On multimodality of obnoxious faclity location models

Obnoxious single facility location models are models that have the aim to find the best location for an undesired facility. Undesired is usually expressed in relation to the so-called demand points that represent locations hindered by the facility. Because obnoxious facility location models as a rule are multimodal, the standard techniques of convex analysis used for locating desirable facilities in the plane may be trapped in local optima instead of the desired global optimum. It is assumed that having more optima coincides with being harder to solve. In this thesis the multimodality of obnoxious single facility location models is investigated in order to know which models are challenging problems in facility location problems and which are suitable for site selection. Selected for this are the obnoxious facility models that appear to be most important in literature. These are the maximin model, that maximizes the minimum distance from demand point to the obnoxious facility, the maxisum model, that maximizes the sum of distance from the demand points to the facility and the minisum model, that minimizes the sum of damage of the facility to the demand points. All models are measured with the Euclidean distances and some models also with the rectilinear distance metric. Furthermore a suitable algorithm is selected for testing multimodality. Of the tested algorithms in this thesis, Multistart is most appropriate. A small numerical experiment shows that Maximin models have on average the most optima, of which the model locating an obnoxious linesegment has the most. Maximin models have few optima and are thus not very hard to solve. From the Minisum models, the models that have the most optima are models that take wind into account. In general can be said that the generic models have less optima than the weighted versions. Models that are measured with the rectilinear norm do have more solutions than the same models measured with the Euclidean norm. This can be explained for the maximin models in the numerical example because the shape of the norm coincides with a bound of the feasible area, so not all solutions are different optima. The difference found in number of optima of the Maxisum and Minisum can not be explained by this phenomenon

Spherical Location Under Restricted Distance

This paper deals with the problem of locating a new facility with respect to n given demand points on earth, with upper bounds imposed on distances between the new facility and each demand points. Distances are measured as the length of the shortest arc of great circle. The proposed algorithm makes use of a Lagrangean relaxation in which the distance constraints, which are not satisfied by the associated unconstrained solution, are incorporated in the economic function. Computational results of a limited number of test problems are presented

Data clustering for circle detection

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples

Data clustering for circle detection

This paper considers a multiple-circle detection problem on the basis of given data. The problem is solved by application of the center-based clustering method. For the purpose of searching for a locally optimal partition modeled on the well-known k-means algorithm, the k-closest circles algorithm has been constructed. The method has been illustrated by several numerical examples

An analysis of minimax facility location problems with area demands /

The unconstrained model, and its solution technique can be easily modified to solve the limiting case where all facilities are fixed points, and also the case when metric constraints are added.Examples are solved to show the impact of assuming area demands, the conflicting nature of the minimax and minisum criteria and to illustrate the solutions techniques developed.A minimax objective function constrained by a bound on the total average cost of servicing all existing facilities (minisum function) is then discussed. Using duality properties, this problem is shown to be equivalent to another model which minimizes the minisum function subject to a bound on the same minimax function. This last problem proves to be easier to solve, and a specialized solution technique is developed. The resulting solutions are nondominated solutions in relation to the two criteria involved. Another way to generate nondominated solutions is by combining the two functions into a weighted sum. The constrained criterion method is shown to be superior both analytically and practically.Most probabilistic facility location problems investigated to date were variations of the generalized Weber formulation. In this research, several single facility minimax location models are analyzed, where both the weights and the locations of the existing facilities are random variables. The demand points are uniformly distributed over rectangular areas, the rectilinear metric is used and the weights are assumed to be independently distributed random variables. Two unconstrained probabilistic models are analyzed and compared to the centroid formulation, it is seen that the probabilistic models are sensitive to deviations from optimal solutions. An expected value criterion formulation is also presented along with lower and upper bound approximating functions

A MODELING FRAMEWORK ON DISTANCE PREDICTING FUNCTIONS FOR LOCATION MODELS IN CONTINUOUS SPACE

Continuous location models are the oldest models in locations analysis dealing with the geometrical representations of reality, and they are based on the continuity of location area. The classical model in this area is the Weber problem. Distances in the Weber problem are often taken to be Euclidean distances, but almost all kinds of the distance functions can be employed. In this survey, we examine an important class of distance predicting functions (DPFs) in location problems all of practical relevance. This paper provides a review on recent efforts and development in modeling travel distances based on the coordinates they use and their applicability in certain practical settings. Very little has been done to include special cases of the class of metrics and its classification in location models and such merit further attention. The new metrics are discussed in the well-known Weber problem, its multi-facility case and distance approximation problems. We also analyze a variety of papers related to the literature in order to demonstrate the effectiveness of the taxonomy and to get insights for possible research directions. Research issues which we believe to be worthwhile exploring in the future are also highlighted