Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach
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Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach Sumit Kumar Maiti1 · Sankar Kumar Roy2 Received: 23 March 2018 / Revised: 10 October 2018 / Accepted: 21 November 2018 © Operations Research Society of China, Periodicals Agency of Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract This paper introduces a ranking function procedure on a bi-level programming for Stackelberg game involving intuitionistic fuzzy parameters. Intuitionistic fuzzy number is considered in many real-life situations, so it makes perfect sense to address decision-making problem by using some specified intuitionistic fuzzy numbers. In this paper, intuitionistic fuzziness is characterized by a normal generalized triangular intuitionistic fuzzy number. A defuzzification method is introduced based on the proportional probability density function associated with the corresponding membership function, as well as the complement of non-membership function. Using the proposed ranking technique, a methodology is presented for solving bi-level programming for Stackelberg game. An application example is provided to demonstrate the applicability of the proposed methodology, and the achieved results are compared with the existing methods. Keywords Bi-level programming · Triangular intuitionistic fuzzy number · Ranking function · Nonlinear programming · Optimal solution Mathematics Subject Classification 90C05 · 90C70 · 90C30
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Sankar Kumar Roy [email protected] Sumit Kumar Maiti [email protected]
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School of Applied Sciences and Humanities, Haldia Institute of Technology, Purba Midnapore, WB 721 657, India
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Department of Applied Mathematics with Oceanology and Computer, Programming, Vidyasagar University, Midnapore, WB 721 102, India
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S. K. Maiti, S. K. Roy
1 Introduction and Literature Review Bi-level programming problem (BLPP), first introduced by H. von Stackelberg [1], is a nested optimization problem including a leader and a follower problem. Making a decision at one level affects the objective function and the decision space of other, and the order is taken from top to bottom. More formally, consider the leader that selects a solution x1 ∈ X 1 ⊆ Rn 1 . Then, the follower’s task is to solve the problem max{Z 2 (x1 , x2 ) : x2 ∈ X 2 (x1 )}. x2
(1.1)
Here, X 2 (x1 ) ⊆ Rn 2 for all x1 ∈ X 1 ⊆ Rn 1 , Z i : X 1 × X 1 → R, i = 1, 2. Let ξ(x1 ) denote the set of optimal solutions of Eq. (1.1) for fixed x1 ∈ X 1 . Then, the leader’s aim is to maximize his/her objective function Z 1 (x1 , x2 ) subject to x2 ∈ ξ(x1 ) and x1 ∈ X 1 . Assume that the set ξ(x1 ) consists of no more than one element for each x1 ∈ X 1 . Then, the BLPP can be formulated as ⎧ max Z 1 (x1 , x2 ) ⎪ ⎪ ⎨x1 ∈X 1 s.t. max Z 2 (x1 , x2 ) ⎪ x2 ⎪ ⎩ s.t. x2 ∈ X 2 (x1 ).
(1.2)
One of the surprising facts in BLPP is that, if a constraint is moved from the upper to the lower-level problem, optimality of a feasible solution can be lost. The following point is of main importance in Stacke
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