Fuzzy portfolio selection problem is a major issue in the financial field and a special case of constrained fuzzy-valued optimization problems (CFOPs). In this respect, the present paper aims to investigate the CFOP with regard to the features of the parametric representation of fuzzy numbers named as convex constraint function (CCF) which is proposed by Chalco-Cano et al. in 2014. Furthermore, relying on this parametric representation, some proper conditions are provided for the existence of solutions to a CFOP. To this end, by the increasing representation of CCF, the main problem is converted to a parametric multiobjective programming problem and some solution concepts from a similar framework in the multiobjective programming are proposed for the CFOP. Eventually to illustrate the proposed results, the fuzzy portfolio selection problem is discussed.

1. Introduction

In fact, because of using the experimental and empirical data for modeling a real world phenomenon, a deterministic mathematical model may not be a perfectly realistic representation. There are several approaches to deal with such real world phenomena, for example, fuzzy techniques, stochastic models, and interval analysis, which differ by their advantages and disadvantages [1, 2]. However, in many practical situations, the uncertainties are not of the statistical or interval type; more precisely, this situation happens mainly through the modeling in terms of linguistic expressions that depend on the human judgment. In other words, an expert perceives exactly which values and parameters are possible and which are not. Therefore, the set of all possible values and parameters can be naturally described as fuzzy numbers by the expert’s knowledge.

Historically, fuzzy set theory was proposed by Zadeh in [3] and developed considerably by many other researchers. This theory provides conceptually powerful techniques to handle the imperfect information related to vagueness and imprecision.

Nowadays, the fuzzy optimization problem is effective in a lot of different disciplines related to optimization such as operations research engineering, economics, and artificial intelligence [47]. It can be said that the fuzzy optimization problem provides an appropriate choice for considering the vagueness and ambiguousness into the formulation and solutions of the multitude of optimization problems. Indeed, there are several motivations to apply fuzzy optimization model; first, it deals with some practical optimization problems more conveniently than conventional optimization model; also, fuzzy optimization model efficiently reduces information loss arising from the traditional optimization model; moreover, it allows the designer to implement linguistic constraints that may not be easily defined using more conventional optimization algorithms; finally, this model may permit managers to have not only one solution but also a set of them, so that the most suitable solution can be applied according to the state of existing decision of the production process at a given time and without increasing delay. Furthermore, accessing a set of solutions enables user to investigate and analyze the system information in more detail.

On the other hand, most of the common portfolio selection models deal with the uncertainty via probabilistic approaches, where those probabilistic approaches only partly capture the reality. In addition, there are some other techniques that manage the uncertainty of the financial markets as the fuzzy set theory. It is noteworthy that the fuzzy portfolio selection model integrates the quantitative and qualitative analysis, experts’ knowledge, and the investors’ opinion in a better manner [7]. Therefore, in this paper, the portfolio selection problem under fuzzy environment based on the constrained fuzzy optimization problem is going to be studied. Many modern computing methodologies can be seen for various fuzzy systems, for example, [814]. It is also worth mentioning that there are several results associated with parametric representations of fuzzy numbers [1517]. Recently, Chalco-Cano et al. [15] have proposed two parametric representations for interval numbers named as “increasing/decreasing convex constraint function” and then explicitly extended the proposed representations to the fuzzy case. The representations have the advantage of allowing flexible and easy-to-control shapes of the fuzzy numbers and it is very simple to implement [17]. Accordingly, this point of view and its increasing parametric representation motivate us here to study the fuzzy portfolio selection problem as an application of the constrained fuzzy-valued optimization problem. To this end, the arithmetic of fuzzy numbers and the calculus of fuzzy-valued functions are developed based on this parametric representation. Then, by parametric representation of fuzzy-valued function, the constrained fuzzy optimization problem is transformed into a deterministic multiobjective problem. Besides, some solution concepts from a similar framework in the multiobjective programming problem are proposed for the constrained fuzzy-valued optimization problem, by converting it to a general constrained optimization problem. Finally, it has been demonstrated that the solution of the general optimization is related to the solution of the main problem.

The rest of the paper is organized as follows. Section 2 is devoted to giving the definitions of fuzzy numbers and some arithmetic that are used later in the development of results in fuzzy environment. The fuzzy-valued functions in the parametric form and their properties, calculus and convexity, are studied in Section 3. In Section 4, the constrained fuzzy-valued optimization problem is discussed and, as an application, the proposed method is considered to the fuzzy-valued quadratic programming problem. In Section 5, two numerical examples are established to confirm the efficacy of the proposed approach; more particularly one of them reveals how to solve the fuzzy portfolio selection problem. At the end, the conclusion is made in Section 6.

2. Fuzzy Numbers and Their Arithmetic

In this section, some basic notations and results on the fuzzy arithmetic are presented; however, it is assumed that the reader is familiar with the fuzzy theory.

Definition 1 (see [18]). Let be a fuzzy set on the set of real numbers . The fuzzy set is a fuzzy number if it is a normal, convex, upper semicontinuous, and compactly supported.

The set of all fuzzy numbers on is denoted by . For all , -level set of any is defined as . The -level set is defined as the closure of the set . By Definition 1, for any and for each , is a compact and convex subset of and . can be recovered from its -level by a well-known decomposition theorem, which states that where denotes the algebraic product of a scalar with the -level set and union on the right-hand side is the standard fuzzy union.

As previously mentioned, fuzzy numbers and their arithmetic can be expressed in terms of parameters in the several models [1517]. Here, from increasing parametric representation [15], each -level of an arbitrary fuzzy number is represented alternatively by its bounds as follows:which is based on the convex combination of upper and lower bounds. Moreover, by the parametric representation (2), the -level of a -dimensional fuzzy vector and a fuzzy matrix can be represented as the set of real-valued vectors and matrices, respectively; that is,The parametric representation (2) helps us to build the fuzzy arithmetic, immediately. Nevertheless, the binary operations between two arbitrary fuzzy number can be defined in terms of parameter as follows.

Definition 2. For , the algebraic operations can be defined aswhere and .

Remark 3. It is clear that in general, where is the Hukuhara difference. However, it can be deduced that if, in (5), and is a nondecreasing function for all .

Definition 4. The product of -dimensional fuzzy vector and a -dimensional real vector is defined as , where it is a fuzzy number.

It is noteworthy that fuzzy numbers are frequently partial ordered. In fact, there are many ways to define the fuzzy order among the set of all fuzzy numbers [1922]. For example, Ramík and ímánek [22] proposed a partial order relation called the fuzzy-max order; Molinari [20] considered a new criterion of choice between generalized triangular fuzzy numbers and so on. In this paper, two specific partial ordering relations on fuzzy numbers using parametric representation are introduced.

Definition 5. For two arbitrary , with parametric representations and , it can be deduced that (i) if , ,(ii) if , .

It is easy to see that and are partial order relations on .

3. Fuzzy-Valued Function and Its Differential Calculus

A fuzzy-valued function is a function with fuzzy values, as , where is a subset of the vector space . Here, the fuzzy-valued functions with fuzzy coefficients are considered which allow us to express their -levels as a set of classical functions, using the parametric representation (2). To this end, let denote the set of all coefficients present in the fuzzy-valued function , respectively. Without loss of generality, one can consider as an ordered set with respect to its presence in the fuzzy-valued function (or as fuzzy vector like ). Then, for a given fuzzy vector , the -level of fuzzy-valued function can be considered asFor every fixed and , is continuous in ; consequently, and exist and

Example 6. Consider the fuzzy-valued function , where , , and , , and . For every , by the parametric representation (3), we have and so Because of the continuity of at , for every and , (9) provides that

Definition 7. Let be a fuzzy-valued function, where is a convex subset of . is called convex on with respect to iffor all and . Moreover, the fuzzy-valued function is convex with respect to , if (13) is valid for .

Remark 8. By Definitions 5 and 7, the convexity of fuzzy-valued function with respect to or can be deduced by respectively, for all and .

For any two arbitrary fuzzy vectors , the definition of algebraic operations between fuzzy-valued functions can be expressed, based on the parametric representation (8), as where .

It is obvious that a metric to define the distance between two arbitrary fuzzy numbers is required to introduce the differential calculus of a fuzzy-valued function in the parametric form. Hereinafter, for two arbitrary fuzzy numbers and , the quantitydescribes the distance between and , where It is easy to see that the upper metric is equivalent to the well-known Hausdorff metric [23].

Definition 9 (limit of fuzzy-valued function [24]). Let be a fuzzy-valued function and , . The limit of as approaches is the fuzzy number and we write , if for every there exists such that , whenever . Here, is the usual (Euclidean) norm in .

Moreover, the fuzzy-valued function is continuous at if and only if, for every , there exists such that

Proposition 10 (see [24]). The limit of fuzzy-valued function exists at , if exists for every andMoreover, is continuous at if and only if is continuous at for every and .

One of the first definitions of differentiability for fuzzy-valued functions is the Hukuhara differentiability, which suffers disadvantages in particular to the point that the inverse subtraction does not exist. The generalized Hukuhara derivative has been attempted to clear these difficulties, which is more general than Hukuhara derivative. Finally, the generalized derivative is proposed based on the generalized difference [23]. Roughly speaking, all these derivatives vary with respect to their corresponding differences. Here, based on Definition 2, the following derivative can be defined in terms of parameter.

Definition 11 (differentiability of fuzzy-valued function). The fuzzy-valued function is said to be differentiable at , if is differentiable at for every and and satisfies the assumptions of Stacking Theorem [23].

In addition, the fuzzy-valued function is said to be differentiable on if it is differentiable for all

Proposition 12 (see [24]). If the fuzzy-valued function is differentiable at , then there exists a fuzzy number such that

Remark 13. Consider the fuzzy-valued function defined as where and . Using gH-derivative and Definition 11, we have respectively, which are different. In fact, the sign of the independent variable is not considered in Definition 11, while the gH-derivative depends on the sign of . Therefore, in general, it cannot be expected that the derivatives of a fuzzy-valued function be equal.

The partial derivative of in the direction at the point can be defined in terms of its -level as provided that, for every , , exist and they are the -level of a fuzzy number. Moreover, the gradient of fuzzy-valued function (i.e., the partial derivatives at the point ) is defined as a fuzzy vector as follows:

Definition 14. Suppose that the second-order partial derivatives , at the point exist and are -levels of fuzzy number (i.e., satisfying the assumption of Stacking Theorem). Then, the Hessian matrix of at the given point is given by where It is apparent that

Definition 15. The fuzzy-valued function is said to be twice continuously differentiable at , if its Hessian matrix at that point (i.e., ) exists and all of its components are continuous functions. Also, is twice continuously differentiable on , if it is twice continuously differentiable for all .

Furthermore, using Definition 14 and Proposition 10, it can be shown that the fuzzy-valued function inherits the twice continuous differentiability of at , when , are -levels of a fuzzy number.

Definition 16. A fuzzy matrix is said to be symmetric if each of its -levels is symmetric. Moreover, by the parametric representation (4), a fuzzy matrix is positive definite (or positive semidefinite) if every is positive definite (or positive semidefinite).

Theorem 17 (see [24]). Let be a twice continuously differentiable fuzzy-valued function on the open convex set . The function is convex with respect to if and only if its fuzzy Hessian matrix is positive semidefinite for all .

4. Constrained Fuzzy-Valued Optimization Problem

Consider the following constrained fuzzy-valued optimization problem:where for and are fuzzy-valued functions with the parametric representationsAccording to the partial orderings as discussed in Definition 5 and the parametric representations (3) and (4), the feasible region of the CFOP can be expressed as

4.1. Solution Concepts and Optimality Conditions

In this section, a novel solution methodology was presented on the CFOP, which has depended on the definition of the corresponding optimal solution. Accordingly, it has been tried to define this concept based on the proposed partial orders. The feasible point is said to be an optimal solution of the CFOP with respect to , if and only if for all . Nevertheless, by the definition of the partial order , the CFOP can be handled via the following multiobjective problem: So, the solution of the CFOP can be interpreted as the solution of , which is conforming to the concept of an efficient solution of a multiobjective problem. Consequently, the solution concept of the CFOP can be determined based on the thought of dominance.

Definition 18. Let be the feasible set of the CFOP
(i) A point is an efficient solution of the CFOP if there is no , where(ii) A point is said to be a properly efficient solution of the CFOP, if it is an efficient solution and there is a real number such that there exists at least one , , with , whereas for some and every with

One of the main advantages of efficient solutions is to enable the decision maker to select one optimal solution that is matched best to his demand. In order to enhance the usefulness, the proposed solution concepts can be typically expanded as follows:(1)CFOP has a weak efficient solution at , whenever relation (32) is established for some .(2)CFOP has a strong efficient solution at , if, for all , relation (32) is valid.(3)CFOP has a strong independent efficient solution at , when is an efficient solution and it is independent of .(4)CFOP has no efficient solution, if there is no such that relation (32) is satisfied for any .Likewise, the concepts of weak, strong, strong independent, and no properly efficient solutions for the CFOP can be defined.

The fundamental idea to handle the considered multiobjective problem with infinity objective is to convert it to the following constrained single-objective optimization problem: where is a weight function and are mutually independent. It can be shown that the solutions of the COP can be related to the CFOP ones.

Theorem 19. If is an optimal solution of the COP, then it is a properly efficient solution of the CFOP.

Proof. By contradiction, assume that is not a properly efficient solution of the CFOP. Therefore, is not an efficient solution of the CFOP or the second part of Definition 18 (ii) is violated. Anyway, for some and with , pick out a continuous weight function . Then, by choosing , we have for all with . Consequently, By integrating with respect to , we have therefore which contradicts the assumption that is an optimal solution of the COP.

It is noteworthy that a CFOP is said to be a constrained fuzzy-valued convex programming problem if are convex functions with respect to or .

Theorem 20. If the CFOP is a constrained fuzzy-valued convex programming problem, then the COP is a constrained convex programming problem.

Proof. It is the same as the proof of Theorem of [25].

4.2. Constrained Fuzzy-Valued Quadratic Programming Problem

The constrained fuzzy-valued quadratic programming (CFQP) problem is a special case of the CFOP, when the fuzzy-valued function is quadratic and the constraints are linear in . Generally, the problem can be formulated as follows:where is a symmetric fuzzy matrix, , , and . According to (29), the feasible region of the CFQP can be obtained by the setIt is self-evident that is a convex set. Moreover, if the fuzzy Hessian matrix is positive semidefinite then, by Theorem 17, the objective function will also be a fuzzy-valued convex function with respect to . Consequently, the CFQP will be a convex optimization problem. On the other hand, its corresponding optimization problem with a weight function can be denoted bywhere , , and . By Theorem 20, the optimization problem (41) is also a constrained convex quadratic programming problem. Therefore, using KKT optimality conditions, its corresponding optimal solution can be obtained which is a properly efficient solution for (39).

Consider the feasible region ; accordingly, the Lagrange function is obtained as where and , , , and So, the KKT optimality conditions are where , , and

Furthermore, if the feasible region takes another form in (40), then the KKT optimality conditions can be determined in a similar way.

5. Numerical Examples

In this section, two examples are given to illustrate the efficiency of the proposed approach. In the first example, the various solutions are discussed in detail and in the second one a special problem, namely, the constrained portfolio selection problem, is expressed.

Example 1. Consider the following constrained fuzzy-valued quadratic programming problem:where , , , , , , and . The corresponding optimization problem with respect to iswhere . The fuzzy-valued function is convex (see Example [24]). Therefore, the constrained quadratic programming problem (46) is a convex programming problem, by Theorem 20. Accordingly, using the obtained result of Section 4.2, the KKT conditions for CQP are For a particular weight function , the above system can be simplified as For each , the solution of problem is . Therefore, the fuzzy-valued optimization problem (45) has a weak properly efficient solution at .

Now, by considering some changes in the fuzzy-valued optimization problem (45), it can be shown that the solution of problem changes.

Case 1. Set instead of . In this case, the following system may be obtained: Consequently, the problem has a strong independent properly efficient solution at .

Case 2. Set instead of , where is a triangular fuzzy number; we have which has no solution, and so the problem has no properly efficient solution.

Case 3. Set instead of . Therefore, the corresponding system is