On Solving Fuzzy Rough Linear Fractional Programming Problem

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International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. 2 Preliminaries,2 1 Triangular Fuzzy Number, Definition 1 For any a fuzzy set the membership function of is written asn a fuzzy set is defined. The of fuzzy set defined as,The support set of a fuzzy set defined as. Definition 2 A fuzzy set is convex if for any we have. Definition 3 A fuzzy set is called normal if The set of all points with. is called core of a fuzzy set,Definition 4 Let there exists. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2100. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. then we say that is triangular fuzzy number written as. In this paper the class of all triangular fuzzy number is called Triangular fuzzy number space. which is denoted by, Definition 5 For any triangular fuzzy number for all we get a crisp. interval by operation defined as, Definition 6 A positive triangular fuzzy number is denoted as.
And we say that the fuzzy number is negative where. Definition 7 For any two fuzzy numbers we say that iff for. 2 2 Basic Operation of Triangular Fuzzy Number,Let and be two triangular fuzzy numbers Then. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2101. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Definition 8 Let and be two triangular fuzzy numbers and then greater. than and less than operations can be defined as follows 12. 2 3 Fuzzy Rough Interval, Definition 9 Let be denote a compact set of real numbers A fuzzy rough interval. is defined as where are fuzzy sets called lower and upper approximation. fuzzy of with, In this paper we denote by the set of all fuzzy rough with triangular fuzzy number in the real number. Suppose we can write,Where triangular fuzzy numbers defined as. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2102. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Proposition 1 For the fuzzy rough the following holds. ii iff and,Definition 9 A fuzzy rough interval,is said to be normalized if are normal.
Definition 10 Let and be two triangular fuzzy rough numbers and then. greater than and less than operations can be defined as follows. Definition 11 Let and, be two fuzzy rough intervals in We write if and only if and. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2103. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. 2 4 Basic Operation for Fuzzy Rough Interval, For any fuzzy rough when we can defined the operation for. fuzzy rough as follows,2 5 Linear Fractional Programming Problem. The general linear fractional programming LFP problem is defined as follows. Subject to,Variable Transformation Method, In this method for solving linear fractional programming we as usual assume that the denominator is positive. everywhere in and make the variable change,With this change the objective function becomes.
2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2104. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. If we make the additional variable changes For all The linear. fractional programming problem with variable changes as formulation. Subject to, With constraints and variables that can be solved by the simplex method. Definition12 A point is said to be optimal solution of the linear fractional programming. problem if there does not exist, 2 6 Multiobjective Linear Fractional Programming Problem. The general multi objective linear fractional programming MOLFP problem may be written as. Subject to, 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2105. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Definition 13 A point,3 Problem Formulation, The fuzzy rough linear fractional programming FRLFP problem is defined as follows. Subject to,Where and defined as,Also is an constraint matrix defined as.
This problem can be writing as the form,Subject to. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2106. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Now using the operation of the rough interval we have. Subject to, Now the fuzzy rough linear fractional programming problem 3 can be reduced as the multi objective fuzzy linear. fractional programming problem as follows,Subject to. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2107. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Let the parameters are the triangular fuzzy,numbers then we have. Suppose that defined as,thus writing as,Now the problem 4 can be written as follows.
Subject to, 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2108. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. is a fuzzy rough with triangular fuzzy number then this. The relation 6 is called bounded constraints, Now using the arithmetic operations and partial ordering relations we decompose the problem 5 as the. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2109. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Subject to,and all decision variables are non negative. From the above decomposition problem we construct the following five crisp linear fractional programming. problems namely Middle Level problem MLP Upper Upper Level problem UULP Lower Upper Level. problem LULP Upper Lower Level problem ULLP and Lower Lower Level problem LLLP as follows. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2110. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Subject to, constraints in the decomposition problem in which at last one decision variable of the MLP occurs and all. decision variables are non negative,Subject to, and all variables in the constraints and objective function in UULP must satisfy the bounded constraints.
replacing all values of the decision variables which are obtained in MLP and all decision variables are non. Where is the optimal objective value of MLP,Subject to. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2111. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. and all variables in the constraints and objective function in LULP must satisfy the bounded constraints. replacing all values of the decision variables which are obtained in MLP UULP and all decision variables are. non negative,Subject to, and all variables in the constraints and objective function in ULLP must satisfy the bounded constraints. replacing all values of the decision variables which are obtained in MLP UULP LULP and all decision variables. are non negative Where is the optimal objective value of UULP. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2112. International Research Journal of Engineering and Technology IRJET e ISSN 2395 0056. Volume 03 Issue 04 Apr 2016 www irjet net p ISSN 2395 0072. Subject to, and all variables in the constraints and objective function in LLLP must satisfy the bounded constraints. replacing all values of the decision variables which are obtained in MLP UULP LULP ULLP and all decision. variables are non negative, Definition14 A point is said to be optimal fuzzy rough solution of the fuzzy rough linear. fractional programming problem if there does not exist such that. Theorem 1 Let be an optimal solution of MLP be an optimal. solution of UULP be an optimal solution of LULP be an. optimal solution of ULLP and be an optimal solution of LLLP where M UU LU UL. and LL sets of decision variables in the MLP UULP LULP ULLP and LLLP respectively. Then is an optimal fuzzy rough solution to the given FRLFP. The proof of this theorem is much similar to the proof theorem 4 1 given by Pandian and Jayalakshmi 3. 2016 IRJET Impact Factor value 4 45 ISO 9001 2008 Certified Journal Page 2113. On Solving Fuzzy Rough Linear Fractional Programming Used a rough set theory a new mathematical knowledge hidden in information systems may be unraveled and

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