Archive
Special Issues

Volume 4, Issue 1-2, January 2015, Page: 20-30
Linear Fractional Multi-Objective Optimization Problems Subject to Fuzzy Relational Equations with the Max-Average Composition
Z‎. ‎ Valizadeh-Gh, Department of Mathematics‎, ‎Roudehen Branch‎, ‎Islamic Azad University‎, ‎Roudehen‎, ‎Iran
E‎. ‎Khorram, Faculty of Mathematics and Computer Science,‎ Amirkabir University of Technology, ‎Tehran‎, ‎Iran
Received: Dec. 21, 2014;       Accepted: Jan. 19, 2015;       Published: Feb. 8, 2015
Abstract
‎In this paper‎, ‎linear fractional multi-objective optimization problems subject to a system of fuzzy relational equations (FRE) using the max-average composition are considered‎. ‎First‎, ‎some theorems and results are presented to thoroughly identify and reduce the feasible set of the fuzzy relation equations‎. ‎Next‎, ‎the linear fractional multi-objective optimization problem is converted to a linear one using Nykowski and Zolkiewski's approach‎. ‎Then‎, ‎the efficient solutions are obtained by applying the improved ε-constraint method‎. ‎‎Finally‎, ‎the proposed method is effectively tested by solving a consistent test problem‎.
Keywords
Fuzzy Relational Equation, The Max-Average Composition, Linear Fractional Multi-Objective Optimization Problems, The Improved ε-Constraint Method‎
Z‎. ‎ Valizadeh-Gh, E‎. ‎Khorram, Linear Fractional Multi-Objective Optimization Problems Subject to Fuzzy Relational Equations with the Max-Average Composition, Applied and Computational Mathematics. Special Issue: New Advances in Fuzzy Mathematics: Theory, Algorithms, and Applications. Vol. 4, No. 1-2, 2015, pp. 20-30. doi: 10.11648/j.acm.s.2015040102.15
Reference
[1]
Abbasi Molai, A., A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints, Computers & Industrial Engineering, 72, 306-314 (2014)
[2]
Abbasi Molai‎, ‎A.‎, ‎Resolution of a system of the max-product fuzzy relation equations using L○U-factorization‎, ‎Information Sciences‎, ‎234‎, ‎86--96 (2013)‎
[3]
Abbasi Molai‎, ‎A.‎, ‎The quadratic programming problem with fuzzy relation inequality constraints‎, ‎Computers & Industrial Engineering‎, ‎62(1)‎, ‎256--263 (2012)‎
[4]
Brouwer‎, ‎R.K.‎, ‎A method of relational fuzzy clustering based on producing feature vectors using Fast Map‎, ‎Information Sciences‎, ‎179(20)‎, ‎3561-3582 (2009)‎
[5]
Di Martino‎, ‎F.‎, ‎& Sessa‎, ‎S.‎, ‎Digital watermarking in coding/decoding processes with fuzzy relation equations‎, ‎Soft Computing‎, ‎10‎, ‎238--243 (2006)‎
[6]
Ehrgott‎, ‎M.‎, ‎Multicriteria Optimization‎, ‎Springer‎, ‎Berlin (2005)‎
[7]
Ehrgott‎, ‎M.‎, ‎& Ruzika‎, ‎S.‎, ‎Improved ε-Constraint Method for Multiobjective Programming‎, ‎Journal of Optimization Theory and Applications‎, ‎138‎, ‎375--396 (2008)‎
[8]
Friedrich‎, ‎T.‎, ‎Kroeger‎, ‎T.‎, ‎& Neumann‎, ‎F.‎, ‎Weighted preferences in evolutionary multi-objective optimization‎, ‎International Journal of Machine Learning and Cybernetics‎, ‎4(2)‎, ‎139--148 (2013)‎
[9]
Ghodousian‎, ‎A.‎, ‎& Khorram‎, ‎E.‎, ‎Linear optimization with an arbitrary fuzzy relational inequality‎, ‎Fuzzy Sets and Systems‎, ‎206‎, ‎89--102 (2012)‎
[10]
Guo‎, ‎F.F.‎, ‎Pang‎, ‎L.P.‎, ‎Meng‎, ‎D.‎, ‎& Xia‎, ‎Z.Q.‎, ‎An algorithm for solving optimization problems with fuzzy relational inequality constraints‎, ‎Information Sciences‎, ‎252‎, ‎20-31 (2013)‎
[11]
Guu‎, ‎S.M.‎, ‎Wu‎, ‎Y.K.‎, ‎& Lee‎, ‎E.S.‎, ‎Multi-objective optimization with a max-t-norm fuzzy relational equation constraint‎, ‎Computers and Mathematics with Applications‎, ‎61‎, ‎1559--1566 (2011)‎
[12]
Khorram‎, ‎E.‎, ‎& Ghodousian‎, ‎A.‎, ‎Linear objective function optimization with fuzzy relation equation constraints regarding max-average composition‎, ‎Applied Mathematics and Computation‎, ‎173‎, ‎872--886 (2006)‎
[13]
Khorram‎, ‎E.‎, ‎& Hassanzadeh‎, ‎R.‎, ‎Solving nonlinear optimization problems subjected to fuzzy relation equation constraints with max-average composition using a modified genetic algorithm‎, ‎Computers & Industrial Engineering‎, ‎55‎, ‎1--14 (2008)‎
[14]
Khorram‎, ‎E.‎, ‎& Zarei‎, ‎H.‎, ‎Multi-objective optimization problems with fuzzy relation equation constraints regarding max-average composition‎, ‎Mathematical and Computer Modelling‎, ‎49‎, ‎856--867 (2009)‎
[15]
Klir‎, ‎G.J.‎, ‎& Folger‎, ‎T.A.‎, ‎Fuzzy Sets‎, ‎Uncertainty and information‎, ‎Prentice-Hall‎, ‎NJ (1988)‎
[16]
Loetamonphong‎, ‎J.‎, ‎Fang‎, ‎S.C.‎, ‎& Young‎, ‎R.E.‎, ‎Multi-objective optimization problems with fuzzy relation equation constraints‎, ‎Fuzzy Sets and Systems‎, ‎127‎, ‎141--164 (2002)‎
[17]
Li‎, ‎P.‎, ‎& Fang‎, ‎S.C.‎, ‎Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm‎, ‎Journal of Systems Science and Complexity‎, ‎22‎, ‎49--62 (2009)‎
[18]
Li, D.-C., & Geng, S.-L., Optimal solution of multi-objective linear programming with inf-→ fuzzy relation equations constraint, Information Sciences, 271, 159-178 (2014)
[19]
Nykowski‎, ‎I.‎, ‎& Zolkiewski‎, ‎Z.‎, ‎A compromise procedure for the multiple objective linear fractional programming problem‎, ‎European Journal of Operational research‎, ‎19(1)‎, ‎91--97 (1985)
[20]
Peeva‎, ‎K.‎, ‎Resolution of fuzzy relational equations‎ -- ‎Method‎, ‎algorithm and software with applications‎, ‎Information Sciences‎, ‎234‎, ‎44--63 (2013)‎
[21]
Sanchez‎, ‎E.‎, ‎Resolution of composite fuzzy relation equations‎, ‎Information and Control‎, ‎30‎, ‎38--48 (1976)‎
[22]
Sandri, S., & Martins-Bedê, F.T., A method for deriving order compatible fuzzy relations from convex fuzzy partitions, Fuzzy Sets and Systems, 239, 91-103 (2014)
[23]
Wang‎, ‎H.F.‎, ‎A multi-objective mathematical programming problem with fuzzy relation constraints‎, ‎Journal of Multi-Criteria Decision Analysis‎, ‎4‎, ‎23--35 (1995)‎
[24]
Wang, X., Cao, X., Wu, C., & Chen, J., Indicators of fuzzy relations, Fuzzy Sets and Systems, 216, 91-107 (2013)
[25]
Wang, X., & Xue, Y., Traces and property indicators of fuzzy relations, Fuzzy Sets and Systems, 246, 78-90 (2014)
[26]
Zhou‎, ‎X.G.‎, ‎& Ahat‎, ‎R.‎, ‎Geometric programming problem with single-term exponents subject to max-product fuzzy relational equations‎, ‎Mathematical and Computer Modelling‎, ‎53(1--2)‎, ‎55--62 (2011)