We consider the n-period economic lot sizing problem, where the cost coefficients are not restricted in sign. In their seminal paper, H. M. Wagner and T. M. Whitin proposed an O(n2) algorithm for the special case of this problem, where the marginal production costs are equal in all periods and the unit holding costs are nonnegative. It is well known that their approach can also be used to solve the general problem, without affecting the complexity of the algorithm. In this paper, we present an algorithm to solve the economic lot sizing problem in O(n log n) time, and we show how the Wagner-Whitin case can even be solved in linear time. Our algorithm can easily be explained by a geometrical interpretation and the time bounds are obtained without the use of any complicated data structure. Furthermore, we show how Wagner and Whitin's and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem.

T his paper presents two different models and algorithms for integrated vehicle and crew scheduling in the multiple-depot case. The algorithms are both based on a combination of column generation and Lagrangian relaxation.Furthermore, we compare those integrated approaches with each other and with the traditional sequential one on randomly generated, as well as real-world, data instances for a suburban/extraurban mass transit system. To simulate such a transit system, we propose a new way of randomly generating data instances such that their properties are the same as for our real-world instances. IntroductionVehicle and crew scheduling are two main problems arising in public transport scheduling. Mostly, these problems are considered separately, where first the vehicle-scheduling problem, and afterward the crewscheduling problem, is solved. In this paper we consider the suburban/extraurban transit system with multiple depots, where we investigate the savings of using an integrated approach instead of a sequential one. It is generally expected that the savings of using an integrated approach in a suburban/extraurban transit system are much more significant than in an urban mass transit system because there is much less opportunity to relieve one driver for another one in such a way that both drivers can enjoy their break or start/finish their duty. These reliefs are only allowed at depots and certain other specified locations, which are much further away from each other than in the urban context. If first an optimal vehicle schedule is constructed, there can be vehicles that do not pass a relief location for hours. Therefore, it is very possible that a feasible crew schedule does not exist at all, or, more probably, that the crew schedule will be very inefficient.In this paper we extend the mathematical model and the solution approach that we developed for the single-depot case in Freling, Huisman, and Wagelmans (2003) to the multiple-depot setting. This solution approach is based on Lagrangian relaxation * In Memoriam: Richard Freling passed away on January 29, 2002, at the age of 34.in combination with column generation. The column generation is used to generate a set of duties, while Lagrangian relaxation is used to solve the master problem. Finally, Lagrangian heuristics are used to compute feasible solutions. Furthermore, we formulate another model that is an extension of the model for the single-depot case proposed by Haase, Desaulniers, and Desrosiers (2001), and we show the relation between both models. We also develop an algorithm for this model that is based on the same ideas as the algorithm for the first model. However, an important difference between the single-depot and multiple-depot cases is that in the latter one the underlying vehicle-scheduling problem is NP-hard (see Bertossi, Carraresi, and Gallo 1987), while in the former it can be solved in polynomial time. Of course, the underlying crew-scheduling problem is NP-hard in both cases (see Fischetti, Martello, and Toth 1989).Although a lot of a...

V ehicle scheduling is the process of assigning vehicles to a set of predetermined trips with fixed starting and ending times, while minimizing capital and operating costs. This paper considers modeling, algorithmic, and computational aspects of the polynomially solvable case in which there is a single depot and vehicles are identical. A quasiassignment formulation is reviewed and an alternative asymmetric assignment formulation is given. The main contributions of the paper are a new two-phase approach which is valid in the case of a special cost structure, an auction algorithm for the quasiassignment problem, a core-oriented approach, and an extensive computational study. New algorithms are compared with the most successful algorithms for the vehicle-scheduling problem, using both randomly generated and real-life data. The new algorithms show a significant performance improvement with respect to computation time. Such improvement can, for example, be very important when this particular vehicle-scheduling problem appears as a subproblem in more complex vehicle-and crew-scheduling problems.The single-depot vehicle-scheduling problem (SDVSP) is defined as follows: Given a depot and a set of trips with fixed starting and ending times, and given the travelling times between all pairs of locations, find a feasible minimum-cost schedule such that (1) each trip is assigned to a vehicle, and (2) each vehicle performs a feasible sequence of trips. All the vehicles are supposed to be identical. A schedule for a vehicle is composed of vehicle blocks, where each block is a departure from the depot, the service of a feasible sequence of trips, and the return to the depot. The cost function is usually a combination of vehicle capital (fixed) and operational (variable) cost. The SDVSP is well known to be solvable in polynomial time.Overviews of algorithms and applications for the SDVSP and some of its extensions can be found in Daduna and Paixão (1995) and in Desrosiers et al. (1995). Several network-flow-type algorithms have been proposed for the SDVSP in the literature. In particular, the SDVSP has been formulated as a linear assignment problem, a transportation problem, a minimum-cost flow problem, a quasiassignment problem, and a matching problem. We briefly discuss the most relevant algorithms. Let n be the number of trips to be covered by vehicles. Paixão and Branco (1987) propose an O n 3 quasiassignment algorithm which is specially designed for the SDVSP. Computational results indicate that this approach significantly outperforms approaches based on transportation and linear-assignment models. The quasiassignment algorithm is a modified version of the Hungarian algorithm for the linear assignment problem, including the improvements proposed by Jonker and Volgenant (1986). An extension of this algorithm can also deal with a fixed number of vehicles (see Paixão and Branco 1988). Dell'Amico (1989) (see also Dell'Amico et al. 1993) proposes an O n 3 successive shortest-path algorithm for the SDVSP, which uses the initializ...

Abstract:In this paper, the problem of integrating vehicle and crew scheduling is considered. Traditionally, vehicle and crew scheduling have been dealt with in a sequential manner, where vehicle schedules are determined before the crew schedules. The few papers that have appeared in the literature have in common that no comparison is made between simultaneous and sequential scheduling, so there is no indication of the benefit of a simultaneous approach. In order to get such an indication before even solving the integrated problem, we propose a method to solve crew scheduling independently of vehicle scheduling. We introduce a mathematical formulation for the integrated problem, and briefly outline algorithms. The paper concludes with computational results for an application to bus scheduling at the public transport company RET in Rotterdam, The Netherlands. The results show that the proposed techniques are applicable In practice. Furthermore, we conclude that the effectiveness of integration as compared to a sequential approach is mainly dependent on the flexibility in changing buses during a duty.

B arter exchange markets are markets in which agents seek to directly trade their goods with each other.Exchanges occur in cycles or in chains in which each agent gives a good to the next agent. Kidney exchange is an important type of barter exchange market that allows incompatible patient-donor pairs to exchange kidneys so the involved patients can receive a transplant. The clearing problem is to find an allocation of donors to patients that is optimal with respect to multiple criteria. To achieve the best possible score on all criteria, long cycles and chains are often needed, particularly when there are many hard-to-match patients. In this paper we show why this may pose difficulties for existing approaches to the optimization of kidney exchanges. We then present a generic iterative branch-and-price algorithm that can deal effectively with multiple criteria, and we show how the pricing problem may be solved in polynomial time for a general class of criteria. Our algorithm is effective even for large, realistic patient-donor pools. Our approach and its effects are demonstrated by using simulations with kidney exchange data from the Netherlands and the United States.