A Modified Vogel s Approximation Method for Obtaining a Good Primal Solution of Transportation Problems

Transcription

1 Annals of Pure and Appled Mathematcs Vol., No., 06, 6-7 ISSN: X (P), (onlne) Publshed on 5 January 06 Annals of A Modfed Vogel s Appromaton Method for Obtanng a Good Prmal of Transportaton Problems M.Wal Ullah, M.Alhaz Uddn * and Rjwana Kawser epartment of Busness Admnstraton, Northern Unversty Bangladesh, epartment of Mathematcs, Khulna Unversty of Engneerng and Technology, Khulna-90, Bangladesh *correspondng author, Emal: alhazuddn@yahoo.com Receved ecember 05; accepted ecember 05 Abstract. etermnng the effcent soluton for large scale of transportaton problems (TPs) s an mportant task n operatons research. Vogel s Appromaton Method (VAM) whch s one of the well-known transportaton methods n the lterature was nvestgated to obtan an ntal transportaton cost (ITC). In ths paper, Vogel s Appromaton Method (VAM) s modfed for obtanng more effcent soluton of a large scale of TPs. The most attractve feature of ths method s that t requres very smple arthmetcal and logcal calculatons and avods large number of teratons. The proposed method s easy to understand and wll be very helpful for those decson makers who are dealng wth logstcs and supply chan related ssues than the other estng methods. One can also easly mplement ths proposed method among the estng methods for obtanng a good prmal soluton of a large scale of TPs. Keywords: Transportaton problems, least cost method, VAM, Modfed Vogel s appromaton method AMS Mathematcs Subject Classfcaton (00): 90C08. Introducton Transportaton problem s a partcular class of lnear programmng, whch s assocated wth day-to-day actvtes n our real lfe. Transportaton models play an mportant role n logstcs and supply chans. It helps n solvng problems on dstrbuton and transportaton of resources from place to another. The problem bascally deals wth the determnaton of a cost plan for transportng a sngle commodty from varous sources to several destnatons. The am of such TPs s to mnmze the total transportaton cost (TTC) of shppng goods from one locaton to another so that the needs of each arrval area are met and every shppng locaton operates wthn ts capacty. TPs can be solved by usng general smple based nteger programmng methods, however t nvolves tmeconsumng computatons. We are gong to propose a specalzed algorthm wth less 6

2 M.Wal Ullah, M.Alhaz Uddn and Rjwana Kawser number of teraton for solvng TPs that are much more effcent than the smple algorthm. The basc steps for solvng TPs are: Step. etermnaton the ntal feasble soluton. Step. etermnaton optmal soluton usng the ntal soluton. The most common method used to determne effcent ntal solutons for solvng TPs (usng a modfed verson of the smple method) s Vogel s Appromaton Method (VAM). Ths method nvolves calculatng the penalty (dfference between the lowest cost and the second-lowest cost) for each row and column of the cost-matr, and then assgnng the mamum number of unts possble to the least-cost cell n the row or column wth the largest penalty. Vogel s appromaton method (VAM) gves appromate soluton whle MOI and Steppng Stone (SS) methods are consdered as standard technques for obtanng optmal soluton of TPs. Goyal [] has mproved Vogel s appromaton method (VAM) for the unbalanced transportaton problems. Ramakrshna [] has dscussed some mprovement to Goyal s Modfed VAM for unbalanced TPs. Moreover, Sultan [], Sultan and Goyal [] have studed ntal basc feasble solutons and resoluton of degeneracy n TPs. ew researchers have tred to gve ther alternate methods for overcomng major obstacles over MOI and SS methods. Adlakha and Kowalsk [5,6] have suggested an alternatve soluton algorthm for solvng certan TPs based on the theory of absolute pont. Shmshak et al. [7] have studed on modfcaton of Vogel's appromaton method through the use of heurstcs. Sharma et al.[8] have studed on uncapactated TP for obtanng a good prmal soluton. Balakrshnan [9] has dscussed Modfed Vogel s appromaton method for unbalanced TP. Ullah and Uddn [0] have developed an algorthmc approach to calculate the mnmum tme of shpment n TPs. Ullah et al. [] have developed a drect analytcal method for fndng an optmal soluton for transportaton problems. Ahmed et al. [] have developed an effectve modfcaton to solve transportaton problems for mnmzng cost. Ukl et al. [] have presented tme manufacturng technque used n probablstc contnuous economc order quantty revew model. Krca and Satr [] have developed a heurstc for obtanng an ntal soluton for the transportaton problems. Sharma and Sharma [5] have presented a new dual based procedure for the transportaton problems. Hakm [6] has developed an alternatve method to fnd ntal basc feasble soluton of a transportaton problem. In ths paper, a smple heurstc approach s proposed (MVAM) for obtanng good prmal soluton of wde range of TPs and the soluton obtaned by the proposed method s often very good n terms of mnmzng TTC than the other estng methods.. Modfed Vogel s Appromaton Method etaled processes of proposed Modfed Vogel s Appromaton Method (MVAM) are gven below: Step : Subtract the largest entry from each of the elements of every row of the transportaton table and place them on the left-top of the correspondng element. Step : Subtract the largest transportaton cost from each of the entres of every column of the transportaton table and wrte them on the left-bottom of the correspondng element. 6

3 A Modfed Vogel s Appromaton Method for Obtanng a Good Prmal of Transportaton Problems Step : orm a reduced matr whose elements are the summaton of left-top and leftbottom elements of step and step. Step : Calculate the dstrbuton by subtractng of the largest and net-tolargest element of each row and each column of the reduced matr and wrte them just after and below of the supply and demand amount respectvely. Step 5: Identfy the hghest dstrbuton ndcator, f there are two or more hghest, choose the hghest ndcator along whch the largest element s present. If there are two or more largest elements present, choose any one of them arbtrarly. Step 6: Allocate = mn( a, b ) on the left bottom of the largest element n the (, j) th cell of the reduced matr. Step 7: If the jth column and readjust a as and jth column. j j b as a < bj, leave the th row and readjust j a = a bj. If a = bj Step 8: Repeat step to step7 untl the rm requrement ehausted. b j = bj a. If a > bj, leave, then leave both the th row Step 9: Put all the allocatons of the postve allocated cells of the reduced matr to the orgnal transportaton table and calculate the TTC, z = m n = j = c j j where j s the total allocaton of the (, j) th cell and c j s the correspondng unt transportaton cost.. Eample Let us consder the followng TP to fnd out the mnmum TTC wth three sources and four destnatons. estnaton Supply 9 a 00 9 a 0 = = = a 70 emand b = 0 b = 0 b = 60 b = (Total) At frst we calculate the row dfferences and column dfferences whch are shown n the net table. 65

4 M.Wal Ullah, M.Alhaz Uddn and Rjwana Kawser estnaton Supply emand (Total) Now we can form the reduced matr as follows estnaton Supply emand (Total) Now we determne the dstrbuton for each row and each column by subtractng the largest and net-to-largest element. Among the dstrbuton, s the hghest one and c s the largest element. We allocate = = mn( a, b ) = mn(70,0) = 0 on the left bottom of c. estnaton Supply Row dstrbuton emand (Total) Column dstrbuton 8 b s ehausted and readjust a as a = a b = 70 0 = 50. Among the dstrbuton n the second step, s the hghest one and c = s the largest element. We allocate = mn( a, b ) = mn(00,80) = 80 on the left bottom of c. 66

5 A Modfed Vogel s Appromaton Method for Obtanng a Good Prmal of Transportaton Problems estnaton Supply Row dstrbuton emand (Total) Column 8 dstrbuton Here, b s ehausted and readjust a as a = a b = = 0. Among the dstrbuton n the thrd step, 0 s the hghest one and c = 0 s the largest element. We allocate = mn( a, b ) = mn(0,0) = 0 on the left bottom of c. estnaton 67 Supply Row dstrbuton emand (Total) Column 8 dstrbuton Here, a s ehausted and readjust b as b = b a = 0 0 = 0. Among the dstrbuton n the net step, 0 s the hghest one and c = 8 s the largest element. We allocate = mn( b, a) = mn(0,0) = 0 on the left bottom of c. estnaton Supply Row dstrbuton emand (Total) Column 8 dstrbuton 8

6 M.Wal Ullah, M.Alhaz Uddn and Rjwana Kawser Here, b s ehausted and readjust a as a = a b = 0 0 = 0. Havng no other alternatves, net two allocatons are automatcally 0 and 50 to the cell wth cost c and c respectvely. estnaton Supply Row dstrbuton emand (Total) Column 8 dstrbuton 8 Now all the rm requrements are satsfed and the ntal basc feasble soluton s = 0, = 80, = 0, = 0, = 0, = 50 whch we allocate to the orgnal transportaton table. estnaton Supply a = a = a = 70 emand b = 0 b = 0 b = 60 b = (Total) Therefore the TTC s, z m n = = c = j = c j j = = 00. Comparson of TTC obtaned n dfferent methods s gven n the followng table: Name of the Methods Prmal No. of Iteratons to Get an Optmal North-West Corner Method 80 5 Least Cost Method 090 Vogel s Appromaton Method 70 MVAM (Proposed) 00 68

7 A Modfed Vogel s Appromaton Method for Obtanng a Good Prmal of Transportaton Problems The optmal soluton obtaned by adoptng Modfed strbuton (MOI) Method s 00. It s seen that the value of the objectve functon obtaned by the proposed MVAM s same as the optmal value obtaned by MOI method. To apply and justfy the effcency of the proposed MVAM, we also have consdered the followng several supply chan (problems -5) TPs from dfferent sources to several destnatons. Problem : Problem : Problem : estnaton E G H Supply A B C emand estnaton E G H Supply A B C emand estnaton G H I Supply A 5 0 B 5 0 C E emand Problem : estnaton E G H I Supply A B C emand

8 Problem 5: M.Wal Ullah, M.Alhaz Uddn and Rjwana Kawser estnaton E G Supply A B C 5 0 emand Comparsons of ntal soluton of the above (-5) TPs obtaned by all procedures are gven n the followng table wth number of teratons: 5 Intal solutons obtaned by all procedures: No. of Problems Methods Optmal NWC M LCM VAM MVAM (Proposed) (MOI) Prmal No. of Iteratons to Get an Optmal 67 Prmal soluton No. of Iteratons to Get an Optmal 968 Prmal soluton No. of Iteratons to Get an Optmal 7 8 Prmal soluton No. of Iteratons to Get an Optmal Prmal soluton No. of Iteratons to Get an Optmal Concluson In ths artcle, North-West Corner Method (NWCM), Least Cost Method (LCM), Vogel s Appromaton Method (VAM) and proposed Modfed Vogel s Appromaton Method (MVAM) are used to fnd the ntal basc feasble soluton and are compared to optmal soluton obtaned by MOI method. It s seen that, the results obtaned by the proposed MVAM s almost same as optmal soluton obtaned by MOI method for several TPs and better than the soluton obtaned by the other estng methods (vz. NWCM, LCM 70

9 A Modfed Vogel s Appromaton Method for Obtanng a Good Prmal of Transportaton Problems and VAM). More effcent ntal soluton s obtaned by the proposed MVAM for a wde range of TPs wthn a few numbers of teratons. REERENCES. S.K. Goyal, Improvng VAM for the unbalanced transportaton problem, Journal of the Operatonal Research Socety, 5 (98) -.. C.S.Ramakrshna, An mprovement to Goyal s modfed VAM for the unbalanced transportaton problem, J. Operatonal Research Socety, 9 (988) A.Sultan, Heurstc for fndng an ntal b. f. s. n transportaton problems, Opsearch, 5 (988) A.Sultan and S.K.Goyal, Resoluton of degeneracy n transportaton problems, Journal of the Operatonal Research Socety, 9 (988) V.Adlakha and K.Kowalsk, An alternatve soluton algorthm for certan transportaton problems, Internatonal Journal of Mathematcal Educaton n Scence and Technology, 0 (999) V.Adlakha, K.Kowalsk and B.Lev, Solvng transportaton problem wth med constrants, Internatonal Journal of Management Scence and Engneerng Management, (006) G.Shmshak, J.A.Kaslk and T..Barclay, A modfcaton of Vogel's appromaton method through the use of heurstcs, Infor nformaton systems and operatonal research, 9 (98) R.R.K.Sharma and S.Prasad, Obtanng a good prmal soluton to the uncapactated transportaton problem, European J. Operatons Research, (00), N.Balakrshnan, Modfed Vogel s appromaton method for unbalanced transportaton problem, Appled Mathematcs Letters, () (990) M.W.Ullah and M.A.Uddn, An algorthmc approach to calculate the mnmum tme of shpment of a transportaton problem, European Journal of Industral and System Engneerng, 0 (0) M.W.Ullah, R.Kawser and M.A.Uddn, A drect analytcal method for fndng an optmal soluton for transportaton problems, J. Mechancs of Contnua and Mathematcal Scences, 9 () (05) 5-.. M.M.Ahmed, A.S.M.Tanvr, S.Sultana, S.Mahmud and M.S.Uddn, An effectve modfcaton to solve transportaton problems: a cost mnmzaton approach, Annals of Pure and Appled Mathematcs, 6() (0) S.I.Ukl, M. M. Ahmed, M.S.A.Jaglul, N.Sultana and M.S.Uddn, An analyss of just n tme manufacturng technque used n probablstc contnuous economc order quantty revew model, Annals of Pure and Appl. Mathematcs, 9() (05) O.Krca and A.Satr, A heurstc for obtanng an ntal soluton for the transportaton problem, Journal of Operatonal Research Socety, (9) (990) R.R.K.Sharma and K..Sharma, A new dual based procedure for the transportaton problem, European Journal of Operatons Research, (000), M.A.Hakm, An alternatve method to fnd ntal basc feasble soluton of a transportaton problem, Annals of Pure and Appled Mathematcs, () (0),