Compact formulations for multi-depot routing problems: Theoretical and computational comparisons

Highlights

• We review and propose new compact formulations for multi-depot routing problems.

• We study two variants, one with depot selection decisions, and one without.

• We provide comprehensive theoretical and practical comparisons of the formulations.

• We offer suggestions on which formulations perform better computationally.

Abstract

Multi-depot routing problems mainly arise in distribution logistics where a fleet of vehicles are used to serve clients from a number of (potential) depots. The problem concerns deciding on the routes of each vehicle and the depots from which the vehicles depart, so as to minimize the total cost of travel. This paper reviews a number of existing compact formulations, and proposes new ones, for two types of multi-depot routing problems, one that includes the depot selection decisions, and the other where depots are pre-selected. The formulations are compared theoretically in terms of the strength of their linear programming relaxation, and computationally in terms of the running time needed to solve the instances to optimality.

Introduction

Multi-depot routing problems (MDRPs) involve routing a fleet of vehicles from several depots at minimal cost (e.g., distance) to serve a number of clients, where each vehicle returns back to the depot it has originated from. The locations of the depots may either be fixed a priori, the case which we will name as the Fixed-MDRP (F-MDRP), or alternatively chosen from a set of potential locations, which we will denote by the Location-MDRP (L-MDRP) in the ensuing exposition. MDRPs are generalizations of single-depot routing problems such as the traveling salesman problem (see, e.g., Lawler et al., 1985, Applegate et al., 2006) and the vehicle routing problem (see, e.g., Toth and Vigo, 2014).

Formulations hitherto proposed for the MDRP contain two types of constraints, one requiring that each client is served in exactly one vehicle route, and the other guaranteeing that each route must contain exactly one depot. The latter is usually modelled by two complementary sets of constraints to ensure that each route contains (i) at least one, and (ii) at most one depot, respectively. If condition (i) is not satisfied, then there exists a route, or a subtour, disconnected from a depot and formed within clients only. Subtours are avoided by using the classical subtour elimination constraints (see, e.g., Öncan et al., 2009, Godinho et al., 2011, Roberti and Toth, 2012 for surveys on the topic). As for condition (ii), if it is not satisfied it implies a vehicle traveling on a path between two different depots. Routes of such nature are prohibited by the use of the so-called path elimination constraints. One such class called chain-barring constraints is described by Laporte et al. (1983), later improved by Laporte et al. (1986). Improvements of chain-barring constraints are described by Belenguer et al. (2011) and Benavent and Martínez-Sykora (2013) later used in the work by Sundar and Rathinam (2017). A new set of multi-cut-like path elimination constraints, which differ from what was previously used in the literature, is described by Bektaş et al. (2017). All the path elimination constraints proposed in the studies just cited are exponential in size and therefore require separation routines for use in algorithms such as branch-&-cut. In contrast, Albareda-Sambola et al., 2005, Bektaş, 2012, Burger et al., 2018 present path elimination constraints that are polynomial in size.

Our motivation for this paper stems from the fact that compact formulations, i.e., those that are polynomial in the number of variables and constraints, are of practical significance as they can be readily used in combination with off-the-shelf optimization software. This is in contrast to other types of formulations described for the MDRP, some of which were described in the previous paragraph, that involve the use of exponentially sized sets of constraints. However, algorithms that are based on such formulations require the use of specialized methods, such as constraint separation, which may not always be easy to understand, implement and use. For this reason, in this paper, we exclusively focus on compact formulations for the MDRP. We extend and build on the MDRP literature by presenting compact sets of path elimination constraints, which we combine with compact sets of subtour elimination constraints to derive several valid formulations for the MDRP, to then compare them both theoretically and computationally. The contributions of this paper may be summarized as follows: (i) we review, propose and compare sets of compact path elimination constraints that can be used for both the F-MDRP and the L-MDRP, (ii) we describe formulations for the two variants of the MDRP obtained by combining different sets of path elimination and subtour elimination inequalities, (iii) we compare the formulations theoretically in terms of the strength of the linear programming relaxation, and computationally in terms of the time requirements for obtaining optimal solutions on a number of benchmark and randomly generated instances.

The following section presents a generic formulation for the MDRP, followed by a description of compact subtour elimination constraints in Section 3 and path elimination constraints in Section 4, the latter including theoretical comparison results. Details of the computational experimentation along with the results are described in Section 5. Extensions of the models and the results to a more general case of the MDRP are discussed in Section 6. The paper concludes in Section 7.