A STUDY ON FUZZY RANDOM PORTFOLIO SELECTION PROBLEMS BASED ON POSSIBILITY AND NECESSITY MEASURES

This paper incorporates fuzzy random variables with a portfolio selection problem based on the single index model. The rate of return on each investment can be represented with a fuzzy random variable. A novel decision-making model based both on possibilistic programming and on stochastic programming. It is shown that the formulated problem is transformed into the deterministic equivalent nonlinear programming problem. The deterministic problem is solved by utilizing the property that the problem is regarded as a convex programming problem including a parameter. 1 Introduction There are two different kinds of decision-making models through math- ematical programming in uncertain circumstances. One is stochastic programming and the other is fuzzy programming. The former is a useful tool in stochastic systems and the latter in fuzzy systems. A comparative study between two programs have also been investigated (1), (2). In real systems, however, fuzzy information and random factor may arise at the same time. Then we are often faced with the case where fuzziness and randomness cannot be separable as information. A fuzzy random variable (3, 4, 5, 6) was defined in order to represent the element containing fuzzy and random information simultaneously. In recent years, some researchers have applied fuzzy random variables to various decision-making problems such as linear programming problem (7, 8) and minimum spanning tree problems (9, 10). In this article we focus on a single index model of a portfolio selection problem. Previ- ous studies (11, 12) express the rate of return on each investment with a random variable. However, in a case where an expert estimates the value of the rate of return, it includes not only randomness but also fuzziness. Since stochastic programming models cannot take into account of fuzzy information by the expert, it is necessary to introduce a framework of new mathematical programming models to portfolio selection problems. In order to pro- vide such a model, we try to incorporate stochastic programming models with possibilistic programming models. Later, it will be shown that the model has an advantage that an optimal solution of the deterministic equivalent problem is relatively easy to solve in spite that the problem is a nonconvex programming problem. The rest of this paper is organized as follows: Section 2 roughly explains a single index model of a portfolio selection problem. Section 3 formulates a fuzzy random portfolio se- lection problem. Section 4 proposes a fuzzy random programming model using the concept of possibility measure and shows the process of transforming the problem including both fuzziness and randomness into the deterministic equivalent problem. Regarding the deter- ministic problem as a convex programming problem including a parameter, we provide a solution method to obtain an optimal solution. In Section 5, we consider a model using a necessity measure. Finally section 6 summaries this paper.