یک مدل فازی محدودیت های نرم تصادفی برای مشکل انتخاب پرتفولیو

The financial market behavior is affected by several non-probabilistic factors such as vagueness and ambiguity. In this paper we develop a multistage stochastic soft constraints fuzzy program with recourse in order to capture both uncertainty and imprecision as well as to solve a portfolio management problem. The results we obtained confirm the studies carried out in literature addressed to integrate stochastic and possibilistic programming.

The largest part of mathematical models used in many areas of decision making has primarily a hard or crisp structure, i.e. the solutions are considered to be either feasible or unfeasible. This dichotomy often forces researchers to represent real behaviors by yes-or-not models unable to cover uncertainty, complexity, and vague or imprecise concepts and variables. This is particularly true if the problem includes: (a) vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence; (b) natural language to be modeled; (c) state variables that can be described only approximately. As stated in Zadeh’s principle of incompatibility: “as the complexity of the system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics” [20]. The growing interest in both uncertainty and vagueness sources and the different approaches by which they could be controlled, has contributed to develop new optimization models most of which have already been applied in many areas and in particular in financial modeling. Sophisticated techniques of stochastic programming and fuzzy theory have been used to solve real portfolio problems [12,8,11, 7,17]. We propose a multistage stochastic soft constraints fuzzy model with recourse to solve a portfolio management problem. The preliminary results confirm the importance of integrating stochastic programming with fuzzy logic in modeling real financial problems. The remaining sections of this paper are organized as follows. Section 2 is devoted to a brief survey of linear fuzzy programming. Section 3 is dedicated to a brief survey on stochastic programming. Section 4 shows the asset/liability management model whereas Section 5 explains our model. Finally, the empirical analysis is listed in Section 6. Section 7 concludes the paper.

The analysis we carry out allows us to make several estimates of the optimal values of the investments on portfolio in an environment with imprecise information. Such estimates hold the possible scenarios derived from a probabilistic approach and the degree of confidence of the decision maker with respect to the scenarios. The examples here proposed develop different relaxation in order to study how the behavior changes according to the degree of fuzziness, but, in reality one fuzzy model is sufficient. The benefit of this approach is to consider objective factors like scenarios in conjunctions with subjective ones and non-neutral attitude with respect to the experts’ estimates. The estimate is influenced by the subjectivity of the expert and consequently has to be handled as an imprecise information. On the other hand the deterministic part of the model could be affected by evaluation errors. Ignoring it could produce the uselessness of the model, with negative outcomes for the optimization. The study realized is to be considered like the first stage of a larger analysis to be carried out, but it gives interesting incentives to keep going. As far as the model is concerned, the fuzzy approach could be extended also to the variable coefficients and further (given the extension of the fuzzy approach to the probability [16] area) to the scenarios probabilities.