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The min-max vehicle routing problem (VRP) is a variant of the classical VRP in which the objective is to minimize the duration of the longest route. Examination of the VRP [...]

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2008.07.015 Byline: Julian Molina (c), Luis V. Santana (a), Alfredo G. Hernandez-Diaz (b), Carlos A. Coello Coello (a), Rafael Caballero (c) Keywords: Multiple-criteria decision making; Interactive methods; Preference information; Reference point Abstract: One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469-486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM's preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273-286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems. Author Affiliation: (a) CINVESTAV-IPN, Computer Science Department, Mexico (b) Pablo de Olavide University, Seville, Spain (c) University of Malaga, Applied Economics (Mathematics), Campus El Ejido s./n., 29071, Malaga, Spain Article History: Received 1 February 2007; Accepted 7 July 2008

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2008.05.013 Byline: Kaisa Miettinen (a), Julian Molina (b), Mercedes Gonzalez (b), Alfredo Hernandez-Diaz (c), Rafael Caballero (b) Keywords: Multiple criteria programming; MCDM; Preference information; Pareto optimality; Visualization; Interactive methods Abstract: Because of the conflicting nature of criteria or objectives, solving a multiobjective optimization problem typically requires interaction with a decision maker who can specify preference information related to the objectives in the problem in question. Due to the difficulties of dealing with multiple objectives, the way information is presented plays a very important role. Questions posed to the decision maker must be simple enough and information shown must be easy to understand. For this purpose, visualization and graphical representations can be useful and constitute one of the main tools used in the literature. In this paper, we propose to use box indices to represent information related to different solution alternatives of multiobjective optimization problems involving at least three objectives. Box indices are an intelligible and easy to handle way to represent data. They are based on evaluating the solutions in a natural and rough enough scale in order to let the decision maker easily recognize the main characteristics of a solution at a glance and to facilitate comparison of two or more solutions in an easily understandable way. Author Affiliation: (a) Department of Mathematical Information Technology, P.O. Box 35 (Agora), FI-40014, University of Jyvaskyla, Finland (b) University of Malaga, Malaga, Spain (c) Pablo de Olavide University, Seville, Spain Article History: Received 23 April 2007; Accepted 17 May 2008

We describe the development and testing of a metaheuristic procedure, based on the scatter-search methodology, for the problem of approximating the efficient frontier of nonlinear multiobjective optimization problems with continuous […]

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.10.017 Byline: Rafael Caballero (a), Mercedes Gonzalez (a), Flor Ma Guerrero (b), Julian Molina (a), Concepcion Paralera (b) Keywords: Location routing problems; Multiobjective combinatorial optimization; Metaheuristics Abstract: In this work we present a multiobjective location routing problem and solve it with a multiobjective metaheuristic procedure. In this type of problem, we have to locate some plants within a set of possible locations to meet the demands of a number of clients with multiple objectives. This type of model is used to solve a problem with real data in the region of Andalusia (Spain). Thus, we study the location of two incineration plants for the disposal of solid animal waste from some preestablished locations in Andalusia, and design the routes to serve the different slaughterhouses in this region. This must be done while taking into account certain economic objectives (start-up, maintenance, and transport costs) and social objectives (social rejection by towns on the truck routes, maximum risk as an equity criterion, and the negative implications for towns close to the plant). Author Affiliation: (a) Faculty of Economics, Department of Applied Economics (Mathematics), University of Malaga, Campus El Ejido, 29007 Malaga, Spain (b) Department of Economy and Business, University Pablo de Olavide, Ctra. de Utrera, Km. 1- 41013 Sevilla, Spain

The application of metaheuristic programming techniques, to perform multiobjective combinatorial optimization of the location routing problems faced by manufacturing firms in Andalusia, is described.

Despite the mathematical properties and algorithmic features of an interactive method, that method's success usually lies in the kind of information it requires from the decision maker. In some cases, this information constrains the decision maker. If she does not find it easy and comfortable to answer the questions posed by the algorithm, then she will very likely give inconsistent answers, and the method will fail to find her most preferred solution. Therefore, it is of interest to find relations among the different kinds of information (local weights, local trade-offs, reference points, etc.) to allow the decision maker to choose what kind of questions he wants to answer, and to provide him with enough information to give such answers. In this paper, we define equivalent information--that is, different kinds of information--that produces the same solution when used in their corresponding interactive schemes. Key words: multiobjective programming; interactive methods; equivalent information, 1. Introduction Among all the multiobjective programming techniques, interactive methods are one of the most widely used approaches. A great number of them have been developed during recent decades, and [...]

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2005.04.040 Byline: Rafael Caballero (a), Francisco Ruiz (a), M. Victoria Rodriguez Uria (b), Carlos Romero (c) Keywords: Goal programming; Meta-goal programming; Interactive; Satisficing Abstract: The concept of meta-goal programming is developed and linked to an interactive framework. An algorithm is proposed, in which the decision maker can establish target values on several achievement functions and use an interactive procedure to update these values. This substantially alleviates the problems associated with assigning to each attribute a target value, in order to build the goals, as well as the selection of a suitable achievement function. The functioning of the proposed interactive approach is illustrated with the help of an example taken from the farm management literature. Author Affiliation: (a) Departamento de Economia Aplicada (Matematicas), Facultad de Ciencias Economicas y Empresariales, Universidad de Malaga, Campus El Ejido s/n, 29071 Malaga, Spain (b) Departamento de Economia Cuantitativa, Facultad de Ciencias Economicas y Empresanales, Universidad de Oviedo, Campus del Cristo, 33006 Oviedo, Spain (c) Departamento de Economia y Gestion Forestal, ETS Ingeniero de Montes Universidad Politecnica de Madrid, Avda, Complutense s/n 28040 Madrid, Spain Article History: Received 8 June 2004; Accepted 26 April 2005

A study develops a meta-goal programming, which can be used by decision makers to fix targets through interactive procedure.

To link to full-text access for this article, visit this link: http://dx.doi.org/10.1016/j.ejor.2004.08.042 Byline: Rafael Caballero, Monica Hernandez Keywords: Goal programming; Fractional programming; Restoration of efficiency Abstract: The problem resulting from a goal programming problem with linear fractional criteria is not easy to solve due to the non-linear constraints inherent in its formulation. This paper introduces a simple and reliable test to establish whether a linear fractional goal programming problem has solutions that verify all goals and, if so, how to find them by solving a linear programming problem. This paper also outlines a new technique for restoring efficiency based on a minimax philosophy. An example is presented. Author Affiliation: Department of Applied Economics (Mathematics), University of Malaga, Campus El Ejido s/n, 29071 Malaga, Spain Article History: Received 4 February 2003; Accepted 30 August 2004

A solution to goal programming problem, using multiple criteria decision making, is presented.

ABSTRACT The monthly variation in the biochemical composition of the digestive gland, the adductor muscle and the gonad and its relationship with the gonadal index (GI) were studied during 2 [...]

The problem of achieving efficient solutions in stochastic multiobjective programming by combining the techniques of both stochastic and multiobjective approaches, in a double transformation of the problem, is investigated. It is identified from the investigation that the order of transformation followed may have its impact on the efficiency of the solutions provided.

Land-based systems of low stocking rate are usually associated with low environmental concerns. However, we hypothesized that traditional unrestructured grazing systems may represent more environmental costs than benefits. Community-based research work was conducted in the 1990s in the cereal--sheep system of Castile--La Mancha, with the objective of assessing the typology of the system, including the grazing structures and social groups involved in grazing management. Results showed that traditional management represented a heterogeneous spatio-temporal distribution of grazing at the regional level, with high variability at the local level. Occasional overgrazing may occur, but a trend towards fewer grazing-days with correspondingly more indoor feeding and widespread underuse of grazing resources was more apparent. Proposals to improve the grazing system reached a high degree of consensus among farmers, but the two social groups (cultivators and landless pastoralists) showed divergent views in some important management issues. The imbalance in group numbers indicated that collective action is difficult to achieve. Implementation of improved grazing practices would require a policy intervention based on the disentangled constraints and consensus among interested farmers. Key words: agro-pastoral systems, environmental implications, grazing management, landless tenure, Mediterranean agriculture

Goal Programming is considered with regard to GP variants in decision making, and the Meta-GP approach is examined.

Farming systems are often complex elaborations of the human societies to which they belong, with many traditional and social implications. Untangling the main social and structural constraints may improve productivity without an increase in environmental costs. Integration of cereal and sheep farming throughout the Mediterranean basin has been traditional. Mutually beneficial relationships between the sheep industry and cereal farming, and the vital role of forage legumes in meeting the modest needs of sheep for nitrogen, are stressed. This agropastoral system, however, is endangered in central Spain mainly because the pastoralist (sheep owner) is land-less, while the cultivator (land owner) has little interest in enforcing the law; fees for grazing rights are very low. Farmers, particularly young farmers, reject the current sheep operation because of the harsh working conditions. Restructuring of the grazing system would require a new policy scenario that would link European Union farm subsidies to structural reforms and would stress cooperative behavior.

The weakness of Goal Programming algorithms in producing dominated solutions have been improved by the use of an interactive algorithm that uses the decision maker's input of priority levels. The interactive algorithm allows for the recycling of generated values of the objective functions, with each recycling generating a Pareto optimal evaluation measure which in the end is evaluated for a single value, representing the optimal solution to the problem.