A new bounding method for single facility location models

This paper develops a new lower bound for single facility location problems withl _p distances. We prove that the method produces superior results to other known procedures. The new bound is also computationally efficient. Numerical results are given for a range of examples with varying numbers of existing facilities andp values.     

Two new examples of sets without medians and centers

Many problems in continuous location theory, reduce to finding a best location, in the sense that a facility must be located at a point minimizing the sum of distances to the points of a given finite set (median) or the largest distances to all points (center). The setting is often assumed to be a Banach space. To have a better understanding concerning the structure of location problems, it may occur also in rather simple cases. In this paper we indicate two simple examples of four-point sets such that one of the two problems indicated has a solution, while the other one has no solution. Also, we list papers containing examples previously given, dealing with this lack of optimal solutions.     

Optimal location and design of a competitive facility

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.Mathematics Subject Classification (2000):90B85, 90C30, 90C29, 91B42Partially supported by Grant PB96-1416-C02-02 of the D.G.E.S. and Grant BFM2002-04525-C02-02 of Ministerio de Ciencia y Tecnología, Spain     

Planar weber location problems with line barriers

The Weber problem for a given finite set of existing facilities in the plane is to find the location of a new facility such that the weithted sum of distances to the existing facilities is minimized.

A variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders or mountain ranges are frequently encountered in practice.

Structural results as well as algorithms for this non-convex optimization problem depending on the distance function and on the number and location of passages through the barrier are presented.     

A Linear Programming Approach to the Solution of Constrained Multi-Facility Minimax Location Problems where Distances are Rectangular

The problem of locating new facilities with respect to existing facilities is stated as a linear programming problem where inter-facility distances are assumed to be rectangular. The criterion of location is the minimization of the maximum weighted rectangular distance in the system. Linear constraints which (a) limit the new facility locations and (b) enforce upper bounds on the distances between new and existing facilities and between new facilities can be included. The dual programming problem is formulated in order to provide for an efficient solution procedure. It is shown that the duLal variables provide information abouLt the complete range of new facility locations which satisfy the minimax criterion.     

Finding efficient solutions for rectilinear distance location problems efficiently

Given n planar existing facility locations, a planar new facility location X is called efficient if there is no other location Y for which the rectilinear distance between Y and each existing facility is at least as small as between X and each existing facility, and strictly less for at least one existing facility. Rectilinear distances are typically used to measure travel distances between points via rectilinear aisles or street networks. We first present a simple arrow algorithm, based entirely on geometrical analysis, that constructs all efficient locations. We then present a row algorithm which is of order n(log n) that constructs all efficient locations, and establish that no alternative algorithm can be of a lower order.     

A BSSS algorithm for the single facility location problem in two regions with different norms

Suppose the plane is divided by a straight line into two regions with different norms. We want to find the location of a single new facility such that the sum of the distances from the existing facilities to this point is minimized. This is in fact a non-convex optimization problem. The main difficulty is caused by finding the distances between points on different sides of the boundary line. In this paper we present a closed form solution for finding these distances. We also show that the optimal solution lies in the rectangular hull of the existing points. Based on these findings then, an efficient big square small square (BSSS) procedure is proposed.     

On a robustness property in single-facility location in continuous space

We consider a single-facility location problem in continuous space—here the problem of minimizing a sum or the maximum of the possibly weighted distances from a facility to a set of points of demand. The main result of this paper shows that every solution (optimal facility location) of this problem has an interesting robustness property. Any optimal facility location is the most robust in the following sense: given a suitable highest admissible cost, it allows the greatest perturbation of the locations of the demand without exceeding this highest admissible chosen cost.     

Planar Weber location problems with barriers and block norms

The Weber problem for a given finite set of existing facilities Ex={Ex₁,Ex₂,...,Ex_M}⊂∝² with positive weights w_m (m=1,...,M) is to find a new facility X^*∈∝² such that Σ _m=1 ^M w_md(X,Ex_m) is minimized for some distance function d. In this paper we consider distances defined by block norms. A variation of this problem is obtained if barriers are introduced which are convex polyhedral subsets of the plane where neither location of new facilities nor traveling is allowed. Such barriers, like lakes, military regions, national parks or mountains, are frequently encountered in practice. From a mathematical point of view barrier problems are difficult, since the presence of barriers destroys the convexity of the objective function. Nevertheless, this paper establishes a discretization result: one of the grid points in the grid defined by the existing facilities and the fundamental directions of the polyhedral distances can be proved to be an optimal location. Thus the barrier problem can be solved with a polynomial algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the structure of the solution set for the single facility location problem with average distances

This paper analyzes continuous single facility location problems where the demand is randomly defined by a given probability distribution. For these types of problems that deal with the minimization of average distances, we obtain geometrical characterizations of the entire set of optimal solutions. For the important case of total polyhedrality on the plane we derive efficient algorithms with polynomially bounded complexity. We also develop a discretization scheme that provides ${\varepsilon}$ -approximate solutions of the original problem by solving simpler location problems with points as demand facilities.     

An efficient solution method for Weber problems with barriers based on genetic algorithms

In this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers. It is assumed that a barrier is a region where neither facility location nor travelling are permitted. The resulting non-convex optimization problem can be reduced to a finite series of convex subproblems, which can be solved by the Weiszfeld algorithm in case of the Weber objective function and Euclidean distances. A solution method is presented that, by iteratively executing a genetic algorithm for the selection of subproblems, quickly finds a solution of the global problem. Visibility arguments are used to reduce the number of subproblems that need to be considered, and numerical examples are presented.     

On worst-case aggregation analysis for network location problems

Network location problems occur when new facilities must be located on a network, and the network distances between new and existing facilities are important. In urban, regional, or geographic contexts, there may be hundreds of thousands (or more) of existing facilities, in which case it is common to aggregate existing facilities, e.g. represent all the existing facility locations in a zip code area by a centroid. This aggregation makes the size of the problem more manageable for data collection and data processing purposes, as well as for purposes of analysis; at the same time, it introduces errors, and results in an approximating location problem being solved. There seems to be relatively little theory for doing aggregation, or evaluating the results of aggregation; most approaches are based on experimentation or computational studies. We propose a theory that has the potential to improve the means available for doing aggregation.This research was supported in part by the National Science Foundation, Grant No. DDM-9023392.     

On the unboundedness of facility layout problems

Facility layout problems involve the location of facilities in a planar arrangement such that facilities that are strongly connected to one another are close to each other and facilities that are not connected may be far from one another. Pairs of facilities that have a negative connection should be far from one another. Most solution procedures assume that the optimal arrangement is bounded and thus do not incorporate constraints on the location of facilities. However, especially when some of the coefficients are negative, it is possible that the optimal configuration is unbounded. In this paper we investigate whether the solution to the facility layout problem is bounded or not. The main Theorem is a necessary and sufficient condition for boundedness. Sufficient conditions that prove boundedness or unboundedness are also given.     

The single facility location problem with time-dependent weights and relocation cost over a continuous time horizon

In this study, we investigate the problem of locating a facility in continuous space when the weight of each existing facility is a known linear function of time. The location of the new facility can be changed once over a continuous finite time horizon. Rectilinear distance and time- and location-dependent relocation costs are considered. The objective is to determine the optimal relocation time and locations of the new facility before and after relocation to minimize the total location and relocation costs. We also propose an exact algorithm to solve the problem in a polynomial time according to our computational results.     

The multi-source Weber problem with constant opening cost

A constant fixed cost of establishing a facility is introduced within the framework of minisum facility location in the continuous space. The solution method developed uses a multi-phase heuristic that first solves a discrete version of the problem by existing methods to obtain an estimate of the optimal number of facilities. Some results are presented for test problems taken from the literature and compared with best-known solutions of the multi-source Weber problem with the addition of the appropriate fixed costs.     

On the Weber facility location problem with limited distances and side constraints

The objective in the continuous facility location problem with limited distances is to minimize the sum of distance functions from the facility to the customers, but with a limit on each of the distances, after which the corresponding function becomes constant. The problem has applications in situations where the service provided by the facility is insensitive after a given threshold distance. In this paper, we propose a global optimization algorithm for the case in which there are in addition lower and upper bounds on the numbers of customers served.     

Isodistant points in competitive network facility location

An isodistant point is any point on a network which is located at a predetermined distance from some node. For some competitive facility location problems on a network, it is verified that optimal (or near-optimal) locations are found in the set of nodes and isodistant points (or points in the vicinity of isodistant points). While the nodes are known, the isodistant points have to be determined for each problem. Surprisingly, no algorithm has been proposed to generate the isodistant points on a network. In this paper, we present a variety of such problems and propose an algorithm to find all isodistant points for given threshold distances associated with the nodes. The number of isodistant points is upper bounded by nm, where n and m are the number of nodes and the number of edges, respectively. Computational experiments are presented which show that isodistant points can be generated in short run time and the number of such points is much smaller than nm. Thus, for networks of moderate size, it is possible to find optimal (or near-optimal) solutions through the Integer Linear Programming formulations corresponding to the discrete version of such problems, in which a finite set of points are taken as location candidates.     

On a Constrained Optimal Location Algorithm

In problems of optimal location, one seeks a position or location that optimizes a particular objective function; this objective function typically relates location and distances to a fixed point set. When one's search is restricted to a given set, we refer to this as a constrained optimal location problem. For a finite point set A, there exist numerous finite algorithms to solve optimal location problems. In this paper we demonstrate how an algorithm, solving optimal location problems in inner-product spaces, can be modified to solve certain constrained optimal location problems. We then apply this modification to a particularly simple (and easily implemented) algorithm and investigate the complexity of the result. In particular we improve a known estimate from exponential to polynomial.     

Cut Generation Algorithm for the Discrete Competitive Facility Location Problem

A competitive facility location model formulated as a bilevel programming problem is considered. A new approach to the construction of estimating problems for bilevel competitive location models is proposed. An iterative algorithm for solving a series of mixed integer programming problems to obtain a pessimistic optimal solution of the model under consideration is suggested.