THREE DIMENSIONAL FIXED CHARGE BI-CRITERION INDEFINITE QUADRATIC TRANSPORTATION PROBLEM *

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1 Yugoslav Joural of Oeratios Research (), Nuber, - THREE DIMENSIONAL FIXED CHARGE BI-CRITERION INDEFINITE QUADRATIC TRANSPORTATION PROBLEM S.R. ARORA Deartet of Matheatics, Has Raj College, Uiversity of Delhi Delhi-, Idia. srarora@yahoo.co Archaa KHURANA Deartet of Matheatics, Uiversity of Delhi, Delhi-, Idia archaa@rediffail.co, archaa@du.ac.i Received: October / Acceted: August Abstract: The three-diesioal fixed charge trasortatio roble is a extesio of the classical three-diesioal trasortatio roble i which a fixed cost is icurred for every origi. I the reset aer three-diesioal fixed charge bi-criterio idefiite quadratic trasortatio roble, givig the sae riority to cost as well as tie, is studied. A algorith to fid the efficiet cost-tie trade off airs i a three diesioal fixed charge bi-criterio idefiite quadratic trasortatio roble is develoed. The algorith is illustrated with the hel of a uerical exale. Keywords: Three diesioal quadratic trasortatio roble, cost-tie trade-off airs, fixed charge, bi-criterio idefiite quadratic trasortatio roble. INTRODUCTION I the classical trasortatio roble the cost of trasortatio is directly roortioal to the uber of uits of the coodity trasorted. But i real world situatios whe a coodity is trasorted, a fixed cost is icurred i the objective fuctio. The fixed cost ay rereset the cost of retig a vehicle, ladig fees i a airort, set u costs for achies i a aufacturig eviroet etc. Matheatics Subject Classificatio: Priary: B; Secodary: C

2 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge The fixed charge trasortatio roble was origially forulated by G.B.Datzig ad W. Hirsch [] i. The i K.G.Murty [] solved the fixed charge roble by rakig the extree oits. After that several rocedures for solvig fixed charge trasortatio robles were develoed. Soeties there ay exist eergecy situatios such as fire services, abulace services, olice services etc. whe the tie of trasortatio is of greater iortace tha the cost of trasortatio. Several ethods [, ] for iiizig the tie of trasortatio are also develoed. I Bhatia [] et.al. rovided the tie-cost trade-off airs i a liear trasortatio roble. Also i Basu et. al.[] develoed a algorith for the otiu tie-cost trade-off i a fixed charge liear trasortatio roble givig sae riority to cost ad tie. The trasortatio roble cosidered i the classical trasortatio roble is geerally a two-diesioal liear trasortatio roble. Haley [] i described the solutio of a liear ulti-idex trasortatio roble where there are three idices. The ethod for solutio reseted by Haley is a extesio of MODI ethod. I, Basu et al. [] rovided a algorith for fidig the otiu solutio of the solid fixed charge liear trasortatio roble. I this aer three diesioal fixed charge bi-criterio idefiite quadratic trasortatio roble, givig the sae riority to cost ad tie, is studied. A algorith to idetify the efficiet cost-tie trade-off airs for the roble is develoed.. PROBLEM FORMULATION Suose i =,,..., are the origis j =,,..., are the destiatios ad k =,,..., are the various tyes of coodities to be trasorted i a three diesioal trasortatio roble. Let x = the aout of kth tye of coodity trasorted for the ith origi to the jth destiatio c = the variable cost er uit aout of the kth tye of coodity trasorted fro the ith origi to the jth destiatio which is ideedet of the aout of the coodity trasorted, so log as x > d = the er uit dereciatio cost (wear ad tear or daaged cost) of the kth tye of coodity trasorted fro the ith origi to the jth destiatio, which is ideedet of the aout of coodity trasorted, so log as x >. A jk = the total quatity of kth tye of the coodity received by jth destiatio fro all the sources B ki = the total quatity of the kth tye of the coodity available at the ith origi to be sulied to all destiatios

3 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge E ij = the total quatity of all tyes of coodities to be sulied fro ith origi to the jth destiatio. The the three diesioal trasortatio roble is defied as i Z subject to c x i= j= k= = (P ) x = Ajk, =,,...,, =,,..., i= j k x = Bki, =,,...,, =,,..., j= k i x = Eij, =,,...,, =,,..., k = i j x, i =,,...,, j =,,...,, k =,,..., Here, there are origis, destiatios ad tyes of coodities to be trasorted. Also, Ajk = Bki, k,,..., j= i= Bki = Eij, i,,..., k= j= Eij = Ajk, j,,..., i= k= = (i) = (ii) jk ki ij j= k= k= i= i= j= = (ii) A = B = E (iv) (i) ilies kth tye of coodity received by all destiatios = kth tye of coodity sulied fro all origis. (ii) ilies differet tyes of coodities sulied by the ith source = aout of coodities received by all destiatios fro the ith source (iii) ilies aout of coodities sulied fro all sources to jth destiatio = differet tyes of coodities received by the jth destiatio. (iv) ilies aout of coodities received by all destiatios of differet tyes of coodities = aout of coodities sulied fro all origis to all destiatios = aout of differet tyes of coodities sulied fro all origis.

4 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Note: (i) to (iv) idicates that the trasortatio roble (P ) cosidered is a balaced trasortatio roble. Now let, F ik = the fixed cost associated with origi i ad the kth tye of coodity. We defie F ik accordig to the aout sulied as where δ F = F δ, i =,,...,, k =,,..., ik j= if x >, = if x =,, i =,,...,, j =,,...,, k =,,..., Now, cosider the three diesioal fixed charge bi-criterio idefiite quadratic trasortatio roble as subject to i c x d x + Fik, ax[ t / x > ] i i= j= k= i= j= k= i= k= j k (P ) x = Ajk, =,,...,, =,,..., i= j k x = Bki, =,,...,, =,,..., j= k i x = Eij, =,,...,, =,,..., k = i j x, i =,,...,, j =,,...,, k =,,..., Also, () Ajk = Bki, k =,,..., j= i= Bki = Eij, i =,,..., k= j= Eij = Ajk, j =,,..., i= k= A = B = E jk ki ij j= k= k= i= i= j=

5 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge I the roble (P ), we eed to iiize the trasortatio cost ad dereciatio cost siultaeously of the kth tye of the roduct to be trasorted fro the ith origi to jth destiatio. Also we eed to iiize the total cost (variable cost + fixed cost) ad the total tie of trasortatio. Therefore we have cosidered the objective fuctio of the for as i roble (P ).. THEORETICAL DEVELOPMENT: To solve the roble (P ) we searate it ito two robles (P' ) ad (P" ) as i Z = c x d x + Fik i= j= k= i= j= k= i= k= subject to () (P' ) i T = ax[ t / x > ] subject to () (P" ) i j k To obtai the set of efficiet cost-tie trade off airs, we first solve (P' ) ad read the tie with resect to the iiu cost Z where tie T is give by the roble (P" ). At the first iteratio, let Z be the iiu total cost of the roble (P' ) ad T be the otial tie of the roble (P" ) with resect to is coleted earlier tha T would cost ore tha Z. So Z, the ay schedule which ( Z, T ) is called the tiecost trade off air at the first iteratio. After odifyig the costs with resect to the tie obtaied, a ew otial cost is obtaied ad tie is read with resect to the ew otial cost. This rocedure is called re-otiizatio rocedure. Let after qth iteratio, the solutio be ifeasible. Thus we get the followig colete set of tie cost trade off airs, ( Z, T ), ( Z, T ), ( Z, T ),, ( Z, T ) q q where Z < Z < Z < < Z q ad T > T > T > >. T q The airs so defied are areto-otial solutios of the give roble. The we idetify the iiu cost Z ad iiu tie T q aog the above trade-off airs. The air ( Z, T q ) with iiu cost ad iiu tie is tered as the ideal solutio which ca ot be achieved i ractical situatios.

6 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Cosider a three diesioal quadratic trasortatio roble as i Z = c x d x i= j= k= i= j= k= subject to (P ) x = Ajk, =,,...,, =,,..., i= j k x = Bki, =,,...,, =,,..., j= k i x = Eij, =,,...,, =,,..., k = i j ad x, i =,,...,, j =,,...,, k =,,..., Theore. Let X = { x } be a basic feasible solutio of roble (P ) with basis atrix B. The it will be a otial basic feasible solutio if where R, cells (, i j, k) B =, cells (, i j, k) B R = θ ( z d )( z c ) Z ( z d ) Z ( z c ) ujk + vki + wij = z, cells (, i j, k) B u + v + = z, cells (, i j, k) B jk ki ij Also ujk + vki + wij = c, cells (, i j, k) B u + v + = d, cells (, i j, k) B () jk ki ij Z be the value of c x i j k at the curret basic feasible solutio corresodig to basis atrix B. Z be the value of d x at the curret basic feasible solutio corresodig i j k to basis atrix B. θ is the level at which a o-basic cell (, i j, k ) eters the basis relacig soe basic cell of B.

7 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Note: u, v, w, u, v, are deteried by usig equatios () ad takig jk ki ij jk ki ij + + of the u jk s or v ki s or w ij s ad u jk s or v ki s or ij s as zero. Proof: Let Z be the objective fuctio value of the roble (P ). Let Z = ZZ Let Ẑ be the value of the objective fuctio at the curret basic feasible solutio Xˆ = { x } corresodig to the basis B obtaied o eterig the cell (, i j, k ) ito the basis. The Z ˆ = [ Z ( )][ ( )] + θ c z Z + θ d z Now, ˆ = [ + θ ( )][ + θ ( )] = ZZ + Zθ ( d z ) + Zθ ( c z ) + θ ( c z )( d z ) ZZ = Zθ ( d z ) + Zθ ( c z ) + θ ( c z )( d z ) = θ[ Z( d z ) + Z( c z ) + θ ( c z )( d z )] Z Z Z c z Z d z Z Z This basic feasible solutio will give a iroved value of Z if Ẑ < Z. i.e., if ˆ Z Z < i.e., if θ[ Z( d z ) + Z( c z ) + θ ( c z ) < ( d z )] < sice θ [ Z ( d z ) + Z ( c z ) + θ ( c z )( d z )] < () Oe ca ove fro oe basic feasible solutio to aother basic feasible solutio o eterig the cell (, i j, k ) ito the basis for which coditio () is satisfied. It will be a otial basic feasible solutio if θ ( z d )( z c ) Z ( z d ) Z ( z c ) or R cells (, i j, k) B where R = θ ( z d )( z c ) Z( z d ) Z( z c ) Also, it ca easily be see that R = cells (, i j, k) B ALGORITHM: Ste : Fid the iitial basic feasible solutio of the roble (P' ). Ste : Calculate the fixed cost of the curret basic feasible solutio ad deote this by F (curret), where F (curret) = i= k= Ste : Calculate R cells (, i j, k) B, B is the curret basis. where R = θ ( z d )( z c ) Z( z d ) Z( z c ) F ik

8 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge ad ujk + vki + wij = z cells (, i j, k) B u + v + = z cells (, i j, k) B jk ki ij Ste : Fid A = R E where A is the chage i cost that occurs o itroducig a o-basic cell (, i j, k ) with value E ( (, i j, k) B) ito the basis. Ste : Fid F (Differece) = F (NB) F (curret) where F (NB) is the total fixed cost ivolved o itroducig the variable x with values ( E ) ( (, i j, k) B) ito the curret basis to for a ew basis. Ste : Calculate = F (Differece) + A ( (, i j, k) B) Ste : If all Let its iiu be the go to ste, otherwise fid i{ /, (,, ) } qr. The cell ( qr,, ) eters the basis. Go to ste. Ste : Let Z be the otial cost of (P' ) ad corresodig to Z. Ste : Fid T = ax{ t / x > } M if t T Ste : Defie c =, c if t < T < i j k B. X be the otial solutio of (P' ) where M is a sufficietly large ositive uber. Ste : Fid a basic feasible solutio of the roble (P' ) with resect to the ew variable costs c. Go to ste ad reeat the rocess. Ste : Let after the qth iteratio, the solutio is ifeasible. The idetify the colete set of efficiet cost tie trade off airs. subject to. NUMERICAL ILLUSTRATION Cosider a three diesioal fixed charge bi-criterio trasortatio roble. i c x d x + Fik, ax[ t / x > ] i i= j= k= i= j= k= i= k= j k (P) i= x x j= = A = B jk ki

9 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge x k = = E ij x where A = B = E jk ki ij j= k= k= i= i= j= The data of variable cost c ad tie t is give Table ad Table ' resectively. c d Table. j = j = j = B ki i = E = E = E = i = E = E = E = i = E = E = E = A jk

10 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge The fixed costs are F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = F =, F =, F = Table '. t i = i = i = j= j= j= Usig the North-West Corer Rule, we fid the iitial basic feasible solutio of roble (P) as give i Table.

11 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Table. j = j = j = F (Basic) c w = () w = - (-) w = - v d () (-) (-) - v (-) i= () () (-) = () () = - () = - () w = () w = w = () (-) () i= () () () = () = () = () u +v +w c (-) w = () w = - () w = u + v + w -d () i= () () () = () = (-) (-) = (-) (-) - - u u -

12 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Here, Z =, Z = Table. (i,j,k) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) A = R E No - No No = - = loo = = = - loo loo F (Diff.) - - = A + F (Diff.) - - Here, i{ / <, ( i, j, k) B} = i{, } = cell (,,) eters the basis. We fid the ew solutio. Reeat the rocess. The otial basic feasible solutio is give i Table.

13 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Table. j = j = j = F (Basic) w = () w = () w = v i= () (-) (-) - v () () () = () = - (-) () = - () - () w = - () w = () w = - (-) - i= () () () (-) - = (-) (-) = - () = () - () w = () w = () w = i= () () () () (-) (-) = = (-) = (-) u u

14 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Here, Z =, Z = Table. (i,j,k) (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) A = R E No No = No = loo = loo loo = F (Diff.) - - = A + F (Diff.) Sice, (, i j, k) B we sto. The otial value of Z = Z = + = ad the corresodig tie T =. The first cost-tie trade off air is (, ). Defie c M, t T = = c, t < T =. ad for the ew three diesioal quadratic trasortatio roble. O solvig this roble the ext trade off air is ( Z, T ) = (, ). Proceedig like this, we get the third cost-tie trade-off air as ( Z, T ) = (, ). After that the roble defied at tie T becoes ifeasible. Hece the tie-cost trade-off airs are (, ), (, ) ad (, ). Coclusio. For fidig efficiet cost-tie trade off airs i a three-diesioal idefiite quadratic trasortatio roble (A) First we iiize total cost (variable cost + fixed cost) ad the iiize tie with resect to iiu cost obtaied ad for the first cost-tie trade-off air. (B) Next after odifyig cost with resect to the tie obtaied i the last result we agai iiize cost, read the corresodig tie ad for the ext cost-tie trade-off air. (C) Therefore we kee o icreasig the cost ad readig the tie o each ste ad fid the efficiet cost-tie trade off airs. (B) is reeated util the roble becoes ifeasible. Reark.. A alterative aroach to solve roble (P ) is to first iiize the tie fuctio ad the read the corresodig cost (variable + fixed) fro the solutio which gives the iiu tie. Thereafter, we kee o icreasig the tie steadily ad readig the cost at each ste. We cotiue till the solutio becoes ifeasible.. Proble (P' ) caot be solved by the ethod develoed for liear trasortatio roble as the variable δ takes the value or. So we first fid the basic feasible solutio of the roble (P ) ad the calculate the corresodig fixed charge.

15 S.R. Arora, A. Khuraa / Three Diesioal Fixed Charge Ackowledgeets:. The authors are grateful to referees for their valuable coets ad suggestios.. The author Ms. Archaa Khuraa is also grateful to Coucil of Scietific ad Idustrial Research, New Delhi for the fiacial assistace. REFERENCES [] Ahuja, A., ad Arora, S.R., Multi-idex fixed charge bi-criterio trasortatio roble, Idia Joural of Pure ad Alied Matheatics, () () -. [] Basu, M., Pal, B.B., ad Kudu, A., A algorith for otiu tie-cost trade-off i fixed charge bi-criterio trasortatio roble, Otiizatio, () -. [] Basu, M., Pal, B.B., ad Kudu, A., A algorith for fidig the otiu solutio of solid fixed-charge trasortatio roble, () -. [] Bhatia, H.L., Swaroo, K., ad Puri, M.C., Tie-cost trade-off i a trasortatio roble, Osearch, () -. [] Bhatia, H.L., Swaroo, K., ad Puri, M.C., A rocedure for tie iiizatio trasortatio roble, Idia Joural of Pure ad Alied Matheatics, () -. [] Haley, K.B, The solid trasortatio roble, Oeratios Research, () -. [] Haley, K.B., The ulti-idex roble, Oeratios Research, () -. [] Haer, Peter, L., Tie-iiizig trasortatio robles, Naval Research Logistics Quarterly, () -. [] Hirsch, W.M., ad Datzig, G.B., Notes o liear rograig: Part XIX, The fixed charge roble, Rad Research Meoradu No., Sata Moica, Califoria,. [] Kuh, H., ad Bauol, W., "A aroxiate algorith for the fixed-charge trasortatio roble", Naval Research Logistics Quarterly, () -. [] Murty, K. G., Solvig the fixed-charge roble by rakig the extree oits, Oeratios Research, () -. [] Raakrisha, C.S, A ote o the tie iiizig trasortatio roble, Osearch, () -. [] Sadagoa, S., ad Ravidra, A., A vertex rakig algorith for the fixed-charge trasortatio roble, Joural of Otiizatio Theory ad Alicatios, () -. [] Sadrock, K., A sile algorith for solvig sall fixed-charge trasortatio robles, Joural of the Oeratioal Research Society, () -. [] Steiberg, D.I., The fixed-charge roble, Naval Research Logistics Quarterly, () -. [] Szwarc, W., Soe rearks o the tie trasortatio roble, Naval Research Logistics Quarterly, () -.