Christian Michelot

Geometrical properties of the Fermat-Weber problem☆

Abstract This paper is devoted to the Fermat-Weber problem with mixed gauges in order to take into account nonsymmetric distances. A geometrical description of the set of solutions is obtained. Then the concept of metric hull is defined; general sufficient conditions are given such that the metric hull of existing facilities intersects the set of optimal locations. Particularly, it is shown that an optimal location can be found in the metric hull, in dimension two, whatever the gauges are and in dimension n with the rectilinear norm. The connection between metric hull, convex hull and octagonal hull is studied.

Sets of efficient points in a normed space

Abstract Sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized; hull and closure properties are obtained. These results are relevant to geometry of normed spaces and are mostly useful in the context of location theory.

A primal-dual algorithm for the fermat-weber problem involving mixed gauges

We give a new algorithm for solving the Fermat-Weber location problem involving mixed gauges. This algorithm, which is derived from the partial inverse method developed by J.E. Spingarn, simultaneously generates two sequences globally converging to a primal and a dual solution respectively. In addition, the updating formulae are very simple; a stopping rule can be defined though the method is not dual feasible and the entire set of optimal locations can be obtained from the dual solution by making use of optimality conditions.When polyhedral gauges are used, we show that the algorithm terminates in a finite number of steps, provided that the set of optimal locations has nonepty interior and a counterexample to finite termination is given in a case where this property is violated. Finally, numerical results are reported and we discuss possible extensions of these results.

On the uniqueness of optimal solutions in continuous location theory

Abstract This paper deals with the uniqueness of an optimal solution in general continuous single facility minisum and minimax location problems. We define the concept of an S-norm and obtain general conditions which guarantee the existence of a unique optimal location. Some consequences for the uniqueness of optimal locations in multi-facility location problems are discussed.

Localization in multifacility location theory

Abstract We present two localization theorems for a multifacility location problem involving linear costs and a single norm in a two-dimensional space. It is shown that optimal locations for all the new facilities can be found in the metric hull of existing facilities, whatever the norm is. In the polyhedral norm case we study if only finite set of intersections points which belong to the metric hull needs to be considered; the result is obtained in particular for the l 1 -norm and a counter example is given for a general polyhedral norm.

On the Set of Optimal Points to the Weber Problem: Further Results

In a recent paper, Z. Drezner and A. J. Goldman address the problem of determining the smallest set among those containing at least one optimal solution to every Weber problem based on a set of demand points in the plane. In the case of arbitrary mixed gauges (i.e., possibly nonsymmetric norms), the authors have shown that the set of strictly efficient points which are also intersection points always meets the set of Weber solutions. As shown by Drezner and Goldman, this set is optimal with the l 1 or l x distance but is not optimal with the l p distance, 1 p p distance case, we disprove a conjecture of Drezner and Goldman about the possibility of extending their result to more than two dimensions. The paper contains a different view of the problem: whereas Drezner and Goldman use algebraic-analytical approach, the authors use a geometrical approach which permits us to obtain more general results and also clarifies the geometric nature of the problem.

A contribution to the linear programming approach to joint cost allocation : Methodology and application

The linear programming (LP) approach has been commonly proposed for joint cost allocation purposes. Within a LP framework, the allocation rules are based on a marginal analysis. Unfortunately, the additivity property which is required to completely allocate joint costs fails in presence of capacity, institutional or environmental constraints. In this paper, we first illustrate that the non allocated part can be interpreted as a type of producers surplus. Then, by using the information contained in the Simplex tableau we propose an original two-stage methodology based on the marginal costs and the production elasticity of input factors to achieve an additive cost allocation pattern. The distinguished feature of our approach is that it requires no more information or iterative computations than what is provided by the final Simplex tableau. A real-type refinery case study is provided.

Duality for constrained multifacility location problems with mixed norms and applications

A dual problem is developed for the constrained multifacility minisum location problems involving mixed norms. General optimality conditions are also obtained providing new algorithms based on the concept of partial inverse of a multifunction. These algorithms which are decomposition methods, generate sequences globally converging to a primal and a dual solution respectively. Numerical results are reported.

Efficiency in constrained continuous location

Abstract We present a geometrical characterization of the efficient, weakly efficient and strictly efficient points for multi-objective location problems in presence of convex constraints and when distances are measured by an arbitrary norm. These results, established for a compact set of demand points, generalize similar characterizations previously obtained for uncontrained problems. They are used to show that, in planar problems, the set of constrained weakly efficient points always coincides with the closest projection of the set of unconstrained weakly efficient points onto the feasible set. This projection property which are known previously only for strictly convex norms, allows to easily construct all the weakly efficient points and provides a useful localization property for efficient and strictly efficient points.

An Extended Multifacility Minimax Location Problem Revisited

We consider the following model of Drezner (1991) for the location of several facilities. The weighted sum of distances to all facilities plus a set-up cost is calculated for each separate demand point. The maximal value among these sums is to be minimized. In this note we show that if the weights used in the model decompose into a product of two factors, one depending only on the demand point, the other only on the new facilities, there exists at least one optimal solution such that all new facilities coincide. We also investigate when a unique optimal solution of coincidence type exists, and obtain a full description of the set of optimal solutions when the weights have this multiplicative structure and the norm is round. An example shows that this kind of coincidence does not necessarily happen when the weights may have any value.