Mariano Luque

Global formulation for interactive multiobjective optimization

Interactive methods are useful and realistic multiobjective optimization techniques and, thus, many such methods exist. However, they have two important drawbacks when using them in real applications. Firstly, the question of which method should be chosen is not trivial. Secondly, there are rather few practical implementations of the methods. We introduce a general formulation that can accommodate several interactive methods. This provides a comfortable implementation framework for a general interactive system. Besides, this implementation allows the decision maker to choose how to give preference information to the system, and enables changing it anytime during the solution process. This change-of-method option provides a very flexible framework for the decision maker.

A classification of the weighting schemes in reference point procedures for multiobjective programming

The reference point-based methods form one of the most widely used class of interactive procedures for multiobjective programming problems. The achievement scalarizing functions used to determine the solutions at each iteration usually include weights. In this paper, we have analysed nine weighting schemes from the preferential point of view, that is, examining their performance in terms of which reference values are given more importance and why. As a result, we have carried out a systematic classification of the schemes attending to their preferential meaning. This way, we distinguish pure normalizing schemes from others where the weights have a preferential interpretation. This preferential behaviour can be either designed (thus, predetermined) by the method, or decided by the decision maker. Besides, several figures have been used to illustrate the way each scheme works. This paper enables the potential users to choose the most appropriate scheme for each case.

Using interactive multiobjective methods to solve DEA problems with value judgements

Data envelopment analysis (DEA) is a performance measurement tool that was initially developed without consideration of the decision maker (DM)s preference structures. Ever since, there has been a wide literature incorporating DEA with value judgements such as the goal and target setting models. However, most of these models require prior judgements on target or weight setting. This paper will establish an equivalence model between DEA and multiple objective linear programming (MOLP) and show how a DEA problem can be solved interactively without any prior judgements by transforming it into an MOLP formulation. Various interactive multiobjective models would be used to solve DEA problems with the aid of PROMOIN, an interactive multiobjective programming software tool. The DM can then search along the efficient frontier to locate the most preferred solution where resource allocation and target levels based on the DMs value judgements can be set. An application on the efficiency analysis of retail banks in the UK is examined. Comparisons of the results among the interactive MOLP methods are investigated and recommendations on which method may best fit the data set and the DMs preferences will be made.

Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection

Graphical abstractDisplay Omitted HighlightsWe consider a constrained three-objective optimization portfolio selection problem.We solve the problem by means of evolutionary multi-objective optimization.New mutation, crossover and reparation operators are designed for this problem.They are tested in several algorithms for a data set from the Spanish stock market.Results for two performance metrics reveal the effectiveness of the new operators. In this paper, we consider a recently proposed model for portfolio selection, called Mean-Downside Risk-Skewness (MDRS) model. This modelling approach takes into account both the multidimensional nature of the portfolio selection problem and the requirements imposed by the investor. Concretely, it optimizes the expected return, the downside-risk and the skewness of a given portfolio, taking into account budget, bound and cardinality constraints. The quantification of the uncertain future return on a given portfolio is approximated by means of LR-fuzzy numbers, while the moments of its return are evaluated using possibility theory. The main purpose of this paper is to solve the MDRS portfolio selection model as a whole constrained three-objective optimization problem, what has not been done before, in order to analyse the efficient portfolios which optimize the three criteria simultaneously. For this aim, we propose new mutation, crossover and reparation operators for evolutionary multi-objective optimization, which have been specially designed for generating feasible solutions of the cardinality constrained MDRS problem. We incorporate the operators suggested into the evolutionary algorithms NSGAII, MOEA/D and GWASF-GA and we analyse their performances for a data set from the Spanish stock market. The potential of our operators is shown in comparison to other commonly used genetic operators and some conclusions are highlighted from the analysis of the trade-offs among the three criteria.

A preference-based evolutionary algorithm for multiobjective optimization: the weighting achievement scalarizing function genetic algorithm

When solving multiobjective optimization problems, preference-based evolutionary multiobjective optimization (EMO) algorithms introduce preference information into an evolutionary algorithm in order to focus the search for objective vectors towards the region of interest of the Pareto optimal front. In this paper, we suggest a preference-based EMO algorithm called weighting achievement scalarizing function genetic algorithm (WASF-GA), which considers the preferences of the decision maker (DM) expressed by means of a reference point. The main purpose of WASF-GA is to approximate the region of interest of the Pareto optimal front determined by the reference point, which contains the Pareto optimal objective vectors that obey the preferences expressed by the DM in the best possible way. The proposed approach is based on the use of an achievement scalarizing function (ASF) and on the classification of the individuals into several fronts. At each generation of WASF-GA, this classification is done according to the values that each solution takes on the ASF for the reference point and using different weight vectors. These vectors of weights are selected so that the vectors formed by their inverse components constitute a well-distributed representation of the weight vectors space. The efficiency and usefulness of WASF-GA is shown in several test problems in comparison to other preference-based EMO algorithms. Regarding a metric based on the hypervolume, we can say that WASF-GA has outperformed the other algorithms considered in most of the problems.

Hierarchial generation of pareto optimal solutions in large-scale multiobjective systems

In this paper, the problem of the determination of Pareto optimal solutions for certain large-scale systems with multiple conflicting objectives is considered. As a consequence, a two-level hierarchical method is proposed, where the global problem is decomposed into smaller multiobjective problems (lower level) which are coordinated by an upper level that has to take into account the relative importance assigned to each subsystem. The scheme that has been developed is an iterative one, so that a continuous information exchange is carried out between both levels in order to obtain efficient solutions for the initial global problem. The practical implementation of the developed scheme allows us to prove its efficiency in terms of processing time.

Multiple objectives decomposition-coordination methods for hierarchical organizations

Abstract In this paper, the integration of goal programming models and hierarchical programming models is analyzed. The systems under study are assumed to consist of interconnected subsystems with multiple goals in each. Three possible cases regarding the number of decision makers will be considered: (1) one decision maker for the overall goals and one decision maker for each subsystem, (2) conflicting decision makers who are interested in their subsystems, and (3) just one decision maker for the overall system. Next, conditions are stated under which the problem of obtaining satisfying solutions for problems (1) and (3) can be reduced to the problem of obtaining satisfying solutions for the case (2). In order to determine such solutions, hierarchical techniques which exploit the structure of a decomposable system are analyzed. The empirical implementation of the two algorithms proposed shows their efficiency in terms of processing time.

Equivalent Information for Multiobjective Interactive Procedures

Despite the mathematical properties and algorithmic features of an interactive method, that methods 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.

An application of reference point techniques to the calculation of synthetic sustainability indicators

Sustainability is nowadays a key factor to analyse the development of the societies. Therefore, measuring sustainability is a main concern of the scientific community. The basic necessity to simultaneously consider the economical, social and environmental aspects make sustainability, by nature, a multicriteria concept, and therefore, multicriteria techniques are to be used to measure it. In this paper, we propose a method to develop synthetic sustainability indicators, based on the double (reservation–aspiration) reference point approach. This scheme is applied to each territorial unit considered, in order to determine, on the base of a given set of indicators, a couple of synthetic indicators that measure the weak and the strong sustainability of the unit.

Global wasf-ga: An evolutionary algorithm in multiobjective optimization to approximate the whole pareto optimal front

In this article, we propose a new evolutionary algorithm for multiobjective optimization called Global WASF-GA (global weighting achievement scalarizing function genetic algorithm), which falls within the aggregation-based evolutionary algorithms. The main purpose of Global WASF-GA is to approximate the whole Pareto optimal front. Its fitness function is defined by an achievement scalarizing function (ASF) based on the Tchebychev distance, in which two reference points are considered (both utopian and nadir objective vectors) and the weight vector used is taken from a set of weight vectors whose inverses are well-distributed. At each iteration, all individuals are classified into different fronts. Each front is formed by the solutions with the lowest values of the ASF for the different weight vectors in the set, using the utopian vector and the nadir vector as reference points simultaneously. Varying the weight vector in the ASF while considering the utopian and the nadir vectors at the same time enables the algorithm to obtain a final set of nondominated solutions that approximate the whole Pareto optimal front. We compared Global WASF-GA to MOEA/D (different versions) and NSGA-II in two-, three-, and five-objective problems. The computational results obtained permit us to conclude that Global WASF-GA gets better performance, regarding the hypervolume metric and the epsilon indicator, than the other two algorithms in many cases, especially in three- and five-objective problems.