The design of the distribution system is a strategic issue for almost every company. The problem of locating facilities and allocating customers covers the core topics of distribution system design. Model formulations and solution algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance. This paper reviews some of the contributions to the current state-of-the-art. In particular, continuous location models, network location models, mixed-integer programming models, and applications are summarized.

The Weber Problem for a given finite set of existing facilities Ex = fEx 1 ; Ex 2 ; : : : ; ExM g ae IR 2 with positive weights wm (m = 1; : : : ; M) is to find a new facility X such that P M m=1 wm d(X; Exm ) is minimized for some distance function d. 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 reduction to convex optimization problems is used to derive efficient algorithms. 1 Introduction Modern life encounters an ever growing concentration in many respects. Growing population, higher integration of electronic circuits or the economical need to choose an optimized site for new facilities have led to planar locat...

Let A be a nonempty finite subset of the plane representing the geographical coordinates of a set of demand points (towns, . . . ), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for a 2 A if the facility is located at x 2 S is proportional to dist(x; a) --- the distance from x to a --- and that demand of point a is given by ! a , minimizing the total transportation cost TC(!;x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector ! is not known, and only an estimator ! can be provided. Moreover the errors in such estimation process may be non-negligible. We propose a new model for this situation: select a threshold value B ? 0 representing the highest admissible transportation cost. Define the robustness ae of a location x as the minimum increase in demand needed to become inadmissible, i.e. ae(x) = min fk! \Gamma !k : TC(! ; x) ? B; ! 0g and find the x maximizin...

The Weber problem for a given finite set of existing facilities Ex = fEx 1 ; Ex 2 ; : : : ; ExM g ae IR 2 with positive weights wm (m = 1; : : : ; M) is to find a new facility X 2 IR 2 such that P M m=1 wm d(X; Exm ) is minimized for some distance function d. In this paper we consider distances defined by polyhedral gauges. A variation of this problem is obtained if barriers are introduced which are convex polygonal 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 gauge distances can be proved to be an optimal locati...

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 weighted 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.

In continuous location problems we are given a set of existing facilities and we are looking for the location of one or several new facilities. In the classical approaches weights are assigned to existing facilities expressing the importance of the new facilities for the existing ones. In this paper, we consider a pointwise defined objective function where the weights are assigned to the existing facilities depending on the location of the new facility. This approach is shown to be a generalization of the median, center and centdian objective functions. In addition, this approach allows to formulate completely new location models. Efficient algorithms as well as structural results for this algebraic approach for location problems are presented. A complexity analysis and extensions to the multifacility and restricted case are also considered. Keywords:Location Theory, Global optimization, Algebraic optimization, Convexity. 1 Introduction In the last three decades a lot of research has ...

Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-world problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may have complicated shapes. It may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems. In this paper we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Also computational tests are included. Keywords: Location Theory, Forbidden Regions, Approximation, Global Optimization, Applications Partially supported by a grant of the Deutsche Forschungsgemeinschaft 1 Introdu...

Given a finite set of points in the plane and a forbidden region R, we want to find a point X 62 int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings -- such as asymmetry -- to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical metho...

The multiple objective median problem (MOMP) involves locating a new facility with respect to a given set of existing facilities so that a vector of performance criteria is optimized. 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. In this paper, theory of an MOMP with line barriers is developed. As this problem is non-convex but specially-structured, a reduction to a series of convex optimization problems is proposed. The general results lead to a polynomial algorithm for finding the set of efficient solutions. The algorithm is proposed for bi-criteria problems with different measures of distance. 1 Introduction Planar location problems have been intensively studied over the last two decades due to their increasing importance in modern life. Growing population and increased economic demand gave rise to studies on choosing an