##### Abstract

The area of facilities location covers a wide variety of problems involving both public and private sector applications. To date, the study of location problems has been restricted primarily to deterministic formulations of the problem. The present research effort investigates the effect of random variation on the location decision.
Three location problems are considered: the single facility location problem, the multifacility location problem, and the emergency service location problem. The first two problems treated are defined as the generalized Weber problem, where the concern is to locate one or more new facilities in the plane relative to several existing facilities such that the expected total cost of item movements is minimized. The total cost function is considered to be a linear function of either the expected rectilinear or the Euclidean distance, as well as a quadratic function of the expected Euclidean distance.
In the generalized Weber problem the locations of the existing facilities and the item movement between facilities are considered to be random variables. Two expected total cost formulations are presented; the first involves the product of the random variables, weight and distance; the second involves the random sum of each individual distance traveled. For each formulation, possible applications are discussed, theoretical properties are developed, and a solution procedure is provided. Each algorithm is programmed and optimal solutions are obtained for several example problems. A comparison between the probabilistic and deterministic solutions is provided. Both discretely and continuously distributed random variables are treated; however, for the case of continuously distributed random variables, the normal distribution is emphasized. Both constrained and unconstrained formulations are considered.
In formulating the emergency service facilities location problems which are studied, random variation is assumed to be present due to the assumption that the location of an incident is a random variable occurring uniformly over a given region. Both discrete space and continuous space formulations are considered. For the discrete case, a covering criterion is employed and the deterministic equivalent problem is solved as a set cover problem. For the continuous case, the problem is solved as a location-allocation problem. In all formulations, the rectilinear norm is used to measure the distance traveled. An example is solved for each case to illustrate the impact of probabilistic aspects on the location decision.