This article describes a tabu search heuristic for the dial-a-ride problem with the following characteristics. Users specify transportation requests between origins and destinations. They may provide a time window on their desired departure or arrival time. Transportation is supplied by a fleet of vehicles based at a common depot. The aim is to design a set of least cost vehicle routes capable of accommodating all requests. Side constraints relate to vehicle capacity, route duration and the maximum ride time of any user. Extensive computational results are reported on randomly generated and real-life data sets.

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The Dial-a-Ride Problem (DARP) consists of designing vehicle routes and schedules for n users who specify pickup and delivery requests between origins and destinations. The aim is to plan a set of m minimum cost vehicle routes capable of accommodating as many users as possible, under a set of constraints. The most common example arises in door-to-door transportation for elderly or disabled people. The purpose of this article is to review the scientific literature on the DARP. The main features of the problem are described and a summary of the most important models and algorithms is provided.

In the pickup and delivery problem with time windows (PDPTW), vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and a delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and a non-elementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the PDPTW are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

In the Berth Allocation Problem (BAP) the aim is to optimally schedule and assign ships to berthing areas along a quay. The objective is the minimization of the total (weighted) service time for all ships, defined as the time elapsed between the arrival in the harbor and the completion of handling. Two versions of the BAP are considered: the discrete case and the continuous case. The discrete case works with a finite set of berthing points. In the continuous case ships can berth anywhere along the quay. Two formulations and a tabu search heuristic are presented for the discrete case. Only small instances can be solved optimally. For these sizes the heuristic always yields an optimal solution. For larger sizes it is always better than a truncated branch-and-bound applied to an exact formulation. A heuristic is also developed for the continuous case. Computational comparisons are performed with the first heuristic and with a simple constructive procedure.

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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...

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