Queueing-location problems on the plane

In this article we present a queueing-location problem where a location of a service station has to be determined. The two main results of this article are a convexity proof for general distances and a theorem that limits the area in the plane where the solution can lie. We also propose some solution procedures.     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.     

Location of regional facilities

In this paper we consider the single-facility and multifacility problems of the minisum type of locating facilities on the plane. Both demand locations and the facilities to be located are assumed to have circular shapes, and demand and service is assumed to have a uniform probability density inside each shape. The expected distance between two facilities is calculated. Euclidean and squared-Euclidean distances are discussed.     

A cover based competitive facility location model with continuous demand

In this paper we propose and solve a competitive facility location model when demand is continuously distributed in an area and each facility attracts customers within a given distance. This distance is a measure of the facility's attractiveness level which may be different for different facilities. The market share captured by each facility is calculated by two numerical integration methods. These approaches can be used for evaluating functional values in other operations research models as well. The single facility location problem is optimally solved by the big triangle small triangle global optimization algorithm and the multiple facility problem is heuristically solved by the Nelder‐Mead algorithm. Extensive computational experiments demonstrate the effectiveness of the solution approaches.     

Location strategies for satellites' orbits

We investigate the problem of covering a given area (or the whole globe) by moving satellites in space. The objective is to find patterns of satellites' orbits that assure the presence of at least one satellite above any prespecified area at all times.     

Minimax and maximin facility location problems on a sphere

The problem dealt with in this article is as follows. There are n “demand points” on a sphere. Each demand point has a weight which is a positive constant. A facility must be located so that the maximum of the weighted distances (distances are the shortest arcs on the surface of the sphere) is minimized; this is called the minimax problem. Alternatively, in the maximin problem, the minimum weighted distance is maximized. A setup cost associated with each demand point may be added for generality. It is shown that any maximin problem can be reparametrized into a minimax problem. A method for finding local minimax points is described and conditions under which these are global are derived. Finally, an efficient algorithm for finding the global minimax point is constructed.     

Sensitivity analysis of the optimal location of a facility

We perform a sensitivity analysis of the Euclidean, single-facility minisum problem, which is also known as the Weber problem. We find the sensitivity of the optimal site of the new facility to changes in the locations and weights of the demand points. We apply these results to get the optimal site if some of the parameters in the problem are changed. We also get approximate formulas for the set of all possible optimal sites if demand points are restricted to given areas, and weights must be within given ranges, which is a location problem under conditions of uncertainty.     

Optimal location of a facility relative to area demands

This paper deals with the Weber single-facility location problem where the demands are not only points but may be areas as well. It provides an iterative procedure for solving the problem with l^p distances when p > 1 (a method of obtaining the exact solution when p = 1 and distances are thus rectangular already exists). The special case where the weight densities in the areas are uniform and the areas are rectangles or circles results in a modified iterative process that is computationally much faster. This method can be extended to the simultaneous location of several facilities.     

On the rectangular p-center problem

The p-center problem involves finding the best locations for p facilities such that the furthest among n points is as close as possible to one of the facilities. Rectangular (sometimes called rectilinear, Manhattan, or l₁) distances are considered. An O(n) algorithm for the 1-center problem, an O(n) algorithm for the 2-center problem, and an O(n logn) algorithm for the 3-center problem are given. Generalizations to general p-center problems are also discussed.