Weighted sum of order p and minisum location models

Abstract

<p>The Weighted Sum of Order p is a norm ([cursive l] bp -norm) that we adopt to estimate distances in a given transportation network. A distance function is used to transform the point coordinate differences of two points into an estimate of the travel distance between them. Distance functions are employed in applications such as location and location-allocation models, transportation and facility layout problems, distribution management software, computational geometry, distance calculations in Geographical Information Systems, accuracy validation of actual road network distance data, service cost quotations, and random problem generation to test algorithms. In this dissertation, we investigate the superiority of the [cursive l] bp -norm in distance estimation accuracy over the well known weighted [cursive l]p distance. We present computational procedures for determining the parameters of the [cursive l]bp -norm for a given transportation network and we provide theoretical and empirical results indicating its higher accuracy. In order to determine the best parameter values of the [cursive l]bp -norm and to measure its accuracy, we utilize an estimation errors function, the sum of Squared Deviations (SD ), as the goodness-of-fit criterion. We develop certain properties of the [cursive l]bp -norm and the SD function, and using these properties we produce a computational procedure for determining the parameters of the [cursive l]bp -norm. We apply the procedure in seventeen geographical regions and find that the [cursive l] bp -norm models the distances with a higher accuracy than its closest competitor, the weighted [cursive l]p -norm. A new method is devised to calculate the confidence intervals for estimated distances. Using this method, the confidence intervals for estimated actual distances are developed for the [cursive l]p norm and [cursive l] bp -norm. Our empirical study in the seventeen geographical regions indicates that better confidence intervals for the unknown actual distances are obtained with the [cursive l]bp -norm than the [cursive l]p -norm. A distance function constitutes an important part of the objective function in continuous location models. A minisum continuous location model is concerned with the determination of one or more new facility locations in a region so that the total transportation cost between the demand points and the new facilities and also between the new facilities is minimized. Total transportation cost is given by the sum of distances weighted by their corresponding demands. Since the model should represent the real situation as accurately as possible, the accuracy of the distance function employed plays a crucial role in terms of the validity and the applicability of the locational decisions. Therefore, we incorporate the new distance function in single-facility and multi-facility continuous location models and develop generalized iterative solution procedures and fixed point optimality conditions. We also investigate the convergence properties of the iterative procedure when it is applied to the single-facility minisum location models. In order to terminate the iterative procedure used to solve the location problem a bounding method is required. We consider a method which involves the solution of a rectangular distance location problem in each iteration, and provide its generalizations to the approximated [cursive l] p and the [cursive l]bp distance location problems. Finally, we develop the lagrangian and the conjugate dual formulations of the most general [cursive l]p -norm multi-facility minisum location model considering both linear and distance constraints, and also generalize our results to the [cursive l]bp -norm location models.</p>