## On the resolution and optimization of a system of fuzzy relational equations with sup-$$T$$ composition.(English)Zbl 1169.90493

Summary: This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-$$T$$ composition, where $$T$$ is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-$$T$$ equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-$$T$$ equations can be reduced into a 0-1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-$$T$$ equations. Further generalizations and related issues are also included for discussion.

### MSC:

 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 90C10 Integer programming
Full Text:

### References:

 [1] Abbasi Molai A. and Khorram E. (2007a). A modified algorithm for solving the proposed models by Ghodousian and Khorram and Khorram and Ghodousian. Applied Mathematics and Computation 190: 1161–1167 · Zbl 1227.90049 [2] Abbasi Molai, A., & Khorram, E. (2007b). Another modification from two papers of Ghodousian and Khorram and Ghorram et al. Applied Mathematics and Computation. doi: 10.1016/j.amc.2007.07.061 . · Zbl 1141.65045 [3] Abbasi Molai, A., & Khorram, E. (2007). An algorithm for solving fuzzy relation equations with max-T composition operator. Information Sciences. doi: 10.1016/j.ins.2007.10.010 . · Zbl 1227.90049 [4] Alsina C., Frank M.J. and Schweizer B. (2006). Associative functions: Triangular norms and copulas. World Scientific, Singarpore · Zbl 1100.39023 [5] Arnould T. and Tano S. (1994a). A rule-based method to calculate the widest solution sets of a max–min fuzzy relational equation. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 2: 247–256 · Zbl 1232.03037 [6] Arnould T. and Tano S. (1994b). A rule-based method to calculate exactly the widest solution sets of a max-min fuzzy relational inequality. Fuzzy Sets and Systems 64: 39–58 · Zbl 1232.03037 [7] Balas E. and Padberg M.W. (1976). Set partitioning: A survey. SIAM Review 18: 710–760 · Zbl 0347.90064 [8] Bellman R.E. and Zadeh L.A. (1977). Local and fuzzy logics. In: Dunn, J.M. and Epstein, G. (eds) Modern uses of multiple valued logic, pp 103–165. Reidel, Dordrecht [9] Bour L. and Lamotte M. (1987). Solutions minimales d’équations de relations floues avec la composition max t-norme. BUSEFAL 31: 24–31 · Zbl 0633.04002 [10] Bourke M.M. and Fisher D.G. (1998). Solution algorithms for fuzzy relational equations with max-product composition. Fuzzy Sets and Systems 94: 61–69 · Zbl 0923.04003 [11] Caprara A., Toth P. and Fischetti M. (2000). Algorithms for the set covering problem. Annals of Operations Research 98: 353–371 · Zbl 0974.90006 [12] Cechlárová K. (1990). Strong regularity of matrices in a discrete bottleneck algebra. Linear Algebra and Its Applications 128: 35–50 · Zbl 0704.15003 [13] Cechlárová K. (1995). Unique solvability of max-min fuzzy equations and strong regularity of matrices over fuzzy algebra. Fuzzy Sets and Systems 75: 165–177 · Zbl 0852.15011 [14] Chen L. and Wang P.P. (2002). Fuzzy relation equations (I): The general and specialized solving algorithms. Soft Computing 6: 428–435 · Zbl 1024.03520 [15] Chen L. and Wang P.P. (2007). Fuzzy relation equations (II): The branch-poit-solutions and the categorized minimal solutions. Soft Computing 11: 33–40 · Zbl 1108.03310 [16] Cheng L. and Peng B. (1988). The fuzzy relation equation with union or intersection preserving operator. Fuzzy Sets and Systems 25: 191–204 · Zbl 0651.04005 [17] Clifford A.H. (1954). Naturally totally ordered commutative semigroups. American Journal of Mathematics 76: 631–646 · Zbl 0055.01503 [18] Cormen T.H., Leiserson C.E., Rivest R.L. and Stein C. (2001). Introduction to algorithms (2nd ed.). MIT Press, Cambridge, MA · Zbl 1047.68161 [19] Cuninghame-Green R.A. (1979). Minimax algebra, Lecture Notes in Economics and Mathematical Systems, Vol. 166. Springer, Berlin · Zbl 0399.90052 [20] Cuninghame-Green R.A. (1995). Minimax algebra and applications. Advances in Imaging and Electron Physics 90: 1–121 [21] Czogała E., Drewiak J. and Pedrycz W. (1982). Fuzzy relation equations on a finite set. Fuzzy Sets and Systems 7: 89–101 · Zbl 0483.04001 [22] De Baets B. (1995a). An order-theoretic approach to solving sup- $${\mathcal{T}}$$ equations. In: Ruan, D. (eds) Fuzzy set theory and advanced mathematical applications, pp 67–87. Kluwer, Dordrecht · Zbl 0874.04005 [23] De Baets, B. (1995b). Oplossen van vaagrelationele vergelijkingen: een ordetheoretische benadering, Ph.D. Dissertation, University of Gent. [24] De Baets, B. (1995c). Model implicators and their characterization. In N. Steele (Ed.), Proceedings of the First ICSC International Symposium on Fuzzy Logic (pp. A42–A49). ICSC Academic Press. [25] De Baets B. (1997). Coimplicators, the forgotten connectives. Tatra Mountains Mathematical Publications 12: 229–240 · Zbl 0954.03029 [26] De Baets, B. (1998). Sup- $${\mathcal{T}}$$ Equations: State of the art. In O. Kaynak, et al. (Eds.), Computational Intelligence: Soft Computing and Fuzzy-Neural Integration with Applications, NATO ASI Series F: Computer and Systems Sciences (Vol. 162, pp. 80–93). Berlin: Springer-Verlag. · Zbl 0927.03078 [27] De Baets, B. (2000). Analytical solution methods for fuzzy relational equations. In D. Dubois & H. Prade, (Eds.), Fundamentals of fuzzy sets, the handbooks of fuzzy sets series (Vol. 1, pp. 291–340). Dordrecht: Kluwer. · Zbl 0970.03044 [28] De Baets B., Van de Walle B. and Kerre E. (1998). A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. Part II: The identity case. Fuzzy Sets and Systems 99: 303–310 · Zbl 1083.91503 [29] De Cooman G. and Kerre E. (1994). Order norms on bounded partially ordered sets. Journal of Fuzzy Mathematics 2: 281–310 · Zbl 0814.04005 [30] Demirli K. and De Baetes B. (1999). Basic properties of implicators in a residual framework. Tatra Mountains Mathematical Publications 16: 31–46 · Zbl 0949.03025 [31] Di Martino, F., Loia, V., & Sessa, S. (2003). A method in the compression/decompression of images using fuzzy equations and fuzzy similarities. In T. Bilgiç, B. De Baets, & O. Kaynak (Eds.), Proceedings of the 10th International Fuzzy Systems Association World Congress, Istanbul, Turkey, pp. 524–527. [32] Di Nola A. (1984). An algorithm of calculation of lower solutions of fuzzy relation equation. Stochastica 3: 33–40 · Zbl 0593.04005 [33] Di Nola A. (1985). Relational equations in totally ordered lattices and their complete resolution. Journal of Mathematical Analysis and Applications 107: 148–155 · Zbl 0588.04006 [34] Di Nola A. (1990). On solving relational equations in Brouwerian lattices. Fuzzy Sets and Systems 34: 365–376 · Zbl 0701.04003 [35] Di Nola A. and Lettieri A. (1989). Relation equations in residuated lattices. Rendiconti del Circolo Matematico di Palermo 38: 246–256 · Zbl 0691.06002 [36] Di Nola A., Pedrycz W. and Sessa S. (1982). On solution of fuzzy relational equations and their characterization. BUSEFAL 12: 60–71 · Zbl 0525.04003 [37] Di Nola A., Pedrycz W. and Sessa S. (1988). Fuzzy relation equations with equality and difference composition operators. Fuzzy Sets and Systems 25: 205–215 · Zbl 0645.04004 [38] Di Nola A., Pedrycz W., Sessa S. and Sanchez E. (1991). Fuzzy relation equations theory as a basis of fuzzy modelling: An overview. Fuzzy Sets and Systems 40: 415–429 · Zbl 0727.04005 [39] Di Nola A., Pedrycz W., Sessa S. and Wang P.Z. (1984). Fuzzy relation equations under triangular norms: a survey and new results. Stochastica 8: 99–145 · Zbl 0581.04002 [40] Di Nola A. and Sessa S. (1983). On the set of solutions of composite fuzzy relation equations. Fuzzy Sets and Systems 9: 275–286 · Zbl 0514.94027 [41] Di Nola A. and Sessa S. (1988). Finite fuzzy relational equations with a unique solution in linear lattices. Journal of Mathematical Analysis and Applications 132: 39–49 · Zbl 0661.04003 [42] Di Nola A., Sessa S., Pedrycz W. and Sanchez E. (1989). Fuzzy relation equations and their applications to knowledge engineering. Kluwer, Dordrecht · Zbl 0694.94025 [43] Drewniak J. (1982). Note on fuzzy relation equations. BUSEFAL 12: 50–51 [44] Drewniak J. (1983). System of equations in a linear lattice. BUSEFAL 15: 88–96 · Zbl 0523.04001 [45] Drossos, C., & Navara, M. (1996). Generalized t-conorms and closure operators. In Proceedings of the Fourth European Congress on Intelligent Techniques and Sof Computing, EUFIT’96, Aachen, Germany, pp. 22–26. [46] Dubois D. and Prade H. (1980). New results about properties and semantics of fuzzy set-theoretic operators. In: Wang, P.P. and Chang, S.K. (eds) Fuzzy sets: Theory and applications to policy analysis and information systems, pp 59–75. Plenum Press, New York [47] Dubois D. and Prade H. (1986). New results about properties and semantics of fuzzy set-theoretic operators. In: Wang, P.P. and Chang, S.K. (eds) Fuzzy sets: Theory and applications to policy analysis and information systems, pp 59–75. Plenum Press, New York [48] Fang S.-C. and Li G. (1999). Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103: 107–113 · Zbl 0933.90069 [49] Fodor J.C. (1991). Strict preference relations based on weak t-norms. Fuzzy Sets and Systems 43: 327–336 · Zbl 0756.90006 [50] Frank M.J. (1979). On the simultaneous associativity of F(x,y) and x + y – F(x,y). Aequationes Mathematicae 19: 194–226 · Zbl 0444.39003 [51] Gavalec M. (2001). Solvability and unique solvability of max-min fuzzy equations. Fuzzy Sets and Systems 124: 385–393 · Zbl 0994.03047 [52] Gavalec M. and Plávka J. (2003). Strong regularity of matrices in general max-min algebra. Linear Algebra and Its Applications 371: 241–254 · Zbl 1030.15016 [53] Ghodousian A. and Khorram E. (2006a). An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition. Applied Mathematics and Computation 178: 502–509 · Zbl 1105.65067 [54] Ghodousian A. and Khorram E. (2006b). Solving a linear programming problem with the convex combination of the max-min and the max-average fuzzy relation equations. Applied Mathematics and Computation 180: 411–418 · Zbl 1102.90036 [55] Ghodousian, A., & Khorram, E. (2007). Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition. Information Sciences. doi: 10.1016/j.ins.2007.07.022 . · Zbl 1149.90189 [56] Goguen J.A. (1967). L-Fuzzy sets. Journal of Mathematical Analysis and Applications 18: 145–174 · Zbl 0145.24404 [57] Golumbic M.C. and Hartman I.B.-A. (2005). Graph theory, combinatorics and algorithms: Interdisciplinary applications. Springer-Verlag, New York · Zbl 1077.05001 [58] Gottwald, S. (1984). T-normen und $${\phi}$$ -operatoren als Wahrheitswertfunktionen mehrtiger junktoren. In G. Wechsung (Ed.), Frege Conference 1984, Proceedings of the International Conference held at Schwerin (GDR), Mathematical Research (Vol. 20, pp. 121–128). Berlin: Akademie-Verlag. · Zbl 0556.03024 [59] Gottwald S. (1986). Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet 22: 67–91 · Zbl 0607.03016 [60] Gottwald S. (1993). Fuzzy sets and fuzzy logic: The foundations of application–from a mathematical point of view. Vieweg, Wiesbaden · Zbl 0782.94025 [61] Gottwald S. (2000). Generalized solvability behaviour for systems of fuzzy equations. In: Novák, V. and Perfilieva, I. (eds) Discovering the world with fuzzy logic, pp 401–430. Physica-Verlag, Heidelberg · Zbl 1006.03033 [62] Guo F.-F. and Xia Z.-Q. (2006). An algorithm for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities. Fuzzy Optimization and Decision Making 5: 33–47 · Zbl 1176.90675 [63] Gupta M.M. and Qi J. (1991). Design of fuzzy logic controllers based on generalized T-operators. Fuzzy Sets and Systems 40: 473–489 · Zbl 0732.93050 [64] Guu S.-M. and Wu Y.-K. (2002). Minimizing a linear objective function with fuzzy relation equation constraints. Fuzzy Optimization and Decision Making 1: 347–360 · Zbl 1055.90094 [65] Han S.C. and Li H.X. (2005). Note on ”pseudo-t-norms and implication operators on a complete Brouwerian lattice” and ”pseudo-t-norms and implication operators: Direct products and direct product decompositions”. Fuzzy Sets and Systems 153: 289–294 · Zbl 1086.03019 [66] Han S.C., Li H.X. and Wang J.Y. (2006). Resolution of finite fuzzy relation equations based on strong pseudo-t-norms. Applied Mathematics Letters 19: 752–757 · Zbl 1121.03075 [67] Han S.R. and Sekiguchi T. (1992). Solution of a fuzzy relation equation using a sign matrix. Japanese Journal of Fuzzy Theory and Systems 4: 160–171 [68] Higashi M. and Klir G.J. (1984). Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems 13: 65–82 · Zbl 0553.04006 [69] Höhle U. (1995). Commutative residuated l-monoids. In: Höhle, U. and Klement, E.P. (eds) Non-classical logics and their applications to fuzzy subsets. A handbook of the mathematical foundations of fuzzy set theory, pp 53–106. Kluwer Academic Publishers, Boston [70] Imai H., Kikuchi K. and Miyakoshi M. (1998). Unattainable solutions of a fuzzy relation equation. Fuzzy Sets and Systems 99: 195–196 · Zbl 0938.03081 [71] Imai H., Miyakoshi M. and Da-Te T. (1997). Some properties of minimal solutions for a fuzzy relation equation. Fuzzy Sets and Systems 90: 335–340 · Zbl 0919.04008 [72] Jenei S. (2001). Continuity of left-continuous triangular norms with strong induced negations and their boundary condition. Fuzzy Sets and Systems 124: 35–41 · Zbl 0989.03059 [73] Jenei S. (2002). Structure of left-continuous t-norms with strong induced negations, (III) Construction and decomposition. Fuzzy Sets and Systems 128: 197–208 · Zbl 1050.03505 [74] Kawaguchi M.F. and Miyakoshi M. (1998). Composite fuzzy relational equations with non-commutative conjunctions. Information Sciences 110: 113–125 · Zbl 0930.03074 [75] Khorram E. and Ghodousian A. (2006). Linear objective function optimization with fuzzy relation equation constraints regarding max-av composition. Applied Mathematics and Computation 173: 872–886 · Zbl 1091.65057 [76] Khorram E., Ghodousian A. and Abbasi Molai A. (2006). Solving linear optimization problems with max-star composition equation constraints. Applied Mathematics and Computation 179: 654–661 · Zbl 1103.65067 [77] Klement E.P., Mesiar R. and Pap E. (1999). Quasi- and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems 104: 3–13 · Zbl 0953.26008 [78] Klement E.P., Mesiar R. and Pap E. (2000). Triangular norms. Kluwer, Dordrecht · Zbl 0972.03002 [79] Klement E.P., Mesiar R. and Pap E. (2004a). Triangular norms. Position paper I: Basic analytical and algebraic properties. Fuzzy Sets and Systems 143: 5–26 · Zbl 1038.03027 [80] Klement E.P., Mesiar R. and Pap E. (2004b). Triangular norms. Position paper II: General constructions and parameterized families. Fuzzy Sets and Systems 145: 411–438 · Zbl 1059.03012 [81] Klement E.P., Mesiar R. and Pap E. (2004c). Triangular norms. Position paper III: Continuous t-norms. Fuzzy Sets and Systems 145: 439–454 · Zbl 1059.03013 [82] Klir G. and Yuan B. (1995). Fuzzy sets and fuzzy logic: Theory and applications. Prentice Hall, Upper Saddle River, NJ · Zbl 0915.03001 [83] Kolesárová A. (1999). A note on Archimedean triangular norms. BUSEFAL 80: 57–60 [84] Krause G.M. (1983). Interior idempotents and non-representability of groupoids. Stochastica 7: 5–10 · Zbl 0572.39006 [85] Lettieri A. and Liguori F. (1984). Characterization of some fuzzy relation equations provided with one solution on a finite set. Fuzzy Sets and Systems 13: 83–94 · Zbl 0553.04004 [86] Lettieri A. and Liguori F. (1985). Some results relative to fuzzy relation equations provided with one solution. Fuzzy Sets and Systems 17: 199–209 · Zbl 0588.04007 [87] Li J.-X. (1990). The smallest solution of max-min fuzzy equations. Fuzzy Sets and Systems 41: 317–327 · Zbl 0731.04006 [88] Li J.-X. (1994). On an algorithm for solving fuzzy linear systems. Fuzzy Sets and Systems 61: 369–371 · Zbl 0826.04004 [89] Li P., Fang, S.-C. (2008). A survey on fuzzy relational equations, Part I: Classification and solvability. Fuzzy Optimization and Decision Making (submitted). [90] Li H.X., Miao Z.H., Han S.C. and Wang J.Y. (2005). A new kind of fuzzy relation equations based on inner transformation. Computers and Mathematics with Applications 50: 623–636 · Zbl 1085.03043 [91] Ling C.M. (1965). Representation of associative functions. Publicationes Mathematicae Debrecen 12: 189–212 · Zbl 0137.26401 [92] Loetamonphong J. and Fang S.-C. (1999). An efficient solution procedure for fuzzy relation equations with max-product composition. IEEE Transactions on Fuzzy Systems 7: 441–445 [93] Loetamonphong J. and Fang S.-C. (2001). Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118: 509–517 · Zbl 1044.90533 [94] Loetamonphong J., Fang S.-C. and Young R. (2002). Multi-objective optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 127: 141–164 · Zbl 0994.90130 [95] Loia V. and Sessa S. (2005). Fuzzy relation equations for coding/decoding processes of images and videos. Information Sciences 171: 145–172 · Zbl 1078.68815 [96] Lu J. and Fang S.-C. (2001). Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119: 1–20 [97] Luo Y. and Li Y. (2004). Decomposition and resolution of min-implication fuzzy relation equations based on S-implication. Fuzzy Sets and Systems 148: 305–317 · Zbl 1060.03077 [98] Luoh L., Wang W.J. and Liaw Y.K. (2002). New algorithms for solving fuzzy relation equations. Mathematics and Computers in Simulation 59: 329–333 · Zbl 0999.03513 [99] Luoh L., Wang W.J. and Liaw Y.K. (2003). Matrix-pattern-based computer algorithm for solving fuzzy relation equations. IEEE Transactions on Fuzzy Systems 11: 100–108 [100] Markovskii A. (2004). Solution of fuzzy equations with max-product composition in inverse control and decision making problems. Automation and Remote Control 65: 1486–1495 · Zbl 1114.90488 [101] Markovskii A. (2005). On the relation between equations with max-product composition and the covering problem. Fuzzy Sets and Systems 153: 261–273 · Zbl 1073.03538 [102] Mayor G. and Torrens J. (1991). On a family of t-norms. Fuzzy Sets and Systems 41: 161–166 · Zbl 0739.39006 [103] Milterson P.B., Radhakrishnan J. and Wegener I. (2005). On converting CNF to DNF. Theoretical Computer Science 347: 325–335 · Zbl 1080.94018 [104] Miyakoshi M. and Shimbo M. (1985). Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems 16: 53–63 · Zbl 0582.94031 [105] Miyakoshi M. and Shimbo M. (1986). Lower solutions of systems of fuzzy equations. Fuzzy Sets and Systems 19: 37–46 · Zbl 0628.04004 [106] Mordeson J.N. and Malik D.S. (2002). Fuzzy automata and languages: Theory and applications. Chapman & Hall/CRC, Boca Raton · Zbl 1046.68068 [107] Mostert P.S. and Shields A.L. (1957). On the structure of semi-groups on a compact manifold with boundary. The Annals of Mathematics 65: 117–143 · Zbl 0096.01203 [108] Noskoá L. (2005). Systems of fuzzy relation equation with inf-composition: solvability and solutions. Journal of Electrical Engineering 12(s): 69–72 [109] Oden G.C. (1977). Integration of fuzzy logical information. Journal of Experimental Psychology, Human Perception and Performance 106: 565–575 [110] Pandey, D. (2004). On the optimization of fuzzy relation equations with continuous t-norm and with linear objective function. In Proceedings of the Second Asian Applied Computing Conference, AACC 2004, Kathmandu, Nepal pp. 41–51. [111] Pappis C.P. and Sugeno M. (1985). Fuzzy relational equations and the inverse problem. Fuzzy Sets and Systems 15: 79–90 · Zbl 0561.04003 [112] Pedrycz, W. (1982a). Fuzzy control and fuzzy systems. Technical Report 82 14, Delft University of Technology, Department of Mathematics. · Zbl 0498.04004 [113] Pedrycz W. (1982b). Fuzzy relational equations with triangular norms and their resolutions. BUSEFAL 11: 24–32 · Zbl 0498.04004 [114] Pedrycz W. (1985). On generalized fuzzy relational equations and their applications. Journal of Mathematical Analysis and Applications 107: 520–536 · Zbl 0581.04003 [115] Pedrycz W. (1989). Fuzzy control and fuzzy systems. Research Studies Press/Wiely, New York, NY · Zbl 0723.93042 [116] Pedrycz W. (1991). Processing in relational structures: Fuzzy relational equations. Fuzzy Sets and Systems 40: 77–106 · Zbl 0721.94030 [117] Pedrycz W. (2000). Fuzzy relational equations: bridging theory, methodolody and practice. International Journal of General Systems 29: 529–554 · Zbl 0965.03066 [118] Peeva K. (1985). Systems of linear equations over a bounded chain. Acta Cybernetica 7: 195–202 · Zbl 0584.68074 [119] Peeva K. (1992). Fuzzy linear systems. Fuzzy Sets and Systems 49: 339–355 · Zbl 0805.04005 [120] Peeva K. (2006). Universal algorithm for solving fuzzy relational equations. Italian Journal of Pure and Applied Mathematics 9: 9–20 · Zbl 1137.03030 [121] Peeva K. and Kyosev Y. (2004). Fuzzy relational calculus: Theory, applications and software. World Scientific, New Jersey · Zbl 1083.03048 [122] Peeva K. and Kyosev Y. (2007). Algorithm for solving max-product fuzzy relational equations. Soft Computing 11: 593–605 · Zbl 1113.65042 [123] Perfiliva I. and Tonis A. (2000). Compatibility of systems of fuzzy relation equations. International Journal of General Systems 29: 511–528 · Zbl 0955.03062 [124] Prévot M. (1981). Algorithm for the solution of fuzzy relations. Fuzzy Sets and Systems 5: 319–322 · Zbl 0451.04004 [125] Rudeanu S. (1974). Boolean functions and equations. Amsterdam, North Holland · Zbl 0321.06013 [126] Rudeanu S. (2001). Lattice functions and equations. Springer, London · Zbl 0984.06001 [127] Sanchez, E. (1974). Equations de relation floues, Thèse de Doctorat, Faculté de Médecine de Marseille. [128] Sanchez E. (1976). Resolution of composite fuzzy relation equation. Information and Control 30: 38–48 · Zbl 0326.02048 [129] Sanchez E. (1977). Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic. In: Gupta, M.M., Saridis, G.N. and Gaines, B.R. (eds) Fuzzy automata and decision processes, pp 221–234. Amsterdam, North-Holland [130] Schweizer B. and Sklar A. (1963). Associative functions and abstract semigroups. Plublicationes Mathematicae Debrecen 10: 69–81 · Zbl 0119.14001 [131] Sessa S. (1984). Some results in the setting of fuzzy relation equations theory. Fuzzy Sets and Systems 14: 281–297 · Zbl 0559.04005 [132] Shi E.W. (1987). The hypothesis on the number of lower solutions of a fuzzy relation equation. BUSEFAL 31: 32–41 · Zbl 0645.04005 [133] Shieh B.-S. (2007). Solutions of fuzzy relation equations based on continuous t-norms. Information Sciences 177: 4208–4215 · Zbl 1122.03054 [134] Stamou G.B. and Tzafestas S.G. (2001). Resolution of composite fuzzy relation equations based on Archimedean triangular norms. Fuzzy Sets and Systems 120: 395–407 · Zbl 0979.03042 [135] Thole U., Zimmermann H.-J. and Zysno P. (1979). On the suitability of minimum and product operators for the intersection of fuzzy sets. Fuzzy Sets and Systems 2: 167–180 · Zbl 0408.94030 [136] Van de Walle, De Baets, B., Kerre, E. (1998). A plea for the use of Łukasiewicz triples in the definition of fuzzy preference structures. Part I: General argumentation. Fuzzy Sets and Systems 97: 349–359 · Zbl 0936.91012 [137] Wagenknecht M. and Hartmann K. (1990). On the existence of minimal solutions for fuzzy equations with tolerances. Fuzzy Sets and Systems 34: 237–244 · Zbl 0687.90094 [138] Wang H. F. (1995). A multi-objective mathematical programming problem with fuzzy relation constraints. Journal of Multi-Criteria Decision Analysis 4: 23–35 · Zbl 0843.90131 [139] Wang X.P. (2001). Method of solution to fuzzy relational equations in a complete Brouwerian lattice. Fuzzy Sets and Systems 120: 409–414 · Zbl 0981.03055 [140] Wang X.P. (2003). Infinite fuzzy relational equations on a complete Brouwerian lattice. Fuzzy Sets and Systems 138: 657–666 · Zbl 1075.03026 [141] Wang X.P. and Xiong Q.Q. (2005). The solution set of a fuzzy relational equation with sup-conjunctor composition in a complete Brouwerian lattice. Fuzzy Sets and Systems 153: 249–260 · Zbl 1073.03539 [142] Wang H.F. and Hsu H.M. (1992). An alternative approach to the resolution of fuzzy relation equations. Fuzzy Sets and Systems 45: 203–213 · Zbl 0761.04006 [143] Wang P.Z., Sessa S., Di Nola A. and Pedrycz W. (1984). How many lower solutions does a fuzzy relation equation have?. BUSEFAL 18: 67–74 · Zbl 0581.04001 [144] Wang, P. Z., & Zhang, D.Z. (1987). Fuzzy decision making, Beijing Normal University Lectures, 1987. [145] Wang P.Z., Zhang D.Z., Sanchez E. and Lee E.S. (1991). Latticized linear programming and fuzzy relation inequalities. Journal of Mathematical Analysis and Applications 159: 72–87 · Zbl 0746.90081 [146] Wengener I. (1987). The complexity of Boolean functions. Wieley, New York [147] Wu, Y.-K. (2006). Optimizing the geometric programming problem with max-min fuzzy relational equation constraints, Technical Report, Vanung University, Department of Industrial Management. [148] Wu Y.-K. (2007). Optimization of fuzzy relational equations with max-av composition. Information Sciences 177: 4216–4229 · Zbl 1140.90523 [149] Wu Y.-K. and Guu S.-M. (2004a). A note on fuzzy relation programming problems with max-strict-t-norm composition. Fuzzy Optimization and Decision Making 3: 271–278 · Zbl 1091.90087 [150] Wu, Y.-K., & Guu, S.-M. (2004b). On multi-objective fuzzy relation programming problem with max-strict-t-norm composition, Technical Report, Yuan Ze University, Department of Business Administration. [151] Wu Y.-K. and Guu S.-M. (2005). Minimizing a linear function under a fuzzy max-min relational equation constraint. Fuzzy Sets and Systems 150: 147–162 · Zbl 1074.90057 [152] Wu Y.-K. and Guu S.-M. (2008). An efficient procedure for solving a fuzzy relational equation with max-Archimedean t-norm composition. IEEE Transactions on Fuzzy Systems 16: 73–84 · Zbl 05516363 [153] Wu Y.-K., Guu S.-M. and Liu J.Y.-C. (2002). An accelerated approach for solving fuzzy relation equations with a linear objective function. IEEE Transactions on Fuzzy Systems 10: 552–558 [154] Wu, Y.-K., Guu, S.-M., & Liu, J. Y.-C. (2007). Optimizing the linear fractional programming problem with max-Archimedean t-norm fuzzy relational equation constraints. In Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 1–6. [155] Xiong Q.Q. and Wang X.P. (2005). Some properties of sup-min fuzzy relational equations on infinite domains. Fuzzy Sets and Systems 151: 393–402 · Zbl 1062.03053 [156] Xu, W.-L. (1978). Fuzzy relation equation. In Reports on Beijing Fuzzy Mathematics Meeting, 1978. [157] Xu W.-L., Wu C.-F. and Cheng W.-M. (1982). An algorithm to solve the max-min fuzzy relational equations. In: Gupta, M. and Sanchez, E. (eds) Approximate reasoning in decision analysis, pp 47–49. North-Holland, Amsterdam [158] Yager R.R. (1982). Some procedures for selecting fuzzy set-theoretic operators. International Journal General Systems 8: 115–124 · Zbl 0488.04005 [159] Yang, J. H., & Cao, B. Y. (2005a). Geometric programming with fuzzy relation equation constraints. In Proceedings of the IEEE International Conference on Fuzzy Systems, pp. 557–560. [160] Yang, J. H., & Cao, B. Y. (2005b). Geometric programming with max-product fuzzy relation equation constraints. In Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society, 2005, 650–653. [161] Yang J.H. and Cao B.Y. (2007). Posynomial fuzzy relation geometric programming. In: Melin, P., Castillo, O., Aguilar, L.T., Kacprzyk, J. and Pedrycz, W. (eds) Proceedings of the 12th International Fuzzy Systems Association World Congress, pp 563–572. Cancun, Mexico · Zbl 1202.90289 [162] Yeh C.-T. (2008). On the minimal solutions of max-min fuzzy relational equations. Fuzzy Sets and Systems 159: 23–39 · Zbl 1176.03040 [163] Zhao C.K. (1987). On matrix equations in a class of complete and completely distributive lattice. Fuzzy Sets and Systems 22: 303–320 · Zbl 0621.06006 [164] Zimmermann H.-J. (2001). Fuzzy set theory and its applications (4th ed.). Kluwer, Boston [165] Zimmermann K. (2007). A note on a paper by E. Khorram and A. Ghodousian. Applied Mathematics and Computation 188: 244–245 · Zbl 1118.65066 [166] Zimmermann H.-J and Zysno P. (1980). Latent connectives in human decision-making. Fuzzy Sets and Systems 4: 37–51 · Zbl 0435.90009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.