Antoon W.J. Kolen

Economic lot sizing: an O ( n log n ) algorithm that runs in linear time in the Wagner-Whitin case

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 Whitins and our algorithm are related to algorithms that solve the dual of the simple plant location formulation of the economic lot sizing problem.

Minimizing the number of tool switches on a flexible machine

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.

The partial constraint satisfaction problem: Facets and lifting theorems

We study the polyhedral structure of the partial constraint satisfaction problem (PCSP). Among the problems that can be formulated as such are the maximum satisfiability problem and a fairly general model of frequency assignment problems. We present lifting theorems and classes of facet defining inequalities, and we provide preliminary experiments.

Genetic local search in combinatorial optimization

Abstract The most common application of genetic algorithms to combinatorial optimization problems has been restricted to the traveling salesman problem. We review some of these ideas and present some new results, especially in the case that severe time constraints are imposed on the running time of the algorithm.

Solving partial constraint satisfaction problems with tree decomposition

In this paper, we describe a computational study to solve hard partial constraint satisfaction problems (PCSPs) to optimality. The PCSP is a general class of problems that contains a diversity of problems, such as generalized subgraph problems, MAX-SAT, Boolean quadratic programs, and assignment problems like coloring and frequency planning. We present a dynamic programming algorithm that solves PCSPs based on the structure (tree decomposition) of the underlying constraint graph. With the use of dominance and bounding techniques, we are able to solve small and medium-size instances of the problem to optimality and to obtain good lower bounds for large-size instances within reasonable time and memory limits.

Finding efficient solutions for rectilinear distance location problems efficiently

Abstract Given n planar existing facility locations, a planar new facility location X is called efficient if there is no other location Y for which the rectilinear distance between Y and each existing facility is at least as small as between X and each existing facility, and strictly less for at least one existing facility. Rectilinear distances are typically used to measure travel distances between points via rectilinear aisles or street networks. We first present a simple arrow algorithm, based entirely on geometrical analysis, that constructs all efficient locations. We then present a row algorithm which is of order n (log n ) that constructs all efficient locations, and establish that no alternative algorithm can be of a lower order.

A linear description of the discrete lot-sizing and scheduling problem

A new integer linear programming formulation for the discrete lot-sizing and scheduling problem is presented. This polynomial-size formulation is obtained from the model with the natural variables by splitting these variables. Its linear programming relaxation is shown to be tight, by reformulating it as a shortest path problem. The latter also provides a dynamic programming formulation for the discrete lot-sizing and scheduling problem.

A dynamic programming algorithm for the local access telecommunication network expansion problem

In this paper we consider the local access telecommunication network expansion problem, in which growing demand can be satisfied by expanding cable capacities and/or installing concentrators in the network. The problem is known to be NP-hard. We prove that the problem is weakly NP-hard, and present a pseudo-polynomial dynamic programming algorithm for the problem, with time complexity OOnB 2 U and storage requirements OOnBU, where n refers to the size of the network, and B to an upper bound on concentrator capacity. The cost structure in the network is assumed to be decomposable, but may be non-convex, non-concave, and node and edge dependent otherwise. This allows for incorporation of many aspects occurring in practical planning problems. Computational results indicate that the algorithm is very eAcient and can solve medium to large scale problems to optimality within (fractions of) seconds to minutes. ” 2000 Elsevier Science B.V. All rights reserved.

Sensitivity analysis of list scheduling heuristics

When jobs have to be processed on a set of identical parallel machines so as to minimize the makespan of the schedule, list scheduling rules form a popular class of heuristics. The order in which jobs appear on the list is assumed here to be determined by the relative size of their processing times; well-known special cases are the LPT rule and the SPT rule, in which the jobs are ordered according to non-increasing and non-decreasing processing time respectively. When all processing times are exactly known, a given list scheduling rule will generate a unique assignment of jobs to machines. However, when there exists a priori uncertainty with respect to one of the processing times, then there will be, in general, several possibilities for the assignment that will be generated once the processing time is known. This number of possible assignments may be viewed as a measure of the sensitivity of the list scheduling rule that is applied. We derive bounds on the maximum number of possible assignments for several list scheduling heuristics, and we also study the makespan associated with these assignments. In this way we obtain analytical support for the intuitively plausible notion that the sensitivity of a list scheduling rule increases with the quality of the schedule produced.

Local Search in Combinatorial Optimization

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