Jacques Desrosiers

A new optimization algorithm for the vehicle routing problem with time windows

The vehicle routing problem with time windows VRPTW is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service. In this paper, we present the development of a new optimization algorithm for its solution. The LP relaxation of the set partitioning formulation of the VRPTW is solved by column generation. Feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming. The LP solution obtained generally provides an excellent lower bound that is used in a branch-and-bound algorithm to solve the integer set partitioning formulation. Our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark VRPTW test problems. The algorithm was capable of optimally solving 100-customer problems. This problem size is six times larger than any reported to date by other published research.

The pickup and delivery problem with time windows

Abstract The vehicle routing problem ( vrp ) involves the design of a set of minimum cost routes for a fleet of vehicles which services exactly once a set of customers with known demands. The pickup and delivery problem with time windows ( pdptw ) is a generalization of the vrp which is concerned with the construction of optimal routes to satisfy transportation requests, each requiring both pickup and delivery under capacity, time window and precedence constraints. This paper presents an exact algorithm which solves the pickup and delivery problem when transporting goods. This algorithm uses a column generation scheme with a constrained shortest path as a subproblem. The algorithm can handle multiple depots and different types of vehicles.

Survey Paper—Time Window Constrained Routing and Scheduling Problems

We have witnessed recently the development of a fast growing body of research focused on vehicle routing and scheduling problem structures with time window constraints. It is the aim of this paper to survey the significant advances made for the following classes of routing problems with time windows: the single and multiple traveling salesman problem, the shortest path problem, the minimum spanning tree problem, the generic vehicle routing problem, the pickup and delivery problem including the dial-a-ride problem, the multiperiod vehicle routing problem and the shoreline problem. Having surveyed the state-of-the-art in this area, we then offer some perspectives on future research.

Routing with time windows by column generation

Consider a set of trips where each trip is specified a priori by a place of origin, a destination, a duration, a cost, and a time interval within which the trip must begin. The trips may include visits to one or more specific points. Our problem is to determine the number of vehicles required, together with their routes and schedules, so that each trip begins within its given time interval, while the fixed costs related to the number of vehicles, and the travel costs between trips, are minimized. The problem is a generalization of the m-traveling salesman problem. We use column generation on a set partitioning problem solved by simplex and branch-and-bound; columns are generated by a shortest path algorithm with time windows on the nodes. Numerical results for several school bus transportation problems with up to 151 trips are discussed.

2-Path Cuts for the Vehicle Routing Problem with Time Windows

This paper introduces a strong valid inequality, the 2-path cut, to produce better lower bounds for the vehicle routing problem with time windows. It also develops an effective separation algorithm to find such inequalities. We next incorporate them as needed in the master problem of a Dantzig-Wolfe decomposition approach. In this enhanced optimization algorithm, the coupling constraints require that each customer be serviced. The subproblem is a shortest path problem with time window and capacity constraints. We apply branch and bound to obtain integer solutions. We first branch on the number of vehicles if this is fractional, and then on the flow variables. The algorithm has been implemented and tested on problems of up to 100 customers from the Solomon datasets. It has succeeded in solving to optimality several previously unsolved problems and a new 150-customer problem. In addition, the algorithm proved faster than algorithms previously considered in the literature. These computational results indicate the effectiveness of the valid inequalities we have developed.

Stabilized column generation

Abstract Column generation is often used to solve large-scale optimization problems, and much research has been devoted to improve the convergence of the solution process. We focus on Kelleys algorithm, which frequently exhibits slow convergence, and propose an algorithm that stabilizes and accelerates the solution process while remaining within the linear programming framework. Preliminary numerical results, obtained on air transportation and location problems, show that the stabilized algorithm can be used to improve the solution times for difficult instances and to solve larger ones.

A Primer in Column Generation

We give a didactic introduction to the use of the column generation technique in linear and in particular in integer programming. We touch on both, the relevant basic theory and more advanced ideas which help in solving large scale practical problems. Our discussion includes embedding Dantzig-Wolfe decomposition and Lagrangian relaxation within a branch-and-bound framework, deriving natural branching and cutting rules by means of a so-called compact formulation, and understanding and influencing the behavior of the dual variables during column generation. Most concepts are illustrated via a small example. We close with a discussion of the classical cutting stock problem and some suggestions for further reading.

A Unified Framework for Deterministic Time Constrained Vehicle Routing and Crew Scheduling Problems

Time constrained routing and scheduling is of significant importance across land, air and water transportation. These problems are also encountered in a variety of manufacturing, warehousing and service sector environments. Their mathematical complexity and the magnitude of the potential cost savings to be achieved by utilizing O.R. methodologies have attracted researchers since the early days of the field. Witness to this are the pioneering efforts of Dantzig and Fulkerson (1954), Ford and Fulkerson (1962), Appelgren (1969, 1971), Levin (1971), Madsen (1976) and Orloff (1976). Much of the methodology developed has made extensive use of network models and algorithms.

A Dynamic Programming Solution of the Large-Scale Single-Vehicle Dial-A-Ride Problem with Time Windows

SYNOPTIC ABSTRACTThe single-vehicle dial-a-ride problem with time window constraints for both pick-up and delivery locations, and precedence and capacity constraints, is solved using a forward dynamic programming algorithm. The total distance is minimized. The development of criteria for the elimination of infeasible states results in solution times which increase linearly with problem size.

Benders Decomposition for Simultaneous Aircraft Routing and Crew Scheduling

Given a set of flight legs to be flown by a single type of aircraft, the simultaneous aircraft routing and crew scheduling problem consists of determining a minimum-cost set of aircraft routes and crew pairings such that each flight leg is covered by one aircraft and one crew, and side constraints are satisfied. While some side constraints such as maximum flight time and maintenance requirements involve only crews or aircraft, linking constraints impose minimum connection times for crews that depend on aircraft connections. To handle these linking constraints, a solution approach based on Benders decomposition is proposed. The solution process iterates between a master problem that solves the aircraft routing problem, and a subproblem that solves the crew pairing problem. Because of their particular structure, both of these problems are solved by column generation. A heuristic branch-and-bound method is used to compute integer solutions. On a set of test instances based on data provided by an airline, the integrated approach produced significant cost savings in comparison with the sequential planning process commonly used in practice. The largest instance solved contains more than 500 flight legs over a 3-day period.