A Priori Performance Measures for Arc-Based Formulations of Vehicle Routing Problem

Abstract

5). It is well known that highly constrained VRPs, such as those with tight capacity or time window constraints, are more amenable to a column generation solution approach because the constraints lead to pruning up front a large number of infeasible paths (6). Therefore, there is clearly a strong relationship between specific algorithms and problem formulation. However, little is known about how the formulation of the problem influences the intrinsic difficulty of the problem and hence the solution run-time of any algorithm for VRPs of a fixed size in practice.This paper studies experimentally the effect of problem formulation on solution run-times for the VRP when solved optimally by a commercial integer programming (IP) solver. The goal of this work then is similar to that of Jones et al., which investigates the effect of problem formulation on decomposition algorithms for multicommodity flow problems (7 ). However, by considering a general purpose solver, the authors aim to identify problem features that can indicate whether a problem formulation is fundamentally easier to solve or not. For this reason, special solution procedures favoring particular formulations are avoided in this study. There is a vast literature of experimental analysis on well-known hard VRP instances combining formulation and solution procedure. Some recent examples are R. Baldacci et al., Ralphs et al.,. Because such studies develop special solution procedures for a given formulation, they do not identify the sole effect of the problem formulation independent from its special solution procedure.When a fleet of vehicles is being routed, being able to identify a VRP formulation that is easier to solve can provide better routing solutions through improved solution methods that use the correct formulations. The appropriate VRP formulations can improve solution methods for two reasons. First, they can help steer algorithm research to focus on good formulations that are amenable to fast solutions, the idea being that research on these fundamentally simpler problem formulations can lead to improved algorithms. Second, general purpose solvers such as the one used in this study have improved considerably over time. These solvers can now be used to obtain exact solutions to small problems and approximate solutions for large problems by stopping the execution after a run-time limit. For problems of a given size, identifying good formulations can help provide solutions within a reasonable time.In this study, the effect of problem formulation is investigated based on a priori performance measures. The focus is on identifying measures based on problem data to describe the difficulty of the problem instance caused by problem formulation, without solving the problem. There are other performance measures proposed in the VRP literature to investigate the effect of problem formulation. However, these measures are mainly posterior and typically depend on the optimal solution. For example, the average number of customers per route The vehicle routing proble...

A Time-Constrained Capacitated Vehicle Routing Problem in Urban E-Commerce Delivery

“…The authors introduced the problem in detail and highlighted its resemblance to the traveling salesman problem (TSP) studied in [9]. Since then, numerous variants of the problem have been studied, and alternative solution approaches have been proposed [10][11][12][13][14][15]. For further information, see recent surveys of the VRP literature [16][17][18].…”

Time-Constrained Capacitated Vehicle Routing Problem in Urban E-Commerce Delivery

“…The authors introduced the problem in detail and highlighted its resemblance to the traveling salesman problem (TSP) studied by Flood (9). Since then, numerous variants of the problem have been studied, and alternative solution approaches have been proposed (10)(11)(12)(13)(14)(15). For further information, see recent surveys of the VRP literature (16)(17)(18).…”

Kernel Search for the Capacitated Vehicle Routing Problem

“…The following mathematical model is taken from [24]. Two-index binary variables x ij are used as decision variables equal to 1 if arc (i, j) belongs to the optimal solution and 0 otherwise.…”