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.
Abstract:In interval scheduling, not only the processing times of the jobs but also their starting times are given. This article surveys the area of interval scheduling and presents proofs of results that have been known within the community for some time. We first review the complexity and approximability of different variants of interval scheduling problems. Next, we motivate the relevance of interval scheduling problems by providing an overview of applications that have appeared in literature. Finally, we focus on algorithmic results for two important variants of interval scheduling problems. In one variant we deal with nonidentical machines: instead of each machine being continuously available, there is a given interval for each machine in which it is available. In another variant, the machines are continuously available but they are ordered, and each job has a given "maximal" machine on which it can be processed. We investigate the complexity of these problems and describe algorithms for their solution.
In vehicle routing problems with time windows, a fixed fleet of vehicles of limited capacity is available at a depot to serve a set of clients with given demands. Each client must be visited within a given time window. We describe a branch-and-bound method that minimizes the total route length, and present some computational results.
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Abstract. This article analyzes a tool switching problem arising in certain flexible manufacturing environments.A batch of jobs have to be successively processed on a single flexible machine. Each job requires a subset of tools, which have to be placed in the tool magazine of the machine before the job can be processed. The tool magazine has a limited capacity, and, in general, the number of tools needed to produce all the jobs exceeds this capacity. Hence, it is sometimes necessary to change tools between two jobs in a sequence. The problem is then to determine a job sequence and an associated sequence of loadings for the tool magazine, such that the total number of tool switches is minimized. This problem has been previously considered by several authors; it is here revisited, both from a theoretical and from a computational viewpoint. Basic results concerning the computational complexity of the problem are established. Several heuristics are proposed for its solution, and their performance is computationally assessed.
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