Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients

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1 Open Science Journal of Mathematics and Application 2016; 4(1): ISSN: (Print); ISSN: (Online) Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients El-Saeed Ammar, Mohamed Muamer Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt address (M. Muamer) To cite this article El-Saeed Ammar, Mohamed Muamer. Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients. Open Science Journal of Mathematics and Application. Vol. 4, No. 1, 2016, pp Received: March 15, 2016; Accepted: March 28, 2016; Published: May 18, 2016 Abstract In this paper, we introduce algorithm for solving multiobjective linear fractional programming problems with a fuzzy rough coefficients in the objective functions (MOFRLFP). All the parameters of the objective functions are assumed to be fuzzy rough with triangular fuzzy number. The first algorithm is follows by use the (a-cut) approach and second algorithm by ranking function to solve the above problem. A numerical example is given for the sake of illustration. Keywords Multiobjective Linear Fractional Programming, Fuzzy Rough Interval, ( ), Ranking Function 1. Introduction We need to fractional linear programming in many realworld problems such as production and financial planning and institutional planning and return on investment, and others. Multiobjective Linear fractional programming problems useful targets in production and financial planning and return on investment.there are several ways to solving the linear fractional programming (LFP) and multiobjective linear fractional programming (MOLFP) problems [1, 2, 3, 4]. Zimmermann Presented a fuzzy approach to multiobjective linear programming problems [5]. Maleki Proposed a new method for solving linear programming problems with fuzzy variables [6]. Tantawy Proposes a new method for solving linear fractional programming problems [7]. Singh, Sharma and Dangwal Proposed a solution concept to MOLFP problem using the Taylor polynomial series at optimal point of each linear fractional function in feasible region [8]. Effati and Pakdaman Introduce an interval valued linear fractional programming (IVLFP) problem, also they convert an IVLFP to an optimization problem with interval valued objective function which its bounds are linear fraction function [9]. Sulaiman and Abulrahim Used transformation technique for solving multiobjective linear fractional programming problems to single objective linear fractional programming problem through a new method using mean and median and then solve the problem by modified simplex method [10, 11]. Guzel Proposes a new solution to the multiobjective linear fractional programming MOLFP problem, Thus MOLFP problem is reduced to linear programming problem [12, 13]. Pawlak Rough set theory is a new mathematical approach to imperfect knowledge [14]. Pal The rough set approach seems to be of fundamental importance to cognitive sciences, especially in the areas of machine learning, decision analysis, expert systems [15]. Pawlak Rough set theory introduced by the author expresses vagueness, not by means of membership, but employing a boundary region of a set. The theory of rough set deals with the approximation of an arbitrary subset of a universe by two definable or observable subsets called lower and upper approximations [16]. Tsumoto Using the concept of lower and upper approximation in rough sets theory, knowledge hidden in information systems may be unraveled and expressed in the form of decision rules [17]. Lu and Huang Theconcept of rough interval will be introduced to represent dual uncertain information of many parameters, and the associated

2 2 El-Saeed Ammar and Mohamed Muamer: Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients solution method will be presented to solve rough interval fuzzy linear programming problems dual uncertain solutions [18]. Zhange et al Developed a general framework for the study of generalized fuzzy rough set models in which both constructive and axiomatic approaches are considered [19]. The method for ranking was first proposed by (Jain, 1976), Yager Proposed four indices which may be employed for the purpose of ordering fuzzy quantities in unit interval [20]. Kaufmann and Gupta Approach is presented for the ranking of fuzzy numbers ranking function [21]. Durge and Dash develop a method for solving multiobjective fuzzy fractional programming problems where all parameters of the objective function and constraints are fuzzy numbers using Ranking function [22, 23]. In this paper, We introduce alogarithm for solving multiobjective linear fractional programming problems with afuzzy rough coefficient in the objective functions (MOFRLFP). All the parameters of the objective functions are triangular fuzzy number. We deal with this problem by two methods ( approach and Ranking function). A numerical example is given for the sake of illustration. 2. Preliminaries 2.1. Triangular Fuzzy Number First of all, we introduce a set of definitions for the problem in this paper. Let R be the real number field and the (R) the family of all fuzzy sets over R. Definition 1. Suppose is the set of all compact intervals in the set of all real numbers R. Let then we write, with and the following holds [6]: i. 0 iff 0 for all. ii. 0 iff 0 for all Basic Operations of Interval Let,,, be two closed interval in R. where 0 and 0 we have: , + 2., 3.,, 0, 0 4., 5.,. Definition 2. (order relation # ) Let,,, be two closed interval in R. we write # if and only if and. Also we write # if and only if and it means That is inferior to or is superior to. Definition 3. For any % (R), the membership function of %is written as (), a fuzzy set % is defined by % &',()( ϵr +,(),0,1.. The of fuzzy set % defined as: / 0: (),2 /, /,,0,1 Where / 3 0:() 2 and / 450:() 2,,0,1. and the support set of a fuzzy set % defined as: sup'%( 0ϵR () > 02. Definition 4. A fuzzy set ; is convex if for any <, R + 3>? 0,1, we have: (? < +(1?) < ),( )2. Definition 5. A fuzzy set % is called normal if < 0:() 12 B. The set of all points ϵr + with () 1 is called core of a fuzzy set%. Definition 6. Let there exists <,, C R such that G < E < < (), F C E D C C And () 0 for each (, < ) ( C,+ ) then we say that % is triangular fuzzy number, written as ; ( <,, C ). In this paper the class of all triangular fuzzy number is called Triangular fuzzy number space, which is denoted by J(K). Definition 7. For any triangular fuzzy number % ( <,, C ), for all,0,1 we get a crisp interval by LMNML operation defined as: ; / < +( < ), C +( C ) < (), C (). Definition 8. A positive triangular fuzzy number ; is denoted as ; ( <,, C ) where < > 0. And we say that the fuzzy number ; ( <,, C ) is negative where C < 0. Definition 9. For any ;,; J(K) we say that % PQ R iff () () for all ϵr Basic Operation of Triangular Fuzzy Number Let ; ( <,, C ) and ; ( <,, C ) be two triangular fuzzy numbers then i. ; ; ( < + <, +, C + C ) ii. ; ; ( < C,, C < ), iii. ; ( <,, C ), VW 0. iv. ; ( C,, < ), VW < 0. ( < <,, C C ) < 0 v. ; ; Y( < C,, C C ) < < 0, C 0 ( < C,, C < ) C < 0 vi. 0 ; ( <,, C ) hm3 ; ; ]^_ `a,^b `b,^a c. `_ We say that ; is relatively less than ;which is denoted by ; PQ ; if and only if 1. < or 2. 3> C < > C < 3., C < C < 3> < + C < < + C. Definition 10. Let ; ( <,, C ) and ; ( <,, C ) be two triangular fuzzy numbers.we write ; ; if and only if < <, 3> C C. Remark. ; PQ ;if and only if ; PQ ;or ; ;. Definition 11. Let % the set of all fuzzy number in R. A function ;:R ;is called a fuzzy function defined as: ;() ( (), (), ()) where (), () and () are real function such that: () () ().

3 Open Science Journal of Mathematics and Application 2016; 4(1): Ranking Function for Triangular Fuzzy Number: See [23] The ranking function for % ( <,, C ) proposed by F. Reuben's defined by: R'%( 1 < 2 j ( < ()+ C ())> k 2.2. Rough Sets and Approximations Let l denoted a finite and non-empty set called the universe, and let m l l denoted an equivalence relation on l, if two elements,n l are in the same equivalence class ( i.m mn). The quotient set of lby the relation mis denoted by l/ mand l/m 0p <,p,p C, p + 2 Wherep is an equivalence class of m, 1,2,,3. The pair ( l,m ) is called an approximation space (pawlak approximation space).a rough set is interpreted by three ordinary set [14, 15, 16]: Reference set: p l Lower approximation: p r 0 l / # X 2 Upper approximation: p r 0 l / # X 2. Where # denoted a category in mcontaining the element l. The lower and upper approximation of p can also be written as: p r 0 n U/R n X 2 p r 0 n l / # X 2. The interval [p r,p r is called a rough set with a reference set X in approximation space ( l,m ), we see the boundary region of p p r p r. Remark: Set p is crisp if the boundary region of p is empty, p is rough set if the boundary region not empty. Rough set are defined by approximation, Approximation have the following properties. Proposition 1 1. p r p p r 2. r r, l r l r l 3. (p x) r p r x r, (p x) r p r x r 4. p x p r x r 3> p r x r. 5. ( p ) r ( p ) r, ( p ) r ( p ) r. Proof: see [14] Rough Interval Definition 12. Let p be denote a compact set of real numbers. A rough interval Χ # is defined as: Χ # p (r{) :p (r{) where p (r{) 3> p (r{) are compact intervals denoted by lower and upper approximation intervals of Χ # with p (r{) # p (r{). Proposition 2. For the rough interval Χ # the following holds: i. Χ # # 0 #, iff p (r{) 0 and p (r{) 0. ii. Χ # # 0 #, iff p (r{) 0 and p (r{) 0. In this paper we denote by # the set of all rough intervals in R. Suppose #, # # we can write # r{ r{ and also # r{ r{ where r{, }, r{, } and, },, } R. Similarly we can defined r{, r{. Definition 13. Let # r{ r{ and # r{ r{ be two rough interval in R. We write # # # if and only if r{ r{ and r{ r{. Als we write # # # if and only if r{ # r{ and r{ # r{ Basic Operation for Rough Interval For any rough interval #, # when # 0 # 3> # 0 # we can defined the operation on rough intervals as follows [18]: 1) # # r{ + r{ r{ + r{ Such that: r{ + r{ +, } + } and r{ + r{ +, } + }. 2) # # r{ r{ r{ r{ Such that: r{ r{ }, } and r{ r{ }, }. 3) # # r{ r{ r{ r{ Such that: r{ r{, } } and r{ r{, } }. 4) # # r{ r{ r{ r{ Such that: r{ r{ }, } and r{ r{ }, } Fuzzy Rough Interval Definition 14. Let p be denote a compact set of real numbers. Afuzzy rough interval pr # is defined as: pr # pr :pr where pr 3> pr are fuzzy number called lower and upper approximation fuzzy numbersof pr # with pr PQ pr. In this paper we denote by % # the set of allfuzzy rough with triangular fuzzy number in R. Suppose % #,R # % # we can write % # % %,R # R : R where %,%,R 3> R triangular fuzzy numbers defined as: % (a <,a,a C ), % (a <,a, a C ), R (b <,b,b C ) and R ( <,, C ) where a < a < a a C a C and b < b < b b C b C. Proposition 3. For thefuzzy rough p; # the following holds: i. pr # 0R #, iff pr 0R and pr 0R ii. pr # 0R #, iff pr 0R and pr 0R. Definition 15. A fuzzy rough interval % # % % is said to be normalized if ; 3> ; are normal. Definition 16. Let % # % % and ; # R R be two fuzzy rough intervals in R. We write % # # R # if and only if% R and % R. Definition 17. The of a fuzzy rough interval % # defined as: (% # ) / % / % / where % / 3> % / are intervals with (% ) / # (% ) / For any fuzzy rough with triangular fuzzy number % # defined as: % # (a <,a,a C ) (a <,a, a C )we can defined (% # ) / as the follows:

4 4 El-Saeed Ammar and Mohamed Muamer: Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients (% # ) / Ša < (),a C (): a < (),a C () Basic Operation for Fuzzy Rough Interval For any fuzzy rough % #,R # when % # # 0R # 3> R # # 0R # we can defined the operation for afuzzy rough as follows: 1) % # (+)R # '% R ( '% R ( 2) % # ( )R # '% R ( '% R ( 3) % # ( )R # '% R ( '% R ( 4) % # ( )R # '% R ( '% R (. Definition 18. [9] Let the set of all closed and bounded intervals in R. A function : R + is called an interval valued function with () (), ()] where for every R +, (), () are real function and () (). Definition 19. Suppose % # the set of allfuzzy rough with triangular fuzzy number in R. A function ;# :R ; # is called fuzzy rough function defined as: %# () % () % () where % () 3> % () are lower and upper approximation fuzzy functions of %# with % () % () Ranking Function for a Fuzzy Rough Interval Assume that R % # R be linear ordered function that maps, each fuzzy rough with triangular fuzzy number in to the real number R, in which % # denotes the all fuzzy rough with triangular fuzzy number in the real number. The ranking function for a fuzzy rough % # % % where % (a <,a,a C ), % (a <,a, a C ) can be defined using convex combination betweenr(% ), R(% ) as suggested by Roubens to get: R(% # ) < < ( < ()+ C ()+ < ()+ C ())> k Accordingly for any two fuzzy rough % # and R # we have: 1. % # R # iff R(% # ) R(R # ) 2. % # R # iff R(% # ) R(R # ) 3. % # R # iff R(% # ) R(R # ) Linear Fractional Programming (LFP) Problem The general linear fractional programming problem is defined as M Q( ) ( ) where K() +, () > + are valued and continuous functions on p 3> > + 0 VW LL p and p 0:, 0 2,,,> R +,, R R, R + Theorem 1. [12] Q( ) Q( ) if and only if ( ) ( ) (, ) 0K() (), p Multi-Objective Linear Fractional Programming Problem The general multi objective linear fractional programming (MOLFP) problem may be written as: () 0 < (), (),.. ()2 p 0:, 02 Where () œ }/ Q ( ), ž }Ÿ ( ),> R +,, R () > 0, VW LL 1,2,.. Definition 20. A point R + is an efficient solution of MOLFP problem if there is no R + such that Q ( ) ( ) Q ( ) ( ), 1,2, k and Q ( ) > Q ( ) ( ) ( ), for at least one. otherwise is inefficient. The set of all efficient points is called efficient set solution. It is helpful at this time to introduce the following notation: 0 4 M M3 2, M M3 MMW@5V3 V 2 Theorem 2. [12] If is an optimal solution of? (K () ( ) ' ()(), p < where is ( ) Q ( ) Q ( ) ( ) ( ) 0, <? 1 1,2, k then is an efficient solution of 0 () œ }/ ž }Ÿ 3. Problem Formulation, 1,2, k, p 2. for all? The multi objective linear fractional programming with fuzzy rough coefficient (MOFRLFP) problems is defined as follows: ªR # () K ; # () ; # () # # (+)5 >% # # (+) 1,2, p 0 R + :, 0 2 (1) Where #, >% #, 5 # and # % #, is 3 constraint matrix, R, 2. Using the operation of a fuzzy rough interval can be convertingthe problem (1) as: MaxªR # () ; }±R ; ² }±; ² ² 1,2, µ ž ; }³ ž ; ² }³ p 0:, 0 2. (2) Where % (c <,c, c C ), >% (d <,d, d C ) 5 (p <,p, p C )and (q <,q, q C )are triangular fuzzy number. Similarly we can defined %,>%,5 and. The objective in problem (2) is quotient of two fuzzy rough functions using the operation of fuzzy rough interval we have:

5 Open Science Journal of Mathematics and Application 2016; 4(1): ªR # ()» % +¼R >% + % +¼R >% + ½ Now using for any 0,1 we have: ªR # ()» c < (α), c C (α) + p < (α), p C (α) d < (α), d C (α+q < (α), q C (α c < (α), c C (α) + p < (α), p C (α) d < (α), d C (α+q < (α), q C (α ½ Using some operation of the intervalvalue we have: ZR À(x)  à Ä_ Å Å (Æ)Ç} È Ä_ (Æ) Å,ÃÄa(Æ)Ç} Å ÈÄa(Æ) É Å Ä_ (Æ)Ç} Ê Å Ä_ (Æ),É Å Äa (Æ)Ç} Ê Å Äa (Æ), Ë Ã Ä_ (Æ)Ç} Ë ÈÄ_ (Æ) Ë,ÃÄa(Æ)Ç} Ë ÈÄa(Æ) É Ë Ä_ (Æ)Ç} Ê Ë Ä_ (Æ),É Ë Äa (Æ)Ç} Ê Ë Ì. Äa (Æ), Using the theorem (2-1) [9] we can write the objective function in the following forma as: ZR À(x) Š Z (x),z } (x) : Z (x),z } (x) Where Z (x),z } (x)z (x) and Z } (x) are multiobjective linear fractional programming defined as: Z (x) à Ä_ Å Å (Æ)Ç} È Ä_ (Æ) É Å Äa (Æ)Ç} Ê Äa Å (Æ), Z } (x) à Äa Z (x) à Ä_ Ë Ë (Æ)Ç} È Ä_ (Æ) É Ë Äa (Æ)Ç} Ê Ë, Z } (x) à Äa Äa (Æ) Å Å (Æ)Ç} È Äa(Æ) É Å Ä_ (Æ)Ç} Ê Å Ä_ (Æ) Ë (Æ)Ç} ÈÄa Ë (Æ) É Ä_ Ë (Æ)Ç } Ê Ä_ Ë (Æ) For all 1,2, Now the problem (1) can be convert into multiobjective rough interval linear fractional programming MORLFP problems defined as: MaxͪR # () Š Z (x),z } (x) : Z (x),z } (x) 1,2,,Î p 0 R + :, 0 2 (3) Thus the MOFRLFP problem (3) decomposes to multiobjective linear fractional programming (MOLFP) problem define as follows: N } Z } () (x) D } () c C (α)x+ p C (α) d < (α)x + q < (α) N Z () (x) D () c < (α)x+ p < (α) d C (α)x + q C (α) N } Z } () (x) D } () c C (α)x+ p C (α) d < (α)x + q < (α) N Z () (x) D () c < (α)x+ p < (α) d C (α)x + q C (α) For all 1,2, p 0 R + :, 0 2 (4) Now using the theorem (2) we can convert the MOLFP problem (4) to the linear programming problem defines as follows: Ñ < Where K } () (ª } ) } ()+K () (ª ) () +K } () (ª } ) } ()+K () (ª ) () p 0 R + :, V 2. (5) (ª } ) K } () } () (ª ) K () () (ª } ) QÒ ( ) Ò ( ) (ª ) Q Ó ( ) Ó ( ) Definition 21. A point R + is an efficient solution of MOFRLFP problem (1) if there is no R + such Q Ô ( ) Ô ( ) Ô ( ) that QÔ ( ), 1,2,, and Q least one Theorem 3. If is an optimal solution of Ô ( ) Q Ô ( ) Ô ( ) Ô ( )? (K # () ( ) ( # ()), p < where is (ª # ) QÔ ( ) Q Ô ( ) Ô ( ) Ô ( ) for all < for at? 0,? 1 1,2, k then is an efficient solution of the MOFRLFP problem (1). The proof of this theorem is much similar to the proof given by Guzel in [12] Ö ØÙ Algorithm for Solving MOFRLFP Problem We propose algorithm for solving a MOFRLFP problem (1) is as follows: Step 1. Convert any problem in the form of MORLFP problem (3). Step 2. Use decompose technique for the MOFRLFP problem (3) to get the MOLFP problem (4). Step 3. Usetheorem (2) for convert the MOLFP problem

6 6 El-Saeed Ammar and Mohamed Muamer: Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients (4) to getthe LP problem as the form problem (5). Step 4. Find the optimal solution of the LP problem (5), thus is the efficient solution of MORLFP problem (1). Numerical example 1. Consider the following MORFLFP problem: Maximize: ÍZR < À (x) Ú ; Û Ç _ Ò Ü;Û Ç b ;Û Ç _ Ò Ý;Û Ç b,zr À (x) ; Û Ç _ Ò C; Û Ç b <;Û Ç _ Ò <;Û Ç b };Û Î < 1, 3 < +2 12, < 2.5, 1.5 <, 0. Where 6R # (5.5,6,6.5 ):(5.2,6,6.8 ), solution 5R# (4.6,5,5.6 ):(4.2,5,5.9 ), 2R # (1.7,2,2.4 ):(1.2,2,2.8 ), 7R # (6.5,7,7.5 ):(6.1,7,7.9 ), 2R # (1.6,2,2.5 ):(1.5,2,2.9 ), 3R # (2.5,3,3.5 ):(2.1,3,3.8), 1R # (0.5,1,1.4 ):(0.2,1,1.5 ), 4R # (3.7,4,4.3 ):(3.5,4,4.5 ). Using the operations of fuzzy rough interval can be writing the objective function as follows: where 6R (5.5,6,6.5 ),6R (5.2,6,6.8 ), 5R (4.6,5,5.6 ),5R (4.2,5,5.9 ) 2R (1.7,2,2.4 ),2R (1.2,2,2.8 ) 7R (6.5,7,7.5 ),7R (6.1,7,7.9 ) 2R (1.6,2,2.5 ),2R (1.5,2,2.9 ) 3R (2.5,3,3.5 ),3R (2.1,3,3.8) 1R (0.5,1,1.4 ),1R (0.2,1,1.5 ) 4R (3.7,4,4.3 ) 4R (3.5,4,4.5 ) The objective function ªR < # (),ªR # () are the a quotient of two rough function, using the operations of fuzzy rough interval we have: ZR < À (x)» 6 R x < + 5R x 2R x < + 7R x 6 R x < + 5R x 2R x < + 7R x ½ ZR À (x) Š ; Å Ç _ } C; Å Ç b <;Å Ç _ }<;Å Ç b } ;Å ; Ë Ç _ } C; Ë Ç b <;Ë Ç _ } <;Ë Ç b } ;Ë. Now using for all triangular fuzzy number where % (α) c < +'c c C (,c C +'c c C ( % (α) c < +'c c C (,c C +'c c C (,,0,1 ZR < À (x) 6 R x < } 5R x 6R x < } 5R x 2R x < } 7R x 2R x < } 7R x ZR À (x) 2R x < } 3R x : 2R x < } 3R x 1R x < } 1R x +4R 1R x < } 1R x +4R We can now choose 0.5 thus can be written ªR < # () and ªR # () on the form in the following: ZR < À (x)» 5.75x < + 4.8x,6.25x < + 5.3x 1.85x < x,2.2x < +7.25x : 5.6x < + 4.6x,6.4x < x 1.6x < x,2.4x < x ½ ZR À 1.8x < x,2.25x < x (x)» 0.75x < x +3.85,1.2x < +1.2x : 1.75x < x,2.45x < + 3.4x o.6x < + 0.6x +3.75,1.25 x < x ½ Now using the theorem (2-1) [9]: ZR À < (x) æâ 5.75x < + 4.8x 6.25x < + 5.3x, Ì Â 5.6x < + 4.6x, 6.4x < x Ìç 2.2x < +7.25x 1.85x < x 2.4x < x 1.6x < x ZR À 1.8x < x (x)æâ 1.2x < +1.2x +4.15, 2.25x < x 0.75x < x Ì: Â 1.75x < x 1.25 x < x +4.25, 2.45x < + 3.4x o.6x < + 0.6x Ìç Now we can decomposition the a bove MORLFP problem to get the MOLFP problem defined as follows: < } () 6.4x < x 1.6x < x, } () 2.45x < + 3.4x o.6x < + 0.6x µ

7 Open Science Journal of Mathematics and Application 2016; 4(1): < () 5.6x < + 4.6x, 2.4x < x () 1.75x < x 1.25 x < x µ } < () 6.25x < + 5.3x, } 1.85x < x () 2.25x < x 0.75x < x µ < () 5.75x < + 4.8x, 2.2x < +7.25x () 1.8x < x 1.2x < +1.2x µ < 1, 3 < +2 12, < 2.5, 1.5 <, 0. Using the theorem (2) for convert the above MOLFP problems to get the LP problem define as follows form: 0Ñ < 1.92 < ? 3.47 < < 1, 3 < +2 12, < 2.5, 1.5 x <,x 0. For Ñ < Ñ 0.5 we have: < < 1, 3 < +2 12, < 2.5, 1.5 <, 0. The optimal solution of the above linear programming is: x < 4, x 0 The efficient solution of MOFRLFP problem is x < 4, x 0 with the maximum objective value are: ZR < À (2.30,3,3.80 ):(1.86,3,5.67 ) ZR À ( 0.16,0.25,0.44 ):(0.14,0.25,0.67 ) 3.2. Ranking Function Algorithm for Solving MOFRLFP Problem The multi objective linear fractional programming problems with fuzzy rough coefficient (MOFRLFP) is defined as follows: ªR # () Q ; Ô ( ) ; œ Ô }è Ô 1,2, µ (4) Ô ( ) žr Ô }³ Ô p 0 R + :, V2. Where #, >% #, 5 # and # % #, is 3 constraint matrix R, 2. Step 1: The multi objective linear fractional programming problems with fuzzy rough coefficient (MOFRLFP) is first convert to multiobjective linear fractional programming (MOLFP) problems using linear Ranking function: R(ªR # () R(Q ; Ô ( )) R'œ Ô ( }R'è Ô ( R] ; Ô ( )c R'žR Ô ( }R'³ Ô ( p 0 R + :, V2. 1,2 (5) Step 2: The multiobjective linear fractional programming (MOLFP) problems can be reduced to linear programming problem: & <? (R(K; # ()) ( ) (R( ; # ()), p. (6) Where ( ) R(Q ; Ô ( )) R( ; R(Q ;Ô ( )) Ô ( )) R( ; 1,2 Ô ( )) Subject to : p 0 R + :, V2. Step 3: solving the linear programming problem (6)for any? 0,1, <? 1 The optimal solution thus obtained shall be efficient solution of the MOFRLFP problems (3). Numerical example 2. Consider the MOFRLFP problemas in Example (1): Solution by ranking function: First using linear Ranking function the multiobjective rough fuzzy linear fractional programming problemcan be written of the form: Maximize: G R( ªR # < ()) R(6 R # ) < + R'5R# (, E R (2R# ) < +R( 7R# ) F R'2R E R( ªR # ( # < + R (3R # ) ê ()) E D R(1R# ) < + R'1R# ( +R ( 4R# )é < 1, 3 < +2 12, < 2.5, 1.5 <, 0. Using R(A; À ) < í (a < +a C )+4a +(a < + a C ), we get: R'6R À (6, R'6R À ( , R'2R À ( , R'7R À ( 7 R'2R À (2.0625, R'3R À (2.9875, R'1R À (0.95, R'4R À ( 4 Now the problem can be written as follows: Max F D Subject to : G z < (x) 6 x < x x < +7 x, ë E z (x) x < x ê 0.95 x < x + 4é x < x 1, 3x < +2x 12,x < 2.5,x 1.5 x <,x 0. ë

8 8 El-Saeed Ammar and Mohamed Muamer: Algorithm for Solving Multi Objective Linear Fractional Programming Problem with Fuzzy Rough Coefficients Which is the multiobjective linear fractional programming (MOLFP) problem can be reduced to the linear programming (LP) problem as follows :? Max < (6 x < x ) 2.98( x < +7 x )+? (2.0625x < x ) 1.29(0.95 x < x + 4) µ x < x 1, 3x < +2x 12, x < 2.5, x 1.5 For? <? 0.5 we have: x <,x 0. Max 00,42 x < 7.03x x < x 1, 3x < +2x 12,x < 2.5,x 1.5 x <,x 0. The optimal solution of LP problem is (4,0 ) Then the efficient solution of (MOFRLFP) problem is (4,0 ) with ZR < À ( ) (2.29,3,3.8 ):( 1.86,3,5.6 ) ZR À ( ) (0.16,0.25,0.43 ):( 0.14,0.25,0.67 ) 4. Conclusion A new approach is proposed for solving multiobjective linear fractional programming with fuzzy rough coefficients (MOFRLFP) problem by two methods (, ranking function). Where a new definition for ranking a fuzzy rough is introduced, that is used for solving the (MOFRLFP) problem. References [1] Borza, M. Rambely, A. S. and Saraj, M.; (2012), A new approach for solving linear fractional programming problems with interval coefficients in the objective function, Applied mathematics Sciences, 6(69) [2] Charnes, A and Cooper, W. W.; (1962), Programming with linear fractional function, Naval Research Logistics Quaterly, [3] Chakraborty, M. and Gupta, S.; (2002), Fuzzy mathematices programming for multiobjective linear fractional programming problem, Fuzzy sets and systems, 125(3), [4] Chankong, V. and Haimes, Y. Y.; (1983), Multiobjective decision making: Theory and Methodology, Elsevier North Holland. [5] Zimmermann, H. J; (1983), Fuzzy mathematical programming, Computer and ops. Res vol 10 (4) [6] Maleki, H. R.at (2000); Linear programming with fuzzy variables, Fuzzy sets and systems [7] Tantawy, S. F.; (2007), A new method for solving linear fractional programming problem, journal of basic and applied sciences, 1(2) [8] Dangwal, R. at el, (2012), Taylor series solution of multiobjective linear fractional programming problem, International journal of fuzzy mathematics and systems, vol. 2, [9] Effati, S. and Pakdaman, M.; (2012), Solving the intervalvalued linear fractional programming problem, AJCM, [10] Sulaiman, N. A., and Abdulrahim, B. K.; (2013), Using Transformation technique to solve multiobjective linear fractional programming problem, IJRRAS, 14(3), [11] Sulaiman, N. A., Sadiq, G. W. and Abdulrahim, B. K.; (2014), Used a new transformation technique for solving multiobjective linear fractional programming problem, IJRRAS (18), 2, [12] Guzel, N.; (2013), A proposal to the solution of multi objective linear fractional programming problem, Abstract and Applied Analysis, Volume 2013, Article ID , 4 pages. [13] Guzel, N., and Sivri, M.; (2005), Proposel of a solution to multobjective linear fractional programming problem, Sigma journal of Engineering and Natural Sciences, vol. 2, [14] Pawlak, Z.; (1982), Rough sets, Int. J. of information and computer sciences,11(5) [15] Pal, S. K.; (2004), Soft data mining computational theory of perceptions and rough fuzzy approach, Information Sciences, 163, (1-3), [16] Pawlak, Z.; (1991), Rough sets: theoretical aspects of reasoning about data, System theory, knowledge engineering and problem solving, vol. 9, Kluwer Academic publishers, Dordrecht, the Netherlands. [17] Tsumoto, S.; (2004), Mining diagnostic rules from clinical databases using rough sets and medical diagnostic model, Inform- Science,162(2) [18] Lu, H., Huang, G., and He, L.; (2011), An inexact rough interval fuzzy linear programming method for generating conjunctive water allocation strategies to agricultural irrigation systems, Applied mathematics modeling, 35, [19] Zhang, W. X; (2003), generalizedfuzzy rough sets, Information Sciences, 151, [20] Yager, RR., (1981), A procedure for ordering fuzzy subsets of the unit interval. Information Sciences, 24(2), [21] Kaufmann, A., Gupta, MM., (1988), Fuzzy mathematical models in engineering and management science. Elsevier Science Publishers, Amsterdam, Netherlands. [22] Fortemps, P. and Roubens, F. (1996), Ranking and defuzzification methods based area compensation, Fuzzy sets and systems, vol.82, [23] Durga, P and Dash, R; (2012), Solving multiobjective fuzzy fractional programming problem, Ultra Scientist, 24(3)