Adaptation of Iterative Methods to Solve Fuzzy Mathematical Programming Problems

Authors: Ricardo C. Silva, Luiza A. P. Cantao, Akebo Yamakami

Abstract:

Based on the fuzzy set theory this work develops two adaptations of iterative methods that solve mathematical programming problems with uncertainties in the objective function and in the set of constraints. The first one uses the approach proposed by Zimmermann to fuzzy linear programming problems as a basis and the second one obtains cut levels and later maximizes the membership function of fuzzy decision making using the bound search method. We outline similarities between the two iterative methods studied. Selected examples from the literature are presented to validate the efficiency of the methods addressed.

Keywords: Fuzzy Theory, Nonlinear Optimization, Fuzzy Mathematics Programming.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1063124

References:

[1] L. A. ZADEH, "Fuzzy sets," Information and Control, vol. 8, pp. 338- 353, 1965.
[2] R. C. SILVA, L. A. P. CANTA˜ O, and A. YAMAKAMI, "Meta-heuristic to mathematical programming problems with uncertainties," in II International Conference on Machine Intelligence, 2005, pp. 306-311.
[3] Y. H. LEE, B. H. YANG, and K. S. MOON, "An economic machining process model using fuzzy non-linear programming and neural network," International Journal Production Research, vol. 37, no. 4, pp. 835-847, 1999.
[4] J.-F. C. TRAPPEY, C. R. LIU, and T.-C. CHANG, "Fuzzy non-linear programming: Theory and application in manufaturing," International Journal Production Research, vol. 26, no. 5, pp. 975-985, 1988.
[5] C. XU, "Fuzzy optimization of structures by the two-phase method," Computers & Structures, vol. 31, no. 4, pp. 575-580, 1989.
[6] H. J. ZIMMERMANN, "Fuzzy mathematical programming," Computer & Operation Research, vol. 10, no. 4, pp. 291-298, 1983.
[7] L. A. P. CANTA╦£ O, "Programac┬©a╦£o na╦£o-linear com para╦åmetros fuzzy," Ph.D. dissertation, FEEC - UNICAMP, Campinas, Maro 2003.
[8] G. J. KLIR and B. YUAN, Fuzzy Sets and Fuzzy Logic: Theory and Applications. New Jersey: Prentice Hall, 1995.
[9] W. PEDRYCS and F. GOMIDE, An Introduction of Fuzzy Sets: Analisys and Design. A Bardford Book, 1998.
[10] R. E. BELLMAN and L. A. ZADEH, "Decision-marking in a fuzzy environment," Management Science, vol. 17, no. 4, pp. B141-B164, 1970.
[11] R. C. SILVA, "Contribuic┬© ╦£oes ao estudo de programac┬© ╦£ao n╦£ao-linear com incertezas," Master-s thesis, FEEC - UNICAMP, Campinas, Maio 2005.
[12] K. SCHITTKOWSKI, More Test Examples for Nonlinear Programming Codes. Spring-Verlag, 1987.