Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR FUZZY LINEAR PROGRAMMING

  • Farhadinia, Bahram (Department of Mathematics, Technical Institute of Higher Education of Quchan)
  • Received : 2010.01.09
  • Accepted : 2010.10.19
  • Published : 2011.01.30
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Abstract

This paper is concerned with deriving necessary and sufficient optimality conditions for a fuzzy linear programming problem. Toward this end, an equivalence between fuzzy and crisp linear programming problems is established by means of a specific ranking function. Under this setting, a main theorem gives optimality conditions which do not seem to be in conflict with the so-called Karush-Kuhn-Tucker conditions for a crisp linear programming problem.

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References

  1. M. S. Bazaraa, J. J. Jarvis and H. D. Sherali, Linear Programming and Network Flows, Second Ed., John Wiely, New York, 1990.
  2. R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management Science 17 (1970), B141-B164.
  3. J. J. Buckley, Possibilistic linear programming with triangular fuzzy number, Fuzzy Sets and Systems 26 (1988), 135-138. https://doi.org/10.1016/0165-0114(88)90013-9
  4. J. J. Buckley, Solving possibilistic linear programming problms, Fuzzy Sets and Systems 31 (1989), 329-341. https://doi.org/10.1016/0165-0114(89)90204-2
  5. C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95 (1998), 307-317. https://doi.org/10.1016/S0165-0114(96)00272-2
  6. D. Dubois and H. Prade, Fuzzy Stes and Systems: Theory and Applications, Academic Press, New York, 1980.
  7. P. Fortemps and M. Roubens, Ranking and defuzzification methods based on area com-pensation, Fuzzy Sets and Systems 82 (1996), 319-330. https://doi.org/10.1016/0165-0114(95)00273-1
  8. R. N. Gasimov and Yenilmez, Solving fuzzy linear programming with linear membership functions, Turk. J. Math. 26 (2002), 375-396.
  9. H. Lee-Kwang and J. H. Lee, A method for ranking fuzzy numbers and its application to decision-making, IEEE Trans. On Fuzzy Systems 7 (1999), 677-685. https://doi.org/10.1109/91.811235
  10. S. Leu, A. T. Chen and C. H. Yang, A GA-based fuzzy optimal model for construction time-cost trade-off, Int. J. Proj. Management 19 (2001), 47-58. https://doi.org/10.1016/S0263-7863(99)00035-6
  11. M. Panigrahi, G. Panda and S. Nanda, Convex fuzzy mapping with differentiability and its application in fuzzy optimization, Eur. J. Oper. Res. 185 (2008), 47-62. https://doi.org/10.1016/j.ejor.2006.12.053
  12. M. R. Safi, H. R. Maleki and E. Zaeimazad, A geometric approach for solving fuzzy linear programming problems, Fuzzy Optim. and Decision Making 6 (2007), 315-336. https://doi.org/10.1007/s10700-007-9016-8
  13. H. Tanaka and K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems 13 (1984), 1-10. https://doi.org/10.1016/0165-0114(84)90022-8
  14. J. F. Tang, D. W. Wang and R. Y. K. Fung, Modeling and method based on GA for nonlinear programming problems with fuzzy objective and resources, Int. J. Sys. Sciences 29 (1998), 907-913. https://doi.org/10.1080/00207729808929582
  15. H. C.Wu, Saddle point optimality conditions in fuzzy optimization problems, Fuzzy Optim. and Decision Making 3 (2003), 261-273.
  16. H. C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function, Eur. J. Oper. Res. 176 (2007), 46-59. https://doi.org/10.1016/j.ejor.2005.09.007
  17. H. C. Wu, Duality theorems and saddle point optimality conditions in fuzzy nonlinear programming problems based on different solution concepts, Fuzzy Sets and Systems 158 (2007), 1588-1607. https://doi.org/10.1016/j.fss.2007.01.004
  18. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
  19. H. J. Zimmermann, Fuzzy Set Theory and its Applications, Fourth Edition, Kluwer Aca-demic Publishers, 1998.
  20. H. J. Zimmermann, Fuzzy mathematical programming, Comput. and Ops. Res. 10 (1983), 291-298. https://doi.org/10.1016/0305-0548(83)90004-7
  21. H. J. Zimmermann, Fuzzy programming and linear programming with second objective functions, Fuzzy Sets and Systems 1 (1978), 45-55. https://doi.org/10.1016/0165-0114(78)90031-3