Applying Genetic Algorithms for Inventory Lot-Sizing Problem with Supplier Selection under Storage Capacity Constraints

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1 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 18 Applying Geneic Algorihms for Invenory Lo-Sizing Problem wih Supplier Selecion under Sorage Capaciy Consrains Chirawa Woarawichai 1, Krisana Kuruvi 1, Paioon Vashirawongpinyo 2 1 Deparmen of Indusrial Engineering, Faculy of Engineering, Raamangala Universiy of Technology Lanna Tak, Thailand 2 Deparmen of Indusrial Engineering, Faculy of Engineering, Siam Universiy Bangkae Bangkok, Thailand Absrac The obecive of his research is o calculae he opimal invenory lo-sizing for each supplier and minimize he oal invenory cos which includes oin purchase cos of he producs, ransacion cos for he suppliers, and holding cos for remaining invenory Geneic algorihms are applied o he muli-produc and muli-period invenory lo-sizing problems wih supplier selecion under sorage capaciy Also a maximum sorage capaciy for he decision maker in each period is considered The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods I is assumed ha demand of muliple producs is known over a planning horizon The problem is formulaed as a mixed ineger programming and is solved wih he Geneic algorihms The deailed compuaion resuls are presened Keywords: Geneic Algorihms, Supplier selecion, Sorage capaciy 1 Inroducion Lo-sizing problems are producion planning problems wih he obecive of deermining he periods where producion should ake place and he quaniies o be produced in order o saisfy demand while minimizing producion and invenory coss [1] Since lo-sizing decisions are criical o he efficiency of producion and invenory sysems, i is very imporan o deermine he righ lo- sizes in order o minimize he overall cos Lo-sizing problems have araced he aenion of researchers The muli-period invenory lo-sizing scenario wih a single produc was inroduced by Wagner and Whiin [2], where a dynamic programming soluion algorihm was proposed o obain feasible soluions o he problem Soon aferwards, Basne and Leung [3] developed he muli-period invenory lo-sizing scenario which involves muliple producs and muliple suppliers The model used in hese former research works is formed by a single-level unconsrained resources indicaing he ype, amoun, suppliers and purchasing ime of he produc This model is no able o consider he capaciy limiaions One of he imporan modificaions we consider in his paper is ha of inroducing sorage capaciy One of he imporan modificaions we consider in his paper is ha of inroducing sorage capaciy consrains Wih he adven of supply chain managemen, much aenion is now devoed o supplier selecion Rosenhal e al [4] sudy a purchasing problem where suppliers offer discouns when a bundle of producs is bough from hem, and one needs o selec suppliers for muliple producs Then a mixed ineger programming formulaion is presened Jayaraman e al [5] proposed a supplier selecion model ha considers qualiy, producion capaciy, lead-ime, and sorage capaciy limis In his paper based on Basne and Leung [3] geneic algorihms (GAs) are applied o he muli-produc and muli-period invenory lo-sizing problem wih supplier selecion under sorage space Also a maximum sorage space for he decision maker in each period is considered The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods The obecive of his research is o calculae he opimal invenory lo-sizing for each supplier and minimize he oal invenory cos This paper is organized as follows: Secion 2 he geneic algorihm approach is applied o problem Secion 3 we describe our model In Secion 4 presens a numerical example of he model Finally, compuaion resuls and conclusions are presened in Secion 5 and 6 Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved

2 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 19 2 Mehodology The geneic algorihms (GAs) approach is developed o find opimal (or near opimal) soluion Deail discussion on GAs can be found in Holland [6], Michalewicz [7], and Gen and Cheng [8] [9] In his secion, we explain GAs procedure is illusraed in Fig 1 o sar he search GAs are iniialized wih a populaion of individuals The individuals are encoded as chromosomes in he search space GAs use mainly wo operaors namely, crossover and muaion o direc he populaion o he global opimum Crossover allows exchanging informaion beween differen soluions (chromosomes) and muaion increases he variey in he populaion Afer he selecion and evaluaion of he iniial populaion, chromosomes are seleced on which he crossover and muaion operaors are applied Nex he new populaion is formed This process is coninued unil a erminaion crierion is me [1] No 3 Formulaion Iniial populaion Evaluaion Selecion Crossover Muaion Terminaion rule Yes Sop Fig 1 The geneic algorihm procedure We also make he following assumpions and mahemaical for he model: 31 Assumpions Demand of producs in period is known over a planning horizon All requiremens mus be fulfilled in he period in which hey occur: shorage or backordering is no allowed Transacion cos is supplier dependen, bu does no depend on he variey and quaniy of producs involved Holding cos of produc per period is producdependen Iniial invenory of he firs period and he invenory a he end of he las period are assumed o be zero Produc needs a sorage space and available oal sorage space is limied Base on he above assumpion of model, Fig 2 shows he behavior of he model considering he scenario of muliperiod invenory lo-sizing problem wih supplier selecion under sorage space and budge consrains Iems from period -1 Supplier 1 Produc 1 Produc 2 Produc i Produc I Iems o period +1 Iems from period -1 Supplier 2 Iems o period +1 Fig 2 Behavior of he model in period 32 Mahemaical modeling Iems from period -1 Supplier Iems o period +1 Supplier J Iems from period -1 Iems o period +1 This paper is buil upon Basne and Leung [3] model We formulae he muli-produc and muli-period invenory lo-sizing problem wih supplier selecion under sorage space and budge consrains using he following noaion: Indices: i = 1,, I index of producs = 1,, J index of suppliers = 1,,T index of ime periods Parameers: D i = demand of produc i in period P i = purchase price of produc i from supplier H i = holding cos of produc i per period O = ransacion cos for supplier w i = sorage space produc i S = oal sorage capaciy Decision variables: X i = number of produc i ordered from supplier in period Y = 1 if an order is placed on supplier in ime period, 0 oherwise Inermediae variable: R i = Invenory of produc i, carried over from period o period + 1 Regarding he above noaion, he mixed ineger programming is formulaed as follows: Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved

3 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 20 Minimize ( TC) Subec o: R i PiXi i i Hi k1 X OY ik k 1 D ik (1) Xik Dik 0 for all i and, (2) k1 T Dik Y k i w i k1 X X k1 i ik Y X i 0 Dik k 1 for all i,, and, S for all, \ (3) (4) 0 or 1 for all and, (5) all 0 for i,, and, (6) The obecive funcion as shown in (1) consiss of hree pars: he oal cos (TC) of 1) purchase cos of he producs, 2) ransacion cos for he suppliers, and 3) holding cos for remaining invenory in each period in + 1 Consrain in (2) all requiremens mus be filled in he period in which hey occur: shorage or backordering is no allowed Consrain in (3) here is no an order wihou charging an appropriae ransacion cos Consrain in (4) each producs have limied capaciy Consrain in (5) is binary variable 0 or 1 and Consrain in (6) is nonnegaiviy resricions on he decision variable According o a large opimal problem, a GAs approach is applied o solve his problem 4 A numerical example In his secion we solved a numerical example of he model using he LINGO We consider a scenario wih hree producs over a planning horizon of five periods whose requiremens are as follows: demands of hree producs over a planning horizon of five periods are given in Table 1 There are hree suppliers and heir prices and ransacion cos, holding cos and sorage space are show in Table 2 and Table 3, respecively Table 1: Demands of hree producs over a planning horizon of five periods ( Di) and Budge of hem ( B ) Planning Horizon (Five Periods) Producs A B C Table 2: Price of hree producs by each of hree suppliers X, Y, Z ( Pi) and ransacion cos of hem ( O ) Price Producs X Y Z A B C Transacion Cos Table 3: Holding cos of hree producs A, B, C ( H i) and sorage space of hem ( w i) Producs A B C Holding Cos Sorage Space The oal sorage capaciy (S) is equal o 200 The resuls of applying he proposed mehod are shown in Table 4 The soluion of his problem (I = 3, J = 3, and T = 5) is o place he following orders All oher X i = 0: Table 4: Order of hree producs over a planning horizon of five periods ( X i) Planning Horizon (Five Periods) Producs A X 111 = 12 X 132 = 15 X 113 = 37 - X 135 = 13 B X 231 = 20 X 232 = 21 X 213 = 22 X 234 = 23 X 235 = 24 C X 321 = 20 X 332 = 19 X 313 = 18 X 334 = 17 X 335 = 16 Cos calculaion for his soluion: Purchase cos for produc 1 from supplier 1, 3 = (37 30) + ( ) 32 = 2,390 Purchase cos for produc 2 from supplier 1, 3 = (22 32) + ( ) 30 = 3,344 Purchase cos for produc 3 from supplier 1, 3 = (18 45) + ( ) 45 = 4,050 Transacion cos from supplier 1, 3 = (1 110) + (4 102) = 518 Holding cos for produc 1 R 13 X D13 = = 20 = H 1 R1 = 1 ( ) = 20 Thus, he oal cos for his soluion = 2, , , = 10,322 Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved

4 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 21 5 Compuaion resuls In his secion he comparison of he wo mehods solved problem size is using a commercially available opimizaion package like LINGO and GAs code is developed in MATLAB Experimens are conduced on a personal compuer equipped wih an Inel Core 2 duo 200 GHz, CPU speeds, and 1 GB of RAM The ransacion coss are generaed from in [50; 200], a uniform ineger disribuion including 50 and 200 The prices are from in [20; 50], he holding coss from in [1; 5], he sorage space from in [10; 50], and he demands are from in [10; 200] The resul in Table 5 shows he GAs comparing wih LINGO for he nine problem sizes A problem size of I; J; T indicaes number of suppliers = I, number of producs = J, and number of periods = T Compuaion ime limi is se a 120 minues For comparison, he percenage error is calculaed by (7) and (8) Percenage error of LINGO Upper bound - Lower bound 100 Upper bound (7) Percenage error of GAs Upper bound LINGO - GAs 100 (8) Upper bound LINGO The soluion ime of LINGO o opimal is a shor ime as he small problem size (wih he problem sizes 3 x 3 x 5; 3 x 3 x 10; 3 x 3 x 15; and 4 x 4 x 10) For large problems sizes LINGO canno obain opimal soluions wihin limi ime due o as he larger problem size (wih he problem sizes 4 x 4 x 15; 5 x 5 x 20; 10 x 10 x 50; 10 x 10 x 80; and 15 x 15 x 50) The GAs can opimally solve when he problem size is small (wih he problem sizes 3 x 3 x 5; 3 x 3 x 10; 3 x 3 x 15; 4 x 4 x 10; 4 x 4 x 15; 5 x 5 x 20; and 10 x 10 x 50) There are wo problems which GAs canno obain opimal soluions (wih he problem sizes 10 x 10 x 80; and 15 x 15 x 50) Nex, we sudy differences in he problem sizes beween soluions from he opimizaion wih LINGO and he GAs The resuls are show in Fig 4, a plo of he problem size versus soluion ime LINGO uses longer compuaion ime more han GAs wih seven problem sizes, bu uses equal ime wih wo problem sizes As show in Fig 5 a plo of he problem size versus % error when he problem size is very large, LINGO used a maximum % error from he opimal soluions is found o be 441% (a he problem size 10 x 10 x 80) which has more % error han GAs The GAs can solve small % error of wo problem sizes (a he problem size 10 x 10 x 80; and 15 x 15 x 50) Fig 6 and Fig 7 show compares resul beween LINGO and GAs in problem size 3 x 3 x 5 Thus, i is eviden ha GAs is an effecive means for solving he problem GAs soluion is opimal when he problem size is small For larger problems GAs can find feasible soluion wihin ime limi for which LINGO fails o find he opimum Table 5: Comparaive resuls of he wo mehods Opimizaion approach wih LINGO Geneic Algorihms (GAs) Problem size Soluion ime Soluion ime Toal cos % Error Toal cos (minue) (minue) % Error 3 x 3 x 5 10, , x 3 x 10 20, , x 3 x 15 30, , x 4 x 10 25, , x 4 x 15 38,154 a, 37,828 b , x 5 x 20 60,218 a, 59,527 b , x 10 x ,344 a, 274,758 b , x 10 x ,494 a, 436,317 b , x 15 x ,800 a, 405,155 b , a LINGO12 = Upper bound, b LINGO12 = Lower bound Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved

5 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 22 However, he GAs provides superior soluions o hose from LINGO ha are close o opimum in a very shor ime, and hus appears quie suiable for realisically sized problems Addiionally, he compuaion ime when using GAs is also shor, making i a very pracical means for solving he muliple producs and muli-period invenory lo-sizing problem wih supplier selecion under sorage capaciy 140 LINGO12 GAs 120 Soluion ime (minue) Fig 6 The bes obecive of LINGO 0 3x3x5 3x3x10 3x3x15 4x4x10 4x4x15 5x5x20 Problem size 10x10x50 10x10x80 15x15x50 Fig 4 Plo of he problem size vs soluion ime (minue) % Error LINGO12 GAs 3x3x5 3x3x10 3x3x15 4x4x10 4x4x15 Problem Size 5x5x20 10x10x50 10x10x80 15x15x50 Fig 5 Plo of he problem size vs % error 6 Conclusions Fig 7 The finess value of GAs In his paper, we presen geneic algorihms (GAs) applied o he muli-produc and muli-period invenory lo-sizing problem wih supplier selecion under sorage capaciy Also a maximum sorage space for he decision maker in each period is considered The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods The mahemaical model is give and he use of he model is illusraed hough a numerical example The problem is formulaed as a mixed ineger programming and is solved wih LINGO and he GAs As compared o he soluion of opimizaion package like LINGO, he GAs soluions are superior Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved

6 IJCSI Inernaional Journal of Compuer Science Issues, Vol 9, Issue 1, No 1, January 2012 wwwijcsiorg 23 References [1] GH Goren, S Tunali, and R Jans, A review of applicaions of geneic algorihms in lo sizing, Journal of Inelligen Manufacuring Springer Neherlands 2008 [2] H M Wagner, and T M Whiin, Dynamic version of he economic lo-size model, Managemen Science 5, pp, 89 96, 1958 [3] C Basne and J M Y Leung, Invenory lo-sizing wih supplier selecion, Compuers and Operaions Research 32, pp, 1 14, 2005 [4] EC Rosenhal, JL Zydiak, and SS Cauchy, Vendor selecion wih bundling, Decision Sciences 26, 1995, [5] V Jayaraman, R Srivasava, and WC Benon, Supplier selecion and order quaniy allocaion: a comprehensive model, The Journal of Supply Chain Managemen 35, 1999, [6] J H Holland, Adapaion in Naural and Arificial Sysems The Universiy of Michigan Press, Ann Arbor 1975 [7] Z Michalewicz, Geneic Algorihms + Daa Srucures = Evoluion Programs AI Series Springer-Verlag, New York 1944 [8] M Gen, and R Cheng, Geneic Algorihms and Engineering Design Wiley, New York 1977 [9] M Gen, and R Cheng, Geneic Algorihms and Engineering Opimizaion Wiley, New York 2000 Copyrigh (c) 2012 Inernaional Journal of Compuer Science Issues All Righs Reserved