We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branchand-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. Wethen discuss computational issues and implementation of column generation, branch-andbound algorithms, including special branching rules and e cient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality. 1

We use a local search method we term Large Neighbourhood Search (LNS) for solving vehicle routing problems. LNS meshes well with constraint programming technology and is analogous to the shuffling technique of job-shop scheduling. The technique explores a large neighbourhood of the current solution by selecting a number of customer visits to remove from the routing plan, and re-inserting these visits using a constraint-based tree search. We analyse the performance of LNS on a number of vehicle routing benchmark problems. Unlike related methods, we use Limited Discrepancy Search during the tree search to re-insert visits. We also maintain diversity during search by dynamically altering the number of visits to be removed, and by using a randomised choice method for selecting visits to remove. We analyse the performance of our method for various parameter settings controlling the discrepancy limit, the dynamicity of the size of the removal set, and the randomness of the choice. We demonst...

: This paper presents a probabilistic technique to diversify, intensify and parallelize a local search adapted for solving vehicle routing problems. This technique may be applied to a very wide variety of vehicle routing problems and local searches. It is shown that efficient first level taboo searches for vehicle routing problems may be significantly improved with this technique. Moreover, the solutions produced by this technique may often be improved by a post-optimization technique presented in this paper too. The solutions of nearly 40 problem instances of the literature have been improved. Key words : Vehicle routing, local searches, parallel algorithms. 1. INTRODUCTION More and more, local search methods are used to find good solutions to combinatorial optimization problems. Throughout the paper, we use the term local search as a synonym of neighbourhood search. Local searches are sometimes restricted to steepest descent algorithms but we also include taboo search and simulated...

We present DRIVE (Dynamic Routing of Independent VEhicles), a planning module to be incorporated in a decision support system for the direct transportation at Van Gend & Loos BV. Van Gend & Loos BV is the largest company providing road transportation in the Benelux with about 1400 vehicles transporting 160,000 packages from thousands of senders to tens of thousands of addressees per day. The heart of DRIVE is a branch-and-price algorithm. Approximation and incomplete optimization techniques as well as a sophisticated column management scheme have been employed to create the right balance between solution speed and solution quality. DRIVE has been tested by simulating a dynamic planning environment with real-life data and has produced very encouraging results.

The Mars system is described which models cooperative scheduling within a society of shipping companies as a multiagent system. Emphasis is placed on the functionality of the system as a whole--- the solution of the global scheduling problem emerges from local decision-making and problem-solving strategies. An extension of the contract net protocol is presented; we show that it can be used to obtain good initial solutions for complex resource allocation problems. By introducing global information based upon auction protocols, this initial solution can be improved significantly. Experimental results are provided evaluating the performance of different cooperative scheduling strategies. Although the concepts for resource scheduling are presented solely for the transportation domain, their abstraction is useful for a broad variety of resource allocation problems. The Mars system solves the dynamic scheduling problem where no complete specification of the problem is available a priori; thus, it is designed as an on-line system based upon anytime algorithms.

A multiagent approach to designing the transportation domain is presented. The Mars system is described which models cooperative order scheduling within a society of shipping companies. We argue why Distributed Artificial Intelligence (DAI) offers suitable tools to deal with the hard problems in this domain. We present three important instances for DAI techniques that proved useful in the transportation application: cooperation among the agents, task decomposition and task allocation, and decentralised planning. An extension of the contract net protocol for task decomposition and task allocation is presented; we show that it can be used to obtain good initial solutions for complex resource allocation problems. By introducing global information based upon auction protocols, this initial solution can be improved significantly. We demonstrate that the auction mechanism used for schedule optimisation can also be used for implementing dynamic replanning. Experimental results are provided ev...

Dantzig-Wolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

This paper is the second part of a work on the application of new search techniques for the vehicle routing problem with time windows. It describes GENEROUS, the GENEtic ROUting System, which is based on the natural evolution paradigm. Under this paradigm, a population of solutions evolves from one generation to the next by "mating" parent solutions to form new offspring solutions that exhibit characteristics inherited from their parents. For this vehicle routing application, a specialized methodology is devised for merging two vehicle routing solutions into a single solution that is likely to be feasible with respect to the time window constraints. Computational results on a standard set of test problems are reported, and comparisons are provided with other heuristics.

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