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.
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Acknowledgements
The authors are very much thankful to the Associate Editor, Prof. Dong-Lei Du and the anonymous reviewers for their valuable comments to increase the novelty and overall quality of the paper.
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Appendices
Appendix
A detailed calculation to show \(s_{1}=\frac{2}{(a_{2}-a_{1})}\) is given below:
Appendix
Another calculation is shown for \(s_{2}\) as follows:
Appendix
Again detailed calculation is presented for \(M_{X}(2)\)
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Maiti, S.K., Roy, S.K. Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach. J. Oper. Res. Soc. China 9, 131–149 (2021). https://doi.org/10.1007/s40305-018-0234-2
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DOI: https://doi.org/10.1007/s40305-018-0234-2
Keywords
- Bi-level programming
- Triangular intuitionistic fuzzy number
- Ranking function
- Nonlinear programming
- Optimal solution