A bibliography for some fundamental problem categories in discrete location science

Abstract

Following a brief taxonomy of the broad field of facility location modeling, this paper provides an annotated bibliography of recent papers in two branches of discrete location theory and modeling. In particular, we review papers related to (1) the median and plant location models and (2) to center and covering models. We show how the contributions of the papers we review are embedded in the field. A summary and outlook conclude the paper.

Introduction

Location models have been studied in various forms for hundreds of years. Even though the contexts in which these models are situated may differ, their main features are always the same: a space including a metric, customers whose locations in the given space are known, and facilities whose locations have to be determined according to some objective function. For more detailed and systematic introductions to the field, see Daskin, 1995, Eiselt and Sandblom, 2004, ReVelle and Eiselt, 2005. Many classification schemes of the field exist, most use the decision maker’s objective or the space of the model as their main criterion and this paper is no exception. We divide the contributions in the literature on location modeling into four broad categories:

• Analytic models are based on a large number of simplifying assumptions. For example, a typical analytic model assumes that demands are uniformly distributed with density ρ over the service region of size a. Models also often assume that the fixed cost of locating a facility is f, no matter where in the service region the facility is located and the cost per unit shipment per unit distance is c. Analytic models admit closed-form expressions for the total cost, typically as a function of the total number of facilities being located. For example, under the assumptions above and with the additional assumption that travel distances follow the Manhattan metric, the total cost associated with using n facilities is $\mathit{fn}+\frac{\rho \mathit{ca}}{3}\sqrt{\frac{2a}{n}}$, which is minimized for $n=a{\left(\frac{c\rho \sqrt{2}}{6f}\right)}^{2/3}$. At this point, the total cost is approximately 1.1447${\mathit{Af}}^{1/3}(c\rho {)}^{2/3}$. While such models provide valuable insight into the relationship between the optimal total cost and number of facilities on the one hand and the key input parameters on the other hand, the stringent assumptions made by this class of models make them of limited value for decision-making for practical location problems. The reader is referred to Leamer, 1968, Geoffrion, 1979 for a discussion of this class of models.

• Continuous models typically assume that facilities can be located anywhere in the service area, while demands are often taken as being at discrete locations. The classic model in this area is the Weber problem of locating a single facility to serve m demands with coordinates $(x_i\text{,}{y}_i)$ with $i=1\text{,}\dots \text{,}m$ and demands (weights) w_i, $i=1\text{,}\dots \text{,}m$. Distances in the Weber problem are often taken to be straight-line or Euclidean distances. The problem is to locate a single facility with coordinates $(x_0\text{,}{y}_0)$ to minimize the demand-weighted total distance. Drezner et al. (2001) provide a review of this model, solution algorithms for the problem and a discussion of important extensions to the problem. Continuous models of this sort can be applied in limited contexts in which it is possible to locate facilities anywhere in the space being considered. For example, it may be possible to use such models in locating video cameras or pollution censors to monitor certain environments.

• Network models assume that the location problem is embedded in a network composed of links and nodes. Demands typically arise on the nodes, though some research has been done on the case in which demands arise on the links and nodes. One practical example in which demands arise on both the nodes and the links is the demand for emergency highway services. Much of the literature in this area is concerned with finding special structures that can be exploited to derive low-order polynomial time algorithms for particular cases of various problems. Goldman’s (1971) O(n) algorithm for finding the 1-median on a tree composed of n nodes (and the often disregarded paper by Hua Lo-Keng et al. (1962) that predated Goldman’s work) is typical of this class of research. In a subsequent paper, Goldman (1972) proposes an O(n) algorithm for the 1-center on a tree.

• Discrete location models assume that there is a discrete set of demands, I, and a discrete set of candidate locations, J. Such problems are often formulated as integer or mixed integer programming problems as shown below. Most such problems are NP-hard on general networks. (Again, much of the research in the network modeling area entails finding special instances of such problems or special graph structures under which the problems admit polynomial time solution algorithms.) As such, there is a sizable body of literature (some of which is summarized below) devoted to finding effective and efficient heuristic algorithms for many discrete problems. Discrete location models have been used in many practical contexts.

  In this paper, we provide an annotated bibliography of papers that have appeared recently in two major areas of discrete location modeling: plant and median problems on the one hand and center and covering problems on the other hand. Section 2 focuses on median and plant location problems, which are both based on minimizing the average demand-weighted distance between a demand node and the facility to which the demand node is assigned. Section 3 addresses covering based models in which there is (are) one (or more) service standard(s) that are to be met or partially met by the facilities being located. In some cases, papers discussed both median and covering objectives. In those cases, the papers are listed in Section 3 after we present a brief overview of the class of covering problems. We present conclusions in Section 4.

  We do not pretend that this annotated bibliography is complete in any sense even within the class of discrete location modeling. First, there are entire classes of papers that we intentionally omitted including much of the literature on obnoxious facility location modeling as well as some of the literature on the issue of aggregation. Also, as is the case in any study of this sort, many papers were published between the time we initially drew up the list of papers to be included in the bibliography and the time we completed the review of the papers. Some of the more recent papers have been included below though there are many that we have undoubtedly omitted. Despite these omissions, we believe that the bibliography below will be a valuable summary of recent research in this clearly dynamic and evolving area.