On possibility-degree formulae for ranking interval numbers

The Role of Metaheuristics as Solutions Generators

Optimization problems are ubiquitous nowadays. Many times, their corresponding computational models necessarily leave out of consideration several characteristics and features of the real world, so trying to obtain the optimum solution can not be enough for a problem solving point of view. The aim of this paper is to illustrate the role of metaheuristics as solutions’ generators in a basic problem solving framework. Metaheuristics become relevant in two modes: firstly because every run (in the case of population based techniques) allows to obtain a set of potentially good solutions, and secondly, if a reference solution is available, one can set up a new optimization problem that allows to obtain solutions with similar quality in the objectives space but maximally different structure in the design space. Once a set of solutions is obtained, an example of an a posteriori analysis to rank them according with decision maker’s preferences is shown. All the problem solving framework steps, emphasizing the role of metaheuristics are illustrated with a dynamic version of the tourist trip design problem (for the first mode), and with a perishable food distribution problem (for the second one). These examples clearly show the benefits of the problem solving framework proposed. The potential role of the symmetry concept is also explored.

Flexibility Degree of Fuzzy Numbers and its Implication to a Group-Decision-Making Model

The theory of fuzzy sets considers that everything exhibits some elasticity and is a matter of degree. When fuzzy numbers are used to evaluate the judgements of decision makers (DMs) in pairwise comparisons of alternatives following the analytic hierarchy process, the flexibility experienced by DMs has been exhibited. In order to capture this aspect of flexibility, it is important to know how to realize the flexibility degree of fuzzy numbers and further present a method of realizing its quantification. In this paper, a definition of the flexibility degree of fuzzy numbers is proposed. Some formulas are proposed to quantify the flexibility and rigidity degrees of interval numbers, triangular fuzzy numbers, and trapezoidal fuzzy numbers. A group decision making (GDM) model is developed under the consideration of the flexibility of DMs. By considering the effects of the applied scale and the reciprocal relation, the flexibility degree of interval multiplicative reciprocal comparison matrices is further defined, which is used to evaluate the flexibility degree of the DM involved in the decision process. An RD-IOWGA operator is proposed to aggregate individual interval multiplicative reciprocal matrices by associating more importance to that with less flexibility. A new algorithm is shown to solve GDM problems with interval multiplicative reciprocal preference relations. Numerical studies are carried out to illustrate the new definitions and offer some comparative analysis. The observations reveal that the developed consensus method can be used to model the GDM with a dominant position.