Max-Min Fuzzy Relation Equations for a Problem of Spatial Analysis

Ferdinando Di Martino, Salvatore Sessa


We implement an algorithm that uses a system of max-min fuzzy  relation equations (SFRE) for solving a problem of spatial analysis. We integrate this algorithm in a Geographical information Systems (GIS) tool. We apply our  process to determine the symptoms after that an expert sets the SFRE with the values of the impact coefficients related to some parameters of a geographic zone under study. We also define an index of evaluation about the reliability of the results.


Fuzzy relation equation; max-min composition; GIS; triangular fuzzy number

Full Text:



Chen, L., Wang, P.: Fuzzy Relational Equations (I): the General and Specialized Solving Algorithms. Soft Computing 6, 428—435 (2002)

De Baets, B.: Analytical Solution Methods for Fuzzy Relational Equations. In: Dubois, D., Prade, H., (eds.) Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Vol. 1, pp. 291—340. Kluwer Academic Publishers, Dordrecht (2000)

Di Martino, F., Loia, V., Sessa, S.: A Fuzzy-Based Tool for Modelization and Analysis of the Vulnerability of Aquifers: a Case Study. International Journal of Approximate

Reasoning 38, 98—111 (2005)

Di Martino, F., Loia, V., Sessa, S., Giordano, M.: An Evaluation of the Reliability of a GIS Based on the Fuzzy Logic in a Concrete Case Study. In: Petry, F.E., Robinson,V.B., Cobb, M.A. (eds.) Fuzzy Modeling with Spatial Information for Geographic Problems, pp. 185—208. Springer, Heidelberg (2005)

Di Martino, F., Loia, V., Sessa, S.: Extended Fuzzy C-Means Clustering Algorithm for

Hotspot Events in Spatial Analysis, International Journal of Hybrid Intelligent

Systems 5 (1), 31—44 (2008)

Di Nola, A., Pedrycz, W., Sessa, S., Sanchez, E.: Fuzzy Relation Equations and Their Application to Knowledge Engineering. Kluwer Academic Press, Dordrecht, 1989

Groenemans, R., Van Ranst, E., Kerre, E.: Fuzzy Relational Calculi in Land Evaluation. Geoderma 77 (2-4), 283—298 (1997)

Hemetsberger, M., Klinger, G., Niederer, S., Benedikt, J.: Risk Assessment of Avalanches - a Fuzzy GIS Application. In: Ruan, D., D’hondt, P., Kerre, E.E. (eds.) Proceedings of 5th International FLINS. Conference Computational Intelligent Systems for Applied Research, pp. 397—402. World Scientific, Singapore (2002)

Higashi, M., Klir, G.J.: Resolution of Finite Fuzzy Relation Equations. Fuzzy Sets and Systems 13 (1), 65—82 (1984)

Kyosev, Y.: Diagnostics of Sewing Process Using Fuzzy Linear Systems. In: Proceedings of 5th International Conference Textile Science TEXSCI 2003. CD ROM Edition. Liberec, (2003)

Li, P., Fang, S.C..: A Survey on Fuzzy Relational Equations, Part I: Classification and Solvability. Fuzzy Optimation and Decision Making 8, 179—229 (2009)

Markovskii, A.V. : On the Relation between Equations with Max-Product Composition and the Covering Problem. Fuzzy Sets and Systems 153, 261—273 (2005)

Miyakoshi, M., Shimbo, M.: Minimal Solutions of Systems of Fuzzy Equations. Fuzzy Sets and Systems 19, (1986) 37—46

Pappis, C.P., Adamopoulos, G.: A Computer Algorithm for the Solution of the Inverse Problem of Fuzzy Systems. Fuzzy Sets and Systems 39, 279—290 (1991)

Pappis, C.P., Sugeno, M.: Fuzzy Relational Equations and the Inverse Problem. Fuzzy Sets and Systems 15, 79—90 (1985)

Peeva, K.: Systems of Linear Equations over a Bounded Chain. Acta Cybernetica 7(2), 195-202 (1985)

Peeva, K.: Fuzzy Linear Systems. Fuzzy Sets and Systems 49, 339—355 (1992)

Peeva, K.: Fuzzy Linear Systems –Theory and Applications in Artificial Intelligence Areas, DSc Thesis, Sofia (2002) (in Bulgarian)

Peeva, K.: Resolution of Min–Max Fuzzy Relational Equations. In: Nikravesh, M., Zadeh , L.A., Korotkikh, V. (eds.) Fuzzy Partial Differential Equations and Relational Equations, pp. 153—166. Springer, Heidelberg (2004)

Peeva, K.: Universal Algorithm for Solving Fuzzy Relational Equations. Italian Journal of Pure and Applied Mathematics 19, 9—20 (2006)

Peeva, K., Kyosev, Y.: Algorithm for Solving Max-product Fuzzy Relational Equations, Soft Computing 11 (7), 593—605 (2007)

Peeva, K., Kyosev, Y.: Fuzzy Relational Calculus: Theory, Applications and Software (with CD-ROM). Series Advances in Fuzzy Systems-Applications and Theory, vol. 22, World Scientific, Singapore (2004)

Sanchez, E.: Resolution of Composite Fuzzy Relation Equations. Information and Control, 30, 38—48 (1976)

Shieh, B.S.: New Resolution of Finite Fuzzy Relation Equations with Max-Min Composition. Internat. J. of Uncertainty, Fuzziness Knowledge Based Systems 16 (1), 19—33 (2008)

Sicat,, R.S., Carranza, E.J.M., Nidumolu, U.B.: Fuzzy modeling of farmers’ kno- wledge for land suitability classification. Agricultural Systems 83, 49—75 (2005)

Wu, Y.K., Guu, S.M.: An efficient procedure for solving a fuzzy relational equati- on with max-Archimedean t-norm composition. IEEE Transactions on Fuzzy Syst- ems 16 (1), 73—84 (2008)



  • There are currently no refbacks.

Copyright (c) 2017 Ferdinando Di Martino, Salvatore Sessa

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.