Fourier Method for Solving Transportation. Problems with Mixed Constraints

Transcription

1 It. J. Ctemp. Math. Scieces, Vl. 5, 200,. 28, Furier Methd fr Slvig Trasprtati Prblems with Mixed Cstraits P. Padia ad G. Nataraja Departmet f Mathematics, Schl f Advaced Scieces V I T Uiversity, Vellre , Tamil Nadu, Idia padia6@rediffmail.cm Abstract A ew methd called, Furier trasprtati algrithm based Mdified Furier Elimiati methd is prpsed fr fidig a ptimal sluti f trasprtati prblems with mixed cstraits. This methd is very easy t uderstad ad apply which ca serve maagers by prvidig the ptimal sluti t a variety f distributi prblems with mixed cstraits. The prpsed methd is illustrated with a umerical example. Mathematics Subject Classificatis: 90C08, 90C90 Keywrds: Trasprtati prblem, Mixed cstraits, Optimal sluti, Furier trasprtati methd. Itrducti The trasprtati prblem is a special class f liear prgrammig prblem, which deals with shippig cmmdities frm surces t destiatis. The bjective f the trasprtati prblem is t determie the shippig schedule that miimize that ttal shippig cst while satisfyig supply ad demad limits. The trasprtati prblem has a applicati i idustry, cmmuicati etwrk, plaig, schedulig, trasprtati ad alltmet etc. I literature, a gd amut f research has available t btai a ptimal sluti fr trasprtati prblems with equality cstraits [2,4,7,2]. I real life, hwever, mst prblems have mixed cstraits accmmdatig may applicatis that g beyd trasprtati related prblems t iclude jb schedulig, prducti ivetry, prducti distributi, allcati prblems, ad ivestmet aalysis. The trasprtati prblems with mixed cstraits are t addressed i the literature because f the rigr required t slve these prblems ptimally. A literature search revealed systematic methd fr fidig a ptimal sluti f trasprtati prblems with mixed cstraits. Gupta et al.[3] ad Arsham [] btaied the mre-fr-less sluti fr the

2 386 P. Padia ad G. Nataraja trasprtati prblem with mixed cstraits by relaxig the cstraits ad by itrducig ew slack variables. While yieldig the best mre-fr-less sluti, their methd is tedius sice it itrduces mre variables ad requires slvig sets f cmplex equatis. Perturbed methd was used fr slvig the trasprtati prblems with cstraits i [5,8,9]. Williams [] develped a methd called the Furier elimiati methd fr slvig liear prgrammig prblems based the Furier elimiati techique which geerates mre additial ad redudat cstraits. Kaiappa ad Thagavel [6] mdified the methd f Furier elimiati methd develped by Williams [] fr slvig liear prgrammig prblems by chsig a variable fr elimiati, which geerates miimum umber f cstraits i each step cmpared t that f the methd f Furier elimiati methd. Recetly, Veea et al.[0] have prpsed a heuristic methd fr slvig trasprtati prblems with mixed cstraits which is based the thery f shadw price. The sluti btaied by the heuristic methd itrduced by Veea et al.[0] is a iitial sluti f the trasprtati prblems with mixed cstraits. I this paper, we develp a ew methd called Furier trasprtati methd fr fidig a ptimal sluti f trasprtati prblems with mixed cstraits based the Mdified Furier s Elimiati methd [6] i sigle stage. This methd is very easy t uderstad ad apply which ca serve maagers by prvidig the ptimal sluti t a variety f distributi prblems with mixed cstraits. The prpsed methd is illustrated with a umerical example. 2 Trasprtati prblems with mixed cstraits Csider the mathematical mdel fr a TP with mixed cstraits. (P) Miimize z = cijxij subject t m xij ai j= xij ai j= x ij = ai j= m xij bj j=, i Q (), i T (2), i S (3), j U (4)

3 Furier methd fr slvig trasprtati prblems 387 m xij bj m x ij = bj, j V (5), j W (6) xij 0, i =,2,..., m ad j =,2,..., ad itegers (7) where Q, T ad S are pairwise disjit subsets f {,2,3,..., } such that Q T S = {,2,3,..., }; U, V ad W are pairwise disjit subsets f{,2,3,..., m } such that U V W = {,2,3,..., m }; c is the cst f shippig e uit frm supply pit i t the demad pit j ; ij a i is the supply at supply pit i ; b is the demad at demad pit j ad j x ij is the umber f uits shipped frm supply pit i t demad pit j. Remark : If Q, T, U ad V are empty, the the prblem (P) becmes the TP with equality cstraits. The trasprtati prblem with mixed cstraits (P) ca be represeted by a matrix f rder m x called the cst matrix. Fr m = 3 ad = 3, the structure f cst matrix is 2 3 Supply l h k = e 2 p q r f 3 x y z g Demad = a b c (XP) The maximizati prblem crrespdig t the prblem (P) is give belw. m Maximize w = cijxij j= subject t () t (7) are satisfied where w = z. The equivalet prblem t the prblem (XP) is give belw. (EP) Maximize w m subject t c x + w 0 j= ad als, () t (7) are satisfied. ij ij

4 388 P. Padia ad G. Nataraja Therem : If { x ij : i =,2,..., m ad j =,2,..., } is a ptimal sluti fr the prblem (EP), the { x ij : i =,2,..., m ad j =,2,..., } is als a ptimal sluti f (P) ad the miimum value f m m z, z = cijxij j= Prf. Sice (EP) is a equivalet prblem f (XP), the { x ij : i =,2,..., m ad j =,2,..., } is a ptimal sluti fr the prblem (XP). Sice (XP) is the maximizati type prblem crrespdig t the prblem (P), the { x ij : i =,2,..., m ad j =,2,..., } is a ptimal sluti fr the prblem (P) ad the miimum value f Hece the therem. m m z, z = cijxij j= Dmiace prperty: Csider the fllwig system f iequalities a x + a x + a x ) d x b2 x2 + b3x3 d2 b.. ( ( 2 + ) d If 2 d d, 2 d d ad 2 d where each pair d 2 ad d b a b2 a2 b3 a3, b ad a, b 2 ad a2 ad b 3 ad a3 is f same sig r zer, we say that the iequality ( ) is dmiated by the iequality ( 2 ). Als, iequality ( ) is called the redudat cstrait f the system. Therem 2: Csider the system a x + a2 x a x b a2 x + a22x a2 x b2 M. (I) am x + am2x am x bm x, x2,..., 0. x x j = 0 If aij 0, fr all i, the. Prf: Nw, elimiatig x j frm the give system usig Furier elimiati methd, we have the fllwig system a x a2x a x + a x a x b + j j j+ j+ a2 x + a22x2 + + a2 j x j + a2 j+ x j b a2x M. (II) amj x j + amj+ x j amx bm am x + am2x2 + x, x2,..., x j, x j+,..., x 0. 2

5 Furier methd fr slvig trasprtati prblems 389 Frm the system (I) ad the system (II), we bserve that there is rle f the variable x j, that is, x j is a iactive variable. Therefre, x j = 0. Hece the therem. Fr easy t uderstad ad t cmpute quickly, we use a ew type f cmputati table called Furier elimiati table. The Furier elimiati table fr a liear system AX B is a rectagular arragemet which ctais variables X ad its cefficiets, the values f B ad equati umbers. The Furier elimiati table fr the fllwig liear system f iequalities a x + a x + a x ) b is give belw : d x b2 x2 + b3 x3 d2 ( 3 ( 4 + ) x x 2 x 3 B Equati umber a a 2 a 3 d ( 3 ) b b 2 b 3 d 2 ( 4 ) 3 Furier trasprtati methd Nw, we prpse a ew methd called Furier trasprtati methd fr fidig a ptimal sluti f a trasprtati prblem with mixed cstraits. The Furier trasprtati methd prceeds as fllws. Step : Write the give trasprtati prblem with mixed cstraits i the frm f a pure iteger liear prgrammig prblem. Step 2: Cvert the pure iteger liear prgrammig prblem btaied frm the Step.it a maximizati prblem. Step 3: Write the maximizati prblem btaied frm the Step 2. havig ly e type f the iequality by elimiatig e variable i a equality cstrait i the give prblem. Step 4: Write the equivalet pure iteger liear prgrammig prblem t the mdified maximizati prblem btaied frm the Step 3. ad the, cstruct the Furier elimiati table fr the equivalet prblem.. Step 5: Select ad remve a variable frm the Furier elimiati table by the Mdified Furier s elimiati methd [6]. Step 6: Frm a reduced Furier elimiati table frm the Furier Elimiati table after deletig the true statemet(s) ad redudat cstraits (if ay) ad als, usig the Therem 2.

6 390 P. Padia ad G. Nataraja Step 7: Repeat the Step 5 t the Step 6 util all variables f except the bjective fucti variable w. x ij ' s are elimiated Step 8: Fid the least upper bud f all maximum pssible values f w. Say d. The ptimum sluti w is d. The values f all backward substituti methd ad basic algebraic methd. Say ad j =,2,,. x ij ' s are cmputed usig x ij, i =,2,,m Step 9: The ptimal sluti fr the trasprtati prblem with mixed cstraits is x ij = xij, i =,2,,m ad j =,2,, ad the miimum value f z, z = d. 4 Numerical Examples The prpsed methd is illustrated by the fllwig examples. Example. Csider the fllwig trasprtati prblem with mixed cstraits. 2 3 Supply 0 7 = Demad = Nw, the pure iteger liear prgrammig prblem f the give prblem is give belw. (P) Miimize z = 0x + x + 7x + 5x + 7x + x + 8x + 9x + x x2 + x3 = 2 + x22 + x x x2 + x3 = x22 + x x23 5, x2, x3, x2, x22, x23, x3, x32, x33 0 subject t x 5 (8) x (9) x (0) x () x (2) x (3) x ad itegers. (4) Nw, the maximizati prblem crrespdig t (P) is give belw. (XP) Maximize w = 0x x2 7x3 5x2 7x22 x23 8x3 9x subject t (8) t (4) are satisfied where w = z Nw, usig ( 8) ad (), (XP) ca be writte as fllws. (XP) Maximize w = 45 4x 6x3 7x22 x23 3x3 9x32 2x33 subject t x + x3 5 ; x 3 + x32 9 ; x 3 + x23 5 ; x + x 8 ; x + x x x 5; x + x x x 2 ; x 33

7 Furier methd fr slvig trasprtati prblems 39 x x, x, x, x, x ad x 0 ad itegers., Nw, the equivalet prblem t the prblem (XP) is give belw. (EP) Maximize w subject t 4x + 6x3 + 7x22 + x23 + 3x3 + 9x32 + 2x33 + w 45; x 3 + x32 9 ; x 3 + x23 5 ; x + x3 x22 x32 5; x + x3 8 ; x + x3 5 ; x + x3 x22 x23 2 ; x x, x, x, x, x ad x 0 ad itegers., By the Therem 2., we have x = 0, x 3 = 0, x 3 = 0, x 23 = 0 ad x 33 = 0 ad therefre, prblem (EP) reduces t (EP) Maximize w subject t 7x x 32 + w 45; x 32 9; x 22 x32 5 ; x 22 ad x 32 0 ad itegers. Nw, we frm the startig Furier elimiati table fr the prblem (EP). x 22 x 32 w B Equati Number (5) (6) (7) (8) (9) Nw, elimiatig x 22 frm the table usig Mdified Furier elimiati methd, we have the fllwig Furier elimiati table after remvig the true statemets ad the redudat cstraits. x 32 w B Equati Number 0 9 (6) 2-80 (20) [(5)+7(7)] (9) By the Therem 2., x 32 = 0. Therefre, we remve the clum f x 32 frm the abve table ad als, remve the true statemets ad the redudat f the result table. The, we have the fllwig table. w B Equati Number -80 (20) The least upper bud f the maximum pssible values f w is Therefre, the miimum trasprtati cst, z = 80. Nw, usig backward substituti methd ad basic algebraic methd, we have x = 0, x 2 = 5, x 3 = 0, x 2 = 8, x 22 = 5, x 23 = 0, x 3 = 0, x 32 = 0, x 33 = 0.

8 392 P. Padia ad G. Nataraja Therefre, the ptimal sluti is x = 0, x 2 = 5, x 3 = 0, x 2 = 8, x 22 = 5, x 23 = 0, x 3 = 0, x 32 = 0, x 33 = 0 ad z = 80. Example 2. Csider the fllwig balaced trasprtati prblem. Nw, the purely iteger liear prgrammig prblem fr the abve balaced trasprtati prblem is give belw. (P) Miimize z = 4x + x2 + 7x3 + 3x2 + 2x22 + 2x23 + 5x3 + 3x32 + 4x33 subject t x + x2 + x3 = 80 (2) x 2 + x22 + x23 = 20 (22) x 3 + x32 = 35 (23) x + x2 + x3 = 60 (24) x 2 + x22 + x32 = 40 (25) x 3 + x23 = 35 (26) x x, x, x, x, x, x, x, x 0 ad itegers. (27), Nw, the maximizati prblem crrespdig t (P) is give belw. (EP) Maximize w = 4x x2 7x3 3x2 2x22 2x23 5x3 3x32 4x33 subject t (2) t (27) are satisfied where w = z. Nw, usig (2) t (27), (EP) ca be writte as fllws. (EP) Maximize w = 540 2x x23 x32 + 4x33 subject t x 22 + x23 20 ; x ; x 22 + x32 40 ; x + x 35 ad x x, x, x 0 ad itegers , The equivalet prblem t the prblem (EP) is give belw. (EP2) Maximize w subject t 2x 22 4x23 + x32 4x33 + w 540 ; x 22 + x23 20 ; x ; x + x 40 ; x + x 35 ad x x, x, x 0 ad itegers Supply Demad , By the Therem 2., we have, x 22 = 0, x 32 = 0 ad therefre, the prblem (EP2) reduces t (EP3) Maximize w

9 Furier methd fr slvig trasprtati prblems 393 subject t x 4x + w 540 ; x 20 ; x 35; , x33 33 x ad x 0 ad itegers. Nw, we frm the startig Furier elimiati table fr the prblem (EP3). x 23 x 33 w B Equati Number (28) (29) (30) 0 35 (3) (32) (33) Nw, elimiatig x 23 frm the first iterati table usig Mdified Furier elimiati methd, we have the fllwig Furier elimiati table after remvig the true statemets ad the redudat cstraits. x 33 w B Equati Number (34) [4(3)+(28)] (35) [4(27)+(28)] 0 35 (30) (33) Nw, elimiatig x 33 frm the first iterati table usig Mdified Furier elimiati methd, we have the fllwig Furier elimiati table after remvig the true statemets ad the redudat cstraits. w B Equati Number -400 (34) The least upper bud f the maximum pssible value f w = 400. Therefre, ttal miimum trasprtati cst z = 400. Nw, usig the backward substituti methd ad the basic algebraic methd, we btai the values f variables: x = 40 ; x 2 = 40 ; x 3 = 0; x 2 = 0 ; x 22 = 0, x 23 = 20 ; x 3 = 20 ; x 32 = 0 ad x 33 = 5. Therefre, the ptimal sluti is x = 40, x 2 = 40, x 3 = 0, x 2 = 0, x 22 = 0, x 23 = 20, x 3 = 20, x 32 = 0 ad x 33 = 5 ad z = 400.

10 394 P. Padia ad G. Nataraja 5 Cclusi We have prvided a methd called the Furier trasprtati methd t fid a ptimal sluti fr trasprtati prblems with mixed cstraits. This methd is very easy t uderstad ad apply. S, the Furier trasprtati methd ca serve maagers by prvidig the ptimal sluti t a variety f distributi prblems with mixed cstraits. I ear future, we exted this methd t geeralized trasprtati prblems ad multibjective trasprtati prblems. REFERENCES [] H. Arsham, Pstptimality aalyses f the trasprtati prblem, Jural f the Operatial Research Sciety, 43 (992), [2] M.S.Bazaraa, J.J.Jaruis ad H.D. Sherali, Liear prgrammig ad Netwrk Flws, Jh Wiley ad Ss, New Yrk, 997. [3] A. Gupta, S. Khaa ad M. Puri, Paradxical situatis i trasprtati prblems, Cahiers du Cetre d Etudes de Recherche Operatiell, 34 (992), [4] S.K.Gupta, Liear prgrammig ad etwrk mdel, Affiliated East-West Press (PVT), New Delhi, 989. [5] Y. Itratr ad B. Lev, Methds fr idetificati f vaishig variables i trasprtati prblems ad its pssible applicatis, Cmputers ad Operatis Research, 3 (976), [6] P.Kaiappa ad K.Thagavel, Mdified Furier s methd fr slvig liear prgrammig prblems, OPSEARCH, 35(998), [7] H.S. Kasaa ad K.D. Kumar, Itrductry Operatis Research Thery ad Applicatis, Spriger Iteratial Editi, New Delhi, [8] B. Lev, A iterative algrithm fr tridiagal trasprtati prblems ad its geeralizati, Jural f Operatis Research Sciety f America, 20(972), [9] B. Lev ad Y. Itratr, Applicatis f vaishig variables methds t special structured trasprtati prblems, Cmputers ad Operatis Research, 4(977), 2 26.

11 Furier methd fr slvig trasprtati prblems 395 [0] Veea Adlakha, Krzysztf Kwalski ad Bejami Lev, Slvig trasprtati prblems with mixed cstraits, Iteratial Jural f Maagemet Sciece ad Egieerig Maagemet, (2006), [] H.P.Williams, Furier s methd f liear Prgrammig ad its Dual, America Mathematical Mthly, 93 (986), [2] W.L.Wist, Itrducti t Mathematical Prgrammig Applicatis ad Algrithms, Duxbury Press, Califria, 99. Received: Jauary, 200