Solving Fuzzy Linear Fractional Programming. Problem Using Metric Distance Ranking

Transcription

1 pplied Matheatical Sciences, Vol. 6, 0, no. 6, Solving Fuzzy inear Fractional Prograing Proble Using Metric Distance anking l. Nachaai Kongu Engineering College, Erode, Tailnadu, India P. Thangara annari Institute o Technology,annari, Tailnadu,India bstract Fuzzy linear prograing is an application o uzzy set theory in linear decision aking probles and ost o those probles are related to linear prograing with uzzy variables. In this paper, a new ethod is introduced to solve uzzy linear ractional prograing [ FFP], where all the paraeters and variables are triangular uzzy nubers. Here we introduce a ethod called ranking ethod based on etric distance to solve FFPP. Keywords: Fuzzy linear prograing, Fuzzy linear ractional prograing, Triangular uzzy nuber, Metric distance ranking. I. INTODUCTION Zierann [8] presented a uzzy approach to ulti obective linear prograing probles. He also studied the duality relations in uzzy linear prograing. H.. Maleki [3] proposed a new ethod or solving linear prograing probles with uzzy variables. S.H.Nasseri [5] proposed a new ethod or solving uzzy nuber linear prograing probles with uzzy nubers. In uzzy decision aking probles the concept o axiizing decision

2 76 l. Nachaai and P. Thangara was proposed by ellan and zadeh []. The concept o uzzy linear prograing on general level was irst proposed by Tanaka etal [7]. Jing Shaing Yao [] used ranking uzzy nubers based on decoposition principle or coparison o uzzy nuber and obtain an optiu solution.. Nagoor Gani [4] introduced a Technique or solving uzzy linear prograing proble using - uzzy nuber. The obective o this paper is to deal with a kind o uzzy linear ractional prograing probles where all the variables are triangular uzzy nuber. In this paper we study the arithetic operations o triangular uzzy nubers and we introduce a new technique o solving FFP based on etric distance ranking o uzzy nubers. This paper is outlined as ollows: asic deinitions, notations and arithetic o triangular uzzy nubers are discussed in the second section. Soe concepts o ranking based on uzzy etric distance are discussed in section three. We have introduced the deinition o uzzy linear ractional prograing proble in section 4 and we have proved a new theore called conditions or uzzy optiality. lso we proposed a new algorith to solve FFPP. In section 5 nuerical proble is solved. II. DEFINITIONS ND OPETIONS In this section soe o the undaental deinitions, operations and concepts o uzzy sets theory initiated by Zadeh are reviewed. Deinition.. uzzy set in is a set o ordered pairs: {( x, μ ( x ))/ } Where μ = x is called the ebership unction o x in which aps to a subset o the nonnegative real nubers whose supreu is inite. Deinition. : The support o a uzzy set on is the crisp set o all such that μ ( x ) 0. > x Deinition.3 : The set o eleents that belong to the uzzy set on at least to = x / μ x α. { } the degree α is called the α - cut set: ( ) α

3 Solving uzzy linear ractional prograing proble 77 Deinition.4 : uzzy set on is convex i μ λ x + λ y in μ x, μ, and 0, y x y λ. ( ( ) ) ( ) ( ) { } [ ] Deinition.5 : uzzy set is called triangular uzzy nuber with peak a, let width a 0 and right width a 3 0 i its ebership unction has the ollowing or μ = 0 ( a x) ( x a ) / a / a 3, i a a x a, i a x a + a 3,, otherwise Deinition.6: uzzy nuber is said to be a - type uzzy nuber i and only i, ( a x) / a, i a a x a μ = ( x a )/ a, i a x a + a 3 0, Otherwise is or let and or right reerence and a is the core o. We write = (a, a, a 3 ). rithetic Operations: and are two triangular uzzy nuber o - type = (a, a, a 3 ), = (b, b, b 3 ) then (i) + = (a, a, a 3 ) + (b, b, b 3 ) = (a + b, a + b, a 3 + b 3, ) (ii) - = (a, a, a 3 ) - (b, b, b 3 ) = (a + b 3, a b, a 3 + b ) (iii) * = (a, a, a 3 ) * (b, b, b 3 ) = (a b + b a, a b, a b 3 + b a 3 ) (iv) / = (a, a, a 3 ) / (b, b, b 3 ) = (a / b + a / b, a / b, a / b 3 + a 3 / b ) Note : uzzy set is convex i all α - cuts are convex. III. METIC DISTNCE NKING n eicient approach or ordering the eleents is to deine a ranking unction D : T () which aps or each uzzy nuber into the real line. We deine T () as ollows:

4 78 l. Nachaai and P. Thangara i and only i D ( ) D ( ) i and only i D( ) D( ) = i and only i D( ) = D( ) N. avi chankar [8] revised a etric distance ethod to rank uzzy nubers. et and be two uzzy nubers deined as ollows: = =, x <, x, x <, x (3.) (3.) Where and are the ean o and. The etric distance between and can be calculated as ollows: D = ( ) + ( ( ) ( )) ] dy g y g y dy (, ) g ( y) g ( y) 0 0 (3.3) Where g,, and g,, g and respectively. g are the inverse unctions o In order to rank the uzzy nubers the etric distance between and O is calculated as ollows. D (,0) ( g ( y) ) dy + ( g ( y ) 0 ) = dy 0 (3.4) triangular uzzy nuber = (a, b, c) can be approxiated as a syetry uzzy nuber S ( μ,σ ), μ denotes the ean o, σ denotes the standard deviation o and the ebership unction o is deined as ollows:

5 Solving uzzy linear ractional prograing proble 79 ( μ -σ ) x, i μ σ x μ σ = ( μ + σ ) x (3.5), i μ x x μ + σ σ Where μ andσ are calculated as ollows. c - a σ = (3.6) a + b + c μ = 3 The inverse unctions g and ollows: g o and (3.7) respectively are shown as ( y) = ( μ σ ) + σ y g (3.8) ( y) = ( μ + σ ) σ y g (3.9) IV. FUZZY INE FCTION POGMMING POEM: The basic deinitions o uzzy linear prograing probles are as ollows. Deinition : 4.: uzzy linear prograing proble is deined as ollows. Maxiize Z = C X X b Subect to X 0 x n n Where T ( ), C T ( ) and b T( ) Deinition 4.: ny X = ( x, x... xn ) T( ), where each x i T( ), which satisies the constraints and non negative restrictions o FPP is said to be uzzy easible solution to FPP.

6 80 l. Nachaai and P. Thangara Deinition 4.3: uzzy basic easible solution X to the FPP is called an optiu uzzy basic easible solution, i Z = C X Z 0, where Z is the value o obective unction or any uzzy easible solution. Kanli swarup [9] introduced an algorith or the solution o FPP without reducing it to PP. Now we introduce soe concepts o uzzy linear ractional prograing proble Deinition 4.4 : Fuzzy linear ractional prograing proble is deined as ( C X + α ) /( d x + β) = X Maxiize ( ) Subect to the constraints x = b x 0 With the additional assuption that the denoinator is positive or all possible solutions. Initial asic Feasible Solution et = X be the initial basic easible solution such that X = b where ( b, b,... ) Further let Z Z b = C = d X X. + α + β Where C and d are the vectors having their coponents as the coeicients associated with the basic variables in the nuerator and the denoination o the obective unction respectively. Theore 4.: [Conditions or uzzy optiality] suicient condition or a easible solution to a FFPP to be uzzy optiu is that i μ > 0 or which the colun vector a in but not in the basis. Proo : et the FFPP is be to deterine X so as to C X + α / d x + β = x axiize ( ) ( ) ( )

7 Solving uzzy linear ractional prograing proble 8 Subect to the constraints X b >= X, 0 Suppose let all μ 0; ( i,... ) i = et us assue that uzzy basic easible solution X to this FFPP Ie, x bi = b i= i= xi bi θ a + θ a = b Since θ a = -θ u b i, we have i= x b θ i i ι = i= u i= i bi. a ( x θ ) b u i + θ a = b i i i + θ a = b When θ > 0 we have x θ u 0 Since by assuption x 0 Thereore τ x u θ i i i θ i,... x u θ is a uzzy easible solution or all θ > 0. Thus the set S is unbounded, which is contrary to our hypothesis o regularity. Hence u i > 0 and

8 8 l. Nachaai and P. Thangara Deinition 4.5: uzzy basic easible x to FFPP is called uzzy unbounded solution i or at least one, or which y i 0, z c and z d are negative, then there does not exit any optiu solution to this FFPP. lgorith or FFPP: We purpose a new algorith to solve a FFPP is as ollows:- Step : Check whether the obective unction o the given FFPP is to be z = ax - z axiized o iniized.. I it is iniized then convert in ( ).Step : Check whether all b i are non negative. I any one o ultiply the corresponding in equation o the constant by-. b i is negative then Step 3 : Convert all the in equations o the constraints into equations by introducing slack and or surplus uzzy variables in the constraints. Take zero or the costs. Step 4: Obtain an initial uzzy basic easible solution to the given FFPP. Step 5: Copute z = d X z z = where = c X + α z + β Step 6: Copute the net evaluations ( z c ) and ( z d ) are calculated by using the relation z c = c y c z d = c y d Step 7: Exaine the sign (i) I all ( z c ) 0 optiu basic easible uzzy solutions (ii) otherwise proceed to the next step. z and Δ, where z ( z c ) = z ( z d ) Δ. Δ using the ethod o etric distance anking. then the initial basic easible uzzy solution x is an

9 Solving uzzy linear ractional prograing proble 83 Step 8:I there are ore than one negative the. et it be Δ or soe = r. r Δ then chose the ost negative o (i) I all y 0 ir, then there is an un bounded solution to the given proble. (ii) I at least one y ir > 0, then the corresponding vector y r enter the basis y. x i Step 9 :Copute, i =,,... and choose iniu o the. et y ir x r iniu o these ratios be. Then the vector y k will leave the basics y. ykr The coon eleent y k r, which in the k th row and r th colun is known as leading (or pivotal) triangular uzzy nuber. Step 0 : Convert the pivotal eleent into unit triangular uzzy nuber [ TFN] by dividing its row by the pivotal TFN itsel and all other eleents in its colun to zero TFN by using the relation. yˆ y and ) y k k i = yi yir yk = y k r yk r Step : Go to step 6. nd repeat the procedure until either an optiu solution is obtained or there is an indication o an unbounded solution. V. NUMEIC EXMPE Solve the ollowing FFPP Maxiize (5,, 3) x + (4,, 6) x / (4, 6, 5) x + (6, 3, 9) x + (,, 6) Subect to the constraints: (3,, ) x + (6, 4, ) x (3, 5, ) (4,, ) x + (6, 5, 4) x (6,3,9)

10 84 l. Nachaai and P. Thangara Table I C D (5,, 3) (4, 6, 5) (4,, 6) (6, 3, 9) c d y X Y Y Y 3 Y S (3, 5, ) (3,, )* (6, 4, ) (0,, 0) 0 0 S (6,3,9) (4,, ) (6, 5, 4) Z = (,, 6) Z = 0 Z = Z / Z = 0 Z c (3, -, 5) Z d (5, -6, 4) Δ (,, ) (6, -, 4) (9, -3, 6) Here y enters the basis and S leaves the basis using Min [D ( Δ ), D ( Δ ) ] = D ( Δ ) = y Min [D(X / y ), D(X / y ) ) ] = D(X / y ) = S (0,, 0) (9,, 8) (3,, )* is the pivotal eleent, convert this eleent as unit triangular nuber and all the other eleents in the colun as zeros. The new table will be Table II C D (5,, 3) (4, 6, 5) (4,, 6) (6, 3, 9) c d y X Y Y Y 3 Y 4 (5,, 3) (4, 6, 5) y (8.,.5, 6) (.,,.5) (4.3,, 4.5) (0.3,0.5, ) 0 0 S (7., -, (4., 0, 6.5) (8.3, -3, 0.5) (0.3,0.5, ) Z = (58., 4, 49.6) Z = (8.4,,.6) Z = Z / Z =(.3, 0.4, 0.87) Here all.6) Z c (0., 0, 0.5) Z d (., 0, 7) Δ (33.8, 0, 565.7) Δ 0 and D( Δ ) 0. The Optiu asic Feasible solution is (5.7, 3, 6.7) (6.3, 5, 3.5) Maxiize Z =(.3, 0.4, 0.87) at X = (8.,.5, 6) and X = (0,0,0) (.8, 0.5, 3.5) (.8, 0.5, 3.5) (0,, 0) 567.4, 64, 65.) (67., 6, 86.4)

11 Solving uzzy linear ractional prograing proble 85 VI. Conclusion In this paper, uzzy linear ractional prograing probles with uzzy variables and uzzy constraints are discussed. anking based on uzzy etric distance to evaluate the sign is disused. We have proposed a new technique and algorith to solve FFPP. Thus the ethod is very useul in the real world probles where the product in uncertain. eerences []..E.ellann,..Zadeh, : Decision aking in a uzzy environent anageent Sci.7 (970). []. Jin-Shing yao, Kweiei Wu, anking uzzy nubers based on decoposition principle and signed distance, Fuzzy sets and systes 6 (000) [3]. H..Maleki, M.Tata, M.Mashichi, inear prograing with uzzy variables, Fuzzy sets and systes 09 (000) -33. [4]..Nagoorgani,C.Duraisay,C.Veeraani. note on uzzy linear prograing proble using. Fuzzy nuber IJCM 009. [5].S.H.Nasseri, E.rdil,.Yazdani and.zaearian Siplex ethod or solving linear prograing probles with uzzy nubers PWSET.Vol 0 (005). [6]. N.avishankar, V.Sireesha, K.ao, N.Vani, Fuzzy critical path ethod based on etric distance ranking o Fuzzy nubers Int ounal o Math.nalysis 00. [7]. H.Tanaka, T.Okuda and K.sai, On uzzy atheatical prograing, The Journal o Cybernatics, 3 (974) [8]. Zierann.H.J: Fuzzy atheatical prograing,coput & ops.esvol Vol 0 No4 (983) eceived: Septeber, 0