Unbalanced Transportation Problems in Fuzzy Environment using Centroid Ranking Technique

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1 Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0 Unbalanced Transportaton Probles n Fuzzy Envronent usng Centrod Rankng Technque R. K. an epartent of Matheatcal cences and Coputer pplcatons, Bundelkhand Unversty, Jhans, Inda tul angal Research cholar Banasthal Unversty, Banasthal, Raasthan, Inda O Prakash epartent of Matheatcs, IIT Patna, Bhar, Inda BTRCT In the present paper a new ethod proposed for the soluton of unbalanced transportaton probles wth Trapezodal haped Generalzed Fuzzy Nubers (T P GFN) usng centrod rankng technque va zero suffx ethod. Nuercal exaples show that ths technque offers effectve way for handng the unbalanced fuzzy transportaton proble wth precse render and requreent condton. The advantage of the proposed ethod over the exstng ethods s to fnd fuzzy optal soluton does not nvolve only the duy destnaton. The ethod has sple algorths for coputed and the study s checked wth nuercal exaples. Keywords Unbalanced Transportaton Probles, Trapezodal haped Generalzed Fuzzy Nuber, Centrod Rankng Technque, Zero uffx Method. M ubect Classfcaton: 90B06, 90C90, 90C70.. INTROUCTION Transportaton probles are solved wth the assuptons that the paraeters of the transportaton proble are specfed n crsp envronent. The paraeters of the transportaton proble are not always exactly known and stable. Ths precson ay follow fro the lack of exact nforaton, uncertanty n udgent etc. Zadeh [] ntroduced the noton of fuzzness, that was renforced by Bellan and Zadeh []. Htchcock [] orgnally developed the basc transportaton proble. ppa [] dscussed several varatons of the transportaton proble. Zerann [7] fuzzy lnear prograng has developed nto seven fuzzy optzaton ethods for solvng transportaton probles. In [0] O hegeartagh proposed a new algorth for the soluton of transportaton probles where the capactes and requreents are fuzzy nubers wth lnear trangular ebershp functons. The concept of optal soluton for the transportaton proble wth fuzzy coeffcents expressed as fuzzy nubers are proposed by Chanas and Kuchta [6], for obtanng the optal soluton by developng an algorth. aad & bbas [] dscussed an algorth for solvng the transportaton probles n fuzzy envronent. By usng all trangular fuzzy nubers as paraeters, as & Baruah [7] proposed Vogel s approxaton ethod to fnd the fuzzy ntal basc feasble soluton of fuzzy transportaton probles. By transforng the fuzzy paraeters nto crsp paraeters, Basrzadeh [] used the classcal algorths to fnd the fuzzy optal soluton of fully fuzzy transportaton probles. To fnd the fuzzy optal soluton of fuzzy transportaton probles, where trapezodal fuzzy nubers represent all the paraeters, Pandan & Natraan [, ] proposed a new algorth, naely fuzzy zero pont ethod. Gan et al [] obtaned the fuzzy optal soluton of fuzzy transportaton probles havng paraeters as trapezodal fuzzy nubers. For ore detals see [, 8, 9, 0,6, 7, 9,,7]. For solvng unbalanced fuzzy transportaton probles, two cases ay arse as ether the total deand s ore than the total supply or vce-versa. In case of excess supply, t ay happen soetes that there s no enough storage place for the excess coodty, we wsh to transport t to the destnatons for the further deand. or n case of excess deand, wsh to transport t for future supply. Here we add a duy destnaton, where the excess avalablty s transported. Because the duy destnaton does not have any exstence n realty, so t s not possble to fnd that the excess avalable product should be transported to whch the destnaton at a nu cost.. PRELIMINRIE In ths secton, soe basc defntons, arthetc operatons and coparson of generalzed trapezodal fuzzy nubers are presented.. efnton Let R s a unversal set of real nubers. fuzzy set defned s sad to be a fuzzy nuber f ts ebershp functon has the followng characterstcs: () : R 0, s contnuous () (x) 0 for all x (,a] [d, ) () ( x) s strctly ncreasng on [a,b] and strctly decreasng on [c,d] (v) (x) for all x [b,c], where a b c d. efnton Let R s a unversal set of real nubers. fuzzy set s sad to be generalzed fuzzy nuber f ts ebershp functon has the followng characterstcs: () : R 0, s contnuous () (x) 0 for all x (,a] [d, ) () ( x) s strctly ncreasng on [a,b] and strctly decreasng on [c,d] (v) (x) = w for all x [b,c], where 0 w. 7

2 Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0. efnton fuzzy set of the unverse of dscourse X s called a noral fuzzy set f there exst at least one x X such that (x) w.. efnton The fuzzy set s convex f and only f, for any x, x X, the ebershp functon of satsfes the nequalty x x n x, x, 0.. efnton (Trapezodal Fuzzy Nuber)[]: trapezodal fuzzy nuber (x), t can be represented by a,b, c,d; as gven by a,b,c,d; (x a), a x b (b a), b x c (d x), c x d ( d c) 0, otherwse Fg. : Trapezodal Fuzzy Nuber a,b, c,d; The α-cut of a fuzzy nuber (x) s defned as (x) x : (x), [0,] Fg. : cut of a, b, c,d;.6 efnton generalzed fuzzy nuber a,b, c,d; w s sad to be a generalzed trapezodal fuzzy nuber f ts ebershp functon s gven by w(x a), a x b (b a) w, b x c (a, b,c,d; w) w(d x), c x d ( d c) 0, otherwse.7 Reark generalzed trapezodal fuzzy nuber a,b, c,d; w can be explaned as follows: () ccordng to decson aker the cost (or deand or supply or proft etc.) of the product wll be greater than unts and less than n unts. () In favor of the cost (or deand or supply or proft etc.), the decson aker s w 00 whch wll be greater than or equal to n unts and less than or equal to p unts. () For the reanng values of cost (or deand or supply or proft etc.), the percentage of the favourness can be obtaned as x (a,b,c,d;w ) 00, where x represents the cost (or deand or supply or proft etc.).. RITHMETIC OPERTION The arthetc operatons between two generalzed trapezodal fuzzy nubers, defned on unversal set of real nubers R s coputed as follows: Let a,b,c,d ;w and B a,b,c,d ;w generalzed trapezodal fuzzy nubers then be two B a a,b b, c c,d d ; nu w, w! B a a,b b, c c, d d ; nu w,w B ( a, b, c, d ; nu(w, w )), where a nu a a, a d, d a, d d, b nu b b, b c, c b, c c, c axu b b, b c, c b, c c, d axu a a, a d, d a, d d () ( a, b, c, d ;w ), 0 ( d, c, b, a ;w ), 0. UE YMBOL : nuber of sources n: nuber of destnatons a : the fuzzy avalablty of the product at source b : the fuzzy deand of the product at destnaton : the fuzzy cost for transportng one unt of the product fro to x : the fuzzy quantty of the product to be transported fro to u : dual varable correspondng to th source at (TP) v : dual varable correspondng to th destnaton at (TP) u : dual varable correspondng to th source at (TP) v : dual varable correspondng to th destnaton at (TP) 8

3 Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0. NEW RNKING TECHNIQUE Here we are takng a new rankng ethod called centod rankng ethod for the generalzed fuzzy nuber for solvng the transportaton probles. Ths approach nvolves sple coputatonal and s easly understandable. The centrod rankng ethod can explaned easly n geoetrcal fro. For understandng ths ethod we ake a fgure: Fg. : Centod Rankng Method Fro the fgure, t s sple that centod ethod depends on trapezodal + trangular fuzzy nubers. We can clearly see that n the above fgure, trangular shaped fgure PB, CQ and rectangular shaped fgure BPQC erged n trapezodal shaped fgure PQ. Let the centrod of PB, rectangle BPQC, and CQ are G, G and G respectvely. The centrod G (suppose) of these centrods G, G and G s taken as the pont of reference, to defne the rankng of generalzed trapezodal fuzzy nuber. The reason for selectng ths pont of reference s that each centrod pont G, of trangle PB, G of rectangle BPQC and G of QB are balancng ponts of each ndvdual plane fgure and the centrod of these centrod ponts s a uch ore balancng pont for a general trapezodal fuzzy nuber. fter calculatng the rankng ndces for the use of new rankng ethod, we apply the followng forula for generalzed trapezodal fuzzy nuber, s. Reark [8]: a,b,c a + 7b + 7c + d 7w R() =. 8 8,d ;w B a,b,c,d ;w Let and two trapezodal fuzzy nubers. Then f, f R( ) f R( ), () p, f R( ) p R( ) () (), f R( ) R( ).. Reark generalzed trapezodal fuzzy nuber a, b, c, d;w be s sad to be zero fuzzy nuber f and only f R() 0 and s denoted by 0.. Reark generalzed trapezodal fuzzy nuber a, b, c, d;w s sad to be non-negatve nuber f and only f R() 0 and s denoted by ± IMPROVE ZERO UFFIX METHO Here we ntroduce an proved zero suffx ethod for fndng on optal soluton to the transportaton proble as: tep Construct the transportaton table. tep ubtract row nu value fro each row entres of the correspondng row n transportaton table. The sae process ust be done for coluns of the transportaton table. tep In the reduced cost atrx, there wll be at least one zero n each row and each colun. Fnd the suffx value of all the zero s n the cost atrx as follows: su of the values (c )of zero s n correspondng rowand colun except tself No. of zeros tep Choose the axu of, f t has one axu value then frst supply to that deand correspondng to the cell. If t has ore equal values then select {a,b } and supply to that deand axu possble. tep fter the above steps, the exhausted deands or supples to be tred. The resultant atrx ust possess at least one zero s each row and colun, else repeat step. tep6 Repeat steps to untl the optal soluton s obtaned. 7. PROBLEM FORMULTION Consder an unbalanced trapezodal shaped generalzed fuzzy transportaton proble (T P GFTP) havng sources wth fuzzy avalablty a ( ) and n destnatons wth fuzzy deand b ( n). Each of the sources transport to any of the n destnatons cost of can at a transportaton c per unt. Let x be the fuzzy quantty of the product that should be transported fro source to destnaton. The transportaton proble (TP ) can be represented as follows: (TP ) n Mnze c x n subect to x ± a, ( ) x ± b, ( ) 7. Exaple Consder a transportaton proble wth sources say,,, and destnatons,,, wth unt costs of transportng the product fro source ( ) to destnaton ( ) The generalzed fuzzy transportaton cost atrx for unt quantty of product fro th sources to th destnatons s as gven below: 9

4 (c ),7,,,,,7,,,9 9,8,,,,6,,9,6, 0,9,8,7,0,,8,9,6, 7,8,9,,,, 8,7,, 6,,9, 7,,,9 8,0,,,,, Fuzzy avalablty (supply) of the product at sources are,,, and 0,9,,, 6,0,7,,,,, and the fuzzy deand of the product to the destnatons are 6,,,0.,,8,, 7,,8,6,,,, and The product can be shpped fro any source to any destnaton. Here we try to fnd the generalzed fuzzy quantty ources Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0 of the product that should be transported fro each source to each destnaton so that the total generalzed fuzzy cost of transportaton s nu. For the above transportaton proble, the total generalzed fuzzy deand s ore than the total generalzed fuzzy supply. o we frst convert t to a balanced proble by addng a duy source ( ) wth the correspondng unt transportaton costs to be zero trapezodal generalzed fuzzy nubers and convert the constrant nequaltes to equatons. The balanced generalzed fuzzy transportaton proble n tabular for as follows: oluton The fuzzy Transportaton probles are gven n Table-,7,,,,,7,,,9 9,8,,,,6,,9,6, 0,9,8,7,0,,8,9,6,7,8,9,,,, 8,7,, 6,,9, 7,,,9 8,0,,,,, eand,,8, 7,,8,6,,, 6,,,0 upply 0,9,, 6,0,7,,,,,,, Table-: Unbalanced generalzed fuzzy transportaton proble In conforaton to odel the fuzzy transportaton proble can be forulated n the followng atheatcal for,7,,,,,7,,,9 6,,, 6,,6,,9,6, R 0,9,8,7 x R,0,,8 x R,9,6,7 x R,8,9, x R,,, x R 8,7,, x R 6,,9, x R 7,,,9 x R 8,0,, x R,,, x MnZ R x R x R x R x R x R x Now for rankng, we take 7 7 7w R() a b c d 8 8 For rankng the fuzzy nubers, we use centrod rankng ethod for w = 0. and applyng as follows: ( ) 7(7 ) 7(.) R(c ) R,7,, Proceedng slarly, the rankng ndces for the cost c are calculated as: ) R )R,,,9 ) ) ) ) R( R,,, , R( R 9,8,,.879, R( R,,6,.87, R( R,9,6,., R( R 0,9,8,7.6, ( 0.07, ) R,,,) c ) 06, R(c ) R(,0,,8).87, R( R,8,9,.6, R( R 8,7,,. R (c ) R(7,,,9).60, R( )R 8,0,,.60, R( )R,,, 0.9. Rank of all eand R,,8,.9, R,,,.0, Rank of all upply R 0,9,,., R,,,. 87, R( c ) R,9,6,7. 7, (c ) R(.6, R( c ) R 6,,9,. 080, R 7,,8,6.9, R 6,,, 0.. 6,0,7,,,, R. 79, R

5 Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0 ources ( 0.9) 0.07 (. 0) ( ).(.6) (. 70) eand upply ( 0. 8) (. 79) ( 0. 7) ources Table-: oluton of balance generalzed fuzzy transportaton proble after rankng,7,,,,,7,,,-,,,9 (,,, ) (- ),,6, (,8,-,-0),9,6, 8,0,,9,9,6,7 ( 6,,, 0),8,9, 6,,9, 8,0,, 0,0,0,0 (,, 0, 0) 0,0,0,0 eand,,8, 7,,8,6 9,8,, 0,9,8,7,0,,8,,, 8,7,, (, 9,, ) 8,0,,,,, (,,, ) 0,0,0,0 0,0,0, 0,,, 6,,,0 Table-: oluton of balance generalzed fuzzy transportaton proble by zero suffx ethod The generalzed fuzzy optal solutons are X = (7, 6,-,-), X = (,,-,-0), X = (,,, ), X = (6, 0, 7, ), X = (6,,-, 0), X = (, 9,,-), X = (,,, ), X = (7,, 0,-0). The total generalzed fuzzy optal cost s,6,). z = (-, PROPOE METHO In ths secton, we proposed a ethod to solve the unbalanced (T P GFTP) n whch total generalzed fuzzy deand s ore than the total generalzed fuzzy supply. We frst prove the followng theore. 8. Theore Let (TP ) be the fuzzy lnear forulaton of balanced (T P GFTP) obtaned by addng a duy source + wth unt fuzzy transportaton costs c ( +) = n( ), n, a duy destnaton n+ wth c(n ) n( ),, to n (TP ). lso, let (u,v) and (u,v) be the optal solutons of the duals of (TP ) and (TP ), respectvely, where (u,v) = (u,u,...u ;v,v,...v ) and n (u,v) = (u,u,...u ;v,v,...v ). Then + n+ and v v, n provded u 0. Proof: The proble s (TP ) n Mnze (c x ), n subect to x a ( ), u,u, x b ( n ), where n x ± 0 a a and b a excess supply. upply 0,9,, 6,0,7,,,,,,,,,0,-0 nce (u, v) s the optal soluton of dual of (TP ), so u v,, n and u± 0,. To confr that the soluton set (u,u,...u ;v,v,...v ) s n also an optal soluton of the dual of (TP ), t s suffcent to show that u v ;, n () () u± 0,. and v± 0, n. nce (u,v) s the optal soluton of the dual to proble (TP ), so u v ;, n. nce u v c we have u v ( ) ; n u v, and n (n) u v ; n v c. () () nce ( u 0) v c ; ( snce c n ( c )) c v ± 0 ( ) u± 0. larly fro u v, t can be proved that bn v± 0 (snce n (n) u 0 so v n 0 as are so chosen that n any feasble soluton x ( )(n ) ust be a basc varable). teps for Proposed Method tep : Balance the gven (T P GFTP) by addng a duy source (( +) th source) wth avalablty equal to total avalablty as well as duy destnaton ((n +) th destnaton) wth deand equal to su of total avalablty and excess supply.

6 Internatonal Journal of Coputer pplcatons ( ) Volue 0 No., January 0 That s a a and b a excess supply The unt n The unt transportaton costs are taken as follows: ources upply eand Table-: Balanced generalzed fuzzy transportaton proble tep : pply the zero suffx ethod to the balanced (T P GFTP) obtaned n step. Let the generalzed fuzzy optal soluton obtaned be x,, n. tep : Fnd the values of all the dual varables u, and v, n by assung u (0,0,0,0;.) and usng the relaton u v c for basc varables. n (c ),, n (c ), n, (n) () n c,, n, and n (0,0,0,0;.). ()(n ) tep: By theore, u u, and v v, n, we fnd those dual varables and u and vwhch have central rank zero. Now, we obtaned the generalzed fuzzy optal soluton of the proble n ters of orgnal sources and destnatons. Let x ( )p for soe p and x q(n ) for soe q be the basc varables n the fuzzy optal soluton obtaned n step. lso, let u and v have rank zero for I and J. Then ncrease the value of the basc varable n the cell wth n by x by x (+)p and the value of I J p q (+)p n by x. vde the te value n nu value(s) q(n+) arbtrarly. If the nu cost cell s non-basc n the optal soluton obtaned n step, then t ay becoe basc n the fnal soluton. ccordng to the proposed ethod the fnal transportaton table wth x s as follows: ources ( 0.0) 0.07(.00) (0.08).(.6) (. 87) ( 0. 7) (0.0) 0.9(. 79) eand upply Table-: oluton of balanced generalzed fuzzy transportaton proble by zero suffx ethod ources,7,,,,,7,,-7,-),,,9 (,6,, 0) 9,8,,,,,9,,6,,9,-,-,9,6, 7,,-, 0,9,8,7,0,,8,9, 6,,9, 6, 7,,,,8,9,,,, 8,7,, 8,7,, 8,0,,,,,,,,,,,,, 9, (,,-,- 8),,,,9,,- 0,0,0,0 7,,8,8 eand,,, 6,,,0 7,,8,8 ( 6,,9,,7,,,,8, 8,0,,,,,7 7,,8,6 The fuzzy optal solutons are X = (,,-7,-), X = (,6,,0), X = (,9,--), X = (7,,-,), X = (,,, ), X = (,,, ), X = (,,-,-8), X = (,9,,-) X = (7,,9,). The total generalzed fuzzy optal cost s Table-6: oluton of balanced generalzed fuzzy transportaton proble z= (-7,- 0,,0). upply 0,9,, 6,0, 7,,,,,,,,,9, 9. CONCLUION The centrod rankng technque to convert generalzed fuzzy nuber nto sple crsp nuber, whch one concludes by exaple, s ore sutable. Moreover the fuzzy transportaton cost and fuzzy optal cost consdered here as generalzed fuzzy nubers s ore effectve by proposed ethod. 0. REFERENCE [] bbasbandy. and Haar T., new approach for rankng of trapezodal fuzzy nubers, Coputers and Matheatcs wth pplcatons, 7 (009), -9. [] ppa G M, The transportaton proble and ts varants. Operaton. Research. Quarterly. (97),

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