Academic journal article Geographical Analysis

Aggregation Decomposition and Aggregation Guidelines for a Class of Minimax and Covering Location Models

Academic journal article Geographical Analysis

Aggregation Decomposition and Aggregation Guidelines for a Class of Minimax and Covering Location Models

Article excerpt

Facility location problems often involve movement between facilities to be located and customers/demand points, with distances between the two being important. For problems with many customers, demand point aggregation may be needed to obtain a computationally tractable model. Aggregation causes error, which should be kept small. We consider a class of minimax location models for which the aggregation may be viewed as a second-order location problem, and use error bounds as aggregation error measures. We provide easily computed approximate "square root" formulas to assist in the aggregation process. The formulas establish that the law of diminishing returns applies when doing aggregation. Our approach can also facilitate aggregation decomposition for location problems involving multiple "separate" communities.

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1. INTRODUCTION

For various sorts of analytical models in geography, there is often a question of how much detail to build into the models. The question is particularly acute for location models, since the underlying problem may involve determining the location of one or more new facilities to serve a large population. For example, if demand is generated by all private residences in a major metropolitan area, there can be hundreds of thousands of demand points. Instead of modeling such a problem with all of its detail, an alternative is to first do demand point aggregation, a process that reduces the level of detail in the model by replacing demand points by aggregate demand points. However, it is well known that this aggregation introduces error, since some of the actual data is not used. In this paper we consider a class of "minimax" location models, the best known being the p-center model, where the error can be quantified and evaluated, thus facilitating the choice of a level of modeling detail. A consequence is that we obtain a theoretical basis for the application of the law of diminishing returns when doing aggregation for these models.

We consider the law of diminishing returns to be an important practical aspect of demand point aggregation. Under certain conditions, as we increase the number of aggregate demand points, the aggregation error decreases, but at a decreasing rate. This means that a small number of aggregate demand points can give a large error, while a large number of aggregate demand points may give an error not appreciably less than a somewhat smaller number of aggregate demand points.

Francis et al. (1999) observe that much of the early recognition of aggregation error occurred in the geography literature. They point out that aggregation gives a smaller, more tractable model (one easier to solve). It can reduce the cost to obtain the data, solve the location problem, and interpret the results. It can also reduce confidentiality concerns, as well as statistical variability in the data. The price paid is that the model is less accurate. Readers are referred to this paper for discussion of such literature, as well as a general discussion of demand point aggregation, and its impact, on location modeling. One gives up accuracy in order to simplify the problem, so in principle there is a tradeoff. To evaluate this tradeoff, the error must be quantified, which is the problem we consider. Recently, Murray (2000) discusses concepts related to those we consider.

Demand point data is becoming widely available. For example, a well-known CDROM phone book for the United States gives latitude and longitude for addresses. Also the U. S. Post Office provides a Delivery Point Validation database with almost 150 million addresses for mail delivery. Demand point data is also available from various commercial organizations at a price. Thus, data availability for some location studies is no longer an issue. However, problem size and its impact on solvability remains a problem.

Let us establish a problem context and notation. We assume we have a well-defined distance function, d(x,y), used to measure distances between any of m demand points, and any new facility locations of interest. …

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