In vehicle routing problems with time windows (VRPTW), a set of vehicles with limits on capacity and travel time are available to service a set of customers with demands and earliest and latest time for servicing. The objective is to minimize the cost of servicing the set of customers without being tardy or exceeding the capacity or travel time of the vehicles. As finding a feasible solution to the problem is NP-complete, search methods based upon heuristics are most promising for problems of practical size. In this paper we describe GIDEON, a genetic algorithm heuristic for solving the VRPTW. GIDEON consists of a global customer clustering method and a local post-optimization method. The global customer clustering method uses an adaptive search strategy based upon population genetics, to assign vehicles to customers. The best solution obtained from the clustering method is improved by a local post-optimization method. The synergy a between global adaptive clustering method and a local route optimization method produce better results than those obtained by competing heuristic search methods. On a standard set of 56 VRPTW problems obtained from the literature the GIDEON system obtained 41 new best known solutions.

Abstract not found

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 Multiple Trips is an extension of the classical Vehicle Routing Problem in which each vehicle may perform several routes in the same planning period. In this paper, an adaptive memory algorithm to solve this problem is proposed. The algorithm was run over a set of benchmark problem instances, consistently finding high-quality solutions.

Abstract—When deployed for long-term tasks, the energy required to support sensor nodes ’ activities is far more than the energy that can be preloaded in their batteries. No matter how the battery energy is conserved, once the energy is used up, the network life terminates. Therefore, guaranteeing longterm energy supply has persisted as a big challenge. To address this problem, we propose a node replacement and reclamation (NRR) strategy, with which a mobile robot or human labor called mobile repairman (MR) periodically traverses the sensor network, reclaims nodes with low or no power supply, replaces them with fully-charged ones, and brings the reclaimed nodes back to an energy station for recharging. To effectively and efficiently realize the strategy, we develop an adaptive rendezvousbased two-tier scheduling (ARTS) scheme to schedule the replacement/reclamation activities of the MR and the duty cycles of nodes. Extensive simulations have been conducted to verify the effectiveness and efficiency of the ARTS scheme. I.

Abstract. We design a simple PTAS for a multi depot capacitated vehicle routing problem. In this problem a set of customers and set of depots are represented by points in the Euclidean plane. Vehicles have capacities expressed in the number of customers that can be visited on one route starting and ending in a depot. The objective is to determine a set of routes such that all customers are visited and the total length of the routes is minimized. Our results extend the results by Haimovich and Rinnooy Kan [6]. 1

In this paper, we propose a novel research scheme to solve the single vehicle pickup and delivery problem (PDPTW) with time window constraints. The family competition genetic algorithm (FCGA) is a modern approach that has been successfully applied to solve the traveling salesman problem. We illustrate the family competition GA and give the experimental results that show the FCGA is a brilliant algorithm for solving the single vehicle PDPTW. Genetic algorithms (GA) have been successful applied to solve the combinatorial computation problems. The family competition will improve the achievements for obtaining optimal solutions and the probability to hit the feasible solutions. By comparing FCGA with traditional GA, this excellent approach does not need enormous resources. Applying FCGA to single vehicle PDPTW, it succeeded in finding feasible solutions for all problems and obtained efficient results in our experimentation.

Abstract- In vehicle routing problems with time windows (VRPTW), a set of vehicles with limits on capacity and travel time are available to service a set of customers with demands and earliest and latest time for servicing. The objective is to minimize the cost of servicing the set of customers without being tardy or exceeding the capacity or travel time of the vehicles. As finding a feasible solution to the problem is NP-complete, search methods based upon heuristics are most promising for problems of practical size. In this paper we describe GIDEON, a genetic algorithm heuristic for solving the VRPTW. GIDEON consists of a global customer clustering method and a local post-optimization method. The global customer clustering method uses an adaptive search strategy based upon population genetics, to assign vehicles to customers. The best solution obtained from the clustering method is improved by a local post-optimization method. The synergy a between global adaptive clustering method and a local route optimization method produce better results than those obtained by competing heuristic search methods. On a standard set of 56 VRPTW problems obtained from the literature the GIDEON system obtained 41 new best known solutions.