Jack Brimberg

The p-Median Problem: A Survey of Metaheuristic Approaches

The p-median problem, like most location problems, is classified as NP -hard, and so, heuristic methods are usually used for solving it. The pmedian problem is a basic discrete location problem with real application that have been widely used to test heuristics. Metaheuristics are frameworks for building heuristics. In this survey, we examine the p-median, with the aim of providing an overview on advances in solving it using recent procedures based on metaheuristic rules.

Improvement and Comparison of Heuristics for Solving the Uncapacitated Multisource Weber Problem

The multisource Weber problem is to locate simultaneouslym facilities in the Euclidean plane to minimize the total transportation cost for satisfying the demand ofn fixed users, each supplied from its closest facility. Many heuristics have been proposed for this problem, as well as a few exact algorithms. Heuristics are needed to solve quickly large problems and to provide good initial solutions for exact algorithms. We compare various heuristics, i.e., alternative location-allocation (Cooper 1964), projection (Bongartz et al. 1994), Tabu search (Brimberg and Mladenovic 1996a),p-Median plus Weber (Hansen et al. 1996), Genetic search and several versions of Variable Neighbourhood search. Based on empirical tests that are reported, it is found that most traditional and some recent heuristics give poor results when the number of facilities to locate is large and that Variable Neighbourhood search gives consistently best results, on average, in moderate computing time.

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.

Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances

This paper considers a general form of the single facility minisum location problem also referred to as the Fermat-Weber problem, where distances are measured by an lp norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean p = 2 distances.

Primal-Dual Variable Neighborhood Search for the Simple Plant-Location Problem

The variable neighborhood search metaheuristic is applied to the primal simple plant-location problem and to a reduced dual obtained by exploiting the complementary slackness conditions. This leads to (i) heuristic resolution of (metric) instances with uniform fixed costs, up to n=15,000 users, and m=n potential locations for facilities with an error not exceeding 0.04%; (ii) exact solution of such instances with up to m=n=7,000; and (iii) exact solutions of instances with variable fixed costs and up to m=n=15,000.

A bicriteria model for locating a semi-desirable facility in the plane

We consider the problem of locating a facility in the plane. Customers want the facility to be close to obtain cheaply the service offered by it. But customers also want the facility to be far away to avoid the pollution from it. We model the situation using two criteria: One is the well-known minisum criterion; in the other we want to minimize the weighted sum of Euclidean distances raised to a negative power. The second criterion is analyzed in some detail, and we state some properties of this part of the model. In the bicriteria model we minimize the weighted sum of the two criteria, with weights adding to one. We outline a method for determining the efficient frontier of facility locations, and provide an illustrative numerical example.

The Fermat-Weber location problem revisited

The Fermat—Weber location problem requires finding a point in ℝN that minimizes the sum of weighted Euclidean distances tom given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfelds algorithm was proven in a seminal paper by Kuhn in 1973. However, since them given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever them given points are not collinear, Weiszfelds algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of ℝN. We resolve this open question by proving that Weiszfelds algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimensionN.

Solving large p-median clustering problems by primal–dual variable neighborhood search

Data clustering methods are used extensively in the data mining literature to detect important patterns in large datasets in the form of densely populated regions in a multi-dimensional Euclidean space. Due to the complexity of the problem and the size of the dataset, obtaining quality solutions within reasonable CPU time and memory requirements becomes the central challenge. In this paper, we solve the clustering problem as a large scale p-median model, using a new approach based on the variable neighborhood search (VNS) metaheuristic. Using a highly efficient data structure and local updating procedure taken from the OR literature, our VNS procedure is able to tackle large datasets directly without the need for data reduction or sampling as employed in certain popular methods. Computational results demonstrate that our VNS heuristic outperforms other local search based methods such as CLARA and CLARANS even after upgrading these procedures with the same efficient data structures and local search. We also obtain a bound on the quality of the solutions by solving heuristically a dual relaxation of the problem, thus introducing an important capability to the solution process.

A New Distance Function for Modeling Travel Distances in a Transportation Network

Continuous location models generally use the Euclidean or rectangular norm to approximate travel in a transportation network. This paper considers a new and more accurate distance measure, termed the weighted one-two norm, which is a positive linear combination of the preceding norms. A directional bias function is introduced to show the equivalence of this distance measure to the well known l p norm. We then formulate a simple linear regression model to fit the parameters of our distance function to a given data set. Some novel applications based on standard statistical tests are derived which provide practical insights into the nature of networks with rectangular bias. The results are readily extended to other types of networks.

A new local search for continuous location problems

This paper presents a new local search approach for solving continuous location problems. The main idea is to exploit the relation between the continuous model and its discrete counterpart. A local search is first conducted in the continuous space until a local optimum is reached. It then switches to a discrete space that represents a discretisation of the continuous model to find an improved solution from there. The process continues switching between the two problem formulations until no further improvement can be found in either. Thus, we may view the procedure as a new adaption of formulation space search. The local search is applied to the multi-source Weber problem where encouraging results are obtained. This local search is also embedded within Variable Neighbourhood Search producing excellent results.