Properties of a multifacility location problem involving euclidian distances

This paper considers a problem of locating new facilities in the plane with respect to existing facilities, the locations of which are known. The problem consists of finding locations of new facilities which will minimize a total cost function which consists of a sum of costs directly proportional to the Euclidian distances among the new facilities, and costs directly proportional to the Euclidian distances between new and existing facilities. It is established that the total cost function has a minimum; necessary conditions for a mimumum are obtained; necessary and sufficient conditions are obtained for the function to be strictly convex (it is always convex); when the problem is “well structured,” it is established that for a minimum cost solution the locations of the new facilities will lie in the convex hull of the locations of the existing facilities. Also, a dual to the problem is obtained and interpreted; necessary and sufficient conditions for optimum solutions to the problem, and to its dual, are developed, as well as complementary slackness conditions. Many of the properties to be presented are motivated by, based on, and extend the results of Kuhn's study of the location problem known as the General Fermat Problem.     

A computation procedure for the exact solution of location-allocation problems with rectangular distances

A method is presented to locate and allocate p new facilities in relation to n existing facilities. Each of the n existing facilities has a requirement flow which must be supplied by the new facilities. Rectangular distances are assumed to exist between all facilities. The algorithm proceeds in two stages. In the first stage a set of all possible optimal new facility locations is determined by a set reduction algorithm. The resultant problem is shown to be equivalent to finding the p-median of a weighted connected graph. In the second stage the optimal locations and allocations are obtained by using a technique for solving the p-median problem.     

Note: Facility location with closest rectangular distances

This paper considers the problem of locating one or more new facilities on a continuous plane, where the destinations or customers, and even the facilities, may be represented by areas and not points. The objective is to locate the facilities in order to minimize a sum of transportation costs. What is new in this study is that the relevant distances are the distances from the closest point in the facility to the closest point in the demand areas. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 77–84, 2000     

Solving multifacility location problems involving euclidean distances

This paper considers the problem of locating multiple new facilities in order to minimize a total cost function consisting of the sum of weighted Euclidean distances among the new facilities and between the new and existing facilities, the locations of which are known. A new procedure is derived from a set of results pertaining to necessary conditions for a minimum of the objective function. The results from a number of sample problems which have been executed on a programmed version of this algorithm are used to illustrate the effectiveness of the new technique.     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

Locating facilities in three-dimensional space by convex programming

We consider the problem of simultaneously locating any number of facilities in three-dimensional Euclidean space. The criterion to be satisfied is that of minimizing the total cost of some activity between the facilities to be located and any number of fixed locations. Any amount of activity may be present between any pair of the facilities themselves. The total cost is assumed to be a linear function of the inter-facility and facility-to-fixed locations distances. Since the total cost function for this problem is convex, a unique optimal solution exists. Certain discontinuities are shown to exist in the derivatives of the total cost function which previously has prevented the successful use of gradient computing methods for locating optimal solutions. This article demonstrates the use of a created function which possesses all the necessary properties for ensuring the convergence of first order gradient techniques and is itself uniformly convergent to the actual objective function. Use of the fitted function and the dual problem in the case of constrained problems enables solutions to be determined within any predetermined degree of accuracy. Some computation results are given for both constrained and unconstrained problems.     

Solving quadratic assignment problems with rectangular distances and integer programming

The problem considered involves the assignment of n facilities to n specified locations. Each facility has a given nonnegative flow from each of the other facilities. The objective is to minimize the sum of transportation costs. Assume these n locations are given as points on a two-dimensional plane and transportation costs are proportional to weighted rectangular distances. Then the problem is formulated as a binary mixed integer program. The number of integer variables (all binary) involved equals the number of facilities squared. Without increasing the number of integer variables, the formulation is extended to include “site costs” Computational results of the formulation are presented.     

Location of facilities with rectangular distances among point and area destinations

This article is concerned with the optimal location of any number (n) of facilities in relation to any number (m) of destinations on the Euclidean plane. The criterion to be satisfied is the minimization of total weighted distances where the distances are rectangular. The destinations may be either single points, lines or rectangular areas. A gradient reduction solution procedure is described which has the property that the direction of descent is determined by the geometrical properties of the problem.     

An iterative algorithm for the multifacility minimax location problem with euclidean distances

An iterative solution method is presented for solving the multifacility location problem with Euclidean distances under the minimax criterion. The iterative procedure is based on the transformation of the multifacility minimax problem into a sequence of squared Euclidean minisum problems which have analytical solutions. Computational experience with the new method is also presented.     

On a rational stopping rule for facilities location algorithms

In the multifacility location problem, a number of new facilities are to be located so as to minimize a sum of weighted distances. Recently, a lower bound on the optimal value was developed, for use in deciding when to stop an iterative solution procedure. We develop a stronger bound that allows some computational savings.     

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.     

Probabilistic formulations of the multifacility weber problem

The problem considered is to locate one or more new facilities relative to a number of existing facilities when both the locations of the existing facilities, the weights between new facilities, and the weights between new and existing facilities are random variables. The new facilities are to be located such that expected distance traveled is minimized. Euclidean distance measure is considered; both unconstrained and chance-constrained formulations are treated.     

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.     

The conditional p-center problem in the plane

An algorithm is given for the conditional p-center problem, namely, the optimal location of one or more additional facilities in a region with given demand points and one or more preexisting facilities. The solution dealt with here involves the minimax criterion and Euclidean distances in two-dimensional space. The method used is a generalization to the present conditional case of a relaxation method previously developed for the unconditional p-center problems. Interestingly, its worst-case complexity is identical to that of the unconditional version, and in practice, the conditional algorithm is more efficient. Some test problems with up to 200 demand points have been solved. © 1993 John Wiley & Sons, Inc.     

An exact branch-and-bound procedure for the quadratic-assignment problem

The quadratic-assignment problem is a difficult combinatorial problem which still remains unsolved. In this study, an exact branch-and-bound procedure, which is able to produce optimal solutions for problems with twelve facilities or less, is developed. The method incorporates the concept of stepped fathoming to reduce the effort expended in searching the decision trees. Computational experience with the procedure is presented.     

An efficient algorithm for the location-allocation problem with rectangular regions

The location-allocation problem for existing facilities uniformly distributed over rectangular regions is treated for the case where the rectilinear norm is used. The new facilities are to be located such that the expected total weighted distance is minimized. Properties of the problem are discussed. A branch and bound algorithm is developed for the exact solution of the problem. Computational results are given for different sized problems.     

A stochastic gradual cover location problem

Covering models assume that a point is covered if it is within a certain distance from a facility and not covered beyond that distance. In gradual cover models it is assumed that a point is fully covered within a given distance from a facility, then cover gradually declines, and the point is not covered beyond a larger distance. Gradual cover models address the discontinuity in cover which may not be the correct approach in many situations. In the stochastic gradual cover model presented in this article it is assumed that the short and long distances employed in gradual cover models are random variables. This refinement of gradual cover models provides yet a more realistic depiction of actual behavior in many situations. The maximal cover model based on the new concept is analyzed and the single facility location cover problem in the plane is solved. Computational results illustrating the effectiveness of the solution procedures are presented. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

A primal simplex algorithm to solve a rectilinear distance facility location problem

This paper considers the problem of locating m new facilities in the plane so as to minimize a weighted rectangular distance between the new facilities and n existing facilities. A special purpose primal simplex algorithm is developed to solve this problem. The algorithm will maintain at all times a basis of dimension m by m; however, because of the triangularity of the basis matrix, it will not be necessary to form a basis inverse explicitly.     

A localization property for facility-location problems with arbitrary norms

In an earlier article we showed that, for facilities-location problems characterized by generalized distance norms and any even number of existing facilities, the optimal location of the new facility is at the intersection of the lines joining the pairs of facilities if these lines intersect at a single point. In this article we extend this concept to show that, for a more general class of problems, the optimal location is one of a set of points which is specified by the existing facilities.     

A linear programming solution to the generalized rectangular distance weber problem

The Weber Problem generalized to the location of several new points with respect to existing points is formulated as a linear programming problem under the assumption that distances are rectangular. The dual problem is then formulated and subsequently reduced to a problem with substantially fewer variables and constraints than required by an existent alternative linear programming formulation. Flows may exist between new as well as between new and existing points. Linear constraints can be imposed to restrict the location of new points. Pairwise constraints limiting distances between new points and between new and existing points can also be accommodated.