American Institute of Mathematical Sciences

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

Linear programming technique for solving interval-valued constraint matrix games

Jiang-Xia Nan 1, and  Deng-Feng Li 2,

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.
Keywords: interval numbers, Game theory, constraint matrix game, linear programming, economics..
Mathematics Subject Classification: Primary: 91A05, 91A40; Secondary: 90C0.
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, Berlin, 2005. Google Scholar

[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), 253-269. doi: 10.1016/S0165-0114(03)00260-4.  Google Scholar

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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), 255-269. Google Scholar

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L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems, 32 (1989), 275-289. doi: 10.1016/0165-0114(89)90260-1.  Google Scholar

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[7]

M. Dresher, Games of Strategy Theory and Applications, Prentice-Hall, New York, 1961. Google Scholar

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D. Dubois and H. Prade, Fuzzy Sets and Systems Theory and Applications, Academic Press, New York, 1980. Google Scholar

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A. Handan and A. Emrah, A graphical method for solving interval matrix games, Abstract and Applied Analysis, (2011), 1-18. Google Scholar

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M. Hladík, Interval valued bimatrix games, Kybernetika, 46 (2010), 435-446.  Google Scholar

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M. Hladík, Support set invariancy for interval bimatrix games, the 7th EUROPT Workshop Advances in Continuous Optimization, EurOPT (2009), July 3-4, Remagen, German. Google Scholar

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M. Larbani, Non cooperative fuzzy games in normal form: A survey, Fuzzy Sets and Systems, 160 (2009), 3184-3210. doi: 10.1016/j.fss.2009.02.026.  Google Scholar

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D. F. Li, Fuzzy Multiobjective Many Person Decision Makings and Games, National Defense Industry Press, Bei Jing, 2003. Google Scholar

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D. F. Li, Lexicographic method for matrix games with payoffs of triangular fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16 (2008), 371-389. doi: 10.1142/S0218488508005327.  Google Scholar

[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), 1112-1128. doi: 10.1109/TFUZZ.2010.2065812.  Google Scholar

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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), 705-737. doi: 10.1142/S021759591100351X.  Google Scholar

[17]

D. F. Li, Linear programming approach to solve interval-valued matrix games, Omega, 39 (2011), 655-666. doi: 10.1016/j.omega.2011.01.007.  Google Scholar

[18]

D. F. Li and C. T. Cheng, Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (2002), 385-400. doi: 10.1142/S0218488502001545.  Google Scholar

[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, Fuzziness and Knowledge-Based Systems, 17 (2009), 585-607. doi: 10.1142/S0218488509006157.  Google Scholar

[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), 1-16. doi: 10.1080/10556781003796622.  Google Scholar

[21]

S. T. Liu and C. Kao, Matrix games with interval data, Computers and Industrial Engineering, 56 (2009), 1697-1700. doi: 10.1016/j.cie.2008.06.002.  Google Scholar

[22]

C. L. Loganathan and M. S. Annie, Fuzzy game value of the interval matrix, International Journal of Engineering Research and Applications, 2 (2012), 250-255. Google Scholar

[23]

R. E. Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, 1979. Google Scholar

[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), 280-289. doi: 10.2991/ijcis.2010.3.3.4.  Google Scholar

[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), 285-305. doi: 10.1142/S0217595909002201.  Google Scholar

[26]

I. Nishizaki and M.Sakawa, Fuzzy and Multiobjective Games for Conflict Resolution, Springer Verlag, Berlin, 2001. Google Scholar

[27]

G. Owen, Game Theory, 2nd edition, Academic Press, New York, 1982. Google Scholar

[28]

V. N. Shashikhin, Antagonistic game with interval payoff functions, Cybernetics and Systems Analysis, 40 (2004), 556-564. doi: 10.1023/B:CASA.0000047877.10921.d0.  Google Scholar

[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), 507-520. doi: 10.3934/jimo.2012.8.507.  Google Scholar

[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), 127-139.  Google Scholar

[31]

Z. H. Wang, W. X. Xing and S. C. Fang, Two-person knapsack game, Journal of Industrial and Management Optimization, 6 (2010), 847-860. doi: 10.3934/jimo.2010.6.847.  Google Scholar

show all references

References:
[1]

C. R. Bector and S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games, Springer Verlag, Berlin, 2005. Google Scholar

[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), 253-269. doi: 10.1016/S0165-0114(03)00260-4.  Google Scholar

[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), 255-269. Google Scholar

[4]

L. Campos, Fuzzy linear programming models to solve fuzzy matrix games, Fuzzy Sets and Systems, 32 (1989), 275-289. doi: 10.1016/0165-0114(89)90260-1.  Google Scholar

[5]

K. W. Chau, Application of a PSO-based neural network in analysis of outcomes of construction claim, Automation in Construction, 16 (2007), 642-646. doi: 10.1016/j.autcon.2006.11.008.  Google Scholar

[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, (editors C. Y. Hu, et al.), Chapter 7, Springer, (2008), 1-19. Google Scholar

[7]

M. Dresher, Games of Strategy Theory and Applications, Prentice-Hall, New York, 1961. Google Scholar

[8]

D. Dubois and H. Prade, Fuzzy Sets and Systems Theory and Applications, Academic Press, New York, 1980. Google Scholar

[9]

A. Handan and A. Emrah, A graphical method for solving interval matrix games, Abstract and Applied Analysis, (2011), 1-18. Google Scholar

[10]

M. Hladík, Interval valued bimatrix games, Kybernetika, 46 (2010), 435-446.  Google Scholar

[11]

M. Hladík, Support set invariancy for interval bimatrix games, the 7th EUROPT Workshop Advances in Continuous Optimization, EurOPT (2009), July 3-4, Remagen, German. Google Scholar

[12]

M. Larbani, Non cooperative fuzzy games in normal form: A survey, Fuzzy Sets and Systems, 160 (2009), 3184-3210. doi: 10.1016/j.fss.2009.02.026.  Google Scholar

[13]

D. F. Li, Fuzzy Multiobjective Many Person Decision Makings and Games, National Defense Industry Press, Bei Jing, 2003. Google Scholar

[14]

D. F. Li, Lexicographic method for matrix games with payoffs of triangular fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16 (2008), 371-389. doi: 10.1142/S0218488508005327.  Google Scholar

[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), 1112-1128. doi: 10.1109/TFUZZ.2010.2065812.  Google Scholar

[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), 705-737. doi: 10.1142/S021759591100351X.  Google Scholar

[17]

D. F. Li, Linear programming approach to solve interval-valued matrix games, Omega, 39 (2011), 655-666. doi: 10.1016/j.omega.2011.01.007.  Google Scholar

[18]

D. F. Li and C. T. Cheng, Fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (2002), 385-400. doi: 10.1142/S0218488502001545.  Google Scholar

[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, Fuzziness and Knowledge-Based Systems, 17 (2009), 585-607. doi: 10.1142/S0218488509006157.  Google Scholar

[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), 1-16. doi: 10.1080/10556781003796622.  Google Scholar

[21]

S. T. Liu and C. Kao, Matrix games with interval data, Computers and Industrial Engineering, 56 (2009), 1697-1700. doi: 10.1016/j.cie.2008.06.002.  Google Scholar

[22]

C. L. Loganathan and M. S. Annie, Fuzzy game value of the interval matrix, International Journal of Engineering Research and Applications, 2 (2012), 250-255. Google Scholar

[23]

R. E. Moore, Method and Application of Interval Analysis, SIAM, Philadelphia, 1979. Google Scholar

[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), 280-289. doi: 10.2991/ijcis.2010.3.3.4.  Google Scholar

[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), 285-305. doi: 10.1142/S0217595909002201.  Google Scholar

[26]

I. Nishizaki and M.Sakawa, Fuzzy and Multiobjective Games for Conflict Resolution, Springer Verlag, Berlin, 2001. Google Scholar

[27]

G. Owen, Game Theory, 2nd edition, Academic Press, New York, 1982. Google Scholar

[28]

V. N. Shashikhin, Antagonistic game with interval payoff functions, Cybernetics and Systems Analysis, 40 (2004), 556-564. doi: 10.1023/B:CASA.0000047877.10921.d0.  Google Scholar

[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), 507-520. doi: 10.3934/jimo.2012.8.507.  Google Scholar

[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), 127-139.  Google Scholar

[31]

Z. H. Wang, W. X. Xing and S. C. Fang, Two-person knapsack game, Journal of Industrial and Management Optimization, 6 (2010), 847-860. doi: 10.3934/jimo.2010.6.847.  Google Scholar

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