Article Contents

2024, Volume 11, Issue 4: 399-421. Doi: 10.3934/jdg.2024010

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Solving matrix game using rough interval payoffs

1.
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India
2.
Faculty of Engineering Management, Poznan University of Technology, Poznan, Poland
3.
Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

^*Corresponding author: Sankar Kumar Roy, Orcid: 0000-0003-4478-1534
^*Corresponding author: Sankar Kumar Roy, Orcid: 0000-0003-4478-1534

Received: September 2023

Revised: February 2024

Early access: April 2024

Published: October 2024

Abstract / Introduction Full Text(HTML) Figure(2) / Table(4) Related Papers Cited by

Abstract

In this paper, we conduct an analysis of a matrix game using rough interval payoffs, which we refer to as the rough interval matrix game (RIMG). We employ three methods to address the problem at hand: linear programming, graphical method, and algebraic method. The proposed approach offers several key advantages over existing methods, which we highlight. We introduce algorithms associated with each of the three mentioned methods to obtain optimal solutions for the RIMG. To validate the effectiveness of our proposed techniques, we present two numerical examples and apply the methodologies to solve them. In conclusion, we summarize the findings of our study and discuss potential avenues for future research based on the insights provided in this paper.

Keywords:

Mathematics Subject Classification: Primary: 91A86; Secondary: 91A80.

Citation:

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Figure 1. Graph of $ \phi_j(t),\; j = 1,2,3 $

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Figure 2. Graph of $ \psi(t) $

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Table 1. Contributions of different authors in relate to matrix game under uncertainty

Author	Payoff Matrix	Approach

Campos (1989)	Fuzzy	Linear programming problem
Li (1999)	Fuzzy	Multi-objective programming
Osman et al. (2011)	Rough interval	Rough programming
Li (2011)	Interval	An effective linear programming problem
Hamzehee et al. (2014)	Rough interval	Linear programming problems with rough approximation concept
Li and Nan (2014)	Interval	Interval programming
Roy and Mondal (2016)	Fuzzy	An effective fuzzy multi-objective linear programming with fuzzy interval payoffs
Roy and Mula (2016)	Rough interval	Genetic algorithm and Linear programming problem
Bhurjee and Panda (2017)	Interval	Normalized matrix game with variable payoffs
Our proposed work	Rough interval	Three traditional methods, namely, AM, GM, and LPP

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Table 2. Optimal value and strategy of players by LPP

Problem	Value of the game	Optimal strategy (Player A)	Optimal strategy (Player B)

Problem-1.1	93.684	(0.474, 0.526)	(0.526, 0.000, 0.474)
Problem-1.2	111.428	(0.524, 0.476)	(0.571, 0.000, 0.429)
Problem-2.1	87.500	(0.500, 0.500)	(0.550, 0.000, 0.450)
Problem-2.2	117.500	(0.500, 0.500)	(0.550, 0.000, 0.450)

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Table 3. Game value using different methods in market share problem

Methods	Value of the game

Using LPP	([93.684,111.428], [87.500,117.500])
Using GM	([97.693,103.076], [92.693,111.153])
Using AM	([97.692,117.692], [92.692,122.692])

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Table 4. Game value using different methods in water resources management problem

Methods	Value of the game

Using LPP	([65.000, 75.000], [60.000, 80.000])
Using GM	([75.000, 79.520], [57.620, 87.620])
Using AM	([53.300, 62.000], [48.300, 67.000])

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