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 purpose of this paper is to present a new deterministic metaheuristic based on a modification of Variable Neighborhood Search of Mladenovic and Hansen (1997) for solving the vehicle routing problem with time windows. Results are reported for the standard 100, 200 and 400 customer data sets by Solomon (1987) and Gehring and Homberger (1999) and two real-life problems by Russell (1995). The findings indicate that the proposed procedure outperforms other recent local searches and metaheuristics. In addition four new best-known solutions were obtained. The proposed procedure is based on a new four-phase approach. In this approach an initial solution is first created using new route construction heuristics followed by route elimination procedure to improve the solutions regarding the number of vehicles. In the third phase the solutions are improved in terms of total traveled distance using four new local search procedures proposed in this paper. Finally in phase four the best solution obtained is improved by modifying the objective function to escape from a local minimum. (Metaheuristics; Vehicle Routing; Time Windows) 1.

We present a state-of-the-art survey of parallel meta-heuristic developments and results, discuss general design and implementation principles that apply to most meta-heuristic classes, instantiate these principles for the three meta-heuristic classes currently most extensively used - genetic methods, simulated annealing, and tabu search, and identify a number of trends and promising research directions.

Abstract. Parallel implementationsof metaheuristicsappear quite naturally asan effective alternative to speed up the search for approximate solutions of combinatorial optimization problems. They not only allow solving larger problems or finding improved solutions with respect to their sequential counterparts, but they also lead to more robust algorithms. We review some trends in parallel computing and report recent results about linear speedups that can be obtained with parallel implementations using multiple independent processors. Parallel implementations of tabu search, GRASP, genetic algorithms, simulated annealing, and ant colonies are reviewed and discussed to illustrate the main strategies used in the parallelization of different metaheuristics and their hybrids. 1. Introduction. Although

This paper is concerned with real-time decision problems. These constitute a generic class of dynamic and stochastic problems. The objective is to provide responses of a required quality in a continuously evolving environment, within a prescribed time frame, using limited resources and information that is often incomplete or uncertain. Furthermore, the outcome of any particular decision may also be uncertain. This paper provides an overview of this class of problems, reviews the relevant Artificial Intelligence literature, proposes a dynamic programming framework, and assesses the potential usefulness of Operations Research approaches for their solution. Throughout the paper, a vehicle dispatching application illustrates the relevant concepts.

Metaheuristic parallel search methods -- tabu search, simulated annealing and genetic algorithms, essentially -- are reviewed, classified and examined not according to particular methodological characteristics, but following the unifying approach of the level of parallelization. It is hoped that by examining the commonalities among parallel implementations across the field of metaheuristics, insights may be gained, trends may be discovered, and research challenges may be identified. Particular attention is paid to applications of parallel metaheuristics to transportation problems.

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This report addresses the issues arising from the use of parallel machines and considers the various techniques used by members of the consortium in this context. Before considering the algorithms in detail, we describe, in section 2, the main types of parallel architecture and survey various attempts at providing a taxonomy. Then, in section 3, we address the difficult issue of the measurement of processor performance in order to quantify any enhancement obtained by implementing an algorithm in parallel. Section 4 presents the main features of in general, and PVM ( ) in particular. The latter is an application that is used to generate distributed versions of sequential algorithms for use on networks of workstations. The parallel implementations of the GA toolkit, , and the associated simulated annealing toolkit, , both developed at UEA, have been produced using PVM. Exact algorithms will always find the optimal solution to a problem given enough time and space. Subject to these constraints, they must always be the preferred method of solution. In practice, the time and space constraints can prevent the use of an exact algorithm and thus the potential of parallelism to reduce these factors becomes an important factor. Total enumeration is embarassingly parallel. With processors it is reasonable to expect an-fold reduction in time to undertake such a thorough search. Such a saving is seldom sufficient to make the method viable so we will concentrate on other exact methods here. In section 5, we review parallel branchand -bound, reprinting a survey paper written by the UEA partners in the consortium and previously published in [1]. Because of the interest in interior point methods for the CALMA project and its widely cited potential for parallelisation, this provides the ...

... This paper presents a survey of the latest research motivated by this recognition. The presentation is focused on multi-stage applications of advanced local search techniques on the VRPTW. Multi-stage algorithms optimize the number of vehicles and travel time independently in order to ensure that the search is directed towards the achievement of the primary objective. Basic features of these algorithms, as well as hybridization strategies are described. For most algorithms, experimental results on Solomon's benchmark test problems are provided and analyzed

Human-Guided Simple Search (HuGSS) allows a human user to guide a combinatorial optimization algorithm (COA). We apply HuGSS to capacitated vehicle routing with time windows (CVRTW)|a common problem dealing with scheduling multiple deliveries using vehicle routing. We study whether arti cial intelligence heuristics (computerized algorithms) can replace the human user and which heuristic performs the best and why. We program the heuristics in C++, integrate them with the COA and analyze their performance by running experiments. The heuristics do not perform on par with human users in HuGSS. HuGSS performs signi cantly better than the heuristics at p = 0:025. This is an important nding, as it ensures that human users are actually doing something "intelligent." Greedy Random (GR) is the top-performing heuristic, signi cantly better than all other heuristics at p =0:025. This heuristic is the rst to use infeasible space (temporary use of invalid solutions) as an optimization technique; GR nds 73.1 % of its improvements from an initial infeasible move. GR performs well because of factors other than infeasibility alone. Variants of GR that only make feasible moves do not have signi cantly di erent performance at a 95 % con dence level. These heuristics can be integrated into HuGSS itself, placing higher-level tools at the disposal of human users. The results of this research can potentially provide signi cant savings for industry.