Abstract
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.
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References
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
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.
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.
Alsina C., Frank M.J. and Schweizer B. (2006). Associative functions: Triangular norms and copulas. World Scientific, Singarpore
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
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
Balas E. and Padberg M.W. (1976). Set partitioning: A survey. SIAM Review 18: 710–760
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
Bour L. and Lamotte M. (1987). Solutions minimales d’équations de relations floues avec la composition max t-norme. BUSEFAL 31: 24–31
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
Caprara A., Toth P. and Fischetti M. (2000). Algorithms for the set covering problem. Annals of Operations Research 98: 353–371
Cechlárová K. (1990). Strong regularity of matrices in a discrete bottleneck algebra. Linear Algebra and Its Applications 128: 35–50
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
Chen L. and Wang P.P. (2002). Fuzzy relation equations (I): The general and specialized solving algorithms. Soft Computing 6: 428–435
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
Cheng L. and Peng B. (1988). The fuzzy relation equation with union or intersection preserving operator. Fuzzy Sets and Systems 25: 191–204
Clifford A.H. (1954). Naturally totally ordered commutative semigroups. American Journal of Mathematics 76: 631–646
Cormen T.H., Leiserson C.E., Rivest R.L. and Stein C. (2001). Introduction to algorithms (2nd ed.). MIT Press, Cambridge, MA
Cuninghame-Green R.A. (1979). Minimax algebra, Lecture Notes in Economics and Mathematical Systems, Vol. 166. Springer, Berlin
Cuninghame-Green R.A. (1995). Minimax algebra and applications. Advances in Imaging and Electron Physics 90: 1–121
Czogała E., Drewiak J. and Pedrycz W. (1982). Fuzzy relation equations on a finite set. Fuzzy Sets and Systems 7: 89–101
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
De Baets, B. (1995b). Oplossen van vaagrelationele vergelijkingen: een ordetheoretische benadering, Ph.D. Dissertation, University of Gent.
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.
De Baets B. (1997). Coimplicators, the forgotten connectives. Tatra Mountains Mathematical Publications 12: 229–240
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.
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.
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
De Cooman G. and Kerre E. (1994). Order norms on bounded partially ordered sets. Journal of Fuzzy Mathematics 2: 281–310
Demirli K. and De Baetes B. (1999). Basic properties of implicators in a residual framework. Tatra Mountains Mathematical Publications 16: 31–46
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.
Di Nola A. (1984). An algorithm of calculation of lower solutions of fuzzy relation equation. Stochastica 3: 33–40
Di Nola A. (1985). Relational equations in totally ordered lattices and their complete resolution. Journal of Mathematical Analysis and Applications 107: 148–155
Di Nola A. (1990). On solving relational equations in Brouwerian lattices. Fuzzy Sets and Systems 34: 365–376
Di Nola A. and Lettieri A. (1989). Relation equations in residuated lattices. Rendiconti del Circolo Matematico di Palermo 38: 246–256
Di Nola A., Pedrycz W. and Sessa S. (1982). On solution of fuzzy relational equations and their characterization. BUSEFAL 12: 60–71
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
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
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
Di Nola A. and Sessa S. (1983). On the set of solutions of composite fuzzy relation equations. Fuzzy Sets and Systems 9: 275–286
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
Di Nola A., Sessa S., Pedrycz W. and Sanchez E. (1989). Fuzzy relation equations and their applications to knowledge engineering. Kluwer, Dordrecht
Drewniak J. (1982). Note on fuzzy relation equations. BUSEFAL 12: 50–51
Drewniak J. (1983). System of equations in a linear lattice. BUSEFAL 15: 88–96
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.
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
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
Fang S.-C. and Li G. (1999). Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems 103: 107–113
Fodor J.C. (1991). Strict preference relations based on weak t-norms. Fuzzy Sets and Systems 43: 327–336
Frank M.J. (1979). On the simultaneous associativity of F(x,y) and x + y – F(x,y). Aequationes Mathematicae 19: 194–226
Gavalec M. (2001). Solvability and unique solvability of max-min fuzzy equations. Fuzzy Sets and Systems 124: 385–393
Gavalec M. and Plávka J. (2003). Strong regularity of matrices in general max-min algebra. Linear Algebra and Its Applications 371: 241–254
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
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
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.
Goguen J.A. (1967). L-Fuzzy sets. Journal of Mathematical Analysis and Applications 18: 145–174
Golumbic M.C. and Hartman I.B.-A. (2005). Graph theory, combinatorics and algorithms: Interdisciplinary applications. Springer-Verlag, New York
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.
Gottwald S. (1986). Characterizations of the solvability of fuzzy equations. Elektron. Informationsverarb. Kybernet 22: 67–91
Gottwald S. (1993). Fuzzy sets and fuzzy logic: The foundations of application—from a mathematical point of view. Vieweg, Wiesbaden
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
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
Gupta M.M. and Qi J. (1991). Design of fuzzy logic controllers based on generalized T-operators. Fuzzy Sets and Systems 40: 473–489
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
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
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
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
Higashi M. and Klir G.J. (1984). Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems 13: 65–82
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
Imai H., Kikuchi K. and Miyakoshi M. (1998). Unattainable solutions of a fuzzy relation equation. Fuzzy Sets and Systems 99: 195–196
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
Jenei S. (2001). Continuity of left-continuous triangular norms with strong induced negations and their boundary condition. Fuzzy Sets and Systems 124: 35–41
Jenei S. (2002). Structure of left-continuous t-norms with strong induced negations, (III) Construction and decomposition. Fuzzy Sets and Systems 128: 197–208
Kawaguchi M.F. and Miyakoshi M. (1998). Composite fuzzy relational equations with non-commutative conjunctions. Information Sciences 110: 113–125
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
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
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
Klement E.P., Mesiar R. and Pap E. (2000). Triangular norms. Kluwer, Dordrecht
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
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
Klement E.P., Mesiar R. and Pap E. (2004c). Triangular norms. Position paper III: Continuous t-norms. Fuzzy Sets and Systems 145: 439–454
Klir G. and Yuan B. (1995). Fuzzy sets and fuzzy logic: Theory and applications. Prentice Hall, Upper Saddle River, NJ
Kolesárová A. (1999). A note on Archimedean triangular norms. BUSEFAL 80: 57–60
Krause G.M. (1983). Interior idempotents and non-representability of groupoids. Stochastica 7: 5–10
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
Lettieri A. and Liguori F. (1985). Some results relative to fuzzy relation equations provided with one solution. Fuzzy Sets and Systems 17: 199–209
Li J.-X. (1990). The smallest solution of max-min fuzzy equations. Fuzzy Sets and Systems 41: 317–327
Li J.-X. (1994). On an algorithm for solving fuzzy linear systems. Fuzzy Sets and Systems 61: 369–371
Li P., Fang, S.-C. (2008). A survey on fuzzy relational equations, Part I: Classification and solvability. Fuzzy Optimization and Decision Making (submitted).
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
Ling C.M. (1965). Representation of associative functions. Publicationes Mathematicae Debrecen 12: 189–212
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
Loetamonphong J. and Fang S.-C. (2001). Optimization of fuzzy relation equations with max-product composition. Fuzzy Sets and Systems 118: 509–517
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
Loia V. and Sessa S. (2005). Fuzzy relation equations for coding/decoding processes of images and videos. Information Sciences 171: 145–172
Lu J. and Fang S.-C. (2001). Solving nonlinear optimization problems with fuzzy relation equation constraints. Fuzzy Sets and Systems 119: 1–20
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
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
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
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
Markovskii A. (2005). On the relation between equations with max-product composition and the covering problem. Fuzzy Sets and Systems 153: 261–273
Mayor G. and Torrens J. (1991). On a family of t-norms. Fuzzy Sets and Systems 41: 161–166
Milterson P.B., Radhakrishnan J. and Wegener I. (2005). On converting CNF to DNF. Theoretical Computer Science 347: 325–335
Miyakoshi M. and Shimbo M. (1985). Solutions of composite fuzzy relational equations with triangular norms. Fuzzy Sets and Systems 16: 53–63
Miyakoshi M. and Shimbo M. (1986). Lower solutions of systems of fuzzy equations. Fuzzy Sets and Systems 19: 37–46
Mordeson J.N. and Malik D.S. (2002). Fuzzy automata and languages: Theory and applications. Chapman & Hall/CRC, Boca Raton
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
Noskoá L. (2005). Systems of fuzzy relation equation with inf-→ composition: solvability and solutions. Journal of Electrical Engineering 12(s): 69–72
Oden G.C. (1977). Integration of fuzzy logical information. Journal of Experimental Psychology, Human Perception and Performance 106: 565–575
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.
Pappis C.P. and Sugeno M. (1985). Fuzzy relational equations and the inverse problem. Fuzzy Sets and Systems 15: 79–90
Pedrycz, W. (1982a). Fuzzy control and fuzzy systems. Technical Report 82 14, Delft University of Technology, Department of Mathematics.
Pedrycz W. (1982b). Fuzzy relational equations with triangular norms and their resolutions. BUSEFAL 11: 24–32
Pedrycz W. (1985). On generalized fuzzy relational equations and their applications. Journal of Mathematical Analysis and Applications 107: 520–536
Pedrycz W. (1989). Fuzzy control and fuzzy systems. Research Studies Press/Wiely, New York, NY
Pedrycz W. (1991). Processing in relational structures: Fuzzy relational equations. Fuzzy Sets and Systems 40: 77–106
Pedrycz W. (2000). Fuzzy relational equations: bridging theory, methodolody and practice. International Journal of General Systems 29: 529–554
Peeva K. (1985). Systems of linear equations over a bounded chain. Acta Cybernetica 7: 195–202
Peeva K. (1992). Fuzzy linear systems. Fuzzy Sets and Systems 49: 339–355
Peeva K. (2006). Universal algorithm for solving fuzzy relational equations. Italian Journal of Pure and Applied Mathematics 9: 9–20
Peeva K. and Kyosev Y. (2004). Fuzzy relational calculus: Theory, applications and software. World Scientific, New Jersey
Peeva K. and Kyosev Y. (2007). Algorithm for solving max-product fuzzy relational equations. Soft Computing 11: 593–605
Perfiliva I. and Tonis A. (2000). Compatibility of systems of fuzzy relation equations. International Journal of General Systems 29: 511–528
Prévot M. (1981). Algorithm for the solution of fuzzy relations. Fuzzy Sets and Systems 5: 319–322
Rudeanu S. (1974). Boolean functions and equations. Amsterdam, North Holland
Rudeanu S. (2001). Lattice functions and equations. Springer, London
Sanchez, E. (1974). Equations de relation floues, Thèse de Doctorat, Faculté de Médecine de Marseille.
Sanchez E. (1976). Resolution of composite fuzzy relation equation. Information and Control 30: 38–48
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
Schweizer B. and Sklar A. (1963). Associative functions and abstract semigroups. Plublicationes Mathematicae Debrecen 10: 69–81
Sessa S. (1984). Some results in the setting of fuzzy relation equations theory. Fuzzy Sets and Systems 14: 281–297
Shi E.W. (1987). The hypothesis on the number of lower solutions of a fuzzy relation equation. BUSEFAL 31: 32–41
Shieh B.-S. (2007). Solutions of fuzzy relation equations based on continuous t-norms. Information Sciences 177: 4208–4215
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
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
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
Wagenknecht M. and Hartmann K. (1990). On the existence of minimal solutions for fuzzy equations with tolerances. Fuzzy Sets and Systems 34: 237–244
Wang H. F. (1995). A multi-objective mathematical programming problem with fuzzy relation constraints. Journal of Multi-Criteria Decision Analysis 4: 23–35
Wang X.P. (2001). Method of solution to fuzzy relational equations in a complete Brouwerian lattice. Fuzzy Sets and Systems 120: 409–414
Wang X.P. (2003). Infinite fuzzy relational equations on a complete Brouwerian lattice. Fuzzy Sets and Systems 138: 657–666
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
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
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
Wang, P. Z., & Zhang, D.Z. (1987). Fuzzy decision making, Beijing Normal University Lectures, 1987.
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
Wengener I. (1987). The complexity of Boolean functions. Wieley, New York
Wu, Y.-K. (2006). Optimizing the geometric programming problem with max-min fuzzy relational equation constraints, Technical Report, Vanung University, Department of Industrial Management.
Wu Y.-K. (2007). Optimization of fuzzy relational equations with max-av composition. Information Sciences 177: 4216–4229
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
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.
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
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
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
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.
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
Xu, W.-L. (1978). Fuzzy relation equation. In Reports on Beijing Fuzzy Mathematics Meeting, 1978.
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
Yager R.R. (1982). Some procedures for selecting fuzzy set-theoretic operators. International Journal General Systems 8: 115–124
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.
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.
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
Yeh C.-T. (2008). On the minimal solutions of max-min fuzzy relational equations. Fuzzy Sets and Systems 159: 23–39
Zhao C.K. (1987). On matrix equations in a class of complete and completely distributive lattice. Fuzzy Sets and Systems 22: 303–320
Zimmermann H.-J. (2001). Fuzzy set theory and its applications (4th ed.). Kluwer, Boston
Zimmermann K. (2007). A note on a paper by E. Khorram and A. Ghodousian. Applied Mathematics and Computation 188: 244–245
Zimmermann H.-J and Zysno P. (1980). Latent connectives in human decision-making. Fuzzy Sets and Systems 4: 37–51
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Li, P., Fang, SC. On the resolution and optimization of a system of fuzzy relational equations with sup-T composition. Fuzzy Optim Decis Making 7, 169–214 (2008). https://doi.org/10.1007/s10700-008-9029-y
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DOI: https://doi.org/10.1007/s10700-008-9029-y