A comparative study of initial basic feasible solution methods for transportation problems

Transcription

1 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) A coparatve study of ntal basc feasble soluton ethods for transportaton probles Abstract Abdul Sattar Sooro 1 Gurudeo Anand Tulara 2 Ghula Murtaa Bhayo 3 dr_sattarsooro@yahoo.co.n, a.tulara@grffth.edu.au, gsndh@yahoo.co 1 Professor of Matheatcs, Insttute of Matheatcs and Coputer Scence, Unversty of Sndh, Jashoro, Sndh, Pakstan 2 Senor Lecturer, Matheatcs and Statstcs, Scence Envronent Engneerng and Technology [ENV], Grffth Unversty, Brsbane Australa 3 Lecturer, Govt. Degree College, Pano Akl, Sukkur, Sndh, Pakstan In ths research three ethods have been used to fnd an ntal basc feasble soluton for the balanced transportaton odel. We have used a new ethod of Mnu Transportaton Cost Method (MTCM) to fnd the ntal basc feasble soluton for the solved proble by Hak [2]. Hak used Proposed Approxaton Method (PAM) to fnd ntal basc feasble soluton for balanced transportaton odel and then copared the results wth Vogel s Approxaton Method (VAM) [2]. The results of both ethods were noted to be the sae but here we have taken the sae transportaton odel and used MTCM to fnd ts ntal basc feasble soluton and copared the result wth PAM and VAM. It s noted that the MTCM process provdes not only the nu transportaton cost but also an optal soluton. Keywords: Transportaton proble, Vogel s Approxaton Method (VAM), Maxu Penalty of largest nubers of each Row 1. Introducton The transportaton proble s a specal lnear prograng proble whch arses n any practcal applcatons n other areas of operaton, ncludng, aong others, nventory control, eployent schedulng, and personnel assgnent [1]. In ths proble we deterne optal shppng patterns between orgns or sources and destnatons. The transportaton proble deals wth the dstrbuton of goods fro the varous ponts of supply, such as factores, often known as sources, to a nuber of ponts of deand, such as warehouses, often known as destnatons. Each source s able to supply a fxed nuber of unts of the product, usually called the capacty or avalablty and each destnaton has a fxed deand, usually called the requreents. The objectve s to schedule shpents fro sources to destnatons so that the total transportaton cost s a nu. There are varous types of transportaton odels and the splest of the was frst presented by Htchcock (1941). It was further developed by developed by Koopans (1949) and Dantzg (1951). Several extensons of transportaton odel and ethods have been subsequently developed. In general, the Vogel s approxaton ethod yelds the best startng soluton and the north-west corner ethod yelds the worst. However, the latter s easer, quck and nvolves the least coputatons to get the ntal soluton [5]. Goyal (1984) proved VAM for the unbalanced transportaton proble, whle Raakrshnan (1988) dscussed soe proveent to Goyal s Modfed Vogel s Approxaton ethod for unbalanced transportaton proble [6]. Adlakha and Kowalsk (2009) suggested a systeatc analyss for allocatng loads to obtan an alternate optal soluton [7]. However, the study on alternate optal solutons s clearly lted n the lterature of transportaton 11

2 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) wth the excepton of Sudhakar VJ, Arunnsankar N, Karpaga T (2012) who suggested a new approach for fndng an optal soluton for transportaton probles [8]. 2. Transportaton proble and General Coputatonal Procedures The transportaton odel of LP can be odeled as follows: Mnze Subject to Z 1 j1 n j1 1 x j x n j x j 0, C j a b x j j ( Total transportaton cost) ( Supply fro sources) ( Deand for all and where Z : Total transportaton cost to be nzed. fro destnatons) C j : Unt transportaton cost of the coodty fro each source to destnaton j. x j : Nuber of unts of coodty sent fro source to destnaton j. a : Level of supply at each source. b j : Level of deand at each destnaton j. j; Supply 1 a Deand 1 b. NOTE: Transportaton odel s balanced f Supply a Deand b. 1 1 Otherwse unbalanced f Supply a Deand b. 1 1 The total nuber of varables s n. The total nuber of constrants s +n, whle the total nuber of allocatons (+n 1) should be n feasble soluton. Here the letter denotes the nuber of rows and n denotes the nuber of coluns. Solvng Transportaton Probles The basc steps for solvng transportaton odel are: Step 1 - Deterne a startng basc feasble soluton. In ths paper we use any one ethod NWCM, LCM, or VAM, to fnd ntal basc feasble soluton. Step 2 - Optalty condton - If soluton s optal then stop the teratons otherwse go to step 3. Step 3 - Iprove the soluton. We use ether optal ethod: MODI or Steppng Stone ethod. Table 1: Transportaton array 12

3 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) D1 DESTINATIONS D2 Dn Supply a S o u r c e s S 1 S S C 11 x 11 C 21 x 21 C 1 x 1 C 12 x 12 C 22 x 22 C 2 x 2 C 1n x 1n C 2n x 2n C n x n a 1 a 2 a Deand b j b 1 b 2 b n Balanced odel 1 a n b j j1 2. Methodology The followng ethods are always used to fnd ntal basc feasble soluton for the transportaton probles and are avalable n alost all text books on Operatons Research [5]. The Intal Basc Feasble Solutons Methods are: () Colun Mnu Method (CMM) () Row Mnu Method (RMM) () North West-Corner Method (NWCM) (v) Least Cost Method (LCM) (v) Vogel s Approxaton Method (VAM) The Optal Methods used are: () Modfed Dstrbuton (MODI) Method or u-v Method () Vogel s Approxaton Method (VAM) 3. Intal Basc Feasble Soluton Methods and Optal Methods There are several ntal basc feasble soluton ethods and optal ethods for solvng transportaton probles satsfyng supplyng and deand. Intal Basc Feasble Soluton Methods We have used followng three ethods to fnd ntal basc feasble soluton of the balanced transportaton proble: Vogel s Approxaton Method (VAM) Proposed Approxaton ethod (PAM) Mnu Transportaton Cost Method (MTCM) For optal ethods we have used the Modfed Dstrbuton (MODI) Method and the Steppng Stone Method Vogel s Approxaton Method (VAM) 13

4 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) Ths ethod provdes a better startng soluton than the North West Corner rule and Least Cost Method. VAM generally yelds an optu or close to optu soluton. Algorth Step 1. Step 2. Copute penalty of each row and a colun. The penalty wll be equal to the dfference between the two sallest shppng costs n the row or colun. Identfy the row or colun wth the largest penalty and assgn hghest possble value to the varable havng sallest shppng cost n that row or colun. Step 3. Cross out the satsfed row or colun. Step 4. Copute new penaltes wth sae procedure untl one row or colun s left out. Note: Penalty eans the dfference between two sallest nubers n a row or a colun. Proposed Approxaton Method (PAM) Ths ethod provdes a better startng soluton than the North West Corner rule and Least Cost Method. The PAM generally yelds optu soluton or close to optu soluton. Algorth Step 1. Identfy the boxes havng axu and nu transportaton cost n each row and wrte the dfference (penalty) along the sde of the table aganst the correspondng row. Step 2. Identfy the boxes havng axu and nu transportaton cost n each colun and wrte the dfference (penalty) aganst the correspondng colun. Step 3. Identfy the axu penalty. If t s along the sde of the table, ake axu allotent to the box havng nu cost of transportaton n that row. If t s below the table, ake axu allotent to the box havng nu cost of transportaton n that colun. Step 4. If the penaltes correspondng to two or ore rows or coluns are equal, select the box where allocaton s axu. Step 5. No further consderaton s requred for the row or colun whch s satsfed. If both the row and colun are satsfed at a te, delete the two or colun s assgned zero supply (or deand). Step 6. Calculate fresh penalty cost for the reanng sub-atrx as n step 1 and allocate followng the procedure of prevous step. Contnue the process untl all rows and coluns are satsfed. Step 7. Copute total transportaton cost for the feasble cost for the feasble allocatons usng the orgnal balanced transportaton cost atrx. Mnu Transportaton Cost Method (MTCM) Ths ethod provdes a better startng soluton than the North West Corner rule and Least Cost Method. The MTCM generally yelds optu soluton or close to optu soluton. Algorth 14

5 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) Step 1. Step 2. Make the table balanced. Copute penalty of each row. The penalty wll be equal to the dfference between the two largest shppng costs n the row. Identfy the row or colun wth the axu penalty and assgn possble value to the varable havng sallest shppng cost n that row. If two or ore rows correspondng equal penalty then select the cell wth nu cost of that axu penalty row. Step 3. Cross out the satsfed row or colun. Step 4. Wrte the reduced table and copute new penaltes wth sae procedure untl one row or colun s left out. Deterne the total nu cost of occuped cells satsfyng +n-1allocatons. Note: Penalty eans the dfference between two largest nubers n a row. Optal Method Modfed Dstrbuton (MODI) Method Ths ethod always gves the total nu transportaton cost to transport the goods fro sources to the destnatons. Algorth 1. If the proble s unbalanced, balance t. Setup the transportaton tableau 2. Fnd a basc feasble soluton. 3. Set u1 0 and deterne u ' s and v j ' s such that u v j c j for all basc varables. 4. If the reduced cost c u v 0 for all non-basc varables (nzaton proble), then the j j current BFS s optal. Stop! Else, enter varable wth ost negatve reduced cost and fnd leavng varable by loopng. 5. Usng the new BFS, repeat steps 3 and The Nuercal Proble We have used three ethods to fnd an ntal basc feasble soluton for the balanced transportaton proble [4]. The proble was developed by Hak [2]. Consder the transportaton proble presented n Table 2 - where there are 4 sources, 6 destnatons; the cost s gven n the cells, and the supply and deand gven n botto and rght hand end row and colun respectvely n Table 2. Table 2: Exaple proble Destnatons Deand Soluton Three ethods have been used here to fnd ntal basc feasble soluton of the above proble and these are presented n turn. 15

6 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) Vogel s Approxaton Method (VAM) Usng VAM the fnal soluton s presented n the Table 3. Table 3: Soluton usng VAM Deand Therefore the total transportaton cost deterned by the Vogel s Approxaton Method s: Mnze Z = (20) (1) + (10) (1) + (20) (2) + (10) (1) + (20) (4) +(20) (2) + (30) (6) + (25) (2) + (20) (1) = = 450 Soluton Proposed Approxaton Method (PAM) The fnal soluton copleted usng PAM s presented n the Table 4. Table 4: Soluton by PAM Deand The total transportaton cost by Proposed Approxaton Method can be gven as: Mnze Z = (30) (1) + (10) (1) + (40) (4) + (20) (4) + (20) (2) +(10) (6) + (25) (2) + (20) (1) = = 450 Mnu Transportaton Cost Method (MTCM) The total transportaton cost by Mnu Transportaton Cost Method s gven n Table 5. Table 5: Soluton by MTCM Deand

7 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) The total transportaton Cost by MCTM s gven as: Mnze Z = (20) (1) + (10) (1) + (20) (2) + (30) (4) + (40) (2) +(10) (6) + (25) (2) + (10) (3) + (10) (4) = = 450 Modfed Dstrbuton (MODI) Method We have found total nu transportaton cost usng MODI ethod by takng ntal basc feasble soluton obtaned by Mnu Transportaton Cost Method (MTCM). The fnal soluton s shown n Table 6. Table 6: Soluton usng MODI Deand The total transportaton cost by Modfed Dstrbuton Method s gven as: Mnze Z = (20) (1) + (10) (1) + (20) (2) + (10) (1) + (20) (4) + (40) (2) + (10) (6) + (25) (2) + (20) (4) = = 430 Table 5: A coparson of the ethods - VAM, PAM and MTCM and MODI Intal Basc Feasble Value of the objectve Mnu Value (cost) ethods functon VAM 450 Sae Result PAM 450 Sae Result MTCM 450 Sae Result MODI Method 430 Optal or nu cost The cost of transportaton shows that the: () Mnu Transportaton Cost Method (MTCM), Vogel s approxaton ethod (VAM), and Proposed Approxaton Method (PAM) provde the sae result, not optal but close to optal; () In MTCM, we have used penalty of axu nubers of each row yet not of each colun; () In VAM and PAM, the penalty of sallest nubers of each row and colun are appled; (v) In MTCM, the penalty of each row akes the proble sple, easy and takes a short te n calculaton; and (v) In VAM and PAM, the penalty of each row and colun akes the proble lengthy and the calculaton te s longer. 17

8 Matheatcal Theory and Modelng ISSN (Paper) ISSN (Onlne) 5. Concluson As transportaton proble s a specal lnear prograng proble havng any practcal applcatons n other areas of operatons, ncludng, aong others, nventory control, eployent schedulng, and personnel assgnent as entoned earler. Here n our research work we have used three ethods, The Mnu Transportaton Cost Method (MTCM), Vogel s Approxaton Method (VAM) and Proposed Approxaton Method (PAM). These were used to fnd an ntal basc feasble soluton for the transportaton balanced odel. The results are noted to be the sae. It s portant to note that we have used only penalty of each row of axu nubers that s a spler opton and thus takes uch less te n the calculaton. In contrast, other ethods usng axu penalty of sallest nubers of each row and colun akes the proble lengthy and the calculaton takes longer. Moreover, the ethod presented here s spler n coparson of other presented ethods earler and can be easly appled to fnd the ntal basc feasble soluton for the balanced and unbalanced transportaton probles. REFERENCES [1] Hady A Taha, Prentce Hall Operatons Research: An ntroducton 7th Edton, p.165 [2] M.A. Hak, An Alternatve Method to Fnd Intal Basc Feasble Soluton of a Transportaton Proble, Annals of Pure and Appled Matheatcs, Vol. 1, No. 2, 2012, [3] S. K Goyal, Iprovng VAM for unbalanced transportaton probles, Journal of Operatonal Research Socety, 35(12) (1984) [4] P. K. Gupta and Man Mohan. (1993). Lnear Prograng and Theory of Gaes, 7th edton, Sultan Chand & Sons, New Delh (1988) [5] Operatons Research by Pre Kuar Gupta and D.S. Hra, Page [6] Goyal (1984) provng VAM for the Unbalanced Transportaton Proble, Raakrshnan (1988) dscussed soe proveent to Goyal s Modfed Vogel s Approxaton ethod for Unbalanced Transportaton Proble. [7] Veena Adlakha, Krzysztof Kowalsk (2009), Alternate Solutons Analyss For Transportaton probles, Journal of Busness & Econocs Research Noveber,Vol 7. [8] Sudhakar VJ, Arunnsankar N, Karpaga T (2012). A new approach for fnd an Optal Soluton for Trasportaton Probles, European Journal of Scentfc Research