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Fuzzy Relational Linear Programming

  • Bing-Yuan CaoEmail author
  • Ji-Hui Yang
  • Xue-Gang Zhou
  • Zeinab Kheiri
  • Faezeh Zahmatkesh
  • Xiao-Peng Yang
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 389)

Abstract

Since Sanchez [1] proposed the resolution to fuzzy relational equations (FREs), many researchers have studied FREs and fuzzy relational inequalities (FRIs) [2, 3, 4, 5, 6, 7]. FRE theory has been applied in many different fields, including fuzzy control [8], fuzzy decision-making [9], fuzzy modeling [10], fuzzy analysis [11], medical diagnosis [12, 13], compression and decompression of images and videos [14, 15, 16, 17, 18], and estimation of flow rates in a chemical plant and pipe network and peak rush hours for transport systems [7].

References

  1. 1.
    Sanchez, E.: Resolution of composite fuzzy relation equation. Inf. Control 30, 38–48 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Han, S.C., Li, H.X., Wang, J.Y.: Resolution of finite fuzzy relation equations based on strong pseudo-t-norms. Appl. Math. Lett. 19, 752–757 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Higashi, M., Klir, G.J.: Resolution of finite fuzzy relation equations. Fuzzy Sets Syst. 13, 65–82 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Loetamonphong, J., Fang, S.C., Young, R.E.: Multi-objective optimization problems with fuzzy relation equation constraints. Fuzzy Sets Syst. 127, 141–164 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Pedrycz, W., Vasilakos, A.V.: Modularization of fuzzy relational equations. Soft Comput. 6, 33–37 (2002)zbMATHCrossRefGoogle Scholar
  6. 6.
    Peeva, K., Kyosev, Y.: Fuzzy Relational Calculus: Theory. Applications and Software. World Scientific, Singapore (2004)zbMATHGoogle Scholar
  7. 7.
    Vasantha Kandasamy, W.B., Smarandache, F.: Some applications of FRE. In: Fuzzy Relational Maps and Neutrosophic Relational Maps, Hexis, Church Rock, pp. 167–220 (2004)Google Scholar
  8. 8.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems-Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  9. 9.
    Bellman, R.E., Zadeh, L.A.: Decision-making in a fuzzy environment. Manag. Sci. 17, 141–164 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Wenstop, F.: Deductive verbal models of organizations. Int. J. Man-Mach. Stud. 8, 293–311 (1976)zbMATHCrossRefGoogle Scholar
  11. 11.
    Pedrycz, W.: An approach to the analysis of fuzzy systems. Int. J. Control 34, 403–421 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Sanchez, E.: Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Gupta, M.M., Saridis, G.N., Gaines, B.R. (eds.) Fuzzy Automata and Decision Processes, pp. 221–234. North-Holland, Amsterdam (1977)Google Scholar
  13. 13.
    Wang, H.F., Wu, C.W., Ho, C.H., Hsieh, M.J.: Diagnosis of gastri ccancer with fuzzy pattern recognition. J. Syst. Eng. 2, 151–163 (1992)Google Scholar
  14. 14.
    Dubois, D., Prade, H.: Operations on fuzzy number. Int. J. Syst. Sci. 9(6), 613–626 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    DiNola, A., Russo, C.: Lukasiewicz transform and it’s application to compression and reconstruction of digital images. Inf. Sci. 177, 1481–1498 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Loia, V., Sessa, S.: Fuzzy relation equations for coding/decoding processes of images and videos. Inf. Sci. 171, 145–172 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nobuhara, H., Pedrycz, W.: Fast solving method of fuzzy relational equation and it application to lossy image compression/reconstruction. IEEE Trans. Fuzzy Syst. 8(3), 325–334 (2000)CrossRefGoogle Scholar
  18. 18.
    Nobuhara, H., Bede, B., Hirota, K.: On various eigen fuzzy sets and their application to image reconstruction. Inf. Sci. 176, 2988–3010 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Czogala, E., Drewniak, J., Pedrycz, W.: Fuzzy relation equations on a finite set. Fuzzy Sets Syst. 7, 89–101 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fang, S.C., Li, G.: Solving fuzzy relation equations with a linear objective function. Fuzzy Sets Syst. 103, 107–113 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Loetamonphong, J., Fang, S.C.: An efficient solution procedure for fuzzy relational equations with max-product composition. IEEE Trans. Fuzzy Syst. 7(4), 441–445 (1999)CrossRefGoogle Scholar
  22. 22.
    Chen, L., Wang, P.P.: Fuzzy relation equation (II): the branch-point-solutions and the categorized minimal solutions. Soft Comput. 11, 33–40 (2007)zbMATHCrossRefGoogle Scholar
  23. 23.
    Shieh, B.S.: Solutions of fuzzy relation equations based on continuous t-norms. Inform. Sci. 177, 4208–4215 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Shieh, B.S.: New resolution of finite fuzzy relation equations with max-min composition. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 16(1), 19–33 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Yeh, C.T.: On the minimal solutions of max-min fuzzy relational equations. Fuzzy Sets Syst. 159, 23–39 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Wu, Y.K., Guu, S.M.: Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets Syst. 150, 147–162 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Wu, Y.K., Guu, S.M., Liu, J.Y.C.: An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Trans. Fuzzy Syst. 10(4), 552–558 (2002)CrossRefGoogle Scholar
  28. 28.
    Nola, A.D., Pedrycz, W., Sessa, S., Zhuang, W.P.: Fuzzy relation equations under a class of triangular norms: a survey and new results. Stochastica 8, 99–145 (1984)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Shimbo, M.M.: Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets Syst. 16, 53–63 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Pedrycz, W.: On generalized fuzzy relational equations and their applications. J. Math. Anal. Appl. 107, 520–536 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Dubois, D., Prade, H.: New results about properties and semantics of fuzzy set-theoratic operators. In: Wang, P.P., Chang, S.K. (eds.) Fuzzy Sets, pp. 59–75. Plenum Press, New York (1986)zbMATHGoogle Scholar
  32. 32.
    Li, J.X.: A new algorithm for minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Sets Syst. 159, 2278–2298 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Oden, G.C.: Integration of fuzzy logical information. J. Exp. Psychol. Hum. Percept. Perform. 106, 565–575 (1977)CrossRefGoogle Scholar
  34. 34.
    Pedrycz, W.: An identification algorithm in fuzzy relation systems. Fuzzy Sets Syst. 13, 153–167 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Thole, U., Zimmermann, H.J., Zysno, P.: On the suit-ability of minimum and product operators for intersection of fuzzy sets. Fuzzy Sets Syst. 2, 167–180 (1979)zbMATHCrossRefGoogle Scholar
  36. 36.
    Xu, C.W., Lu, Y.Z.: Fuzzy model identification and self-learning for dynamic systems. IEEE Trans. Syst. Man Cybernet 17, 683–689 (1987)zbMATHCrossRefGoogle Scholar
  37. 37.
    Zimmermann, H.J., Zysno, P.: Latent connectives in human decision-making. Fuzzy Sets Syst. 4, 37–51 (1980)zbMATHCrossRefGoogle Scholar
  38. 38.
    Yager, R.R.: Some procedures for selecting fuzzy set-theoretic operators. Internat. J. Gen. Syst. 8, 235–242 (1982)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Bourke, M.M., Fisher, D.G.: Solution algorithms for fuzzy relational equations with max-product composition. Fuzzy Sets Syst. 94, 61–69 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Di Martino, F., Loia, V., Sessa, S.: Fuzzy transforms for compression and decompression of color videos. Inf. Sci. 180(20), 3914–3931 (2010)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Qu, X.B., Wang, X.P.: Minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations. Inf. Sci. 178, 3482–3490 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Loetamonphong, J., Fang, S.C.: Optimization of fuzzy relational equations with max-product composition. Fuzzy Sets Syst. 118, 509–517 (2001)zbMATHCrossRefGoogle Scholar
  43. 43.
    Wang, P.Z., Zhang, D.Z., Sanchez, E., Lee, E.S.: Latticized linear programming and fuzzy relation inequalities. J. Math. Anal. Appl. 159(1), 72–87 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)zbMATHCrossRefGoogle Scholar
  45. 45.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Yager, R.R.: A characterization of the extension principle. Fuzzy Sets Syst. 18, 205–217 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Zimmermann, H.J.: Fuzzy Sets Theory and Its Application. Kluwer Academic Publishers, Boston (1991)CrossRefGoogle Scholar
  48. 48.
    Li, P., Fang, S.C.: Latticized linear optimization on the unit interval. IEEE Trans. Fuzzy Syst. 17(6), 1353–1365 (2009)CrossRefGoogle Scholar
  49. 49.
    Yang, S.J.: An algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition-min composition. Fuzzy Sets Syst. 255, 41–51 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Chang, Ch-W, Shieh, B.S.: Linear optimization problem constrained by fuzzy maxmin relation equations. Inform. Sci. 234, 71–79 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Abbasi Molai, A.: Fuzzy linear objective function optimization with fuzzy-valued max-product fuzzy relation inequality constraints. Math Comput Model. 51, 1240–1250 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Bing-Yuan Cao
    • 1
    • 2
    • 3
    Email author
  • Ji-Hui Yang
    • 4
  • Xue-Gang Zhou
    • 5
  • Zeinab Kheiri
    • 6
  • Faezeh Zahmatkesh
    • 6
  • Xiao-Peng Yang
    • 7
  1. 1.University of FoshanFoshanChina
  2. 2.University of GuangzhouGuangzhouChina
  3. 3.Guangzhou Vocational and Technical University of Science and TechnologyGuangzhouChina
  4. 4.College of ScienceShenyang Agricultural UniversityShenyangChina
  5. 5.School of Financial Mathematics and StatisticsGuangdong University of FinanceGuangzhouChina
  6. 6.Higher Education Mega CenterGuangzhou UniversityGuangzhouChina
  7. 7.Department of Mathematics and StatisticsHanshan Normal UniversityChaozhouChina

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