Location Modeling Utilizing Maximum Service Distance Criteria

The location set-covering problem (LSCP) and the maximal covering location problem (MCLP) have been the subject of considerable interest. As originally defined, both problems allowed facility placement only at nodes. This paper deals with both problems for the case when facility placement is allowed anywhere on the network. Two theorems are presented that show that when facility placement is unrestricted, for either the LSCP or MCLP at least one optimal solution exists that is composed entirely of points belonging to a finite set of points called the network intersect point set (NIPS). Optimal solution approaches to the unrestricted site LSCP and MCLP problems that utilize the NIPS and previously developed solution methodologies are presented. Example solutions show that considerable improvement in the amount of coverage or the number of facilities needed to insure total coverage can be achieved by allowing facility placement along arcs of the network. In addition, extensions to the arc-covering model and the ambulance-hospital model of ReVelle, Toregas, and Falkson are developed and solved.     

Evaluation of the Poisson Approximation To Measures of the Random Pattern in the Square

Theoretical values for spacing between points of the Poisson process are frequently used to evaluate the hypothesis that the locations within a region of pointlike objects have a random pattern. These theoretical spacing values are an approximation to spacing values for a uniform process in which the objects are uniformly and independently located in the region. The adequacy of the Poisson approximation to this uniform process for a square region is evaluated by analytic and numeric methods. The approximation is close for a small number of points when spacing between objects is measured by toroidal distance.