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...

This paper documents our investigation into various heuristic methods to solve the vehicle routing problem with time windows (VRPTW) to near optimal solutions. The objective of the VRPTW is to serve a number of customers within predefined time windows at minimum cost (in terms of distance travelled), without violating the capacity and total trip time constraints for each vehicle. Combinatorial optimisation problems of this kind are non-polynomial-hard (NP-hard) and are best solved by heuristics. The heuristics we are exploring here are mainly third-generation artificial intelligent (AI) algorithms, namely simulated annealing (SA), Tabu search (TS) and genetic algorithm (GA). Based on the original SA theory proposed by Kirkpatrick and the work by Thangiah, we update the cooling scheme and develop a fast and efficient SA heuristic. One of the variants of Glover's TS, strict Tabu, is evaluated and first used for VRPTW, with the help of both recency and frequency measures. Our GA implementation, unlike Thangiah's genetic sectoring heuristic, uses intuitive integer string representation and incorporates several new crossover operations and other advanced techniques such as hybrid hill-climbing and adaptive mutation scheme. We applied each of the heuristics developed to Solomon's 56 VRPTW 100-customer instances, and yielded 18 solutions better than or equivalent to the best solution ever published for these problems. This paper is also among the first to document the implementation of all the

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.

In this paper we formally introduce a generic real-time multi-vehicle truckload pick-up and delivery problem. The problem includes the consideration of various costs associated with trucks ’ empty travel distances, jobs ’ delayed completion times, and job rejections. Although very simple, the problem captures most features of the operational problem of a real-world trucking fleet that dynamically moves truckloads between different sites according to customer requests that arrive continuously over time. We propose a mixed integer programming formulation for the off-line version of the problem. We then consider and compare five rolling horizon strategies for the real-time version. Two of the policies are based on a repeated reoptimization of various instances of the off-line problem, while the others use simpler local (heuristic) rules. One of the re-optimization strategies is new while the other strategies have recently been tested for similar real-time fleet management problems. The comparison of the policies is done under a general simulation framework. The analysis is systematic and consider varying traffic intensities, varying degrees of advance information, and varying degrees of flexibility for job rejection decisions. The new re-optimization policy is shown to systematically outperform the others under all these conditions.

The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

We consider a pickup and delivery vehicle routing problem commonly encountered in real-world logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the driver's work rules prescribed by the Department of Transportation which specify legal working hours of ...

With greater availability of real-time information systems, algorithms are needed to support commercial fleet operators in their decisions to assign ve-hicles and drivers to loads in a dynamic environment. We present a rolling horizon framework for the dynamic assignment and sequencing of trucks to jobs consisting of picking up and delivering full truckloads when requests for 1 Submitted for presentation at the 78th Annual Meeting of the Transportation Research Board, January 1999, Washington, D.C. and publication in Transportation Research Board. 1 service arise on a continuous basis. A mathematical formulation of the problem faced at each stage is presented; its solution allows for the dynamic reassign-ment of trucks to loads, including diversion to a new load of a truck already en-route to pick up another load, as well as for the dynamic resequencing of the order in which loads are to be served, as new loads arrive and conditions unfold. Loads have associated time windows for pick up and delivery, and the objective function includes explicit penalty cost for not serving a particular load. A so-lution algorithm is presented and implemented, and computational results are presented, yielding insight into various operational trade-offs in dynamic fleet operations. Because applicability of the solution algorithm is at present limited to relatively small problems, and given the stochastic dynamic nature of these systems, numerical experiments are performed to compare the quality of the solution obtained using this approach to the performance of simpler and less computationally demanding local rules.

This paper presents a new genetic algorithm to solve Vehicle Routing Problem with Time Windows (VRPTW) to near optimal solutions. The objective of the problem is to serve a number of customers within predefined time windows at minimum cost, without violating the capacity and total trip time constraints for each vehicle. Combinatorial optimization problems of this kind are NP-hard and are best solved by heuristics[6]. Our GA implementation uses intuitive integer string representation and incorporates several new crossover operations and other techniques such as hybrid hill-climbing and adaptive mutation scheme using statistical measures, and achieve good results, especially in clustered datasets.

Neighborhood search algorithms are a wide class of improvement algorithms where at each iteration an improving solution is found by searching the “neighborhood ” of the current solution. This paper discusses neighborhood search algorithms where the size of the neighborhood is “very large” with respect to the size of the input data. For large problem instances, it is impractical to search these neighborhoods explicitly, and one must either search a small portion of the neighborhood or else develop efficient algorithms for searching the neighborhood implicitly. We concentrate on a very large scale neighborhood (VLSN) search technique based on compounding independent moves (CIM) such as 2-opts, swaps, and insertions. We demonstrate that the search for an improving neighbor can be done by finding a negative cost path on an auxiliary graph. In this paper we study CIM algorithms for the vehicle routing problem with capacity and distance constraints. We present results of the computational study which indicates that the CIM algorithms for the capacitated vehicle routing problem are competitive with the current state of the art heuristics.

This paper describes a Savings Based algorithm for the Extended Vehicle Routing Problem. This algorithm is compared with a Sequential Insertion algorithm on reallife data. Besides the traditional quality measures such as total distance traveled and total workload, we compare the routing plans of both algorithms according to nonstandard quality measures that help to evaluate the "visual attractiveness" of the plan. Computational results show that, in general, the Savings Based algorithm not only performs better with respect to these non-standard quality measures, but also with respect to the traditional measures. Keywords: distribution; road transport; vehicle routing 1 1. Introduction Ever since Dantzig and Ramser [4] first studied the vehicle routing problem (VRP) in 1959, researchers have spent a lot of time and effort on developing methods to tackle this problem. This is due to the fact that the VRP plays a central role in distribution management. In its most simple form, the pr...