Amercan urnal f Operatns Research,,, 58-588 Publshed Onlne Nvember (http://www.scrp.rg/urnal/ar) http://dx.d.rg/.46/ar..655 Lnear Plus Lnear Fractnal Capactated Transprtatn Prblem wth Restrcted Flw Kavta Gupta, Shr Ram Arra Department f Mathematcs, Ramas Cllege, Unversty f Delh, Delh, Inda Department f Mathematcs, Hans Ra Cllege, Unversty f Delh, Delh, Inda Emal: gupta_kavta@yah.cm, srarra @yah.cm Receved Octber 6, ; revsed Nvember 6, ; accepted Nvember, Cpyrght Kavta Gupta, Shr Ram Arra. Ths s an pen access artcle dstrbuted under the Creatve Cmmns Attrbutn Lcense, whch permts unrestrcted use, dstrbutn, and reprductn n any medum, prvded the rgnal wrk s prperly cted. Abstract In ths paper, a transprtatn prblem wth an bectve functn as the sum f a lnear and fractnal functn s cnsdered. The lnear functn represents the ttal transprtatn cst ncurred when the gds are shpped frm varus surces t the destnatns and the fractnal functn gves the rat f sales tax t the ttal publc expendture. Our bectve s t determne the transprtatn schedule whch mnmzes the sum f ttal transprtatn cst and rat f ttal sales tax pad t the ttal publc expendture. Smetmes, stuatns arse where ether reserve stcks have t be kept at the supply pnts, fr emergences r there may be extra demand n the markets. In such stuatns, the ttal flw needs t be cntrlled r enhanced. In ths paper, a specal class f transprtatn prblems s studed where n the ttal transprtatn flw s restrcted t a knwn specfed level. A related transprtatn prblem s frmulated and t s shwn that t each basc feasble slutn whch s called crner feasble slutn t related transprtatn prblem, there s a crrespndng feasble slutn t ths restrcted flw prblem. The ptmal slutn t restrcted flw prblem may be btaned frm the ptmal slutn t related transprtatn prblem. An algrthm s presented t slve a capactated lnear plus lnear fractnal transprtatn prblem wth restrcted flw. The algrthm s supprted by a real lfe example f a manufacturng cmpany. Keywrds: Transprtatn Prblem; Lnear Plus Lnear Fractnal; Restrcted Flw; Crner Feasble Slutn. Intrductn Transprtatn prblems wth fractnal bectve functn are wdely used as perfrmance measures n many real lfe stuatns such as the analyss f fnancal aspects f transprtatn enterprses and undertakng, and transprtatn management stuatns, where an ndvdual, r a grup f peple s cnfrnted wth the hurdle f mantanng gd rats between sme mprtant and crucal parameters cncerned wth the transprtatn f cmmdtes frm certan surces t varus destnatns. Fractnal bectve functn ncludes ptmzatn f rat f ttal actual transprtatn cst t ttal standard transprtatn cst, ttal return t ttal nvestment, rat f rsk assets t captal, ttal tax t ttal publc expendture n cmmdty etc. Gupta, Khanna and Pur [] dscussed a paradx n lnear fractnal transprtatn prblem wth mxed cnstrants and establshed a suffcent cndtn fr the exstence f a paradx. an and Saksena [] studed tme mnmzng transprtatn prblem wth fractnal bttleneck bectve functn whch s slved by a lexcgraphc prmal cde. Xe, a and a [] develped a technque fr duratn and cst ptmzatn fr transprtatn prblem. In addtn t ths fractnal bectve functn, f ne mre lnear functn s added, then t makes the prblem mre realstc. Ths type f bectve functn s called lnear plus lnear fractnal bectve functn. Khuranaand Arra [4] studed lnear plus lnear fractnal transprtatn prblem fr restrcted and enhanced flw. Capactated transprtatn prblem fnds ts applcatn n a varety f real wrld prblems such as telecmmuncatn netwrks, prductn-dstrbutn systems, ral and urban rad systems where there s scarcty f resurces such as vehcles, dcks, equpment capacty etc. Many researchers lke Gupta and Arra [5], Msra and Das [6] have cntrbuted t ths feld. an and Arya [7] studed the nverse versn f capactated transprtatn prblem. AOR
58 K. GUPTA, S. R. ARORA Many researchers lke Arra and Gupta [8], Khurana, Thrwan and Arra [9] have studed restrcted flw prblems. Smetmes, stuatns arse when reserve stcks are t be kept at surces fr emergences. Ths gves rse t restrcted flw prblem where the ttal flw s restrcted t a knwn specfed level. Ths mtvated us t develp an algrthm t slve a lnear plus lnear fractnal capactated transprtatn prblem wth restrcted flw.. Prblem Frmulatn I (P): mn z r x subect t s x I I tx a x A; I () b x B; () I l x u and ntegers I, () x P mn A, B (4) I I I,,, m s the ndex set f m rgns.,,, n s the ndex set f n destnatns. x = number f unts transprted frm rgn t destnatn. r = per unt transprtatn cst when shpment s sent frm th rgn t the th destnatn. s = the sales tax per unt f gds transprted frm th rgn t the th destnatn. t = the ttal publc expendture per unt f gds transprted frm th rgn t the th destnatn. l and u are the bunds n number f unts t be transprted frm th rgn t th destnatn. a and A are the bunds n the avalablty at the th rgn, I b and B are the bunds n the demand at the th destnatn, It s assumed that tx fr every feasble I slutn X satsfyng (), (), () and (4) and all upper bunds u ;, I are fnte. Smetmes, stuatns arse when ne wshes t keep reserve stcks at the rgns fr emergences, there by restrctng the ttal transprtatn flw t a knwn spec- fed level, say P mn A, B. Ths flw cn- I strant n the prblem (P) mples that a ttal A P I f the surce reserves has t be kept at the varus surces and a ttal B P f destnatn slacks s I t be retaned at the varus destnatns. Therefre an extra destnatn t receve the surce reserves and an extra surce t fll up the destnatn slacks are ntrduced. In rder t slve the prblem (P) we cnvert t n t related prblem (P) gven belw. s y I (P): mn z r y t y subect t I I y A I I l y u,, I y B b m, y A a I n, y A A I, m, n m, A B P r r, I,, y B B B, B A P n r, r, I,, r, M m n t t, I,, m n t m, t, n I,, t m, n M s s, I,, s, s, I, ; s, M m n where M s a large pstve number. I m n I m m,,,, n, n,,,,. Theretcal Develpment Therem: A feasble slutn X x I f prb- S lem (P) wth bectve functn value R wll T be a lcal ptmum basc feasble slutn ff the fllwng cndtns hlds. T s z S t z r z ;, N AOR
K. GUPTA, S. R. ARORA 58 T s z S t z r z ;, N and f X s an ptmal slutn f (P), then where ;, and ;, N R r x, S s x, I I N T t x, I B dentes the set f cells (, ) whch are basc and N and N dentes the set f nn-basc cells (, ) whch are at ther lwer bunds and upper bunds respectvely. u, u, u, v, v, v ; I, are the dual varables such that u v r ; u v z ; u v s ; u v z ; u v t ;, u v z B ; ;, B. Prf: Let X x be a basc feasble slutn I f prblem (P) wth equalty cnstrants. Let z be the crrespndng value f bectve functn. Then sx I I tx I z r x R S T say r u v l r u v u u v x, N, N I s u v x u v x I I r u v x u v x I I t u v x u v x I I, N, N I, N, N I s u v l s u v u u v x t u v l t u v u u v x r z l r z u au b v, N, N I, N, N I, N, N I s z l s z u au b v t z l t z u au b v Let sme nn-basc varable x N underges change by an amunt where s gven by mn u l ; x l fr all basc cells, wth a entry n lp; u x fr all basc cells, wth a entryn lp Then new value f the bectve functn ẑ wll be gven by T t z S s z zˆ R r z T t z S s z S zˆ z R r z R T T s z S t z T T t z r z say AOR
584 K. GUPTA, S. R. ARORA Smlarly, when sme nn-basc varable x N underges change by an amunt then Hence X wll be lcal ptmal slutn ff ;, N and ;, N. If X s a glbal ptmal slutn f (P), then t s an ptmal slutn and hence the result fllws. Defntn: Crner feasble slutn: A basc feasble slutn y I, t(p) s called a crner feasble slutn (cfs) f ym, n Therem. A nn-crner feasble slutn f (P) cannt prvde a basc feasble slutn t (P). Prf: Let y be a nn-crner feasble slu- I tn t (P). Then y m, n Thus Therefre, I Nw, fr I, y y y n, n, m, n I I n, T s z S t z zˆ z r z y A P n, I I I y A P y A A y I I The abve tw relatns mples that A I y P Ths mples that ttal quantty transprted frm all the surces n I t all the destnatns n s P P, a cntradctn t the assumptn that ttal flw s P and hence y cannt prvde a feasble slutn t I (P). Lemma: There s a ne-t-ne crrespndence between the feasble slutn t (P) and the crner feasble slutn t (P). Prf: Let be a feasble slutn f (P). S x I x I wll satsfy () t (4). Defne by the fllwng transfrmatn y I y x, I, y A x I n,, y B x m,, I y m, n It can be shwn that y s defned s a cfs t I (P) Relatn () t () mples that l y u fr all I, y A a, I n, y B b, m, y, m, n Als fr I y y y, n x A x A A Fr m ym, y ym, n B x I B x B P A m I y A; I Smlarly, t can be shwn that y B; Therefre, y I Cnversely, let I s a cfs t (P). y I be a cfs t (P). Defne x, I, by the fllwng transfrmatn. x y, I, It mples that l y u, I, Nw fr I, the surce cnstrants n (P) mples y A A y y A, n a y A (snce yn, A a, I ). Hence, a x A, I Smlarly, fr, b x B Fr = m +, I y A B P m, m AOR
K. GUPTA, S. R. ARORA 585 (because m, n y B P m, y ) Nw, fr the destnatn cnstrants n (P) gve y y B I m, Therefre, y y, B m I y B y P m, I I x P Therefre x I s a feasble slutn t (P) Remark : If (P) has a cfs, then snce c m, n M and d m, n M, t fllws that nn crner feasble slutn cannt be an ptmal slutn f (P). Lemma : The value f the bectve functn f prblem (P) at a feasble slutn s equal t x I the value f the bectve functn f (P) at ts crrespndng cfs and cnversely. y I Prf: The value f the bectve functn f prblem (P) at a feasble slutn s y I r r, I, s s, I, t t, I, sy sx x y, I, I I z r y rx because I t y I t x r, n r m, ; I, I I s n, s m, ; I, t n, t m, ; I, y m, n x the value f the bectve functn f P at the crrespndng feasble slutn I The cnverse can be prved n a smlar way. Lemma : There s a ne-t-ne crrespndence between the ptmal slutn t (P) and ptmal slutn t the crner feasble slutn t (P). Prf: Let x be an ptmal slutn t (P) I yeldng bectve functn value z and y be the I crrespndng cfs t (P).Then by Lemma, the value yelded by y s z. If pssble, let y be I I nt an ptmal slutn t (P). Therefre, there exsts a cfs y say, t (P) wth the value z < z Let x be the crrespndng feasble slutn t (P).Then by Lemma, s x I z r x I tx, I a cntradctn t the assumptn that x s an p- I tmal slutn f (P).Smlarly, an ptmal crner feasble slutn t (P) wll gve an ptmal slutn t (P). Therem : Optmzng (P) s equvalent t ptmz- ng (P) prvded (P) has a feasble slutn. Prf: As (P) has a feasble slutn, by Lemma, there exsts a cfs t (P). Thus by Remark, an ptmal slutn t (P) wll be a cfs. Hence, by Lemma, an ptmal slutn t (P) can be btaned. 4. Algrthm Step : Gven a lnear plus lnear fractnal capactated transprtatn prblem (P), frm a related transprtatn prblem (P). Fnd a basc feasble slutn f prblem (P) wth respect t varable cst nly. Let B be ts crrespndng bass. Step : Calculate, u, u, u, v, v, v, z, z, z ; I, such that u v r ; u v s ; u v t,, B; u v z ; u v z ; u v z,, B = level at whch a nn-basc cell (,) enters the bass replacng sme basc cell f B. u, u, u, v, v, v; I, are the dual varables whch are determned by usng the abve equatns and takng ne f the u s r v s. as zer. AOR
586 K. GUPTA, S. R. ARORA Step : Calculate R, S, T where R r x, S s x, T t x I I I Step 4: Fnd ;, and ;, where and N N T s z S t z r z, N T s z S t z r z, N ; ; where N and N dentes the set f nn-basc cells (,) whch are at ther lwer bunds and upper bunds respectvely. If ;, N and ;, N then the current slutn s btaned s the ptmal slutn t (P) and subsequently t (P). Then g t step (5). Otherwse sme, N fr whch r sme, N fr whch wll enter the bass. G t Step. S Step 5: Fnd the ptmal value f z R T 5. Prblem f the Manager f a Cell Phne Manufacturng Cmpany ABC cmpany prduces cell phnes. These cell phnes are manufactured n the factres () lcated at Haryana, Punab and Chandgarh. After prductn, these cell phnes are transprted t man dstrbutn centres () at Klkata, Chenna and Mumba. The cartage pad per cell phne s, and 4 respectvely when the gds are transprted frm Haryana t Klkata, Chenna and Mumba. Smlarly, the cartage pad per cell phne when transprted frm Punab t dstrbutn centres at Klkata, Chenna and Mumba are 6, and respectvely whle the fgures n case f transprtatn frm Chandgarh s, 8 and 4 respectvely. In addtn t ths, the cmpany has t pay sales tax per cell phne. The sales tax pad per cell phne frm Haryana t Klkata, Chenna, Mumba are 5, 9 and 9 respectvely. The tax fgures when the gds are transprted frm Punab t Klkata, Chenna and Mumba are 4, 6 and respectvely. The sales tax pad per unt frm Chandgarh t Klkata, Chenna and Mumba are, and respectvely. The ttal publc expendture per unt when the gds are transprted frm Haryana t Klkata, Chenna and Mumba are 4, and respectvely whle the fgures fr Punab are, 7 and 4. When the gds are transprted frm Chandgarh t dstrbutn centres at Klkata, Chenna and Mumba, the ttal publc expendture per cell phne s, 9 and 4 respectvely. Factry at Haryana can prduce a mnmum f and a maxmum f cell phnes n a mnth whle the factry at Punab can prduce a mnmum f and a maxmum f 4 cell phnes n a mnth. Factry at Chandgarh can prduce a mnmum f and a maxmum f 5 cell phnes n a mnth. The mnmum and maxmum mnthly requrementf cell phnes at Klkata are 5 and respectvely whle the fgures fr Chenna are 5 and respectvely and fr Mumba are 5 and respectvely. The bunds n the number f cell phnes t be transprted frm Haryana t Klkata, Chenna and Mumba are (, ), (, ) and (, 5) respectvely. The bunds n the number f cell phnes t be transprted frm Punab t Klkata, Chenna and Mumba are (, 5), (, 5) and (, ) respectvely. Thebunds n the number f cell phnes t be transprted frm Chandgarh t Klkata, Chenna and Mumba are (, ), (, ) and (, 5) respectvely. The manager keeps the reserve stcks at the factres fr emergences, there by restrctng the ttal transprtatn flw t 4 cell phnes. He wshes t determne the number f cell phnes t be shpped frm each factry t dfferent dstrbutn centres n such a way that the ttal cartage plus the rat f ttal sales tax pad t the ttal publc expendture per cell phne s mnmum. 5.. Slutn The prblem f the manager can be frmulated as a lnear plus lnear fractnal transprtatn prblem (P) wth restrcted flw as fllws. Let O and O and O dentes factres at Haryana, Punab and Chandgarh. D, D and D are the dstrbutn centres at Klkata, Chenna and Mumba respectvely. Let the cartage be dented by r s ( =,, and =, and ). The sales tax pad per cell phne when transprted frm factres () t dstrbutn centres () s dented by s. The ttal publc expendture per cell phne fr I =, and and =, and s dented by t. Then r, r, r 4, r 6, r, r, r, r 8, r 4 s 5, s 9, s 9, s 4, s 6, s, s, s, s t 4, t, t, t, t 7, t 4, t, t 9, t 4 Let x be the number f cell phnes transprted frm the th factry t the th dstrbutn centre. AOR
K. GUPTA, S. R. ARORA 587 Snce factry at Haryana can prduce a mnmum f and a maxmum f cell phnes n a mnth, t can be frmulated mathematcally as x. Smlarly, x x. 4, 5 Snce the mnmum and maxmum mnthly requrement f cell phnes at Klkata are 5 and respectvely, t can be frmulated mathematcally as 5 x. Smlarly, x x 5, 5. The restrcted flw s P = 4. The bunds n the number f cell phnes transprted can be frmulated mathematcally as x, x, x 5, x 5, x 5, x, x, x, x 5 The abve data can be represented n the frm f Table as fllws. Intrduce a dummy surce and a dummy destnatn n Table wth and and 4 4 B A P 4 8, A B P8 4 4 r4 s4 t4,,, r4 s4 t4,,, r44 s44 t44 M. Table. Prblem (P). D D D A 5 9 4 9 O 4 O O 6 4 6 4 7 4 8 4 5 9 4 B Nte: values n the upper left crners are r s and values n upper rght crners are s s and values n lwer rght crners are t s fr I =,, and =,,. Als we have x4 7, x4, x4 4, x 5, x 5, x 5 4 4 4 Nw we fnd an ntal basc feasble slutn f prblem (P) whch s gven n Table belw. S 57 R 5 5.95 T 67 R 5, S 57, T 67 Snce ;, N and ;, N as shwn n Table, the slutn gven n Table s an ptmal slutn t prblem (P) and subsequently t S 57 (P). Therefre mn z R 5 5.94 T 67 Therefre, the cmpany shuld send cell phne frm Haryana t Klkata, unts frm Haryana t Chenna. The number f cell phnes t be shpped frm factry at Punab t Chenna and Mumba centres are 5 and 5 O O O O 4 Table. A basc feasble slutn f prblem (P). D D D D 4 5 9 4 9 4 7 6 4 6 5 7 5 4 8 4 7 9 4 M M 5 M v v v 4 u u u Nte: Entres f the frm a and b represent nn basc cells whch are at ther lwer and upper bunds respectvely. Entres n bld are basc cells. Table. Calculatn f and. NB O D O D O D O D O D O D O D O 4 D 7 4 7 4 r z 5 7 - s t z 7 7 4 z 5 7 7.4 6.5 8.5 5.4.879 7.976.8.967 AOR
588 K. GUPTA, S. R. ARORA respectvely. Factry at Chandgarh shuld send 7 unts t Klkata nly. The ttal cartage pad s 5, ttal sales tax pad s 57 and ttal publc expendture s 67. 6. Cnclusn Ths paper deals wth a lnear plus lnear fractnal transprtatn prblem where n the ttal transprtatn flw s restrcted t a knwn specfed level. A related transprtatn prblem s frmulated and t s shwn that t exted an ptmal slutn. An algrthm s presented and tested by a real lfe example f a manufacturng cmpany. 7. Acknwledgements We are thankful t the referees fr ther valuable cmments wth the help f whch we are able t present ur paper n such a nce frm. References [] A. Gupta, S. Khanna and M. C. Pur, A Paradx n Lnear Fractnal Transprtatn Prblems wth Mxed Cnstrants, Optmzatn, Vl. 7, N. 4, 99, pp. 75-87. http://dx.d.rg/.8/99884896 [] M. an and P. K. Saksena, Tme Mnmzng Transprtatn Prblem wth Fractnal Bttleneck Obectve Functn, Yugslav urnal f Operatns Research, Vl., N.,, pp. -6. [] F. Xe, Y. a and R. a, Duratn and Cst Optmzatn fr Transprtatn Prblem, Advances n Infrmatn Scences and Servce Scences, Vl. 4, N. 6,, pp. 9-. http://dx.d.rg/.456/ass.vl4.ssue6.6 [4] A. Khurana and S. R. Arra, The Sum f a Lnear and Lnear Fractnal Transprtatn Prblem wth Restrcted and Enhanced Flw, urnal f Interdscplnary Mathematcs, Vl. 9, N. 9, 6, pp. 7-8. http://dx.d.rg/.8/975.6.745 [5] K. Gupta and S. R. Arra, Paradx n a Fractnal Capactated Transprtatn Prblem, Internatnal urnal f Research n IT, Management and Engneerng, Vl., N.,, pp. 4-64. [6] S. Msra and C. Das, Sld Transprtatn Prblem wth Lwer and Upper Bunds n Rm Cndtns A Nte, New Zealand Operatnal Research, Vl. 9, N., 98, pp. 7-4. [7] S. an and N. Arya, An Inverse Capactated Transprtatn Prblem, IOSR urnal f Mathematcs, Vl. 5, N. 4,, pp. 4-7. http://dx.d.rg/.979/578-5447 [8] S. R. Arra and K. Gupta, Restrcted Flw n a Nn- Lnear Capactated Transprtatn Prblem wth Bunds n Rm Cndtns, Internatnal urnal f Management, IT and Engneerng, Vl., N. 5,, pp. 6-4. [9] A. Khurana, D. Thrwan and S. R. Arra, An Algrthm fr Slvng Fxed Charge B Crtern Indefnte Quadratc Transprtatn Prblem wth Restrcted Flw, Internatnal urnal f Optmzatn: Thery, Methds and Applcatns, Vl., N. 4, 9, pp. 67-8. AOR