Solving the multi-period fixed cost transportation problem using LINGO solver

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1 Inernaonal Journal of Pure and Appled Mahemacs Volume 119 No , ISSN: (on-lne verson) url: hp:// Specal Issue pam.eu Solvng he mul-perod fxed cos ransporaon problem usng LINGO solver N. Bala 1, A.N.Bala 2 1 Professor, Deparmen of Mechancal Engneerng, Sr Krshna College of Engneerng and Technology, Combaore ,Tamlnau,Inda. 2 Professor, Deparmen of Mechancal Engneerng, KLN College of Engneerng Svaganga , Tamlnadu, Inda Tel.: ; Fax: E-mals: balan@skce.ac.n, rabala2001@gmal.com Absrac The expermenal oucomes whle solvng a mul-perod fxed cos ransporaon problem usng an Operaonal Research sofware ool LINGO solver were addressed n hs paper. The obecve was o compare he solver me wh respec o problem sze. The proposed solver was esed wh fory es problems. The resuls dsclosed he performance of he proposed solver. In addon, he expermen revealed he dffculy of solvng large scale mul-perod fxed cos ransporaon problem. Keywords: Mul-perod fxed cos ransporaon problem; LINGO solver; Solver me

2 Inernaonal Journal of Pure and Appled Mahemacs Specal Issue 1. Inroducon Tradonal ransporaon problems are confned o a parcular perod. Bu n realy, he dsrbuon decsons are prolonged for more han one me frame, because provdes a chance o ake advanage n lo szng. Ths model s referred as mul-perod ransporaon problem. The ncluson of fxed coss n he problem ransformed he problem as mul-perod fxed cos ransporaon problem. The fxed cos ransporaon problems perss o be a crucal area of ransporaon research (Palekar e al 1990, Sun e al 1998, Km and Pardalos 1999, Km e al 2000, Adlakha and Kowalsk 2003, Poh e al 2005, Adlakha e al 2007, Jawahar and Bala 2009, Jawahar and Bala 2010, Jawahar and Bala 2011, Jawahar and Bala 2012). The leraure survey uncovered he followngs: Fxed cos ransporaon problems are hard o solve due o he presence of fxed coss; Branch and bound echnques are exensvely proposed o solve fxed cos problems bu sll s a very compuaonal nensve procedure; Several approxmae algorhms have been proposed for some specfc ypes of problems bu here s sll a need for a more effecve soluon approach o more general problems; The problem specfc heurscs provde a good foundaon for solvng small sze problems. Few arcles (Jawahar and Bala 2010, Jawahar and Bala 2011, 2012) have red o solve mul-perod problems approxmaely by annealng/genec based mea heursc algorhms and by LINGO solver by relaxng he nonlneares. LINGO s a well-known and powerful Operaonal Research sofware ool avalable oday desgned o make buldng and solvng problems easer and more effcen. I s capable o solve any knd of nonlnear problems. I ncludes a se of negral solvers o handle a wde range of problems. All of he LINGO solvers are drecly assocaed wh he modelng suaon. I comprses a number of ways o fnd locally or globally opmal soluons o nonlnear models. LINGO has an oponal nonlnear solver based on he prncple of generalzed reduced graden algorhm. The key benefs of LINGO are easy model expresson, convenen daa opons, creae urn-key applcaons and exensve documenaon and help (Lngo User s Gude 1999). In hs paper, he mul-perod fxed cos ransporaon problem has been aemped o solve usng LINGO solver ncludng he nonlneares. The expermenal oucomes are addressed by solvng a se of fory es problems usng LINGO solver. The res of he paper s prearranged as follows: The problem envronmen and mahemacal deals of he mul-perod fxed cos ransporaon problem are gven n secon 2. Secon 3 dscusses he performance parculars of LINGO solver whle solvng es problems of mul-perod fxed cos ransporaon problem. A summary of mul-perod fxed cos ransporaon es problems solved by LINGO solver are concluded n secon

3 Inernaonal Journal of Pure and Appled Mahemacs Specal Issue 2. Mul-perod fxed cos ransporaon problem A number of realsc ransporaon problems can be represened as mulperod fxed cos ransporaon problem. Jawahar e al (2011) and Jawahar and Bala (2012), formulaed a more generalsed mul-perod fxed cos ransporaon problem ncludng varable and fxed cos alled wh ransporaon, nvenory ha can be sored a any of he suppler s or cusomer s places wh varable nvenory sorng coss and unflled demands me as backorders o any of he cusomers wh varable backorder penaly coss. The mahemacal model expressed as a Pure Ineger Non-Lnear Programmng problem as shown below. I negraes he decsons on ransporaon, nvenores and backorders. The decson varables along wh model are shown below. Decson varables X SI CI BL Mahemacal Model - Shpmen from Suppler o Cusomer durng perod - h Suppler nvenory durng perod - h Cusomer nvenory durng perod - h Cusomer Backorder durng perod Mn Z T m n ( * X FC * ) SH * SI T 1 m 1 1 T 1 n 1 1 T 1 n C CH * CI BC * BL (1) 1 1 Subec o P SI 1 D BL 1 n 1 CI X 1 SI m 1 X BL CI X *( 1) 0 (4) X 0 (5) 0 X 0, SI 0, CI 0, BL 0 and neger; (6), 1... m,, 1... n &, 1... T, The oal cos of ransporaon, suppler sde nvenory, cusomer sde nvenory and cusomer sde back order for he enre perod T s shown n obecve equaon 1. Frs consran se equaes he maeral balance a he suppler sde. In he same way, second consran se balances he maerals a he cusomer sde. I also ensures he presence of nvenory or backorder on boh sdes of he equaon. Consrans 4 and 5 send back he value of based on he value of consran declares he non-negavy of he decson varables. X (2) (3). Las

4 T*m*n No of Varables No of Consrans T*m*n No of Varables No of Consrans Inernaonal Journal of Pure and Appled Mahemacs Specal Issue 3. Performance of LINGO solver In hs chaper, here are fory es problems wh dfferen dmensons are aemped o solve opmally usng LINGO solver. Table 1 demonsraes he oupu of he es problems wh he maxmum run me of 350 seconds. Fgure 1 llusraes he solver me for provdng he soluons o he es problems. Table 1: Tes problems oupus usng LINGO solver S.No. Problem dmensons Solver Tme (secs) Toal Cos S.No. Problem dmensons Solver Tme (secs) Toal Cos (Noe: N-ONon Opmal) 1. 2*2* , *5* , *2* , *5* , *2* , *2* , *2* , *2* , *2* , *2* , *2* , *3* , *3* , *3* , *3* , *3* ,105 N:O 9. 2*3* , *3* ,070 N:O 10. 2*3* , *4* , *3* , *4* , *3* , *4* ,230 N:O 13. 2*4* , *4* ,825 N:O 14. 2*4* , *5* , *4* , *5* , *4* , *2* , *4* , *2* , *5* , *2* , *5* , *3* , *5* , *3* ,145 N:O

5 Inernaonal Journal of Pure and Appled Mahemacs Specal Issue Solver ends and resuls n nonopmal soluons Solver Tme (secs) Problem s seral number Fgure 1: LINGO solver compuaonal me for fory es problems 4. Conclusons In hs paper, a more generalsed mul-perod fxed cos ransporaon problem s consdered for expermenaon. An Operaonal Research sofware ool LINGO (8) solver s used o evolve he opmal soluons. The proposed solver s esed wh fory es problems. The resuls reveal he followng; The solver s able o solve mos of he problems opmally n a reasonable good compuaonal me. I s unable o provde opmal soluon o ceran large scale problems (problems wh S.No. 28, 5429, 32, 33 and 40). Above all, he expermen reveals ha solver me grows along wh he problem sze and valdaes he dffculy of solvng large scale mul-perod fxed cos ransporaon problems. I s a challengng sudy o develop an effcen ool o ackle he mul-perod fxed cos ransporaon problem n he presen supply chans neworks. References 1. Adlakha V, Kowalsk K A smple heursc for solvng small fxed-charge ransporaon problems. Omega: he Inernaonal Journal of Managemen Scence. 31:

6 Inernaonal Journal of Pure and Appled Mahemacs Specal Issue 2. Adlakha V, Kowalsk K., Vemugan R.R, and Lev B More-forless algorhm for fxed-charge ransporaon problems. Omega: he Inernaonal Journal of Managemen Scence. 35: Ehsan Goodarz, Mna Zae and Edward Za Hossenpour Opmzaon Analyss Usng LINGO and MATLAB. Tex book on Inroducon o Opmzaon Analyss n Hydrosysem Engneerng Jawahar N, Angappa Gunasekaran and Bala N A smulaed annealng algorhm based heursc o he mul-perod fxed charge dsrbuon problem. Inernaonal Journal of Producon Research. 50(9): Jawahar N and Bala N Consrucve heurscs o he mulperod fxed charge ransporaon problem. Inernaonal Journal of Busness Performance and Supply Chan Modelng. 2(2): Jawahar, N. and Bala, A. N A genec algorhm for he wosage supply chan ransporaon problem assocaed wh a fxed charge. European Journal of Operaons Research. 194: Jawahar N and Bala N A genec algorhm based heursc o he mul-perod fxed charge dsrbuon problem. Appled Sof Compung. 12: Km, D., Pardalos, P.M A soluon approach o he fxed charge nework flow problem usng a dynamc slope scalng procedure. Operaons Research Leer. 24: Km.J.U, and Km.Y.D A Lagrangan relaxaon approach o mul-perod nvenory/dsrbuon plannng. Journal of he Operaonal Research Socey. 51: Klose, A Algorhms for solvng he sngle-snk fxed-charge ransporaon problem. Compuers & Operaons Research. 35(6): Lngo User s Gude. Chcago, LINDO Sysems Inc., Palekar, U.S., Karwan, M.H., Zons, S A branch-and-bound mehod for he fxed charge ransporaon problem. Managemen Scence. 36(9): Poh.K.L, Choo.K.W, Wong.C.G A heursc approach o he mul-perod mul-commody ransporaon problem. Journal of he Operaonal Research Socey. 56: Sun, M., Aronson, J. E., McKeown, P. G. and Drnka, D A abu search heursc procedure for he fxed charge ransporaon problem. European Journal of Operaons Research. 106:

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