In pickup and delivery problems vehicles have to transport loads from origins to destinations without transshipment atintermediate locations. In this paper, we discuss several characteristics that distinguish them from standard vehicle routing problems and present a survey of the problem types and solution methods found in the literature.

We present HAS-SOP, a new approach to solving sequential ordering problems. HAS-SOP combines the ant colony algorithm, a population-based metaheuristic, with a new local optimizer, an extension of a TSP heuristic which directly handles multiple constraints without increasing computational complexity. We compare different implementations of HASSOP and present a new data structure that improves system performance. Experimental results on a set of twenty-three test problems taken from the TSPLIB show that HAS-SOP outperforms existing methods both in terms of solution quality and computation time. Moreover, HAS-SOP improves most of the best known results for the considered problems.

We present a new local optimizer called SOP-3-exchange for the sequential ordering problem that extends a local search for the traveling salesman problem to handle multiple constraints directly without increasing computational complexity. An algorithm that combines the SOP-3-exchange with an Ant Colony Optimization algorithm is described and we present experimental evidence that the resulting algorithm is more effective than existing methods for the problem. The best-known results for many of a standard test set of 22 problems are improved using the SOP-3-exchange with our Ant Colony Optimization algorithm or in combination with the MPO/AI algorithm (Chen and Smith 1996).

Routing with time windows (VRPTW) has been an area of research that have attracted many researchers within the last 10 { 15 years. In this period a number of papers and technical reports have been published on the exact solution of the VRPTW. The VRPTW is a generalization of the well-known capacitated routing problem (VRP or CVRP). In the VRP a eet of vehicles must visit (service) a number of customers. All vehicles start and end at the depot. For each pair of customers or customer and depot there is a cost. The cost denotes how much is costs a vehicle to drive from one customer to another. Every customer must be visited exactly ones. Additionally each customer demands a certain quantity of goods delivered (know as the customer demand). For the vehicles we have an upper limit on the amount of goods that can be carried (known as the capacity). In the most basic case all vehicles are of the same type and hence have the same capacity. The problem is now for a given scenario to plan routes for the vehicles in accordance with the mentioned constraints such that the cost accumulated on the routes, the xed costs (how much does it cost to maintain a vehicle) or a combination hereof is minimized. In the more general VRPTW each customer has a time window, and between all pairs of customers or a customer and the depot we have a travel time. The vehicles now have to comply with the additional constraint that servicing of the customers can only be started within the time windows of the customers. It is legal to arrive before a time window \opens" but the vehicle must wait and service will not start until the time window of the customer actually opens. For solving the problem exactly 4 general types of solution methods have evolved in the literature: dynamic programming, Dantzig-Wolfe (column generation), Lagrange decomposition and solving the classical model formulation directly. Presently the algorithms that uses Dantzig-Wolfe given the best results (Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis shows promising results for using the classical model formulation directly. In this Ph.D. project we have used the Dantzig-Wolfe method. In the Dantzig-Wolfe method the problem is split into two problems: a \master problem" and a \subproblem". The master problem is a relaxed set partitioning v vi problem that guarantees that each customer is visited exactly ones, while the subproblem is a shortest path problem with additional constraints (capacity and time window). Using the master problem the reduced costs are computed for each arc, and these costs are then used in the subproblem in order to generate routes from the depot and back to the depot again. The best (improving) routes are then returned to the master problem and entered into the relaxed set partitioning problem. As the set partitioning problem is relaxed by removing the integer constraints the solution is seldomly integral therefore the Dantzig-Wolfe method is embedded in a separation-based solution-technique. In this Ph.D. project we have been trying to exploit structural properties in order to speed up execution times, and we have been using parallel computers to be able to solve problems faster or solve larger problems. The thesis starts with a review of previous work within the eld of VRPTW both with respect to heuristic solution methods and exact (optimal) methods. Through a series of experimental tests we seek to dene and examine a number of structural characteristics. The rst series of tests examine the use of dividing time windows as the branching principle in the separation-based solution-technique. Instead of using the methods previously described in the literature for dividing a problem into smaller problems we use a methods developed for a variant of the VRPTW. The results are unfortunately not positive. Instead of dividing a problem into two smaller problems and try to solve these we can try to get an integer solution without having to branch. A cut is an inequality that separates the (non-integral) optimal solution from all the integer solutions. By nding and inserting cuts we can try to avoid branching. For the VRPTW Kohl has developed the 2-path cuts. In the separationalgorithm for detecting 2-path cuts a number of test are made. By structuring the order in which we try to generate cuts we achieved very positive results. In the Dantzig-Wolfe process a large number of columns may be generated, but a signicant fraction of the columns introduced will not be interesting with respect to the master problem. It is a priori not possible to determine which columns are attractive and which are not, but if a column does not become part of the basis of the relaxed set partitioning problem we consider it to be of no benet for the solution process. These columns are subsequently removed from the master problem. Experiments demonstrate a signicant cut of the running time. Positive results were also achieved by stopping the route-generation process prematurely in the case of time-consuming shortest path computations. Often this leads to stopping the shortest path subroutine in cases where the information (from the dual variables) leads to \bad" routes. The premature exit from the shortest path subroutine restricts the generation of \bad" routes signi cantly. This produces very good results and has made it possible to solve problem instances not solved to optimality before. The parallel algorithm is based upon the sequential Dantzig-Wolfe based algorithm developed earlier in the project. In an initial (sequential) phase unsolved problems are generated and when there are unsolved problems enough vii to start work on every processor the parallel solution phase is initiated. In the parallel phase each processor runs the sequential algorithm. To get a good workload a strategy based on balancing the load between neighbouring processors is implemented. The resulting algorithm is eÆcient and capable of attaining good speedup values. The loadbalancing strategy shows an even distribution of work among the processors. Due to the large demand for using the IBM SP2 parallel computer at UNIC it has unfortunately not be possible to run as many tests as we would have liked. We have although managed to solve one problem not solved before using our parallel algorithm.

Département d’informatique et de recherche opérationnelle and Centre de recherche sur les transports, Université de Montréal

We present an improvement heuristic for vehicle routing problems. The heuristic finds complex customer interchanges to improve an initial solution. Our approach is modular, thus it is easily adjusted to different side constraints such as time windows, backhauls and a heterogeneous vehicle fleet. The algorithm is well suited for parallelization. We report on a parallel implementation of the Simulated Trading heuristic on a cluster of workstations using PVM. The computational results were obtained using two sets of vehicle routing problems which differ in the presence of time windows. Our results show that Simulated Trading is better suited for problems with time windows.

Pickup and delivery problems constitute an important class of vehicle routing problems in which objects or people have to be collected and distributed. This paper introduces a general framework to model a large collection of pickup and delivery problems, as well as a three-field classification scheme for these problems. It surveys the methods used for solving them.

this article basic principles for implementing local search techniques and metaheuristics in Constraint Programming are presented. These principles have been applied to Tabu Search, with experimental results on various symmetric TSPs and various VRPs (with capacities, time windows, deadlines). This article is organized as follows: After describing the types of VRP considered here, a general Constraint Programming model for the VRP is given in section 3 that allows the description of various routing problems, ranging from TSPs to PDPs. In particular, this model allows to take into account various side constraints expressed as standard Constraint Programming constraints. The solving method is then described in section 4, which consists in an adaptation of the standard two-phase approach. A feasible solution is first computed using a basic construction heuristic. It is subsquently improved using local search. A way to implement local search in Constraint Programming is thus given, along with some algorithmic improvements specific to the case of the VRP. Section 5 and 6 are devoted to the main choices that were made with respect to the Tabu Search metaheuristic. Finally, experimental results are given in Section 7 that show that performance of the proposed approach. Sophia-Antipolis, France, July 21-24, 1997 INRIA & PRiSM-Versailles 2 2ND INTERNATIONAL CONFERENCE ON METAHEURISTICS - MIC97 2 Vehicle Routing Problems

. Constraint programming is an appealing technology to use for vehicle routing problems. Traditional linear programming models do not have the flexibility or generality required by businesses wishing to model complex side constraints. This paper describes how a constraint programming framework for vehicle routing problems was implemented using ILOG Solver. A method for incorporating local search into a monotonic constraint programming toolkit is then described. Techniques for accelerating the method via filtering processes are also presented. Computational tests using the constraint programming framework and a greedy search method were performed. Results indicate that moderately sized VRPs (100--200 nodes) can be solved to within 10% of the best known solution in seconds. Keywords: Constraint Programming, Combinatorial problems, Iterative Improvement Techniques, Vehicle Routing Problem. 1 Introduction Problems such as the Traveling Salesman Problem (TSP) or the Vehicle Routing Proble...

Introduction Ant Colony Optimization (ACO) is a paradigm for designing metaheuristic algorithms for combinatorial optimization problems. The first algorithm which can be classified within this framework was presented in 1991 [21 , 13] and, since then, many diverse variants of the basic principle have been reported in the literature. The essential trait of ACO algorithms is the combination of a priori information about the structure of a promising solution with a posteriori information about the structure of previously obtained good solutions. Metaheuristic algorithms are algorithms which, in order to escape from local optima, drive some basic heuristic: either a constructive heuristic starting from a null solution and adding elements to build a good complete one, or a local search heuristic starting from a complete solution and iteratively modifying some of its elements in order to achieve a better one. The metaheuristic part permits the lowlevel heuristic to obtain solutions better