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The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely di cult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is 1 8. An exact algorithm for lling a single bin is developed, leading to the de nition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: it is shown that many instances can be solved to optimality within a reasonable time limit.

The aim of this paper is to present a survey of recent optimization models for the most commonly studied rail transportation problems. For each group of problems, we propose a classification of models and describe their important characteristics by focusing on model structure and algorithmic aspects. The review mainly concentrates on routing and scheduling problems since they represent the most important portion of the planning activities performed by railways. Routing models surveyed concern the operating policies for freight transportation and railcar fleet management, whereas scheduling models address the dispatching of trains and the assignment of locomotives and cars. A brief discussion of analytical yard and line models is also presented. The emphasis is on recent contributions, but several older yet important works are also cited.The rail transportation industry is very rich in terms of problems that can be modeled and solved using mathematical optimization techniques. However, the related literature has experienced a slow growth and, until recently, most contributions were dealing with simplified models or small instances failing to incorporate the characteristics of real-life applications. Previous surveys by ASSAD (1980bASSAD ( , 1981 and HAGHANI (1987) suggest that optimization models for rail transportation were not widely used in practice and that carriers often resorted to simulation. This situation is somewhat surprising given the considerable potential savings and performance improvements that may be realized through better resource utilization. It is also contrasting with the rapid penetration of optimization methods in other fields such as air transportation (YU, 1998).In fact, the development of optimization models for train routing and scheduling was for a long time hindered by the large size and the high difficulty of the problems studied. Important computing capabilities were needed to solve the proposed models, and even the task of collecting and organizing the relevant data required installations that very few railroads could afford. As a result, practical implementations of optimization models often had a limited success, which deterred both researchers and practitioners from pursuing the effort.In the last decade however, a growing body of advances concerning several aspects of rail freight and passenger transportation has appeared in the operations research literature. The strong competition facing rail carriers, the privatization of many national railroads, deregulation, and the ever increasing speed of computers all motivate the use of optimization models at various levels in the organization. In addition, recently proposed models tend to exhibit an increased level of realism and to incorporate a larger variety of constraints and possibilities. In turn, this convergence of theoretical and practical standpoints results in a growing interest for optimization techniques. Hence, although simulationbased approaches are still widely used to evaluate and compare different scenarios, one witn...

We describe a new variant, called granular tabu search, of the well-known tabu-search approach. The method uses an effective intensification/diversification tool that can be successfully applied to a wide class of graph-theoretic and combinatorial-optimization problems. Granular tabu search is based on the use of drastically restricted neighborhoods, not containing the moves that involve only elements that are not likely to belong to good feasible solutions. These restricted neighborhoods are called granular, and may be seen as an efficient implementation of candidate-list strategies proposed for tabu-search algorithms. Results of computational testing of the proposed approach on the well-known symmetric capacitated and distance-constrained vehicle-routing problem are discussed, showing that the approach is able to determine very good solutions within short computing times.

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