Multi objective programming problem in the hesitant fuzzy environment
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Multi objective programming problem in the hesitant fuzzy environment F. F. Rouhbakhsh1 · M. Ranjbar2 · S. Effati2,3 · H. Hassanpour1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we introduce a hesitant fuzzy multi-objective programming problem, in which the evaluation information provided by the decision makers is expressed in a hesitant fuzzy environment. For this purpose a new solution concept, namely hesitant fuzzy Pareto optimal solution to the problem is introduced, and two methods are proposed to obtain it. Then it is shown that the optimal solutions of these methods are the hesitant fuzzy Pareto optimal solutions. Finally, these methods are implemented on some illustrative examples and comparative analysis of our methodology is taken with other extensions of fuzzy sets. Keywords Multi-objective programming · Hesitant fuzzy sets · Hesitant fuzzy Pareto optimal solution
1 Introduction In real world there exist many problems called multiobjective programming (MOP) problems, which contain two or more objective functions under given constraints. They can be considered as follows: minimizex∈X (f1 (x), f2 (x), . . . , fp (x))
(1)
where X is a feasible set. x¯ ∈ X is said to be a Pareto optimal solution (efficient solution) to the MOP problem (1) S. Effati
[email protected] F. F. Rouhbakhsh fateme [email protected] M. Ranjbar [email protected]; [email protected] H. Hassanpour [email protected] 1
Faculty of Mathematical Sciences and Statistics, University of Birjand, Birjand, Iran
2
Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
3
Center of Excellence of Soft Computing and Intelligent Information Processing (SCIIP), Ferdowsi University of Mashhad, Mashhad, Iran
if there does not exist another x ∈ X such that fi (x) ≤ ¯ for all i and fj (x) = fj (x) ¯ for at least one j (see [1]). fi (x) The necessity of dealing with uncertainty in real world problems has been a long-term research challenge that has originated different methodologies and theories. Fuzzy sets along with their extensions, such as type-2 fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, and hesitant fuzzy sets (HFSs) have provided a wide range of tools which are able to deal with different types of uncertainties in many problems. For this problem, Zimmermann [2] extended his fuzzy linear programming approach to the multi-objective linear programming problem. He expressed objective functions by a fuzzy set with membership functions increasing linearly from 0 to 1. Sakawa [3, 4] introduced fuzzy multi-objective programming (FMOP) as a natural extension of his fuzzy linear programming. Lai and Hwang [5] presented some fuzzy multiple-objective decision making-methods and applications. Also, Loetamonphong et al [6] studied a class of multiple-objective optimization problems subject to a set of fuzzy relation equations, then proposed a genetic-based algorithm to obtain the Pareto optimal solutions. Rouhbakhsh et al. [7]
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