Modified Vogel s Approximation Method For Solving Transportation Problems

Transcription

1 Modfed Vogel s pproxaton Method For Solvng Transportaton Probles bdl Sattar Sooro 1 Mhaad Jnad Grdeo nand Tlara 3 dr_sattarsooro@yahoo.co.n haadnad.rapt@yahoo.co,,a.tlara@grffth.ed.a 1 Professor of Matheatcs, Insttte of Matheatcs and opter Scence, Unversty of Sndh, Jashoro, Sndh, Pakstan M.Phl Scholar, Insttte of Matheatcs and opter Scence, Unversty of Sndh, Jashoro, Sndh, Pakstan 3 Senor Lectrer, Matheatcs and Statstcs, Scence Envronent Engneerng and Technology [ENV], Grffth Unversty, rsbane strala bstract In ths research, we have Modfed Vogel s pproxaton Method (MVM) to fnd an ntal basc feasble solton for the transportaton proble whenever VM was developed n Three ethods North West orner Method (NWM), least ost Method (LM) and Vogel s pproxaton Method (VM) have been sed to fnd ntal basc feasble solton for the transportaton odel. We have taken sae transportaton odels and sed MVM to fnd ts ntal basc feasble solton and copared ts reslt wth above three ethods, bt MVM gves n transportaton cost and also optal and n soe probles the reslt of MVM s sae as VM bt better than NWM and LM. Keywords: Transportaton proble, Vogel s pproxaton Method (VM), Max Penalty of largest nbers of each Row, Mn Penalty of sallest nbers of each coln. 1. Introdcton One of the earlest and ost portant applcatons of lnear prograng has been the forlaton and solton of the transportaton proble as a lnear prograng proble. In ths proble we deterne optal shppng schedle of a sngle coodty between sorces and destnatons. The obectve s to deterne the nber of nts to be shpped fro the sorce to the destnaton, so that the total deand at the destnatons s copletely satsfed and the cost of transportaton s n. Let x 0 be the qantty shpped fro the sorce to the destnaton. The atheatcal forlaton of the proble s as follows: 3

2 ISSN (Paper) ISSN 5-05 (Onlne) Mnze Sbect to n Z x ( Total transportaton cost) 1 1 n x 1 a ( Spply fro sorces) x b 1 ( Deand fro destnatons) x 0, for all and. where Z : Total transportaton cost to be nzed. : Unt transportaton cost of the coodty fro each sorce to destnaton. x : Nber of nts of coodty sent fro sorce to destnaton. a : Level of spply at each sorce. b : Level of deand at each destnaton. NOTE: Transportaton odel s balanced f Spply a Deand b. 1 1 Otherwse nbalanced f Spply a Deand b. 1 1 The total nber of varables s n. The total nber of constrants s +n, whle the total nber of allocatons (+n 1) shold be n feasble solton. Here the letter denotes the nber of rows and n denotes the nber of colns. General optatonal Procedre for Transportaton Model The basc steps for solvng transportaton odel are: Step 1 - Deterne a startng basc feasble solton. We se any one ethod NWM, LM, or VM, to fnd ntal basc feasble solton. Step -Optalty condton. If solton s optal then stop the teratons otherwse go to step 3. Step 3 - Iprove the solton. We se any one optal ethod MODI or Steppng Stone ethod. 33

3 Table 1: Transportaton array D1 DESTINTIONS D Dn Spply a S o r c e s S 1 S.... S 11 x 11 1 x 1 1 x 1 1 x 1 x x 1n x 1n n x n n x n a 1 a a Deand b b 1 b b n alanced odel 1 a n b 1. Methodology The followng ethods are always sed to fnd ntal basc feasble solton for the transportaton probles and are avalable n every text book of Operatons Research [1]. Intal asc Feasble Soltons Methods () oln Mn Method (MM) () Row Mn Method (RMM) ()North West-orner Method (NWM) (v) Least ost Method (LM) (v) Vogel s pproxaton Method (VM) Optal Methods () Modfed Dstrbton (MODI) Method or -v Method () Steppng Stone ethod. 3. Intal asc Feasble Solton Methods and Optal Methods There are several ntal basc feasble solton ethods and optal ethods for solvng transportaton probles satsfyng spply and deand. Intal asc Feasble Solton Methods We have sed followng for ethods to fnd ntal basc feasble solton of the transportaton proble: 34

4 North West-orner Method (NWM) Least ost Method (LM) Vogel s pproxaton Method (VM) Modfed Vogel s pproxaton Method (MVM) Optal Methods For optal solton we have sed the Modfed Dstrbton (MODI) Method. 4. LGORITHMS OF INITIL SI FESILE SOLUTION METHODS 1. North-West orner Method (NWM) lgorth Step 1. Select the North-West (pper left-hand) corner cell of the transportaton table and allocate nts accordng to the spply and deand. Step. If the deand for the frst cell s satsfed, then ove horzontally to the next cell n the second coln. Step 3. If the spply for the frst row s exhasted, then ove down to the frst cell n the second row. Step 4. ontne the process ntl all spply and deand vales are exhasted.. Least ost Method (LM) lgorth Step 1. Frst exane the cost atrx and choose the cell wth n cost and then allocate there as ch as possble. If sch a cell s not nqe, select arbtrary any one of these cells. Step. ross ot the satsfed row or a coln. If ether a coln or a row s satsfed sltaneosly, only one ay be crossed ot. Step 3.Wrte the redced transportaton table and repeat the process fro step 1 to step, ntl one row or one coln s left ot. 3. Vogel s pproxaton Method (VM) lgorth Step 1. opte penalty of each row and a coln. The penalty wll be eqal to the dfference between the two sallest shppng costs n the row or coln. 35

5 Step. Identfy the row or coln wth the largest penalty and assgn hghest possble vale to the varable havng sallest shppng cost n that row or coln. Step 3. ross ot the satsfed row or coln. Step 4. opte new penaltes wth sae procedre ntl one row or coln s left ot. Note: Penalty eans the dfference between two sallest nbers n a row or a coln. 4. Modfed Vogel s pproxaton Method (MVM) lgorth Step 1. opte penalty of each row and a coln. The penalty of each row wll be eqal to the dfference between the two largest shppng costs bt the penalty of each coln s eqal to the dfference between sallest costs. Step. Identfy the row or coln wth the largest penalty and assgn n possble vale to the varable havng sallest shppng cost n that row or coln. Step 3. ross ot the satsfed row or coln. Step 4. opte new penaltes wth sae procedre ntl one row or coln s left ot. Optal Method Modfed Dstrbton (MODI) Method Ths ethod always gves the total n transportaton cost to transport the goods fro sorces to the destnatons. lgorth 1. If the proble s nbalanced, balance t. Setp the transportaton tablea. Fnd a basc feasble solton. 3. Set 1 0 and deterne ' s and v ' s sch that v c for all basc varables. 4. If the redced cost c v 0 for all non-basc varables (nzaton proble), then the crrent FS s optal. Stop! Else, enter varable wth ost negatve redced cost and fnd leavng varable by loopng. 5. Usng the new FS, repeat steps 3 and 4. 5 The Nercal Probles 36

6 Factory Matheatcal Theory and Modelng Exaple 1: onsder the followng transportaton proble: Table : ost Matrx for the Nercal exaple Destnaton D E F G Spply Deand pplyng the algorth of the Row Mn Method (RMM), we obtan the followng allocatons: Table 3: Intal asc Feasble Solton sng Row Mn Method D E F G Spply Deand z pplyng the algorth of the oln Mn Method (MM), we obtan the followng allocatons: 37

7 Table 4: Intal asc Feasble Solton sng oln Mn Method D E F G Spply Deand z pplyng the algorth of the North West orner Method (NWM), we obtan the followng allocatons: Table 5: Intal asc Feasble Solton sng North West orner Method D E F G Spply Deand z pplyng the algorth of the Least ost Method (LM), we obtan the followng allocatons: 38

8 oln Penalty Matheatcal Theory and Modelng Table 6: Intal asc Feasble Solton sng Least ost Method D E F G Spply Deand z pplyng the algorth of the Vogel s pproxaton Method (VM), we obtan the followng allocatons: Table 7: Intal asc Feasble Solton sng Vogel s pproxaton Method D E F G Spply Row Penalty (13) (13) () () () (5) (5) (5) 17 Deand 60 Z= =350 (4) (15) (8) (3) (4) -- (8) (3) (8) -- (11) (8) (8) (8) z pplyng the algorth of the Modfed Vogel s pproxaton Method (VM), we obtan the followng allocatons: 39

9 oln Penalty Matheatcal Theory and Modelng Table 8: Intal asc Feasble Solton sng Modfed Vogel s pproxaton Method D E F G Spply Row Penalty () (13) (16) (13) (15) (17) (5) (1) (17) 17 Deand Z= =3460 (4) (15) (8) (3) (4) -- (8) (3) (4) (3) (4) (3) z Optalty Test of Exaple 1 Takng ntal asc Feasble Solton de to proposed ethod, we now proceed for optalty sng Modfed Dstrbton Method. Here we calclate and v for occped basc cells sng v c and assng 0 : 1 0 v 1 0 v 0 1 v1 c11 1 v c1 1 0 v v 1 v4 c14 0 v 4 4 v 4 v3 c3 4 3 v 3 9 v 6 v4 c v4 c

10 alclaton of opportnty cost for non basc varable sng c v : (1, 3) c v (, 1) c v (, ) c v (3, 1) c v (3, ) c v (3, 3) c v Snce all opportnty costs are postve, the basc feasble solton obtaned by the proposed ethod s an optal solton. Table 9: coparson of the ethods Proble s Sze of a proble NWM LM VM MODI MVM Rearks 1.* 3X Opt. 3x Opt 3. 3x Opt 4. 3x Opt 5. 3x Opt 6. 3x Near to Opt 7. 3x Opt 8. 3x Opt 9.* 3x Opt 10. 3x Opt The cost of transportaton shows that the: () Modfed Vogel s pproxaton Method (MVM) and Vogel s approxaton ethod (VM), provde the sae reslt bt alost optal bt n soe probles very close to optal. () () In (MVM), all reslts are better than both ethods NWM and LM. In MVM, we have sed penalty of ax nbers of each row bt kept sae penalty of n nbers of each coln as n VM. 41

11 (v) (v) In Proble No1 the reslt of MVM s better than the reslt of VM. In MVM and VM, the penalty of each row akes the proble sple, easy and takes a sae te n calclaton. 6. onclson We have sed here for ethods North West orner Method (NWM), Least ost Method (LM), Vogel s pproxaton Method (VM) and Modfed Vogel s pproxaton Method (MVM) to fnd an ntal basc feasble solton for the transportaton odel. The reslts of MVM and VM are alost sae optal bt better than NWM and LM. In soe probles the reslt of MVM s better than VM. However, t s portant to note that we have sed penalty of each row of ax nbers bt kept sae penalty of n nbers of each coln as n VM. Ths or ethod s also easly appled to fnd the ntal basc feasble solton for the balanced and nbalanced transportaton probles. REFERENES [1] Operatons Research by Pre Kar Gpta and D.S. Hra, Page [] M.. Hak, n lternatve Method to Fnd Intal asc Feasble Solton of a Transportaton Proble, nnals of Pre and ppled Matheatcs, Vol. 1, No., 01, [3] S. K Goyal, Iprovng VM for nbalanced transportaton probles, Jornal of Operatonal Research Socety, 35(1) (1984) [4] P. K. Gpta and Man Mohan. (1993). Lnear Prograng and Theory of Gaes, 7th edton, Sltan hand & Sons, New Delh (1988) [6] Goyal (1984) provng VM for the Unbalanced Transportaton Proble, Raakrshnan (1988) dscssed soe proveent to Goyal s Modfed Vogel s pproxaton ethod for Unbalanced Transportaton Proble. [7] bdl Sattar Sooro, (014). coparatve stdy of ntal basc feasble solton ethods for transportaton probles, Matheatcal Theory and Modelng, Vol.4, No.1, 014,