A New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems

Transcription

1 Appled Mathematcal Scences, Vol. 4, 200, no. 2, A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence and Humantes VIT Unversty, Vellore-4, Taml Nadu, Inda Correspondng author. pandan6@redffmal.com ( P. Pandan ) Abstract A new algorthm namely, fuzzy zero pont method s proposed for fndng a fuzzy optmal soluton for a fuzzy transportaton problem where the transportaton cost, supply and demand are trapezodal fuzzy numbers. The optmal soluton for the fuzzy transportaton problem by the fuzzy zero pont method s a trapezodal fuzzy number. The soluton procedure s llustrated wth numercal example. Mathematcs Subect Classfcaton: 90C08, 90C90 Keywords: Fuzzy transportaton problem, Trapezodal fuzzy numbers, Optmal soluton, Fuzzy zero pont method Introducton The transportaton problem s one of the earlest applcatons of lnear programmng problems. Transportaton models have wde applcatons n logstcs and supply chan for reducng the cost. Effcent algorthms have been developed for solvng the transportaton problem when the cost coeffcents and the supply and demand quanttes are known exactly. The occurrence of randomness and mprecson n the real world s nevtable owng to some unexpected stuatons. There are cases that the cost coeffcents and the supply and demand quanttes of a transportaton problem may be uncertan due to some uncontrollable factors. To deal quanttatvely wth mprecse nformaton n makng decsons, Bellman and Zadeh [2] and Zadeh [9] ntroduced the noton of fuzzness.

2 80 P. Pandan and G. Nataraan A fuzzy transportaton problem s a transportaton problem n whch the transportaton costs, supply and demand quanttes are fuzzy quanttes. The obectve of the fuzzy transportaton problem s to determne the shppng schedule that mnmzes the total fuzzy transportaton cost whle satsfyng fuzzy supply and demand lmts. Snce the transportaton problem s essentally a lnear programmng problem, one straght forward dea s to apply the exstng fuzzy lnear programmng technques [3,4,9,0,2,4,5,7] to solve the fuzzy transportaton problem. Unfortunately, most of the exstng technques [3,4,9,2,5,7] provde only crsp solutons for the fuzzy transportaton problem. Julen [] and Parra et al. [4] proposed a method for solvng fuzzy transportaton problems and also, to fnd the possblty dstrbuton of the obectve value of the transportaton problem provded all the nequalty constrants are of types or types. Chanas et al. [6] developed a method for solvng transportaton problems wth fuzzy supples and demands va the parametrc programmng technque usng the Bellman Zadeh crteron [2]. Chanas and Kuchta [5] ntroduced a method for solvng a transportaton problem wth fuzzy cost coeffcents by transformng the gven problem to a bcrteral transportaton problem wth crsp obectve functon whch provdes only crsp soluton to the gven transportaton problem. Lu and Kao [6] developed a soluton procedure for computng the fuzzy obectve value of the fuzzy transportaton problem, where at least one of the parameters are fuzzy numbers usng the Zadeh s extenson prncple [8, 9,20]. Nagoor Gan and Abdul Razak [3] obtaned a fuzzy soluton for a two stage cost mnmzng fuzzy transportaton problem n whch supples and demands are trapezodal fuzzy numbers usng a parametrc approach. In ths paper, we propose a new algorthm namely, fuzzy zero pont method for fndng a fuzzy optmal soluton for a fuzzy transportaton problem where all parameters are trapezodal fuzzy numbers. The optmal soluton for the fuzzy transportaton problem by the fuzzy zero pont method s a trapezodal fuzzy number. The soluton procedure s llustrated wth numercal example. When we use the fuzzy zero pont method for fndng an optmal soluton for a fuzzy transportaton problem, we have the followng advantages. We do not use lnear programmng technques. We do not use goal and parametrc programmng technques. The optmal soluton s a fuzzy number and The proposed method s very easy to understand and to apply. 2 Fuzzy number and Fuzzy transportaton problem We need the followng mathematcal orentated defntons of fuzzy set, fuzzy number and membershp functon whch can be found n Zadeh [6].

3 Algorthm for fndng a fuzzy optmal soluton 8 Defnton 2. Let A be a classcal set and μ A (x ) be a functon from A to [0,]. A fuzzy set A wth the membershp functon μ ) s defned by A (x A = {( x, μ A( x)) : x A and μ A( x) [0,] }. Defnton 2.2 A real fuzzy number a = ( a, a, a, ) s a fuzzy subset from the 2 3 a4 a ( a real lne R wth the membershp functon μ ) satsfyng the followng condtons: () μ ( ) s a contnuous mappng from R to the closed nterval [0, ], a a a ( a a ( a a ( a a ( a a ( a () μ ) = 0 for every a, a ], ( () μ ) s strctly ncreasng and contnuous on a, ], (v) μ ) = for every a a 2, a ], [ 3 [ a 2 (v) μ ) s strctly decreasng and contnuous on a, ] and (v) μ ) = 0 for every a [ a 4, + ]. [ 3 a 4 Defnton 2.3 A fuzzy number a s a trapezodal fuzzy number denoted by ( a, where a, a2, a3 and a4 are real numbers and ts member shp functon μ ( x) s gven below. a 0 ( x a) /( a2 a) μ ( ) a x = ( a4 x) /( a4 a3) 0 for x a for a x a for a2 x a for a3 x a for x a We need the followng defntons of the basc arthmetc operators on fuzzy trapezodal numbers based on the functon prncple whch can be found n [6,7]. Defnton 2.4 Let ( a, and ( b, b2, b3, b4 ) be two trapezodal fuzzy numbers. Then () ( a, ( b, b2, b3, b4 ) = ( a + b, a2 + b2, a3 + b3, a4 + b4 ). () ( a, Θ ( b, b2, b3, b4 ) = ( a b4, a2 b3, a3 b2, a4 b ). () k ( a, = ( ka, k k k, for k 0. (v) k ( a, = ( ka 4, k k ka ), for k < 0. (v) a, a, a, ) b, b, b, ) = t, t t, ) ( 2 3 a4 ( 2 3 b4 ( 2 3 t4 where mnmum{ ab,ab 4,a4b,a4b4 } 2 mnmum a2b2,a2b3,a3b2,a3b3 3 maxmum a2b2,a2b3,a3b2,a3b3 4 maxmum ab,ab 4,a4b,a4b4 t = ; t = ; { } { } { } t = and t =.

4 82 P. Pandan and G. Nataraan We need the followng defnton of the defuzzfed value of a fuzzy number based on graded mean ntegraton method whch can be found n [7]. Defnton 2.5 If a = ( a, s a trapezodal fuzzy number, then the defuzzfed value or the ordnary (crsp) number of a, a s gven below. a + 2a2 + 2a3 + a4 a =. 6 We need the followng defntons of orderng on the set of the fuzzy numbers based on the magntude of a fuzzy number whch can be found n []. Defnton 2.6 The magntude of the trapezodal fuzzy number u ( x σ, x, y, y + β ) wth parametrc form u = ( u( r), u( r )) where = o o o o r) = x o σ + σr u( and u( r) = y o + β βr s defned as Mag (u ) = ( ( u( r) + u( r) + x yo) rdr) where r [0,]. o, Remark 2. The magntude of the trapezodal fuzzy number u = ( a, b, c, d ) s gven by 5 5 ( a + b + c + d Mag u ) =. 2 Defnton 2.7 Let u and v be two trapezodal fuzzy numbers. The rankng of u and v by the Mag (.) on E, the set of trapezodal fuzzy numbers s defned as follows: () Mag (u ) > Mag (v ) f and only f u f v ; () Mag (u ) < Mag (v ) f and only f u p v and () Mag (u ) = Mag (v ) f and only f u v. Defnton 2.8 The orderng f and p between any two trapezodal fuzzy numbers u and v are defned as follows: () u f v f and only f u f v or u v and () u p v f and only f u p v or u v. Note 2. () u = ( a, b, c, d) 0 f and only f Mag ( u ) = 0; () u = ( a, b, c, d) f 0 f and only f Mag ( u ) 0 and () u = ( a, b, c, d) p 0 f and only f Mag ( u ) 0. Defnton 2.9 Let {,,2,..., n } a = be a set of trapezodal fuzzy numbers. If Mag( a ) ( k Mag a ), for all, then the fuzzy number a k s the mnmum of a, =,2,..., n. { }

5 Algorthm for fndng a fuzzy optmal soluton 83 Defnton 2.0 Let {,,2,..., n } a = be a set of trapezodal fuzzy numbers. If Mag( a ( t ) Mag a ), for all, then the fuzzy number a t s the maxmum of a, =,2,..., n. { } Consder the followng fuzzy transportaton problem (FTP) havng fuzzy costs, fuzzy sources and fuzzy demands, (FTP) Mnmze m n z = c x = subect to n x a, for =,2,,m () = m x b, for =,2,,n (2) x f 0, for =,2,,m and =,2,,n, (3) where m = the number of supply ponts; n = the number of demand ponts; x ( x, x, x, x ) s the uncertan number of unts shpped from supply pont to demand pont ; 2 3 c (,,, 4 c c c c ) s the uncertan cost of shppng one unt from supply pont to the demand pont ; 2 3 a ( a, a, a, a 4 ) s the uncertan supply at supply pont and 2 3 (,,, 4 b b b b b ) s the uncertan demand at demand pont. The above problem can put n a table namely, fuzzy transportaton table gven below. c K c n Supply b M M M M c K m c mn b n Demand a K a n 3 Fuzzy zero pont method We, now ntroduce a new algorthm called the fuzzy zero pont method for fndng a fuzzy optmal soluton for fuzzy transportaton problems n sngle stage.

6 84 P. Pandan and G. Nataraan The zero pont method proceeds as follows. Step. Construct the fuzzy transportaton table for the gven fuzzy transportaton problem and then, convert t nto a balanced one, f t s not. Step 2. Subtract each row entres of the fuzzy transportaton table from the row mnmum. Step 3. Subtract each column entres of the resultng fuzzy transportaton table after usng the Step 2. from the column mnmum. Step 4. Check f each column fuzzy demand s less than to the sum of the fuzzy supples whose reduced costs n that column are fuzzy zero. Also, check f each row fuzzy supply s less than to sum of the column fuzzy demands whose reduced costs n that row are fuzzy zero. If so, go to Step 7. (Such reduced table s called the allotment table). If not, go to Step 5. Step 5. Draw the mnmum number of horzontal lnes and vertcal lnes to cover all the fuzzy zeros of the reduced fuzzy transportaton table such that some entres of row(s) or / and column(s) whch do not satsfy the condton of the Step4. are not covered. Step 6: Develop the new revsed reduced fuzzy transportaton table as follows: () Fnd the smallest entry of the reduced fuzzy cost matrx not covered by any lnes. () Subtract ths entry from all the uncovered entres and add the same to all entres lyng at the ntersecton of any two lnes. and then, go to Step 4. Step 7. Select a cell n the reduced fuzzy transportaton table whose reduced cost s the maxmum cost. Say ( α, β ). If there are more than one, then select anyone. Step 8. Select a cell n the α -row or/ and β column of the reduced fuzzy transportaton table whch s the only cell whose reduced cost s fuzzy zero and then, allot the maxmum possble to that cell. If such cell does not occur for the maxmum value, fnd the next maxmum so that such a cell occurs. If such cell does not occur for any value, we select any cell n the reduced fuzzy transportaton table whose reduced cost s fuzzy zero. Step 9. Reform the reduced fuzzy transportaton table after deletng the fully used fuzzy supply ponts and the fully receved fuzzy demand ponts and also, modfy t to nclude the not fully used fuzzy supply ponts and the not fully receved fuzzy demand ponts. Step 0. Repeat Step 7 to Step 9 untl all fuzzy supply ponts are fully used and all fuzzy demand ponts are fully receved. Step. Ths allotment yelds a fuzzy soluton to the gven fuzzy transportaton problem.

7 Algorthm for fndng a fuzzy optmal soluton 85 Now, we prove the followng theorems whch are used to derve the soluton to a fuzzy transportaton problem obtaned by the fuzzy zero pont method s a fuzzy optmal soluton to the fuzzy transportaton problem. Theorem 3. Any optmal soluton to the fuzzy problem ( P ) where ( P ) Mnmze * m n z = ( c ΘuΘv ) x = subect to () to (3) are satsfed, where u and v are some real trapezodal fuzzy numbers, s an optmal soluton to the problem (P) where (P) Mnmze m n z = c x = subect to () to (3) are satsfed. Proof. Now, * m n z c x m n Θ u x Snce m = b u = m b z Θ u and n = v a n Θ = = Θ m n = v x v a. ( from () and (2) ) are ndependent of x, for all and, we can conclude that any optmal soluton to the problem ( P ) s also a fuzzy optmal soluton to the problem (P). Hence the theorem. Theorem 3.2 If { x, =,2,,m and =,2,,m } s a feasble soluton to the problem (P) and ( c Θ uθv ) f 0, for all and where u and v are some real o trapezodal fuzzy numbers, such that the mnmum o () to (3) are satsfed, s fuzzy zero, then { x a fuzzy optmum soluton to the problem (P). Proof. From the Theorem 3., the result follows. Hence the theorem. m n = ( c Θu Θv ) x subect to, =,2,,m and =,2,,m } s Now, we prove that the soluton to a fuzzy transportaton problem obtaned by the fuzzy zero pont method s a fuzzy optmal soluton to the fuzzy transportaton problem. Theorem 3.3 A soluton obtaned by the zero pont method for a fuzzy transportaton problem wth equalty constrants (P) s a fuzzy optmal soluton for the fuzzy transportaton problem (P). Proof. We, now descrbe the fuzzy zero pont method n detal.

8 86 P. Pandan and G. Nataraan We construct the fuzzy transportaton table [ c ] for the gven fuzzy transportaton problem and then, convert t nto a balanced one f t s not balanced. Let u be the mnmum of -th row of the table [ c ]. Now, we subtract u from the -th row entres so that the resultng table s [ c Θ u]. Let v be the mnmum of -th column of the resultng table [ c Θ u]. Now, we subtract v from the -th column entres so that the resultng table s [ c Θ uθv ]. It may be noted that c Θ uθv f 0, for all and and each row and each column of the resultng table c Θ u Θv ] has atleast one fuzzy zero entry. [ Each column fuzzy demand of the resultng table [ c Θ uθv] s less than to the sum of the fuzzy supply ponts whose reduced costs n the column are fuzzy zero. Further, each row fuzzy supply of the resultng table [ c Θ uθv] s less than to sum of the column fuzzy demand ponts whose reduced costs n the row are fuzzy zero (If not so, as per drecton gven n the Step 5 and 6 n the zero pont method, we can make t that). The current resultng table s the allotment table. We fnd a cell n the allotment table [ c Θ uθv] whose reduced cost s maxmum. Say ( α, β ). We allot the maxmum possble to a cell n the α -row or/ and β column whch s the only cell n the α -row or/ and β column whose reduced cost s fuzzy zero. The resultng fuzzy transportaton table s reformed after deletng the fully used fuzzy supply ponts and the fully receved fuzzy demand ponts. Also, the not fully used fuzzy supply ponts and the not fully receved fuzzy demand ponts are modfed. We repeat the above sad procedure tll the total fuzzy supply are fully used and the total fuzzy demand are fully receved. Fnally, we have a soluton { x, =,2,,m and =,2,,n } for the reduced fuzzy transportaton problem whose cost matrx s [ c Θ uθv] such that x 0 for c Θ uθv f 0 and x f 0 for c ΘuΘv 0. m n Therefore, the mnmum ( c ΘuΘv ) x subect to () to (3) are = satsfed, s fuzzy zero. Thus, by the Theorem 3.2, the soluton { x, =,2,,m and =,2,,n } s a fuzzy optmal soluton to the fuzzy transportaton problem (P). Hence the theorem.

9 Algorthm for fndng a fuzzy optmal soluton 87 4 Numercal Example The proposed method called the fuzzy zero method s llustrated by the followng example. Example 4. Consder the followng fuzzy transportaton problem. Supply (,2,3,4) (,3,4,6) (9,,2,4) (5,7,8,) (,6,7,2) (0,,2,4) (-,0,,2) (5,6,7,8) (0,,2,3) (0,,2,3) (3,5,6,8) (5,8,9,2) (2,5,6,9) (7,9,0,2) (5,0,2,7) Demand (5,7,8,0) (,5,6,0) (,3,4,6) (,2,3,4) Now, the total fuzzy supply, S = (6,7,2,32 ) and the total fuzzy demand, D = (8,7,2,30). Snce Mag ( S ) = Mag( D ), the gven problem s a balanced one. Now, usng the Step 2 to the Step 3 of the fuzzy zero pont method, we have the followng reduced fuzzy transportaton table. Supply 0 (-3,0,2,5) (-4,,5,0) (-3,2,6,2) (,6,7,2) (-2,0,2,5) (0,,2,3) 0 (-3,2,4,9) (-5,2,6,3) (-5,,5,) (5,0,2,7) Demand (5,7,8,0) (,5,6,0) (,3,4,6) (,2,3,4) Now, usng the Step 4 to the Step 6 of the fuzzy zero pont method, we have the followng allotment table. Supply (-23,-5,7,25) 0 0 (-44,-0,4,49) (,6,7,2) (-37,-6,6,48) (-9,-,5,3) 0 (-23,-5,7,25) (0,,2,3) 0 (-33,-7,9,34) 0 0 (5,0,2,7) Demand (5,7,8,0) (,5,6,0) (,3,4,6) (,2,3,4) Now, usng the allotment rules of the fuzzy zero pont method, we have the allotment. Supply (,5,6,0) (-9,0,2,) (,6,7,2) (0,,2,3) (0,,2,3) (5,7,8,0) (-9,-,3,) (,2,3,4) (5,0,2,7) Demand (5,7,8,0) (,5,6,0) (,3,4,6) (,2,3,4)

10 88 P. Pandan and G. Nataraan Therefore, the fuzzy optmal soluton for the gven fuzzy transportaton problem s x 2 = (,5,6,0), x 3 = ( 9,0,2,), x 23 = (0,,2,3 ), x 3 = (5,7,8,0), x = ( 9,,3,) 23 and x 34 (,2,3,4 ) wth the fuzzy obectve value z = ( 274, 58,88, 575) and the crsp value of the optmum fuzzy transportaton cost for the problem, z s Concluson Some prevous studes have provded an optmal soluton for fuzzy transportaton problems whch s not fuzzy number, but t s a crsp value. If the obtaned results are crsp values, then t mght lose some helpful nformaton. The fuzzy zero pont method provdes that the optmal value of the obectve functon and shppng unts are fuzzy trapezodal fuzzy numbers for the fuzzy transportaton problem wth the unt shppng costs, the supply quanttes and the demand quanttes are trapezodal fuzzy numbers. Ths method s a systematc procedure, both easy to understand and to apply and also, t can serve as an mportant tool for the decson makers when they are handlng varous types of logstc problems havng fuzzy parameters. References [] S.Abbasbandy and T.Haar, A new approach for rankng of trapezodal fuzzy numbers, Computers and Mathematcs wth Applcatons, 57( 2009), [2] R.E. Bellman and L.A. Zadeh, Decson-makng n a fuzzy envronment, Management Scence, 7 (970), B4 B64. [3] J.J. Buckly, Possblstc lnear programmng wth trangular fuzzy numbers, Fuzzy Sets and Systems, 26 (988), [4] J.J. Buckly, Solvng possblstc programmng problems, Fuzzy Sets and Systems, 3(988), [5] S. Chanas and D. Kuchta, A concept of the optmal soluton of the transportaton problem wth fuzzy cost coeffcents, Fuzzy Sets and Systems, 82 (996), [6] S. Chanas, W. Kolodzeczyk and A. Macha, A fuzzy approach to the transportaton problem, Fuzzy Sets and Systems, 3 (984), 2 22.

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12 90 P. Pandan and G. Nataraan [9] L.A. Zadeh, Fuzzy sets as a bass for a theory of possblty, Fuzzy Sets and Systems, (978), [20] H.J. Zmmermann, Fuzzy set theory and ts applcatons, Kluwer-Nhoff, Boston, 996. Receved: May, 2009