# zbMATH — the first resource for mathematics

Single-variable term semi-latticized fuzzy relation geometric programming with max-product operator. (English) Zbl 1390.90614
Summary: Considering the practical application background in peer-to-peer network system, we investigate the single-variable term semi-latticized fuzzy relation geometric programming with max-product operator in this paper. Based on the characteristics of the objective function and feasible domain, we develop a matrix approach to deal with the proposed problem. A step-by-step algorithm is proposed to find an optimal solution without solving all the minimal solutions of the constraint. Two examples are given to illustrate the feasibility and efficiency of the algorithm.

##### MSC:
 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 90C30 Nonlinear programming 90C35 Programming involving graphs or networks
Full Text:
##### References:
 [1] Sanchez, E., Equations de Relations Floues, Thèse Biologie Humaine Faculté de Médecine de Marseille, (1974), Marseille, France [2] Sanchez, E., Resolution of composite fuzzy relation equations, Inf. Control, 30, 38-48, (1976) · Zbl 0326.02048 [3] Sanchez, E., Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic, (Gupta, M. M.; Saridis, G. N.; Gaines, B. R., Fuzzy Automata and Decision Processes, (1977), North-Holland Amsterdam), 221-234 [4] Czogala, E.; Drewniak, J.; Pedrycz, W., Fuzzy relation equations on a finite set, Fuzzy Sets Syst., 7, 89-101, (1982) · Zbl 0483.04001 [5] Li, P.; Fang, S.-C., On the unique solvability of fuzzy relational equations, Fuzzy Decis. Mak. Optim., 10, 115-124, (2011) · Zbl 1213.03065 [6] Luoh, L.; Wang, W. J.; Liaw, Y. K., New algorithms for solving fuzzy relation equations, Math. Comput. Simul., 59, 329-333, (2002) · Zbl 0999.03513 [7] Li, X.; Ruan, D., Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: part III, Fuzzy Sets Syst., 109, 355-362, (2000) · Zbl 0956.68131 [8] Chen, L.; Wang, P. P., Fuzzy relation equations (I): the general and specialized solving algorithms, Soft Comput., 6, 428-435, (2002) · Zbl 1024.03520 [9] Luoh, L.; Wang, W.-J.; Liaw, Y.-K., Matrix-pattern-based computer algorithm for solving fuzzy relation equations, IEEE Trans. Fuzzy Syst., 11, 1, 100-108, (2003) [10] Wang, P. Z.; Sessa, S.; Nola, A. D.; Pedrycz, W., How many lower solutions does a fuzzy relation equation have?, Busefal, 18, 67-74, (1984) · Zbl 0581.04001 [11] Loetamonphong, J.; Fang, S.-C., An efficient solution procedure for fuzzy relation equations with MAX-product composition, IEEE Trans. Fuzzy Syst., 7, 441-445, (1999) [12] Di Nola, A.; Sessa, S.; Pedrycz, W.; Sanchez, E., Fuzzy Relation Equations and their Applications to Knowledge Engineering, (1989), Kluwer Academic Publishers Dordrecht, Boston, London · Zbl 0694.94025 [13] Loia, V.; Sessa, S., Fuzzy relation equations for coding/decoding processes of images and videos, Inf. Sci., 171, 145-172, (2005) · Zbl 1078.68815 [14] Nobuhara, H.; Bede, B.; Hirota, K., On various eigen fuzzy sets and their application to image reconstruction, Inf. Sci., 176, 2988-3010, (2006) · Zbl 1102.68697 [15] Nobuhara, H.; Pedrycz, W.; Sessa, S.; Hirota, K., A motion compression/reconstruction method based on MAX t-norm composite fuzzy relational equations, Inf. Sci., 176, 2526-2552, (2006) · Zbl 1102.68698 [16] Nola, A. D.; Russo, C., Lukasiewicz transform and its application to compression and reconstruction of digital images, Inf. Sci., 177, 1481-1498, (2007) · Zbl 1114.06009 [17] Wang, P. Z.; Zhang, D. Z.; Sanchez, E.; Lee, E. S., Latticized linear programming and fuzzy relation inequalities, J. Math. Anal. Appl., 159, 1, 72-87, (1991) · Zbl 0746.90081 [18] Loetamonphong, J.; Fang, S.-C., Optimization of fuzzy relation equations with MAX-product composition, Fuzzy Sets Syst., 118, 509-517, (2001) · Zbl 1044.90533 [19] Wu, Y.-K.; Guu, S.-M.; Liu, J. Y.-C., An accelerated approach for solving fuzzy relation equations with a linear objective function, IEEE Trans. Fuzzy Syst., 10, 4, 552-558, (2002) [20] Ghodousian, A.; Khorram, E., Solving a linear programming problem with the convex combination of the MAX-MIN and the MAX-average fuzzy relation equations, Appl. Math. Comput., 180, 411-418, (2006) · Zbl 1102.90036 [21] Wu, Y.-K.; Guu, S.-M., Minimizing a linear function under a fuzzy MAX-MIN relational equation constraint, Fuzzy Sets Syst., 150, 147-162, (2005) · Zbl 1074.90057 [22] Guo, F.-F.; Pang, L.-P.; Meng, D.; Xia, Z.-Q., An algorithm for solving optimization problems with fuzzy relational inequality constraints, Inf. Sci., 252, 20-31, (2013) · Zbl 1330.90139 [23] Chang, C.-W.; Shieh, B.-S., Linear optimization problem constrained by fuzzy MAX-MIN relation equations, Inf. Sci., 234, 71-79, (2013) · Zbl 1284.90103 [24] Lu, J.; Fang, S.-C., Solving nonlinear optimization problems with fuzzy relation equations constraints, Fuzzy Sets Syst., 119, 1-20, (2001) [25] Khorram, E.; Hassanzadeh, R., Solving nonlinear optimization problems subjected to fuzzy relation equation constraints with MAX-average composition using a modified genetic algorithm, Comput. Ind. Eng., 55, 1-14, (2008) [26] Hassanzadeh, R.; Khorram, E.; Mahdavi, I.; Mahdavi-Amiri, N., A genetic algorithm for optimization problems with fuzzy relation constraints using MAX-product composition, Appl. Soft Comput., 11, 551-560, (2011) [27] Khorram, E.; Ezzati, R.; Valizadeh, Z., Solving nonlinear multi-objective optimization problems with fuzzy relation inequality constraints regarding Archimedean triangular norm compositions, Fuzzy Decis. Mak. Optim., 11, 299-335, (2012) · Zbl 1254.90209 [28] Cao, B.-Y., Optimal Models and Methods with Fuzzy Quantities, (2010), Springer-Verlag Berlin Heidelberg [29] Hu, B.-Q., Foundament of Fuzzy Theory, (2010), Wuhan University Press Wuhan, China [30] Yang, J.-H.; Cao, B.-Y., Geometric programming with fuzzy relation equation constraints, Proceedings of IEEE International Conference on Fuzzy Systems, 557-560, (2005) [31] Yang, J.-H.; Cao, B.-Y., Monomial geometric programming with fuzzy relation equation constraints, Fuzzy Decis. Maki. Optim., 6, 337-349, (2007) · Zbl 1152.90677 [32] Shivanian, E.; Khorram, E., Monomial geometric programming with fuzzy relation inequality constraints with MAX-product composition, Comput. Ind. Eng., 56, 1386-1392, (2009) [33] Wu, Y.-K., Optimizing the geometric programming problem with single-term exponents subject to MAX-MIN fuzzy relational equation constraints, Math. Comput. Model., 47, 352-362, (2008) · Zbl 1171.90572 [34] Zhou, X. G.; Ahat, R., Geometric programming problem with single-term exponents subject to MAX-product fuzzy relational equations, Math. Comput. Model., 53, 55-62, (2011) · Zbl 1211.90330 [35] Peeva, K.; Kyosev, Y., Algorithm for solving MAX-product fuzzy relational equations, Soft Comput., 11, 7, 593-605, (2007) · Zbl 1113.65042 [36] Cao, B. Y., Fuzzy geometric programming optimum seeking in power supply radius of transformer substation, Proceedings of IEEE International Fuzzy Systems Conference, Korea, vol. 3, 1749-1753, (1999) [37] Cao, B. Y., The more-for-less paradox in fuzzy posynomial geometric programming, Inf. Sci., 211, 81-92, (2012) · Zbl 1250.90118 [38] Li, P.; Fang, S.-C., Latticized linear optimization on the unit interval, IEEE Transactions on Fuzzy Systems, 17, 1353-1365, (2009) [39] Li, H.; Wang, Y., A matrix approach to latticized linear programming with fuzzy-relation inequality constraints, IEEE Trans. Fuzzy Syst., 21, 781-788, (2013) [40] Peeva, K., Resolution of fuzzy relational equations - method, algorithm and software with applications, Inf. Sci., 234, 44-63, (2013) · Zbl 1284.03249 [41] Yang, X.-P.; Zhou, X.-G.; Cao, B.-Y., Multi-level linear programming subject to addition-MIN fuzzy relation inequalities with application in peer-to-peer file sharing system, J. Intell. Fuzzy Syst., 28, 2679-2689, (2015) · Zbl 1352.90113
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.