# American Institute of Mathematical Sciences

2014, 10(4): 1059-1070. doi: 10.3934/jimo.2014.10.1059

## Linear programming technique for solving interval-valued constraint matrix games

 1 School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China 2 School of Management, Fuzhou University, Fujian 350108, China

Received  January 2012 Revised  September 2013 Published  February 2014

The purpose of this paper is to propose an effective linear programming technique for solving matrix games in which the payoffs are expressed with intervals and the choice of strategies for players is constrained, i.e., interval-valued constraint matrix games. Because the payoffs of the interval-valued constraint matrix game are intervals, its value is an interval as well. In this methodology, the value of the interval-valued constraint matrix game is regarded as a function of values in the payoff intervals, which is proven to be monotonous and non-decreasing. By the duality theorem of linear programming, it is proven that both players always have the identical interval-type value and hereby the interval-valued constraint matrix game has an interval-type value. A pair of auxiliary linear programming models is derived to compute the upper bound and the lower bound of the value of the interval-valued constraint matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. Validity and applicability of the linear programming technique proposed in this paper is demonstrated with a numerical example of the market share game problem.
Citation: Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059
##### References:
 [1] C. R. Bector and S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games,, Springer Verlag, (2005). [2] C. R. Bector, S. Chandra and V. Vijay, Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs,, Fuzzy Sets and Systems, 146 (2004), 253. doi: 10.1016/S0165-0114(03)00260-4. [3] C. R. Bector, S. Chandra and V. Vijay, Matrix games with fuzzy goals and fuzzy linear programming duality,, Fuzzy Optimization and Decision Making, 4 (2004), 255. [4] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games,, Fuzzy Sets and Systems, 32 (1989), 275. doi: 10.1016/0165-0114(89)90260-1. [5] K. W. Chau, Application of a PSO-based neural network in analysis of outcomes of construction claim,, Automation in Construction, 16 (2007), 642. doi: 10.1016/j.autcon.2006.11.008. [6] W. D. Collins and C. Y. Hu, Application of a PSO-based neural network in analysis of outcomes of construction claim,, in Knowledge Processing with Interval and Soft Computing, (2008), 1. [7] M. Dresher, Games of Strategy Theory and Applications,, Prentice-Hall, (1961). [8] D. Dubois and H. Prade, Fuzzy Sets and Systems Theory and Applications,, Academic Press, (1980). [9] A. Handan and A. Emrah, A graphical method for solving interval matrix games,, Abstract and Applied Analysis, (2011), 1. [10] M. Hladík, Interval valued bimatrix games,, Kybernetika, 46 (2010), 435. [11] M. Hladík, Support set invariancy for interval bimatrix games,, the 7th EUROPT Workshop Advances in Continuous Optimization, (2009), 3. [12] M. Larbani, Non cooperative fuzzy games in normal form: A survey,, Fuzzy Sets and Systems, 160 (2009), 3184. doi: 10.1016/j.fss.2009.02.026. [13] D. F. Li, Fuzzy Multiobjective Many Person Decision Makings and Games,, National Defense Industry Press, (2003). [14] D. F. Li, Lexicographic method for matrix games with payoffs of triangular fuzzy numbers,, International Journal of Uncertainty, 16 (2008), 371. doi: 10.1142/S0218488508005327. [15] D. F. Li, Mathematical-programming approach to matrix games with payoffs represented by Atanassov's interval-valued intuitionistic fuzzy sets,, IEEE Transactions on Fuzzy Systems, 18 (2010), 1112. doi: 10.1109/TFUZZ.2010.2065812. [16] D. F. Li, Note on Linear programming technique to solve two person matrix games with interval pay-offs,, Asia-Pacific Journal of Operational Research, 28 (2011), 705. doi: 10.1142/S021759591100351X. [17] D. F. Li, Linear programming approach to solve interval-valued matrix games,, Omega, 39 (2011), 655. doi: 10.1016/j.omega.2011.01.007. [18] D. F. Li and C. T. Cheng, Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers,, International Journal of Uncertainty, 10 (2002), 385. doi: 10.1142/S0218488502001545. [19] D. F. Li and J. X. Nan, A nonlinear programming approach to matrix games with payoffs of Atanassov's intuitionistic fuzzy sets,, International Journal of Uncertainty, 17 (2009), 585. doi: 10.1142/S0218488509006157. [20] D. F. Li, J. X. Nan and M. J. Zhang, Interval programming models for matrix games with interval payoffs,, Optimization Methods and Software, 27 (2012), 1. doi: 10.1080/10556781003796622. [21] S. T. Liu and C. Kao, Matrix games with interval data,, Computers and Industrial Engineering, 56 (2009), 1697. doi: 10.1016/j.cie.2008.06.002. [22] C. L. Loganathan and M. S. Annie, Fuzzy game value of the interval matrix,, International Journal of Engineering Research and Applications, 2 (2012), 250. [23] R. E. Moore, Method and Application of Interval Analysis,, SIAM, (1979). [24] J. X. Nan, D. F. Li and M. J. Zhang, A lexicographic method for matrix games with payoffs of triangular intuitionistic fuzzy numbers,, International Journal of Computational Intelligence Systems, 3 (2010), 280. doi: 10.2991/ijcis.2010.3.3.4. [25] P. K. Nayak and M. Pal, Linear programming technique to solve two person matrix games with interval pay-offs,, Asia-Pacific Journal of Operational Research, 26 (2009), 285. doi: 10.1142/S0217595909002201. [26] I. Nishizaki and M.Sakawa, Fuzzy and Multiobjective Games for Conflict Resolution,, Springer Verlag, (2001). [27] G. Owen, Game Theory,, 2nd edition, (1982). [28] V. N. Shashikhin, Antagonistic game with interval payoff functions,, Cybernetics and Systems Analysis, 40 (2004), 556. doi: 10.1023/B:CASA.0000047877.10921.d0. [29] L. J. Sun, Z. Y. Gao and Y. J. Wang, A Stackelberg game management model of the urban public transport,, Journal of Industrial and Management Optimization, 8 (2012), 507. doi: 10.3934/jimo.2012.8.507. [30] C. F. Wang and H. Yan, Optimal assignment of principalship and residual distribution for cooperative R and D,, Journal of Industrial and Management Optimization, 8 (2012), 127. [31] Z. H. Wang, W. X. Xing and S. C. Fang, Two-person knapsack game,, Journal of Industrial and Management Optimization, 6 (2010), 847. doi: 10.3934/jimo.2010.6.847.

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##### References:
 [1] C. R. Bector and S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games,, Springer Verlag, (2005). [2] C. R. Bector, S. Chandra and V. Vijay, Duality in linear programming with fuzzy parameters and matrix games with fuzzy pay-offs,, Fuzzy Sets and Systems, 146 (2004), 253. doi: 10.1016/S0165-0114(03)00260-4. [3] C. R. Bector, S. Chandra and V. Vijay, Matrix games with fuzzy goals and fuzzy linear programming duality,, Fuzzy Optimization and Decision Making, 4 (2004), 255. [4] L. Campos, Fuzzy linear programming models to solve fuzzy matrix games,, Fuzzy Sets and Systems, 32 (1989), 275. doi: 10.1016/0165-0114(89)90260-1. [5] K. W. Chau, Application of a PSO-based neural network in analysis of outcomes of construction claim,, Automation in Construction, 16 (2007), 642. doi: 10.1016/j.autcon.2006.11.008. [6] W. D. Collins and C. Y. Hu, Application of a PSO-based neural network in analysis of outcomes of construction claim,, in Knowledge Processing with Interval and Soft Computing, (2008), 1. [7] M. Dresher, Games of Strategy Theory and Applications,, Prentice-Hall, (1961). [8] D. Dubois and H. Prade, Fuzzy Sets and Systems Theory and Applications,, Academic Press, (1980). [9] A. Handan and A. Emrah, A graphical method for solving interval matrix games,, Abstract and Applied Analysis, (2011), 1. [10] M. Hladík, Interval valued bimatrix games,, Kybernetika, 46 (2010), 435. [11] M. Hladík, Support set invariancy for interval bimatrix games,, the 7th EUROPT Workshop Advances in Continuous Optimization, (2009), 3. [12] M. Larbani, Non cooperative fuzzy games in normal form: A survey,, Fuzzy Sets and Systems, 160 (2009), 3184. doi: 10.1016/j.fss.2009.02.026. [13] D. F. Li, Fuzzy Multiobjective Many Person Decision Makings and Games,, National Defense Industry Press, (2003). [14] D. F. Li, Lexicographic method for matrix games with payoffs of triangular fuzzy numbers,, International Journal of Uncertainty, 16 (2008), 371. doi: 10.1142/S0218488508005327. [15] D. F. Li, Mathematical-programming approach to matrix games with payoffs represented by Atanassov's interval-valued intuitionistic fuzzy sets,, IEEE Transactions on Fuzzy Systems, 18 (2010), 1112. doi: 10.1109/TFUZZ.2010.2065812. [16] D. F. Li, Note on Linear programming technique to solve two person matrix games with interval pay-offs,, Asia-Pacific Journal of Operational Research, 28 (2011), 705. doi: 10.1142/S021759591100351X. [17] D. F. Li, Linear programming approach to solve interval-valued matrix games,, Omega, 39 (2011), 655. doi: 10.1016/j.omega.2011.01.007. [18] D. F. Li and C. T. Cheng, Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers,, International Journal of Uncertainty, 10 (2002), 385. doi: 10.1142/S0218488502001545. [19] D. F. Li and J. X. Nan, A nonlinear programming approach to matrix games with payoffs of Atanassov's intuitionistic fuzzy sets,, International Journal of Uncertainty, 17 (2009), 585. doi: 10.1142/S0218488509006157. [20] D. F. Li, J. X. Nan and M. J. Zhang, Interval programming models for matrix games with interval payoffs,, Optimization Methods and Software, 27 (2012), 1. doi: 10.1080/10556781003796622. [21] S. T. Liu and C. Kao, Matrix games with interval data,, Computers and Industrial Engineering, 56 (2009), 1697. doi: 10.1016/j.cie.2008.06.002. [22] C. L. Loganathan and M. S. Annie, Fuzzy game value of the interval matrix,, International Journal of Engineering Research and Applications, 2 (2012), 250. [23] R. E. Moore, Method and Application of Interval Analysis,, SIAM, (1979). [24] J. X. Nan, D. F. Li and M. J. Zhang, A lexicographic method for matrix games with payoffs of triangular intuitionistic fuzzy numbers,, International Journal of Computational Intelligence Systems, 3 (2010), 280. doi: 10.2991/ijcis.2010.3.3.4. [25] P. K. Nayak and M. Pal, Linear programming technique to solve two person matrix games with interval pay-offs,, Asia-Pacific Journal of Operational Research, 26 (2009), 285. doi: 10.1142/S0217595909002201. [26] I. Nishizaki and M.Sakawa, Fuzzy and Multiobjective Games for Conflict Resolution,, Springer Verlag, (2001). [27] G. Owen, Game Theory,, 2nd edition, (1982). [28] V. N. Shashikhin, Antagonistic game with interval payoff functions,, Cybernetics and Systems Analysis, 40 (2004), 556. doi: 10.1023/B:CASA.0000047877.10921.d0. [29] L. J. Sun, Z. Y. Gao and Y. J. Wang, A Stackelberg game management model of the urban public transport,, Journal of Industrial and Management Optimization, 8 (2012), 507. doi: 10.3934/jimo.2012.8.507. [30] C. F. Wang and H. Yan, Optimal assignment of principalship and residual distribution for cooperative R and D,, Journal of Industrial and Management Optimization, 8 (2012), 127. [31] Z. H. Wang, W. X. Xing and S. C. Fang, Two-person knapsack game,, Journal of Industrial and Management Optimization, 6 (2010), 847. doi: 10.3934/jimo.2010.6.847.
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