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Continuous approximation for selection routing problems

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Continuous approximation for selection routing problems

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Content Continuous Approximation for Selection Routing Problems by MohammadJavad Azizi 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) August 2022 Copyright 2022 MohammadJavad Azizi Dedication I dedicate this thesis to Beardwood-Halton-Hammersley (BHH), for their never ending mysteries. ii Acknowledgements Throughout my Ph.D. years at University of Southern California, I have been so fortunate to be supported by many amazing people. I would like to give my deepest gratitude to my advisor, Professor John Gunnar Carlsson, to whom I am indebted forever; I could not have done any of this without his guidance. I was always amazed by his astonishing research abilities, great personality, impressive mentorship, and enthu- siasm towards science and life. His passion for research is very inspiring for me. I feel so fortunate to meet Professor Carlsson and learn mathematics from such a scientist. I would like to thank my amazing committee members - Professor Sheldon Ross, Professor Meisam Razaveyin, Professor Ashutosh Nayyar, and Professor Andres Gomez Escobar. They provided very in- sightful suggestions during my qualifying exam, which greatly helped on this dissertation. It was my privilege to be a part of Professor Carlsson’s research group. This group is full of talented and fun scientists who always gave me useful advice and left me with unforgettable moments throughout the years. I would like to thank all my labmates; Bo Jones, Jiachuan Chen, Ying Peng, Shannon Sweitzer, Haochen Jia, and Julien Yu. I also appreciate the aid and help from other ISE students and friends that I met. With them, life has been so meaningful and joyful. I also thank my apartment staffs and coordinators, specially Shelly Lewis, Roxanna Carter, and Grace Owh, for their support during my PhD years. They were always my reference for all the academic and official matters. iii Last but not least, I would like to take this opportunity to thank my parents and all my family, who have been so supportive and encouraging all the way along. It is their unconditional love and care that helped me go through all the ups and downs, to which I am indebted forever. iv TableofContents Dedication ii Acknowledgements iii ListofTables vii ListofFigures viii Abstract xi Chapter1: Introduction 1 1.1 Contributions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Generalized TSP (GTSP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Continuous approximation models . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter2: SelectionRoutingProblems 16 2.1 One-of-a-subset TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Sparse Subset TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 The "m-of-a subset" TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Sparse Subset GTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Random K GTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Chapter3: SyntheticExperimentalResults 49 3.1 One-of-a-subset TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Sparse Subset TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 m-of-a subset TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Sparse Subset GTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Random K GTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Chapter4: CaseStudy: RandomStowWarehouse 54 4.1 The Warehouse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Discrete Event Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter5: ConclusionandFutureWork 64 v Bibliography 65 AppendixA 72 Further Technicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.1 Technical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.2 Preliminary Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.3 Further Proof related to the Approximation Lemma . . . . . . . . . . . . . . . . . . . . . . 73 AppendixB 77 Further Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 B.0.1 Inventory level results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.1 Discrete Event Simulation Detail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 vi ListofTables 1.1 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Summary of the results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 vii ListofFigures 2.1 The gray area isR and the arrows showξ i ’s, the dashed rectangles represent the ′ i all with areaA ′ i = 1 s , The darker gray shows largera i and note thatA i a i =1/s . . . . . . . . 22 2.2 Assuming that L(Y ′ p : Y ′ p ⊂ S s i=j ′ i ) consists of the dashed lines connecting TSP i = TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) with the thin lines of theTSP i ’s, the idea is that the length of the dashed lines is constant w.r.tn, then the thick lines insideTSP i ’s are the extra part that makes the inequality valid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 A collection of Riemann rectangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Connecting a family of Riemann rectangles to form a loop . . . . . . . . . . . . . . . . . . 40 3.1 (Left) is the tour length for one-of-a-subset with uniform distribution instances for n=10 to 100 clusters, (Right) shows the tour length forone-of-a-subset withtruncated normal distribution instances,n=110 to 190 clusters each withk∈{2,3,...,6}. . . . . 50 3.2 (Left) SSTSP tour length for uniform distribution, n ∈ {150,200,...,950,1140} and p∈{0.1,0.25,0.5,0.75,0.9}, (Right) tour length forSSTSP instances,truncatednormal distribution,n=10 to 460 points andp∈{0.25,0.5,0.75,0.9} . . . . . . . . . . . . . . . 51 3.3 Tour length form-of-a-subset instances foruniform distribution,n=50 to 100 clusters, m∈{2,3,4}. (Left) figure has k =5 and (Right) figure has k =4 . . . . . . . . . . . . . . 52 3.4 Tour length form-of-a-subset instances of atruncatednormal distribution,n = 50 to 190 clusters withm∈{1,...,k}, (Left) figure has k =5 while (Right) figure has k =6 . . 52 3.5 Tour length for Sparse Subset GTSP instances of a uniform distribution, n = 10 to 70 clusters withp∈{0.2,0.5,0.8} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6 Tour length for Random K GTSP instances of a uniform distribution,n=10 to 70 clusters withp∈{0.2,0.5,0.8} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1 The item locations (Top two for RS, Bottom two for CL) at the first and last orders, ord# is the order number,n is the number of items,tl is the tour length. . . . . . . . . . . . . . 57 4.2 Random stow (rs) vs CL (cl) picking average tour length (tls) . . . . . . . . . . . . . . . 58 viii 4.3 Random stow (rs) vs classic (cl) average packing tour length (tls) . . . . . . . . . . . . . . 58 4.4 Random stow (rs) vs classic (cl) picking + average packing tour length (tls) . . . . . . . . 59 4.5 Random stow (rs) picking tour length/ √ nk/(k+1) and classic (cl) picking tour length / √ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.6 Here λ = 5,c = 20,m = 20,N = 100 2 ,¯ρ = 0.3,p 1 = p 2 = 10,r 1 = 10, ¯k = 10. RS is stable since Eq. (4.1) holds sincer 1 ≥ λm √ N(1− ρ ) ≃ 1.2. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃7. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃22. . . . . . . . . . . . . . 60 4.7 Hereλ = 5,c = 20,m = 10,N = 100 2 ,¯ρ = 0.38,p 1 = p 2 = 10,r 1 = 8, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.63. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃2. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃11. . . . . . . . . . . . . . 60 4.8 Here λ = 5,c = 10,m = 10,N = 100 2 ,¯ρ = 0.4,p 1 = p 2 = 8,r 1 = 10, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.64. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃2.8. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃15.8. . . . . . . . . . . 60 4.9 Here λ = 5,c = 10,m = 10,N = 100 2 ,¯ρ = 0.4,p 1 = p 2 = 8,r 1 = 12, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.64. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃2.8. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃15.8. . . . . . . . . . . 60 4.10 Warehouse inventory level where CL fails and RS is stable. The fill shows the standard deviation of 10 simulations for 100 orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.11 Hereλ = 20,c = 10,m = 10,N = 10 2 ,¯ρ = 0.26,p 1 = p 2 = 8,r 1 = 10, ¯k = 12. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃23.2. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃18. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. . . . . . . . . . 61 4.12 Hereλ = 20,c = 10,m = 10,N = 10 2 ,¯ρ = 0.31,p 1 = p 2 = 8,r 1 = 12, ¯k = 14. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃ 24. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃16. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. . . . . . . . . . 61 4.13 Hereλ =20,c =10,m=10,N =10 2 ,¯ρ =0.3,p 1 =p 2 =8,r 1 =15, ¯k =13. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃23. Also, Eq. (4.2) does not hold since p 1 < λm √ ck ≃17. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. . . . . . . . . . . . . 61 4.14 Hereλ = 20,c = 10,m = 10,N = 10 2 ,¯ρ = 0.3,p 1 = p 2 = 10,r 1 = 12, ¯k = 13. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃ 23. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃17. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. . . . . . . . . . 61 ix 4.15 Warehouse inventory level where CL and RS both fail. The fill shows the standard deviation of 10 simulations for 100 orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.16 Here λ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.47,p 1 = p 2 = 8,r 1 = 15, ¯k = 33. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃ 0.13. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃0.55. For CL, Eq. (4.3) holds too, sincep 2 ≥ λm √ c ≃3.16. . . . . . . . . . . . . 62 4.17 Here λ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.14,p 1 = p 2 = 10,r 1 = 8, ¯k = 11. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃ 0.1. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃0.95. For CL, Eq. (4.3) holds too, sincep 2 ≥ λm √ c ≃3.16. . . . . . . . . . . . . 62 4.18 Here λ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.14,p 1 = p 2 = 15,r 1 = 12, ¯k = 37. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃ 0.1. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃0.51. For CL, Eq. (4.3) holds too, sincep 2 ≥ λm √ c ≃3.16. . . . . . . . . . . . . 62 4.19 Here λ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.16,p 1 = p 2 = 15,r 1 = 15, ¯k = 31. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃ 0.1. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃0.56. For CL, Eq. (4.3) holds too, sincep 2 ≥ λm √ c ≃3.16. . . . . . . . . . . . . 62 4.20 Warehouse inventory level where CL and RS both are stable. The fill shows the standard deviation of 10 simulations for 100 orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 B.1 Tour length for one-of-a-subset instances. n = 150 to 1000 clusters. All with k ∈{2,3,...,6} points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 B.2 Further inventory level experiments. The fill shows the standard deviation of 10 simulations. 78 B.3 Order arrival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 B.4 Inventory load arrival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 x Abstract We study the probabilistic behavior of routing optimization problems in which one is given a collection of points and the goal is to find the shortest tour that visits a subset of those points that meets certain criteria. Examples of such problems, which we call “selection routing problems”, include the ”one-of-a-subset” trav- elling salesman problem (TSP), and generalized TSP (GTSP), the orienteering problem, the prize collecting travelling salesman problem, and the distance-constrained generalized vehicle routing problem. We derive continuous approximation formulas for several such problems under the assumption that point-to-point distances are Euclidean and all points are independent and identical samples of a probability density on a compact planar region. Numerical simulations demonstrate that our formulas provide accurate estimates of routing costs. Finally, we analyze the random stow warehouse model using our results and show its efficiency compared to classical warehouse policies. xi Chapter1 Introduction The traveling salesman problem (TSP) is an NP-hard problem in computer science and combinatorial op- timization literature. This problem is concerned with finding the optimal Hamiltonian cycle which has the minimum sum of edge weights. In the Euclidean TSP, the edge weights are the Euclidean distance of the vertices in a Euclidean space. There are many problems in the combinatorial optimization literature that root in TSP, such as the traveling repairman problem (TRP), the traveling purchaser problem (TPP), and the vehicle routing problem (VRP), that all have important applications in engineering and computer science problems. This family of problems, especially TSP, arise in applications such as optimal routing in communication networks, planning, logistics, manufacturing of microchips, transportation, and delivery systems. There are many efficient heuristics and exact algorithms to attack various versions of these problems. However, these algorithms are not flexible enough for large-scale real-world applications where points are usually distributed under different probability laws. Having a comprehensive and large-scale insight into these problems can be very constructive; this motivates the study of an asymptotic and high probability analysis. There is a long-standing history of studies on asymptotic and probabilistic bounds over different graph structures. These works study many graph structures over Euclidean points in the context of the Beardwood-Halton-Hammersley (BHH) theorem and its extensions. This theorem is originally stated in [6] 1 and later further developed by Steele in [67]. In this dissertation, we are interested in the optimal (shortest) tour length over random sample points embedded in Euclidean spaces. According to the BHH theorem, the strong law of large numbers (LLN) holds for specific spanning graphs over the random points. The TSP, minimal spanning tree (MST), and nearest neighbor graphs (k-NNG) are some examples of such graph structures. The sort of problems we look at has additional attributes on top of these problems, which make them challenging and interesting. We study the asymptotic and probabilistic behavior of what we will call selection routing problems (SRP). One of the first instances of this problem is the generalized traveling salesman problem (GTSP) or one-of-a-subset (or m-of-a-subset) problem. In this problem, given a collection of point sets,X 1 ,...,X n , we seek the optimal ordering and selection of points that visit1 (respectivelym) point of each point setX i . GTSP is a well-known combinatorial problem in the literature and arises in a wide variety of applications. For instance, analyzing the trade-off between localized and centralized provision of services and goods. In the localized provision case, specialization of the business and proximity of consumer supplier is more achievable. On the other hand, in the centralized version, the economies of scale, density, and agglomera- tion become beneficial. This trade-off is aligned with the comparison between efficiency in transportation along a backbone network versus direct trips between locations. These fundamental dichotomies consists an important body of the research in transportation and logistics [10]. The e-commerce business such as delivery services [43] need to consider the aforementioned trade- off. Examples are Google Shopping Express, Amazon Prime, Instacart, Walmart To Go, and Ubereats. Especially, after the Covid-19 pandemic, the delivery services demand surged. A big challenge in this business is having a mathematical model that is tractable enough to give useful and also truthful insights. This trade-off also shows up in household problems like running routine errands, called multi-stop trips [12]. For instance, one might want to pick a bank that is farther but next to a shopping center, rather than going to a closer bank and then driving to a shop. 2 Last-mile delivery businesses [8] can apply GTSP models in their analysis; for instance, we can use the socially networked systems to improve on the delivery systems. That is to offer multiple possible locations for delivery point to the customers (as opposed to their doorstep). Then each customer would represent a point setX i . Another motivation for studying selection routing problems, like GTSP, is their applications in warehouse design problems. For instance, Amazon employs the random stow strategy in their warehouse and believes in its efficiency in helping the fast-paced delivery [17]. The idea is to disperse the stock keeping units (SKU) randomly across the warehouse. Then since an order often consists of multiple items, the spread of the SKU could maximise the chance of them being closer to each other. We analyse the performance of the random stow using a GTSP model and compare it with a classic warehouse. The "cardinality constrained" TSP, or sparse subset TSP (SSTSP) [2], is another instance of SRP in which we seek the optimal tour visiting⌈pn⌉ ofn points. Note that the TSP is an instance of this problem with p = 1. The paper [2] proved an asymptotic approximation for SSTSP in L 2 . They also proved the existence of a positive constant such that SSTSP/p √ n → c(0 + ) in L 2 . Analysing SSTSP can help us develop models for cumulative TSP and traveling repairman problem as well. We study the asymptotic bounds of this problem in a high probability setting. We also drive asymptotic bounds for Sparse Subset GTSP which is a combination of GTSP and SSTSP in which we seek to optimally select⌈pn⌉ point sets and visit a point from each of them. RandomK GTSP is another instance of SRPs that we bound its tour length. In this problem the cardinality of the point sets vary based on a distribution and is not necessarily the same (known) for all of them. 3 1.1 Contributions: Table 1.2 shows a summary of the bounds we obtain in this dissertation with the notation based on Ta- ble 1.1. The notation isP(x) = RR x ′ :f(x ′ )≤ f(x) f(x ′ )dx ′ . We validate these bounds in Chapter 3 for syn- thetic instances. Also, we model and analyze random stow in Chapter 4 and validate our findings using simulation experiments in Section 4.3. 1(A) Indicator function of eventA A rectangle inR 2 R A compact planar region of unit volume inR 2 X A point set with size|X|=k X A point in aX n Number of point sets k i (=k) The cardinality ofX i (we assumek i =k in our work) m Number of points to visit from eachX i f A absolutely continuous probability distribution defined on R Table 1.1: Notational conventions 4 Problem Uniform Non-uniform One-of-a-subset TSP √ nk k+1 √ nk R R p f(x)P(x) k− 1 dx m-of-a-subset TSP m p n k - Cardinality-constrained TSP (SSTSP) p √ n √ n R R p f(x)1(P(x)≥ m n )dx Sparse Subset GTSP p √ nK/(k+1) - RandomK GTSP p √ nEK - Table 1.2: Summary of the results. 1.2 Relatedworks In this section, we review the previous research in the area of selection routing problems and other re- lated problems. The examples are TSP, GTSP, Traveling purchaser problem, Orienteering problem, Prize- collecting TSP, cumulative TSP, Time dependent TSP and TSP with neighborhoods. We also visit the related works on continuous approximation models. 1.2.1 TSP The traveling salesman problem (TSP) is one of the most widely studied NP-hard combinatorial optimiza- tion problems which has links with many fields of Mathematics and computer science, e.g. combinatorial optimization problems, graph theory, and integer programming. The real-world applications of TSP are prevalent in many different settings. Also, many practical problems can be formulated as a TSP, yet it is hard to solve it efficiently. These applications include examples such as drilling of printed circuit boards, overhauling gas turbine engines, X-ray crystallography, computer wiring, order-picking in warehouses, vehicle routing, and mask plotting in PCB production [50]. 5 There is a rich literature on devising efficient algorithms and performing probabilistic analysis on TSP. Christofides’ paper [15] gave a polynomial-time, O(n 3 ) for n points, approximation algorithm for TSP with a solution that in the worst-case is 50% longer than the optimal solution. This algorithm consists of two steps, first calculating the shortest spanning tree in the graph and then finding a minimum cost perfect matching of the induced sub-graph. Since then many researchers analyzed the bounds of this algorithm (e.g. [16]) and also tried to devise better algorithms. Recently, the paper [41] devised, for someϵ> 10 − 36 a3/2− ϵ approximation algorithm for metric TSP, which is the state-of-art at the moment. Formulating TSP as an integer programming problem is simple [50]. These formulations can be based on the Hamiltonian cycle properties, flow based, Local-global formulations, and etc [54]. This problem then is solved by branch-and-bound (B&B) algorithm. B&B iteratively tightens the upper and lower bound on the solution until there is an optimal feasible solution. Linear programming relaxations produce upper bounds and cuts can be added into the constraints gradually as in [54] and [4]. To reduce the time complexity of the exact formulations and improve on their scalability, many re- searchers also used heuristic algorithms to solve this problem. Heuristic algorithms normally give sub- optimal solutions but with more reasonable computational complexity. The solution quality of these algo- rithms potentially depends on their initialization and hyper-parameters. Lin and Kernighan’s algorithm in [47] have almost the best quality solutions, especially the variant proposed in [32]. Many other algo- rithms can approximately solve the TSP as well, including; Genetic algorithm as in [37] which uses a cycle crossover operator, simulated annealing [76], reinforcement learning algorithms like Ant-Q [24], Tabu search based algorithms [48]. For many applications, it is beneficial to assume a probabilistic distribution law on the points of the graph. There is a long history of studies on asymptotic and high probability bounds over various graph structures. These works study many graph structures over Euclidean sample points in the context of the Beardwood-Halton-Hammersley (BHH) theorem and its extensions. The strong LLN establishes the 6 backbone of the BHH theorem and its extensions [38]. The motivation for developing this analysis is devising an effective and truthful approximation with a small computational price, i.e. constant time given the distribution. 1.2.2 GeneralizedTSP(GTSP) GTSP is one of the most famous variants of TSP. In this work, we have two versions of GTSP, one-of- a-subset or TSP(n,k,1) and m-of-a-subset or TSP(n,k,m) for m ≤ k. The former is more related to the well known GTSP. Actually, in the literature GTSP stands for the optimal (shortest) tour that visits at least one point from each point set. Many researchers refer to ourone-of-a-subset as E-GTSP. For instance in [21], E-GTSP means the tour that visits exactly one point in each point set where E denotes Equality. However, in the Euclidean space, due to the triangle inequality, GTSP is equivalent to E-GTSP. In this problem, given a collection of point sets,X 1 ,...,X n , we seek the optimal tour that visits1 (or m) point of each point setX i . One of the first researchers who worked on GTSP is Henry-Labordere. He first introduced the asymmetrical GTSP in [1] and proposed a dynamic programming approach for solving it. The application in their paper is sequencing computer files. GTSP has more constraints compared to TSP because it allows visitingm of each point set, as such it has one extra constraint for each point set. The GTSP is an NP-Hard optimization problem with potential applications in warehousing, distribu- tion, and scheduling. As such it is important to solve it to a good level of optimality. This problem adds the ability of selection on top of TSP which is just a sequencing problem. Saksena [60] studied the routing of welfare clients through governmental agencies as a symmetric GTSP. Other applications include vehicle dispatching [26], locating problems like the plant location [58], and other problems such as the warehouse order picking with multiple stock locations, airport selection and routing for courier planes, and certain types of flexible manufacturing scheduling. For more on the applications refer to papers like [21], [40]. 7 In the sense of the trade-off between a backbone structure versus a localized business that we men- tioned in the introduction, GTSP can be used in solving problems like the Truck and a Drone [13]. This problem is related to [73], which calculates the net CO2 emissions that result from introducing grocery delivery services vs pickups. It is also similar to [74], which computes the optimal layout of a set of facility locations that are, themselves, connected with a backbone network. Many researchers studied solving particular GTSP instances. These works are often under one of the following three categories; (1) Applying transformations to turn GTSP into TSP, (2) exact solutions, and of course (3) heuristic algorithm on top of the exact algorithms. In the following, we discuss the main contributions in each of these groups. Transformations: One way of finding near-optimal solutions for GTSP is to transform it to other prob- lems that have more efficient solutions. The state-of-the-art transformation which has the same number of points as GTSP is the one in [52] by Noon et al. which transforms GTSP into an asymmetric TSP. This transformation got more attention when the well-known solver GLKH [33] used it. GLKH algorithm first uses the above transformation to build a TSP corresponding to the original GTSP. Then this TSP is solved using LKH [32] solver. This solver has an effective implementation of the Lin-Kernighan heuristic [47]. At the same time as Noon et al., Lien et al. [46] introduced a transformation that turns a GTSP into a TSP instance with more than 3 times number of points. This transformation is not that efficient. After a while, Dimitrijevic and Saric [18] developed a transformation that is more efficient with less number of points. Their transformation has twice as many points as the original problem. Considering the GTSP polytope, a non-exact transformation is developed in [22]. Laporte and Semet [44] gave a transformation with dou- ble number of vertices and also report the results of extensive computational experiments relative to this transformation. As we mentioned, the transformations introduce at least a larger problem which would need more computational power to solve it. Gutin et al. [29] mentioned another limitation of these transformations 8 and introduced a memetic algorithm in their paper. That is the transformations might not preserve the optimality of sub-optimal solutions. In other words, the conversions of near-optimal TSP tours may well result in infeasible GTSP solutions. In other words, the sub-optimal solution of the transformed TSP is not necessarily feasible in the corresponding GTSP. Thus, transformation approaches may not allow us to find a sub-optimal GTSP solution quickly. This is a motivation for the developing special GTSP heuristics, which we discuss below. Also, another limitation of the transformations is the use of big M edge cost or minus big M edge cost. This affects the stability of the solver solutions and increases the optimality gap. Also, exact solution models can solve GTSP but obviously, they do not scale well enough. Heuristics: Heuristic algorithms can help solving the exact mathematical programming model. Noon and Bean [51] proposed an exact model for the asymmetric GTSP problem. They developed a Lagrangian based heuristic to bound the solution from above and below. Then, they remove the edges and points that are certainly not in the optimal solution. For this, they use a greedy heuristic based on relative costs. One set of constraints in their model corresponds to the network flow conservation constraints and ensure that a solution tour is uninterrupted and continuous. They bring these constraints to the objective function using a Lagrangian relaxation which gives an upper bound for the problem. In the end, an Enumeration procedure divides the feasibility space into components that construct the basis for a branch-and-bound method to find the optimal solution. Their experimental results include up to n = 100 points in total. In another study, Laporte et al. [45] formulated GTSP as an integer programming model and solved it as a problem of finding Hamiltonian tour on a collection of sets. The branch-and-cut algorithm of Fischetti et al. [21] solves an LP relaxation of the original integer programming model at each stage. They propose a separation algorithm (separation of GTSP polytopes sub-spaces) that helps the branch-and-bound method for the exact solution of GTSP. Their experimental results include instance with up to 442 points. Due to the limitations of the transformations and exact algorithms such as scalability and optimality preservation problems, researchers developed many direct heuristic algorithms for GTSP as well. Many 9 of these heuristics obtain a solution within 1% optimality except some large test cases. There are many instances of these algorithms which we mention in the following. Gutin and Karapetyan [29] proposed a memetic algorithm in their paper using a combination of local/global search along with a genetic algorithm. Renaud et al. [56] proposed a composite heuristic, called GI 3 (Generalized Initialization, Insertion, and Improvement). This heuristic includes three phases: first the construction of an initial partial solution. In this phase, we construct an initial greedy tour that tries to visit one point of each point set. The second phase is the insertion of a node from each non-visited point set. Finally, a solution improvement phase which applies 2-opt and 3-opt operations to improve the tour. This algorithm is based on their own I 3 heuristic [57] which solves the symmetric TSP. Particle swarm optimization (PSO)-based algorithms are also effective for solving GTSP. For example, the work in [61] which gives a PSO-based algorithm for TSP and GTSP. Another paper, [71], in the same year combined a discrete PSO with a local search and variable neighborhood descend algorithm. Other types of heuristics also have been used. Snyder et al. [65] combined a genetic algorithm with a local search heuristic and encoded the solutions with a random- key algorithm. Ant colony heuristic is also applied for solving GTSP; for instance, the work in [28] and [75]. Smith and Imesn [64] proposed a heuristic called GLNS based on adaptive large neighborhood search. Their algorithm includes a general insertion mechanism that contains nearest, farthest, and random insertion heuristics. This algorithm recursively eliminates the points from the tour and inserts unvisited nodes into the tour in a guided way. Approximation methods: We can also attack the GTSP problem using approximation methods. This class of work is related to our work in a randomized setting. Slavik in his first papers [62] and [63] devel- oped approximation algorithms for GTSP (one-of-a-subset), including a 3/2ρ − approximation algorithm for the GTSP, whereρ is the number of points in the largest cluster. As we can see, the error bound of this algorithm depends on the largest cluster’s cardinality, which can make it a very weak approximation. For instance, in the case that k i = k, the error bound is 3/2k which is not very desirable. Later, Garg et al. 10 [25] improved on this approximation and proposed aO(log 2 nkloglognklogn)− approximation for the case in which the edge costs satisfy the triangle inequality. A new family of work considers the case where the clusters are defined inside grid systems. In particular, [7] proposed a ( 1.5+8 √ 2+ϵ )-approximation polynomial-time algorithm for clusters inside grid systems. This algorithm’s time complexity is of polyno- mial order in number of clusters and number of points. Following this work, Khachay et al. [42] proposed (1+ϵ )-approximation schemes in polynomial time whenn andk has special relations order-wise. 1.2.3 Continuousapproximationmodels The motivation for studies on continuous approximation paradigm is the replacement of combinatorial quantities that are difficult to compute with simpler mathematical expressions, which (under certain con- ditions) provide accurate estimations. This type of approximation is common for combinatorial problems. Examples include TSP approximations as in [6] and [19], facility location problem as in [31], [34], and [53], and basically anysubadditiveEuclideanfunctional such as a minimum spanning tree [68], k-medians [35], Steiner tree [68], or matching [66] and other papers such as [55], [67], and [69] papers. We consider a continuous approximation model for the selection routing problems. In this way our work is similar to papers like [9], which studies the trade-offs between inventory and transportation costs. Also, the work in [36] studies how to route relief vehicles following a disaster in a time-sensitive manner. Chowdhury et al. [14] showed how to optimally partition the disaster-affected region for emergency drone distribution. Another similar work is [39] which develops a geometric model to find the optimal long-term vehicle fleet composition for distribution activities. Finally for more refere to Franceschetti et al. who reviewed many studies in the same vein in their paper [23]. In this work, we assume that the points are distributed based on a density function, i.e. stochastically. The paper [11] studies a partitioning algorithm in which the client locations are independent and identi- cally distributed samples from a given probability density function. Goodson [27] describes a set of rollout 11 policies based on fixed routes to obtain dynamic solutions to the vehicle routing problem with stochastic demand and duration limits. For more on the literature of stochastic vehicle routing models refer to the survey in [59]. 1.3 Preliminaries In this section, we start with few preliminary results we need for the main theorems. Theorem 1 is the famous Beardwood-Halton-Hammersley (BHH) Theorem proved in [6]. Theorem 1 (BHH Theorem). There is a constant β d such that, for almost any sequence of independent random variables{X i } sampled from an absolutely continuous density f onR d with compact support, we have lim n→∞ TSP(X 1 ,...,X n ) n (d− 1)/d =β d Z R d f(x) (d− 1)/d dx with probability one. Ford = 2, although the exact value ofβ 2 is unknown, it has been shown that0.6250≤ β ≤ 0.9204; see [3]. The next result states the additivity properties of TSP tour length. Lemma 2 (Super- and Sub-additivity of the TSP). LetR ⊂ R 2 be a compact Lebesgue measurable set, partitioned into piecesP 1 ,...,P m whose common boundaries (i.e. ∂P i ∩ ∂P j ) have finite length. There existsaconstantC thatdependsonlyonthepartitionsuchthat,foranysetofpointsX ={x 1 ,...,x n }⊂R , we have − C + m X i=1 TSP(X∩P i )≤ TSP(X)≤ C + m X i=1 TSP(X∩P i ) See Lemma 2.4.1 of [69], for the proof. The lemma below is useful for bounding the length of a TSP tour from below. 12 Lemma 3. LetX 0 be the origin inR d and letX 1 ,...,X n be a collection of independent, uniform samples drawn from a regionR of unit volume inR d . Then Pr(TSP(X 0 ,X 1 ,...,X n )≤ ℓ)≤ Γ( n+1)· 2π d/2 Γ( d/2) ! n · Γ( d) n Γ( dn+1) · ℓ dn . Ford=2 the RHS is justn! (2πℓ 2 ) n (2n)! . The following lemma allows us to approximate a probability density function with a step function in a way that preserves certain quantities of interest. Lemma 4 (Approximation With a Step Function). Let f be a probability density function with compact supportR⊂ R 2 whose level sets have Lebesgue measure zero, letk be a positive integer, and define P(x):=Pr(f(X)≤ f(x))= Z x ′ :f(x ′ )≤ f(x) f(x ′ ) dx ′ . For anyϵ> 0, there exists a step density functionϕ (x):= P s i=1 a i 1(x∈ i ) and corresponding Π( x):=Pr(ϕ (X)≤ ϕ (x))= Z x ′ :ϕ (x ′ )≤ ϕ (x) ϕ (x ′ ) dx ′ such that the following conditions hold: 1. R R |ϕ (x)− f(x)| dx≤ ϵ , 2. P(x) k− 1 − Π( x) k− 1 ≤ ϵ for allx∈R, 3. |1(P(X)≥ p)− 1(Π( X)≥ p)|≤ ϵ for allx∈R, 4. All of the components ofϕ have the same mass, i.e. a i Area( i )=1/s. Proof of Lemma 4. The requirement that the level sets have measure zero is not actually necessary, but we find it useful for keeping notation consistent throughout this paper (this requirement is flagrantly violated 13 whenf is a uniform distribution, which is our base case for all of the various TSP instances in this paper anyway). For a large integerq, define contour sets S i ={x : (i− 1)/q < P(x)≤ i/q}. For eachS i , we can approximate the restriction off toS i (i.e. f(x)1(x∈S i )) to arbitrary precisionϵ ′ by a step function ψ i (x)= P j a ij 1{x∈ ij } (this is a classical result of measure theory; see e.g. Theorem 2.4(ii) of [70], i.e. step functions are dense inL 1 (R d )). Appendix A.3 establishes that for alli andj, we can assume without loss of generality that • ij ⊂S i for alli andj (i.e. the support ofψ i is contained inS i ), • a ij 0, we setq =⌈1/ϵ ⌉ andϵ ′ = ϵ/q in the above construction. The functionψ := P q i=1 ψ i is therefore a step density approximation off whose aggregate error overR is at mostϵ , so condition 1 is satisfied. If we define Π ′ (x)= R x ′ :ψ (x ′ )≤ ψ (x) ϕ (x) dx, then condition 2 and 3 are satisfied as well (using triangular inequality and add subtract Π ′ ); indeed, this is the purpose of the initial decomposition ofR into theS i ’s. For ease of notation, we now re-index all of the components of ψ (i.e. we disregard the fact that ψ decomposes into a sum ofψ i ’s) so that we simply haveψ (x)= P j b j 1{x∈ j }, whereb j andArea( j ) are rational. If we takeδ to be the lowest common denominator over allb j Area( j ), then we can write b j Area( j )=z j δ , withz j a positive integer. To satisfy the last condition, all that remains is to decompose each j intoz j pieces of equal area, and letϕ denote the step function resulting thereof, which completes the proof. Finally, some technical lemmas we use in the proofs. The first two are textbooks 14 Lemma5 (Stirling’s Formula). ThegammafunctionΓ( x)satisfies logΓ( x+1)=xlogx− x+ 1 2 logx+ 1 2 log2+ 1 2 logπ +O(1/x) asx→∞. Lemma6 (Borel-Cantelli, [70]). Let{E n }beasequenceofeventsinasamplespace. Thenif P ∞ n=1 Pr(E n )< ∞, we havePr(E n occurs infinitely often )=0, or equivalently Pr(limsup n→∞ E n ):=Pr( ∞ \ n=1 ∞ [ m=n E m )=0. Lemma7. Letℓ > 0 and letD⊂ R dn denote the set of alln-tuples(u 1 ,...,u n ) of points inR d such that P n i=1 ∥u i ∥≤ ℓ. The volume ofD,Vol(D), satisfies Vol(D)= 2π d/2 Γ( d/2) ! n · Γ( d) n Γ( dn+1) · ℓ dn . (1.1) Ford=2 this is just (2πℓ 2 ) n (2n)! . Proof. The proof is nothing more than the integral Z B d (ℓ) Z B d (ℓ−∥ un∥) ··· Z B d (ℓ− P n i=3 ∥u i ∥) Z B d (ℓ− P n i=2 ∥u i ∥) 1du 1 du 2 ... du n− 1 du n , which we can compute using a standard inductive argument applying Lemma 21. 15 Chapter2 SelectionRoutingProblems In this section, we state the main results for different selection routing problems. These results are stated as asymptotic bounds on the tour length of the problems. The main theorems consist of matching upper and lower bounds. 2.1 One-of-a-subsetTSP Throughout this section, we study a version of generalized TSP where one point from each point set is visited. We refer to this problem as the One-of-a-subset TSP. In particular, we let GTSP(n,k,1) := L(X 1 ,...,X n ) denote the length of the shortest tour that visits one element from each point setX i , which has|X i |=k≥ 2 for alli. Theorem8 (Uniform One-of-a-subset TSP). Letk≥ 2 be fixed and let X 1 ,...,X n be point sets of cardi- nalityk that are all drawn independently and uniformly at random in a region of unit area inR 2 . Then for all fixed k≥ 2, we have 0.4839=:λ 1 <liminf n→∞ L(X 1 ,...,X n ) √ nk/(k+1) ≤ limsup n→∞ L(X 1 ,...,X n ) √ nk/(k+1) <µ 1 :=1.8408 (2.1) with probability one. 16 We sketch the proof it in the following. Proof Sketch. For the lower bound, we apply a union bound to Lemma 3 over all possible ways of selecting one point from each point set. Then we find the largest constant λ 1 such that the limiting behavior happen. For the upper bound, Theorem 1 along with choosing the right-most point from each point set suffices. Proof of Theorem 8. To derive the lower bound, letE n be the event thatL(X 1 ,...,X n )<c √ nk/(k+1) for fixed c andk. Applying the union bound to Lemma 3 for the cased = 2 and using the fact that there arek n different possible ways to select one member from each set X i , we see that Pr(E n )≤ k n · Γ( n+1) Γ(2 n+1) 2πc 2 nk (k+1) 2 n (2.2) ≤ k n · Γ( n+1) Γ(2 n+1) 2πc 2 n k n =⇒ logPr(E n )≤ (1+2logc− log2+logπ )n−O (1) (2.3) where we have applied Lemma 5. We see that Eq. (2.3)→−∞ if and only if the coefficient of n is negative: 0> 1+2logc− log2+logπ ⇕ c< r 2 πe ≈ 0.48393. (2.4) Furthermore, this guarantees thatPr(E n )≤ a − n for somea > 1, so that P ∞ n=1 Pr(E n ) <∞. We apply Lemma 6 to obtainλ 1 <liminf n→∞ L(X 1 ,...,X n ) √ nk/(k+1) with probability one as desired. 17 The upper bounding constant is conceptually very simple: first, take the special case where R is the unit square, and from each point setX i , let point X i be the member ofX i that lies the farthest to the right. The pointsX i follow the probability distribution f(x=(x 1 ,x 2 ))=kx k− 1 1 and Theorem 1 says that a TSP tour of a collection of points following this distribution must satisfy (with probability one) lim n→∞ TSP(X 1 ,...,X n ) √ n =β 2 Z R p f(x)dx= Z 1 0 Z 1 0 q kx k− 1 1 dx 1 dx 2 = 2β 2 √ k k+1 (2.5) whereβ 2 is the BHH constant, and so we merely apply the bound ofβ 2 <0.9204 from Section 8.5 of [20] which gives the desired value ofµ 1 . WhenR is an arbitrary region (of unit area), the intuition is the same, but we exploit the fact that the selection of the rightmost point in the previous construction was arbitrary. LetR 1 ,...,R s be a partition ofR into measurable pieces with area 1/s, and for each point setX i , let X i be the member ofX i that belongs to the piece of the highest index (breaking ties arbitrarily). We have Pr(X i ∈ j [ j ′ =1 R j ′)= j s k =⇒ Pr(X i ∈R j )= j s k − j− 1 s k =:p j 18 withp∈∆ s− 1 , the usual probability simplex. TheX i ’s follow a step distributionϕ (x)=s P s j=1 p j 1{x∈ R j }, and so Theorem 1 says that a TSP tour of a collection of points following this distribution must satisfy (with probability one) lim n→∞ TSP(X 1 ,...,X n ) √ n =β 2 Z R p ϕ (x)dx= β 2 s s X j=1 √ sp j →β 2 Z 1 0 √ kt k− 1 dt ass→∞ = 2β 2 √ k k+1 which completes the proof. Here note thatp j < kj k− 1 +k 2 j k− 2 s k and thus s X j=1 √ sp j s < s X j=1 r kj k− 1 +k 2 j k− 2 s k+1 < s X j=1 √ k s ( j s ) (k− 1)/2 + r k s ( j s ) (k− 2)/2 → √ k Z 1 0 √ t k− 1 dt ass→∞ Next we want to generalize our result to the case where the distribution is not necessarily uniform. For this case we require the following consequence of Theorem 8. Corollary 9 (Tour length in a subset with uniform demand). LetX 1 ,...,X n be independent uniform samples of cardinalityk drawn from a compact regionR with volume1 and letS ⊂R . Then liminf n→∞ L(X i :X i ⊂S ) √ n ≥ λ 1 p Area(S) k+1 k k+1 withprobabilityone,whereX i ⊂S meansthatallkelementsofX i lieinS andλ 1 =0.4839fromTheorem8. We prove Corollary 9 using an scaling and binomial probability law application. 19 Proof of Corollary 9. Let{Y i } denote a sequence of uniform samples of cardinalityk drawn fromS (not R). Certainly, by scaling areas, we have liminf n→∞ L(Y 1 ,...,Y n ) √ n ≥ λ 1 p Area(S)k/(k+1). We now define N(n) =|i∈{1,...,n} :X i ⊂S| andp = Area(S) k to be the probability thatX i ⊂S . We have liminf n→∞ L(X i :X i ⊂S ) √ n =liminf n→∞ L(Y 1 ,...,Y N(n) ) √ n =liminf n→∞ L(Y 1 ,...,Y N(n) ) √ n · s N(n) N(n) =liminf n→∞ L(Y 1 ,...,Y N(n) ) p N(n) · r N(n) n = liminf n→∞ L(Y 1 ,...,Y N(n) ) p N(n) ! lim n→∞ r N(n) n ! > λ 1 p Area(S)k/(k+1) √ p=λ 1 q Area(S) k+1 k/(k+1) as desired. We next prove the non-uniform convergence result for a special case of the general form distribution. the density is a step function. Lemma 10 (Tour length from a step density). Let ϕ (x) = P s i=1 a i 1(x ∈ i ) be a step density function with compact supportR such thata 1 ≥···≥ a s anda i Area( i )=1/s for alli (so thatArea(R)=1). If Y 1 ,...,Y n are independent samples fromϕ having cardinalityk andΠ( x) is defined as in Lemma 4, then liminf n→∞ L(Y 1 ,...,Y n ) √ n ≥ λ 1 √ k 2 Z R q ϕ (x)Π( x) k− 1 dx with probability one, whereλ 1 =0.4839 from Theorem 8. 20 As the proof of this Lemma is very insightful we present it here. The main idea is to scale each i to get areas where the distribution is uniform. Then we use Corollary 9 along with Lemma 2 to calculate the asymptotic tour length. Proof of Lemma 10. We have L(Y 1 ,...,Y n )= min Y i ∈Y i TSP(Y 1 ,...,Y n ) = min Y i ∈Y i s X i=1 TSP(Y 1 ,...,Y n ∩ i )+O(1) from Lemma 2. For each i, define Ψ i : i → R 2 by Ψ i (y) = √ a i y +ξ i , where the ξ i are selected so that the imagesΨ i ( i ) are all disjoint (their specific values are irrelevant). Let Ψ: R→R 2 be the union of all the Ψ i ’s (i.e. for any y ∈ R, we have Ψ( y) = Ψ i (y), where i ∋ y). The significance of this construction is that the image Ψ( R) has area 1 and Ψ( Y 1 ),...,Ψ( Y n ) becomes a uniform collection of samples inΨ( R), as shown in Fig. 2.1. For notational compactness, define Y ′ i =Ψ( Y i ),Y ′ i =Ψ( Y i ), and ′ i =Ψ( i )=Ψ i ( i ). 21 Figure 2.1: The gray area isR and the arrows showξ i ’s, the dashed rectangles represent the ′ i all with areaA ′ i = 1 s , The darker gray shows largera i and note thatA i a i =1/s Basic scaling arguments tell us that TSP({Y 1 ,...,Y n }∩ i )= 1 √ a i TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ). Since thea i ’s are decreasing, we can define increasing terms b i = 1/ √ a i and we can also construct non- negativec j ’s so thatb i = P i j=1 c j . This tells us that min Y i ∈Y i s X i=1 TSP({Y 1 ,...,Y n }∩ i )= min Y ′ i ∈Y ′ i s X i=1 1 √ a i TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) = min Y ′ i ∈Y ′ i s X i=1 b i TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) = min Y ′ i ∈Y ′ i s X i=1   i X j=1 c j   TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) = min Y ′ i ∈Y ′ i s X j=1 c j s X i=j TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) 22 and now note that for allj, s X i=j TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i )≥ L(Y ′ p :Y ′ p ⊂ s [ i=j ′ i )−O (1), as shown in Fig. 2.2. Figure 2.2: Assuming thatL(Y ′ p :Y ′ p ⊂ S s i=j ′ i ) consists of the dashed lines connecting TSP i =TSP({Y ′ 1 ,...,Y ′ n }∩ ′ i ) with the thin lines of theTSP i ’s, the idea is that the length of the dashed lines is constant w.r.tn, then the thick lines insideTSP i ’s are the extra part that makes the inequality valid. For notational simplicity, we define ′ ≥ j = S s i=j ′ i , which has area (s− j + 1)/s; the samples Y ′ p :Y ′ p ⊂ ′ ≥ j are independently and uniformly distributed within ′ ≥ j . Therefore, Corollary 9 says that liminf n→∞ 1 √ n L(Y ′ p :Y ′ p ⊂ ′ ≥ j )≥ λ 1 s s− j+1 s k+1 k (k+1) 23 and so if we adopt the convention thatb 0 =0, we have liminf n→∞ 1 √ n min Y i ∈Y i s X j=1 c j s X i=j TSP({Y ′ 1 ,...,Y ′ n }∩ √ a i i ) ≥ λ 1 √ k (k+1) s X j=1 c j s s− j+1 s k+1 = λ 1 √ k (k+1) s X j=1 (b j − b j− 1 ) s s− j+1 s k+1 ≥ λ 1 √ k (k+1) √ s k+1 s X j=1 b j q (s− j+1) k+1 − q (s− j) k+1 and we have q (s− j+1) k+1 − q (s− j) k+1 ≥ k+1 2 q (s− j) k− 1 for allj≤ s andk, thus 24 s X j=1 λ 1 √ kb j q (s− j+1) k+1 − q (s− j) k+1 (k+1) √ s k+1 ≥ λ 1 √ k 2 √ s k+1 s X j=1 b j q (s− j) k− 1 = λ 1 √ k 2s s X j=1 1 √ a j s s− j s k− 1 = λ 1 √ k 2 s X j=1 √ a j Area( j ) s s− j s k− 1 ≥ λ 1 √ k 2 s+1 X j=2 √ a j Area( j ) s s− j+1 s k− 1 (2.6) = λ 1 √ k 2 s X j=2 √ a j Area( j ) s s− j+1 s k− 1 = λ 1 √ k 2   s X j=1 √ a j Area( j ) s s− j+1 s k− 1 − √ a 1 Area( 1 )   ≥ λ 1 √ k 2   s X j=1 √ a j Area( j ) s s− j+1 s k− 1 − 1 s p Area(R)   (2.7) = λ 1 √ k 2 Z R q ϕ (x)Π( x) k− 1 dx− λ 1 √ k 2s p Area(R) | {z } (∗ ) (2.8) as desired. Here Eq. (2.6) holds for large enoughs since √ a 1 Area( 1 ) r ( s− 1 s ) k− 1 ≥ s X j=2 √ a j Area( j ) r ( s− j+1 s ) k− 1 − r ( s− j s ) k− 1 = s X j=2 1 s √ a j r ( s− j+1 s ) k− 1 − r ( s− j s ) k− 1 where lettings→∞, left hand side is √ a 1 Area( 1 ), while the right hand side is 0. Also Eq. (2.7) uses the fact thata 1 Area( 1 ) = 1/s anda 1 is the largest of all thea i ’s. The desired result follows from the 25 fact that we can makes as large as we like by breaking each component i ofϕ into multiple components of equal area without changing the functionϕ , which allows us to drop the term(∗ ). To attack the non-uniform case of Theorem 8, we require one more observation as follows. Claim11. If R R |g|dx≤ δ , then R R p |g|dx≤ p Area(R)δ . Proof of Claim 11. If we maximize R R √ gdx subject to the constraint that R R gdx≤ δ , the solutiong ∗ is uniform withg ∗ (x)=δ/ Area(R) everywhere. It is now a simple matter to apply Lemma 4 to prove the following result for arbitrary distributed points. Theorem12 (Non-uniform One-of-a-subset TSP). Letf,R,andP beasinLemma4. Withprobabilityone, we have λ 1 2 <liminf n→∞ L(X 1 ,...,X n ) √ nk RR R p f(x)P(x) k− 1 dx ≤ limsup n→∞ L(X 1 ,...,X n ) √ nk RR R p f(x)P(x) k− 1 dx < µ 1 2 (2.9) withλ 1 ,µ 1 as in Theorem 8. Proof Sketch. First we use Lemma 4 to approximatef byϕ . Then using a coupling argument we coupleX withY such thatY ∼ ϕ andPr(X ̸=Y)≤ ϵ/k for anyϵ . Now we can apply Lemma 10 to sampled point sets fromY we can lower bound the tour length for anyϵ . Therefore the limiting lower bound holds. For the upper bound, we construct a tour by visiting the densest area. This is intuitive because it is desirable to select those points that belong to more densely populated areas. Finally, by Theorem 1 we can approximate the resulting tour length which is an upper bound. The detailed proof is as follows. Proof of Theorem 12. Let ϕ be the approximation of f from Lemma 4. By a standard coupling argument (for example theγ coupling of [49]), there is a joint distribution for random variables(X,Y) such thatX 26 has densityf,Y has densityϕ , andPr(X ̸=Y)≤ ϵ/k for anyϵ ; this means that ifX =(X 1 ,...,X k ) is a collection ofk independent samples off andY =(Y 1 ,...,Y k ) is a collection ofk independent samples ofϕ , thenPr(X ̸=Y )<ϵ . 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 )−O (1) =⇒ liminf n→∞ L(X 1 ,...,X n ) √ n ≥ liminf n→∞ L(Y 1 ,...,Y n ) √ n − α 2 p Area(R)ϵn √ n (2.10) =liminf n→∞ L(Y 1 ,...,Y n ) √ n − α 2 p ϵ Area(R) ≥ λ 1 √ k 2 Z R q ϕ (x)Π( x) k− 1 dx− α 2 p ϵ Area(R) where we have applied Lemma 10 in the last inequality. Also to get Eq. (2.10) we used the almost sure bound in [72]. Finally, note that we can select our approximation ϕ, Π arbitrarily closely so that, by Lemma 4 (compactifying our notation momentarily), ϵ ≥ max x P(x) k− 1 − Π( x) k− 1 + Z R |ϕ (x)− f(x)| dx ≥ Z ϕ P k− 1 − Π k− 1 + Z |ϕ − f|P k− 1 ≥ Z ϕ Π k− 1 − ϕP k− 1 +ϕP k− 1 − fP k− 1 = Z ϕ Π k− 1 − fP k− 1 =⇒ p Area(R)ϵ ≥ Z q |ϕ Π k− 1 − fP k− 1 |≥ Z p ϕ Π k− 1 − Z p fP k− 1 =⇒ Z p ϕ Π k− 1 ≥ Z p fP k− 1 − p Area(R)ϵ 27 and therefore, we ultimately conclude that liminf n→∞ L(X 1 ,...,X n ) √ n ≥ λ 1 √ k 2 Z R q ϕ (x)Π( x) k− 1 dx− α 2 p ϵ Area(R) ≥ λ 1 √ k 2 Z q f(x)P(x) k− 1 dx− p Area(R)ϵ − α 2 p ϵ Area(R) ≥ λ 1 √ k 2 Z q f(x)P(x) k− 1 dx− p ϵ Area(R) λ 1 √ k 2 − α 2 ! | {z } (∗ ) (2.11) which completes the lower bound, since Eq. (2.11) can be made as small as desired by choosing small values ofϵ . The upper bounding argument is very simple: for each point setX i , let pointX i be the member ofX i wheref is the densest, i.e.X i =argmax x∈X i f(x). This is intuitive because it is desirable to select those points that belong to more densely populated areas. The density function on theX i ’s is ˜ f(x)=kf(x)P(x) k− 1 and by Theorem 1, the length of a tour of thoseX i ’s satisfies lim n→∞ TSP(X 1 ,...,X n ) √ n =β 2 Z R q ˜ f(x)dx=β 2 Z R q kf(x)P(x) k− 1 dx≤ µ 1 2 Z R q kf(x)P(x) k− 1 dx whereβ 2 is the BHH constant, which completes the proof. 28 2.2 SparseSubsetTSP Throughout this section, we letSSTSP(n,m/n) := L(X 1 ,...,X n ;m) denote the length of the shortest tour that visitsm of the pointsX 1 ,...,X n . We call this a Sparse Subset TSP (SSTSP) or Cardinality Con- strainedTSP (CCTSP). Whenm is non-integer, we round it upwards. The first result is akin to Theorem 8. Theorem13. LetX 1 ,...,X n be independent uniform samples drawn from a region of unit area inR 2 . For all fixed 0 plog(1− p)+2plogc− plog2+plogπ − log(1− p)+p ⇕ c< r 2 πe · (1− p) (1− p)/p . the right hand side of the above expression is convex and increasing inp, and satisfies lim p→0 + r 2 πe · (1− p) (1− p)/p ≥ √ 2π e >0.2935=:λ 2 . 30 Furthermore, this guarantees thatPr(E n )≤ a − n for somea > 1, so that P ∞ n=1 Pr(E n ) <∞. We apply Lemma 6 to obtainλ 2 <liminf n→∞ L(X 1 ,...,X n ;pn)/(p √ n) with probability one as desired. The upper bounding constant is simple: take a TSP tourT of all of the points, whose length satisfies TSP(X 1 ,...,X n )/ √ n→ β 2 with probability one (whereβ 2 is the BHH constant from Theorem 1). Fix an orientation of the tour and letT i denote the subtour ofT that begins at pointi and traversesT until it has visited⌈pn⌉ points. Certainly, n X i=1 length(T i )=(⌈pn⌉− 1)length(T) =⇒ min i length(T i )≤ 1 n n X i=1 length(T i )= (⌈pn⌉− 1) n length(T) =⇒ limsup n→∞ min i length(T i ) √ n ≤ lim n→∞ (⌈pn⌉− 1) n · length(T) √ n =β 2 p<µ 2 p as desired, where we apply the bound ofβ 2 <0.9204 from Section 8.5 of [20]. For approaching an arbitrarily distributed set of points we need the following lemmas. The next lemma as a consequence of the previous theorem. Corollary 14 (Tour length in a subset with uniform demand). Let X 1 ,...,X n be independent uniform samples drawn from a compact regionR with area1 and letS ⊂R , withArea(S)=q. Then liminf n→∞ L({X 1 ,...,X n }∩S;pn) p √ n ≥            λ 2 ifp≤ q ∞ otherwise. 31 Proof of Corollary 14. This is simple: if p > q then the LLN says that|{X 1 ,...,X n }∩S|/n → q with probability one, soL({X 1 ,...,X n }∩S;pn) does not exist (we would have to visit more points than are contained inS). On the other hand, ifp≤ q, then we merely observe that L({X 1 ,...,X n }∩S;pn)≥ L(X 1 ,...,X n ;pn) and apply Theorem 13. The non-uniform equivalent of Theorem 13 for a step functions follows: Lemma 15 (Tour length from a step density). Let ϕ (x) = P s i=1 a i 1(x ∈ i ) be a step density function with compact supportR such thata 1 ≥···≥ a s anda i Area( i )=1/s for alli (so thatArea(R)=1). If Y ={Y i } is a sequence of independent samples fromϕ ,0p). Proof of Theorem 16. Letϕ be the approximation off from Lemma 4 andX = (X 1 ,...,X k ) be a collec- tion ofk independent samples off. By a coupling argument, similar to proof of Theorem 12, define Y =(Y 1 ,...,Y k ) as a collection ofk independent samples ofϕ , thenPr(X ̸=Y )<ϵ . We have L(X 1 ,...,X n ;pn)≥ L(X 1 ,...,X n :X i =Y i ;pn)=L(Y 1 ,...,Y n :X i =Y i ;pn) ≥ L(Y 1 ,...,Y n ;pn)− L(Y 1 ,...,Y n :X i ̸=Y i ;pn)−O (1) =⇒ liminf n→∞ L(X 1 ,...,X n ;pn) √ n ≥ liminf n→∞ L(Y 1 ,...,Y n ;pn) √ n − α 2 p Area(R)ϵpn √ n (2.18) =liminf n→∞ L(Y 1 ,...,Y n ;pn) √ n − α 2 p pϵ Area(R) ≥ λ 2 Z R p ϕ (x)1(P(X)≥ p)dx− α 2 p pϵ Area(R) 35 where we have applied Lemma 10 in the last inequality. Also to get Eq. (2.10) we used the almost sure bound in [72]. Finally, note that we can select our approximationϕ, Π arbitrarily closely so that, by Lemma 4 (with a compact notation), ϵ ≥ max x |1(P ≥ p)− 1(Π ≥ p)|+ Z R |ϕ − f| dx ≥ Z ϕ |1(P ≥ p)− 1(Π ≥ p)|+ Z |ϕ − f| 1(P ≥ p) ≥ Z |ϕ 1(Π ≥ p)− ϕ 1(P ≥ p)+ϕ 1(P ≥ p)− f 1(P ≥ p)| = Z |ϕ 1(Π ≥ p)− f 1(P ≥ p)| Now by Claim 11 we have p Area(R)pϵ ≥ Z p |ϕ 1(Π ≥ p)− f 1(P ≥ p)|≥ Z p ϕ 1(Π ≥ p)− Z p f 1(P ≥ p) =⇒ Z p ϕ 1(Π ≥ p)≥ Z p f 1(P ≥ p)− p pϵ Area(R) and therefore liminf n→∞ L(X 1 ,...,X n ;pn) √ n ≥ λ 2 Z R p ϕ (x)1(Π( x)≥ p)dx− α 2 p pϵ Area(R) ≥ λ 2 Z p f(x)1(P(x)≥ p)dx− p pϵ Area(R) − α 2 p pϵ Area(R) ≥ λ 2 Z p f(x)1(P ≥ p)dx− p pϵ Area(R)(λ 2 − α 2 ) | {z } (∗ ) (2.19) which completes the lower bound, since Eq. (2.19) shrinks to 0 by choosing small values ofϵ . 36 The upper bounding argument is very simple: let pointX i ’s be the points wheref is the densest and at least p, i.e. X i = argmax x f(x)1(P(x) > p). This is intuitive because it is desirable to select those points that belong to more densely populated areas. The density function on theX i ’s is ˜ f(x)=f(x)/p1(P(x)>p) and by Theorem 1, the length of a tour of thoseX i ’s satisfies lim n→∞ L(X 1 ,...,X n ;pn) √ np =β 2 Z R q ˜ f(x)dx=β 2 Z R p f(x)/p1(P(x)>p)dx ≤ µ 2 Z R p f(x)/p1(P(x)>p)dx which completes the proof. 2.3 The"m-of-asubset"TSP In this section, we letGTSP m (n,k,m) := L m (X 1 ,X 2 ,...,X n ) denote the length of the shortest tour that visitsm element of each point setX i , with|X i |≥ 2, ∀i. Firstly, we can assert that this tour length for uniformly distributed points is O(m p n/k). The lower bounding argument is similar to Theorem 8, while for the upper bound we employ a zigzag technique [12]. In this technique we take a zigzag tour through the region and visitm closest point of each point set to the main tour. Then optimizing over the number of zigzags yields the asymptotically tight bound. 37 Theorem 17 (Uniform demand m). LetX 1 ,X 2 ,...,X n all be point sets drawn independently and uni- formly at random in a region of unit area inR 2 , where|X i |=k≥ 2, ∀i. Then almost surely λ 1 <liminf n→∞ L m (X 1 ,...,X n ) m p n/k ≤ limsup n→∞ L m (X 1 ,...,X n ) m p n/k <µ 2 (2.20) Proof of Theorem 17. For the lower bound, let E n,m stand for the event that L m (X 1 ,X 2 ,...,X n ) < cm p n/k. Similar to Theorem 8 apply union bound to Lemma 3 with d = 2 and note that there are k m n different possible ways to select m member from each setX i to get Pr(E n,m )≤ k m n · Γ( nm+1) Γ(2 nm+1) 2πc 2 nm 2 k nm =⇒ logPr(E n )≤ (log k m +mlog πc 2 m 2k +m)n+O(1) (2.21) where we used Lemma 5. We can see that Eq. (2.21)→ −∞ again if and only if the coefficient of n is negative. For all k ≥ m ≥ 1 the same constant λ 1 for c similar to Theorem 8 works. Thus we know Pr(E n,m )≤ a − n for somea > 1 and anyk ≥ m≥ 1, as such P ∞ n=1 Pr(E n,m ) <∞ and we can apply Borel-Cantelli lemma to get thatE n,m does not occur infinitely often and thus λ 1 ≤ lim inf n→∞ L m (X 1 ,X 2 ,...,X n ) m p n/k Now for the upper bound let’s assume R is a unit square and use a zigzag tour, P, as in [12] by traversing the width ofR horizontally a total of q times, starting at the upper leftmost corner ofR and moving downward by an amount 1/(q− 1). Now perturbP to get pathP ′ by visiting the m closest points of each point set to the main pathP. Now note that if d i,ℓ is the distance ofℓth closest point from point seti to the main path, using order statistics for uniform distribution we haveE[ℓ i,j ] = ℓ 2(q− 1)(k+1) . 38 The objective now is to minimize the tour length ofP ′ which isL(P ′ ) = q+2+2 P n i=1 P m ℓ=1 d i,ℓ . By exchanging the finite summations order and strong LLN we know that L(P ′ )=q+2+2 m X ℓ=1 n X i=1 d i,ℓ →q+2+2 m X ℓ=1 nℓ 2(q− 1)(k+1) ≤ q+2+2 nm 2(q− 1)(k+1) almost surely. Now minimizing overq the optimal tour length is3+ q 2nm(m+1) k+1 , thus lim n→∞ L m (X 1 ,...,X n ) √ n ≤ r 2m(m+1) k+1 ≈ m p 2/k Now for the general unit area region, we take a zigzag approach as in Remake 6 of [13]. Simply speaking, we construct a zigzag in the region that includes q line segments build based on the Riemann rectangles and their distance is 1 q− 1 . Fig. 2.4 shows an example of this construction from [13]. The distance between any point to this path is uniform[0, 1 2(q− 1) ]. The line segments are connected (to build the zigzag) such that the tour length isq+2P whereP is the perimeter ofR (constant). Now we perturb this path to get a similar pathP ′ , as in the unit square case, which has length L(P ′ )=q+2P +2 m X ℓ=1 n X i=1 d i,ℓ →q+2P +2 m X ℓ=1 nℓ 2(q− 1)(k+1) ≤ q+2P +2 nm 2(q− 1)(k+1) which by minimizing overq gives the desired result. 39 Figure 2.3: A collection of Riemann rectangles Figure 2.4: Connecting a family of Riemann rectangles to form a loop We try to bound theGTSP m tour for a non-uniform distribution. First, the following lemma bound the tour length in the subsets of the region. Lemma18 (Tour length in a subset with uniform demand,m case). LetX 1 ,...,X n be independent uni- form samples of cardinalityk drawn from a compact regionR with volume1 and letS ⊂ R be with volume t. Then r 2ta πe exp(− b/a)≤ liminf n→∞ L m (X i :|X i ∩S|>0) √ n ≤ limsup n→∞ L m (X i :|X i ∩S|>0) √ n ≤ r k m Z t 0 q F k− m,m (x)dx withprobabilityone,whereX i ∩S isthesetofpointsthatlieintoS frompointsetX i ,F k− m,m (x)isaBeta distribution with parametersα =k− m,β =m, and a= k X i=q p i (i− q), b= k X i=q p i log i i− q 40 withp i =P(|X j ∩S|=i) for anyj andq =k− m. The proof starts off by noting the following. We know given a uniform distribution, the number of points form each point set lying in a region of area t ≤ 1 is binomial β (k,t). Then by Lemma 15 and considering the valid points selection, we calculate the total number of possible selection and orderings. The rest is analogous to Theorem 8. For the upper bound, sample the rightmost of each point set in the region and use Theorem 1. Proof of Lemma 18. We know given a uniform distribution, the number of points form each point set j lying in a region of area t is binomial β (k,t). If X j points of point set j lie inS, then you have to visit (X j − q) + of them, whereq =k− m. This is because we assume the points outside ofS are free to visit as used in the proof of Lemma 10. Letp i = Pr(X = i) for any point set andn i be the number of point sets which havei points in theS. The number of valid selection of points to visit is n Y j=1 X j X j − q = k Y i=1 i i− q n i and total number of points in the tour that connects all of the selected points is n X j=1 (X j − q)= k X i=q n i (i− q) As such, the total number of possible selection and orderings of all the points in the tour is k Y i=1 i i− q n i   k X i=q n i (i− q)   ! 41 If you have selectedN = P k i=q n i (i− q) points inS, then the probability that their length does not exceedℓ is the same as asking if we hadN points in a region with area 1 and we did not want the length to exceedℓ/ √ t, which by Lemma 7 is (2π (ℓ/ √ t) 2 ) N (2N)! (2.22) LetE n,m be the event that the length of the tour visiting the selected points, call itL m (X 1 ,··· ,X n ), is smaller thanℓ/ √ t, then by union bound and Eq. (2.22) we get P(E n,m )≤ k Y i=1 i i− q n i   k X i=q n i (i− q)   !× (2π (ℓ/ √ t) 2 ) N (2N)! Then taking alog we get logP(E n,m )≤ k X i=1 n i log i i− q +log   k X i=q n i (i− q)   !+log (2π (ℓ/ √ t) 2 ) N (2N)! Whenn→∞ by LLN we replacen i withn× p i . Leta = P k i=q p i (i− q) andb = P k i=q p i log i i− q and use the Stirling’s formula to get logP(E n,m )≤ [nb]+ nalogna− na+ 1 2 logna+o(1+ 1 na ) + nalog(2π )+2nalogℓ− nalogt− 2nalog(2na)+2na− 1 2 log(2na)− o(1+ 1 2na ) Letℓ=c √ n, then we get logP(E n,m )≤ n(2alogc− alog(t)+b− aloga+a− alog2+alog(π ))+ ¯ f(n) 42 where ¯ f(n) is a function that does not depend onn. Now if the coefficient of n is negativelogP(E n,m )→0 asn→∞ and by Borel-Cantelli lemma we get a lower bound, i.e. if we set 2alogc− alog(t)+b− aloga+a− alog2+alog(π )<0 ⇒c< r 2ta πe exp(− b a ) then if we setc= q 2ta πe exp(− b a )− ϵ for someϵ ∈(0,1) we have liminf n→∞ L m (X 1 ,...,X n ) √ n ≥ c For the upper bound, the proof is same as in Theorem 8 except that the distribution is different. In the case of a unit square region, we sample the rightmost of each point set inS. The density is f(x)= k m F k− m,m (x) whereF k− m,m follows a beta distribution with parametersα = k− m,β = m. Based on Theorem 1 the TSP tour length is upper bounded with r k m Z t 0 q F k− m,m (x)dx which completes the proof. 43 2.4 SparseSubsetGTSP In this section, we letSSGTSP(n,k,p):=L(X 1 ,...,X n ;1,p) denote the length of the shortest tour that visits one element from⌈pn⌉ of then point sets, i.e.⌈p× n⌉ points are visited in total, each from a different point set. The case where the points and point sets are distributed uniformly at random has the following property. Theorem19 (SSGTSP, Uniform demand). Letk≥ 2 andp∈ (0,1) be fixed and let X 1 ,...,X n be point setsofcardinalityk thatarealldrawnindependentlyanduniformlyatrandominaregionofunitareainR 2 . Then for all fixed k,p, we have 0.2935=λ 2 <liminf n→∞ L(X 1 ,...,X n ;1,p) p √ nk/(k+1) ≤ limsup n→∞ L(X 1 ,...,X n ;1,p) p √ nk/(k+1) <µ 1 =1.8408 (2.23) with probability one. The analysis of Theorem 13 easily generalize to calculate the results for Theorem 19. Proof of Theorem 19. To derive the lower bound, letE n be the event thatL(X 1 ,...,X n ;1,p)<cp √ nk/(k+ 1) for fixed c andk. Applying the union bound to Lemma 3 for the cased=2 and using the fact that there are n ⌈np⌉ k ⌈np⌉ different possible ways to select the points, we see that Pr(E n )≤ n ⌈np⌉ (⌈np⌉)!k ⌈np⌉ (2πc 2 p 2 n/k) ⌈np⌉ (2⌈np⌉)! logPr(E n )≤ plog(1− p)+2plogc− plog2+plogπ − log(1− p)+p n − 1 2 logp− 1 2 logn− 1 2 log(1− p)− log2− 1 2 logπ +O(1/n) (2.24) 44 by Lemma 5 and Eq. (2.13), same as in proof of Theorem 13. Thus we get 0.2935=λ 2 <liminf n→∞ L(X 1 ,...,X n ;1,n)/(p √ nk/(k+1)) with probability one as desired. For the upper bound, take a TSP tourT of arbitrary points of each point set, i.e. GTSP tour, then length ofT satisfies limsup n→∞ length(T) √ nk/(k+1) <µ 1 probability one by Theorem 8. Fix an orientation of the tour and letT i denote the subtour ofT that begins at pointi and traversesT until it has visited⌈pn⌉ points. Certainly, n X i=1 length(T i )=(⌈pn⌉− 1)length(T) =⇒ min i length(T i )≤ 1 n n X i=1 length(T i )= (⌈pn⌉− 1) n length(T) =⇒ limsup n→∞ min i length(T i ) √ nk/(k+1) ≤ lim n→∞ (⌈pn⌉− 1) n · length(T) √ nk/(k+1) <µ 1 p as desired. 2.5 RandomKGTSP In this section, we introduce the Random K GTSP (RKGTSP). Assume that we have a discrete probability distribution on random variableK and sizes of the point sets,{k 1 ,k 2 ,··· ,k n }, are i.i.d samples from this 45 distribution. The goal is to find the shortest tour that within point set i visits⌊pk i ⌉ points. We letβ be a Rademacher random variable (1 with probability 0.5 and -1 otherwise) then define ⌊pk i ⌉ to be ⌊x⌉=                                ⌊x⌋ x−⌊ x⌋<0.5 ⌈x⌉ x−⌊ x⌋>0.5 ⌊x⌋ x−⌊ x⌋=0.5&β =− 1 ⌈x⌉ x−⌊ x⌋=0.5&β =1 (2.25) LetRGTSP(n,K,p) := L(X 1 ,...,X n ;K,p) denote the length of the shortest such tour. We prove the next theorem using the fact that RGTSP(n,k,p) ≥ SSTSP(∪ i X i ,¯p) where ¯p = P i ⌊pk i ⌉ P i k i . Then Theorem 13 yields the lower bound. The upper bounding argument is simpler and uses a zigzag technique along with the Law of Large Numbers. Theorem 20 (RGTSP, Uniform demand). Let K have a discrete distribution, ν , withEK = µ . Also, let p∈(0,1)befixedandlet X 1 ,...,X n bepointsetsofcardinalityk 1 ,k 2 ,··· ,k n whicharei.i.dsampleofν . The points in each point set are all drawn independently and uniformly at random in a region of unit area in R 2 . Then for all fixed p, we have 0.2935=:λ 2 <liminf n→∞ L(X 1 ,...,X n ;K,p) p √ nEK ≤ limsup n→∞ L(X 1 ,...,X n ;K,p) p √ nEK <2 (2.26) with probability one. Proof of Theorem 20. Lower bound: Let ¯p= P i ⌊pk i ⌉ P i k i . A lower bound is taking a SSTSP tour that visits ¯p of all the∪ i X i points. Therefore RGTSP(n,K,p)≥ SSTSP(∪ i X i ;¯p) (2.27) 46 as it visits the same number of points but it is not constrained to visit exactly⌊pk i ⌉ points fromX i . By LLN we can replace|∪ i X i | withnµ asn→∞ and since P i ⌊pk i ⌉− pk i P i k i = (2.28) P i:⌊pk i ⌉>pk i ⌊pk i ⌉− pk i + P i:⌊pk i ⌉≤ pk i ⌊pk i ⌉− pk i P i k i ≤ (2.29) 0.5|{i:⌊pk i ⌉>pk i }|− 0.5|{i:⌊pk i ⌉≤ pk i }| P i k i n→∞,LLN −→ (2.30) 0.5n/2− 0.5n/2 nµ =0 (2.31) then ¯p = P i ⌊pk i ⌉ P i k i = p+ P i ⌊pk i ⌉− pk i P i k i n→∞ −→ 0+p. Also, by continuity of L(X 1 ,...,Xn;K,p) ¯p √ |∪ i X i | and L(X 1 ,...,Xn;¯p) ¯p √ |∪ i X i | in ¯p and|∪ i X i | , Eq. (2.27), and Theorem 13 liminf n→∞ L(X 1 ,...,X n ;K,p) p √ nµ =liminf n→∞ L(X 1 ,...,X n ;K,¯p) ¯p p |∪ i X i | ≥ liminf n→∞ L(X 1 ,...,X n ;¯p) ¯p p |∪ i X i | ≥ λ 2 Upper bound: We take a zigzag tour argument as in Theorem 17. Then if the perimeter of the area is P and we haveq Riemann rectangles we get the following tour length for the modified zigzag tour P ′ length(P ′ )=q+2P + 1 q− 1 n X i=1 ⌊pk i ⌉ X ℓ=1 ℓ k i +1 which similar to Theorem 4 in [12] for largen if we take derivative w.r.tq and try to find the stationary point of it, we get dlength(P ′ ) dq =1− 1 (q− 1) 2 n X i=1 ⌊pk i ⌉ X ℓ=1 ℓ k i +1 =0 n→∞ −→ q ∗ ≃ v u u t n X i=1 ⌊pk i ⌉ X ℓ=1 ℓ k i +1 47 and the optimal tour length ofP ′ is length(P ′ )≃2 v u u t n X i=1 ⌊pk i ⌉ X ℓ=1 ℓ k i +1 <2 v u u t n X i=1 ⌊pk i ⌉ X ℓ=1 ℓ k i ≤ 2 v u u t n X i=1 ⌊pk i ⌉ 2 k i ≤ 2p v u u t n X i=1 k i + n X i=1 1/k i n→∞ −→ 2p √ nµ which taking thelimsup n→∞ dividing by √ n gives the result. 48 Chapter3 SyntheticExperimentalResults In this section we provide examples illustrating the bounds for simulated instances. We use Google’s OR- tools for the routing problems to solve the randomly generated instances of different sizes. We study tour lengths arising from uniform and truncated normal distributions with variance σ 2 = 0.01 centered at (0.5,0.5) in a unit square. 3.1 One-of-a-subsetTSP For the first instance we look into uniform samples of points in a unit square. Fig. 3.1(left) shows the exper- imental results for 10 runs for each pair of(n,k) wheren∈{10,15,20,...,100} andk∈{2,3,...,6}. Our bounds contain the tour length, which is increasing inn and decreasing ink. In the figures we show the lower bounds (LB), the average tour length of 10 samples (L), and upper bounds (UB). Note that our bounds are even valid when the number of points are small. This can motivate an study for high proba- bility non-asymptotic bounds as well. Appendix B includes similar experiments for larger instances up to n=1000. Now for the non-uniform distribution, we take truncated bi-variate Gaussian distribution in the unit square. Fig. 3.1(Right) shows the tour length forn∈{110,130,...,190} andk∈{3,4,5,6} instances of 49 Figure 3.1: (Left) is the tour length forone-of-a-subset withuniform distribution instances forn=10 to 100 clusters, (Right) shows the tour length forone-of-a-subset withtruncatednormal distribution instances,n=110 to 190 clusters each withk∈{2,3,...,6}. this distribution. As we can see, depending on the distribution, the upper bound can be weaker than the uniform case and could be improved in the future work. 3.2 SparseSubsetTSP In this case, we run the experimental results forn∈{150,200,...,950,1140} andp∈{0.1,0.25,0.5,0.75,0.9} and Fig. 3.2(Left) shows the results confirming the bounds. For the non-uniform case of SSTSP we can see a set of examples in the Fig. 3.2(Right). The bounds for this case are quite tight. 3.3 m-of-asubsetTSP For the m-of-a-subset we have Fig. 3.3 for different m and k ∈ {4,5}. Fig. 3.4 shows the bounds for a non-uniform case of m-of-a-subset problem. Here we used our conjectured upper bound. As we can see, 50 Figure 3.2: (Left)SSTSP tour length foruniform distribution,n∈{150,200,...,950,1140} and p∈{0.1,0.25,0.5,0.75,0.9}, (Right) tour length forSSTSP instances,truncatednormal distribution, n=10 to 460 points andp∈{0.25,0.5,0.75,0.9} all the experimental results show that the upper and lower bounds are valid for even smalln. 3.4 SparseSubsetGTSP We test the bounds in Section 2.4 with finite n using synthetic problems. Fig. 3.5 confirms our findings as expected. 3.5 RandomKGTSP The experimental results confirm our bounds of Section 2.5 under even finite n as shown in Fig. 3.6. 51 Figure 3.3: Tour length form-of-a-subset instances foruniform distribution,n=50 to 100 clusters, m∈{2,3,4}. (Left) figure has k =5 and (Right) figure has k =4 Figure 3.4: Tour length form-of-a-subset instances of atruncatednormal distribution,n=50 to 190 clusters withm∈{1,...,k}, (Left) figure has k =5 while (Right) figure has k =6 52 Figure 3.5: Tour length for Sparse Subset GTSP instances of a uniform distribution,n=10 to 70 clusters withp∈{0.2,0.5,0.8} Figure 3.6: Tour length for Random K GTSP instances of a uniform distribution,n=10 to 70 clusters withp∈{0.2,0.5,0.8} 53 Chapter4 CaseStudy: RandomStowWarehouse In this section, we study the application of our formulas to random stow (RS) models. Random stow, also called chaotic stow, is used by Amazon in its warehouses. In this scheme, items are not arranged by size, shipment date, or any other system. They are shelved randomly, wherever workers can find a place to stash them. Our goal is to show that this model is better than organized classic (CL) model of the warehousing. As a related work, [5] used the CGTSP (clustered GTSP) model for AS/RS robots warehousing systems. They only consider picking up the order items from the warehouse and multiple orders are picked up together by the robot to fill a pallet. The item locations are clustered by orders. These clusters include sub-clusters where each one is a SKU type in the corresponding order. 4.1 TheWarehouseModel We will assume that we haven different SKU types and they arrive into the warehouse (from the retailers) in loads of sizem at rateλ per unit time; the material is entering the warehouse at rateλm . We assume the warehouse has N total slots and N ≫ nm. We take picker and packers’ capacity to both be equal toc. The warehouse is assumed to be a unit square. As the inventory loads come in, packers policy is as follows: 54 • Classic packing: grab as many SKUs of the same type as possible, and pile them in a part of the grid that is designated for that SKU type. • Random Stow packing: grab as many SKUs of different types as possible, and place them randomly in a consecutive block of open locations; that is, distribute the random collection of SKUs in consecutive grid cells. For packing the items into the shelves, in the CL model we assume the packing tour length is zero as it is constant w.r.tn andk. In the RS model, we suppose there areN slots in the unit square, and the gap be- tween each one is1/ √ N. Assume that the average utilization of the warehouse isρ ∈[0,1], so in average there areN(1− ρ ) empty slots. A packing tour seeks the shortest tour that fills c of these empty slots. Based on SSTSP results in Section 2.2, the length of a packing tour is of order c N(1− ρ ) p N(1− ρ ) = c √ N(1− ρ ) in average. Therefore, the rate at which items are packed into the warehouse shelves is( c √ N(1− ρ )c ) − 1 = p N(1− ρ ). Now if there arer 1 packers, then in order for the warehouse’s usage not to explode, we need to have r 1 p N(1− ρ )≥ λm ⇒ r 1 ≥ λm p N(1− ρ ) . (4.1) As the input rate is equal to the output, the total output rate of the pickers should be λm as well. Therefore, w.l.o.g we assume the orders are sizem and arrive with rateλ . If we assume there arek items of each SKU in the warehouse then by Theorem 8 the picking tour length is of order √ ck/(k+1) for an order of sizec. Hence, for each item it is √ k √ c(k+1) . If there arep 1 pickers then the rate of material leaving the warehouse isp 1 √ c(k+1) √ k , then we need this to be at leastλm so p 1 √ c(k+1) √ k ≥ p 1 √ ck≥ λm ⇒ p 1 ≥ λm √ ck (4.2) 55 This is while for the CL order picking tour it takes √ c for the same order, again by Theorem 8. Thus if there arep 2 pickers we similarly get p 2 √ c≥ λm ⇒ p 2 ≥ λm √ c . (4.3) Based on the above relationships, we analyze the regime under which random picking is better than non- random (CL), so by comparing Eq. (4.1)+Eq. (4.2) vs Eq. (4.3) we show that the number of pickers plus packers for RS is smaller than the CL model as follows: λm √ ck + λm p N(1− ρ ) ≤ λm √ c ⇒ ρ ≤ 1− ck N( √ k− 1) 2 which is almost always true since we assumeN ≫ c and therefore ck N( √ k− 1) 2 ≃ 0. Therefore, as all the other conditions were the same for both models, we can conclude that in average the RS model is more efficient than the CL model. 4.2 DiscreteEventSimulation The empirical results here aim on validating the findings in Section 4.1. We build a discrete event simulation model to compare RS model with the Classic model. We simulate a warehouse order picking and packing operation for a given number of orders, N o . For each order we generate a multinomial of size O z with uniform probability of1/n for each SKU type. We also generate random times with exponential rateλ for each order to arrive after the previous one. The event diagrams of our simulation model are reported in Appendix B.1. Fig. 4.1 show an instance of the item locations for RS and CL model picking, respectively. As we can see after many orders are picked up the tour length increase for the RS model but for the CL model remains the same. This is because in the RS model the picking procedure is so efficient that the items become scarce 56 as they are picked up faster. However, in Section 4.3 we show that in average RS has shorter tour length, complementing our analysis in Section 4.1. Figure 4.1: The item locations (Top two for RS, Bottom two for CL) at the first and last orders, ord# is the order number,n is the number of items,tl is the tour length. 57 4.3 ExperimentalResults We present the results of simulations based on Section 4.2 here. The simulation runs for different policy (Random Stow, rs, and CL, cl) for the same sequence of random orders. Then we compare the average tour length of picking tours and packing tours aggregated over the orders. Fig. 4.2 (Fig. 4.3) shows the RS picking (packing) tour length vs CL tour length for two different number of item types in the warehouse. For each order size (n) we experiment with different "starting number of items for each SKU" ( k) and average the results together. We can observe that the RS picking tour length is shorter than CL policy for different n, while the packing tour length of RS is longer. Also, the packing tour length is almost for different order size. However, as depicted in Fig. 4.4 the total tour length of picking plus packing is shorter for the RS policy compared to the CL policy. Figure 4.2: Random stow (rs) vs CL (cl) picking average tour length (tls) Figure 4.3: Random stow (rs) vs classic (cl) average packing tour length (tls) Based on the BHH Theorem the picking tour length for the CL policy must be of order β √ n where β ≃ 0.73. We depict the ratio of the tour length over √ n for averagen in the picking tours in Fig. 4.5 in 58 Figure 4.4: Random stow (rs) vs classic (cl) picking + average packing tour length (tls) dashed lines. On the other hand, based on Theorem 8, the tour length of a RS picking tour over √ nk/(k+1) must be betweenλ 1 = 0.4839 andµ 1 = 1.8408. This ratio is confirmed in the experiments as shown in solid lines in Fig. 4.5. We can see that for 100 item types, the ratio is contained better compared to 64 items by the bounds as the bound is asymptotic. Figure 4.5: Random stow (rs) picking tour length/ √ nk/(k+1) and classic (cl) picking tour length/ √ n Inventory Level Stability: Fig. 4.10 shows the warehouse inventory level for RS and CL model under the same conditions (same order arrival and inventory arrival rate, same number of picker and packer). For RS, we can predict the stability of the inventory level by checking Eq. (4.1) and Eq. (4.2) for RS, and Eq. (4.3) for CL. We examine these conditions in the caption of each plot using average values of the parameters, and show our analysis are consistent with the simulations. As we can see since CL model is not as efficient as RS in picking the orders, the inventory level keeps exploding in the CL model while it is steady in the RS model under the same conditions. There are also 59 examples where both of RS and CL are stable or instable. However, there are no examples where CL is stable and RS is not. These results also compliment our analysis in Section 4.1 showing CL requires more pickers and pack- ers. Further experimental results in Appendix B.0.1 also confirm this. 0 10 20 30 40 50 Orders 1000 1200 1400 1600 Inventory level # picker 10, # packer 10, order size 20 RS CL Figure 4.6: Here λ = 5,c = 20,m = 20,N = 100 2 ,¯ρ = 0.3,p 1 = p 2 = 10,r 1 = 10, ¯k = 10. RS is stable since Eq. (4.1) holds sincer 1 ≥ λm √ N(1− ρ ) ≃1.2. Also, Eq. (4.2) holds sincep 1 ≥ λm √ ck ≃ 7. For CL, Eq. (4.3) does not hold, since p 2 < λm √ c ≃22. 0 20 40 60 80 100 Orders 2800 3000 3200 Inventory level # picker 10, # packer 8, order size 10 RS CL Figure 4.7: Here λ = 5,c = 20,m = 10,N = 100 2 ,¯ρ = 0.38,p 1 = p 2 = 10,r 1 = 8, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.63. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃ 2. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃11. 0 20 40 60 80 100 Orders 3000 3200 3400 Inventory level # picker 8, # packer 10, order size 10 RS CL Figure 4.8: Here λ = 5,c = 10,m = 10,N = 100 2 ,¯ρ = 0.4,p 1 = p 2 = 8,r 1 = 10, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.64. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃ 2.8. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃15.8. 0 20 40 60 80 100 Orders 3000 3200 3400 Inventory level # picker 8, # packer 12, order size 10 RS CL Figure 4.9: Here λ = 5,c = 10,m = 10,N = 100 2 ,¯ρ = 0.4,p 1 = p 2 = 8,r 1 = 12, ¯k = 30. RS is stable since Eq. (4.1) holds since r 1 ≥ λm √ N(1− ρ ) ≃ 0.64. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃ 2.8. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃15.8. Figure 4.10: Warehouse inventory level where CL fails and RS is stable. The fill shows the standard devia- tion of 10 simulations for 100 orders. 60 Fig. 4.15 shows the cases where both CL and RS could fail. The main differences are that the order arrival rate, λ = 20, is larger and the warehouse size, N = 100, is small, whilep 1 ,p 2 , andr 1 are in the same order as Fig. 4.10. 0 20 40 60 80 100 Orders 1000 1200 1400 1600 Inventory level # picker 8, # packer 10, order size 10 RS CL Figure 4.11: Hereλ =20,c=10,m=10,N = 10 2 ,¯ρ = 0.26,p 1 = p 2 = 8,r 1 = 10, ¯k = 12. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃ 23.2. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃18. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. 0 20 40 60 80 100 Orders 1000 1200 1400 1600 Inventory level # picker 8, # packer 12, order size 10 RS CL Figure 4.12: Hereλ =20,c=10,m=10,N = 10 2 ,¯ρ = 0.31,p 1 = p 2 = 8,r 1 = 12, ¯k = 14. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃ 24. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃16. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. 0 20 40 60 80 100 Orders 1000 1200 1400 1600 Inventory level # picker 8, # packer 15, order size 10 RS CL Figure 4.13: Hereλ =20,c=10,m=10,N = 10 2 ,¯ρ = 0.3,p 1 = p 2 = 8,r 1 = 15, ¯k = 13. RS is not stable since Eq. (4.1) is violated as r 1 < λm √ N(1− ρ ) ≃ 23. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃17. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. 0 20 40 60 80 100 Orders 1000 1200 1400 Inventory level # picker 10, # packer 12, order size 10 RS CL Figure 4.14: Hereλ =20,c=10,m=10,N = 10 2 ,¯ρ = 0.3,p 1 = p 2 = 10,r 1 = 12, ¯k = 13. RS is not stable since Eq. (4.1) is violated asr 1 < λm √ N(1− ρ ) ≃ 23. Also, Eq. (4.2) does not hold sincep 1 < λm √ ck ≃17. For CL, Eq. (4.3) does not hold, sincep 2 < λm √ c ≃63. Figure 4.15: Warehouse inventory level where CL and RS both fail. The fill shows the standard deviation of 10 simulations for 100 orders. 61 Fig. 4.20 shows the cases where both CL and RS could keep an stable inventory level. The main differ- ences are that the order arrival rate,λ = 1, is lower and the warehouse size,N = 100 2 , is large enough, whilep 1 ,p 2 , andr 1 are in the same order as Figs. 4.10 and 4.15. 0 10 20 30 40 50 60 70 80 Orders 2900 2950 3000 3050 3100 Inventory level # picker 8, # packer 15, order size 10 RS CL Figure 4.16: Hereλ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.47,p 1 = p 2 = 8,r 1 = 15, ¯k = 33. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃ 0.13. Also, Eq. (4.2) holds since p 1 ≥ λm √ ck ≃ 0.55. For CL, Eq. (4.3) holds too, sincep 2 ≥ λm √ c ≃3.16. 0 10 20 30 40 50 60 70 80 Orders 800 850 900 950 1000 Inventory level # picker 10, # packer 8, order size 10 RS CL Figure 4.17: Hereλ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.14,p 1 = p 2 = 10,r 1 = 8, ¯k = 11. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃0.1. Also, Eq. (4.2) holds sincep 1 ≥ λm √ ck ≃ 0.95. For CL, Eq. (4.3) holds too, since p 2 ≥ λm √ c ≃3.16. 0 10 20 30 40 50 60 70 80 Orders 2900 2950 3000 3050 Inventory level # picker 15, # packer 12, order size 10 RS CL Figure 4.18: Hereλ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.14,p 1 = p 2 = 15,r 1 = 12, ¯k = 37. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃0.1. Also, Eq. (4.2) holds sincep 1 ≥ λm √ ck ≃ 0.51. For CL, Eq. (4.3) holds too, since p 2 ≥ λm √ c ≃3.16. 0 10 20 30 40 50 60 70 80 Orders 2900 2950 3000 3050 Inventory level # picker 15, # packer 15, order size 10 RS CL Figure 4.19: Hereλ = 1,c = 10,m = 10,N = 100 2 ,¯ρ = 0.16,p 1 = p 2 = 15,r 1 = 15, ¯k = 31. RS is stable since Eq. (4.1) holds as r 1 ≥ λm √ N(1− ρ ) ≃0.1. Also, Eq. (4.2) holds sincep 1 ≥ λm √ ck ≃ 0.56. For CL, Eq. (4.3) holds too, since p 2 ≥ λm √ c ≃3.16. Figure 4.20: Warehouse inventory level where CL and RS both are stable. The fill shows the standard deviation of 10 simulations for 100 orders. 62 ConclusionofRandomStowExperiments: The analytical model in Section 4.1 and the experiments in Section 4.3 assert that as long as the size of warehouse is larger than the order sizes (N ≫ c) and order arrival rate is low enough, RS is better than CL (see Figs. 4.2 to 4.4 and 4.10. However, if order arrival is too high, both RS and CL fail to keep an steady inventory level and explode (see Fig. 4.15). We can observe that there are cases where both RS and CL are stable (see Fig. 4.20), but we could not find instances where RS is not stable while CL is. This actually confirms the validity of our analysis in Section 4.1, in particular, the fact that if Eq. (4.3) holds (our conditions for CL stability), then Eq. (4.2) also holds as k ≥ 1 (one of our conditions for RS stability). We intend to consider the correlation between the SKUs in the warehouse in the future work. That is to consider the fact that most of related items are usually packed close to each other like Gin and Tonic which are ordered usually together. 63 Chapter5 ConclusionandFutureWork We drove the asymptotic bounds for several selection routing problems and validated them using extensive experimental studies. Our experiments and analysis in Chapter 4 show how we can use our bounds to correctly predict the behavior of Random Stow and Classic warehousing policies. For arbitrary distribution in Section 2.3, our conjecture is that there is an upper and lower bound of O(m √ nk). As a future work we will be trying to prove our conjecture. Also, our experimental results confirm this upper bound. Take a look at Fig. 3.4. We intend to use the continuous approximations we developed here for random K GTSP to analyse the RS model further. For this we intend to use empirical distribution of number of items of each SKU as the distribution of K. Extending this work to other variants of TSP problems with higher dimensions (d > 2) is also an in- teresting direction of research. For instance, we can apply the SSTSP results to Prize collecting problem to drive required results. Another interesting and different enough paradigm of work is the capacitated ver- sion of TSP/VRP [30]. We believe that continuous approximation methods are applicable to these problems too. 64 Bibliography [1] Henrylab. AL. “Record balancing problem-a dynamic programming solution of a generalized traveling salesman problem”. In: Revue Francaise D Informatique De Recherche Operationnelle 3 (1969). [2] D. Aldous and M. Krikun. “Percolating paths through random points”. In: arXiv preprint math/0509492 (2005). [3] D. Applegate, W. Cook, D. S. Johnson, and N. J. A. Sloane. Using large-scale computation to estimate the Beardwood-Halton-Hammersley TSP constant. Presentation at 42 SBPO. 2010. [4] Egon Balas and Matteo Fischetti. “Polyhedral Theory for the Asymmetric Traveling Salesman Problem”. In: The Traveling Salesman Problem and Its Variations. Boston, MA: Springer US, 2007, pp. 117–168.isbn: 978-0-306-48213-7.doi: 10.1007/0-306-48213-4_3. [5] Pouya Baniasadi, Mehdi Foumani, Kate Smith-Miles, and Vladimir Ejov. “A transformation technique for the clustered generalized traveling salesman problem with applications to logistics”. In: European Journal of Operational Research 285.2 (2020), pp. 444–457.issn: 0377-2217.doi: https://doi.org/10.1016/j.ejor.2020.01.053. [6] J. Beardwood, J. H. Halton, and J. M. Hammersley. “The shortest path through many points”. In: Mathematical Proceedings of the Cambridge Philosophical Society 55.4 (1959), pp. 299–327. [7] Binay Bhattacharya, Ante Ćustić, Akbar Rafiey, Arash Rafiey, and Vladyslav Sokol. “Approximation Algorithms for Generalized MST and TSP in Grid Clusters”. In: Combinatorial Optimization and Applications. Ed. by Zaixin Lu, Donghyun Kim, Weili Wu, Wei Li, and Ding-Zhu Du. Cham: Springer International Publishing, 2015, pp. 110–125.isbn: 978-3-319-26626-8. [8] Nils Boysen, Stefan Fedtke, and Stefan Schwerdfeger. “Last-mile delivery concepts: a survey from an operational research perspective”. In: Or Spectrum 43.1 (2021), pp. 1–58. [9] Lawrence D. Burns, Randolph W. Hall, Dennis E. Blumenfeld, and Carlos F. Daganzo. “Distribution Strategies that Minimize Transportation and Inventory Costs”. In: Operations Research 33.3 (1985), pp. 469–490.doi: 10.1287/opre.33.3.469. eprint: https://doi.org/10.1287/opre.33.3.469. 65 [10] James F. Campbell and Morton E. O’Kelly. “Twenty-Five Years of Hub Location Research”. In: Transportation Science 46.2 (2012), pp. 153–169.doi: 10.1287/trsc.1120.0410. eprint: https://doi.org/10.1287/trsc.1120.0410. [11] John Gunnar Carlsson. “Dividing a Territory Among Several Vehicles”. In: INFORMS Journal on Computing 24.4 (2012), pp. 565–577.doi: 10.1287/ijoc.1110.0479. eprint: https://doi.org/10.1287/ijoc.1110.0479. [12] John Gunnar Carlsson, Mehdi Behroozi, Raghuveer Devulapalli, and Xiangfei Meng. “Household-level economies of scale in transportation”. In: Operations Research 64.6 (2016), pp. 1372–1387. [13] John Gunnar Carlsson and Siyuan Song. “Coordinated Logistics with a Truck and a Drone”. In: Management Science 64.9 (2018), pp. 4052–4069.doi: 10.1287/mnsc.2017.2824. eprint: https://doi.org/10.1287/mnsc.2017.2824. [14] Sudipta Chowdhury, Adindu Emelogu, Mohammad Marufuzzaman, Sarah G. Nurre, and Linkan Bian. “Drones for disaster response and relief operations: A continuous approximation model”. In: International Journal of Production Economics 188 (2017), pp. 167–184.issn: 0925-5273. doi: https://doi.org/10.1016/j.ijpe.2017.03.024. [15] Nicos Christofides. “worst-case analysis of a new heuristic for the traveling salesman problem”. In: Managenecv Sciences (1976). [16] G. Cornuejols and G. L. Nemhauser. “Tight bounds for christofides’ traveling salesman heuristic”. In: Mathematical Programming, North-Holland Publishing Company (1978). [17] Sohpie Curtis. Amazon at 15: The technology behind amazon-uk’s success. A Telegraph web resource. https://www.telegraph.co.uk/technology/amazon/10377834/Amazon-at-15-the-technology- behind-Amazon-UKs-success.html. Accessed: 2020-11-04. 2018. [18] Vladimir Dimitrijevic and Zoran Saric. “An efficient transformation of the generalized traveling salesman problem into the traveling salesman problem on digraphs”. In: Information Sciences 102.1 (1997), pp. 105–110.issn: 0020-0255.doi: https://doi.org/10.1016/S0020-0255(96)00084-9. [19] L. Few. “The shortest path and the shortest road throughn points”. In: Mathematika 2 (1955), pp. 141–144. [20] S.R. Finch. Mathematical Constants. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 2003.isbn: 9780521818056. [21] M. Fischetti, J. J. Salazar G., and P. Toth. “A branch-and-cut algorithm for the symmetric generalized traveling salesman problem”. In: Operations Research 45.3 (1997), pp. 378–394. [22] Matteo Fischetti, Juan Jose Salazar Gonzalez, and Paolo Toth. “The symmetric generalized traveling salesman polytope”. In: Networks 26.2 (1995), pp. 113–123.doi: https://doi.org/10.1002/net.3230260206. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/net.3230260206. 66 [23] Anna Franceschetti, Ola Jabali, and Gilbert Laporte. “Continuous approximation models in freight distribution management”. en. In: TOP 25.3 (Oct. 2017), pp. 413–433.issn: 1863-8279.doi: 10.1007/s11750-017-0456-1. (Visited on 11/15/2020). [24] Luca M. Gambardella and Marco Dorigo. “Ant-Q: A Reinforcement Learning approach to the traveling salesman problem”. In: Machine Learning Proceedings 1995. Ed. by Armand Prieditis and Stuart Russell. San Francisco (CA): Morgan Kaufmann, 1995, pp. 252–260.isbn: 978-1-55860-377-6. doi: https://doi.org/10.1016/B978-1-55860-377-6.50039-6. [25] Naveen Garg, Goran Konjevod, and R. Ravi. “A Polylogarithmic Approximation Algorithm for the Group Steiner Tree Problem”. In: Journal of Algorithms 37.1 (2000), pp. 66–84.issn: 0196-6774.doi: https://doi.org/10.1006/jagm.2000.1096. [26] Michel Gendreau, Francois Guertin, Jean-Yves Potvin, and Rene Seguin. “Neighborhood search heuristics for a dynamic vehicle dispatching problem with pick-ups and deliveries”. In: Transportation Research Part C: Emerging Technologies 14.3 (2006), pp. 157–174.issn: 0968-090X. doi: https://doi.org/10.1016/j.trc.2006.03.002. [27] Justin C. Goodson, Jeffrey W. Ohlmann, and Barrett W. Thomas. “Rollout Policies for Dynamic Solutions to the Multivehicle Routing Problem with Stochastic Demand and Duration Limits”. In: Operations Research 61.1 (2013), pp. 138–154.doi: 10.1287/opre.1120.1127. eprint: https://doi.org/10.1287/opre.1120.1127. [28] Saban Gulcu, Mostafa Mahi, Omer Baykan, and Halife Kodaz. “A parallel cooperative hybrid method based on ant colony optimization and 3-Opt algorithm for solving traveling salesman problem”. In: Soft Computing (Nov. 2016).doi: 10.1007/s00500-016-2432-3. [29] G. Gutin and D. Karapetyan. “A memetic algorithm for the generalized traveling salesman problem”. In: Natural Computing 9.1 (2009), pp. 47–60.issn: 1567-7818.doi: 10.1007/s11047-009-9111-6. eprint: 0804.0722. [30] M. Haimovich and A. H. G. Rinnooy Kan. “Bounds and Heuristics for Capacitated Routing Problems”. In: Mathematics of Operations Research 10.4 (1985), pp. 527–542.issn: 0364765X, 15265471.url: http://www.jstor.org/stable/3689422. [31] M. Haimovich and Thomas L. Magnanti. “Extremum Properties of Hexagonal Partitioning and the Uniform Distribution in Euclidean Location”. In: SIAM J. Discrete Math. 1 (1 1988), pp. 50–64. [32] Keld Helsgaun. “An effective implementation of the Lin-Kernighan traveling salesman heuristic”. In: European Journal of Operational Research 126.1 (2000), pp. 106–130.issn: 0377-2217.doi: https://doi.org/10.1016/S0377-2217(99)00284-2. [33] Keld Helsgaun. “Solving the equality generalized traveling salesman problem using the Lin-Kernighan-Helsgaun Algorithm”. In: Mathematical Programming Computation 7 (2015), pp. 269–287. [34] D. S. Hochbaum. “When are NP-hard location problems easy?” In: Annals of Operations Research 1 (3 1984), pp. 201–214.issn: 0254-5330.url: http://dx.doi.org/10.1007/BF01874389. 67 [35] Dorit Hochbaum and J. Michael Steele. “Steinhaus’s Geometric Location Problem for Random Samples in the Plane”. In: Advances in Applied Probability 14.1 (1982), pp. 56–67.issn: 00018678. url: http://www.jstor.org/stable/1426733. [36] Michael Huang, Karen R. Smilowitz, and Burcu Balcik. “A continuous approximation approach for assessment routing in disaster relief”. In: Transportation Research Part B: Methodological 50 (2013), pp. 20–41.issn: 0191-2615.doi: https://doi.org/10.1016/j.trb.2013.01.005. [37] Abid Hussain, Yousaf Shad Muhammad, M. Nauman Sajid, Ijaz Hussain, Alaa Mohamd Shoukry, and Showkat Gani. “Genetic Algorithm for Traveling Salesman Problem with Modified Cycle Crossover Operator”. In: Computational Intelligence and Neuroscience (2017).doi: https://doi.org/10.1155/2017/7430125. [38] Sung Jin Hwang, Steven B. Damelin, and Alfred O. Hero III au2. Shortest Path through Random Points. 2015. arXiv: 1202.0045[math.PR]. [39] Ola Jabali, Michel Gendreau, and Gilbert Laporte. “A continuous approximation model for the fleet composition problem”. In: Transportation Research Part B: Methodological 46.10 (2012), pp. 1591–1606.issn: 0191-2615.doi: https://doi.org/10.1016/j.trb.2012.06.004. [40] D. Karapetyan and G. Gutin. “Lin-Kernighan heuristic adaptations for the generalized traveling salesman problem”. In: European Journal of Operational Research 208.3 (2011), pp. 221–232.issn: 0377-2217.doi: https://doi.org/10.1016/j.ejor.2010.08.011. [41] Anna R. Karlin, Nathan Klein, and Shayan Oveis Gharan. A (Slightly) Improved Approximation Algorithm for Metric TSP. 2020. arXiv: 2007.01409[cs.DS]. [42] Michael Khachay and Katherine Neznakhina. “Complexity and approximability of the Euclidean generalized traveling salesman problem in grid clusters”. en. In: Annals of Mathematics and Artificial Intelligence 88.1 (Mar. 2020), pp. 53–69.issn: 1573-7470.doi: 10.1007/s10472-019-09626-w. (Visited on 11/15/2020). [43] Maja Kiba-Janiak, Jakub Marcinkowski, Agnieszka Jagoda, and Agnieszka Skowrońska. “Sustainable last mile delivery on e-commerce market in cities from the perspective of various stakeholders. Literature review”. In: Sustainable Cities and Society 71 (2021), p. 102984. [44] G. Laporte and F. Semet. “Computational evaluation of a transformation procedure for the symmetric generalized traveling salesman problem”. In: INFOR 37.2 (1999), pp. 114–120. [45] Gilbert Laporte, Yves Nobert, and Serge Taillefer. “Solving a Family of Multi-Depot Vehicle Routing and Location-Routing Problems”. In: Transportation Science 22.3 (1988), pp. 161–172.doi: 10.1287/trsc.22.3.161. eprint: https://doi.org/10.1287/trsc.22.3.161. [46] Yao-Nan Lien, Eva Ma, and Benjamin W.-S. Wah. “Transformation of the generalized traveling-salesman problem into the standard traveling-salesman problem”. In: Information Sciences 74.1 (1993), pp. 177–189.issn: 0020-0255.doi: https://doi.org/10.1016/0020-0255(93)90133-7. 68 [47] S. Lin and B. W. Kernighan. “An Effective Heuristic Algorithm for the Traveling Salesman Problem”. In: Operations research (1973), pp. 498–516. [48] Yu Lin, Zheyong Bian, and Xiang Liu. “Developing a dynamic neighborhood structure for an adaptive hybrid simulated annealing - tabu search algorithm to solve the symmetrical traveling salesman problem”. In: Applied Soft Computing 49 (2016), pp. 937–952.issn: 1568-4946.doi: https://doi.org/10.1016/j.asoc.2016.08.036. [49] T. Lindvall. The coupling method. John Wiley, 1992. [50] Rajesh Matai, Surya Singh, and Murari Lal Mittal. “Traveling Salesman Problem: an Overview of Applications, Formulations, and Solution Approaches”. In: Traveling Salesman Problem. Ed. by Donald Davendra. Rijeka: IntechOpen, 2010. Chap. 1.doi: 10.5772/12909. [51] C. E. Noon and J. C. Bean. “A Lagrangian based approach for the asymmetric generalized traveling salesman problem”. In: Operations Research 39.4 (1991), pp. 623–632. [52] Charles E. Noon and James C. Bean. “An Efficient Transformation Of The Generalized Traveling Salesman Problem”. In:INFOR:InformationSystemsandOperationalResearch 31.1 (1993), pp. 39–44. doi: 10.1080/03155986.1993.11732212. eprint: https://doi.org/10.1080/03155986.1993.11732212. [53] C. H. Papadimitriou. “Worst-case and probabilistic analysis of a geometric location problem”. In: SIAM Journal on Computing 10 (1981), p. 542. [54] Petrica C. Pop. “New Integer Programming Formulations of the Generalized Travelling Salesman Problem”. In: American Journal of Applied Sciences (2007).url: https://dl.acm.org/doi/10.5555/2209505.2209517. [55] C. Redmond and J. E. Yukich. “Limit Theorems and Rates of Convergence for Euclidean Functionals”. English. In: The Annals of Applied Probability 4.4 (1994), pp. 1057–1073.issn: 10505164.url: http://www.jstor.org/stable/2245079. [56] Jacques Renaud and Fayez F. Boctor. “An efficient composite heuristic for the symmetric generalized traveling salesman problem”. In: European Journal of Operational Research 108.3 (1998), pp. 571–584.issn: 0377-2217.doi: https://doi.org/10.1016/S0377-2217(97)00142-2. [57] Jacques Renaud, Fayez F. Boctor, and Gilbert Laporte. “A Fast Composite Heuristic for the Symmetric Traveling Salesman Problem”. In: INFORMS Journal on Computing 8.2 (1996), pp. 134–143.doi: 10.1287/ijoc.8.2.134. eprint: https://doi.org/10.1287/ijoc.8.2.134. [58] Charles S. Revelle and Gilbert Laporte. “The Plant Location Problem: New Models and Research Prospects”. In: Operations Research 44.6 (1996), pp. 864–874.doi: 10.1287/opre.44.6.864. eprint: https://doi.org/10.1287/opre.44.6.864. [59] Ulrike Ritzinger, Jakob Puchinger, and Richard F. Hartl. “A survey on dynamic and stochastic vehicle routing problems”. In: International Journal of Production Research 54.1 (2016), pp. 215–231. doi: 10.1080/00207543.2015.1043403. eprint: https://doi.org/10.1080/00207543.2015.1043403. 69 [60] J. P. Saskena. “Mathematical Model of Scheduling Clients through Welfare Agencies”. In:Journalof the Canadian Operational Research Society (1970). [61] X.H. Shi, Y.C. Liang, H.P. Lee, C. Lu, and Q.X. Wang. “Particle swarm optimization-based algorithms for TSP and generalized TSP”. In: Information Processing Letters 103.5 (2007), pp. 169–176.issn: 0020-0190.doi: https://doi.org/10.1016/j.ipl.2007.03.010. [62] Petr Slavik. “On the approximation of the generalized traveling salesman problem”. In: Technical (1997). [63] Petr Slavík. Approximation Algorithms For Set Cover And Related Problems. 1998. [64] Stephen L. Smith and Frank Imeson. “GLNS: An effective large neighborhood search heuristic for the Generalized Traveling Salesman Problem”. In: Computers & Operations Research 87 (2017), pp. 1–19.issn: 0305-0548.doi: https://doi.org/10.1016/j.cor.2017.05.010. [65] Lawrence V. Snyder and Mark S. Daskin. “A random-key genetic algorithm for the generalized traveling salesman problem”. In: European Journal of Operational Research 174.1 (2006), pp. 38–53. issn: 0377-2217.doi: https://doi.org/10.1016/j.ejor.2004.09.057. [66] Timothy Law Snyder and J. Michael Steele. “Worst-case greedy matchings in the unit d-cube”. In: Networks 20.6 (1990), pp. 779–800.doi: https://doi.org/10.1002/net.3230200607. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/net.3230200607. [67] J. M. Steele. “Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability”. English. In: The Annals of Probability 9.3 (1981), pp. 365–376.issn: 00911798.url: http://www.jstor.org/stable/2243524. [68] J. Michael Steele. “Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges”. In: Ann. Probab. 16.4 (Oct. 1988), pp. 1767–1787.doi: 10.1214/aop/1176991596. [69] J.M. Steele. Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, 1987.isbn: 9780898713800. [70] E. M Stein and R. Shakarchi.Realanalysis:measuretheory,integration,andHilbertspaces. Princeton University Press, 2009. [71] Mehmet Fatih Tasgetiren, P. N. Suganthan, and Quan-Qe Pan. “A Discrete Particle Swarm Optimization Algorithm for the Generalized Traveling Salesman Problem”. In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation. GECCO 07. London, England: Association for Computing Machinery, 2007, pp. 158–167.isbn: 9781595936974.doi: 10.1145/1276958.1276980. [72] Eric W Weisstein. Traveling Salesman Constants. A Wolfram web resource. https://mathworld.wolfram.com/TravelingSalesmanConstants.html. Accessed: 2020-10-29. 2020. 70 [73] E. Wygonik and A. Goodchild. “Evaluating the efficacy of shared-use vehicles for reducing greenhouse gas emissions: a US case study of grocery delivery”. In: Journal of the Transportation Research Forum. Vol. 51. 2012. [74] Weijun Xie and Yanfeng Ouyang. “Optimal layout of transshipment facility locations on an infinite homogeneous plane”. In: Transportation Research Part B: Methodological 75 (2015), pp. 74–88.issn: 0191-2615.doi: https://doi.org/10.1016/j.trb.2015.02.001. [75] J. Yang, X. Shi, M. Marchese, and Y. Liang. “An ant colony optimization method for generalized TSP problem”. In: Progress in Natural Science 18.11 (2008), pp. 1417–1422. [76] Shi-hua Zhan, Juan Lin, Ze-jun Zhang, and Yi-wen Zhong. “List-Based Simulated Annealing Algorithm for Traveling Salesman Problem”. In: Computational Intelligence and Neuroscience (2016).doi: https://doi.org/10.1155/2016/1712630. 71 AppendixA FurtherTechnicalities A.1 TechnicalTools The first four results are stated without proof and are standard textbook material. Lemma 21. Let f : R → R be a real-valued function and letB d (r) ⊂ R d be a ball of radius r centered about the origin. We have Z B d (r) f(∥x∥)dx= Z r 0 S d− 1 (t)f(t)dt, whereS d− 1 (t) is the surface area of a(d− 1)-sphere of radiust, which isS d− 1 (t)= 2π d/2 Γ( d/2) t d− 1 . Lemma22. The volume of ad-dimensional ball of radiusr isπ d/2 r d /Γ( d/2+1). Corollary 23. Letℓ > 0 and letD ′ ⊂ R dn denote the set of alln-tuples(x 1 ,...,x n ) of points inR d such that∥x 1 ∥+ P n i=2 ∥x i − x i− 1 ∥≤ ℓ. The volume ofD ′ ,Vol(D ′ ), satisfies Vol(D ′ )= 2π d/2 Γ( d/2) ! n · Γ( d) n Γ( dn+1) · ℓ dn . (A.1) Proof. This is just Cavalieri’s principle applied to Lemma 7 72 A.2 PreliminaryProofs ExampleA.2.1. By Lemma 5 we have log x y =logΓ( x+1)− logΓ( y+1)− logΓ( x− y+1) =xlogx− 2x+ 1 2 logx− ylogy+ 1 2 logy +(y− x)log(x− y)− 1 2 log(x− y)+O(1+1/x− 1/y− 1 x− y ) =O(xlogx) Proof of Lemma 3. It is easy to see that Pr ∥X 1 ∥+ n X i=2 ∥X i − X i− 1 ∥≤ ℓ ! | {z } (∗ ) ≤ 2π d/2 Γ( d/2) ! n · Γ( d) n Γ( dn+1) · ℓ dn ; (A.2) this is because we can regard the samplesX 1 ,...,X n as being asingle sample drawn uniformly fromR n , so that the probability of interest(∗ ) is simply the probability that this single sample lies in the domainD ′ described in Corollary 23. This probability is of course equal toVol(D ′ ∩R n )≤ Vol(D ′ ), which gives us the desired inequality Eq. (A.2). We obtain our corollary by applying the union bound to Eq. (A.2) over all n!=Γ( n+1) permutations ofX 1 ,...,X n . A.3 FurtherProofrelatedtotheApproximationLemma By reducing our use of subscripts, in order to justify our assumption in the proof of Lemma 4, it will suffice to prove the following: 73 Claim 24. Let g be a non-negative measurable function with bounded support S whose level sets have Lebesgue measure zero and assume that R S g is rational. For any ϵ > 0, there exists a step function ap- proximationψ (x)= P j a j 1(x∈ j ) ofg such that R S |g− ψ |≤ ϵ , and • ψ (x)=0 wheneverx / ∈S, • inf x∈S g(x) l and u ′ < u be sufficiently small rational numbers that l ′ Area(S) < R g < u ′ Area(S), 74 (l ′ − l)Area(S) ≤ ϵ/ 8, and (u− u ′ )Area(S) ≤ ϵ/ 8. For each component coefficient b j of σ , define ˜ b j =min{max{b j ,l ′ },u ′ }. The function ˜ σ (x)= P j ˜ b j 1(x∈s j ) satisfies Z |σ − ˜ σ |= X j:b j <l ′ (l ′ − b j )Area(s j )+ X j:b j >u ′ (b j − u ′ )Area(s j ) ≤ X j:b j <l ′ (l ′ − l)Area(s j )+ X j:b j >u ′ (u− u ′ )Area(s j ) ≤ max{l ′ − l,u− u ′ }Area(S)≤ ϵ/ 8 and therefore R |g− ˜ σ |≤ ϵ/ 4. We can approximate each pieces j to arbitrary precision with a finite collection of rectangles ⊞ j , all of which are contained inS; this is just the Lebesgue inner measure. If each collection⊞ j is chosen so that Area(s j \⊞ j )≤ ϵ/ (8 ˜ b j #˜ σ ), where#˜ σ denotes the number of components of ˜ σ , then the resulting step functionφ= P j ˜ b j 1(x∈⊞ j ) satisfies Z |˜ σ − φ|= X j ˜ b j Area(s j \⊞ j )≤ ϵ/ 8 and therefore R |g− φ|≤ 3ϵ/ 8. For ease of notation, we re-index the entries of φ and write φ(x) = P j c j 1(x ∈ j ), where each j is an individual rectangle (the identification of the sets ⊞ j is no longer of any relevance to us). The penultimate step is to construct a further approximation ˜ φ(x) = P j ˜ c j 1(x ∈ j ) such that ˜ c j and Area( j ) are rational. This is straightforward; choose δ > 0 sufficiently small so that δ ≤ ϵ (16 R φ) − 1 75 and (1− δ )c j > l ′ for all j such that c j > l ′ . For each j, let j ⊂ j have rational endpoints with Area( j )≥ (1− δ )Area( j ), and let˜ c j be rational withc j ≥ ˜ c j ≥ (1− δ )c j . We have Z |φ− ˜ φ|= X j c j Area( j )− ˜ c j Area( j ) ≤ X j c j Area( j )− (1− δ ) 2 c j Area( j ) ≤ 2δ X j c j Area( j )=2δ Z φ≤ ϵ/ 8 and so the step function ˜ φ(x)= P j ˜ c j 1(x∈ j ) satisfies R |g− ˜ φ|≤ ϵ/ 2. The last step is to define ψ t (x) = min{max{˜ φ(x) + t,l ′ },u ′ } (this simply amounts to shifting ˜ φ vertically by an amountt but truncating everything belowl ′ or aboveu ′ ). Clearly, the functionρ (t)= R ψ t is continuous and monotonically increasing, and there exist values t − and t + such that ρ (t − ) < R g < ρ (t + ), and thereforeρ (t ∗ ) = R g for somet ∗ . Because ˜ c j andArea( j ) are rational, we know thatt ∗ is rational as well, and therefore the step functionψ ≡ ψ t ∗ satisfies all four bullet points that we required. The last step is to show that R |˜ φ− ψ |≤ ϵ/ 2, whence R |g− ψ |≤ ϵ . Since ˜ φ andψ differ only on a vertical translation, we have Z |˜ φ− ψ |= Z (˜ φ− ψ ) = Z ˜ φ− Z ψ = Z ˜ φ− Z g ≤ Z |˜ φ− g|≤ ϵ/ 2 which completes the proof. 76 AppendixB FurtherExperiments One-of-a-subsetTSP: Fig. B.1 shows the analogous of Fig. 3.1 but for larger instance ofn∈{150,200,...,1000} andk∈{2,3,...,6}. Figure B.1: Tour length forone-of-a-subset instances. n=150 to 1000 clusters. All withk ∈{2,3,...,6} points 77 0 20 40 60 80 100 Orders 3000 3200 3400 Inventory level # picker 8, # packer 12, order size 10 RS CL 0 20 40 60 80 100 Orders 2800 3000 3200 Inventory level # picker 10, # packer 8, order size 10 RS CL 0 20 40 60 80 100 Orders 3000 3200 3400 Inventory level # picker 10, # packer 12, order size 10 RS CL 0 20 40 60 80 100 Orders 3000 3200 3400 Inventory level # picker 10, # packer 15, order size 10 RS CL 0 20 40 60 80 100 Orders 2800 2900 3000 3100 3200 Inventory level # picker 12, # packer 8, order size 10 RS CL 0 20 40 60 80 100 Orders 2900 3000 3100 3200 3300 Inventory level # picker 12, # packer 12, order size 10 RS CL 0 20 40 60 80 100 Orders 2900 3000 3100 3200 3300 Inventory level # picker 12, # packer 15, order size 10 RS CL Figure B.2: Further inventory level experiments. The fill shows the standard deviation of 10 simulations. 78 B.0.1 Inventorylevelresults B.1 DiscreteEventSimulationDetail Fig. B.3 shows the event diagram for when an order arrives or is de-queued from the orders waiting queue. This queue is meant for the orders to wait for pickers to get assigned to them or for the inventory to arrive if there is not enough in stock. We assume that orders have a patience time (randomly generated from a uniform distribution) and if it takes longer than that to start picking them up, they renege. The model in Section 4.1 does not include this detail so we set all the patience to infinity for the experiments. When an order is finished we start de-queuing the inventory loads that have been waiting in the inventory queue for the space to open up. Figure B.3: Order arrival While there are orders not finished (or reneged) we keep generating inventory loads coming in with the same rate as the orders. Fig. B.3 shows the event diagram of inventory load arrival. This is same as when a load is de-queued from the inventory waiting queue. This queue is meant for the loads to wait for the space to open up or for the packers to pack them. We also assume that inventories have a patience similar to the orders. When an inventory is finished packing we start de-queuing the orders that have been waiting in the order queue for the inventories to arrive. We also check to see if we need to order more inventory loads. 79 Figure B.4: Inventory load arrival 80

Abstract (if available)

Abstract We study the probabilistic behavior of routing optimization problems in which one is given a collection of points and the goal is to find the shortest tour that visits a subset of those points that meets
certain criteria. Examples of such problems, which we call ``selection routing problems'', include the ''one-of-a-subset'' travelling salesman problem (TSP), and generalized TSP (GTSP), the orienteering problem, the prize collecting travelling salesman problem, and the distance-constrained generalized vehicle routing problem. We derive continuous approximation formulas for several such problems under the assumption that point-to-point distances are Euclidean and all points are independent and identical samples of a probability density on a compact planar region. Numerical simulations demonstrate that our formulas provide accurate estimates of routing costs. Finally, we analyze the random stow warehouse model using our results and show its efficiency compared to classical warehouse policies.

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Asset Metadata

Creator Azizi, MohammadJavad (author)

Core Title Continuous approximation for selection routing problems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2022-08

Publication Date 06/10/2022

Defense Date 05/12/2022

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

Tag BHH theorem,continuous approximation,generalized traveling salesman problem,inventory systems,OAI-PMH Harvest,sparse subset

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Carlsson, John (committee chair), Gomez Escobar, Andres (committee member), Nayyar, Ashutosh (committee member), Razaveyin, Meisam (committee member), Ross, Sheldon (committee member)

Creator Email azizim@usc.edu,mja1374.azizi@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111339637

Unique identifier UC111339637

Document Type Dissertation

Format application/pdf (imt)

Rights Azizi, MohammadJavad

Internet Media Type application/pdf

Type texts

Source 20220613-usctheses-batch-946 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Repository Email cisadmin@lib.usc.edu