Elsevier

Fuzzy Sets and Systems

Volume 404, 1 February 2021, Pages 23-37
Fuzzy Sets and Systems

Optimization of fuzzy-valued functions using Zadeh's extension principle

https://doi.org/10.1016/j.fss.2020.07.007Get rights and content

Abstract

This paper proposes a fuzzy optimization problem whose objective function is Zadeh's extension of a function with respect to a parameter and the independent variable. Making use of a partial order relation, the extension of a function mapping each parameter to the corresponding local minimizing point is proven to provide the local smallest fuzzy value for the extended function. Deploying the usual order relation in the literature, the same point is proven to provide the minimal value for the function. Examples illustrate the results.

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 x0”, 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 xˆ that locally minimizes fˆ(xˆ,λˆ), where the fuzzy number λˆ is a fixed parameter and fˆ is Zadeh's extension of a classical function f:(x,λ)f(x,λ) where x is the independent variable and λ is a parameter. The fuzzy-number-valued function fˆ 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 uˆ be a fuzzy subset of R and denote by μuˆ:R[0,1] the membership function of uˆ. The support of uˆ is the classical set suppuˆ={xR:μuˆ(x)>0}. The α-levels are defined by [uˆ]α={xR|μuˆ(x)α} for all α(0,1] and [uˆ]0=suppuˆ.

Definition 1

(See [17].) A fuzzy subset uˆ is a fuzzy number if μuˆ:R[0,1] and

  • 1.

    all α-levels of uˆ are nonempty, with 0

Results

The following results provide means to demonstrate the main theorem. Theorem 2 connects the left endpoints of [fˆ]α with the endpoints of [xˆ]α and [λˆ]α.

Theorem 2

Let f be a continuous function and monotonic in the second argument λ such that the local minimum function ρ:ΛΩϵ exists, λˆF(R) and [λˆ]0Λ. Let xˆ be Zadeh's extension of ρ at λˆ, that is, xˆ=ρˆ(λˆ).

  • (i)

    If f is increasing in the second argument and ρ is increasing, then fˆ(xˆ,λˆ)Lα=f(xLα,λLα) for all α[0,1].

  • (ii)

    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 λˆ, the local minimum point xˆ=ρˆ(λˆ), 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).

References (35)

  • M. Panigrahi et al.

    Convex fuzzy mapping with differentiability and its application in fuzzy optimization

    Eur. J. Oper. Res.

    (2008)
  • R. Osuna-Gómez et al.

    Necessary and sufficient conditions for fuzzy optimality problems

    Fuzzy Sets Syst.

    (2016)
  • J. Ramík et al.

    Inequality relation between fuzzy numbers and its use in fuzzy optimization

    Fuzzy Sets Syst.

    (1985)
  • Y. Chalco-Cano et al.

    On the Newton method for solving fuzzy optimization problems

    Fuzzy Sets Syst.

    (2015)
  • L.A. Zadeh

    Fuzzy sets

    Inf. Control

    (1965)
  • D. Dubois et al.

    The legacy of 50 years of fuzzy sets: a discussion

    Fuzzy Sets Syst.

    (2015)
  • M.T. Mizukoshi et al.

    Fuzzy differential equations and the extension principle

    Inf. Sci.

    (2007)
  • Cited by (0)

    1

    This work was partially supported by the Brazilian National Council for Scientific and Technological Development CNPq, grant number 160747/2013-9.

    View full text