Discrete Optimization
Balanced vehicle routing: Polyhedral analysis and branch-and-cut algorithm

https://doi.org/10.1016/j.ejor.2018.08.034Get rights and content

Highlights

  • A branch and cut algorithm is described for the balanced routing problem.

  • A polyhedral analysis is presented and new facet-inducing inequalities are described.

  • Separation routines are presented for some of the inequalities.

  • The algorithm outperform an existing exact procedure on 100 node instances.

  • The algorithm solves instances of the problem with 151 nodes for the first time.

Abstract

This paper studies a variant of the unit-demand Capacitated Vehicle Routing Problem, namely the Balanced Vehicle Routing Problem, where each route is required to visit a maximum and a minimum number of customers. A polyhedral analysis for the problem is presented, including the dimension of the associated polyhedron, description of several families of facet-inducing inequalities and the relationship between these inequalities. The inequalities are used in a branch-and-cut algorithm, which is shown to computationally outperform the best approach known in the literature for the solution of this problem.

Introduction

The Capacitated Vehicle Routing Problem (CVRP) is concerned with designing routes for a fleet of capacitated vehicles to serve a number of customers. Each customer is in a geographical location and demands a commodity that must be transported by a vehicle from a specific location called the depot. Each vehicle must start from and end at the depot, transporting the demands of a subset of customers without violating the capacity limitations. Travel costs between any pair of locations are known and symmetric, and might be related to the distance between the two locations. The CVRP consists of determining a minimum cost set of routes for the vehicles to serve the demand of each customer with exactly one visit. The capacity limitations in the CVRP relate to the maximum load that the vehicle can carry upon leaving the depot; see e.g., Letchford and Salazar-Gonzalez (2015) for different CVRP formulations. For the variant of the problem where the demand of each customer is equal to one (often called the unit-demand CVRP), the capacity limitation can be interpreted as the maximum number of customers that can be visited in each route.

Under the traditional objective of minimizing the travel cost, optimal solutions may contain imbalanced routes, in the sense that some vehicles visit the maximum allowed number of customers while others visit only a few customers. One way to avoid this situation is to impose an additional limitation on the minimum number of customers that should be served on each route. With such a constraint, optimal solutions may result in routes serving similar numbers of customers. We will refer to this extension as the Balanced Vehicle Routing Problem (BVRP).

The BVRP has been studied by Gouveia, Riera-Ledesma, and Salazar-González (2013); Gouveia and Salazar-González (2013) under the name of “vehicle routing problem with lower bound on the number of customers per route”. The work in Gouveia et al. (2013) describes a branch-and-cut algorithm based on a class of so-called Reverse Multi-Star (RMS) inequalities, which are obtained by projecting out the flow variables from a single commodity flow formulation of the problem. The RMS inequalities are related to the Multi-Star (MS) inequalities considered in Araque, Hall, and Magnanti (1990). A multi-depot variant of the problem has been studied in Bektaş (2012), describing several formulations and Benders decomposition algorithms.

Alternative approaches described in the literature for finding balanced routes include the use of a multi-objective function, as opposed to enforcing additional constraints in the formulation. Since our paper follows the latter approach to solve the BVRP, we refer the interested reader to Jozefowiez, Semet, and Talbi (2008) for a survey on the former.

The main contribution of our paper is to provide the first investigation of the BVRP polyhedron extending known results on the CVRP polytope. One of the earliest works that conducts a polyhedral study for the cardinality-constrained minimum spanning tree problem and the unit-demand CVRP is Araque et al. (1990). The authors study a variety of MS inequalities, divided into “Large”, “Intermediate” and “Small”. They present two additional sets of inequalities called Ladybug and Partial MS inequalities, and conclude by analyzing the clique inequalities. The number of routes in a solution is not fixed in their work. Further polyhedral analysis for the unit-demand CVRP where the number of routes is fixed can be found in Campos, Corberan, and Mota (1991), which presents results on the dimension of the associated polyhedron and facet-inducing properties of the trivial inequalities and the capacity constraints. Most of the proofs presented in Campos et al. (1991) are based on what is known as the “indirect method”, which consists of concluding the unique representation of the inequality; see e.g. Grötschel and Padberg (1979). Cornuejols and Harche (1993) extend the polyhedral analysis by studying additional inequalities and extensions beyond the unit-demand case. They exploit the fact that the polyhedron of the Graphical Vehicle Routing Problem (GVRP) is a full-dimensional and includes the CVRP polytope as a face. Most of the analysis in that article are about inequalities of the GVRP polyhedron. A generalization of these inequalities for the general-demand CVRP can be found in Letchford, Eglese, and Lysgaard (2002); Letchford and Salazar-Gonzalez (2006).

Our work aims to contribute to the efforts above and present a polyhedral analysis of the BVRP, which includes the unit-demand CVRP as a special case when the lower limit for the vehicle load is equal to one. We study the dimension of the associated polyhedron and some facet-inducing properties. These results are exploited in a branch-and-cut algorithm to solve the BVRP. Computational experiments show that our implementation is able to solve much larger-scale BVRP instances than previous approaches in the literature.

The rest of the paper is organized as follows. Section 2 provides a formal description of the problem, introduces the notation, and presents a mathematical formulation. Section 3 studies the dimension of the BVRP polyhedron and presents some of its facets. A branch-and-cut algorithm using the new inequalities, together with computational results, are detailed in Section 4. Conclusions are given in Section 5.

Section snippets

Problem description

The BVRP is defined on an undirected graph G=(V,E) where V={1,,n} is the set of vertices, each one representing a location, and E={(i,j)|i<j,iV,jV} is the set of edges. Each edge (i, j) ∈ E has a travel cost cij, which is often a function of the distance between the locations. Node 1 represents the depot and the set V={2,,n} represents the customers, each of which has a unit demand. The minimum and maximum numbers of customers allowed to be served by each vehicle are denoted by Q and Q¯,

Polyhedral analysis

In this section, we first provide results on the dimension of the BVRP polyhedron, defined asPBVRP=Convex.Hull{xR|E||xsatisfies(2.2)--(2.6)},and then present a number of facet-defining inequalities.

Theorem 3.1

dim(PBVRP)=|E|n.

A detailed proof of Theorem 3.1 is presented in Appendix A. It is based on a partition of V′ into subsets Hi (i=1,,m), each containing a number of customers between Q and Q¯, and then exploit known results for the polytope associated to the Travelling Salesman Problem (TSP). Note

Branch-and-cut algorithm and computational analysis

This section first describes a branch-and-cut algorithm that uses the inequalities studied in the previous sections, and then presents computational results on solving the BVRP using benchmark instances from the literature.

Conclusions

This paper presented polyhedral results and an exact algorithm for a unit-demand vehicle routing problem with lower and upper bounds on the number of customers that can be visited in each route. The computational results showed that the proposed algorithm outperformed a previously described algorithm on instances with up to 100 customers, and was able to solve instances with up to 150 customers for the first time in the literature. The computational tests suggested that all the inequalities

Acknowledgments

The authors thank two anonymous reviewers for their constructive comments on an earlier version of this paper.

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This work has been partially supported by the research project MTM2015-63680-R (MINECO/FEDER).

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