Multi-objective enhanced interval optimization problem
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Multi-objective enhanced interval optimization problem P. Kumar1
· A. K. Bhurjee2
Accepted: 7 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider a multiple objective optimization problem whose decision variables and parameters are intervals. Existence of solution of this problem is studied by parameterizing the intervals. A methodology is developed to find the tω-efficient solution of the problem. The original problem is transformed to an equivalent deterministic problem and the relation between solutions of both is established. Finally, the methodology is verified in numerical examples. Keywords Non-linear optimization problem · Interval valued function · Interval optimization problem · Efficient solution Mathematics Subject Classification 90C25 · 90C29 · 90C30
1 Introduction In a conventional multi-objective decision-making problem, the coefficients in the objective functions and constraints are usually fixed real numbers. However, in most of the real-life situations these parameters are not exactly known because relevant data are inconsistent, or scarce, or difficult to obtain, or estimate. In an optimization model, such types of uncertainties are usually measured by probability theory or possibility theory. However, in some cases, it is very difficult to specify the probability/possibility distributions of these parameters. To overcome these difficulties, the uncertain parameters may be assumed to be closed intervals. Existing literature shows that, multi-objective linear programming problems with interval coefficients have been studied by Bitran (1980), Inuiguchi and Sakawa (1996), Urli and Nadeau (1992), Oliveira and Antunes (2007), Hladik (2008), Oliveira and Antunes (2009), Rivaz and Yaghoobi (2013) and Roy et al. (2017). Using deterministic multi-objective programming, Chanas and Kuchta (1996) discussed the generalized perceptions of the solution
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P. Kumar [email protected] A. K. Bhurjee [email protected]
1
Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chennai 603203, India
2
VIT Bhopal University, Bhopal-Indore Highway, Kothrikalan, Sehore, Madhya Pradesh 466114, India
123
Annals of Operations Research
methodology of the linear programming problem with interval parameters in the objective function based on preference relations between intervals. In which they have been generalized the preference relations of intervals by considering two parameters t0 , t1 ∈ [0, 1]. Nonlinear multi-objective interval optimization problem (MIOP) has been studied by Wu (2009), Soares et al. (2009), Gong et al. (2013) and Wu (2009) has examined the conditions for the existence of solution of an MIOP whose objective functions are interval valued functions, and whose all constraints are real valued functions. Gong et al. (2013) has been established an interactive evolutionary algorithm to obtain a set of non-dominated solutions. Rivaz et al. (2016) established a methodology for the mini-max regret solutio
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