𝕋-Relative Fuzzy Linear Programming

Kelly Osawaru; Owin Olowu

doi:https://doi.org/10.14445/22315373/IJMTT-V69I9P503

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 9 | Year 2023 | Article Id. IJMTT-V69I9P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I9P503

𝕋-Relative Fuzzy Linear Programming

Kelly Osawaru, Owin Olowu

Received	Revised	Accepted	Published
13 Jul 2023	19 Aug 2023	04 Sep 2023	27 Sep 2023

Citation :

Kelly Osawaru, Owin Olowu, "𝕋-Relative Fuzzy Linear Programming," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 9, pp. 13-22, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I9P503

Abstract

We introduce and develop in this article linear programming in the context of the 𝕋-Relative fuzzy sets that were introduced by Osawaru, Olaleru and Olaoluwa [3]. Here, the objective function and (or) its set of constraints expressed as a function of a parameter (or variable), say time, is considered. By relaxing the pretenses of optimization using a subjective gradation relative to the parameter, we model and obtain optimal solutions by employing the tools of the relative fuzzy membership functions. Thus, fuzzy optimal values obtained are expressed relative to the parameter of control defining the dynamics of the fuzziness. The results of this study generalize the results obtained for fuzzy linear programming in literature.

Keywords

𝕋-Relative fuzzy sets, 𝕋-Relative fuzzy mapping, 𝕋-Relative fuzzy linear programming, Fuzzy sets.

References

[1] R.E. Bellman, and L.A. Zadeh, “Decision-Making in a Fuzzy Environment,” Management Sciences, vol. 17, no. 4, 1970.
[CrossRef] [Google Scholar] [Publisher Link]
[2] Stanisław Heilpern, “Fuzzy Mappings and Fixed Point Theorem,” Journal of Mathematical Analysis and Applications, vol. 83, no. 2, pp. 566-569, 1981.
[CrossRef] [Google Scholar] [Publisher Link]
[3] K.E. Osawaru, J.O. Olaleru, and Hallowed Olaoluwa, “Relative Fuzzy Sets,” Journal of Fuzzy Set Valued Analysis, vol. 2, pp. 86-110, 2018.
[Publisher Link]
[4] Shizuo Kakutani, “A Generalization of Brouwer’s Fixed Point Theorem,” Duke Mathematical Journal, vol. 8, no. 3, pp. 457-459, 1941.
[CrossRef] [Google Scholar] [Publisher Link]
[5] Samuel Eilenberg, and Drane Montgomery, “Fixed Point Theorems for Multivalued Transformations,” American Journal of Mathematics, vol. 68, no. 2, pp. 214-222, 1946.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Wayman L. Strother, “On an Open Question Concerning Fixed Points,” Proceedings of the American Mathematical Society, vol. 4, no. 6, pp. 988-993, 1953.
[CrossRef] [Google Scholar] [Publisher Link]
[7] Robert L. Plunkett, “A Fixed Point Theorem for Continuous Multivalued Transformations,” Proceedings of the American Mathematical Society, vol. 7, no. 1, pp. 160-163, 1956.
[Google Scholar] [Publisher Link]
[8] L.E. JR. Ward, “Characterization of the Fixed Point Property for a Class of Set-Valued Mappings,” Fundamenta Mathematicae, vol. 50, pp. 159-164, 1961.
[CrossRef] [Google Scholar] [Publisher Link]
[9] J.R. Sam Bernard Nadler, “Multivalued Contraction Mappings,” Pacific Journal of Mathematics, vol. 30, no. 2, pp. 475-488, 1969.
[CrossRef] [Google Scholar] [Publisher Link]
[10] J.R. Sam Bernard Nadler, Some Results on Multivalued Contraction Mappings, Fleischman, W.M. (eds) Set-Valued Mappings, Selections and Topological Properties of 2x, Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, vol. 171, pp. 1-2, 2006.
[Google Scholar] [Publisher Link]
[11] R.K. Bose, and D. Sahani, “Fuzzy Mappings and Fixed Point Theorems,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 53-58, 1987.
[CrossRef] [Google Scholar] [Publisher Link]
[12] Singh, Shyam Lal, and Talwar, Rekha, “Fixed Points of Fuzzy Mappings,” Soochow Journal of Mathematics, vol. 19, no. 1, pp. 95-102, 1993.
[Google Scholar] [Publisher Link]
[13] Tanmoy Som, and R.N. Mukherjee, “Some Fixed Point Theorems for Fuzzy Mappings,” Fuzzy Sets and Systems, vol. 33, no. 2, pp. 213- 219, 1989.
[CrossRef] [Google Scholar] [Publisher Link]
[14] Pu Pao-Ming, and Liu Ying-Ming, “Fuzzy Topology. I. Neighborhood Structure of a Fuzzy Point and Moore-Smith Convergence,” Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571-599, 1980.
[CrossRef] [Google Scholar] [Publisher Link]
[15] Byung Soo Lee, and Sung Jin Cho, “A Fixed Point Theorem for Contractive-Type Fuzzy Mappings,” Fuzzy Sets and Systems, vol. 61, no. 3, pp. 309-312, 1994.
[CrossRef] [Google Scholar] [Publisher Link]
[16] Jong Yeoul Park, and Jae Ug Jeong, “Fixed Point Theorems for Fuzzy Mappings,” Fuzzy Sets and Systems, vol. 87, no. 1, pp.111-116, 1997.
[CrossRef] [Google Scholar] [Publisher Link]
[17] L.A. Zadeh, “Fuzzy Set,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[CrossRef] [Google Scholar] [Publisher Link]
[18] Ming-Po Chen, and Mau-Hsiang Shih, “Fixed Point Theorems for Point-to-Point and Point-to-Set Maps,” Journal of Mathematical Analysis and Applications, vol. 71, no. 2, pp. 515-524, 1979.
[CrossRef] [Google Scholar] [Publisher Link]
[19] R.K. Bose, and R.N. Mukherjee, “Common fixed points of some set-valued mappings”, Tamkang Journal of Mathematics, vol. 8, no. 2, pp. 245-249, 1977.
[20] Ismat Beg, and Akbar Azam, “Fixed Points of Asymptotically Regular Multivalued Mappings,” Journal of the Australian Mathematical Society, vol. 53, no. 3, pp. 313-326, 1992.
[CrossRef] [Google Scholar] [Publisher Link]
[21] Vicente D. Estruch, and Anna Vidal, “A Note on Fixed Fuzzy Points for Fuzzy Mappings,” In: Rendiconti dell’Istituto di Matematica dell’Università di Trieste, An International Journal of Mathematics, vol. 32, no. 2, pp. 39-45, 2001.
[Google Scholar] [Publisher Link]
[22] Stefan Banach, “On Operations in Abstract Sets and their Application to Integral Equations,” Fundamenta Mathematicae, vol. 3, no. 1, pp. 133-181, 1992.
[Google Scholar] [Publisher Link]
[23] K.E. Osawaru, J.O. Olaleru, and H. Akewe, “T-Relative Fuzzy Maps and Some Fixed Point Results,” Journal of Mathematics Statistics, vol. 13, no. 1, pp. 61-88, 2020.
[Google Scholar] [Publisher Link]
[24] Rafail R. Gasimov, and Kürşat Yenilmez, “Solving Fuzzy Linear Programming with Linear Membership Functions,” Turkish Journal of Mathematics, vol. 26, no. 4, pp. 375-396, 2002.
[Google Scholar] [Publisher Link]
[25] Hideo Tanaka, Tetsuji Okuda, and Kiyoji Asai, “On Fuzzy-Mathematical Programming,” Journal of Cybernetics, vol. 3, no. 4, pp. 37-46, 1974.
[CrossRef] [Publisher Link]
[26] Hans-Jürgen Zimmermann, “Fuzzy Programming and Linear Programming with Several Objective Functions,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 45-55, 1978.
[CrossRef] [Google Scholar] [Publisher Link]
[27] Hans-Jürgen Zimmermann, Fuzzy Set Theory-and its Applications, 4th ed., New York: Springer Science Business Media, 2002.
[CrossRef] [Google Scholar] [Publisher Link]
[28] Hans-Jürgen Zimmermann, “Fuzzy Programming and Linear Programming with Several Objective Functions,” Fuzzy Sets and Systems, vol. 1, no. 1, pp. 45-55, 1978.
[CrossRef] [Google Scholar] [Publisher Link]
[29] K. Zimmermann, “Fuzzy Set Covering Problem,” International Journal of General Systems, vol. 20, no. 1, pp. 127-131, 1991.
[CrossRef] [Google Scholar] [Publisher Link]
[30] Brigitte Werners, “An Interactive Fuzzy Programming System,” Fuzzy Sets and Systems, vol. 23, no. 1, pp. 131-147, 1987.
[CrossRef] [Google Scholar] [Publisher Link]
[31] M.J. Hwang, C.I. Chiang, and Y.H. Liu, “Solving a Fuzzy Set-Covering Problem,” Mathematical and Computer Modelling, vol. 40, no. 7-8, pp. 861-865, 2004.
[CrossRef] [Google Scholar] [Publisher Link]