Median problems in networks

The P-median problem is a classical location model “par excellence”. In this paper we, first examine the early origins of the problem, formulated independently by Louis Hakimi and Charles ReVelle, two of the fathers of the burgeoning multidisciplinary field of research known today as Facility Location Theory and Modelling. We then examine some of the traditional heuristic and exact methods developed to solve the problem. In the third section we analyze the impact of the model in the field. We end the paper by proposing new lines of research related to such a classical problem.P-median, location modelling

Inverse median problems

AbstractThe inverse p-median problem consists in changing the weights of the customers of a p-median location problem at minimum cost such that a set of p prespecified suppliers becomes the p-median. The cost is proportional to the increase or decrease of the corresponding weight. We show that the discrete version of an inverse p-median problem can be formulated as a linear program. Therefore, it is polynomially solvable for fixed p even in the case of mixed positive and negative customer weights. In the case of trees with nonnegative vertex weights, the inverse 1-median problem is solvable in a greedy-like fashion. In the plane, the inverse 1-median problem can be solved in O(nlogn) time, provided the distances are measured in l1- or l∞-norm, but this is not any more true in R3 endowed with the Manhattan metric

Solving Discrete Ordered Median Problems with Induced Order

Ordered median functions have been developed to model flexible discrete location problems. A weight is associated to the distance from a customer to its closest facility, depending on the position of that distance relative to the distances of all the customers. In this paper, the above idea is extended by adding a second type of facility and, consequently, a second weight, whose values are based on the position of the first weights. An integer programming formulation is provided in this work for solving this kind of models

Data aggregation for p-median problems

In this paper, we use a pseudo-Boolean formulation of the p-median problem and using data aggregation, provide a compact representation of p-median problem instances. We provide computational results to demonstrate this compactification in benchmark instances. We then use our representation to explain why some p-median problem instances are more difficult to solve to optimality than other instances of the same size. We also derive a preprocessing rule based on our formulation, and describe equivalent p-median problem instances, which are identical sized instances which are guaranteed to have identical optimal solutions

Approximation Algorithms for Geometric Median Problems

In this paper we present approximation algorithms for median problems in metric spaces and xed-dimensional Euclidean space. Our algorithms use a new method for transforming an optimal solution of the linear program relaxation of the s-median problem into a provably good integral solution. This transfor- mation technique is fundamentally di erent from the methods of randomized and deterministic rounding [Rag, RaT] and the methods proposed in [LiV] in the following way: Previous techniques never set variables with zero values in the fractional solution to 1. This departure from previous methods is crucial for the success of our algorithms

Location Games and Bounds for Median Problems

We consider a two-person zero-sum game in which the maximizer selects a point in a given bounded planar region, the minimizer selects K points in that region,.and the payoff is the distance from the maximizer's location to the minimizer's location closest to it. In a variant of this game, the maximizer has the privilege of restricting the game to any subset of the given region. We evaluate/approximate the values (and the saddle point strategies) of these games for K = 1 as well as for K + , thus obtaining tight upper bounds (and worst possible demand distributions) for K-median problems

p-Median problems in a fuzzy environment

In this paper a formulation for the fuzzy p-median model in a fuzzy environment is presented. The model allows to find optimal locations of p facilities and their related cost when data related to the node demands and the edge distances are imprecise and uncertain and also to know the degree of certainty of the solution. For the sake of illustration, the proposed model is applied in a reduced map of Kinshasa (Democratic Republic of Congo) obtaining results which are rather than realistic one

Structural Iterative Rounding for Generalized k-Median Problems

This paper considers approximation algorithms for generalized k-median problems. This class of problems can be informally described as k-median with a constant number of extra constraints, and includes k-median with outliers, and knapsack median. Our first contribution is a pseudo-approximation algorithm for generalized k-median that outputs a 6.387-approximate solution with a constant number of fractional variables. The algorithm is based on iteratively rounding linear programs, and the main technical innovation comes from understanding the rich structure of the resulting extreme points. Using our pseudo-approximation algorithm, we give improved approximation algorithms for k-median with outliers and knapsack median. This involves combining our pseudo-approximation with pre- and post-processing steps to round a constant number of fractional variables at a small increase in cost. Our algorithms achieve approximation ratios 6.994 + ? and 6.387 + ? for k-median with outliers and knapsack median, respectively. These both improve on the best known approximations

Erweiterung des 'generalized' p-Median-Problems

Die vorliegende Masterarbeit beschäftigt sich mit den MINISUM-Modellen auf einem Graphen. Die Eigenschaften des „generalized“ p-Median-Problem werden neben den Eigenschaften des ordinären p-Median-Problems untersucht. Dabei kommt folgende Diskrepanz zum Vorschein: Obwohl das „generalized“ p-Median-Problem eine unendliche Anzahl an potenziellen Lösungsmöglichkeiten besitzt und der optimale Standort bei einer derartigen Problemstellung sowohl im Knoten als auch auf der Kante des Graphen liegen kann, wird der Median oft ausschließlich in den Knoten des Graphen gesucht. Dadurch entsteht das Risiko, dass beim Lösen des Problems der optimale Standort von Anfang an nicht mitberücksichtigt wird. Die Forschungsaufgabe dieser Arbeit ist, das „generalized“ p-Median-Problem so zu erweitern, dass aus einem Problem mit unendlicher Anzahl an Lösungsmöglichkeiten ein endliches Problem wird, welches optimal mit einer diskreten Methode gelöst werden kann. Im ersten Schritt werden die potenziellen Standorte auf den Kanten (die sogenannten fiktiven Knoten) ermittelt. Sie werden mit den Knoten des Graphen gleichgestellt und bei der Auffindung des kostenminimalen Standortes einkalkuliert. Damit sind alle potenziellen Standorte abgedeckt und das Problem erhält eine endliche Anzahl an Lösungsmöglichkeiten. Eine weitere Herausforderung liegt in der unkonventionellen Formulierung des Kostenparameters, der beim „generalized“ p-Median-Problem zusätzlich berücksichtigt wird. Die Kosten stellen eine logarithmische Kostenfunktion dar, die von der Verteilung der Nachfrage auf die Mediane abhängig ist. Diese Variable wird als Zuteilung bezeichnet und muss zusätzlich vor der Formulierung des Optimierungsproblems bestimmt werden. Die Zuteilung ist für die Ermittlung der Kosten zuständig und fließt in das Modell nur indirekt mit ein. Abschließend wird die Funktionsfähigkeit des neuen Modells überprüft und dem ursprünglichen Modell (dem umformulierten Warehouse Location Problem) gegenübergestellt. Tatsächlich werden bei dem erweiterten Modell durch die Platzierung der Mediane auf die Kante zusätzliche Kosten eingespart. Die vorliegende Arbeit zeigt das Prinzip, wie das „generalized“ p-Median-Problem erweitert werden kann, und liefert den Beweis über die Funktionstüchtigkeit dieser Methode.The following master’s thesis deals with the MINISUM models on a graph. In this regard the properties of the generalized p-median problem have been investigated alongside the properties of the ordinary p-median problem. In the course of the investigation, the following discrepancy comes to the fore: although the generalized p-median problem has an infinite number of potential solutions, and the optimal location for such a problem may lie in both the vertex and on the edge of the graph, the median is often searched for exclusively in the vertex of the graph. This creates the risk that, upon attempting to find a solution, the optimal location to place the median may not be taken into consideration right from the start. The goal of the following thesis is to extend the generalized p-median problem so that a problem with an infinite number of possible solutions becomes a finite problem which can best be solved with a discrete method. In the first step, all potential locations along the edges (the so-called fictitious vertices) are determined using an empirical-analytical approach. They are equated with the vertices of the graph and taken into account when locating the minimum cost location. This covers all potential locations and through this method the problem receives a finite number of possible solutions. Another challenge lies in the unconventional formulation of the cost parameter, which is additionally taken into account in the generalized p-median problem. The cost represents a logarithmic cost function that depends on the distribution of demand on the median. In the following work, this variable shall be called the allocation and must first be determined in order to formulate the optimization problem framework. The allocation is responsible for determining the costs and is included only indirectly in the model. Finally, the functionality of the new model is checked and compared with the original model, the rewritten warehouse location problem. In fact, the placement of medians on an edge saves additional costs in the extended model. The following elaboration shows the principle of how the generalized p-median problem can be extended, and provides proof of the functionality of this extension