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Branch-And-Cut and Branch-And-Cut-And-Price Algorithms for Solving Vehicle Routing Problems

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Branch-And-Cut and Branch-And-Cut-And-Price Algorithms for Solving Vehicle Routing Problems Downloaded from orbit.dtu.dk on: Oct 05, 2021 Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems Jepsen, Mads Kehlet Publication date: 2011 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Jepsen, M. K. (2011). Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems. Technical University of Denmark. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. PHD Thesis Branch-and-cut and Branch-and-Cut-and-Price Algorithms for Solving Vehicle Routing Problems Author: Mads Kehlet Jepsen Supervisor: Professor David Pisinger min cexe e E X2 s:t: x = 2 i V e 8 2 c e δ(i) 2X x = 2 K e j j e δ(0) 2X x 2r(S) S V ; S 2 e ≥ 8 ⊆ c j j ≥ e δ(S) 2X x 0; 1; 2 e δ(0) e 2 f g 8 2 x 0; 1 e E δ(0) e 2 f g 8 2 n min cpλp p P X2 s.t α λ = 1 i V ijp p 8 2 c p P (i;j) δ+(i) X2 X2 λ 0; 1 p P p 2 f g 8 2 min c x p y e e − i i e E i N X2 X2 x = 2y i V e i 8 2 e δ(i) 2X y0 = 1 x 2y i S; S V ; S 2 e ≥ i 8 2 8 ⊆ c j j ≥ e δ(S) 2X d y Q i i ≤ i N X2 x 0; 1 e E e 2 f g 8 2 y 0; 1 i V: i 2 f g 8 2 8. Juli 2011 Preface This Ph.D thesis was started at the Department of Computer Science, Uni- versity of Copenhagen (DIKU) in collaboration with Microsoft Development Center Copenhagen (MDCC). It was continued and finished at the Depart- ment of Management Engineering, Danish Technical University (DTU Man) at the OR-section. I would like to thank the University of Copenhagen, Microsoft Development Center Copenhagen and the Danish Technical Uni- versity for granting me the opportunity to study a Ph.D. I would like to thank my old colleagues at DIKU and MDCC and my colleagues at DTU Man for creating an excellent working environment dur- ing the past three year. I would like to thank Berit Løfsted, Christian E. M. Plum, Line Blander Reinhardt, Stefan Røpke and Mikkel M. Sigurd for the interesting discussion during the creation of the articles I have written with you. I would especially like to thank Bjørn Petersen, Simon Spoorendonk and my supervisor David Pisinger for the many discussions, we have had during the construction of our papers. Finally I like to thank my wife Line for a lot of proof reading throughout the entire thesis. Thanks i ii Contents 1 Introduction 1 I Academic Papers 31 2 Subset-Row Inequalities Applied to the Vehicle Routing Prob- lem with Time Windows 33 3 A Branch-and-Cut Algorithm for the Symmetric Two-echelon Capacitated Vehicle Routing Problem 68 4 A Branch-and-Cut Algorithm for the Capacitated Profitable Tour Problem 99 5 Partial Path Column Generation for the Vehicle Routing Problem 138 6 Partial Path Column Generation for the Elementary Short- est Path Problem with a Capacity Constraints 163 7 Conclusion 173 II Appendix 177 A The vehicle routing problem with edge set costs 179 B A Path Based Model for a Green Liner Shipping Network Design Problem 197 C Partial Path Column Generation for the Vehicle Routing Problem with Time Windows 215 D Danish Summary 224 iii Chapter 1 Introduction The Vehicle Routing Problem was introduced by Dantzig and Ramser [19] under the name the truck dispatching problem. The problem consists of finding m routes covering n customers with a common point called the depot. The goal of the optimization is to find the set of routes with the minimal overall distance. As time has passed, many variants of the problem have been proposed. Today there probably exist more than 250 different variants and at every major conference several new variants are proposed. The main reason for this rapid growth in the number of problems are, in my opinion be attributed to the importance of the problems in the real world and the excellent solution methods that have been developed. For exact solution methods the two problems that have been the center of the attention is the Capacitated Vehicle Routing Problem (cvrp) where each costumer has a demand and the vehicle has a fixed capacity, and the Vehicle Routing Problem with Time Windows (vrptw) which is a cvrp problem where each customer can only be serviced within a predefined time window. Before 1980 very few exact algorithms for cvrp and vrptw had been proposed, but in the early 1980s two new exact methods where proposed. From this point the history of exact methods for cvrp and vrptw can be divided into three phases. The first phase was the introduction of the Set Partition and the development of Branch-and-Cut-and-Price (bp) algo- rithms using a relaxed pricing problem. The second was the development of Branch-and-Cut (bac) algorithms. In the current phase the pricing problem is no longer relaxed and cuts in the master problem of the Branch-and-Cut- and-Price algorithms is used. The first two phases where started at the same point in time and there is still development on the algorithms in the context of cvrp and vrptw. The algorithms from these two phases are also used on several other variants of the Vehicle Routing Problem. The third phase was started in the middle of the 2000s and the algorithms from this phase are currently the best overall performing algorithms. The main idea of this chapter is to review some of the known concepts 1 Chapter 1 and methods from the literature. Whenever a method or concept is cen- tral for the thesis it will be explained in more detail, while other concepts or methods are only briefly mentioned. Section 1.1 is dedicated the cvrp and vrptw, while section 1.2 is dedicated the Capacitated Location Rout- ing Problem. For the cvrp and vrptw section 1.1.1 will review the basic Branch-and-Cut algorithms and state two of the cutting planes which has been of great importance for this thesis. Section 1.1.3 will review the Branch- and-Cut-and-Price algorithms for the cvrp and vrptw. The section will contain information about how cuts are handled and it will review the so- lution methods for the Resource Constraint Shortest Path Problem and Elementary Resource Constraint Shortest Path Problem, which are some of the possible pricing problems. Either the bac and bcp principles for the cvrp and vrptw or both serves as inspiration in all of the academic papers in Part I and Part II. Section 1.2 will introduce some mathematical models for the Capacitated Location Routing Problem and some of the fundamental cutting for this problem. Both the models and cutting planes have served as inspiration for the solution approach for the article in Chapter 3. Finally section 1.3 will give a short summary of the academic papers in the thesis. 1.1 cvrp and vrptw The symmetric capacitated vehicle routing problem can be formulated as follows. Let V be a vertex set that consists of a set of customers Vc and the depot set V = 0 and K be a set of vehicles. With each node i V there 0 f g 2 c is associated a positive integer di, which we refer to as the demand. Let E be the set of edges connecting the vertices and let c ; e E be the cost of e 2 traversing the edge. A feasible solution to cvrp is K cycles starting in the j j depot, each customer is visited once and the sum of the customers visited on the cycles does not exceed the capacity C. In the optimization version we are seeking the feasible solution where the sum of the edge weights of the selected cycles is minimal. cvrp is often defined on a complete graph G(V; E) and test instances are available at www.branchandcut.org. Rather than referring to a solution as a set of cycles the abbreviation routes is often used and we will therefore use this. A natural extension of cvrp is the inclusion of time windows. A time window is defined as a time interval [a ; b ] for which a customer i V can be i i 2 c serviced by the vehicle. Furthermore, a service time si is associated with the customer i V . To include the time windows an asymmetric formulation 2 c is often used. Let A be a set of arcs, where each arc (i; j) A has a cost 2 cij and a travel time τij associated.
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