Stefan Ropke

A general heuristic for vehicle routing problems

We present a unified heuristic which is able to solve five different variants of the vehicle routing problem: the vehicle routing problem with time windows (VRPTW), the capacitated vehicle routing problem (CVRP), the multi-depot vehicle routing problem (MDVRP), the site-dependent vehicle routing problem (SDVRP) and the open vehicle routing problem (OVRP). All problem variants are transformed into a rich pickup and delivery model and solved using the adaptive large neighborhood search (ALNS) framework presented in Ropke and Pisinger [An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows. Transportation Science, to appear]. The ALNS framework is an extension of the large neighborhood search framework by Shaw [Using constraint programming and local search methods to solve vehicle routing problems. In: CP-98, Fourth international conference on principles and practice of constraint programming, Lecture notes in computer science, vol. 1520, 1998. p. 417-31] with an adaptive layer. This layer adaptively chooses among a number of insertion and removal heuristics to intensify and diversify the search. The presented approach has a number of advantages: it provides solutions of very high quality, the algorithm is robust, and to some extent self-calibrating. Moreover, the unified model allows the dispatcher to mix various variants of VRP problems for individual customers or vehicles. As we believe that the ALNS framework can be applied to a large number of tightly constrained optimization problems, a general description of the framework is given, and it is discussed how the various components can be designed in a particular setting. The paper is concluded with a computational study, in which the five different variants of the vehicle routing problem are considered on standard benchmark tests from the literature. The outcome of the tests is promising as the algorithm is able to improve 183 best known solutions out of 486 benchmark tests. The heuristic has also shown promising results for a large class of vehicle routing problems with backhauls as demonstrated in Ropke and Pisinger [A unified heuristic for a large class of vehicle routing problems with backhauls. European Journal of Operational Research, 2004, to appear].

A unified heuristic for a large class of Vehicle Routing Problems with Backhauls

The Vehicle Routing Problem with Backhauls is a generalization of the ordinary capacitated vehicle routing problem where goods are delivered from the depot to the linehaul customers, and additional goods are brought back to the depot from the backhaul customers. Numerous ways of modeling the backhaul constraints have been proposed in the literature, each imposing different restrictions on the handling of backhaul customers. A survey of these models is presented, and a unified model is developed that is capable of handling most variants of the problem from the literature. The unified model can be seen as a Rich Pickup and Delivery Problem with Time Windows, which can be solved through an improved version of the large neighborhood search heuristic proposed by Ropke and Pisinger [An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows, Technical Report, DIKU, University of Copenhagen, 2004]. The results obtained in this way are comparable to or improve on similar results found by state of the art heuristics for the various variants of the problem. The heuristic has been tested on 338 problems from the literature and it has improved the best known solution for 227 of these. An additional benefit of the unified modeling and solution method is that it allows the dispatcher to mix various variants of the Vehicle Routing Problem with Backhauls for the individual customers or vehicles.

Large Neighborhood Search

Heuristics based on large neighborhood search have recently shown outstanding results in solving various transportation and scheduling problems. Large neighborhood search methods explore a complex neighborhood by use of heuristics. Using large neighborhoods makes it possible to find better candidate solutions in each iteration and hence traverse a more promising search path. Starting from the large neighborhood search method, we give an overview of very large scale neighborhood search methods and discuss recent variants and extensions like variable depth search and adaptive large neighborhood search.

Branch and Cut and Price for the Pickup and Delivery Problem with Time Windows

In the pickup and delivery problem with time windows vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and nonelementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the pickup and delivery problem with time windows are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

Horizontal cooperation among freight carriers: request allocation and profit sharing

In modern transportation systems, the potential for further decreasing the costs of fulfilling customer requests is severely limited while market competition is constantly reducing revenues. However, increased competitiveness through cost reductions can be achieved if freight carriers cooperate in order to balance their request portfolios. Participation in such coalitions can benefit the entire coalition, as well as each participant individually, thus reinforcing the market position of the partners. The work presented in this paper uniquely combines features of routing and scheduling problems and of cooperative game theory. In the first part, the profit margins resulting from horizontal cooperation among freight carriers are analysed. It is assumed that the structure of customer requests corresponds to that of a pickup and delivery problem with time windows for each freight carrier. In the second part, the possibilities of sharing these profit margins fairly among the partners are discussed. The Shapley value can be used to determine a fair allocation. Numerical results for real-life and artificial instances are presented.

The Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations

With the increase of public programs to encourage the use of electric vehicles and the shift of social reception towards so called energy efficient solutions the research in the field of vehicle routing using low-emission vehicles has increased as well. We therefore introduce a new problem, the so-called Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows (EFSMVRPTW). The problem covers real world applications where an optimal mix of different available battery powered (and conventional) vehicles has to be found.

Scheduling technicians and tasks in a telecommunications company

This paper proposes a construction heuristic and an adaptive large neighborhood search heuristic for the technician and task scheduling problem arising in a large telecommunications company. This problem was solved within the framework of the 2007 challenge set up by the French Operational Research Society (ROADEF). The paper describes the authors’ entry in the competition which tied for second place.

Formulations and Branch-and-Cut Algorithms for the Generalized Vehicle Routing Problem

The generalized vehicle routing problem (GVRP) consists of finding a set of routes for a number of capacitated vehicles on a graph where the vertices are partitioned into clusters with given demands, such that the total cost of travel is minimized and all demands are met. This paper describes and compares four new integer linear programming formulations for the GVRP, two based on multicommodity flow and the other two based on exponential-size sets of inequalities. Branch-and-cut algorithms are proposed for the latter two. Computational results on a large set of instances are presented.

Recent Models and Algorithms for One-to-One Pickup and Delivery Problems

In one-to-onePickup and Delivery Problems (PDPs), the aim is to design a set of least cost vehicle routes starting and ending at a common depot in order to satisfy a set of pickup and delivery requests between location pairs, subject to side constraints. Each request originates at one location and is destined for one other location. These requests apply to the transportation of goods or people, in which case the problem is often called the dial-a-ride problem. In recent years, there have been several significant developments in the area of exact and heuristic algorithms for PDPs. The purpose of this chapter is to report on these developments. It contains two main sections devoted to single vehicle and multi-vehicle problems, respectively. Each section is subdivided into two parts, one on exact algorithms and one on heuristics.

A Branch-and-Cut Algorithm for the Symmetric Two-Echelon Capacitated Vehicle Routing Problem

This paper presents an exact method for solving the symmetric two-echelon capacitated vehicle routing problem, a transportation problem concerned with the distribution of goods from a depot to a set of customers through a set of satellite locations. The presented method is based on an edge flow model that is a relaxation and provides a valid lower bound. A specialized branching scheme is employed to obtain feasible solutions. Out of a test set of 93 instances the algorithm is able to solve 47 to optimality, surpassing previous exact algorithms.