Kybernetika 52 no. 1, 153-168, 2016

Solving multi-objective fuzzy matrix games via multi-objective linear programming approach

Abha Aggarwal and Imran KhanDOI: 10.14736/kyb-2016-1-0153


A class of multi-objective fuzzy matrix games is studied and it is shown that solving such a game is equivalent to solving a pair of multi-objective linear programming problems. This work generalizes an earlier study of Fernandez et al. \cite{Fernandez} from crisp scenario to fuzzy scenario on the lines of Bector et al. \cite{Chan2}. Further certain difficulties with similar studies reported in the literature are also discussed.


multi-objective game, Pareto-optimal security strategies, security level, multi-objective linear programming


90C70, 91A40


  1. R. E. Bellman and L. A. Zadeh: Decision making in a fuzzy environment. Management Sci. 17 (1970), 141-164.   DOI:10.1287/mnsc.17.4.b141
  2. D. Blackwell: An analog of the minimax theorem for vector payoff. Pacific J. Math. 6 (1956), 1-8.   DOI:10.2140/pjm.1956.6.1
  3. C. R. Bector, S. Chandra and V. Vijay: Matrix games with fuzzy goals and fuzzy linear programming duality. Fuzzy Optim. Decision Making 3 (2004), 263-277.   DOI:10.1023/b:fodm.0000036866.18909.f1
  4. C. R. Bector and S. Chandra: Fuzzy Mathematical Programming and Fuzzy Matrix Games. Springer-Verlag, Berlin 2005.   DOI:10.1007/3-540-32371-6
  5. W. D. Cook: Zero-sum games with multiple goals. Naval Research Logistics Quarterly 23 (1976), 615-622.   DOI:10.1002/nav.3800230406
  6. S. C. Corley: Games with vector payoffs. J. Optim. Theory Appl. 47 (1985), 463-475.   DOI:10.1007/bf00942194
  7. F. R. Fernandez and J. Puerto: Vector linear programming in zero-sum multicriteria matrix games. J. Optim. Theory Appl. 89 (1996), 115-127.   DOI:10.1007/bf02192644
  8. D. Ghose and U. R. Prasad: A solution concepts in two-person multicriteria games. J. Optim. Theory Appl. 63 (1989), 167-189.   DOI:10.1007/bf00939572
  9. N. Gaskó, M. Suciu, R. I. Lung and D. Dumitrescu: Pareto-optimal Nash equilibrium detection using an evolutionary approach. Acta Univ. Sapientiae 4 (2012), 2, 237-246.   CrossRef
  10. G. Mavrotas: Generation of Efficient Solutions in Multiobjective Mathematical Programming Problems Using GAMS. Effective Implementation of the $\epsilon$-constraint Method. Technical Report: (2007), 167-189.   CrossRef
  11. I. Nishizaki and M. Sakawa: Fuzzy and Multiobjective Games for Conflict Resolution. Kluwer Academic Publishers 2003.   DOI:10.1007/978-3-7908-1830-7
  12. M. Sakawa and I. Nishizaki: Max-min solutions for fuzzy multi-objective mattrix games. Fuzzy Sets and Systems 67 (1994), 53-69.   DOI:10.1016/0165-0114(94)90208-9
  13. R. E. Steuer: Multiple Criteria Optimization: Theory, Computation and Application. John Wiley, New York 1986.   CrossRef
  14. L. S. Shapely: Equilibirum points in games with vector payoff. Naval Research Logistics Quarterly 6 (1959), 57-61.   DOI:10.1002/nav.3800060107
  15. V. Vijay, A. Mehra, S. Chandra and C. R. Bector: Fuzzy matrix games via a fuzzy relation approach, Fuzzy Optim. Decision Making 6 (2007), 299-314.   DOI:10.1007/s10700-007-9015-9
  16. M. Zeleny: Games with multiple payoff. Int. J. Game Theory 4 (1975), 179-191.   DOI:10.1007/bf01769266
  17. H. J. Zimmermann: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems1 (1978), 45-55.   DOI:10.1016/0165-0114(78)90031-3