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In this paper, an axiomatic approach to Pareto set reduction problem is considered. The reduction is based on accounting for preferences of a decision maker which are modeled with the use of a type-2 fuzzy binary relation. This relation is only partially known through a set of so-called information quanta. Since these quanta are provided by the decision maker, it must be verified that they are consistent with the requirements of the axiomatic approach — the axioms of rational choice. Several theorems giving necessary and sufficient conditions of quanta consistency are proved. Ideas for dealing with inconsistencies are discussed with a few examples.

In this paper, we propose a genetic algorithm based design procedure for a radial-basis function neural network. A Hierarchical Rank Density Genetic Algorithm (HRDGA) is used to evolve the neural network's topology and parameters simultaneously. Compared with traditional genetic algorithm based designs for neural networks, the hierarchical approach addresses several deficiencies highlighted in literature. In addition, the rank-density based fitness assignment technique is used to optimize the performance and topology of the evolved neural network to deal with the confliction between the training performance and network complexity. Instead of producing a single optimal solution, HRDGA provides a set of near-optimal neural networks to the designers so that they can have more flexibility for the final decision-making based on certain preferences. In terms of searching for a near-complete set of candidate networks with high performances, the networks designed by the proposed algorithm prove to be competitive, or even superior, to three other traditional radial-basis function networks for predicting Mackey–Glass chaotic time series.

Fuzzy spatial models map a substantial degree of preference indifference. It has been shown that different definitions of covering result in different elements in the uncovered set when preference indifference is present. We consider several of the most frequently used definitions of covering relations found in the literature. The first definition that we examine yields an uncovered set, some of the elements of which are not Pareto efficient. Given that there is no reason to expect a set of players comprising a majority to settle for a Pareto deficient outcome, the remainder of the paper considers the ability of alternative definitions to avoid such a result.

We examine the effect of indifference on the existence of a majority rule maximal set. In our setting, it is shown in all but a limited number of cases that the maximal set is empty in an n-dimensional spatial model if and only if the Pareto set contains a union of cycles. The elements that constitute the exception are completely characterized.