Martin Desrochers

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.

Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints

This paper shows how the subtour elimination constraints developed by Miller, Tucker and Zemlin for the traveling salesman problem can be improved and extended to various types of vehicle routing problems. (A)

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.

A Column Generation Approach to the Urban Transit Crew Scheduling Problem

The urban transit crew scheduling problem arises in mass transit organizations which have to create minimal cost bus driver schedules respecting both the collective agreement with labor unions and the bus schedule. We propose a column generation approach to solve the transit crew scheduling problem. The column generation approach decomposes the problem into two parts. The set covering problem chooses a schedule from already known feasible workdays. The second subproblem is a shortest path problem with resource constraints and is used to propose new feasible workdays to improve the current solution of the set covering problem. The approach has been successfully tested on real-life problems.

A Generalized Permanent Labelling Algorithm For The Shortest Path Problem With Time Windows

AbstractThe shortest path problem with time windows (SPPTW) consists of finding the least cost route between a source and a sink in a network G = (N, A) while respecting specified time windows [ai, bi] at each visited node. The duration dij of each arc is restricted to positive values while the cost Cij of each arc (i, j) Є A is unrestricted.This article presents an efficient generalized permanent labelling algorithm to solve this problem. This new algorithm is based on the definition of the concept of a generalized bucket and on a specific order of handling the labels. The algorithm runs in pseudo-polynomial time. Problems with up to 2500 nodes and 250,000 arcs have heen solved.

Optimal Routing under Capacity and Distance Restrictions

This paper describes an integer linear programming algorithm for vehicle routing problems involving capacity and distance constraints. The method uses constraint relaxation and a new class of subtour elimination constraints. Two versions of the algorithm are presented, depending upon the nature of the distance matrix. Exact solutions are obtained for problems involving up to sixty cities.

A NEW BRANCHING STRATEGY FOR TIME CONSTRAINED ROUTING PROBLEMS WITH APPLICATION TO BACKHAULING

In this paper, we explore a new branching strategy for branch-and-bound approaches based on column generation for the vehicle routing problems with time windows. This strategy involves branching on resource variables (time or capacity) rather than on network flow variables. We also examine criteria for selecting network nodes for branching. To test the effectiveness of the branching strategy, we conduct computational experiments on time window constrained vehicle routing problems where backhauling is permitted only after all the shipments to clients have been made. The branching method proved very effective. In cases where time was the more binding constraint, time-based branching succeeded in decreasing the number of nodes explored by two thirds and the total computation time by more than half when compared to flow-based branching. The computational results also show that the overall algorithm was successful in optimally solving problems with up to 100 customers. It produced an average cost decrease of almost 7% when backhauling was permitted as compared to the cost involved when the client and the distributor routes were distinct.

Two exact algorithms for the distance-constrained vehicle routing problem

This paper considers a version of the vehicle routing problem in which all vehicles are identical and where the distance travelled by any vehicle may not exceed A prespecified upper bound. The problem is first formulated as an integer program which is solved by means of a constraint relaxation procedure. Two exact algorithms are developed: one based on Gomory cutting planes and one on branch and bound. Numerical results are reported. (Author/TRRL)

On the Distance Constrained Vehicle Routing Problem

We analyze the vehicle routing problem with constraints on the total distance traveled by each vehicle. Two objective functions are considered: minimize the total distance traveled by vehicles and minimize the number of vehicles used. We demonstrate a close relationship between the optimal solutions for the two objective functions and perform a worst case analysis for a class of heuristics. We present a heuristic that provides a good worst case result when the number of vehicles used is relatively small.

A reoptimization algorithm for the shortest path problem with time windows

Abstract The shortest path problem with time windows (SPPTW) occurs in the construction of vehicle routes and schedules for which time window constraints must be satisfied. The SPPTW is repeatedly solved to produce routes with disjoint schedules which form the solution of the original problem. The cost of repeatedly solving the SPPTW can be reduced by reusing part of the solution of the preceding problem. A primal-dual reoptimization method which runs in pseudo-polynomial time is proposed. It can solve problems with up to 2500 nodes.