IJRSS Volume 2, Issue 2 ISSN:

Transcription

1 IJRSS Volume, Issue ISSN: An Algorthm To Fnd Optmum Cost Tme Trade Off Pars In A Fractonal Capactated Transportaton Problem Wth Restrcted Flow KAVITA GUPTA* S.R. ARORA** _ Abstract: Ths paper presents an algorthm to fnd optmum cost - tme trade off pars n a fractonal capactated transportaton problem wth bounds on total avalabltes at sources and total destnaton requrements. The obectve functon s a rato of two lnear functons consstng of varable costs and profts respectvely. Sometmes, stuatons arse where ether reserve stocks have to be kept at the supply ponts say, for emergences or there s a shortfall n the producton level. In such stuatons, the total flow needs to be curtaled. In ths paper, a specal class of transportaton problems s studed where n the total transportaton flow s restrcted to a known specfed level. A related transportaton problem s formulated and the effcent cost- tme trade off pars to the gven problem are shown to be dervable from ths related transportaton problem. Moreover, t s establshed that specal type of feasble soluton called corner feasble soluton of related transportaton problem bear one to one correspondence wth the feasble soluton of the gven restrcted flow problem. The optmal soluton to restrcted flow problem may be obtaned from the optmal soluton to related transportaton problem. Numercal llustraton s ncluded n support of theory. Keywords: Capactated transportaton problem, restrcted flow, fxed charge, related problem, corner feasble soluton, trade off pars. * Department of Mathematcs, Jagan Insttute of Management Studes, Insttutonal Area, Sector-5, Rohn, Delh, Inda. ** Ex-Prncpal, Hans Ra College, Unversty of Delh, Delh-7, Inda. 48

2 IJRSS Volume, Issue ISSN: Introducton: There s a wde scope of capactated transportaton problem wth bounds on rm condtons. It can be used extensvely n telecommuncaton networks, producton dstrbuton systems, ral and urban road system when there s a lmted capacty of resources such as vehcles, docks, equpment capacty etc. These are bounded varable transportaton problems. Many researchers lke Dahya and Verma [], Msra and Das [8] have contrbuted n ths feld. Another class of transportaton problem s a non lnear programmng problem where the obectve functon to be optmzed s a rato of two lnear functons. Optmzaton of a rato of crtera often descrbes some knd of an effcency measure for a system. Fractonal programs fnds ts applcaton n a varety of real world problems such as stock cuttng problem, resource allocaton problems, routng problem for shps and planes, cargo loadng problem, nventory problem and many other problems. Dahya and Verma [4] studed paradox n a non lnear capactated transportaton problem. Arora et.al [7] studed ndefnte quadratc transportaton problems. Khurana et. al.[5] studed restrcted and enhanced flow n the sum of a lnear and lnear fractonal transportaton problem n 6. Verma and Pur [] studed paradox n a lnear fractonal transportaton problem n 99. In 994, Basu et.al. [] developed an algorthm for the optmum cost- tme trade off pars n a fxed charge lnear transportaton problem gvng same prorty to cost as well as tme. In 4, Arora et.al.[]also studed tme cost trade off pars n a three dmensonal fxed charge ndefnte quadratc transportaton problem. Many researchers lke Arora [6], Thrwan [9] have studed restrcted flow problems. Sometmes, stuatons arse when reserve stocks are to be kept at sources for emergences. Ths gves rse to restrcted flow problem where the total flow s restrcted to a known specfed level. Ths motvated us to develop an algorthm to fnd the optmum cost - tme trade off pars n a fractonal capactated transportaton problem wth restrcted flow. Problem Formulatons: Consder a fractonal capactated transportaton problem gven by 49

3 IJRSS Volume, Issue ISSN: (P) : subect to cx mn,max t / x dx, a x A ; () b x B ; () l x u and ntegers, () x P mn A, B (4) I = {,, m} s the ndex set of m orgns. J = {,,, n} s the ndex set of n destnatons. x = number of unts transported from orgn to the destnaton. c = per unt plferage cost when shpment s sent from th orgn to the th destnaton. d = the varable proft per unt amount transported from th orgn to the th destnaton. l and u are the bounds on number of unts to be transported from th orgn to th destnaton. a and A are the bounds on the avalablty at the th orgn, I b and B are the bounds on the demand at the th destnaton, J t s the tme of transportng goods from th orgn to the th destnaton. It s assumed that dx for every feasble soluton X satsfyng (),(),() and (4) and all upper bounds u ; (,) I J are fnte. Sometmes, stuatons arse when one wshes to keep reserve stocks at the orgns for emergences, there by restrctng the total transportaton flow to a known specfed level, say P mn A, B J.Ths flow constrant n the problem (P) mples that a total 4

4 IJRSS Volume, Issue ISSN: A P of the source reserves has to be kept at the varous sources and a total B P of destnaton slacks s to be retaned at the varous destnatons. Therefore an extra destnaton to receve the source reserves and an extra source to fll up the destnaton slacks are ntroduced. In order to solve the problem (P), we separate t n to two problems (P) and (P) where (P): mnmze the cost functon cx dx subect to (), (), () and (4). (P): mnmze the tme functon max t / x subect to (), (), () and (4)., In order to solve the problem (P) we convert t n to related problem (P ) gven below. (P ): mn z subect to cy dy y A (5) y B (6) l y u, I J (7) ym, B b y, n A a ym, n A = A, A m+ = B -P, B = B, B n+ = A -P c = c,,, c m+, = c,n+ =,, c m+,n+ = M 4

5 IJRSS Volume, Issue ISSN: d = d,, d m+, = d,n+ =, ; d m+,n+ = M where M s a large postve number. I = {,, m,m+}, J = {,, n, n+} In order to solve the problem (P), we convert t to the related problem (P ) gven below. (P ): mn T max t / y and subect to y A y B l y u, I J ym, B b y, n A a ym, n A = A, A m+ = B P, B = B, B n+ = A P c = c,,, c m+, = c,n+ =,, c m+,n+ = M d = d,, d m+, = d,n+ =, d m+,n+ = M t = t, I J,t m+, = t,n+ =, t m+,n+ > max t / x, To obtan the set of effcent cost - tme trade off pars, we frst solve the problem (P ) and read the tme wth respect to the mnmum cost Z where tme T s gven by the problem (P ) At the frst teraton, let Z * be the mnmum total cost of the problem (P ). Fnd all alternate solutons.e. solutons havng the same value of Z = Z *. Let these solutons be X,X,..X n.. Correspondng to these solutons, fnd the tme T = * mn max t / x.then (Z *, T * ) s called the frst cost tme trade off par. Modfy X,X...Xn, 4

6 IJRSS Volume, Issue ISSN: the cost wth respect to the tme so obtaned.e. defne c = M f t T c f t < T * * and form the new problem (P ) and fnd ts optmal soluton and all feasble alternate solutons. Let the new value of Z be Z * and the correspondng tme s T *, then ( Z *, T *) s the second cost tme trade off par. Repeat ths process. Suppose that after q th teraton,the problem becomes nfeasble. Thus, we get the followng complete set of cost- tme trade off pars. ( Z *, T *), ( Z *, T *),( Z *,T *),.( Z q *, T q *) where Z * Z * Z *.. Z q * and T * > T * > T *..> T q *.The pars so obtaned are pareto optmal solutons of the gven problem. Then we dentfy the mnmum cost Z * and mnmum tme T q * among the above trade off pars. The par (Z *, T q *) wth mnmum cost and mnmum tme s termed as the deal par whch can not be acheved n practcal stuatons. Theoretcal Development: Theorem: A feasble soluton X = {x } I J of problem (P) wth obectve functon value D N wll be a local optmum basc feasble soluton ff the followng condtons holds. D (c z ) N (d z ) ; (, ) N D D (d z ) D (c z ) N (d z ) ; (, ) N D D (d z ) and f X s an optmal soluton of (P),then ; (, ) N and ; (, ) N where N c x, D dx, B denotes the set of cells (,) whch are basc and N and N denotes the set of non basc cells (,) whch are at ther lower bounds and upper bounds respectvely. u,u, v, v ;, are the dual varables such that u v c, (, ) B; u v d, (, ) B ; u v z, (, ) B ; u v z, (, ) B 4

7 IJRSS Volume, Issue ISSN: Proof: Let X = {x } I J be a basc feasble soluton of problem (P) wth equalty constrants. Let z be the correspondng value of obectve functon. Then z cx dx = N D (say) = (c u v )x (u v )x (d u v )x (u v )x (c u v )l (c u v )u (u v )x = (, ) N (, ) N (d u v )l (d u v )u (u v )x (, ) N (, ) N (c z )l (c z )u a u b v = (, ) N (, ) N (d z )l (d z )u au b v (, ) N (, ) N Let some non basc varable x N undergoes change by an amount rswhere rs s gven by u rs l rs mn x l for all basc cells (, ) wth a ( ) entry n loop u x for all basc cells (, ) wth a ( ) entry n loop Then new value of the obectve functon ẑ wll be gven by ẑ N (c z ) D (d z ) rs rs rs rs rs rs ẑ z N (c z ) N D (d z ) D rs rs rs rs rs rs = D (c z ) N (d z ) rs rs rs rs rs rs D D rs(drs z rs ) (say) 44

8 IJRSS Volume, Issue ISSN: Smlarly, when some non basc varable xpq N undergoes change by an amount then pq D (c z ) N (d z ) ẑ z (say) pq pq pq pq pq pq D D pq (dpq z pq) Hence X wll be local optmal soluton ff ; (, ) N and ; (, ) N.If X s a global optmal soluton of (P), then t s an optmal soluton and hence the result follows. Defnton: Corner feasble soluton : A basc feasble soluton {y }, to(p ) s called a corner feasble soluton (cfs) f y m+,n+ = Theorem. A non corner feasble soluton of (P ) cannot provde a basc feasble soluton to (P). Proof: Let {y } I xj be a non corner feasble soluton to (P ).Then y m+,n+ = (>) Thus y, n y, n y m, n = y, n = A P Therefore, y, n A (P ) (8) Now, for, y A A y A (9) (8) and (9) mples that y P Ths mples that total quantty transported from all the sources n I to all the destnatons n J s P + > P, a contradcton to the assumpton that total flow s P and hence {y } I xj cannot provde a feasble soluton to (P). 45

9 IJRSS Volume, Issue ISSN: Lemma : There s a one to-one correspondence between the feasble soluton to (P) and the corner feasble soluton to (P ). Proof: Let {x } I xj be a feasble soluton of (P).So {x } I xj wll satsfy () to (4). Defne {y } I xj by the followng transformaton y = x,, y,n+ = A - x, y m+, = B - x, J y m+,n+ = It can be shown that {y } I xj so defned s a cfs to (P ) Relaton () to () mples that l y u for all, y, n A a, ym, B b, y m+,n+, Also for I y y y, n x A x A A For = m+ ym, y y m, n (B x ) = B x = B P = A m+ y A ; Smlarly, t can be shown that y B ; 46

10 IJRSS Volume, Issue ISSN: Therefore, {y } I xj s a cfs to (P ). Conversely, let {y } I xj be a cfs to (P ).Defne x,, by the followng transformaton. x = y,, It mples that l x u,, Now for I, the source constrants n (P ) mples y A A y y, n A a y A (snce y,n+ A a, ) Hence, a x A, Smlarly, for, b x B For = m+, y A B P m, m ym, B P (because y m+,n+ = ) Now, for J the destnaton constrants n (P ) gve y ym, B Therefore, y ym, B y B ym, P x P Therefore {x } I xj s a feasble soluton to (P) Remark :If (P ) has a cfs,then snce c m+,n+=m and d m+,n+= M, t follows that non corner feasble soluton can not be an optmal soluton of (P ). 47

11 IJRSS Volume, Issue ISSN: Lemma : The value of the obectve functon of problem (P) at a feasble soluton {x } I x J s equal to the value of the obectve functon of (P ) at ts correspondng cfs {y } I xj and conversely. Proof: The value of the obectve functon of problem (P ) at a feasble soluton {y } I xj s z cy dy cx dx because c = c,, d = d,, x = y,, c = c = ;,,n+,n+ m,n m+, d = d = ;, m+, y = the value of the obectve functon of (P) at the correspondng feasble soluton {x } I xj The converse can be proved n a smlar way. Lemma : There s a one to-one correspondence between the optmal soluton to (P) and optmal soluton to the corner feasble soluton to (P ). Proof: Let {x } I J be an optmal soluton to (P) yeldng obectve functon value z and {y } I J be the correspondng cfs to (P ).Then by Lemma, the value yelded by {y } I J s z.. If possble, let {y } I J be not an optmal soluton to (P ). Therefore, there exsts a cfs {y }say,to(p ) wth the value z < z. Let (P).Then by lemma, {x }be the correspondng feasble soluton to 48

12 IJRSS Volume, Issue ISSN: z cx dx, a contradcton to the assumpton that {x } I J s an optmal soluton of (P).Smlarly, an optmal corner feasble soluton to (P ) wll gve an optmal soluton to (P). Theorem : Optmzng (P ) s equvalent to optmzng (P) provded (P) has a feasble soluton. Proof: As (P) has a feasble soluton, by lemma,there exsts a cfs to (P ).Thus by remark,an optmal soluton to (P ) wll be a cfs. Hence, by lemma,an optmal soluton to (P) can be obtaned. 4 Algorthm: Step : Gven a fractonal capactated transportaton problem (P),separate the problem (P) n to two problems (P) and (P).Form the related transportaton problems (P ) and (P ).Fnd a basc feasble soluton of problem (P ) wth respect to the varable cost only. Let B be ts correspondng bass. Step : Calculate, u v c (, ) B u v d (, ) B u v z (, ) N and N u v z (, ) N and N u,u, v, v,z,z ;, such that = level at whch a non basc cell (,) enters the bass replacng some basc cell of B. N and N denotes the set of non basc cells (,) whch are at ther lower bounds and upper bounds respectvely. u, v,u, v are the dual varables whch are determned by usng the above equatons and takng one of the u,s or v,s as zero. 49

13 IJRSS Volume, Issue ISSN: Step(a):Calculate N, D wheren cx, D dx Step(b):Calculate A and A where A (c z ); (, ) B and A (d z ); (, ) B. Step 4(a): Fnd Step 4(b): Fnd D (c z ) N (d z ); (, ) B ; (, ) N and ; (, ) N where ; (, ) N and D D A ; (, ) N where N and N denotes the set of non basc cells (,) whch D D A are at ther lower bounds and upper bounds respectvely. If ; (, ) N and ; (, ) N then the current soluton so obtaned s the optmal soluton to (P ) and subsequently to (P).Then go to step (5). Otherwse some (, ) Nfor whch or some (, ) Nfor whch wll enter the bass. Go to step. Step 5: Let Z be the optmal cost of (P ) yelded by the basc feasble soluton {y }. Fnd all alternate solutons to the problem (P ) wth the same value of the obectve functon. Let these solutons be X,X,..X n and T = mn max t / x.then the correspondng par (Z, X,X...Xn, T ) wll be the frst tme cost trade off par for the problem (P).To fnd the second cost- tme trade off par, go to step 6. Step6: Defne c = M f t T c f t < T where M s a suffcently large postve number. Form the correspondng capactated fxed charge quadratc transportaton problem wth varable cost c.repeat the above process tll the problem becomes nfeasble. The complete set of cost- tme trade off pars of (P) at the end of q th teraton s gven by (Z, T ),(Z,T ).(Z q, T q ) where Z Z.. Z q and T >T > > T q. 4

14 IJRSS Volume, Issue ISSN: Remark : The par (Z, T q ) wth mnmum cost and mnmum tme s the deal par whch can not be acheved n practce except n some trval case. Convergence of the algorthm: The algorthm wll converge after a fnte number of steps because we are movng from one extreme pont to another extreme pont and the problem becomes nfeasble after a fnte number of steps. 5 Numercal Illustraton: Consder a x fractonal capactated transportaton problem wth restrcted flow.table gves the values of c, d, A,B for =,, and =,, Table : cost matrx of problem (P) D D D A O O O B Note: values n the upper left corners are c,s and values n lower left corners are d,s for =,,.and =,,. Also, x, x 4, x 5, 5 x, 4

15 IJRSS Volume, Issue ISSN: x, 5 x x, x, x 5, x 5, x 5, x, x, x, x 5 Table gves the values of t, s for =,, and =,, Table : Tme matrx of problem (P) D D D O 5 8 O 4 O 9 Let the restrcted flow be P = 4 where P = 4 < mn A, B 8 Introduce a dummy orgn and a dummy destnaton n Table wth c 4 = = d 4 for all =,, and c 4 = = d 4 for all =,,. c 44 =d 44 =M where M s a large postve number. Also we have x 4 7, x 4, x 4 4, x 4 5, x 4 5, x 4 5 and F 4 = for =,,,4 In ths way, we form the problem (P ).Smlarly on ntroducng a dummy orgn and a dummy destnaton n Table wth t 4 = for =,,and t 4 = for =,,, t 44 > max t / x,,we form problem (P ). Also, B 4 = A P=-4 = 8 and A 4 = B P= 8-4 = 4 Now we fnd an ntal basc feasble soluton of problem (P ) whch s gven n table below. Table : A basc feasble soluton of problem (P ) D D D D 4 O O u 4 u

16 IJRSS Volume, Issue ISSN: O 4 O M M v v Note: entres of the form a and b represent non basc cells whch are at ther lower and upper bounds respectvely. Entres n bold are basc cells. N = 86, D = Table 4: Calculaton of and NB O D O D O D O D O D O D 4 O D O 4 D c z d z A A Snce ; (, ) N and ; (, ) N, the soluton gven n table s an optmal soluton to problem (P ).Therefore Z =.87 and correspondng tme s T = 5. Hence the frst cost tme trade off par s (.87,5). Defne c = (.87,4) M f t T 5 c f t < T 5 and solvng the resultng problem, the next trade off par s 4

17 IJRSS Volume, Issue ISSN: Defne c = M f t T 4 c f t < T 4 A basc feasble soluton to the new cost problem s gven n table 5 below. Table 5: A basc feasble soluton to the new cost problem O M M O 4 6 O O 4 D D D D M M M M v - v -4 u u N = 94, D = 4, Z=.496 Table 6: Calculaton of and NB O D O D O D O D 4 O D O D 5 9 c z d A A z

18 IJRSS Volume, Issue ISSN: Snce ; (, ) N and ; (, ) N, the soluton gven n table 5 s an optmal soluton to problem (P ).Therefore Z =.496 and correspondng tme s T =. Hence the thrd tme cost trade off par s (.496,). Defne c = M f t T c f t < T and solvng the resultng problem, the next trade off par s (.496,) Defne c 4 M f t c f t and on solvng, the problem becomes nfeasble. Hence the cost tme trade off pars are (.87,5), (.87,4),(.496, ), (.496, ). Concluson: In order to solve a capactated fxed charge b-crteron fractonal transportaton problem wth restrcted flow, gven problem s separated n to two problems. A related transportaton problem s formulated and the effcent cost- tme trade off pars to the gven problem are shown to be dervable from ths related transportaton problem. After calculatng cost, correspondng tme s read. Ths s the frst tme cost trade off par. Proceedng lke ths, we get the varous trade off pars. References: 45

19 IJRSS Volume, Issue ISSN: Arora, S.R., Khurana, A., Three dmensonal fxed charge b crteron ndefnte quadratc transportaton problem, Yugoslav Journal of Operatons Research, 4 () (4) 8-97 Basu, M, Pal, B.B and Kundu, A., An algorthm for the optmum tme cost trade off n a fxed charge b-crteron transportaton problem, Optmzaton,, (994),5-68 Dahya, K., Verma,V., Capactated transportaton problem wth bounds on rm condtons, Europeon Journal of Operatonal Research, 78(7)78-7 Dahya, K., Verma,V., Paradox n non lnear capactated transportaton problem, Yugoslav Journal of Operatons Research, 69()(6)89- Khurana, A., Arora, S.R., The sum of a lnear and lnear fractonal transportaton problem wth restrcted and enhanced flow, Journal of Interdscplnary Mathematcs, 9(6) 7-8 Khurana, A., Thrwan, D. and Arora, S.R., An algorthm for solvng fxed charge b crteron ndefnte quadratc transportaton problem wth restrcted flow, Internatonal Journal of Optmzaton : Theory, Methods and Applcatons,(9)67-8 Khurana, A., Arora, S.R., Fxed charge b crteron ndefnte quadratc transportaton problem wth enhanced flow, Revsta Investgacon operaconal, ()-45 Msra, S., Das, C., Sold transportaton problem wth lower and upper bounds on rm condtons a note, New Zealand Operatonal Research, 9()(98)7-4 Thrwan, D. and Arora, S.R., and Khanna, S., An algorthm for solvng fxed charge b crteron transportaton problem wth restrcted flow, Optmzaton, 4, (997)9-6 Verma,V.and Pur, M.C., On a paradox n lnear fractonal transportaton problem n S.Kumar(ed.), Recent Developments n Australan Socety of Operatonal Research, Gordan and Breach Scence Publshers, (99)