An Effective Modification to Solve Transportation Problems: A Cost Minimization Approach

Transcription

1 Annals of Pure and Appled Mathematcs Vol. 6, No. 2, 204, ISSN: X (P), (onlne) Publshed on 4 August Annals of An Effectve Modfcaton to Solve Transportaton Problems: A Cost Mnmzaton Approach Mollah Mesbahuddn Ahmed, Abu Sadat Muhammad Tanvr 2, Shrn Sultana 3 Sultan Mahmud 4 and Md. Sharf Uddn 5 Department of Scence & Humantes, Mltary Insttute of Scence and Technology (MIST), Dhaka, Bangladesh. Emal: mesbah_972@yahoo.com 2 Department of Cvl Engneerng, Mltary Insttute of Scence and Technology (MIST) Dhaka, Bangladesh. Emal: sadafryan666@gmal.com 3 Department of Natural Scence, Daffodl Internatonal Unversty, Dhaka. Bangladesh Emal: shrn.ns@daffodlvarsty.edu.bd 4 Department of Mathematcs, Jahangrnagar Unversty. Savar, Dhaka, Bangladesh Emal: Sultanbnm9@gmal.com 5 Department of Mathematcs, Jahangrnagar Unversty. Savar, Dhaka, Bangladesh Emal: msharfu@yahoo.com Receved 23 July 204; accepted 3 August 204 Abstract. It s well-known that Lnear Programmng Problem (LPP) s one of the most potental mathematcal tools for effcent allocaton of operatonal resources. Many problems n real stuaton can be formulated as LPP. When a stuaton can be entrely modeled as a network, very effcent algorthms est for the soluton of the optmzaton problem whch s many tmes more effcent than the soluton methods of LPP. Transportaton problems (TP), as s known, are a basc network problem whch can be formulated as a LPP. The man obectve of TP s to mnmze the transportaton cost of dstrbutng a product from a number of sources (e.g. factores) to a number of destnatons (e.g. ware houses). It s to be mentoned that Balanced TP and Unbalanced TP are the types of TP. If the sum of the supples of all the sources s equal to the sum of the demands of all the destnatons, the problem s termed as a balanced transportaton problem. Agan, f the sum of the supples of all the sources s not equal to the sum of the demand of all the destnatons, the problem s termed as unbalanced transportaton problem. Here we have developed a new method of fndng an Intal Basc Feasble Soluton (IBFS) for both the Balanced TP and Unbalanced TP. Keywords: LPP, TP, Transportaton Cost, IBFS AMS Mathematcal Subect Classfcaton (200): 05C78. Introducton The frst algorthm n solvng the TP was developed by G. B. Dantzg [6] and referred as North West Corner Method (NWCM) by Charnes and Cooper. Ths s the method of 99

2 M.M.Ahmed, A.S.Muhammad Tanvr, S.Sultana, S.Mahmud and Md. S.Uddn fndng an IBFS of TP whch consder the north-west-corner cost cell at every stage of allocaton. Then the Least Cost Method (LCM) [2,9] conssts n allocatng as much as possble n the lowest cost cell of the Transportaton Table (TT) n makng allocaton n every stage. Vogel s Appromaton Method (VAM) [7,0,,4] and Etremum Dfference Method (EDM) [8] provdes comparatvely better Intal Basc Feasble Soluton. The problem of mnmzng transportaton cost has been studed snce long and s well known [7,8,,3,4]. In ths work we have added a new algorthm that provdes a better IBFS, for both the balanced and unbalanced TP, than those algorthms ust mentoned. TP n general are concerned wth dstrbutng any sngle commodty from any group of supply centre, called sources, to any group of recevng centre, called destnatons. A destnaton can receve ts demand from one or more sources. Each source has a fed supply of unts, where the entre supply must be dstrbuted to the destnatons. Smlarly, each destnaton has fed demand of unts, where the entre demand must be receved from the sources. 2. Cost mnmzaton TP A cost mnmzaton TP s formulated as Mnmze: subect to m z = = n = m = n = c a b ; =,2,......, m ; =,2,......, n 0 for all and. where =, 2,, m s the set of orgns. =, 2,, n s the set of destnatons. = the quantty transported from the -th orgn to the -th destnaton. c = per unt cost n transportng goods from -th orgn to the -th destnaton. a = the amount avalable at the -th orgn. b = the demand of the -th destnaton. 200

3 An Effectve Modfcaton to Solve Transportaton Problems: A Cost Mnmzaton Approach 3. Proposed method of fndng IBFS Idea of ths method s developed from VAM. The proposed method can be appled to solve all types of TP. Procedure of fndng an IBFS usng ths method s llustrated below. Step Step 2 Step 3 Step 4 Step 5 Step 6 Subtract each of the elements of every row from the largest entry of that row of the Transportaton Table and place them on the rght-top of the correspondng elements. Apply the same operaton on each of the elements of every column and place them on the rght-bottom of the correspondng elements. Form a modfed transportaton table whose elements reman same and place the magntude of the subtracton of rght-top and rght-bottom entry of Step and Step 2 on the rght-bottom of the correspondng elements. Place the row dstrbuton ndcators (RDI) and the column dstrbuton ndcators (CDI) ust after and below the supply lmts and demand requrements respectvely wthn frst bracket, whch are the dfference of the largest and nearest-to-largest rght bottom entres. If there are two or more largest entres, the result s to be consdered as zero. Identfy the hghest dstrbuton ndcator. Choose the lowest element along the hghest dstrbuton ndcator. If there are two or more hghest ndcators, choose the hghest ndcator along whch the smallest cost unt s present. If there are two or more number of smallest elements, choose any one of them arbtrarly. Allocate = mn(a, b ) on the left top of the smallest entry n the (, )-th cell (TT of Step-3). Step 7 If a < b, leave the -th row and readust b as b If a > b, leave the -th column and readust a as / = b a / a. = a b. If a = b, leave ether -th row or -th column but not both. Step 8 Repeat Steps 4 to 7 untl the rm requrements are satsfed. Step 9 Calculate z = c m = n =, z beng the mnmum transportaton cost. For an unbalanced TP we shall solve the problem by followng the above algorthm wthout dong any operaton on dummy destnaton/supply elements. Durng the operaton of Step 4, f we fnd a sngle entry, that entry wll be consdered as row or 20

4 M.M.Ahmed, A.S.Muhammad Tanvr, S.Sultana, S.Mahmud and Md. S.Uddn column dstrbuton ndcator. Fnally we shall allocate the demand/supply of the dummy elements by followng smple arthmetc calculaton. 4. Numercal llustratons 4.. Eample (Balanced TP): A company manufactures motor tyres and t has four factores F, F 2, F 3 and F 4 whose weekly producton capactes are 5, 8, 7 and 4 thousand peces of tyres respectvely. The company supples tyres to ts three showrooms located at D, D 2 and D 3 whose weekly demand are 7, 9 and 8 thousand peces respectvely. The transportaton cost per thousand peces of tyre s gven below n the TT: Showrooms D D 2 D 3 Supply F Factores F F F Demand Table 4... An eample of balanced TP We want to schedule the shftng of tyres from factores to showrooms wth a mnmum cost. Snce the factores demand 34 unts equals the total supply 34 unts, the gven problem s a balanced TP. Applyng the algorthm of the Proposed Method; the row dfferences and the column dfferences are shown on the rght top and rght bottom respectvely to each of the elements. Factores Showrooms Supply D D 2 D 3 F F F F Demand Table Modfed TT The dstrbuton s made accordng to proposed algorthm s 202

5 An Effectve Modfcaton to Solve Transportaton Problems: A Cost Mnmzaton Approach Factores D Showrooms D 2 D 3 Supply RDI F (2) (2) -- F (0) F (2) (2) (2) F (0) (0) (0) Demand (0) (3) (3) (0) () () () () () CDI Table Intal soluton tableau of proposed method Therefore, the soluton for the gven problem s = 5, 8 23=, 32 = 7, = 4 2, = 2 42 and 43 = 0. and the total transportaton cost s z = = = Eample 2 (Unbalanced TP): A company has four plants at locatons A, B, C and D, whch supply to warehouses located at E, F, G, H and I. Monthly plant capactes are 300, 500, 825 and 375 unts respectvely. Monthly warehouse requrements are 350, 400, 250, 50 and 400 unts respectvely. Unt transportaton costs are gven below. Warehouses Plants E F G H I Capactes A B C D Requrements Table An eample of unbalanced TP Determne a dstrbuton plan for the company n order to mnmze the total transportaton cost. Snce the warehouse requrements 550 unts s less than the total plant capactes 2000 unts, the gven problem s an unbalanced TP. We ntroduce a dummy warehouse W havng all the transportaton costs equal to zero to make the TP balanced.

6 M.M.Ahmed, A.S.Muhammad Tanvr, S.Sultana, S.Mahmud and Md. S.Uddn Applyng the algorthm of the Proposed Method; the row dfferences and the column dfferences are shown on the rght top and rght bottom respectvely to each of the elements ecludng the dummy one. Plants Warehouses E F G H I W Capactes A B C D Requrements Table Modfed TT Accordng to our algorthm, the ntal dstrbuton s made wthout dong any operaton on dummy destnaton/supply elements s Plants Warehouses E F G H I W Capactes RDI A (2) (2) (2) B (0) (0) (2) (2) (2) - C D Requr ements CDI (2) (2) (2) (2) (2) (2) (0) (4) (2) (2) (2) (2) (4) (2) (2) (4) (0) - - (2) (2) (4) (0) - - (2) (2) - (0) - - (4) (4) - (0) (4) - (0) (4) - (0) - Table Intal dstrbuton for unbalanced TP Now, the fnal allocaton s, 204

7 An Effectve Modfcaton to Solve Transportaton Problems: A Cost Mnmzaton Approach Plants A Warehouses E F G H I W 0 B C D Capactes Requrements Table Intal soluton tableau of proposed method Therefore, the soluton for the gven problem s =300, = 350, =, 32 = 00, 33 = 00, 35 = 400, 36 = 225, = and 46 = 225. and total transportaton cost s z = = = 5. Comparson of Transportaton Cost Obtaned by Dfferent Methods A comparatve study among the results obtaned by proposed method, estng methods and optmal soluton s also carred out n the table 5... Method Eample Eample 2 NWCM LCM VAM EDM Proposed Method Optmal Soluton Table 5... Comparatve study of the Results 5. Concluson In today s hghly compettve market the pressure s ncreasng rapdly to the organzatons to determne the better ways to delver goods to the customers. That s why dfferent organzatons want to delver products to the customers n a cost effectve way 205

8 M.M.Ahmed, A.S.Muhammad Tanvr, S.Sultana, S.Mahmud and Md. S.Uddn and thus market becomes compettve. For ths, Transportaton model provdes a powerful framework to meet ths challenge. The proposed method of fndng an IBFS for the mnmzaton of transportaton cost s llustrated numercally. It s observed that proposed algorthm provdes comparatvely a better IBFS soluton than those obtaned by the tradtonal algorthms whch s ether optmal or near to optmal soluton. We would fnally conclude that our developed algorthm provdes a remarkable IBFS by ensurng mnmum transportaton cost whch may be an attractve alternatve to the tradtonal transportaton problem soluton methods. REFERENCES. Abdur Rashd, An effectve approach for solvng transportaton problems, Jahangrnagar J. Mathematcs and Mathematcal Scences, 26 (20) R.K.Amnur, Analyss and resoluton of the transportaton problem: an algorthmc approach, M.Phll. Thess, Dept. of Mathematcs, Jahangrnagar Unversty, R.K.Amnur, A resoluton of the transportaton problem: an algorthmc approach, Jahangrnagar Unversty Journal of Scence, 34(2) (20) I.Amrul, R.K.Amnur, M.S.Uddn and M.A.Malek, Determnaton of basc feasble soluton of transportaton problem: a new approach, Jahangrnagar Unversty Journal of Scence, 35() (202) F.L.Htchcock, The dstrbuton of a Product from Several Sources to Numerous Localtes, J. Math. Phys., (94) G.B.Dantzg, Lnear Programmng and Etensons, Prnceton Unversty Press, Prnceton, N.J., H.H.Shore, The transportaton problem and the Vogel s appromaton method, Decson Scence, (3-4) (970) H.S.Kasanan and K.D.Kumar, Introductory Operaton Research: Theory and Applcatons, Sprnger, Md. Amrul Islam, Cost and Tme Mnmzaton n Transportaton and Mamzaton of Proft: A Lnear Programmng Approach, Ph.D. thess, Dept. of Mathematcs, Jahangrnagar Unversty, M.Mathraan and B.Meenaksh, epermental analyss of some varants of Vogel s appromaton method, Asa-Pacfc Journal of Operatonal Research, 2(4) (2004) N.Balakrshnan, Modfed Vogel s appromaton method for unbalance transportaton problem, Appled Mathematcs Letters, 3(2) (2000) P.Pandan and G.Nataraan, A new approach for solvng transportaton problems wth med constrants, Journal of Physcal Scences, 4 (200) R.R.K.Sharma and S.Prasad, Obtanng a good prmal soluton to the uncapactated Transportaton Problem, European Journal of Operaton Research, 44 (2003) S.Korukoglu and S.Ball, An Improved Vogel s Appromaton Method for the Transportaton Problem, Assocaton for Scentfc Research, Mathematcal and Computatonal Applcaton, 6(2) (20) T.C.Koopmans, Optmum utlzaton of the transportaton system, Econometrca, Vol XVII,