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A continuous approximation model for the parallel drone scheduling traveling salesman problem

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A continuous approximation model for the parallel drone scheduling traveling salesman problem

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Content A continuous approximation model for the parallel drone scheduling traveling salesman problem by Haochen Jia A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (INDUSTRIAL AND SYSTEMS ENGINEERING) December 2022 Copyright 2022 Haochen Jia Acknowledgements During my latest four years at the University of Southern California, I got so much help from people there. I would like to express my deepest appreciation to my supervisor, Professor John Gunnar Carlsson. He has a brilliant personality, amazing research ability, and passion for his work and life. I am so fortunate that I was guided by Professor Carlsson in my 4-year Ph.D. life. I would not be where I am today without him. I am also grateful to committee members - Professor Vishal Gupta, Professor Andres Gomez, Professor Qiang Huang, and Professor Ashutosh Nayyar. Their invaluable insights and feedback during my qualify- ing exam contribute a lot to this thesis. Thanks should also go to my genius labmates Jiachuan Chen, Bo Jones, Mohammadjavad Azizi, Ying Peng, Shannon Sweitzer-Siojo, and Yue Yu. I also appreciate the help from other ISE people and my friends. Last but not least, I would like to take this opportunity to thank my patients and my families for their support. At the same time, I want to thank my girlfriend Amy Tang, who always uses her unconditional care, love, and help. Meeting her is the most beautiful thing that has ever happened to me. ii TableofContents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Drone-based TSP Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Related Problem of PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Exact solutions for drone-assisted delivery problem . . . . . . . . . . . . . . . . . . . . . . 5 2.3.1 Exact solutions for the PDSTSP problem . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3.2 Exact solutions for the FSTSP problem . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Heuristic approaches for drone-assisted delivery problems . . . . . . . . . . . . . . . . . . 7 2.4.1 Heuristic approaches for PDSTSP problem . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.2 Heuristic approaches for FSTSP problems . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Continuous Approximation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Chapter 4: Continuous approximation analysis of PDSTSP . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 A minor refinement for model proportionality . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Bounds for (4.3) with uniform demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2.1 Proof of the upper bound (4.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2.2 Proof of the lower bound (4.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.3 Bounds independent of the shape of service region . . . . . . . . . . . . . . . . . . 23 4.3 Bounds for (4.3) with arbitrarily distributed demand . . . . . . . . . . . . . . . . . . . . . . 25 4.3.1 Proof of the upper bound (4.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3.2 Proof of the lower bound (4.11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 5: Supplemental analysis of PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1 Impact of customer distribution to PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.1.1 Lower bound of PDSTSP and TSP ratio . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.1.2 Upper bound of PDSTSP and TSP Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 39 iii 5.2 Optimal depot selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Chapter 6: PDSTSP with truck-only points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.1 Modification of Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Upper bound of PDSTSP with truck only points . . . . . . . . . . . . . . . . . . . . . . . . 46 6.3 Lower bound of PDSTSP with truck only point . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 7: Min-sum PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1 Modification of Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Proof of upper bound (7.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.3 Proof of lower bound (7.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.4 Converting min-sum PDSTSP to TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.4.1 Converting min-sum PDSTSP to prize collecting TSP . . . . . . . . . . . . . . . . . 57 7.4.2 Converting PCTSP to the generalized TSP . . . . . . . . . . . . . . . . . . . . . . . 58 7.4.3 Convert generalized TSP problem to TSP . . . . . . . . . . . . . . . . . . . . . . . 59 Chapter 8: PDSTSP variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.1 Multiple depots PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.2 Multiple trucks PDSTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Chapter 9: Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.1 Simulated data within a square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 9.2 Real-world applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Chapter 10: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Proof of Theorem 8 for a continuous density function . . . . . . . . . . . . . . . . . . . . . . . . 82 iv ListofTables 2.1 Summary of related works (Part 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Summary of related works (Part 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 v ListofFigures 2.1 The comparison between TSP tour, PDSTSP tour, and FSTSP tour for serving 30 customers by 3 drones. Figure (a) shows a traveling salesman tour of a set of a customer set and a central depot. It is the shortest Hamiltonian cycle that starts from the central depots, visits all customers, and back to the central depot. Figure (b) shows a PDSTSP tour, with the collaboration between drones and the truck. drones serve a group of customers and the truck does a TSP tour to serve the rest customers. This design could reduce the number of customers served by the truck. The truck reduced about a third of customers. Figure (c) shows an FSTSP tour. drones depart from the truck at certain customer points, make a delivery, and reunite with the truck after delivery. This design needs to coordinate the time between the truck and the drones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4.1 The functionp7→ 2plogα +p− (1− p)log(1− p)− plog2+plogπ is negative for all 00. Define L(X 1 ,...,X n )= min 0∈S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    to be the makespan of a PDSTSP tour of the pointsX 1 ,...,X n . We have min 0≤ p≤ 1 max ( αp, 2 t ZZ ∥x∥≤ r(p) ∥x∥dx ) ≤ liminf n→∞ L(X 1 ,...,X n ) √ n (4.4) ≤ limsup n→∞ L(X 1 ,...,X n ) √ n (4.5) ≤ min 0≤ p≤ 1 max ( βp, 2 t ZZ ∥x∥≤ r(p) ∥x∥dx ) , (4.6) where we define α =0.2935,β is the constant from Theorem 2, andr(p) is the solution to the equation area({x∈R :∥x∥≤ r(p)})=1− p. Note that the upper and lower bounds are guaranteed to lie within a factor of β/α < 3.15 of one another, sinceβ < 0.9204. 19 4.2.1 Proofoftheupperbound(4.6) Suppose that customer points are all uniform samples in a compact regionR having area 1. A natural strategy is anapriori districting approach, meaning that we identify a sub-regionS ⊂R that contains the origin (i.e. the depot) and assigns all points withinS to the truck, and all points outsideS to the drone. The objective value of this strategy is max    TSP({X i }∩S), 2 t √ n X X i / ∈S ∥X i ∥    and it is routine to analyze the limiting behavior of the two components above as n → ∞: it is a straightforward consequence of the Theorem 2 that lim n→∞ 1 √ n TSP(X∩S)=β area(S) with probability one, and the law of large numbers tells us that lim n→∞ 1 n X x i / ∈S ∥x i ∥= ZZ R\S ∥x∥dx with probability one. Therefore, for any regionS, the objective value obj associated with districting satisfies lim n→∞ obj √ n =max ( β area(S), 2 t ZZ R\S ∥x∥dx ) with probability one. Since the first term β area(S) only depends on the area ofS and not its shape, it is obvious that the optimalR\S should be a disk centered at the origin (the depot). For any fraction 0≤ p≤ 1, adopting the functionr(p) as in the statement of Theorem 6, we have 20 max ( β area(S), 2 t ZZ R\S ∥x∥dx ) =max ( βp, 2 t ZZ ∥x∥≤ r(p) ∥x∥dx ) , which establishes the upper bound (4.6). 4.2.2 Proofofthelowerbound(4.4) To obtain a lower bound of the problem (4.3), we find it is helpful to explicitly take the cardinality |S| into consideration. Of course, we are welcome to rewrite (4.3) with an additional variablep that represents the fraction of nodes assigned to the truck: minimize 0≤ p≤ 1,0∈S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    s.t. |S|≥ pn, and we will relax this problem by decoupling the two terms in themax{·,·} operator: minimize 0≤ p≤ 1;S 1 ,S 2 ⊂{ X 1 ,...,Xn} max    TSP(S 1 ), 2 t √ n X x i ∈S 2 ∥x i ∥    s.t. (4.7) |S 1 |≥ pn |S 2 |≥ (1− p)n. 21 The above problem is clearly a relaxation because we have not made any restrictions betweenS 1 and S 2 (in particular, they are allowed to overlap). For fixed p, we will divide by √ n as in the preceding section, and determine the limiting behavior of the two terms separately; that is, we will study liminf n→∞ 1 √ n min S 1 :|S 1 |≥ pn TSP(S 1 ) and 2 t liminf n→∞ 1 n min S 2 :|S 2 |≥ (1− p)n X x i ∈S 2 ∥x i ∥. The second term is straightforward because the law of large numbers guarantees that lim n→∞ 1 n min S 2 :|S 2 |≥ (1− p)n X x i ∈S 2 ∥x i ∥= ZZ ∥x∥≤ r(p) ∥x∥dx wherer(p) is defined as in Theorem 6. To address the first term, we require the following additional observation: Lemma 7. Let{X i } be a sequence of independent uniform samples are drawn from a planar region of the unit area. For all fixed 0≤ p≤ 1, we have liminf n→∞ min S:|S|≥ pn TSP(S) √ n >αp (4.8) with probability one, whereα :=0.2935, whereS denotes a subset of{X 1 ,...,X n }. Proof. LetE n be the event thatmin S:|S|≥ pn TSP(S)<αp √ n. We will apply the union bound to Corollary 5. Note that the number of possible subsets of cardinality⌈pn⌉ is n ⌈pn⌉ , which satisfies 22 n ⌈pn⌉ = n! ⌈pn⌉!(n−⌈ pn⌉)! =⇒ log n ⌈pn⌉ ≤ log(n!)− log(⌈pn⌉!)− log((n−⌈ pn⌉)!) =− (plogp+qlogq)n+O(logn) by Stirling’s approximation (Lemma 4), whereq =1− p. Therefore, the union bound says (together with Lemma 5) that Pr(E n )≤ n ⌈pn⌉ Pr(TSP(X 0 ,X 1 ,...,X ⌈pn⌉ )≤ αp √ n) ≤ n ⌈pn⌉ ⌈pn⌉!(2π ) ⌈pn⌉ (αp √ n) 2⌈pn⌉ (2⌈pn⌉)! =⇒ logPr(E n )≤ (2plogα +p− qlogq− plog2+plogπ )n+O(logn). The above expression approaches−∞ asn→∞ because the coefficient of n is negative for allp; see Figure 4.1. Furthermore, this guarantees that Pr(E n ) ≤ a − n for some a > 1, so that P ∞ n=1 Pr(E n ) < ∞. The Borel-Cantelli Lemma then establishes that α 2 < liminf n→∞ L(X 1 ,...,X n ;pn)/(p √ n) with probability one, which completes the lower bound (4.4), and therefore the proof of Theorem 6. 4.2.3 Boundsindependentoftheshapeofserviceregion We were unable to derive a purely algebraic closed-form expression in Theorem 6 because our bounds depend on the shape ofR, through the function r(p). If the service regionR is a disk of the unit area centered at the origin, or if the depot is centered in the middle ofR and the optimal value ofp is small, then it is sensible to approximater(p)= p p/π , meaning that the ball of radiusr(p) is contained entirely 23 0 0.2 0.4 0.6 0.8 1 − 1 − 0.8 − 0.6 − 0.4 − 0.2 0 Figure 4.1: The function p 7→ 2plogα + p− (1− p)log(1− p)− plog2 + plogπ is negative for all 00. Define L(X 1 ,...,X n )= min 0∈S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    (4.10) to be the cost of a PDSTSP tour of the pointsX 1 ,...,X n . We have max 0≤ λ ≤ 1 ZZ R min λα p f(x), 2(1− λ ) t ∥x∥f(x) dx≤ liminf n→∞ L(X 1 ,...,X n ) √ n (4.11) ≤ limsup n→∞ L(X 1 ,...,X n ) √ n ≤ max 0≤ λ ≤ 1 ZZ R min λβ p f(x), 2(1− λ ) t ∥x∥f(x) dx, (4.12) 26 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 t Objective value c =α c =β Figure 4.4: Bounds for the objective value of (4.9), obtained by setting p = p ∗ and using the fact that α ≤ c≤ β . whereα andβ are as in Theorem 6. As in Theorem 6, the upper and lower bounds are guaranteed to lie within a factor ofβ/α < 3.15 of one another, sinceβ < 0.9204. 4.3.1 Proofoftheupperbound(4.12) We derive the upper bound (4.12) using a similar a priori districting approach to Section 4.2.1. Suppose that we again divide the service regionR into a "truck region”S and a "drone region"R\S. The objective valueobj associated with such a districting strategy satisfies lim n→∞ obj √ n =max ( β ZZ S p f(x)dx, 2 t ZZ R\S ∥x∥dx ) . For notational compactness, we will set u(x) = β p f(x) and v(x) = 2 t ∥x∥f(x); the lemma below de- scribes the optimal shape ofS: 27 Lemma9. For any sub-regionS ⊂R and any0≤ λ ≤ 1, we have max ( ZZ S u(x)dx, ZZ R\S v(x)dx ) ≥ ZZ R min{λu (x), (1− λ )v(x)}. (4.13) Moreover, there exists a pairingS ∗ , λ ∗ such that equality holds, whereS ∗ = {x ∈ R : λ ∗ u(x) ≤ (1− λ ∗ )v(x)}. Proof. This is a standard duality argument from vector space optimization; we provide direct proof here. For anyλ , define R u ={x∈R :λu (x)≤ (1− λ )v(x)} and let R v ={x∈R :λu (x)> (1− λ )v(x)} be its complement. Following directly from the definitions, we have max ( ZZ S u(x)dx, ZZ R\S v(x)dx ) = max 0≤ λ ′ ≤ 1 λ ′ ZZ S u(x)dx+(1− λ ′ ) ZZ R\S v(x)dx ≥ λ ZZ S u(x)dx+(1− λ ) ZZ R\S v(x)dx = ZZ S λu (x)dx+ ZZ R\S (1− λ )v(x)dx ≥ ZZ Ru λu (x)dx+ ZZ Rv (1− λ )v(x)dx = ZZ R min{λu (x), (1− λ )v(x)} dx which yields the inequality (4.13). To find the pairing S ∗ ,λ ∗ , we note that the function λ 7→ ZZ R min{λu (x), (1− λ )v(x)} dx 28 is differentiable in λ becausef is continuous, and its derivative is equal to d dλ ZZ R min{λu (x), (1− λ )v(x)} dx = ZZ R d dλ min{λu (x), (1− λ )v(x)} dx = ZZ R            u(x) ifx∈R u − v(x) otherwise            dx = ZZ Ru u(x)dx− ZZ Rv v(x)dx. LetR ∗ u andR ∗ v denote the subsetsR u ,R v associated withλ ∗ . Since the above derivative has to be zero at optimality, we know that RR R ∗ u u(x)dx = RR R ∗ v v(x)dx, yielding an objective value of ZZ R min{λ ∗ u(x), (1− λ ∗ )v(x)} dx =λ ∗ ZZ R ∗ u u(x)dx+(1− λ ∗ ) ZZ R ∗ v v(x)dx = ZZ R ∗ u u(x)dx. We find that setting S ∗ = R ∗ u completes the proof, because the two terms in the max{·,·} operator below are equal: max ( ZZ S ∗ u(x)dx, ZZ R\S ∗ v(x)dx ) =max ( ZZ R ∗ u u(x)dx, ZZ R ∗ v v(x)dx ) = ZZ R ∗ u u(x)dx This completes the proof of the upper bound (4.12). 29 4.3.2 Proofofthelowerbound(4.11) To derive the lower bound (4.11), we will first prove the desired result for the case where the density f is a step function; a standard coupling argument will then complete the analysis. Letϕ (x)= P s i=1 a i 1(x∈ ⊡ i ) be a step function withs disjoint rectangular components such that all components have equal mass, i.e. a i area(⊡ i ) = 1/s. For an independent collection of samplesX 1 ,...,X n , letP n i denote the fraction of points in component⊡ i that are assigned to the drone in an optimal solution to (4.10). The s-tuples P n :=(P n 1 ,...,P n s ) all lie in the compact set[0,1] s , and therefore, asn→∞, they have (with probability one) a convergent subsequence with a limitP n → p. The reason this is useful to us is because we are now free to constrain the number of points per component that is assigned by the drone, by solving the problem minimize 0∈S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x j / ∈S ∥x j ∥    s.t. (4.14) |S∩⊡ i |≤ (p i +ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (4.15) |S∩⊡ i |≥ (p i − ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (4.16) for a fixed p∈[0,1] s andϵ> 0 (if constraints (4.15) and (4.16) are infeasible, we set the objective value to ∞, though this is not of any consequence since the law of large numbers guarantees that each component ⊡ i contains∼ n/s points asn→∞). We next recall Lemma 3, which tells us that TSP(S)≥ s X i=1 TSP(S∩⊡ i )−O (1) 30 for allS. In addition, for each rectangular component⊡ i , letr i =min x∈⊡ i ∥x∥ be the minimum distance between⊡ i and the depot, and let δ :=max i max x,x ′ ∈⊡ i ∥x− x ′ ∥ be the maximum diameter of all the⊡ i ’s, which can be made arbitrarily small. Re-indexing in terms of ⊡ i ’s, it is certainly true that X x j / ∈S ∥x j ∥= s X i=1 X x j ∈⊡ i x j / ∈S ∥x j ∥ (4.17) ≥ s X i=1 |(X\S)∩⊡ i |r i , (4.18) and of course, the proportionality constraints (4.15) and (4.16) tell us that |(X\S)∩⊡ i |≥ (1− p i − ϵ )|X∩⊡ i | ∼ (1− p i − ϵ )a i area(⊡ i )n asn→∞ with probability one for alli. Thus, in summary, our problem of interest is now written as the following variation of (4.14): minimize 0∈S⊂{ X 1 ,...,Xn} max ( s X i=1 TSP(S∩⊡ i ), 2 √ n t s X i=1 a i area(⊡ i )(1− p i − ϵ )r i ) s.t. |S∩⊡ i |≤ (p i +ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i |S∩⊡ i |≥ (p i − ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i. 31 The only work remaining is to study the termsTSP(S∩⊡ i ): each component⊡ i satisfies |X∩⊡ i |/n→ 1/s with probability one asn→∞, and hasarea(⊡ i ) = 1/(sa i ). If we visit at least a fractionp i − ϵ of X∩⊡ i with a TSP tour, then Lemma 7 together with routine scaling arguments establish that liminf n→∞ TSP(S∩⊡ i ) √ n ≥ α √ a i area(⊡ i )(p i − ϵ ) whereα =0.2935. Hence, we find that liminf n→∞ 1 √ n max ( s X i=1 TSP(S∩⊡ i ), 2 √ n t s X i=1 a i area(⊡ i )(1− p i − ϵ )r i ) (4.19) ≥ max ( α s X i=1 √ a i area(⊡ i )(p i − ϵ ), 2 t s X i=1 a i area(⊡ i )(1− p i − ϵ )r i ) (4.20) ≥ max ( α s X i=1 √ a i area(⊡ i )p i , 2 t s X i=1 r i (1− p i )a i area(⊡ i ) ) − ϵ s X i=1 (α √ a i +r i a i )area(⊡ i ) | {z } (∗ ) (4.21) where(∗ ) can be made negligible by selecting a sufficiently small ϵ . The problem of selecting an optimal proportion vectorp ∗ is the linear program minimize 0≤ p≤ 1 max ( α s X i=1 √ a i area(⊡ i )p i , 2 t s X i=1 r i (1− p i )a i area(⊡ i ) ) 32 whose dual problem is equivalent to maximize 0≤ λ ≤ 1 s X i=1 min λα √ a i area(⊡ i ), (1− λ ) 2 t r i a i area(⊡ i ) = s X i=1 min λα √ a i , (1− λ ) 2 t r i a i area(⊡ i ) = Z R min ( λα p ϕ (x),(1− λ ) 2 t s X i=1 r i a i 1(x∈⊡ i ) ) dx ≥ Z R min λα p ϕ (x),(1− λ ) 2 t ∥x∥ϕ (x) dx− 2δ/t |{z} (∗ ) where, as before, (∗ ) can be made negligible by selecting a sufficiently small δ . This completes the proof of the lower bound (4.11) for the case where f is a step function. The remaining work for general f is a standard coupling argument and is essentially identical to the proof of Lemma 7.3 in [51], or Step 2 of the proof of Theorem 2.4.2 in [46]; we include the argument in Section A of the Appendix for the sake of completeness. 33 Chapter5 SupplementalanalysisofPDSTSP According to our previous PDSTSP assumptions, we found that the makespan of PDSTSP depends on the following parameters: the truck speed, the drone speed, the distribution of the customer, the number of the customer, and the distance function. In the previous section, we discussed the impacts of speeds and the number of customers. In this section, we will discuss the impact of the customer distribution and the depot location on the entire PDSTSP problem. 5.1 ImpactofcustomerdistributiontoPDSTSP Assume we have a fixed value n of customer quantity and a fixed value t for drone speed. Then we will discuss the ratio between the PDSTSP and TSP. According to (4.11) and (4.12), we could conclude that∃c∈[α,β ], PDSTSP = √ n max λ ∈[0,1] ZZ R min(λc p f(x), 2(1− λ ) t ∥x∥f(x))dx 34 By Theorem 2, the target metric (i.e. the ratio between the performance with and without a drone) is: PDSTSP TSP = max λ ∈[0,1] RR R min(λc p f(x), 2(1− λ ) t ∥x∥f(x))dx RR R β p f(x)dx (5.1) 5.1.1 LowerboundofPDSTSPandTSPratio Theorem10. Assume the diameter of theR isD. Then exists density function f is an absolutely continuous function with bounded support such that lower bound of (5.1) max λ ∈[0,1] RR R min(λc p f(x), 2(1− λ ) t ∥x∥f(x))dx RR R β p f(x)dx (5.2) could still not converge to 1 with a sufficiently small t. Intuitively, let us consider the feature of f which could make (5.2) as small as possible. When the speed of drone is fixed, (5.2) will depend on ∥x∥ function andf function. We should find out the density relationship between the density function and distance function. Lemma11. One of the optimal distributionf for (5.1) has the following necessary condition: ∀x,y∈R,∥x∥<∥y∥→f(x)>f(y) (5.3) ∀x,y∈R,∥x∥=∥y∥→f(x)=f(y) (5.4) 35 Proof. Let us prove (5.3) first: Assume∃x,y∈R,∥x∥<∥y∥ andf(x)≤ f(y), then consider the density function ˆ f(z)=                      f(z) z̸=x,z̸=y f(x) z =y f(y) z =x In this way, the TSP makespan of customers with distributionf is equivalent to TSP makespan of customers with distribution ˆ f. The PDSTSP makespan of customers with distribution f is always no better than customers with distribution ˆ f which has a non-higher drone time cost. Therefore, we could always construct a distribution that optimizes (5.1) and satisfies (5.3). Now let us prove (5.4) : Assume∃x,y ∈ R,∥x∥ = ∥y∥ and f(x) ̸= f(y). Without generality, we assume f(x) < f(y). If ∥x∥ =∥y∥ = 0, then clearlyf(x) = f(y) which implies contradiction. Otherwise, let us construct a ball B centered at y. It must exist z ∈ B such that 0 < ∥z∥ < ∥y∥ and therefore f(x) > f(z) according to (5.3). Consider the segment yz, because f is absolutely continuous, we could find a point w where f(w)=f(x),∥w∥<∥x∥ which contradict to (5.3). According to (5.4), we could rewrite (5.2) in the following way: min f max 0≤ λ ≤ 1 R Ω RR ∥x∥=m min n λc p f(x), 2(1− λ ) t mf(x) o dxdm R Ω RR ∥x∥=m β p f(x)dxdm 36 where Ω is the domain of the distance function, which is the range between [0,D]. According to (5.4), we could assume p f(x) = g(m) where∥x∥ = m. h(m) = RR ∥x∥=m 1dx Then, the target metric will become: min f max 0≤ λ ≤ 1 R Ω h(m)min n λcg (m), 2(1− λ ) t mg 2 (m) o dm R Ω h(m)βg (m)dm (5.5) To achieve optimality, the partial derivative of (5.5) overΩ is zero for all possiblem∈Ω . Ifg(m)∝ 1 m , or saymg(m)=ρ , the target metric will become max 0≤ λ ≤ 1 R Ω h(m)min n λcg (m), 2(1− λ ) t ρg (m) o dm R Ω h(m)βg (m)dm with optimalλ ∗ = 2ρ 2ρ +ct . This result matches the intuition of the gravity hypothesis in economics, migra- tion, transport ([45],[17],[42]) Now, let us consider a distribution f δ (x)=            τ ∥x∥ 2 D≥∥ x∥≥ σ 0 Otherwise over a circle. TakeD =1 ,clearlyτ = − 1 2π logt . Now the TSP cost is: ZZ x:δ ≤∥ x∥≤ 1 β p f δ (x)dx = ZZ x:δ ≤∥ x∥≤ 1 β r τ ∥x∥ 2 = √ 2πβ 1− δ √ − logδ The consider the cost if all customers are served by the drone only : ZZ x:δ ≤∥ x∥≤ 1 2 t ∥x∥f δ (x)dx = 2 t ZZ x:δ ≤∥ x∥≤ 1 τ ∥x∥ dx = 2π (1− δ ) − tπ logδ 37 Figure 5.1: An example of a lower bounding distribution, where the density of the function is getting sparse and sparse with∥x∥ increasing Because PDSTSP is the conical combination of TSP and Drone-only time cost, therefore the target metric PDSTSP TSP ∝ Drone-only time cost TSP ∝O(1) √ 2t √ − logδ β √ π Even thought could be very small, then we could also makeσ small enough to make the target metric not converge to 1. It shows no matter how small the drone speed is, properly choosing the density function will make the drone improve the delivery process. 38 Figure 5.2: The chart is the comparison between the x-axis thet value and the y-axis the value of the target metric. The blue and orange curves are the performance based on the previous function withσ = 0.015. The green and red curves are based on uniform distribution. 5.1.2 UpperboundofPDSTSPandTSPRatio The upper bound of (5.1) is given by max f max λ ∈[0,1] RR R min(λc p f(x), 2(1− λ ) t ∥x∥f(x))dx RR R β p f(x)dx (5.6) It is quite intuitive that (5.6) is bounded by max f max λ ∈[0,1] RR R λc p f(x)dx RR R β p f(x)dx which is c β . Such an upper bound could be achieved whenλ ∗ for (5.6) is 1. Lemma12. One of the optimal distributionf for (5.6) has the following necessary condition: ∀x,y∈R,∥x∥>∥y∥→f(x)≥ f(y) Proof. Similar to the proof of (11) 39 Figure 5.3: An example of all customers have equidistance to the depot. The truck time cost is constant, the drone time cost is proportional to √ n. The drone is useless Similar to what we did in the previous section, we could simply construct a distribution in whichf(x) is monotonically increasing with∥x∥ increasing. Therefore, consider the distribution where f(x)=            0 ∥x∥<D M ∥x∥=D whereM is sufficiently large (i.e. All customers are even distributed to the collection of points which are the furthest to the depot). In this way, theλ ∗ in (5.6) converges to 1, which causes the upper bound to be c β of the target metric. 5.2 Optimaldepotselection In this section, we will discuss how depot location choices impact the objective value of PDSTSP. 40 Lemma 13. PDSTSP is not a convex problem for certain continuous distributions f respecting to the depot location. Proof. Assume the distributionf is symmetric and is centered at the center ofR, sayp 0 . Consider a ball B with a small radiusr centered atp 0 , then we defined the distribution f as f(x)=            1 1− πr 2 |x− p 0 |>r 0 otherwise Now consider a pointA which is not the center. Then, for given drone speedt √ n, the partition based the centerp 0 , which isR t (p 0 ),R d (p 0 ), and the partition-based the centerA, which isR t (A),R d (A). If the optimal PDSTSP result centered at p 0 is better than the optimal PDSTSP result centered at A, then the truck time cost forR t (p 0 ) is less thanR t (A), and ZZ Rt(p 0 ) 1dx< ZZ Rt(A) 1dx That implies ZZ R d (p 0 ) 1dx> ZZ R d (A) 1dx According to the intuitions from (4.2.1),R d (A) should be a disk centered atA andR d (p 0 ) should also be a disk centeredp 0 . For regionR d (p 0 ), because the density function withinB is zero, and therefore E(∥x∥|R d (p 0 ))>E(∥x∥|R d (A)) 41 Figure 5.4: The chart is an example of the non-convexity with certainf. PDSTSP objective value centered inA andA ′ is less than PDSTSP objective value centered inp 0 . which implies ZZ R d (p 0 ) ∥x∥dx> ZZ R d (A) ∥x∥dx and contradict the assumption that PDSTSP centered inp 0 is better than PDSTSP centered inA. Therefore PDSTSP centered inA perform better than PDSTSP centered inp 0 . Now consider another pointA ′ which is symmetric toA , it is easy to conclude that the PDSTSP centered inp 0 , which is the midpoint of segment A,A ′ , has the higher PDSTSP objective value centered inA and centered inA ′ . Therefore PDSTSP is not convex corresponding to the central depot. 42 Regarding the non-convex depot location of PDSTSP, applying brunch and bound methods could be a good choice, such as The big triangle small triangle method [15] 43 Chapter6 PDSTSPwithtruck-onlypoints In the real world, due to the limitation of drone delivery (e.g. the package is too heavy to be delivered by the drone), some customers are unable to be served by drones. Now we define PDSTSP with truck-only points based on (4.3) as the following. Definition14. LetC ={0,1,...,n} denote a collection ofn customers together with a central depot at index0. LetF ⊂ C such thatF is the collection of customers who could only be served by truck only. Let t T ij denote the travel time between nodesi andj with a truck, and lett U i denote the round-trip travel time (back and forth) between the central depot and customer nodei with any of a fleet of m identical drones. The objective of the PDSTSP with truck-only points (PDSTSP) is to visit alln customers in such a way to minimize the makespan, i.e. the time to completion minimize S 0 ,S 1 ,...,Sm max    TSP(S 0 ), max i∈{1,...,m} X j∈S i t U i    s.t. (6.1) S i ∩S j =∅ m [ i=0 S i =F 44 where TSP(S 0 ) denotes the length of a TSP tour of the points S 0 ⊂ C, with lengths taken respect to t T ij , and the constraints merely impose thatS 0 ,S 1 ,...,S m form a partition ofC so that each customer is visited exactly once by either the truck or a drone. 6.1 ModificationofSection4.1 AssumeX is a set of iid samples from the certain distributionf andF is a subset ofX which is generated by the Bernoulli distribution with success rate q. Now, F will be the set of truck-only points. In this section, we will discuss the performance of the following function minimize F⊂∈ S⊂ X max    TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    (6.2) as|X| =n→∞. Theorem15. Let{X i }denoteasequence of independentpoints sampled froma distributionf havingcom- pact supportR,F denote a a subsequence of{X i } generated by Bernoulli distribution with parameterq,and fix t>0. Define L(X 1 ,...,X n )= min F⊂ S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x i / ∈F ∥x i ∥    (6.3) to be the cost of a PDSTSP tour of the pointsX 1 ,...,X n . We have max 0≤ λ ≤ 1 ZZ R min λ 2 α p f(x), λ 2 α p qf(x)+ 2(1− λ ) t (1− q)∥x∥f(x) dx≤ liminf n→∞ L(X 1 ,...,X n ) √ n (6.4) ≤ limsup n→∞ L(X 1 ,...,X n ) √ n ≤ max 0≤ λ ≤ 1 ZZ R min λβ p f(x), λβ p qf(x)+ 2(1− λ ) t (1− q)∥x∥f(x) dx (6.5) 45 whereα andβ are as in Theorem 6. As in Theorem 6, the upper and lower bounds are guaranteed to lie within a factor of2β/α < 7.3 of one another, sinceβ < 0.9204. 6.2 UpperboundofPDSTSPwithtruckonlypoints Similar to the idea of Section 4.3.1, we will consider the following partition strategy: Suppose that we again divide the service regionR into a "truck region”S and a "drone region"R\S. Now, for every customer withinS will be served by the truck, and for customers, withinR\S,q fraction of them will be served by the truck, and other customers will be served by the drone. The truck time cost within the drone region is β s qn ZZ R\S f(x)dx ZZ R\S s f(x) RR R\S f(x)dx dx = √ n ZZ R\S p qf(x)dx The objective valueobj associated with such a districting strategy satisfies lim n→∞ obj √ n =max ( β ZZ S p f(x)dx+β ZZ R\S √ q p f(x)dx, 2 t ZZ R\S (1− q)∥x∥f(x)dx ) . For notational compactness, we will setu(x)=β p f(x) andv(x)= 2 t ∥x∥f(x). Here, becausec is a value between the lemma below describes the optimal shape ofS: Lemma16. For any sub-regionS ⊂R and any0≤ λ ≤ 1, we have max ( ZZ S u(x)dx + ZZ R\S √ qu(x)dx, ZZ R\S (1− q)v(x)dx ) (6.6) ≥ Z R min{λu (x),λ √ qu(x)+(1− λ )(1− q)v(x)}dx. (6.7) 46 Moreover,thereexistsapairingS ∗ ,λ ∗ suchthatequalityholds,whereS ∗ ={x∈R :{λ ∗ u(x),λ ∗ √ qu(x)+ (1− λ ∗ )(1− q)v(x)}. Proof. This is a standard duality argument from vector space optimization; we provide direct proof here. For anyλ , define R u ={x∈R :λu (x)≤ λ √ qu(x)+(1− λ )(1− q)v(x)} and let R v ={x∈R :λu (x)>λ √ qu(x)+(1− λ )(1− q)v(x)} max ( Z S u(x)dx + Z R\S √ qu(x)dx, Z R\S (1− q)v(x)dx ) ≥ t ( Z S u(x)dx + Z R\S √ qu(x)dx ) +(1− t) Z R\S (1− q)v(x)dx = Z S tu(x)dx + Z R\S t √ qu(x)+(1− t)(1− q)v(x)dx ≥ Z Ru tu(x)dx+ Z Rv t √ qu(x)+(1− t)(1− q)v(x)dx = Z R min{tu(x),t √ qu(x)+(1− t)(1− q)v(x)}dx which yields the inequality (6.7). To find the pairing S ∗ ,λ ∗ , we note that the function λ 7→min{λu (x),λ √ qu(x)+(1− λ )(1− q)v(x)}dx 47 is differentiable in λ becausef is continuous, and its derivative is equal to d dλ ZZ R min{λu (x),λ √ qu(x)+(1− λ )(1− q)v(x)}dx = ZZ R d dλ min{λu (x),λ √ qu(x)+(1− λ )(1− q)v(x)}dx = ZZ R            u(x) ifx∈R u √ qu(x)− (1− q)v(x) otherwise            dx = ZZ Ru u(x)dx+ ZZ Rv √ qu(x)− (1− q)v(x)dx. LetR ∗ u andR ∗ v denote the subsetsR u ,R v associated withλ ∗ . Since the above derivative has to be zero at optimality, we know that RR R ∗ u u(x)dx = RR R ∗ v (1− q)v(x)− √ qu(x)dx, yielding an objective value of ZZ R min{λ ∗ u(x),λ ∗ √ qu(x)+(1− λ ∗ )(1− q)v(x)}dx =λ ∗ ZZ R ∗ u u(x)dx+λ ∗ ZZ R ∗ v √ qu(x)dx+(1− λ ∗ ) ZZ R ∗ v (1− q)v(x)dx = ZZ R ∗ u u(x)dx. We find that setting S ∗ =R ∗ u completes the proof, because the two terms in themax{·,·} operator below are equal: max ( ZZ S ∗ u(x)dx+ ZZ R\S ∗ √ qu(x)dx, ZZ R\S ∗ (1− q)v(x)dx ) = ZZ R ∗ u u(x)dx This completes the proof of the upper bound (6.5). 48 6.3 LowerboundofPDSTSPwithtruckonlypoint To derive the lower bound of (6.3), we need to do some transformation on (6.3). minimize 0∈S⊂ X\F max    TSP(S∪F), 2 t √ n X x i ∈X\(S∪F) ∥x i ∥    (6.8) At here, we try to split the truck points into two types. The first type of point is S, which indicates the non truck-only points we choose to visit. The second type point isF , which is theq fraction of truck-only points we have to visit. Let us consider the truck termTSP(S∪F): TSP(S∪F) ≥ max(TSP(S),TSP(F)) = 1 2 (2max(TSP(S),TSP(F)) ≥ 1 2 (TSP(S)+TSP(F)) Therefore, (6.8) will be lower bounded by 1 2 minimize 0∈S⊂ X\F max    TSP(S)+TSP(F), 4 t √ n X x i ∈X\(S∪F ∥x i ∥    (6.9) Now, similar to (4.3.2), we will also prove the desired result for the case where the densityf is a step function; a standard coupling argument will then complete the analysis. Letϕ (x) = P s i=1 a i 1(x∈ i ) be a step function with s disjoint rectangular components such that all components have equal mass. For an independent collection of samplesX 1 ,...,X n , letP n i denote the fraction of points in component⊡ i that are assigned to the S (4.10). The s-tuplesP n := (P n 1 ,...,P n s ), therefore, as n → ∞, they have (with probability one) a convergent subsequence with a limitP n →p. For fixed p, we will construct the 49 following relaxed problem by only restricting the cardinality of each⊡ as what we did in Section 4.3.2. We pickϵ> 0 such that 1 2 minimize 0∈S⊂ X\F max    TSP(S)+TSP(F), 4 t √ n X x j / ∈S∪F ∥x j ∥    s.t. (6.10) |S∩⊡ i |≤ (p i (1− q)+ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (6.11) |S∩⊡ i |≥ (p i (1− q)− ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (6.12) (6.13) and if any constraint is infeasible, we could still set the objective value to∞ By applying the similar method as what we did in (4.17), we could establish that X x j / ∈S∪F ∥x j ∥= s X i=1 X x j ∈⊡ i x j / ∈S∪F ∥x j ∥ ≥ s X i=1 |(X\(S∪F))∩⊡ i |r i ≥ s X i=1 (1− p i − ϵ )(1− q)|X∩⊡ i |r i ∼ (1− p i − ϵ )(1− q)a i area(⊡ i )nr i According to the super- and subaddivity of TSP as stated in Lemma 3, (6.13) could be relaxed to 50 1 2 minimize 0∈S⊂ X\F max    s X i=0 (TSP(S∩⊡ i )+TSP(F ∩⊡ i )), 4 t √ n X x j / ∈S∪F ∥x j ∥    s.t. |S∩⊡ i |≤ (p i (1− q)+ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i |S∩⊡ i |≥ (p i (1− q)− ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i withO(1) cost. Recall the (4.3.2) liminf n→∞ TSP(S∩⊡ i ) √ n ≥ α p a i (1− q)area(⊡ i )(p i − ϵ √ 1− q ) and we cond easy construct the similar lower bound ofTSP(F ∩⊡ i ) withα =0.2935: s X i=1 liminf n→∞ TSP(F ∩⊡ i ) √ n → s X i=1 β √ a i area(⊡ i ) √ q≥ s X i=1 α √ a i area(⊡ i ) √ q Therefore, liminf n→∞ 1 2 √ n max ( s X i=1 (TSP(S∩⊡ i )+TSP(F ∩⊡ i )), 4 √ n t s X i=1 a i area(⊡ i )(1− p i − ϵ )(1− q)r i ) ≥ max ( α 2 s X i=1 √ a i area(⊡ i )(p i p 1− q+ √ q− ϵ ), 2 t s X i=1 a i area(⊡ i )(1− p i − ϵ )(1− q)r i ) ≥ max ( 1 2 α s X i=1 √ a i area(⊡ i )(p i +(1− p i ) √ q), 2 t s X i=1 r i (1− p i )a i area(⊡ i )(1− q) ) − ϵ s X i=1 1 2 α √ a i +2r i a i (1− q) area(⊡ i ) | {z } (∗ ) 51 where(∗ ) can be made negligible by selecting a sufficiently small ϵ . The problem will be equivalent to: minimize 0≤ p≤ 1 max ( 1 2 α s X i=1 √ a i area(⊡ i )(p i +(1− p i ) √ q), 2 t s X i=1 r i (1− p i )a i area(⊡ i )(1− q) ) which corresponding dual is maximize 0≤ λ ≤ 1 s X i=1 min 1 2 λα √ a i area(⊡ i ), 1 2 λα √ q √ a i area(⊡ i )+(1− λ )(1− q) 2 t r i a i area(⊡ i ) = s X i=1 min 1 2 λα √ a i , 1 2 λα √ q √ a i +(1− λ ) 2 t r i (1− q)a i area(⊡ i ) = Z R min 1 2 λα p ϕ (x), 1 2 λα √ q p ϕ (x)+(1− λ ) 2 t (1− q)a i 1(x∈⊡ i ) dx ≥ Z R min 1 2 λα p ϕ (x), 1 2 λα √ q p ϕ (x)+(1− λ )(1− q) 2 t ∥x∥ϕ (x) dx− 2δ/t |{z} (∗ ) where, as before,(∗ ) can be made negligible by selecting a sufficiently small δ . This complete the proof of (6.4) by applying the coupling argument methods stated in Section 4.3.2 52 Chapter7 Min-sumPDSTSP Another type of PDSTSP problem is the cost-based PDSTSP problem. Assume the unit time cost of the truck and unit time cost of the drone is given, and the target is to minimize the total cost of the entire delivery process. 7.1 ModificationofSection4.1 Based on the definition of Section 4.1, we could assume the unit time cost of the truck is ψ 0 and the unit time cost of the drone is ψ 1 . Then, by varying the (4.3), the Min-sum PDSTSP problem will become the following: minimize 0∈S⊂ X ψ 0 TSP(S) + 2ψ 1 t √ n X x i / ∈S ∥x i ∥ Noting that ψ 0 and ψ 0 are fixed values, without loss of generality, we could assume the t value will be come to mϕ 1 ψ 0 ϕ 0 ψ 1 according to the definition of Section4.1. The Min-sum PDSTSP problem will be equivalent to 53 minimize 0∈S⊂ X TSP(S) + 2 t √ n X x i / ∈S ∥x i ∥ (7.1) as|X|→∞. Then, in this section, we will compose the behavior of the Min-sum problem: Theorem17. Let{X i }denoteasequence of independentpoints sampled froma distributionf havingcom- pact supportR, and fix t>0. Define L(X 1 ,...,X n )= min 0∈S⊂{ X 1 ,...,Xn} TSP(S) + 2 t √ n X x i / ∈S ∥x i ∥ (7.2) to be the cost of a min-sum PDSTSP tour of the pointsX 1 ,...,X n . We have ZZ R min α p f(x), 2 t ∥x∥f(x) dx≤ liminf n→∞ L(X 1 ,...,X n ) √ n (7.3) ≤ limsup n→∞ L(X 1 ,...,X n ) √ n ≤ ZZ R min β p f(x), 2 t ∥x∥f(x) dx, (7.4) whereα andβ are as in Theorem 6. 7.2 Proofofupperbound(7.4) Similar to what we did in Section 4.3.1, we still apply the idea of the partition to get the upper bound. Suppose that we again divide the service regionR into a "truck region"S and a "drone region"R\S. The objective valueobj associated with such a districting strategy satisfies lim n→∞ obj √ n =β ZZ S p f(x)dx + 2 t ZZ R\S ∥x∥dx 54 Then, by assumingu(x)=β p f(x) andv(x)= 2 t ∥x∥f(x), the lemma below describes the optimal shape ofS: Lemma18. For any sub-regionS ⊂R we have ZZ S u(x)dx + ZZ R\S v(x)dx≥ ZZ R min{u(x), v(x)}. (7.5) Moreover, there exists a pairingS ∗ , such that equality holds, whereS ∗ ={x∈R :u(x)≤ v(x)}. Proof. This proof is simple: ZZ S u(x)dx + ZZ R\S v(x)dx = ZZ R u(x)dx + ZZ R\S (v(x)− u(x))dx because RR R u(x)dx is a constant for given distributionf, Then, it is obvious that the truck region is R u ={x∈R :u(x)≤ v(x)} and the corresponding drone region is R v ={x∈R :u(x)>v(x)} In this way, ifS =R v , the (7.5) will be an equal sign, which shows the upper bound of min-sum PDSTSP will be RR R min{u(x), v(x)} which finish the proof of (7.4) 55 7.3 Proofoflowerbound(7.3) The proof of the lower bound of the min-sum PDSTSP is almost identical to Section 4.3.2 because the dual of min-sum PDSTSP is simply a special case of the dual PDSTSP with λ = 1 2 . Therefore, with a similar process, we could construct the cardinality relaxed problem withp,ϵ,ϕ (x) has the exact the same definition as Section 4.3.2 minimize 0∈S⊂{ X 1 ,...,Xn} TSP(S)+ 2 t √ n X x j / ∈S ∥x j ∥ s.t. (7.6) |S∩⊡ i |≤ (p i +ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (7.7) |S∩⊡ i |≥ (p i − ϵ )|{X 1 ,...,X n }∩⊡ i | ∀i (7.8) And then, the proof will be the same, but only replace (4.19) with the following lower bound: liminf n→∞ 1 √ n ( s X i=1 TSP(S∩⊡ i ) + 2 √ n t s X i=1 a i area(⊡ i )(1− p i − ϵ )r i ) ≥ maxα s X i=1 √ a i area(⊡ i )(p i − ϵ ) + 2 t s X i=1 a i area(⊡ i )(1− p i − ϵ )r i ≥ maxα s X i=1 √ a i area(⊡ i )p i + 2 t s X i=1 r i (1− p i )a i area(⊡ i )− ϵ s X i=1 (α √ a i +r i a i )area(⊡ i ) | {z } (∗ ) 56 By taking a smallϵ , the(⋆) will be negligible and the corresponding dual will become: s X i=1 min α √ a i area(⊡ i ), 2 t r i a i area(⊡ i ) = s X i=1 min α √ a i , 2 t r i a i area(⊡ i ) = Z R min ( α p ϕ (x), 2 t s X i=1 r i a i 1(x∈⊡ i ) ) dx ≥ Z R min α p ϕ (x), 2 t ∥x∥ϕ (x) dx− 2δ/t |{z} (∗ ) By taking a small δ , the (⋆) will be negligible. By applying the coupling argument stated in (4.3.2), the proof is finished. 7.4 Convertingmin-sumPDSTSPtoTSP 7.4.1 Convertingmin-sumPDSTSPtoprizecollectingTSP Let us consider the Prize Collecting TSP Problem(PCTSP) defined by [3], which states for every point with X ={X 1 ··· X n } has a penalty valueq i . The PCTSP problem will pick a subset of customers to visit which minimize the summation of the trip cost plus the total penalty score of customer who is not visiting: minimize 0∈S⊂ X TSP(S) + X x i / ∈S q i it is easy to find out that PCTSP problem is equivalent to min-sum PDSTSP problem if ∀i,q i = 2 t √ n ∥x i ∥. As a side product, we could borrow some properties of min-sum PDSTSP to do an asymptotic analysis of PCTSP problems. 57 Theorem19. Let{X i }denoteasequence of independentpoints sampled froma distributionf havingcom- pactsupportR,ForpointX i inlocationx i ,thecorrespondingpenaltyisq i . Assumethereexistsanabsolutely continuous mappingg :R→R + such that∀i,g(x i )=q i Define L(X 1 ,...,X n )= min 0∈S⊂{ X 1 ,...,Xn} TSP(S) + X x i / ∈S q i (7.9) to be the cost of a PCTSP tour of the pointsX 1 ,...,X n . We have ZZ R min n α p f(x), g(x)f(x) o dx≤ liminf n→∞ L(X 1 ,...,X n ) √ n (7.10) ≤ limsup n→∞ L(X 1 ,...,X n ) √ n ≤ ZZ R min n β p f(x), g(x)f(x) o dx, (7.11) whereα andβ are as in Theorem 6 with|X|→∞ Proof. Such proof is very simple: Let us go back to (7.3) and (7.4), if we simply replace the∥x∥ function to some other absolutely continuous function, sayg(x), we will get the same result because the proofs of (7.10) and (7.11) are identical to the previous min-sum PDSTSP upper bound and lower bound proofs. Therefore, we could establish a method to convert the min-sum PDSTSP problem to the PCTSP problem in both directions. 7.4.2 ConvertingPCTSPtothegeneralizedTSP Now, let us look at another variation of the TSP problem: The generalized TSP problem, which is proposed in [29]. It states for pointsX = {X 1 ··· X n } belongs to m disjoint sets V 1 ,...,V m , and we want the shortest tour that visits one member of each set: minimize S={X 1 ∈V 1 ,X 2 ∈V 2 ··· X m ∈Vm} TSP(S) 58 Now we could apply the following method to convert the PCTSP problem into a Generalized TSP problem: • For everyX i ∈X, we will construct a duplicate pointX ′ i • We will buildn discrete setsV 1 ··· V n such that∀i,V i ={X i ,X ′ i } • If the pointi is visited, it will be assigned byX i otherwise it will be assigned byX ′ i 1. Connect allX i →X j with costdist(X i ,X j ) as usual 2. Connect allX i →X ′ j with costdist(X i ,X 0 ) 3. Connect allX ′ i →X ′ j with costq i , which is the penalty ofX i in PCTSP 4. Connect allX ′ i →X 0 with costq i , which is the penalty ofX i in PCTSP In this way, PCTSP could be converted to a special type of Generalized TSP problem. 7.4.3 ConvertgeneralizedTSPproblemtoTSP It is well-known [5] that the Generalized TSP can be solved as an instance of (asymmetric) TSP: • Assume the distance matrix of Generalized TSP is D, and we’ll turn this into an asymmetric TSP with distance matrixD ′ . • Place the points in each cluster in some arbitrary directed cycle and insert edges of cost− M between consecutive points • For every pair of pointsp u ∈ V i andp v ∈ V j in distinct sets, letp w be the point that comes afteru according to the arbitrary cycle, and setD ′ uv =D wv This shows that min-sum PDSTSP can be solved as a TSP. 59 Chapter8 PDSTSPvariations 8.1 MultipledepotsPDSTSP Let us assume that there arek central depotsD = {dp 1 ,dp 2 ,dp 3 ··· dp k } for launch and collect drones and the truck. Assume if customeri is served by a drone,y i is the depot of launching and collecting that drone. y 0 is the depot for launching and collecting the truck. Now let us go back to the ( 4.3), and the multiple depots’ case of PDSTSP problem will become: minimize S⊂ X,y 0 ∈D,∀x i ̸∈S,y i ∈D max    TSP(S∪y 0 ), 2 t √ n X x i / ∈S ∥x i − y i ∥ 2    with|X| =n→∞. We will discuss the truck time part first. It is easy to show TSP S⊂ X (S∪y 0 ) = TSP S⊂ X (S)+O(1) = TSP 0∈S⊂ X (S)+O(1) . By Theorem 2,TSP(S∪y 0 )∝ √ n, then we can safely assume, with|X| =n→ ∞, TSP S⊂ X (S∪y 0 )= TSP 0∈S⊂ X (S) 60 And then, the problem will become minimize 0∈S⊂ X max    TSP(S), min ∀x i ̸∈S,y i ∈D 2 t √ n X x i / ∈S ∥x i − y i ∥ 2    Now, we will discuss the drone time part. For the termmin ∀x i ̸∈S,y i ∈D 2 t √ n P x i / ∈S ∥x i − y i ∥ 2 with a given S, it’s minimal will be achieved at 2 t √ n P x i / ∈S min y∈D ∥x i − y∥ 2 . In this way, the multiple depot PDSTSP problems can be formulated as minimize 0∈S⊂ X max    TSP(S), 2 t √ n X x i / ∈S min y∈D ∥x i − y∥ 2    which is exactly the PDSTSP formulation by replacing∥x i ∥ tomin y∈D ∥x i − y∥ 2 . Figure 8.1 is an example of a 3-depot PDSTSP with customers uniformly distributed on a unit square. Figure 8.1: A multiple-depot example where the red, green, and purple dots are depots 61 8.2 MultipletrucksPDSTSP Assume we have a finite-size fleet of homogeneous trucks T = {T 1 ··· T l } with speed 1 instead of one truck to make the delivery. {S 1 ··· S l } are sets of customers served by the corresponding truck. Let us assumeS = S l i=1 S i , the multiple truck PDSTSP will become to minimize S⊂ X max    p(S)TSP(S∪0), 2 t √ n X x i / ∈S ∥x i ∥    s.t. S 1 ,S 2 ··· S l ⊂ S p(S)TSP(S∪0)= max S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0) S i ∩S j =∅ with|X| =n→∞, the optimal value will only be achieved when∀i,|S i |∝|S|∝|X|. Now, we need to inquire about the functionp(S). Lemma 20. AssumeS is a realization of regionR(S), with area greater than 0 and density functiong(·), LetS 1 ,S 2 ··· S l ⊂ S,S i ∩S j =∅, and|S i |,|S|→∞ ,then min S 1 ··· S l max S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0)→ 1 l TSP(S∪0) whereα,β are as in Theorem 6. 62 Proof. It is clear thatTSP(S∪0)→ RR R(S) p g(x)dx. Let us consider the upper bound ofmax S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0) first. If we split the service region within l small region,R(S) 1 ,···R (S) l , andS i =S∩R(S) i . The target equation min S 1 ··· S l max S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0)≤ 1 l ZZ R(S) p g(x)dx→ 1 l TSP(S∪0) when∀i, RR R(S) i p g(x)dx are all equal to 1 l RR R(S) p g(x)dx. As for the lower bound, min S 1 ··· S l max S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0)≥ min S 1 ··· S l 1 l X S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0) = min S 1 ··· S l 1 l X S 0 ∈{S 1 ··· S l } TSP(S 0 )+O(1) ≥ 1 l TSP( l [ j=1 S j )+O(1) → 1 l TSP(S∪0)+O(1) Because|S|→∞ and 1 l TSP(S∪0)→∞, we can conclude that min S 1 ··· S l max S 0 ∈{S 1 ··· S l } TSP(S 0 ∪0)→ 1 l TSP(S∪0) From Lemma 20, p(S) is a constant 1 l .The multiple truck PDSTSP problem could be refined as the following: minimize 0∈S⊂ X max    1 l TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    63 Figure 8.2: A example of a 4-truck case, which shows the increase of truck will have a linear improvement of the truck delivery efficiency 64 Chapter9 Computationalresults The purpose of this section is to compare the performance of two heuristics of the PDSTSP problem and to determine their predictive capability. The first heuristic is to split the region based on the ∥x∥ p f(x) function, and use the google- OR tour to get the heuristic of the TSP. The second heuristic is to apply the min-sum PDSTSP method, such that converting the PDSTSP to min-sum PDSTSP first, and then using the Google OR-Tools package to solve the corresponding equivalent Prize Collecting TSP. In this case, we run simulations with points drawn from the distribution f in a unit square. In this case, we will calculate the relationship betweenϕ and makespan under the assumption that the speed of the truck is always 1. We did 5 draws of each map, and make the comparison of two heuristics Because the min-sum conversion method could not generate a PDSTSP solution with a deterministic speed. Therefore, in order to properly compare the performances, we will provide a series ofϕ for min-sum PDSTSP problems, solve speeds for the corresponding PDSTSP problem and finally use the region partition method to solve the PDSTSP problem. At here, we taken=1000. The local search time of the or-tool for solving the region partition method is 300 seconds whereas the local search time for the min-sum PDSTSP problem time is 2000 seconds because that is difficult to converge. 65 For the min-sum PDSTSP heuristic, we simply make the penalty of the Prize Collecting TSP for cus- tomeri to be 2 ϕ ∥p i ∥, and solve the corresponding truck timespanx ϕ ,and the drone timespany ϕ . Therefore, the corresponding PDSTSP problem will be the speed y ϕ ϕ x ϕ . For the region partition heuristic, we could split the square toψ (R) square bins such that each bin has the same area. Then, we could simulate the distribution from the step function mentioned in 4.3.2. Then, we sort bins with the key ˆ r i q ˆ f i from low to high. As we states before, the truck region prefers a higher ˆ r i q ˆ f i bins whereas the drone region prefers lowerˆ r i q ˆ f i . Therefore, we track the truck timespan and the drone timespan simultaneously. If the truck timespan is greater than the drone timespan, we will pop out the bin with a idle bin with the highest ˆ r i q ˆ f i to the truck and vise versa. In this way, we could ensure that the truck timespan will be close to the drone timespan. As for unknown parameterc, we simply apply β =0.712 here. 9.1 Simulateddatawithinasquare When the density function f is 1,R is a square with size 1000× 1000. In this case, we could make the comparison between two heuristics when customers are uniformly distributed. We make the comparison of the performance between the min-sum PDSTSP heuristic and region par- tition heuristic with 3 different maps which share the same density function: Now, we could discuss the expected time savings, and simulated time-saving under the uniform case. There are two important factors: the first factor is the time-saving ratio which is time saved with the involvement of drones to the time cost without using drones. In other words, TimeSavingRatio =1− Region Partition PDSTSP time TSP Time 66 (a)ϕ =0.408 (b)ϕ =0.408 (c)ϕ =0.733 (d)ϕ =0.733 (e)ϕ =1.031 (f)ϕ =1.031 Figure 9.1: A comparison between PDSTSP estimation based on min-sum PDSTSP solver (left), and based on region partition method (right) 67 (a)ϕ =1.611 (b)ϕ =1.611 (c)ϕ =2.097 (d)ϕ =2.097 (e)ϕ =2.941 (f)ϕ =2.941 Figure 9.2: A comparison between PDSTSP estimation based on min-sum PDSTSP solver (left), and based on region partition method (right) 68 Figure 9.3: Green lines are the region partition PDSTSP simulated truck times, and orange lines are the re- gion partition PDSTSP simulated drone times. Blue lines are the min-sum estimation simulated results.The red lines are the expected optimal value of PDSTSP calculated by 4.9 whenc = 0.712. The range of red curve is not equivalent to other curves because the red curve is the result of the 69 The second factor is the proportion of drone-serving customers to all customers, which is DroneRatio = |V d | n Intuitively, The expected Time-Saving-Ratio should converge to the expected Drone-Ratio under the uniform case. As we mentioned in Section 4.2.3, the drone region should be a circle, which has the radiusr. Let us assume the selections ofr are150,200,250,300,350,400,450,500 and calculate the average Time- Saving-Ratio, and the average Drone-Ratio within 5 different maps, and compare those results. We expect to see the time-saving ratio and the proportion of drone customers are sublinear and converge to each other. In addition, we expected to see the objective value from our simulation converge to our estimation in (4.9). From Figure 9.4, we can see the ratio of (4.9) to our estimated value is always about 0.9. The reason for such a gap may be due to we takeβ instead ofc for partition. Also, the TSP solution of two heuristics may not be optimal. Figure 9.4: The relationship betweenϕ and time savings metrics.The x-axis is theϕ value 9.2 Real-worldapplications We simulate the truck driving time and the drone flying time of 1500 discrete customers within Downtown Los Angeles, which is the regionR. In this case, the drone walking time between different customers is proportional to the Euclidean distance between those customers. However, the truck driving time between different customers is not always proportional to the Euclidean distance between those customers because 70 of the traffic. To find the PDSTSP tour on this map, we need to solve three problems: First, the Region Partition PDSTSP depends on the density function of customers. However, it could not be obtained directly from the discrete customer distribution. We need to find a good estimation of the density function. Second, because there are freeways within the Downtown Los Angeles area, the triangle inequality may not always hold because detouring some of the long-distance freeways may save time. However, current TSP heuristic solvers do not perform well for TSP without triangle inequality cases. Finally, our PDSTSP implication requires Theorem 2, which requires the Euclidean distance assumption. To solve previous problems, we decide to solve the problem with the idea of one step function. Let us split the region in Downtown Los Angeles into s rectangle bins,⊡ 1 ··· ⊡ s with equal area. For each of the bins, we could assume that customers are uniformly iid distributed within the region. Therefore, for the first problem the density function f could be estimated by one step function ˆ f from an empirical distribution where ˆ f(x)= s X i=1 ˆ a i 1(x∈⊡ i ) whenˆ a i = |X∩⊡ i | n . For the second problem, if we could properly choose the value ofs, where each bin does not contains both the entrance and the exit of a long-distance highway simultaneously. This will ensure the triangle inequality holds within the bin. According to Lemma 3 , the target problem could be rewritten as minimize S⊂ X max    s X i=1 TSP(S∩⊡ i )+O(s), 2 t √ n X x i / ∈S ∥x i ∥    as|X| =n→∞. Let us go back to (4.12) and (4.11),∃c∈[α,β ] such that PDSTSP could be rewritten max 0≤ λ ≤ 1 ZZ R min λc p f(x), 2(1− λ ) t ∥x∥f(x) dx 71 if we apply the one-step function off, the expression will become max 0≤ λ ≤ 1 s X i=1 min{ λc β ZZ ⊡ i β p ˆ a i dx, ZZ ⊡ i 2(1− λ ) t ∥x∥ˆ a i dx} = 1 √ n max 0≤ λ ≤ 1 s X i=1 min{ λc β TSP(⊡ i ), 2(1− λ ) t √ n X x i ∈⊡ i ∥x i ∥} (9.1) Because∀i, we could estimateTSP(⊡ i ) byTSP(⊡ i ∩C)with the OR-tools solver, and estimate P x i ∈⊡ i ∥x i ∥ by P x i ∈⊡ i ∩C ∥x i ∥. Therefore, we could simulate the DTLA case by solving the maximizerλ ∗ . In this way, one of the heuristics for the solution is to sort bins with the key t i d i . Bins with the higher key value will be assigned to the truck, and with the lower key value will be assigned to drones. At the same time, we should keep the summation of the drone time approximate to the summation of the truck time. At the same time, we observed that if Euclidean distance assumptions hold, ands value is very large, the such heuristic will be similar to the upper bound of the PDSTSP problem as we stated in Section 4.3.1 In Figure 9.5, we display the result of 1500 empirical distributed customers with different ϕ and the number of depots. The truck driving time between different customers is not exactly proportional to the Euclidean distance of those customers From Figure 9.5, we could find out that there are some intersections within the truck path. That is rea- sonable because, in the real world application, the triangle inequality does not always hold. For example, the time cost of detouring a freeway often outperforms the time cost of direct running locally. Therefore, we are considering the following metric: 72 (a)ϕ =0.5 (b)ϕ =1.0 (c)ϕ =1.5 (d)ϕ =2.0 (e)ϕ =2.5 (f)ϕ =3.0 (g)ϕ =3.5 (h)ϕ =4.0 Figure 9.5: Region Partition PDSTSP tours for 1500 customers from the downtown LA real-time traffic data with different values of ϕ . We expect to see the effect of increment of ϕ to be sub-linear 73 • Get the objective of the TSP tour TSP and assume the distribution of customers is uniformly dis- tributed. • Sort all customers with the distance to the depot. Then,∥x 1 ∥≤∥ x 2 ∥···∥ x n ∥ • For given speedϕ , and the number of drone customers ism, the drone time cost is at least 2 ϕ P m i=1 ∥x i ∥, which is them-nearest neighborhood of the depot. • The truck time cost is proportional to n− m n TSP as the result of the region partition. • The estimated objective value is min m max{ n− m n TSP, 2 ϕ m X i=1 ∥x i ∥} (9.2) Figure 9.6: The truck time cost and the drone time cost of PDSTSP in DTLA. The Blue curve (Predicted objective value of function 9.2) is almost the same line as the Orange curve (The truck time cost) In Figure 9.6, we could see that, on one hand, the PDSTSP truck tour, which is the TSP of n− m n furthest points, is well estimated by n− m n proportion of the TSP tour without applying drone. That makes the result of our simulation result 9.4 from a uniform distribution convincing. on the other hand, the 74 existence of the freeway, which makes the triangle inequality not hold (detouring to the freeway will cost less time than directly going through locally), will not impact the objective value a lot. According to our partition method of splitting the map into small bins as (9.1) stated, the loss of objective value isO(1) due to Lemma 3. That is the reason why the region partition method performs well on the Real world map. 75 Chapter10 Conclusions PDSTSP problems and their variations can potentially be useful tools to model some real-world problems that apply several types of serving vehicles for collaboration. In this thesis, we have proved a few theoret- ical results of asymptotic analysis on PDSTSP, truck-only PDSTSP, and min-sum PDSTSP problems. We have also applied such models to real-world scenarios. There are several directions that still need to work in the future, in theoretical aspects. The first field to be researched is the scenario where the truck has limited capacity for delivery. In real- world practice, the capacity of the delivery truck is limited and needs to be refilled frequently. If a method could be established to solve such restrictions, the model of the PDSTSP problem will be more convincing. The current approach in this dissertation to such restriction is still the partition method. By optimally splitting the truck region into several small pieces to ensure each of the small regions will not exceed the capacity. However, there are potential issues within such thought: because the number of customers of each region is iid distribution, it is possible that the quantity of customers within each small region is still exceeding the truck capacity. Solving such an issue may need the redundancy of capacity for each region and cause the result sub-optimal. Another possible approach to such an issue is to make the PDSTSP into a two-stage optimization problem. The first step is to do region partition and the second step is to apply MILP to solve the problem. Wisely choosing the heuristic method in the second step could still achieve a 76 good balance between the performance and solving time. The second field to be researched is the method of finding optimal depot locations. As we stated in Section 5.2, the PDSTSP problem is not convex with respect to the depot. Therefore the traditional itera- tion methods may converge to local optima. What we could know is that the depot location depends on the speed of the drone and the density function. 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Springer, 2006. 81 AppendixA ProofofTheorem8foracontinuousdensityfunction It is a standard result of measure theory (see e.g. Theorem 2.4(ii) of [47]) that for any continuous density f on a compact regionR, there exists a step function ϕ (x) = P s i=1 a i 1(x ∈ ⊡ i ) that approximates f arbitrarily well in theL 1 sense, i.e. that for anyϵ we have Z R |ϕ (x)− f(x)|dx<ϵ ; it is also routine to constructϕ (x) that additionally satisfies α Z R p ϕ (x)− p f(x) dx<ϵ and 2 t Z R ∥x∥|f(x)− ϕ (x)| dx<ϵ, which guarantees thatϕ also approximates the objective value off well, in the sense that max λα p f(x), 2(1− λ ) t ∥x∥f(x) − max λα p ϕ (x), 2(1− λ ) t ∥x∥ϕ (x) <ϵ 82 for all 0 ≤ λ ≤ 1. Recall that the step function ϕ from Section 4.3.2 had the further property that all of its components had equal mass; this is also straightforward because we could (by the density of the rationals) require that all the components⊡ i have rational endpoints and thata i be rational as well, and then subdivide the components into smaller pieces based on the lowest common denominator of all the a i ’s. LetL denote the same cost function we have used throughout, namely L(X 1 ,...,X n )= min 0∈S⊂{ X 1 ,...,Xn} max    TSP(S), 2 t √ n X x i / ∈S ∥x i ∥    . It is easy to verify by definition that L is sub-additive: for any sets X = {X 1 ,...,X n } and Y = {Y 1 ,...,Y m }, we have L(X∪Y)≤ L(X)+L(Y). It is also true (deterministically, and for fixed R andt) that there exists a constantC such thatL(X 1 ,..., X n )≤ C √ n, for all{X 1 ,...,X n }⊂R ; for example, if we use the truck exclusively, thenL(·) is merely TSP tour length, for which such constants are extensively studied. As a final remark, standard coupling arguments (e.g. the “ γ coupling” of ) guarantee that for any ϵ , there is a joint the distribution between the random variablesX ∼ f andY ∼ ϕ such thatPr(X ̸=Y)≤ ϵ . The various observations made above are all we need to prove the desired result because we have L(X 1 ,...,X n )≥ L(X 1 ,...,X n :X i =Y i ) =L(Y 1 ,...,Y n :X i =Y i ) ≥ L(Y 1 ,...,Y n )− L(Y 1 ,...,Y n :X i ̸=Y i ) ≥ L(Y 1 ,...,Y n )− C p |{Y 1 ,...,Y n :X i ̸=Y i }| 83 and if we divide all the above expressions by √ n and take limits, we have liminf n→∞ L(X 1 ,...,X n ) √ n ≥ liminf n→∞ L(Y 1 ,...,Y n ) √ n | {z } (∗ ) − C r |{Y 1 ,...,Y n :X i ̸=Y i }| n | {z } (∗∗ ) ≥ max 0≤ λ ≤ 1 ZZ R min λα p f(x), 2(1− λ ) t ∥x∥f(x) dx− ϵ (∗ ) − C √ ϵ (∗∗ ) which completes the proof, since− ϵ − C √ ϵ can be made arbitrarily small. 84

Abstract (if available)

Abstract Drones are taking a more and more important position in last-mile delivery problems. Their collaboration with traditional truck delivery is anticipated to sharply improve the efficiency of the delivery process, and become a new class of logistical problem that generalizes the traditional traveling salesman problem (TSP). In this dissertation, we will formulate a continuous approximation model for determining the upper and lower bounds of the parallel drone scheduling traveling salesman problem (PDSTSP) by assigning customers based on region partitioning methods. Previous research on the PDSTSP was based on MILP and iteration methods, limiting the scale of customers. This research bounds the PDSTSP problem based on the demand distribution of customer locations and their distance to a central depot. That makes our approach perform well on large-size instances. When considering instances with more than 1000 points, the result is still encouraging.

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Creator Jia, Haochen (author)

Core Title A continuous approximation model for the parallel drone scheduling traveling salesman problem

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2022-12

Publication Date 09/27/2022

Defense Date 08/31/2022

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

Tag asymptotic optimality,continuous approximation,OAI-PMH Harvest,parallel drone scheduling traveling salesman problem,traveling salesman problem

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Advisor Carlsson, John Gunnar (committee chair), Gómez, Andrés (committee member), Gupta, Vishal (committee member)

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