System of bipolar max-drastic product fuzzy relation equations with a drastic negation

Document Type : Research Paper


School of Mathematics and Computer Sciences, Damghan University, P.O. Box 36715-364, Damghan, Iran


This paper investigates the consistency of bipolar max-$ T_{D} $ Fuzzy Relation Equations (FREs) where $ T_{D} $ is the drastic product with a specific drastic negation operator. We firstly study the bipolar max-$ T_{D} $ fuzzy relation equation. Then the special characterizations of its feasible domain and its maximal and minimal solutions are presented. Furthermore, some necessary conditions and sufficient conditions are proposed for the solvability of a system of bipolar max-$ T_D $ FREs. Some examples are also given to illustrate them.


[1] S. Aliannezhadi and A. Abbasi Molai, Geometric programming with a single term exponent subject to bipolar
max-product fuzzy relation equation constraints, Fuzzy Sets Syst. 397 (2020), 61–83.
[2] S. Aliannezhadi, A. Abbasi Molai and B. Hedayatfar, Linear optimization with bipolar max-parametric Hamacher
fuzzy relation equation constraints, Kybernetika 52 (2016), no. 4, 531–537.
[3] S. Aliannezhadi, S.S. Ardalan and A. Abbasi Molai, Maximizing a monomial geometric objective function subject
to bipolar max-product fuzzy relation constraints, J. Intell. Fuzzy Syst. 32 (2017), no. 1, 337–350.
[4] M. Baczynskia and B. Jayaram, QL-implications: Some properties and intersections, Fuzzy Sets Syst. 161 (2010),
[5] A. Ciaramella, R. Tagliaferri, W. Pedrycz and A. Di Nola, Fuzzy relational neural network, Int. J. Approx. Reason.
41 (2006), no. 2, 146–163.
[6] M.E. Cornejo, D. Lobo and J. Medina, Bipolar fuzzy relation equations systems based on the product t-norm,
Math. Methods Appl. Sci. 42 (2019), no. 17, 5779–5793.
[7] M.E. Cornejo, D. Lobo and J. Medina, On the solvability of bipolar max-product fuzzy relation equations with the
standard negation, Fuzzy Sets Syst. 410 (2021), 1–18.
[8] M.E. Cornejo, D. Lobo and J. Media, On the solvability of bipolar max-product fuzzy relation equation with the
product negation, J. Comput. Appl. Math. 354 (2019), 520–534.
[9] J.D. Dombi, A special class of fuzzy operators and its application in modelling effects and decision problems, Ph.D.
thesis, University of Szeged, September 2013.
[10] S. Freson, B. De Baets and H.D. Meyer, Linear optimaization with bipolar max-min constraints, Inf. Sci. 234
(2013), 3–15.
[11] M. Higashi and G.J. Klir, Resolution of finite fuzzy relation equations, Fuzzy Sets Syst. 13 (1984), 65-–82.
[12] P. Li and S.-C. Fang, On the resolution and optimization of a system of fuzzy relational equations with sup-t
composition, Fuzzy Optim. Decis. Mak. 7 (2008), no. 2, 169–214.
[13] P. Li and Q. Jin, On the resolution of bipolar max-min equations, Kybernetika 52 (2016), no. 4, 514–530.
[14] P. Li and Y. Liu, Linear optimization with bipolar fuzzy relational equation constraints using the Lukasiewicz
triangular norm, Soft Comput. 18 (2014), 1399–1404.
[15] C.C. Liu, Y.Y. Lur and Y.K. Wu, Linear optimization of bipolar fuzzy relational equations with max-Lukasiewicz
composition, Inf. sci. 360 (2016), 149–162.
[16] V. Loia and S. Sessa, Fuzzy relation equations for coding/decoding processes of immages and videos, Inf. Sci. 171
(2005), no. 1-3, 145–172.
[17] L. Luoh, W.-J. Wang and Y.-K. Liaw, New algorithms for solving fuzzy relation equations, Math. Comput. Simul.
59 (2002), 329–333.
[18] H. Nobuhara, W. Pedrycz, S. Sessa and K. Hirota, A motion compression reconstruction method based on max
t-norm composite fuzzy relational equations, Inf. Sci. 176 (2006), no. 17, 2526–2552.
[19] K. Peeva, Universal algorithm for solving fuzzy relational equations, Ital. J. Pure Appl. Math. 19 (2006), 169—188.
[20] W. Pedrycz, On generalized fuzzy relational equations and their applications, J. Math. Anal. Appl. 107 (1985),no. 2, 520–536.
[21] E. Sanchez, Resolution of composite fuzzy relation equations, Inf. Control 30 (1976), no. 1, 38–48.
[22] E. Sanchez, Inverses of fuzzy relations, application to possibility distribtions and medical diagnosis, Fuzzy Sets
Syst. 2 (1985), no. 1, 520–536.
[23] F. Sun, X.-B. Qu, X.-P. Wang and L. Zhu, On pre-solution matrices of fuzzy relation equations over complete
Brouwerian lattices, Fuzzy Sets Syst. 384 (2020), 34–53.
[24] V.L. Tiwari and A. Thapar, Covering problem for solutions of max-archimedean bipolar fuzzy relation equations,
Int. J. Uncertain. Fuzz. 28 (2020), 613–634.
[25] E. Turunen, Necessary and sufficient conditions for the existence of solution of generalized fuzzy relation equations
A ⇔X = B, Inf. Sci. 536 (2020), 351–357.
[26] X.P. Yang, X.G. Zhou and B.-Y. Cao, Latticized linear programming subject to max-product fuzzy relation inequalitties with application in wireless communication, Inf. Sci. 358-359 (2016), 44–55.
[27] X.P. Yang, Resolution of bipolar fuzzy relation equation with max-Lukasiewicz composition, Fuzzy Sets Syst. 397
(2020), 41–60.
[28] X.P. Yang, Solutions and strong solutions of min-product fuzzy relation inequalities with application in supply
chain, Fuzzy Sets Syst. 384 (2020), 54–74.
[29] H.-J. Zimmermann, Fuzzy set theory and its applications, Fourth Edition, Kluwer Academic Publishers, 2001.
Volume 13, Issue 2
July 2022
Pages 2095-2107
  • Receive Date: 11 January 2021
  • Revise Date: 14 March 2021
  • Accept Date: 18 April 2021