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.

In recent years new insights and algorithms have been obtained for the classical, deterministic, vehicle routing problem as well as for natural stochastic and dynamic variations of it. These new developments are based on theoretical analysis, combine probabilistic and combinatorial modelling and lead to (1) new algorithms that produce near optimal solutions and (2) a deeper understanding of uncertainty issues in vehicle routing. In this paper we survey these new developments with an emphasis on the insights gained and on the algorithms proposed. Research supported in part by ONR contract N00014-90-J-1649, NSF contracts DDM-8922712, DDM9014751, and by a Presidential Young Investigator award DDM-9158118 with matching funds from Draper Laboratory. y Sloan School of Management, MIT, Cambridge, MA 02139. z Dept. of Industrial Engineering and Operations Research, Columbia University, NY, NY, 10027 and Department of Operations Research and Management Sciences, Northwestern Universi...

We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY.

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.

The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem. The method starts by solving its linear programming relaxation, via column generation, and then uses a branch and bound strategy to find an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient. In this paper, we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases.

We develop a new local search approach based on a network flow model that is used to simultaneously evaluate several customer ejection and insertion moves. We use this approach and a direct customer swap procedure to solve the well-known Vehicle Routing Problem. The capacity constraints are relaxed using penalty terms whose parameter values are adjusted according to time and search feedback. Tabu Search is incorporated into the procedure to overcome local optimality. More advanced issues such as intensification and diversification strategies are developed to provide effective enhancements to the basic tabu search algorithm. Computational experience on standard test problems is discussed and comparisons with best-known solutions are provided.

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

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.

We consider the capacitated vehicle routing problem (VRP): a fixed fleet of delivery vehicles of uniform capacity must service at minimal transit cost from a common depot known customer demand for a single commodity. This problem is modeled as a side-constrained traveling salesman problem (TSP). Among the additional constraints are the vehicle capacity restrictions, analogous to TSP subtour elimination constraints. We present several variants of a branch-and-cut algorithm for this VRP model, with separation methodology for the capacity restrictions based on the TSP polyhedral relaxation. Specifically, when standard procedures fail to separate a candidate point, we attempt to separate it via decomposition into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity restrictions, and if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. Computational results are given for an implementation within the parallel branch-and-cut framework COMPSys; the platform is the IBM SP2 of the

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