- Research
- Open Access
Duality in nonlinear programming problems under fuzzy environment with exponential membership functions
- S. K. Gupta^{1}Email author,
- D. Dangar^{2},
- I. Ahmad^{3} and
- S. Al-Homidan^{3}
https://doi.org/10.1186/s13660-018-1812-x
© The Author(s) 2018
- Received: 11 April 2018
- Accepted: 13 August 2018
- Published: 22 August 2018
Abstract
In this paper, we have established appropriate duality relations for a general nonlinear optimization problem under fuzzy environment, taking exponential membership functions and using the aspiration level approach. A numerical example has also been shown to justify the results presented in the paper.
Keywords
- Fuzzy optimization
- Duality results
- Exponential membership function
- Mangasarian dual
MSC
- 90C26
- 90C30
- 90C70
- 49N15
1 Introduction
Zadeh in 1965 introduced fuzzy set theory by publishing the first article in this area. He generalized the classical notion of a set and a proposition to accommodate fuzzyness. This has been applied in diverse fields such as machine learning, multi-attribute decision making, supply chain problems, management sciences, etc. Fuzzy control, which directly uses fuzzy rules, is the important application in fuzzy theory. Fuzzy set theory is also applicable in the real life case like controlling smart traffic light. The controller is designed in such a way that it changes the cycle time depending upon the densities of cars behind red and green lights.
The fuzzy set theory provides various logical operators that allow the aggregation of several criteria to just one criterion. These operators can be evaluated with respect to axiomatic requirements, numeric efficiency robustness, degree of compensation among the criteria, and ability to model expert behavior.
Bellman and Zadeh [1] proposed the idea of decision making in fuzzy environment. After the pioneering work on fuzzy linear programming problems (FLPP) in Tanaka et al. [2] and Zimmermann [3], several kinds of (FLPP) along with the different solution methodologies have been discussed in the literature. Many researchers, including Lai and Hwang [4], Shaochang [5], Buckley [6, 7], and Negi [8], have considered the problems where all parameters are fuzzy. Lai and Hwang [4] assumed that the parameters have a triangular possibility distribution. Using multiobjective linear programming methods, they provided an auxiliary model related to it.
Rodder and Zimmermann [9] were the first who studied the duality of (FLPP), considering the economic interpretation of the dual variables. After that, many interesting results regarding the duality of (FLPP) have been investigated by several researchers [10–18]. Zhang et al. [19] investigated the duality theory in fuzzy mathematical programming problems with fuzzy coefficients. Ovchinnikov [20] characterized Zadeh’s extension principle in terms of the duality principle. Introducing the concept of convex fuzzy variables for fuzzy constrained programming, Yang [21] proved a convexity theorem with convex fuzzy parameters and a duality theorem for fuzzy linear constrained programming. Later on, Farhadinia and Kamyad [22] extended the duality theorems for the crisp conic optimization problems to the fuzzy conic programming problems based on the convexity-like concept of fuzzy mappings and the parameterized representation of fuzzy numbers.
The paper is organized as follows. In Sect. 2, we construct a general fuzzy nonlinear programming problem and formulate its Mangasarian type dual. Further, we prove duality theorems using exponential membership functions under convexity assumptions. In the next section, we illustrate a numerical example.
2 Definitions and preliminaries
In the crisp sense, a general nonlinear primal-dual pair can be expressed as follows:
Let the aspiration levels corresponding to the objective function of primal (MP) and dual (MD) be denoted by \(z_{0}\) and \(w_{0}\), respectively.
Now, the above crisp pair (MP) and (MD) can be described in the fuzzy sense as the following pair (MP̃) and (DD̃):
Theorem 2.1
For each feasible point of the problem (MP̃) there exists ξ, \(0 \leq\xi\leq1\) such that \((x,\xi)\) satisfies the constraints (1)–(3) of (PP-1).
Proof
Let \(x\in\widetilde{D}\). Then there exist some \(\hat {p}_{0}\), \(0 \leq\hat{p}_{0} \leq p_{0}\), \(\hat{p}_{i}\), \(0 \leq\hat{p}_{i} \leq p_{i}\), \(i=1,2,\ldots,m\), such that \(x\in \mathit{FR}(\hat{p}_{0},\hat{p})\) and its membership value is given by \(\mu_{\widetilde{\mathit{FR}}}(\mathit{FR}(\hat{p}_{0},\hat {p}))=\min\{\hat{\mu}_{0},\ldots,\hat{\mu}_{m}\}\).
Theorem 2.2
Suppose \(x_{0}\) and \((x_{0},u_{0})\) are the feasible solutions of (MP) and (MD), respectively. If the corresponding objective value of (MP) fully (partially) satisfies the goal \(z_{0}\), then the weak duality theorem between (MP̃) and (DD̃) holds (partially holds). That is, \(z_{0}\geq w_{0}\) (\(z_{0}+p_{0}\geq w_{0}-q_{0}\)).
Proof
Theorem 2.3
(Modified weak duality)
Proof
Theorem 2.4
Proof
3 Numerical illustration
Declarations
Acknowledgements
The authors wish to thank the referees for several valuable suggestions which have considerably improved the presentation of the paper.
Funding
This research is supported by the King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, under the Internal Project No. IN161058.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970) MathSciNetView ArticleMATHGoogle Scholar
- Tanaka, H., Asia, K.: Fuzzy solution in fuzzy linear programming problems. IEEE Trans. Syst. Man Cybern. Syst. 14, 325–328 (1984) View ArticleMATHGoogle Scholar
- Zimmermann, H.J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978) MathSciNetView ArticleMATHGoogle Scholar
- Lai, Y.J., Hwang, C.L.: A new approach to some possibilistic linear programming problem. Fuzzy Sets Syst. 49, 121–133 (1992) MathSciNetView ArticleGoogle Scholar
- Shaocheng, T.: Interval number and fuzzy number linear programming. Fuzzy Sets Syst. 66, 301–306 (1994) MathSciNetView ArticleGoogle Scholar
- Buckley, J.J.: Possibilistic linear programming with triangular fuzzy numbers. Fuzzy Sets Syst. 26, 135–138 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Buckley, J.J.: Solving possibilistic linear programming problems. Fuzzy Sets Syst. 31, 329–341 (1989) MathSciNetView ArticleMATHGoogle Scholar
- Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming Methods and Applications. Springer, Berlin (1992) View ArticleMATHGoogle Scholar
- Rodder, W., Zimmermann, H.J.: Duality in fuzzy linear programming. In: Internat. Symp. on Extremal Methods and Systems Analysis, University of Texas at Austin, pp. 415–427 (1980) View ArticleGoogle Scholar
- Bector, C.R., Chandra, S.: On duality in linear programming under fuzzy environment. Fuzzy Sets Syst. 125, 317–325 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Bector, C.R., Chandra, S., Vidyottama, V.: Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optim. Decis. Mak. 3, 255–269 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Bector, C.R., Chandra, S., Vijay, V.: Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs. Fuzzy Sets Syst. 146, 253–269 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Liu, Y., Shi, Y., Liu, Y.H.: Duality of fuzzy MC^{2} linear programming: a constructive approach. J. Math. Anal. Appl. 194, 389–413 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Ramik, J.: Duality in fuzzy linear programming: some new concepts and results. Fuzzy Optim. Decis. Mak. 4, 25–39 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Verdegay, J.L.: A dual approach to solve the fuzzy linear programming problems. Fuzzy Sets Syst. 14, 131–141 (1984) MathSciNetView ArticleMATHGoogle Scholar
- Wu, H.C.: Duality theory in fuzzy linear programming problems with fuzzy coefficients. Fuzzy Optim. Decis. Mak. 2, 61–73 (2003) MathSciNetView ArticleGoogle Scholar
- Richardt, J., Karl, F., Miller, C.: A fuzzy dual decomposition method for large-scale multiobjective nonlinear programming problems. Fuzzy Sets Syst. 96, 307–334 (1998) View ArticleGoogle Scholar
- Sakawa, M., Yano, H.: A dual approach to solve the fuzzy linear programming problems. Fuzzy Sets Syst. 67, 19–27 (1994) View ArticleMATHGoogle Scholar
- Zhang, C., Yuan, X.H., Lee, E.S.: Duality theory in fuzzy mathematical programming problems with fuzzy coefficients. Comput. Math. Appl. 49, 1709–1730 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Ovchinnikov, S.: The duality principle in fuzzy set theory. Fuzzy Sets Syst. 42, 133–144 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Yang, X.: Some properties for fuzzy chance constrained programming. Iran. J. Fuzzy Syst. 8, 1–8 (2011) MathSciNetMATHGoogle Scholar
- Farhadinia, B., Kamyad, A.V.: Weak and strong duality theorems for fuzzy conic optimization problems. Iran. J. Fuzzy Syst. 1, 143–152 (2013) MathSciNetMATHGoogle Scholar