacity = 2 V ehicle Capacity = 3 Day Number o f Vehicles Requests Hybrid Requests F1,F2 Hybrid Requests F1.F2 1 6 76 15 0.6,1.1 13 0.65,1 2 12 155 30 0.65.1.2 27 0.6,0.85 3 10 150 29 0.65,0.95 30 0.7,1.05 4 10 135 18 0.75,0.9 18 0.75,0.9 5 10 138 17 0.75,1 15 0.75,0.8 6 10 103 19 0.65.0.85 19 0.65.1.1 7 12 139 22 0.6.0.8 19 0.7,0.85 8 4 42 14 0.6.0.85 14 0.6,0.85 9 9 91 27 0.65,0.9 27 0.65,0.85 Total Percent 83 1029 191 18.6% 182 17.7% 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 2 0 FI =0.7 510 I n 500 • 490 F1=0.75 O 480 FI =0.65 g 470 460 450 0.8 0.85 0.95 1.15 1.2 1.25 0.75 0.9 1 1.05 1.1 F2 Figure 7.1. Sensitivity analysis (vehicle distance). 30 29.5 FI =0.6 29 F1=0.7 28.5 28 F1=0.65, 27.5 F1=0.75 27 26.5 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 F2 Figure 7.2. Sensitivity analysis (customer time). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.2 shows the on-demand vehicle distance and customer trip time when the on-demand vehicle capacity equals two for each day for each routing heuristic. For the nine days, the distance traveled by the on-demand vehicles in the hybrid methods (Insertion and Improvement) is always less than it is in the door-to- door delivery methods (Door-to-Door and Manual). Table 7.3 shows the on-demand vehicle distance and customer trip time when the vehicle capacity equals three for each day for each routing heuristic. As it is in the previous table, for the nine days, the distance traveled by the on-demand vehicles in the hybrid methods (Insertion and Improvement) is always less than it is in the door-to-door delivery methods (Door-to-Door and Manual). Note that increasing the vehicle capacity reduces the vehicle distance, however, at the expense o f longer customer trip times due to increased ridesharing. Table 7.4 summarizes the results o f the nine days. As previously noted, as we increase the on-demand vehicle capacity, the on-demand vehicle distance decreases and the customer trip time increases since with more vehicle capacity, the more opportunity their will be for ridesharing which consequently leads to shorter on-demand vehicle distance and longer custom er trip time. Overall, the hybrid delivery methods yield routes that minimize the on-demand vehicle distances with the Improvement heuristic requires significantly more computational time (see later in chapter). In terms o f total trip time, the Improvement heuristic outperformed their manual schedule and is very close to the Door-to-Door heuristic. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.3. Daily vehicle distance and customer time (capacity = 3) Insertion Improvement Door-to-Door Manual Day Vehicle Distance (miles) Customer Time (hours) Vehicle Distance (miles) Customer Time (hours) Vehicle Distance (miles) Customer Time (hours) Vehicle Distance (miles) Customer Time (hours) 1 466 30.1 461 29.7 472 30.2 604 28.6 - > 866 67.5 862 66.6 957 69.9 1.143 68.0 3 878 68.8 856 66.0 905 74.4 1.032 66.0 4 715 53.3 708 52.7 750 56.0 843 52.0 5 837 68.6 829 60.8 903 62.1 985 60.6 6 668 56.5 634 48.7 667 45.7 820 40.0 7 891 77.6 884 75.2 933 74.4 1.140 75.8 8 294 16.3 292 16.1 319 18.6 316 19.5 9 545 40.7 540 39.1 576 40.4 752 39.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 7.4. Total vehicle distance and custom er tim e Rule Capacity Vehicle Distance (m iles) Customer Tim e (hours) (+/-) % or Manual Distance (+/-) % o f Manual Tim e Insertion 2 6.737 442.5 -11.8% -1.6% Insertion 3 6,160 479.4 -19.3% 6.6% Improvement 2 6,593 422.7 -13.6% -6.0% Improvement 3 6,066 454.9 -20.6% 1.2% Door-to-Door 2 7.321 417.5 -4.1% -7.2% Door-to-Door 3 6,482 471.7 -15.1% 4.9% Manual 7.635 449.7 0.0% 0.0% We next examine the total customer trip time in more detail for the hybrid requests. Note that the previous reported trip times included all requests (i.e., hybrid and strictly door-to-door). For hybrid requests, the total trip time includes the time on the on-demand vehicles, time waiting at the entry and exit bus stops, and time on the fixed bus routes. Table 7.5 and 7.6 show the breakdown o f the total trip time for the schedules generated by the Improvement heuristic. For example, in Day I, of the 28 total hours o f custom er trip time 12.6 is for the hybrid requests. The last four columns show the breakdown into the various components. The entry bus stop is stop I and the exit bus stop is stop 2. In the table, we also list the percentage above the manual schedule for the hybrid custom er trip times. W hen considering all the customers, we previously showed that the Improvement 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. heuristic generated schedules with lower total customer trip times than the manual schedule. However, when only counting the hybrid requests. Table 7.5 and 7.6 show that for these passengers their trip times will increase moderately (except for day 6, which has many identical requests arise at the same time). This is to be expected since these passengers require two modes of transportation to satisfy their requests. Another important observation from these two tables is that the custom er waiting time at the exit bus stop is significantly less than the customer waiting time at the entry bus stop. This is due to the fact that there are time windows associated with the exit bus stop since it is considered a pickup point. Thus these windows place a restriction on the waiting time at the exit bus stop. Since the entry bus stop is considered a destination point, there are no time windows associated with these locations. We attempt to minimize the waiting time at the entry bus stops by inserting vehicle idle time if possible in order to have the passenger arrive as close as possible to the departure time o f the fixed bus line. Overall, this analysis showed that shifting some o f the demand to fixed bus lines route (18.6% of the requests) reduces the on-demand vehicle distance by 13.6% and customer trip time by 6.0% over the manual schedule when the vehicle capacity is 2. However, for these customers who take the hybrid delivery method (18.6% o f the requests), their trip time will increase on average by 7.2% (this increase is 16.5% when we use the Insertion heuristic). 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 7.5. C om ponents o f custom er trip tim e for im provem ent heuristic Capacity = 2 Day All Requests Total Time Door-to- Door Requests Total Tim e Hybrid Requests Total Time Hybrid Requests Tim e in Vehicle Hybrid Requests Tim e in Bus Hybrid Requests Tim e at Bus Stopl Hybrid Requests Time at Bus Stop2 1 28.0 15.4 12.6 (-2.4%)' 4.6 6.6 1.2 0.2 2 62.6 38.0 24.6 (+1.8% ) 9.0 11.2 3.5 0.9 3 60.5 34.6 25.9 (+3.5% ) 6.9 11.1 6.4 1.5 4 47.8 35.2 12.6 (+3.4% ) 4.9 5.3 1.7 0.7 5 55.0 42.8 12.2 (+4.8% ) 4.4 5.3 1.2 1.3 6 43.6 27.5 16.1 (+39.9% ) 7.5 5.8 2.1 0.7 7 72.5 54.3 18.2 (+19.2% ) 5.0 8.5 3.8 0.9 8 16.4 7.2 9.2 (-4.3%) 2.8 4.0 1.7 0.7 9 36.3 16.7 19.6 (+5.4% ) 6.2 9.6 2.5 1.3 Total 422.7 271.7 151.0 (+7.2% ) 51.3 67.4 24.1 8.2 1 Percentage above manual. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 7.6. C om ponents o f custom er trip tim e for im provem ent heuristic Capacity = 3 Day All Requests Total Time Door-to- Door Requests Total Tim e Hybrid Requests Total Time Hybrid Requests Tim e in Vehicle Hybrid Requests Time in Bus Hybrid Requests Tim e in Bus Stopl Hybrid Requests Time in Bus Stop2 1 29.7 19.4 10.3 (+3.9% )‘ 3.4 5.0 1.6 0.3 2 66.6 45.2 21.4 (+3.8% ) 5.4 11.3 3.2 1.5 3 66.0 37.3 28.7 (+2.3% ) 11.2 10.0 6.1 1.4 4 52.7 38.7 14.0 (+14.9% ) 5.2 5.3 1.9 1.6 5 60.8 48.8 12.0 (+20.3% ) 4.2 5.0 2.1 0.7 6 48.7 31.3 17.4 (+51.2% ) 8.7 5.7 1.4 1.6 7 75.2 60.5 14.7 (+20.0% ) 5.4 6.0 2.5 0.8 8 16.1 7.0 9.1 (-5.3% ) 2.9 4.1 1.6 0.5 9 39.1 22.1 17.0 (+2.3% ) 5.1 8.7 2.2 1.0 Total 454.9 310.3 144.6 (+10.6% ) 51.5 61.1 22.6 9.4 1 Percentage above manual. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. All the above experiments were done to test the performance o f the heuristics while using the same number o f on-demand vehicles as used by AVTA. We were interested in increasing the ability o f serving more requests by the same number of vehicles while maintaining fair service level. Consequently, our objective was to find effective solutions in terms of on-demand vehicle distance and customer trip time. However, by using the hybrid delivery methods, reasonable vehicle distance and customer trip time might be obtained with less number of on-demand vehicles in each working day. This sensitivity analysis for the Improvement heuristic is shown in Tables 7.7 and 7.8 for vehicle capacity o f two and three, respectively. Note that in all days it is possible to meet the day’s demand with less on-demand vehicles using the Improvement heuristic. Also observe that usually the on-demand vehicle distance and the custom er trip time increase as the num ber o f on-demand vehicles decreases. Moreover, in these tables, we show the CPU times required to execute the Improvement heuristics (CPU time to implement the Insertion is less than one second in all cases). 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 7.7. (continued) Improvement Heuristic Day V ehicles Vehicle Distance Custom er Tim e (+/-)% o f Manual (m iles) (hours) Distance (+/-) % o f Manual Tim e C PU Tim e (seconds) 6 10 657 43.6 -19.9% 9.0% 15 9 663 45.4 -19.1% 13.5% 16 6 723 48.4 -11.8% 21.0% 16 5 759 49.6 -7.4% 24.0% 16 7 12 1.007 72.5 -11.7% -4.4% 34 11 995 75.8 -12.7% 0.0% 33 10 993 75.4 -12.9% -0.5% 33 9 1,016 75.3 -10.9% -0.7% 34 8 1,083 83.8 -5.0% 10.6% 34 8 4 295 16.4 -6.6% -15.9% 3 3 331 16.8 4.7% -13.8% 3 9 9 553 36.3 -26.5% -7.4 19 8 569 36.3 -24.3% -7.4% 19 7 591 38.7 -21.4% -1.3% 20 6 598 38.5 -20.5% -1.8% 19 5 633 39.9 -15.8% 1.8% 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 7 .8 . Sensitivity analysis on num ber o f vehicles (capacity = 3) Improvement Heuristic Vehicle Distance Custom er Tim e (+ /-)% o f Manual (+/-)% o f CPU Tim e Day V ehicles (miles) (hours) Distance Manual Tim e (seconds) 6 462 29.7 -23.5% 3.8% 7 5 469 30.6 -22.4% 7.0% 8 4 497 33.0 -17.7% 15.4% 8 12 862 66.6 -24.6% -2.1% 55 11 866 68.2 -24.2% 0.3% 55 10 890 69.2 -22.1% 1.8% 55 9 881 70.4 -22.9% 3.5% 55 S 931 72.6 -18.5% 6.8% 55 7 976 74.6 -14.6% 9.7% 56 6 990 82.6 -13.4% 21.5% 56 10 856 66.0 -17.1% 0.0% 30 9 862 68.3 -16.5% 3.5% 30 8 904 68.1 -12.4% 3.2% 31 7 905 71.4 -12.3% 8.2% 31 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T abic 7.S. (continued) Improvement Heuristic V ehicle Distance Customer Tim e (+ /-)% o f Manual (+ /-)% o f CPU Time Day V ehicles (m iles) (hours) Distance Manual Tim e (seconds) 4 10 708 52.7 -16.0% 1.3% 9 9 739 57.8 -12.3% 11.2% 9 8 759 56.9 -10.0% 9.4% 9 7 776 58.4 -7.9% 12.3% 9 5 10 829 60.8 -15.8% 0.3% 10 9 860 68.3 -12.7% 12.7% 10 6 10 634 48.7 -22.7% 21.8% 18 9 652 52.4 -20.5% 31.0% 19 8 652 52.2 -20.5% 30.5% 20 7 661 53.5 -19.4% 33.8% 20 6 681 55.2 -17.0% 38.0% 20 5 702 56.0 -14.4% 40.0% 21 7 12 884 75.2 -22.5% -0.8% 18 11 901 76.7 -21.0% 1.2% 18 10 901 75.7 -21.0% -0.1% 18 9 917 77.5 -19.6% 2.2% 18 8 962 87.6 -15.6% 15.6% 18 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 7.8. (continued) Improvement Heuristic Day V ehicles V ehicle Distance (m iles) Customer Tim e (+/-) % o f Manual (hours) Distance (+/-) % o f Manual Time CPU Time (seconds) 8 4 292 16.1 -7.6% -17.4% 3 3 327 16.9 3.5% -13.3% 3 2 347 18.8 9.8% -3.6% 3 9 9 540 39.1 -28.2% -0.3% 16 8 571 39.5 -24.1% 0.8% 16 7 586 39.6 -22.1% 1.0% 17 6 588 40.6 -21.8% 3.6% 17 5 625 43.5 -16.9% 11.0% 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 8 SYSTEM-WIDE STRATEGIES 8.1 Introduction Although the Improvement phase addresses some deficiencies o f the Insertion procedure, better solutions can still be obtained. Below we introduce two strategies to improve the vehicle and customer schedule, which are Re-sequencing strategy and Re-assigning strategy. During these strategies, we are concerned about four constraints: 1. Coupling constraint. For every request or hybrid leg, the same vehicle which pickups a passenger must drop him/her off at his/her delivery point. 2. Precedence constraint. For every request or hybrid leg, the pickup point must be visited before the delivery point. 3. Time window constraint. For every request or hybrid leg, the pickup point must be visited within the time interval specified. For every first hybrid leg, in addition to satisfying the pickup point time window, the delivery point must be visited before the bus departure time, which is obtained from the bus schedule. 4. On-demand vehicle capacity constraint. The number o f passengers on-board the vehicle must be less than or equal to the on-demand vehicle capacity. The Re-sequencing strategy resorts the sequence within a vehicle while the Re- 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. assigning searches to remove a request from a particular vehicle to insert it to another vehicle. The two strategies aim to detect better vehicle and customer schedules. In summary, the proposed methodology to solve the hybrid-scheduling problem is to implement the Insertion heuristic followed by the Improvement heuristic. Two system-wide strategies, which are the Re-sequencing and the Re assigning, then are executed to find better solutions obtained by the Improvement heuristic. Figure 8.1 displays the proposed methodology. Note that we iterate between the Re-sequencing and Re-assigning procedures because we try to find a better sequence when a new request to vehicle assignment is found. Insertion Re-assigning Improvement Re-sequencing Figure 8.1. Proposed methodology. 8.2 Re-sequencing In this strategy, we use a Tabu Search technique in order to find a better requests sequence (which leads to shorter vehicle distance and shorter custom er trip time) in every vehicle while holding the request to vehicle assignment fixed. The outcome problem from this technique will be a single vehicle pickup and delivery problem with hard time windows. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the case of a strictly door-to-door delivery method, requests are entirely satisfied at the delivery point. So the schedule of the vehicles is going to be always feasible if we just satisfy the pickup time windows of the requests during re sequencing. Therefore, it is not obligatory to set a delivery time window for the requests as well as the pickup time window. However, this is not true for the hybrid delivery method where hybrid requests have two delivery points (two legs) done by two different vehicles. Also, due to the dependency between the two trips of the hybrid request, it is mandatory to set a delivery as well as a pickup time window of every first leg o f a hybrid request in each vehicle schedule in order to maintain the feasibility in every vehicle schedule. To clarify this issue, we present the following example: Suppose there is a hybrid request, which has the following schedule (7:00 - 7:07) from origin point to BS1 in vehicle 1, (7:10 - 7:25) from BS1 to BS2 in the bus, and (7:27 - 7:32) from BS2 to final destination in vehicle 2. By doing the re sequencing o f vehicle I schedule without adding a delivery time window, the passenger might end up w ith a delivery time of 7:11 to BS1. In this case he/she will miss the 7:10 bus and use the 7:30 bus, which arrives at the exit bus stop (BS2) at 7:45. Therefore, ignoring the delivery time o f the first leg o f a hybrid request in one on-demand vehicle m ight end up with an infeasible schedule in another on- demand vehicle. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the Re-sequencing strategy, the schedule is improved by moving individual predecessor (pickup point) and/or successor (delivery point) forward and/or backward in their corresponding route. Three conditions need to be satisfied while moving the pickup and delivery pair in order to have a feasible schedule. The first one is the precedence constraint where the pickup point of any request must be visited before the delivery point o f the same request. The second is the on- demand vehicle capacity constraint. The third is the pickup time window (and the delivery time window for the first leg o f the hybrid requests). Since the problem in this technique is a single vehicle PDP, the coupling constraint cannot be violated. The coupling constraint is that once the vehicle pickups a passenger, the same vehicle must drop him/her off at his/her final destination point. Changing the request sequence within the routes will investigate other possible orderings to determine the best possible sequence for the routes. The advantage of this neighborhood search is particularly obvious when large time windows are prevailing. In these circumstances, numerous feasible solutions can be discovered. To minimize the search space and consequently the Tabu memory, the search is further restricted by what we called min (Sn) and max (En) request positions. These tell how far, the request can be moved backward and forward without causing guaranteed infeasibility (Barnes & Carlton, 1995). Below we show an illustration o f the above concept for two possible cases. Figure 8.2.a shows a single vehicle schedule where the first and the last nodes 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fortunately, different techniques for combining nonimproving moves in improving search without repeating solutions, such as Tabu Search and Simulating Annealing, are available in the literature. These techniques have shown to be effective in solving a variety of routing problems. We use the most popular one o f these techniques, which is the Tabu Search, to implement the Re-sequencing strategy. The main idea behind the Tabu Search is to maintain a list o f forbidding moves by which solutions are prevented to be visited again and consequently not allowing falling in infinite cycles. We use a tabu search description similar to the one introduced by Rardin (1998) to illustrate both the Re-sequencing and Re-assigning strategies. The following are the steps o f the Re-sequencing strategy: 1. Initialization. The initial schedule (SCH(O)) is obtained by the improvement heuristic and the initial sequence is added to the Tabu list. Let the current best schedule (SCH*) equal SCH(O) and let the iteration index (t) equals 0. 2. Stopping. If there is no nontabu sequence leading to a feasible solution neighbor o f the current schedule SCH (t) or t = tm ax (tm ax - 10), then stop. The current SCH* is an approximate optimum schedule. 3. Move. Select a nontabu feasible sequence neighbor to the current schedule SCH(t) and find SCH(t-t-l). 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4. Criterion. In this sequence selection step, all the feasible moves as shown in Figure 8.1 are evaluated. The sequence that leads to the best value o f the objective function is selected. 5. Compare. If the objective function value of SC H (t+l) is superior to that o f SCH*, Let SCH* = SCH (t+l). 6. Tabu List. Remove from the list o f forbidden tabu sequences any that have been on it for a sufficient number o f iterations (5 iterations), and add the current sequence o f SCH(t+l) to the list. 7. Increment. Let t = t + 1, and go back to step 2. 8.3 Re-assigning In this strategy, we try to search for a better request-vehicle assignment using a tabu search technique to produce a better solution. The search space o f this method is larger than the one for the Re-sequencing method. Therefore, only few requests with a high potential for improvement are considered to be moved from their current vehicle schedule to another vehicle schedule. The main idea behind the Re-assignment is to move to undiscovered regions and try to search for better solutions there. In the Re-assigning approach, two questions need to be answered: (a) W hich request should be removed from its current vehicle schedule?, and (b) 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Where should it be inserted? To answer the first question, vve set the following criterion: I. Remove the request, which has the maximum saving distance if it is removed, which equals the distance of the removed arcs minus the distance o f the added arcs. Figure 8.3 illustrates the above criterion. In Figure 8.3(b), the request that will be removed from the current on-demand vehicle schedule is request “n” which has adjacent pickup and delivery points. The eliminated arcs are Xo.pn, Xon.p and Xpn.Dn- The arc Xo.pn represents the distance required to reach the pickup point of the request from the previous adjacent point ”a ”. The arc Xon.p represents the distance from the request delivery point to the next adjacent point ”P”. The arc Xpn.Dn represents the distance from the request pickup point to the delivery point. The new arc that will be added after removing the request ”n” is Xo.p which represents the distance from a to p. The double lines in Figure 8.3 are the eliminated arcs while the dotted lines are the new added arcs. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Pn.Dii ■Dn.(J (b ) Figure 8.3. Illustration o f the saving concept. Saving = Xa p„ + Xon.p + Xpn.Dn ~ Xa.p (8 .1 ) In Figure 8.3(b), the request that will be removed “n” does not have adjacent pickup and delivery points. Following the same notation concept, the saving distance will be: Saving = Xa pn + Xon .p + Xp,uy + X r,D n — Xa.y — X Y i p (8.2) We use the saving function as the criterion for selecting the node removal. The other question, which needs an answer, is where this request should be inserted? The same concept of removing is followed for insertion. We will evaluate the insertion o f this request on all other on-demand vehicles. The feasible schedule that has the minimum increment o f distance will be chosen. The time 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w indow and capacity constraints need to be satisfied in order to call the schedule feasible. Below we describe a tabu search for Re-assigning strategy. 8.3.1 Tabu Search for Re-assigning Up to this stage, three algorithms are implemented to find the best final on- demand vehicle and customer schedules. These are Insertion, Improvement and Re-sequencing. In this section, we try to improve the final solution identified by these three algorithms by changing the vehicle-customer assignment by using a tabu search. The following are the steps of the Re-assigning strategy: 1. Initialization. The initial schedule (SCH(O)) is obtained by the final solution o f the Tabu sequencing procedure and the Tabu List is empty. Let the current best schedule (SCH*) equal SCH(O) and let the iteration index (t) equals 0. 2. Stooping. If there is no nontabu move leading to a feasible solution neighbor o f the current schedule SCH(t) or t = tm ax ( W = 10), then stop. The current SCH* is an approximate optimum schedule. 3. M ove. Select a nontabu feasible move neighbor to the current schedule SCH(t) and find SCH(t+l). a. Removing criterion. Remove the request from vehicle (k l) that results in the largest decrease in the total distance traveled. b. Insertion criterion. Insert the request in the vehicle (k2) that results in the smallest increase in the total distance traveled. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. c. Re-sequencing: Re-sequence k l and k2.usingT abu Search. 4. Compare. If the objective function value of SCH (t+l) is superior to that o f SCH*, Let SCH* = SCH(t+l). 5. Tabu list. Remove from the list of tabu of forbidden requests any that have been on it for a sufficient number o f iterations (5 iterations), and add the current request o f SCH (t+l) to the list. 6. Increment. Let t = t + 1, and go back to step 2. 8.4 Results of the System-wide Strategies In this section, we present the results of the computational experiments of the two system-wide strategies introduced above in terms o f solution quality and computational time. All the experiments done in this section assume homogeneous on-demand vehicles with capacity o f two. The Re-sequencing strategy is called TABU-S and the Re-assigning strategy is called TABU-A in this section. Table 8.1 presents a general comparison between all the rules used so far in terms o f total vehicle distance and total custom er time. It is shown that the hybrid scheduling delivery method (TABU-S) decreases both the total vehicle distance and total custom er time by 16.6% and 8.7% respectively comparing to the manual schedule that uses strictly the door-to-door delivery method. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 8.1. T otal vehicle distance and custom er tim e Rule V ehicle Distance (m iles) Customer Time (hours) (+/-) % o f Manual Distance (+/-) % o f Manual Tim e CPU Tim e (seconds) Insertion 6,737 442.5 -11.8% -1.6% 9 Improvement 6,593 422.7 -13.6% -6.0% 231 TA BU-S' 6,365 410.6 -16.6% -8.7% 623 TA B U -A 6,338 411.7 -17.0% -8.5% 1.357 Door-to-Door 7,321 417.5 -4.1% -7.2% N /A Manual 7,635 449.7 0.0% 0.0% N /A 1 Recommended Rule Table 8.2 shows the on-demand vehicle distance and custom er trip time for each day for each routing heuristic and Table 8.3 shows the computational time needed to implement these routing heuristics. For the nine days, as the distance traveled by the on-demand vehicles decreases as the CPU time needed to implement the heuristic increases, which shows the common trade off between the solution quality and the computational time. Also, it is shown that by applying TABU-A after TABU-S, the solutions are insignificantly improved at the expense of relatively large CPU time. 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T abic 8.3. C PU Tim e (Seconds) to im plem ent the heuristics Day Insertion Improvement TA B U -S TA B U -A 1 1 13 45 151 2 1 72 128 206 3 1 46 1 17 279 4 I 10 57 119 5 1 15 62 160 6 1 17 53 92 7 1 33 79 110 8 1 5 26 95 9 1 20 56 145 Total 9 231 623 1.357 As before, we next examine the total customer trip time in more detail for the hybrid requests. Tables 8.4 and 8.5 show the breakdown o f the total trip time for the schedules generated by TABU-S and TABU-A heuristics respectively. 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 8.4. C om ponents o f custom er trip lim e lor TA B U -S Day All Requests Total Time Door-to-Door Requests Total Time Hybrid Requests Total Time Hybrid Requests Tim e in Vehicle Hybrid Requests Time in Bus Hybrid Requests Tim e at Bus Stopl Hybrid Requests Tim e at Bus Stop2 1 26.6 14.65 11.95 (-7.4% )' 4.1 6.6 1.2 0.05 2 61.5 37.00 24.5 (+1.3% ) 8.9 11.2 3.4 1.00 3 59.3 33.90 25.4 (+1.5% ) 6.7 1 l.l 6.6 1.00 4 46.5 34.10 12.4 (+1.8% ) 4.5 5.3 1.8 0.80 5 53.4 41.00 12.4 (+6.5% ) 4.5 5.3 1.3 1.30 6 42.8 27.10 15.7 (+36.5% ) 7.3 5.8 1.9 0.70 7 69.4 51.40 18.0 (+17.9% ) 5.1 8.5 3.6 0.80 8 15.7 7.20 8.5 (-11.6% ) 2.4 4.0 1.8 0.30 9 35.4 15.70 19.7 (+5.9% ) 6.3 9.6 3.0 0.80 Total 410.6 262.05 148.55 (+5.4% ) 49.8 67.4 24.6 6.75 1 Percentage above Manual See Chapter 7 for more detailed explanation about the hybrid request time components. The two tables show similar results in term of the custom er trip time. However, TABU-S generates a schedule with less hybrid requests total time, which is only 5.4% more compared to the manual door-to-door delivery method. The other observations from the tables are sim ilar to those seen in Tables 7.5 and 7.6. 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 8.5. C om ponents o f custom er trip tim e for T A B U -A Day All Requests Total Time D oor-lo-Door Hybrid Requests Total Requests Total Tim e Tim e Hybrid Requests Tim e in Vehicle Hybrid Requests Time in Bus Hybrid Requests Tim e at Bus Slop I Hybrid Requests Tim e at Bus Stop2 1 25.9 13.95 11.95 (-7.4% )' 4.1 6.6 1.2 0.05 2 61.4 36.8 24.6 (+1.8% ) 8.8 11.2 3.6 1.0 3 59.3 33.7 25.6 (+2.3% ) 6.7 11.1 7.1 0.7 4 47.1 34.1 13.0 (+6.7% ) 4.6 5.3 2.1 1.0 5 53.4 41.0 12.4 (+6.5% ) 4.5 5.3 1.3 1.3 6 42.8 27.1 15.7 (+36.5% ) 7.3 5.8 1.9 0.7 7 69.4 50.9 18.5 (+21.2% ) 5.4 8.5 3.5 1.1 8 15.7 7.2 8.5 (-11.6% ) 2.4 4.0 1.8 0.3 9 36.7 16.7 20.0 (+7.6% ) 6.2 9.6 3.5 0.7 Total 411.7 261.45 150.2 (+6.6% ) 50 67.4 26 6.85 1 Percentage above Manual The results show that the Re-sequencing strategy with the help o f the Tabu Search technique has proven to be effective in improving the solution obtained by the Insertion and Improvement heuristics, within a reasonable amount of computational time. Also, it is shown that the Re-assigning strategy with the help o f the Tabu Search technique insignificantly improved the solution obtained by the 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Re-sequencing strategy with relatively large computational lime. Therefore, the recommended methodology o f solving the hybrid problem is to implement the Re sequencing heuristic. We would like to emphasize here that the input schedule o f the Re sequencing Tabu Search plays an important role in achieving a superior output schedule. Therefore, applying the Improvement heuristic between the Insertion heuristic and the Re-sequencing is essential to have a better solution. Table 8.6 shows this result by comparing two scenarios. In the first one, the Insertion heuristic is the input schedule o f the Re-Sequencing while in the second scenario the Improvement heuristic is the input schedule of the Re-Sequencing. The results show that the second scenario performed better than the first one. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tabic S.6. Daily vehicle distance and customer time T A B U -S (Insertion) TA B U-S (Improvement) Day V ehicle Distance (m iles) Custom er Tim e (hours) Vehicle Distance Customer Time (miles) (hours) 1 475 28.5 459 26.6 2 961 65.5 949 61.5 3 889 60.5 882 59.3 4 793 46.4 787 46.5 5 929 56.8 897 53.4 6 676 48.7 640 42.8 7 961 71.3 946 69.4 8 274 15.5 273 15.7 9 543 36.6 532 35.4 Total 6,501 429.8 6.365 410.6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 9 NETWORK DESIGN MODEL 9.1 Introduction Previous chapters were studying the hybrid DARP from an operational point o f view by developing algorithms to improve the scheduling o f a hybrid system. Another integrated aspect is to study the hybrid DARP from a strategic point o f view. In this chapter, we develop a model to aid decision-makers in designing a hybrid network that consists of two modes o f transportation. The transportation requests (trips) arrive in a unit square (service area) at a constant rate L The trips’ origin and destination locations are uniformly distributed within the square. Origin and destination locations are assumed to be independent. The square (service area) is partitioned into n2 zones, each a square o f side 1/n. The system consists o f two components: demand responsive service for pickup/delivery within each zone, and fixed route service for travel between zones. We assume that the number o f on-demand vehicles in each zone is the same. The purpose for the on-demand vehicles is to move passengers within the zone. The fixed route service follows a grid structure, as shown in Figure 9.1. The purpose o f the fixed route service is to transfer passengers traveling between zones. The objective is to develop an analytical model to determine the optimal num ber o f service zones n2. 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T l/n i . Figure 9.1. Hybrid network (n = 3). Note that if n = 1, there will be only one zone with no need for a fixed route service. As n increases, the more likely it will be for a passenger to require a fixed bus line to travel to their final destination. Clearly, from a passenger perspective, they would prefer less zones, thereby minimizing the need for a transfer. However, since typically the cost per passenger mile to an agency is much higher for an on- demand trip than for a fixed route trip, a transit agency from a cost perspective would prefer more zones. Our analytical model trades-off these two factors to determine the optimal number o f zones. First, we introduce some characteristics about the given system. Then, we state the model assumptions. Next, we present the relevant parameters and notation o f the model. W e then derive the relevant formulas and we optimize, n, for a defined set o f parameter values. Finally, we run a simulation to validate the variables derived in the analytical model. 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.2 N etw ork Analysis In this particular network, there are some characteristics that need to be taken into account before starting to solve the problem. These characteristics are related to the decision variable, n (Figure 9 .1). For example: 1. rc represents the number o f service zones (and the number o f bus stops) in the network. 2 . l/n 2 represents the area of each service zone in the network. 3. l/n represents the distance between every adjacent bus stops that are connected by a common fixed bus line. 4. 2n represents the total number o f fixed bus lines in the network. 5. n represents the number of bus stops on each fixed bus line. The above relationships play an important role in determining the total cost function in terms o f n. The decision variable n, which needs to be optimized in this problem, affects the network in different ways. For example, by increasing n, we will have: 1. More and smaller zones which leads to more bus stops. 2. More fixed bus lines which leads to more buses (number o f fixed bus lines = 2n). 3. Shorter distance between bus stops (distance = l/n). 4. Shorter average traveling distance from origin points to entry bus stops and from exit bus stops to destination points. 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5. Less likelihood to have trips with their pickup and delivery points located in the same zone. From the above facts, we conclude that there is a trade-off between two important desires, which are a better customer service and an inexpensive system. 9.3 Assumptions The assumptions o f the model are as follows: 1. Maximum number o f transfers for a customer during an entire trip is three. That is, a customer can only ride at most two on-demand vehicles and two buses to reach his/her final destination. 2. The network is symmetric and the distance between any two points in each zone is the length o f the straight line connecting them. 3. Ridesharing on an on-demand vehicle is not considered in the model. 4. The number o f on-demand vehicles in each zone is the same. 5. The trips are classified into three types: a- Type 1. Origin and destination are located in the same zone. These trips are served using strictly the dem and responsive (curb-to-curb) service. The probability of having this type o f trip is Pi. b. Type 2. Origin and destination are located in two zones that have a common fixed bus line. These trips can be satisfied by exactly one 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fixed bus line and two different on-demand vehicles, one at each zone. Also, in this type, the customer waits for the bus only once, which is at the entry bus stop. The probability o f having this type of trip is P?. c. Type 3. Origin and destination are located in two zones that do not have a common fixed bus line. These trips can be satisfied by exactly two fixed bus lines and two on-demand vehicles. Moreover, this kind o f trip has two alternative paths. In this type, the customer waits for the bus twice, first at the entry bus stop and second at the connection bus stop. The probability o f having this type o f trip is Pj. 6 . Based on assumption 5, the passenger can be in three states: in an on-demand vehicle, waiting at a bus stop, or in a fixed route bus. We assume the different states of a passenger may have a different cost to a passenger. That is, the inconvenience to a passenger for waiting at a stop for a bus may be higher than traveling on a bus. 9.4 Parameters and Notations The following are the parameters o f the model. A Arrival rate (customers/day) av Cost o f traveler time in vehicle ($/customer-unit time) cib Cost o f traveler time in bus ($/customer-unit time) as Cost o f traveler time at bus stop ($/customer-unit time) 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Expected total daily distance traveled on demand responsive vehicles in each zone The total cost of designing the network consists of three m ajor components: customer cost, on-demand vehicle cost and bus cost. The custom er cost has only a variable cost, which depends on the traveler time. However, each state of a customer has a different variable cost. The on-demand vehicle cost has both a variable and fixed cost component. The bus cost has only a fixed cost component. In computing the total miles traveled by an on-demand vehicle within a zone, we assume after a drop-off, a vehicle moves directly to pickup another passenger. Thus, for each pickup/drop-off trip within a zone (custom er type I) has an expected distance o f 2dv2, one trip empty to pickup the passenger and one trip with the passenger to drop them off. For customer types 2 and 3, the expected on- demand vehicle travel distance for each request is (3dv/+dv2). These calculations are an approximation since the pickup requests are not entirely uniformly distributed within each zone due to the fact that transferring passengers (types 2 and 3) are dropped off by the fixed route bus at the center o f the zone. Thus, the demand is no longer uniform with each zone. Figure 9.2 shows an illustration for computing the expected on-demand vehicle travel distance on a hybrid network consisting o f 4 zones (n =2). In this figure, we show two different requests. Customer type 1 needs to travel only within zone II. Custom er type 2 needs to travel from zone I to zone III. The circles 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mark the current location of the on-demand vehicles. As the figure shows, the expected on-demand vehicle travel distance is 2dvj for custom er type 1 and (3dvi+dv2) for customer type 2. The on-demand vehicle distance requirement for customer type 3 is similar to that for type 2. Q Bus stop □ Origin point ^ V ehicle location A Destination point Figure 9.2. Expected vehicle travel distance. Total Cost = Customer Cost + On-demand Vehicle Cost + Bus Cost Customer Cost = Type I Cost + Type 2 Cost + Type 3 Cost On-demand Vehicle Cost ~ Fixed Cost + Variable Cost Bus Cost = Fixed Cost 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Below we show that the total cost function unit is $/dciy. „ , _ „ customer _ . $ Total Cost Rate = *tirne*------ day customer — time $ . vehicle * --------------- * zone + vehicle zone — day $ ^ vehicle time $ . bus vehicle time day bus day = $ / day Total Cost = Pi *A *(i/v?/sv+/W v . + P 2 *&*[(2dvi/sv+2rv ) *av+( ivb)*as+( db/sb+rb) *ab ] + Pj* A.*[(2dvi/sv+2rv)*ai+(2wb)*as+(2db/sb+2rb)*ab] + F * ( t]) *(n2)+b*?i* f(l-P/)*((3dvi+ dv2 )/sv +2rv )+Pi*((2dv2)/sv+rv )] +Fb*(2n) (9.1) 9.5 Derivation of the Terms Used in the Total Cost Function In this section, we derive the terms used in the total cost function in terms o f n. These terms are d vt, dv2 , Pi, P2 , Pj, db, and \vb. 9.5.1 Average On-demand Vehicle Distance (dvi and dv2) In each zone, we would like to find the expected distances dv/ and dv2 as a function o f n. dvj is the expected distance from the bus stop, which represents the 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. center o f the zone, to any other point in the zone w hile dvj is the expected distance between any tw o points in the zone (Figure 9.3). l/n dv l/n Figure 9.3. Average distances dv( and dvi. Larson and Odoni (1981) showed that: dv, = 0.383*(l/n) (9.2) dv 2 = 0.5214*(l/n) (9.3) 9.5.2 Probability of the Three Customer Types Pi, P^, and P3 In this section, we find the probability of each type o f trip in terms o f n. The probability o f each custom er type equals its num ber o f outcomes divided by the total number of outcomes. Next, the number o f outcomes is determined with respect to the value o f n. Figure 9.4 shows different shapes o f the hybrid network according to different values for n. I. Outcome = Trip (origin zone, destination zone) (e.g. (2,3) o f a trip means that the origin point is in zone 2 and the destination point is in zone 3). 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.5.3 Average Bus Distance (db) Below we find the average distance traveled by the customers using the bus on one fixed bus line. This average distance is for customer types 2 and 3. Note that customer type 1 does not use buses, customer type 2 uses one fixed bus line (db), and customer type 3 uses two fixed bus lines (2*db). The tables below show the methodology o f calculating the average distance for different values o f n. Table 9 .1 classifies the bus trips for each value o f n according to the number o f bus stops that the passenger may visit and counts the number for each category. For example, when n equals 3, there are only two types o f bus trips, which are “ 1” when the passenger visits only one bus stop and “2 ” when the passenger visits exactly two bus stops. Also, note that there are 4 possible outcomes of the “ 1” type and 2 possible outcomes o f the “2” type. To illustrate the idea in more detail, the set o f all the possible outcomes o f the “2” type when n equals 3 is {(1,2), (2,1), (2,3), (3,2), (1,3), (3,1)}. Note that outcome (1,2) o f a trip means that the pickup point is located in zone 1 and the delivery point is located in zone 2. The following are new notations: Nn All possible outcomes of type 2 requests on a fixed bus line with n zones D„ Cumulative distance for all possible outcomes o f type 2 requests on a fixed bus line with n zones 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the previous step, the objective is to only count the number of each type of bus trip. These trips are distinct from each other in terms of the distance. In Table 9.2, the number o f each bus trip type is converted into distance by applying equation 9.9. Table 9.2. Length of bus trips # o f Visited Stops n = 2 n = 3 n = 4 n - 5 n = 6 n = 7 n - 8 n = 9 n = 10 1 1 1.333 1.5 1.6 1.667 1.714 1.75 1.778 1.8 2 1.333 2.0 2.4 2.667 2.857 3.00 3.111 3.2 3 1.5 2.4 3.000 3.429 3.75 4.000 4.2 4 1.6 2.667 3.429 4.00 4.444 4.8 5 1.667 2.857 3.75 4.444 5.0 6 1.714 3.00 4.000 4.8 7 1.75 3.111 4.2 8 1.778 3.2 9 1.8 D„ 1 2.667 5.0 8.0 11.667 16.000 21.00 26.667 33.0 By applying equation 9.10, Table 9.3 shows the average distance traveled by the passenger using the fixed route buses for different values o f n. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T abic 9.3. A verage bus trip # o f V isiled Stops n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 D n 1 2.667 5.000 8.0 1.667 16.000 21.000 26.667 33.000 2 6.000 12.000 20.0 30.000 42.000 56.000 72.000 90.000 dbn 0.5 0.444 0.^17 0.4 0.389 0.381 0.375 0.370 0.367 9.5.4 Average Waiting Time at the Bus Stop (wb) The expected waiting time o f passengers at the bus stop wb, who arrive at random times and are independent o f the bus schedule, has been derived by a number of authors including Welding (1957), Holroyed and Scraggs (1966), and Osuna and Newell (1972). The expected length o f time passengers will have to wait before a bus arrives is: wb = H/2*( 1 +Cv ) (9.1 1A) where H is the mean headway (service interval) and C V is the coefficient o f variation of the headways. Assuming that C V is zero, the expected passenger waiting time can be represented by one half o f the service interval at each bus stop. Assuming that every fixed bus line has only one bus moving on either direction on the line, a bus arrives at each bus stop every 2(n-l)/n unit distance. Therefore, 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w b = (n-1 )/(n*sb) {9.1 IB) However, if we have more than one bus in the fixed line and the time between these buses are equal, the average waiting time at the bus stop is (rt- l)/(n*Sb) divided by the number o f buses in the fixed line. 9.5.5 Fixed Cost of On-demand Vehicles To find the fixed cost o f the on-demand vehicles, we need to compute the expected total daily on-demand vehicle travel miles in each zone 8, which depends on the arrival rate and the probability o f the three custom er types. Equation 9.12 states that the vehicles fixed cost equals the cost o f having an on-demand vehicle times the number o f on-demand vehicles per zone times the number of zones. Equation 9.13 states that the number o f vehicle per zone, 7 7, equals the expected total daily on-demand vehicle travel miles in each zone divided by the maximum number o f miles that can be traveled by a vehicle. Although 7 7 is a discrete variable, we will assume it is a continuous variable in order to get a continuous function for n. Vehicles Fixed Cost = Fv*T}*n2 r\=F8/)il (9.12) (9.13) 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Below we determ ine the expected total daily on-dem and vehicle travel miles in each zone, 8: 8 = miles required to serve type 1 customers + miles required to serve type 2 customers + miles required to serve type 3 customers _ X * P, * ( 2 d v 2 ) X * P2 * (3z/v, + d v 2 ) X * P~ * Q d v x + d v 2 ) - l i ; n~ n n~ _ X * Px * (2 r/v,) ^ X * (1 — P ,) * (3d v x + d v 2 ) _ X * P{ * (2 * /v ,) X * (3 f/v ,) + X * (dv2) - X * P{ * (3 dvi )—X*Pt * (dv2) _ | . n~ n" _X* (3 dvx - f - dv2 )—X*Pi * (dv | + dv2) - ) n" _X*(3dv{ + dv2) 4- X*(\ln2) *'(dvx +dv2) n" * 1.6704* (X I n ) - 0.9044 *(X/n3) Thus, o = ----------------------;-------------------- (9.14) 9.5.6 Total Cost Function After finding all the terms needed in terms of n, the total cost rate function (Equation 9.1) can be stated as follows: 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Total Cost = (?HSr4 P S * } 0.766 n * s v + 2 r. k . + n*sh ( i < 1 -1 i, + n ,= <= i «-i ■+r,. * ^ 2 / n - - 2 n + \ f 0.766 _ + 2r, n * s v J < = i < I < > - i 2 ( n - l ) ) n s h +2 - £ 2 /(„ _ !) « ,=l — + r/ > I * < = i /r * 1.6704*(A /n) — 0.9044* (A/n3) ' bk n + 2 r, + — * n~ 1.0428 n * s v ■ + r, / - i + Ffi(2«) (9.15) 9.5.7 Convex Function Realizing the fact that any local minimum o f a convex function is also a global minimum, in this section we show the condition when the total cost function will be convex based on the parameter values. A function o f a single variable is convex if the second derivative is greater than or equal to zero. Also, for a convex function, the global minimum can be found using the first derivative by identifying the value o f the decision variable that makes the first derivative equal zero. df(n ) = 0 at n = n* (9.16) n 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the purpose o f sim plicity, vve set the average distance d b to its ending value o f 0.34 (Figure 9.5). N ote that dbioo~dbiootfd).?>4. 0.600 0.500 0.400 db 0.300 0.200 0.100 0.000 0 5 10 15 20 25 30 35 n Figure 9.5. db vs. n Below, we show the first and second derivative of the total cost function, which is simplified first and to have the following format: 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. /(«) = df{n) dn [X*av) ' X* a ' 0.5214 j ; ^ n3*s n2 \ V / + (**«,■) ' 2 *0.766 2 * 0.766 4r, 4r, N + —------^ n ' * sv n ■ ’ * sv n n" ' 0.68 0.68 2rf c 2r., X A " «- n J / _ ^ , n* sb n~*sh n n' {X*av) f X*a. > 0.766 2*0.766 0.766 . 4rv . 2rv . -------------- 1 ----------- 1 - 2 r ------ H — 7- n*.^. n2 *sv n3*sv v n n2 -f [ « n ' «3 J „ ■/0.68 2*0.68 0.68 . 4rf t 2rf e (A*ah) — ------ - t — + -T T- + 2rh - — + - f n nm n~sb n~ *sit ' X * Fv V 1.6704 0.9044 " j . /< A n ',i J+ (a *4 j6704 1 ,6 7 0 4■ 1 0 4 2 8 2r' - r> + I n*sv n3*sv n3*sv -+2r ------ + — ' V n 1 * 7 n~ + (2«) (9.17) : ( A * « i - ^ I 4 - 4 I n *s„ n X*a + (A*aJ ) 4*0.766 6*0.766 4rt . 8rt. -------------1 -------------------L-|----L 3 a. 4 ^ 2 3 n s n *s n n — T - 4 + 4 - 4 - V ( ;i*‘i. h ^ n~ n n ) /, * J 0.766 4*0.766 3*0.766 4r, 4rv ) (A *av) -----------—-----+ - $ - -----f- I n * sv n *sv n *sv n n , sb n n ) Y 2*0.68 2 * 0.68 , 4rh 4r„ V i H i * 3 * 2 3 [n *sb n *sh n n ^ (X* Fv 1.6704 | 3*0.9044 j | , J 1.6704 3 * 1.6704 3 * 1.0428 4rv 2r, — TT_ + — ri; — + ~ ir— r l rt *sv n *sv n *sv n n r 0.68 2*0.68 2rh 4rh N + . , + 3 n~ *sh n3 *sh rt~ n~ + 2 Fh (9-18) 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d 2J \ n ) d n 1 d Zf{n) dn2 = (**«>•) ' X*a X*av) 12 *0.5214 6 /;. N — -------+ “ T n * s t. n \ 1 / ( 12*0.766 24 *0.766 8r. 24r„ «4 * A',. n 5 * 5. . + Y 4 2 4 24 |+ (A * « J ( 2*0.68 6*0.68 4/) 12 r h \ — — TH r + - r - 4 \ n n n J n * s b n n / 2*0.766 12*0.766 12*0.766 8/;. I2rv - + ----:------------- ^ + ----r- n 3 * s.. « 4 *s„ n 5 *s„ + ' X*as V 12 36 _24^j s„ n 3 n 4 n 5 ) ' 4*0.68 6*0.68 8r„ 12rf c N H : ------------ r- + ----7- n 3 * s b « 4 n n ' X* F, V 2 * 1.6704 12 * 0.9044 A , ^ A ” 3 «s J+ [X*b) 2*1.6704 12*1.6704 12*1.0428 I2r( . 6 rv ----------------------------------------------H --------------------------------- — -H ------ i- /i3 * 5„ n5 * s v n5 * s v n4 n 4 (9.19) 8Xavrv 4Xar 1.36A# - + 4Xahrh + sb sh 12Aa. 4*0.68Aa 2*0.766^,, ■8Xa,.r„ Sv 12*0.766Xav 24Xa, 6*0.68Xah N 6Xavrv + --------------- 1 — 24Xavrv ------------------- s v Sf, Sb 12*0.766Xav 3 6 ^ , 6*0.68Aa6 - 12Xabrb ------------------ -+ \2 X a vrv +-------^ + ------------- - S v s h S b + 12Xab rb - 12Xbrv + 6 Xbrv 12*0.5214Xav 24*0.766Xav 24Xa, 12*0.766^, ----------------- s : ----------------------- — + ---------------- - 247m 12*0.9044AF 12*l.6704A£ 12*1.0428A2> £-------------------------------- i-------------------------------- + --------------------------- s b fi s v s v (9.20) 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. d-f{n) dn2 8 Aii, \36?uih - 4 Xahn h 'b 1.532k/, 3.3416XFV 3.3416Ab + — + 1 - + ------------- s v 12Aas x — 6Aavrv H -------: — 6Abrt 2.93522^, 12 * 0.9044AF, 7.531 V b ' (9.21) d'-fjn) dn2 1.5322a,. 3.3416AF, 3.341622? !22at ) + ------;----- — H ----- ;------ + — ‘ n3 jU n3 sv nA sh '82a, | 1 .36Aiih | 4Aahrh | 6Aiivrv ( 6A /?rt. ' n3Jf c n s. 2.93522a 10.8528AF, 7.5312 A 6 + z - + z - + - nss.. n5fx n \ (9.22) As mentioned earlier, the above total cost rate function is a convex if the second derivative is greater than or equal to zero for all the domain values of n. In other words, the function o f equation 9.22 is convex when the right-hand side (positive) is greater than or equal to the left-hand side (negative). Therefore, we can clear up the above equation by assigning values to the parameters in it. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.6 C om putational Experim ent In this section, we show a computational experiment for a given set o f parameter values. The purpose o f this experiment is to validate the model as well as to obtain the optimal value o f n. We define the parameter values. Then, we show a summary table, which contains the three types of probabilities and marginal costs for each value o f n. Finally, we present the final chart o f the total cost by which the optimal values are determined (Tables 9.4). The second derivative o f the total cost function will be represented by equation 9.23 if we assume a v = 0.25, cib = 0.5, as = 0.75, sv = 1.5, sb = 1, b - 0.25, rv = rb = 0.025, Fv = 100 and fi = 10. Note that A is neglected since it is a common factor. It is obvious that it will be greater than zero for all the values o f n. The domain of n starts from two since it is the minimum number that will provide a hybrid system. v „ £ 2 (9 .23) n ‘ n n n Therefore, the total cost function is a convex function based on the parameter values and any local minimum in the function is automatically a global minimum. Also, the first derivative can be used to determine the minimum values. However, since we are interested in a discrete value o f n and due to the approximation o f db in the function (Figure 9.5), we may determine the global 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T able 9.5. T otal cost values n Probability Marginal Cost Total Cost P. P: Pj Customer V ehicle Bus 2 0.250 0.500 0.250 $707 $7,359 $1,200 $9,266 3 0.111 0.444 0.444 $1,032 $5,334 $1,800 $8,166 4 0.063 0.375 0.563 $1,218 $4,115 $2,400 $7,733 5 0.040 0.320 0.640 $1,337 $3,336 $3,000 $7,673 6 0.028 0.278 0.694 $1,420 $2,800 $3,600 $7,820 7 0.020 0.245 0.735 $1,480 $2,412 $4,200 $8,092 8 0.016 0.219 0.766 $1,527 $2,117 $4,800 $8,444 9 0.012 0.198 0.790 $1,563 $1,887 $5,400 $8,850 10 0.010 0.180 0.810 $1,582 $1,702 $6,000 $9,283 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. According to Figure 9.6, the optimal value o f n is 5. Therefore, we need to design a network consisting o f 25 service zones and 10 fixed bus line. 9500 i 9000 8500 8000 7500 - 7000 0 1 2 3 4 5 6 7 8 9 10 11 12 n Figure 9.6. Total cost function vs. n. 9.7 Simulation The primary objective o f implementing the simulation is to validate the results obtained by the analytical model. Specifically, we would like to validate the probability o f each type o f custom er trip, the expected distances traveled by the on- demand vehicle within each zone dvi and dv2, and the expected distance traveled by the custom er on the fixed route bus db. Figure 9.7 shows a sample o f the simulated 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Use these random num bers to determ ine the (x (, y l) coordinate o f the origin point and the (x2, y2 ) coordinate o f the delivery point. a. Example: X|=Ui * 30, yi=U2 * 30, X 2=U 3 * 30, y2=U4 * 30 4. According to each point’s coordinate, identify the corresponding zone of each point. 5. Find the request type based on its pickup and delivery zones. 6 . Find the distance between each point and the bus stop in its zone dv/ and the distance between the origin and destination points o f type 1 customers dv2 using dis tan ce = y f(x [ - x 2 )2 + (y, - y 2 )2 7. For each request, Find the distance between its two zones’ bus stops on the fixed bus lines (not the direct distance). 8 . Run the simulation 10 times and find the required output. 9. Find the average and standard deviation o f the output. 10. Using a significance level of 5%, find the confidence interval o f the outputs. 11. Compare the simulation output with the theory output. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.7.1 C om parison Between Sim ulation and A nalytical M odel This section compares the results obtained by the simulation with those obtained by the analytical model when the value of n equals 3 (9 zones) and the number of requests equals 1000. 9.7.1.1 Analytical model results. The following are the values obtained from the analytical model: a. P, = 1/n2 = 1/9= 11.1% b. Pi - 2(n-l)/n2 = 4/9 = 44.4% c. P3 = [nz-2(n-l )-l ]/n2 = 4/9 = 44.4% d. dv, = 3.83 dv 2 = 5.214 db = 13.32 9.7.1.2 Simulation results. Table 9.6 shows a summary o f the simulation results. A 95% confidence interval o f each value is given at the bottom o f table. Note that the probability values are in a percentage format (e.g., we are 95% confident that Pi falls within the interval from 10.6% to 11.4%). 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 9.6. Sum m ary o f the sim ulation results Run P, P2 Pj dv, dv2 db 1 11.60 4 5 .5 0 4 2.90 3.93 5.73 13.36 2 10.80 4 5.20 44.00 3.85 5.40 13.40 3 12.20 4 4 .6 0 43.20 3.84 5.16 12.85 4 10.80 43.10 46.10 3.89 5.42 13.16 5 11.70 45.50 4 2.80 3.80 5.33 13.27 6 10.50 43.20 4 6.30 3.87 5.16 13.42 7 10.80 4 6 .6 0 42.60 3.83 5.06 13.48 8 10.80 4 4.30 44.90 3.84 5.24 13.18 9 10.30 4 3.40 4 6.30 3.89 5.54 13.39 10 10.50 4 4 .7 0 44.80 3.71 5.57 13.15 Average 11.00 44.61 44.39 3.85 5 .36 13.27 St. Dev. 0.62 1.14 1.50 0.06 0.21 0.19 KO.025,9) 2.26 2.26 2.26 2.26 2.26 2.26 Lower value 10.60 4 3 .8 0 43.30 3.80 5.20 13.10 Upper value 11.40 4 5 .4 0 45.50 3.89 5 .50 13.40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A PTER 10 C O N CLU SIO N S 10.1 C ontribution Since most dial-a-ride programs for the transport of elderly and disabled persons are heavily subsidized programs, the increased usage o f these curb-to-curb services has put significant budget pressures on most transit agencies. In this dissertation, we have investigated a hybrid system consisting of both on-demand vehicles and fixed route lines for servicing this type o f request. We developed a heuristic solution approach for scheduling a hybrid system. O ur initial solution is first derived using an Insertion procedure. The Insertion procedure consists o f two phases. In the first phase, all the candidate routes/paths that meet a certain criterion for each request are identified. In the second phase, a feasible path from the candidates’ list that has the shortest on- demand vehicle distance is selected and inserted into the vehicle schedule. The solution o f the Insertion procedure is fed into the Improvement procedure. In this procedure, we try to identify an alternative path for requests that have multiple hybrid paths that can satisfy the demand. The solution from the Improvement procedure is the initial solution in the Tabu Search. W e tested our heuristics on actual data from A VTA. The most effective heuristic in terms o f the trade-off between solution quality and computation time 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was TABU-S. That is, the improvement o f the solution found from the Tabu Method for re-sequencing over the initial solution found by the Improvement procedure justified the extra computation time for TABU-S. However, the extra computation time needed to evaluate the re-assignment of requests to vehicles does not justify the marginal improvement in the solution quality. Overall, this analysis showed that shifting some of the demand to a hybrid service route (18.6% of the requests) reduces the on-demand vehicle distance by 16.6% and the overall customer trip time by 8.7% over the manual schedule using the TABU-S heuristic. However, for these customers who take the hybrid delivery method (18.6% o f the requests), their trip time will increase on average by 5.4%. In addition to developing algorithms to improve the scheduling o f a hybrid system, we also develop a model that aids decision-makers in designing a hybrid network. That is, we determine the optimal number o f zones in an area where each zone is served by a number o f on-demand vehicles, which transfer passengers to a fixed route line if the destination is in a different zone or to its final destination if it is within the same zone. 10.2 Future Research There are many directions to extend this research. One o f the directions is to consider the stochastic process in solving the hybrid problem. The uncertainty in the DARP comes from many sources such as the traffic conditions, driver 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. familiarity with addresses, passenger time o f showing up and getting in and out of vehicle, weather conditions, and vehicle breakdowns. Therefore, a number of variables in the problem may be changed to be stochastic variables instead o f being deterministic. The following are some o f these variables: 1. The trip time of the on-demand vehicle between any two points in the network. This variable is the most important one because it is affected by many conditions and at the same time play a significant role in determining the problem solution. 2. The arrival time o f the buses at the bus stops. The problem is going to be more complicated by considering the uncertainty of these two variables together. 3. In many instances, passengers are not home when an on-demand vehicle arrives to pick them up. Hence, we need to consider the probability that the customer is present when the vehicle arrives. 4. The time needed for a passenger to get in and out from a vehicle. This time is significant due to the fact that the passengers are elderly and disabled. Another direction is to solve the dynamic hybrid DARP where the transportation requests may be received during the service process. Thus, the on- demand vehicle schedule can be adjusted after dispatching. This kind o f real time scheduling has the potential to improve the system productivity and/or reduce the operating cost. It can be implemented with the aid o f deploying Intelligent 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Transporting Systems such as mobile data terminaJs (MDT), automatic vehicle location devices (AVL), and advanced wireless communications. One more direction to extend this work is to develop an exact approach that can be used to find the optimum solution for the hybrid DARP. Although, the exact approach can be only employed for small sized problems, it has two significant aspects. The first one is the theoretical analysis o f the problem by providing lower bounds to the problem. The second is the ability o f using it for practical sized problems when they can be divided, with the help o f clustering techniques, into small sized problems. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fiala Timlin, M. T., & Pulleyblank, W. R. (1990). Precedence constrained routing and helicopter scheduling: Heuristic design. Institute for Computer Research, University of Waterloo, Ontario, Canada, UW/ICR 90-02. Haines, G. H., & Wolff, R. N. (1982). Alternative approaches to demand responsive scheduling algorithms. Transportation research, Part A, 16, 43- 54. Hickman, M., & Blume, K. (2000). A method for scheduling integrated transit service. 8lh International Conference on Computer-Aided Scheduling o f Public Transport (CASPT). Higgins, L., Laughlin, J. B., & Turnbull, K. (2000). Automatic vehicle location and advanced paratransit scheduling at Houston METROLife. Proceeding of the 2000 Transportation Research Board Conference, W ashington, D.C. Ioachim, I., Desrosiers, J., Dumas, Y., Solomon, M. M., & Villeneuve, D. (1995). “A request clustering algorithm for door-to-door handicapped transportation. Transportation Science, 2 9 ,63-78. Jaw, J. J., Odoni, A. R., Psaraftis H. N., & Wilson, N. H. M. (1986). A heuristic algorithm for the multi-vehicle many-to-many advance-request dial-a-ride problem with time windows. Transportation Research, Part B, 20, 243- 257. Kalantari, B. Hill, A. V., & Arora, S. R. (1985). An algorithm for the traveling salesman problem with pickup and delivery customers. European Journal o f Operations Research, 22, 377-386. Lave, R. E., Teal, R., & Piras, P. (1996). A handbook for acquiring demand- responsive transit software. Transit Cooperative Research Program Report #18, Transportation Research Board, Washington, D. C. Levine, C. J. (1997). ADA and the demand for paratransit. Transportation Quarterly, 5/(1), 29-43. Liaw, C., W hite, C., & Bander, J. (1996). A decision support system for the bimodal dial-a-ride problem. IEEE, 2 6 ,552-565. Madsen, O. B. G., Raven, H. F., & Rygaard, J. M. (1995). A heuristic algorithm for a dial-a-ride problem with time windows, multiple capacities, and multiple objectives. Annals o f Operations Research 60, 193-208. 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Nanry, W. P., & Barnes, J. W. (2000). Solving the pickup and delivery problem with time windows using reactive tabu search. Transportation Science, Part B, 34, 107-121. Osuna, E. E., & Newell G. F. (1972). Control strategies for an idealized public transportation system. Transportation Science, 6, 57-72. Psaraftis, H. (1980). A dynamic programming solution to the single vehicle many- to-many immediate request dial-a-ride problem. Transportation Science, 14, 130-154. Psaraftis, H. (1983a). An exact algorithm for the single vehicle many-to-many dial-a-ride problem with time windows. Transportation Science, 17, 351- 357. Psaraftis, H. (1983b). Analysis o f an 0 (N 2) heuristic for the single vehicle many- to-many Euclidean dial-a-ride problem. Transportation Research, 17B, 33- 145. Psaraftis, H. (1986). Scheduling large-scale advance-request dial-a-ride systems. American Journal o f M athematical and Management Science, 6, 327-367. Rardin, R. (1998). Optimization in operations research. Upper Saddle River, NJ: Prentice Hail. Rochat, Y., & Taillard, E. (1995). Probabilistic diversification and intensification in local search for vehicle routing. Journal o f Heuristics, 1, 147-167. Ruland, K. S., & Rodin, E. Y. (1997). The pickup and delivery problem: Faces and branch-and-cut algorithm. Computers Mathematics Application, 33, l- 13. Sexton, T., & Bodin, L. (1985a). Optimizing single vehicle many-to-many operations with desired delivery times: I Scheduling. Transportation Science, 19, 378-410. Sexton, T., & Bodin, L. (1985b). Optimizing single vehicle many-to-many operations with desired delivery times: II Routing. Transportation Science, 1 9 ,411-435. Stein, D. M. (1978). An asymptotic probabilistic analysis of a routing problem. M athematics o f Operations Research, 3, 89-101. 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Aldaihani, Majid Mohammad (author)

Core Title Hybrid scheduling methods for the general routing problem

School Graduate School

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, industrial,OAI-PMH Harvest,operations research

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Dessouky, Maged (committee chair), Bukkapatnam, Satish (committee member), McBride, Richard (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-210339

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Legacy Identifier 3073738.pdf

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Rights Aldaihani, Majid Mohammad

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