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The Set of Optimal Solutions of Geometric Programming Problem with Max-Product Fuzzy Relational Equations Constraints

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Abstract

In this paper, we consider how to get the set of optimal solutions of geometric programming problem with single-term exponents subject to a system of fuzzy relational equations about max-product composition. The feasible domain of this problem is nonconvex. Firstly, we propose some algorithms to illustrate how to get the set of optimal solutions based on three cases. Secondly, we show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of optimal lower solutions. If we know the optimal value, the maximum optimal solution can be easily computed. However, solving all optimal lower solutions remains as a challenging problem, since finding all the potential minimal solutions of max-product fuzzy relational equations is an NP-hard problem. Finally, four numerical examples are provided to illustrate validity of the proposed method.

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References

  1. Sanchez, E.: Resolution of composite fuzzy relational equations. Inf. Control 30, 38–48 (1976)

    Article  MATH  Google Scholar 

  2. Bourke, M.M., Fisher, D.G.: Solution algorithms for fuzzy relational equations with max-product composition. Fuzzy Sets Syst. 94, 61–69 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Di Nola, A., Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Applications in Knowledge Engineering. Kluwer Academic Press, Dordrecht (1989)

    Book  MATH  Google Scholar 

  4. Czogala, E., Drewniak, J., Pedrycz, W.: Fuzzy relation equations on a finite set. Fuzzy Sets Syst. 7, 89–101 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Higashi, M., Klir, G.J.: Resolution of finite fuzzy relation equations. Fuzzy Sets Syst. 13, 65–82 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Markovskii, A.V.: On the relation between equations with max-product composition and the covering problem. Fuzzy Sets Syst. 153, 261–273 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, L., Wang, P.P.: Fuzzy relation equations (I): the general and specialized solving algorithms. Soft Comput. 6, 428–435 (2002)

    Article  MATH  Google Scholar 

  8. Wang, S., Fang, S.C., Nuttle, H.L.W.: Solution sets of interval-valued fuzzy relational equations. Fuzzy Optim. Decis. Making 2(1), 41–60 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, P., Fang, S.C.: A survey on fuzzy relational equations, part I: classification and solvability. Fuzzy Optim. Decis. Making 8, 179–229 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gupta, M.M., Qi, J.: Design of fuzzy logic controllers based on generalized T-operators. Fuzzy Sets Syst. 40, 473–489 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pedrycz, W.: On generalized fuzzy relational equations and their applications. J. Math. Anal. Appl. 107, 520–536 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pedrycz, W.: s-t fuzzy relational equations. Fuzzy Sets Syst. 59, 189–195 (1993)

    Article  MathSciNet  Google Scholar 

  13. Tzeng, H.-W.: Fuzzy decomposition method by mapping analysis. Int. J. Fuzzy Syst. 12(1), 33–47 (2010)

    Google Scholar 

  14. Wang, P.Z., Zhang, D.Z., Sanchez, E., Lee, E.S.: Latticized linear programming and fuzzy relation inequalities. J. Math. Anal. Appl. 159, 72–87 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  15. Loetamonphong, J., Fang, S.-C.: Optimization of fuzzy relational equations with max-product composition. Fuzzy Sets Syst. 118, 509–517 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wu, Y.K., Guu, S.M.: A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optim. Decis. Mak. 3, 271–278 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, P., Fang, S.C.: On the resolution and optimization of a system of fuzzy relational equations with sup-T composition. Fuzzy Optim. Decis. Mak. 7(2), 169–214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, J.H., Cao, B.Y.: Geometric programming with fuzzy relation equation constraints. In: Proceedings of IEEE International Conference on Fuzzy Systems, pp. 557–560. IEEE (2005)

  19. Wu, Y.K.: Optimizing the geometric programming problem with single-term exponents subject to max-min fuzzy relational equation constraints. Math. Comput. Model. 47, 352–362 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shivanian, E., Khorram, E.: Monomial geometric programming with fuzzy relation inequality constraints with max-product composition. Comput. Ind. Eng. 56, 1386–1392 (2009)

    Article  Google Scholar 

  21. Zhou, X.G., Ahat, R.: Geometric programming problem with single-term exponents subject to max-product fuzzy relational equations. Math. Comput. Model. 53, 55–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are grateful to the responsible editor and the anonymous referees for their valuable comments and suggestions, which have improved the earlier version of this paper. The authors acknowledge the support of the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (No. S2013040012506), the China Postdoctoral Science Foundation Funded Project (2014M56 2152), the Innovation Capability of Independent Innovation to Enhance the Class of Building Strong School Projects of Colleges of Guangdong Province (20140207), and the GuangZhou Postdoctoral Science Foundation Funded Project (gzhubsh2013006).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, Guangdong, China

    Xue-Gang Zhou, Bing-Yuan Cao & Xiao-Peng Yang

  2. Department of Applied Mathematics, Guangdong University of Finance, Guangzhou, 510521, Guangdong, China

    Xue-Gang Zhou

  3. Guangzhou Vocational College of Science and Technology, Guangzhou, 510550, Guangdong, China

    Bing-Yuan Cao

  4. Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, 521041, Guangdong, China

    Xiao-Peng Yang

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  1. Xue-Gang Zhou
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  2. Bing-Yuan Cao
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Correspondence to Xue-Gang Zhou.

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Zhou, XG., Cao, BY. & Yang, XP. The Set of Optimal Solutions of Geometric Programming Problem with Max-Product Fuzzy Relational Equations Constraints. Int. J. Fuzzy Syst. 18, 436–447 (2016). https://doi.org/10.1007/s40815-015-0083-7

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  • DOI: https://doi.org/10.1007/s40815-015-0083-7

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