On The Use of Transportation Techniques to Determine the Cost of Transporting Commodity

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1 IOSR Joural of Matheatcs (IOSR-JM) e-issn: ,p-ISSN: X, Volue 6, Issue 4 (May - Ju 213), PP wwwosrjouralsorg O The Use of Trasportato Techques to Detere the Cost of Trasportg Coodty Ajbade, AD (1); Babarde, SN (2) 1 Departet of Matheatcs & Statstcs, The Polytechc Ibada, Ngera 2 Departet of Busess Adstrato, The Polytechc Ibada, Ngera Abstract: Ths paper as at detfyg a effectve ad approprate ethod of calculatg the cost of trasportg goods fro several supply ceters to several dead ceters out of ay avalable ethods Trasportato algorths of North-West corer ethod (NWCM), Least Cost Method (LCM), Vogel s Approxato Method (VAM) ad Optalty Test were carred out to estate the cost of trasportg produced ewspaper fro producto ceter to ware-houses usg Statstcal software called TORA The results revealed that: NWCM = 36,689,, LCM = 55,2,34, VAM = 29,97,7 ad Optal soluto = 19,566,332 It was dscovered that Vogel s Approxato ethod gves the trasportato cost that closer to optal soluto Also, the study revealed that a producto ceter should be created at orther part of Ngera to replace the duy supply ceter used the aalyss, so as to ake producto capacty equal to requreet Key Words: algorth, trasportato, optal soluto, degeeracy, duy I Itroducto I the theory of producto, what to produce ad for who to produce are crtcal probles ecouter by ay producers developg coutres such as Ngera After solvg the stated probles, cost of producto s aother ecooc proble cotrbutg to the prce of a produced coodty Ths s revealg fro ecooc perspectve that says prce of coodty greatly rely o the cost of producg such coodty Oe of the cofoudg deterats of producto cost s cost of trasportg aufactured goods fro supply ceters to dead ceters Ths paper attepted at provdg a approprate ethod of calculatg the cost of trasportg a hoogeeous product fro aufacturg ceters to cosupto ceters so as to gude the copay zg the cost of ovg her products to dead areas The study of trasportato proble helps to detfy optal trasportato routes alog wth uts of coodty to be shpped order to ze total trasportato cost (JK Shara 29) Trasportato proble arses where there exst the eed to ove a set of tes fro the uber of dfferet sources (wth each source havg certa capacty) to a uber of destato (wth each destato requrg a certa capacty) Oe of the odels that explaed the study of trasportato proble s Lear Prograg (LP), but portat applcato of ths lear prograg s the area of physcal dstrbuto of goods ad servces after producto to the cosupto ceters (Rchard Broso et al ) At tes, trasportato proble ca be expressed ters of LP odel whch ca be solved by the splex ethod or other ethods of solvg a syste of lear equatos There are varous types of trasportato odels ad the splest of the was frst preseted by FL Htchcock (1941) ad developed by TC Koopas (1949) ad GB Datzg (1951) Later, several odels ad ethods have bee subsequetly developed Trasportato proble soluto attepts to ze total trasportato cost ad at the sae te satsfy the destato requreet gve the trasportato cost per ut The trasportato proble s a for of lear prograg proble, therefore, s a allocato proble for whch a specal techque or ethod of provdg soluto to a proble has bee developed However, the trasportato algorth s specally desg to provde optu soluto to probles arse as a result of trasportg product or allocato of servces or delvery fro several orgs to varous destatos, where the total requreets at each destato are kow quattes, thus, decdg alog whch routes are the best they should be dspatched I busess, sources could be factores ad destatos could be warehouses or custoer s shops It should be oted that trasportato probles are usually zato probles The ltato of lear prograg odel s volveet of a large uber of varables ad costrats that ay takes a log te to solve aually Ths proble s solved by the researchers, through the help of statstcal software called TORA II Soe Useful Cocepts Used Ubalaced Trasportato Proble: A trasportato proble s sad to be ubalaced whe the total dead s ot equal to the total supply I ths stuato, there wll be eed to create a duy ad wwwosrjouralsorg 23 Page

2 O The Use of Trasportato Techques to Detere the Cost of Trasportg Coodty ths depeds o the excess or shortage betwee the dead ad the supply That s there wll be eed to create duy dead f supply s greater tha dead or duy supply f the dead s greater tha supply wth zero cost of trasportato Balaced Trasportato Proble: A trasportato proble s sad to be balaced whe the total dead s equal to the total supply That s 1 a b j11 I ths paper, we have ubalaced trasportato proble, because the total dead s greater tha total supply However, duy supply wth zero cost of trasportato was created to ake total supply equal total dead ad trasportato algorth of North-West Corer Method, Least Cost Method, Vogel s Approxato Method as well as Optzato test were adopted to calculate the cost of trasportg a hoogeous coodty (Newspaper) fro supply ceters to dead ceters Later, the result of NWCM, LCM ad VAM are copared wth the result of optalty test to detfy the ethod that gves u value of the total cost of trasportg the product uder study (ewspaper) v Degeeracy: A basc feasble soluto of ay stadard trasportato proble wth sources of supply ad dead destato requre that allocato be ade (colu + row 1) Degeeracy of a soluto occurs whe the resources of a row are exhausted ad the requreets of a colu are satsfed by a sgle allocato So, f ths happes, t s portat to calculate the shadow cost ad the oly soluto s to treat oe of the uoccuped cells as f t was occuped cell or as f t had a zero allocato Optalty: I trasportato proble, a optal soluto s a soluto obtaed ad there s o other set of trasportato routes (allocatos) that wll further reduce the total trasportato cost III Materal ad Methods Data used for ths research was provded by Afrca Newspapers of Ngera Plc (Ngera Trbue) The data covered te years of supplyg Newspapers fro 3 supply ceters or plats (Ibada, Abuja ad Port- Harcourt) to 12 dead ceters (Abeokuta, Ado-Ekt, Ilor, Osogbo, Kao, Sokoto, Jos, Warr, Be, Owerr, Eugu ad Makurd) betwee 21 ad 21 A trasportato proble volves M sources, each of whch has avalable a (=1,2,) uts of a hoogeeous product ad destato, each of whch requres b j (j=1,2,) uts of the product (Newspaper) The ubers a j ad b j are postve tegers The cost C j of trasportg oe ut of product fro the th destato s gve for each ad j It s assued that total supply ad total dead are equal that s 1 a b j11 The above equato s guarateed by creatg ether a fcttous destato wth a dead equal to the surplus, f total dead s less tha the total supply, or a fcttous source wth a supply equal to the shortage f total dead exceeds total supply If x j represet the uber of uts to be shpped fro sources to j, atheatcal odel for the proble s wwwosrjouralsorg 24 Page

3 Layout of trasportato proble: S 1 C 11 D 1 D 2 D Supply X 11 S 2 C 21 X 21 S 3 C 31 S X 31 C 1 X 1 C 12 X 12 C 22 X 22 C 32 X 32 C 2 X 2 C 1 X 1 C 2 X 2 C 3 X 3 Dead b 1 b 2 b C X Table 1: where: C j s the cost of trasportg ut fro depot to destato j X j s the quatty allocated to destato j fro depot S s the source of depot D j s the destato j a s the capacty of source or dstrbuto cetre b j s the dead or requreet of destato j 1 a b j11 s the total a 1 a 2 a 3 a 1 a b j11 Algorths The frst approxato or coputato syste s always tegral ad therefore serves as optal soluto, rather tha deterg the approxato by a drect applcato of the splex ethod specalzed to detect the optal soluto, trasportato algorths of NWCM, LCM, ad VAM are eployed To use ay oe of the stated algorths, oe ust follow the followg steps: 1 Coputato of tal basc feasble soluto 2 Testg the optalty of the soluto step 1 3 Iprovg the soluto establshed step1 whe t s ot optal 4 Repeatg steps (2) ad (3) utl the optal soluto s obtaed Data presetato, aalyss ad terpretato of results The data collected fro Ngera Trbue s suarzed o table 2 ad the results of aalyses are preseted o table 3, 4, 5 ad 6: Supply/dead IBADAN ABUJA PORT DUMMY REQUIREMNT ceters N N HARCOURT N N Abeokuta Ado Ilor Osogbo Kao Sokoto Jos Warr Be wwwosrjouralsorg 25 Page

4 Owerr Eugu Makurd CAPACITY 673, 53, ,5942 Table 2 North - West Corer Method (NWCM) Table:3 OBJECTIVE FUNCTION: 36,689, ABE OKU TA IBADA 673 N ABUJA 1946 PORT HARC OURT DUMM Y DEMA 8676 ND ADO ILO RIN OSO GBO KA NO SOK OTO JOS WA BENI N OWE ENI UGU MA RK UR DI SUPP LY Least Cost Method (LCM) Table:4 OBJECTIVE FUNCTION: 55,2,34 IBAD AN ABUJ A PORT HARC OURT DUM MY DEMA ND ABE OK UTA AD O ILO RIN OS OG BO KA NO SO KO TO JO S WA BEN IN OW ERR I ENI UG U MA RK UR DI SUPP LY Vogel s Approxato Method (VAM) Table:5 OBJECTIVE FUNCTION: 29,97,7 ABE OKU TA IBADA 673 N ABUJA 1946 PORT HARC OURT DUMM Y DEMA 8676 ND ADO ILO RIN OSO GBO KA NO SOK OTO JOS WA BENI N OW ERR I ENI UG U MA RK UR DI SUPP LY wwwosrjouralsorg 26 Page

5 Optal Soluto Fro To At Shpped Obj Coeff Obj Cotrb S1: IBADAN D1: ABEOKUTA S2: ABUJA D5: KANO S2: ABUJA D7: JOS S3: D1: OWE PORTHARCOURT S4: DUMMY D1: 1946 ABEOKUTA S4: DUMMY D2: ADO S4: DUMMY D3: ILORIN S4: DUMMY D4: OSOGBO S4: DUMMY D6: SOKOTO S4: DUMMY D7: JOS S4: DUMMY D8: WA S4: DUMMY D9: BENIN S4: DUMMY D1: OWE S4: DUMMY D11: ENUGU S4: DUMMY D12: MARKUDI Table: 6 OBJECTIVE VALUE : 19,566,332 IV Dscusso The vogel s approxato ethod gves the total cost (29,97,7:) that s close to the real optal soluto (19,566,332) whch s u cost of trasportato Also, the cost estated through the orth-west corer ethod (36,689,) s better tha the oe obtaed through least cost ethod (55,2,34) The etwork dagra below shows all the routes that ca be used other to ze the total cost of trasportato V Cocluso The data o the supply ceter capacty per year ad the ut cost of trasportg ewspaper fro each of the three (3) sources to twelve (12) destatos was aalyzed ths research work, so as to obta the optu trasportato cost ad optu trasportato cost estated s N19, 566, 332 wwwosrjouralsorg 27 Page

6 Also, Vogel s approxato ethod gave the value (N29, 97, 7) that s closer to optu soluto However, VAM s the best ethod out of suggested ethods A duy supply ceter wth te years capacty of 12,893,542 s created to ake trasportato proble a balaced oe Also, the study revealed that a producto ceter should be created at orther part of the coutry (Ngera) to replace the duy supply ceter used the aalyss, so as to ake producto capacty equal to requreet V Refereces [1] Gass, SI (199) O solvg the trasportato proble Joural of Operatoal Research Socety, 41(4), [2] Lucey T (1981) Quattatve Techques, Ashford Colour Press (Ffth edto), Great Brta [3] Moskowlt, H (1979) Operato Research Techques for Maageet, Maclla Publcato C I (secod edto), New York, USA [4] Ose B A ad Ayasola O A (211) Quattatve Aalyss Made Easy, Hghlad Publshers), Ngera [5] Prof G Srvasa (29) Fudaetals of Operatos Research (A YouTube Vdeo Tutoral) Departet of Maageet Studes, IIT Madras [6] Rchard Broso et al () Theory ad Probles of Operato Research, Publsher: McGraw- [7] Hll USA [8] Shara JK (29) Operato Research (Theory ad Applcato) 4 th Edto, Maclla [9] Publsher, Ida [1] Sudhakar V J ad Arusakar N (212) A ew approach for fdg a Optal Soluto for Trasportato Proble (Euro Jourals Publshg) Ida pp [11] Taha H A (23) Operato Research A Itroducto, Pretce Hall of Ida (7 th Edto), Ida [12] Wahab A (22) Basc Cocept of Operato Research a Idustral Tool Cobat Publcato UI, (Ffth Edto) Ngera [13] Wkpeda Ole Ecyclopeda, (www Wkpedaco) wwwosrjouralsorg 28 Page

Keywords - Fuzzy sets, Fuzzy numbers, Fuzzy transportation problem, Fuzzy ranking, Fuzzy arithmetic

Keywords - Fuzzy sets, Fuzzy numbers, Fuzzy transportation problem, Fuzzy ranking, Fuzzy arithmetic M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: 2248-9622 www.era.co Vol. 3, Issue 1, Jauary -February 2013, pp.1416-1424 A Novel Approach for the fuzzy optal