Optimization of fuzzy-valued functions using Zadeh's extension principle
Introduction
Fuzzy optimization concerns many applications and theoretical approaches. Bellman and Zadeh's classical paper [1] on decision-making proposes a method for finding a value for a variable x with goals and constraints in a fuzzy environment. Considering x a crisp value and goals such as “x should be in the vicinity of ”, fuzziness models the non-sharpness of the equality symbol. Many other studies follow this approach in the framework of fuzzy mathematical programming [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Essentially, fuzzy sets are used to model vagueness and ambiguity. In the first case, known as flexible programming, fuzziness corresponds to flexibility in the goals and elasticity in the constraints (see [14]). The second case deals with ambiguity in the coefficients of the objective function and the constraints, class of problem known as possibilistic programming (see [15]).
Some authors have been studying optimization of fuzzy-set-valued functions with real domain. In that line, different concepts of fuzzy differentiability are employed (see [16], [17], [18] for more information on fuzzy calculus). For instance, [19] obtains KKT optimality conditions to classes of problems where the objective functions are level-wise differentiable and H-differentiable and the constraint functions are real-valued. The solutions are non-dominated and they use a partial order, also known as fuzzy-max order. A similar result is found in [20], in which the authors also explore the relationship between convexity and minimum points. Following the same line, [21], [22] develop analogous theory making use of more generalized types of differentiability (generalized Hukuhara and strongly generalized differentiabilities) and the same partial order that [19] used. Prior to it, [23] had already used this partial order, however, his objective was to optimize real-valued function where partial order was employed only in the formulation of fuzzy constraints. As for numerical methods, [24] improves the Newton method previously proposed by [25] to find non-dominated solutions to a class of problems where the objective functions are generalized Hukuhara differentiable and the constraint functions are real-valued.
Zadeh's extension principle is a fuzzy version of united extension [26], that is, it extends a point-to-point function to a set-to-set function. Since its definition in [27], it has been widely used in the development of theory and applications. For instance, this principle underlies the four operations of arithmetic on fuzzy numbers [28]. As a consequence, it plays an important role in areas such as systems of linear equations with fuzzy coefficients, fuzzy random variables for the handling of linguistic or imprecise statistical data and fuzzy regression methods [29]. Fuzzy differential equations also makes exhaustive use of this tool (see [16], [17], [18], [30], [31], [32]). The authors in [32] argue that Zadeh's extension of the classical solution is the best approach to solve a fuzzy differential equation modeling a linguistic concept evolving over time and exemplify with human growth data, comparing the results with Hukuhara derivative approach. We propose a method for optimizing fuzzy-valued functions given by Zadeh's extension principle. Here, such functions do not necessarily represent evolving concepts but the theory we develop also apply for them as particular case, what has never been done before.
The problem proposed in this study is a kind of optimization approach to find a fuzzy number that locally minimizes , where the fuzzy number is a fixed parameter and is Zadeh's extension of a classical function where x is the independent variable and λ is a parameter. The fuzzy-number-valued function is the extension of f with respect to x and λ and is minimized using fuzzy order relations.
Similarities occur with the possibilistic programming approaches in [5], [15], such as the use of Zadeh's extension to restrict/evaluate the objective function and the search for a fuzzy set to obtain a solution (called possibly optimal solution set). As for the differences, our approach is not restricted to linear functions. Also, [5] is interested in finding distributions to indicate which is the most suitable crisp solution according to the risk the decision maker is willing to accept to maximize profit. On the other side, we look to the whole fuzzy subset, as in [32]. Not being interested in choosing a singular crisp number, we propose the use of an order relation to obtain the optimal fuzzy set, while [15] uses indices defined by possibility and necessity measures to evaluate crisp solutions. In the latter, fuzziness means vagueness or ambiguity, whereas here it may mean membership degree only.
This study also takes a substantially different direction from those presented in [19], [20], [21], [22], [23], [24], [25]. Firstly, we are interested in finding a fuzzy value - not a crisp one - that minimizes the objective function. Secondly, we use Zadeh's extension principle to define the fuzzy-valued function. No fuzzy differentiability is used. Finally, we deploy a unusual partial order relation to find the smallest element of the fuzzy-number-valued function. The proposed way of resolution is based on optimality conditions proved in Section 3. As a consequence, the same minimizing point provides the minimal value in the usual fuzzy-max order relation.
Section snippets
Preliminaries
This section provides notation, definitions and results required for the development of the theory and the main result regarding optimization of a fuzzy function.
Let be a fuzzy subset of and denote by the membership function of . The support of is the classical set . The α-levels are defined by for all and .
Definition 1 (See [17].) A fuzzy subset is a fuzzy number if and all α-levels of are nonempty, with
Results
The following results provide means to demonstrate the main theorem. Theorem 2 connects the left endpoints of with the endpoints of and .
Theorem 2 Let f be a continuous function and monotonic in the second argument λ such that the local minimum function exists, and . Let be Zadeh's extension of ρ at , that is, . If f is increasing in the second argument and ρ is increasing, then for all . If f is increasing in the second
Examples
In what follows one function satisfying the conditions of Theorem 3 and two functions satisfying the conditions of Corollary 2 will illustrate the results of the previous section. For each function the conditions will be checked and the local minimum point function ρ will be calculated in order to provide the local minimizing point and the respective fuzzy function value. Graphics will illustrate the membership functions of the parameter , the local minimum point , the optimal value
Conclusion
We have proposed a fuzzy optimization problem different from those in the literature. The various already existing problems and methods were briefly reviewed in the Introduction. Although there are similarities with the possibilistic programming approach, as the evaluation of a set of solutions, our problem is quite different, e.g., instead of employing possibility and necessity measures to choose one solution, we use partial order for fuzzy numbers to obtain a fuzzy subset as optimal solution.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors thank the anonymous reviewers for their valuable suggestions which improved the quality of the paper. The first author also acknowledges the Brazilian National Council for Scientific and Technological Development CNPq for having supported this research (grant number 160747/2013-9).
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This work was partially supported by the Brazilian National Council for Scientific and Technological Development CNPq, grant number 160747/2013-9.