A decomposition approach for the p-median problem on disconnected graphs

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of pk, and the global solution is obtained by finding the best combination of pk medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time

Measuring Diversity Perceptions: A Qualitative Research

Having diversity inside the organization is getting more and more important. Because of the tough competition in the external world which is changing in dynamic way, companies need to find adaptation strategies. When a company’s diversity capacity increases, its potential for survival and adaptation also increases. To gain the advantages of diversity, the most important thing is to understand people’s diversity perceptions. Then it will be possible to make a decision, if that organization is suitable for diversity management or not. In this research the main aim is to understand the key words about diversity perceptions and how it differentiate. There were 25 participants and the data was collected by face to face. While analyzing the data, some key words were detected which is valid for the whole group and then also analyzed according to gender parameter

A general solution framework for component commonality problems

Component commonality, the use of the same version of a component across multiple products, is increasingly considered as a promising way to offer high external variety while retaining low internal variety in operations. However, increasing commonality has both positive and negative cost effects, so that optimization approaches are required to identify an optimal commonality level. As a more or less of components influences nearly every process step along the supply chain, it is not astounding that a multitude of diverging commonality problems is investigated in literature, each of which developing a specific algorithm designed for the respective commonality problem considered. The paper on hand aims at a general framework, flexible and effcient enough to be applied to a wide range of commonality problems. Such a procedure basing on a two-stage graph approach is presented and tested. Finally, flexibility of the procedure is shown by customizing the framework to account for different types of commonality problems.Product variety, Component commonality, Optimization, Graph approach

O problema da p-mediana aplicado ao problema da gestão óptima da diversidade

Mestrado em Matemática e AplicaçõesNeste trabalho aborda-se o problema da p-mediana e a sua aplicação a um problema da gestão óptima da diversidade na indústria automóvel. O problema da p-mediana é aplicado em diversas situações práticas reais, nomeadamente na localização de equipamentos públicos, industriais, comerciais e de telecomunicações. Na sua forma geral o problema da p-mediana é NP-difícil e na sua resolução são implementados métodos heurísticos. Primeiramente é feita uma apresentação do problema, e são descritos os principais métodos heurísticos de resolução. De seguida, é descrito o problema da gestão óptima da diversidade e apresentada uma aplicação deste problema à indústria automóvel. Por último, é efectuado um estudo computacional de modo a avaliar o desempenho de diferentes versões do algoritmo híbrido e, simultaneamente, comparar essas versões com o algoritmo guloso.This work presents the p-median problem and its application to an optimal diversity management problem in the automobile industry. The p-median problem is applied in different real practical situations, particularly in public, industrial, commercial and telecommunications equipments location. In its general form, it’s a NP-hard problem and in its resolution heuristic methods are implemented. First, there is a presentation of the problem, and the main methods of heuristic resolution are described. Next it is described the optimal diversity management problem and an application of this problem is submitted to the car industry. Finally, it is reported a computational study to assess the performance of different versions of hybrid algorithm and compare these versions simultaneously with the greedy algorithm

A Discrete Optimization Model to Minimize Organ Recovery Time Using Heuristic Algorithms

This study proposes a discrete optimization model to minimize the organ recovery time in an Organ Procurement Organization (OPO) by grouping its associated hospitals and transplant centers into several clusters, based on their available organ recovery groups. Typically, the OPO covers a relatively large geographical area to recover organs from donors and deliver them to the recipients. Organs and/or tissues need to be transplanted within their viable time. Therefore, a discrete optimization model is proposed, based on the -median approach to identify optimal locations of the organ recovery groups to recover the organs within a desired time interval. Three heuristic solution approaches, such as Multi-start Fast Interchange (MFI), Simulated Annealing (SA), and Lagrangian Relaxation Algorithm (LRA), are applied to solve the -median clustering problems. Numerical examples are tested to identify a better solution approach in terms of a set of key performance indicators, such as elapse time, Silhouette index, and objective function value. The experimental results indicate that the MFI approach is effective finding an initial solution in the shortest possible time. To find a non-dominant optimal solution, the LRA outperformed the initial solution. In the future, the experimental results will be compared with real data to ensure the effectiveness of the proposed model

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

Solving the p -Median Problem with a Semi-Lagrangian Relaxation

Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median proble

O problema da gestão ótima da diversidade de cablagens

Mestrado em Matemática e AplicaçõesNeste trabalho, aborda-se o problema da gestão ótima da diversidade, que pode ser considerado um caso particular do problema da p-mediana, aplicado na indústria automóvel. Em primeiro lugar, descreve-se o problema e estudam-se algumas das suas caraterísticas combinatórias. Sendo um problema de otimização combinatória classificado como NP-difícil, usualmente, aplicam-se na sua resolução métodos heurísticos. Nesta tese, estuda-se um algoritmo Greedy com detalhe, de modo a inferir estratégias que o tornem mais eficiente na resolução do problema. Assim, estudam-se algumas caraterísticas combinatórias desse algoritmo Greedy, para um caso particular, e apresentam-se algumas propriedades do mesmo. De seguida, dado que na realidade o problema da gestão ótima da diversidade de cablagens pode ser decomposto em vários subproblemas, mais fáceis de resolver, apresenta-se a abordagem que consiste na aplicação dum algoritmo Greedy em duas etapas, numa primeira etapa o algoritmo é aplicado a cada subproblema e, numa segunda etapa, é aplicado para determinar a melhor forma de combinar as soluções dos vários subproblemas. Por último, é feito um estudo estatístico, com uma amostra de 5 problemas reais de uma empresa de cablagens, de modo a analisar a existência de caraterísticas comuns na resolução dos mesmos, através dum algoritmo Greedy. A partir de alguns resultados observados nesse estudo e pelo facto de, para problemas de grande dimensão, esse algoritmo demorar mais tempo que o desejado a encontrar uma solução, consideram-se duas variantes do mesmo que resultam de restrições na pesquisa que é feita pelo algoritmo original, de forma a torná-lo mais rápido. É apresentado um pequeno estudo computacional para comparar as diferentes variantes do algoritmo.This study covers the optimal diversity management problem, which can be seen as a particular case of p-median problem, applied to automotive industry. The starting point is the problem description as well as a study of some of its combinatorial characteristics. As a combinatorial optimization problem classified as NP-hard, heuristic methods are often used on its resolution. In this thesis, a Greedy algorithm is studied in deep detail, in order to find some strategies to improve it for a more efficient problem resolution. Thus, some particular combinatorial characteristics of this Greedy algorithm are studied, for a particular case, and some of its properties are presented. Thereafter, since optimal diversity management problem can be decomposed into several sub problems – easier to solve – an approach which consists in the application of a Greedy algorithm in two steps is presented: a first step where the algorithm is applied to each sub problem, and a second step where it is applied to determine the best way to combine the several sub problem solutions. Finally, a statistical study is made with a sample of 5 real problems of a wiring harness company in order to analyze the existence of common characteristics in their resolutions, by applying a Greedy algorithm. Based on some results of the previous study and since for real world applications this algorithm may take longer than desired to find a solution, two variants of this algorithm are considered, those resulting from restrictions in the search made by the original algorithm in order to make it faster. A small computational study is presented in order to compare the several variants of the algorithm

A Computational Study of the Pseudo-Boolean Approach to the p-Median Problem Applied to Cell Formation

Abstract. In this study we show by means of computational experiments that a pseudo-Boolean approach leads to a very compact presentation of p-Median problem instances which might be solved to optimality by a general purpose solver like CPLEX, Xpress, etc. Together with p-Median benchmark instances from OR and some other libraries we are able to solve to optimality many benchmark instances from cell formation in group technology which were tackled in the past only by means of different types of heuristics. Finally, we show that this approach is flexible to take into account many other practically motivated constraints in cell formation