We present a survey of over 50 representative problems in location research. Our goal is not to review all variants of different location models or to describe solution results, but rather to provide a broad overview of major location problems that have been studied, indicating briefly how they are formulated and how they relate to one another. We review standard problems such as median, center, and warehouse location problems, as well as less traditional location problems which have emerged in recent years. Our primary focus is on problems for which operations research-type models have been developed. Most of the problems we review have been formulated as optimization problems.location theory

This paper considers the problem of determining optimal paths for routing an undesirable vehicle on a network embedded on an Euclidean plane. A typical application is the transporting of hazardous materials. Demand points, or population centers, are discretely distributed at nodes and continuously distributed on straight-line links of the network. The objective is to find, without incorporating the probability of accidental leakage of hazardous material, a path that minimizes the weighted sum of lengths over which this vehicle is within a threshold distance λ of population centers. By appropriately redefining link lengths, we can use a shortest-path algorithm to solve the problem. Special properties of the objective function allow us to efficiently calculate the modified link lengths. We discuss the properties of the optimal routing strategy and the objective value, and offer an economic interpretation of the case when λ varies. Operating within the framework of risk analysis, we integrate the probability of accidental release of hazardous material into our basic model. The routing objective becomes the minimization of expected damage, where accidental leakage of hazardous materials can inflict damage within a neighborhood λ of the accident site. We also discuss our computational experience in using the model.

This paper extends Hakimi's one-median problem by embedding it in a general queueing context. Demands for service that arise solely on the nodes of a network G occur in time as a Poisson process. A single mobile server resides at a facility located on G. The server, when available, is dispatched immediately to any demand that occurs. When a demand finds the server busy with a previous demand, it is either rejected (model 1) or entered into a queue that is depleted in a firstcome, first-served manner (model 2). It is desired to locate the facility on G so as to minimize average cost of response, which is either a weighted sum of mean travel time and cost of rejection (model 1) or the sum of mean queueing delay and mean travel time. For model 1, one finds that the optimal location reduces to Hakimi's familiar nodal result. For model 2, nonlinearities in the objective function can yield an optimal solution that is either at a node or on a link. Properties of the objective function for model 2 are utilized to develop efficient finite-step procedures for finding the optimal location. Ever since Hakimi's work in 1964 [ 1] and 1965,[2] there has been considerable interest in the problem of optimally locating one or more facilities on a network.

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This paper develops an efficient iterative algorithm to calculate the steady-state distribution of nearly all irreducible discrete-time Markov chains. Computational experiences suggest that, for large Markovian systems (more than 130 states), the proposed algorithm can be ten times faster than standard Gaussian elimination in finding solutions to an accuracy of 0.1%. The proposed algorithm is developed in three stages. First, we develop a very efficient algorithm for determining steady-state distributions of a restricted class of Markovian systems. A second result establishes a relationship between a general irreducible Markovian system and a system in the restricted class of Markovian systems. Finally, we combine the two results to produce an efficient, iterative algorithm to solve Markov systems. The paper concludes with a discussion of the observed performance of the algorithm.