An Inventory Replenishment Model for Deteriorating Items with Time-varying Demand and Shortages using Genetic Algorithm

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1 An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm ohd Omar*, Noor Hanah oin, and Ivan Yeo Iniue of ahemaical Science, Univeriy of alaya, 563 Kuala Lumpur, alayia. * Abrac In hi paper, an invenory replenihmen model for deerioraing iem i developed. Demand for he iem varie wih ime over a finie planning horizon, during which horage are allowed and are compleely back-ordered. he obecive i o deermine a replenihmen policy ha minimize he oal invenory co. A earch procedure baed on Geneic Algorihm (GA i preened and illuraed wih ome numerical reul. Keyword: Invenory; Replenihmen; imevarying demand; Deerioraion; Shorage; Geneic Algorihm. Inroducion When developing invenory model, many implifying aumpion are made o find he opimal replenihmen policie. he Economic Order Quaniy (EOQ model aume a conan demand rae over an infinie planning horizon and minimize he oal invenory co per uni of ime. However, pracical demand rae rarely behave hi way. o iem experience able demand only during he auraion phae of heir life cycle, while demand i increaing or decreaing during heir growh or decline phae repecively. In addiion, mo model aume ha iem have infinie helf-life while in orage. hi aumpion, while applicable o iem wih low deerioraion rae, eem unrealiic for volaile and radioacive maerial, blood bank, food uff, elecronic, ec, ha coninually loe heir uiliy while in ock. Shorage may be economically deirable in many iuaion uch a when orage co are high compared o back-order co or when orage pace i limied, i anoher imporan apec ha need o be given pecial aenion. Hariga (994 developed an opimal procedure o opimize an invenory model wih ime-varying demand and horage. A imilar procedure wa adoped by Hariga and Benkherouf (994 o opimize an invenory model wih ime-varying demand and deerioraion, bu wihou horage. Laer, Benkherouf (995 exended he laer model o include horage a well. Alhough hi model conidered only decreaing demand, i wa menioned ha a ligh modificaion would allow he opimal chedule for increaing demand o be found. Oher conribuor o hi model include Wee (995, Hariga and Al-Alyan (997, Giri e al. (2, and Chu and Chen (22. o lieraure on hi model, a far a he auhor are aware of, conidered only analyical or heuriical approache o direcly find or approximae i opimal oluion. In hi paper, we hall conider a ochaic approach by uing Geneic Algorihm (GA. Our model operae over a finie planning horizon o aify a ime- Inernaional Journal of he Compuer, he Inerne and anagemen Vol.6. N.o.2 (ay-augu, 28 pp

2 ohd Omar, Noor Hanah oin, and Ivan Yeo varying demand and aume ha deerioraion occur a a conan rae and horage are allowed and are compleely back-ordered. he obecive i o deermine a replenihmen policy ha minimize he oal invenory co. hi paper i organied a follow: In he nex ecion, we preen he mahemaical formulaion of our model. In Secion 3, we decribe a GA-baed earch procedure o approximae he opimal oluion. In Secion 4, we illurae hi procedure wih ome numerical example. We alo compare our reul wih ome opimal and heuriical reul. Finally, Secion 5 ummarize our finding followed by appendice decribing an opimal and everal heuriical procedure. 2. ahemaical Formulaion he following aumpion are made in developing our mahemaical model: A ingle ype of iem i held in ock over a finie planing horizon H uni of ime long. Replenihmen occur a an infinie rae wih zero lead ime and i charged wih a co of c per replenihmen. Invenory holding co i charged only o good uni wih a co of c 2 per uni per uni of ime. Shorage are allowed and are compleely back-ordered. Uni hor are charged wih a co of c 3 per uni per uni of ime unil hey are cleared by back-order. he iem deeriorae a a fixed rae θ. he deerioraion of he uni i conidered only afer heir receip ino orage and he deerioraed uni are no replaced or repaired during he planing horizon, bu are charged wih a co of c 4 per uni. he demand rae i a coninuou, deerminiic and ime-varying funcion and > for H. Suppoe ha n replenihmen (including he final backorder are made during he planning horizon. Alo, uppoe ha he oal ime elaped up o and including he h replenihmen cycle ( =,2, K, n i given by. We define = and n = H. oreover, uppoe ha he oal ime elaped up o he ar of horage in he h cycle i given by. o ar wih, conider he h cycle. Iniially, he ock i zero. Replenihmen occur a = and he invenory level reache i maximum immediaely. he poiive invenory level i repreened by y (. hi amoun of invenory i depleed over ime by demand and deerioraion unil i reache zero a = ( < <. Now, a horage occur and he negaive invenory level i repreened by y ( 2. he horage peak a = and i immediaely cleared by a back-order. he cycle hen repea ielf. We noe ha he back-order made during he (, cycle and he replenihmen made during he (, + cycle incur a ingle replenihmen co ince hey boh happen a he ame ime. Hence, he oal invenory co, which i defined a he um of replenihmen, holding, deerioraion and horage co, i given by n- W = nc + ( c2 + c4θ y( d + c3 y2( d =. - ( he variaion of y ( and y ( 2 wih repec o ime in he (, cycle i governed by he following linear differenial equaion: dy ( d + θ y ( =,, (2 8

3 An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm dy2 ( =,, (3 d wih he boundary condiion y ( = and y2 ( =. he oluion for equaion (2 and (3 i y( = exp( θ exp( θu udu,, (4 2 y ( = udu,. (5 Subequenly, he amoun of invenory carried during he, cycle i given by ( y( d = θ (exp[ θ( ] d. (6 For he amoun of uni hor during he (, cycle, we inegrae he poiive expreion of y ( over he inerval, : 2 ( 2 d y = udu d. ( (7 I follow ha he oal invenory co for an n-replenihmen policy i given by W = nc + c a n θ = c3 (exp[ θ( where c a = c + c. 2 4θ udu d ] d +, (8 Our problem i o minimize W by finding he opimal number of replenihmen, n, and he opimal replenihmen poin, 2, K, n, ubec o he following conrain: = < < L < n < n = H, n i an ineger, are a real number. i For a fixed n, he opimal and can be obained from he following e of equaion: W / =, =,2, K, n, (9 W / =, =,2, K, n 2. ( Noe ha, by defaul, n = H. = and A he opimal i deermined in he GA procedure, hen W / = give u ca ( exp[ θ ( ] =. ( c θ 3 3. Geneic Algorihm (GA Search Procedure Geneic Algorihm (GA i a ochaic earch echnique ha mimic naural biological evoluion. GA explore he problem domain by mainaining a populaion of individual, each repreening a poible oluion o he problem, ieraively unil ome opping crieria are aified. For hi ype of invenory model, mo oluion procedure in lieraure find he opimal or near opimal oluion for a fixed n; while he opimal n i found by evaluaing a range of W(n. However, our GA procedure will conider n a a variable and evolve i along wih he replenihmen poin. Each individual in a populaion encode he ineger value of n and i aociaed realvalued replenihmen poin, (, 2, K, n 2 ha aifie he following conrain: n, = < < K < <. 2 n 2 n = H Noe ha he amoun of replenihmen poin i dependan on n, hence no fixed. Conequenly, hi reul in a populaion of individual wih variable lengh. In order o creae he iniial populaion, we randomly Inernaional Journal of he Compuer, he Inerne and anagemen Vol.6. N.o.2 (ay-augu, 28 pp

4 ohd Omar, Noor Hanah oin, and Ivan Yeo generae a erie of n, whoe bound are arbirarily fixed, and hen he appropriae amoun of replenihmen poin for each n. We e he fir individual o have n = 2 and no replenihmen poin o repreen he ingle replenihmen policy wih one backorder. hu, he populaion array ake he form 2 NaN NaN K NaN NaN NaN n n2 nm, 2, m,,2 2,2 m,2 K K O K, n 2, n2 2 m, nm 3 NaN 2, n2 m, nm 2 NaN NaN m, nm (2 where NaN i a non-numeric conan ha doe no paricipae in he procedure. We evaluae he obecive value of a populaion uing (8. Oberve ha in he mahemaical model decribed in he preceding ecion, he oal invenory co i a funcion of (,, and ielf i a funcion of (, where he opimal can be obained from each pair of (, by olving (. We employ he Sochaic Univeral Sampling mehod (Baker, 987 in he elecion proce and we aign he fine of each individual uing he Nonlinear Ranking mehod (Chipperfield e al., 993. We reric he croover o be performed beween individual of he ame lengh only, imilar o he muli-populaion GA whereby he informaion exchange i confined o hoe individual wihin he ame ubpopulaion only. We elec he Dicree Recombinaion operaor (Chipperfield e al., 993 o perform he croover. We perform he proce of muaion in wo age. In he fir age, we perform muaion on he ineger n wih a given probabiliy where i i muaed o any ineger greaer han 2 bu bounded by a fixed ceiling. Since he amoun of replenihmen poin i dependen on n, we devie a procedure whereby if n increae, hen exra replenihmen poin, generaed randomly, will be added accordingly. hen, we or he new e of replenihmen poin in an acending order. If n decreae, hen we remove he la replenihmen poin from he e. hi eliminae he need o or he e again. Our limied experimen have hown ha hi procedure doe no have a dramaic effec on he convergence of he algorihm. In he econd age, we perform muaion on he replenihmen poin uing he Breeder GA (ühlenbein and Schierkamp-Vooen, 993. We muae each ordering poin wih a probabiliy of /( n-2 where (n-2 correpond o he amoun of replenihmen poin aociaed wih each individual. 4. Numerical Example Our procedure wa wrien uing he alab high-level language and wa run on a Penium 4 IB compaible machine running a.5 GHz in a alab 6. environmen. hroughou all our experimen, he following value are ued for he GA parameer (Chipperfield e al., 993: number of individual = 3, generaion gap =.9, recombinaion rae =.9, n muaion rae =.5, inerion rae =.9, elecive preure = 2, and maximal number of generaion =. We illurae our procedure uing four example cae: wo linear marke wih he demand funcion D ( = a + b and wo exponenial marke wih he demand funcion D ( = a exp( b. he parameer value for hee cae are hown in able. For each cae, we run our procedure ime and he reul are hown in able 2. able 3 how he opimal W from an opimal procedure (accurae o 5 decimal place and he W * from five heuriical procedure; hee are decribed in he Appendice. 2

5 An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm able : Parameer value of he example cae Demand cae a b H c c 2 c 3 c 4 θ Linear Linear Exponenial Exponenial able 2: Reul of he example cae uing he GA procedure Demand cae Be W Wor W Average W Linear Linear Exponenial Exponenial able 3: Reul of he example cae uing an opimal procedure and five heuriic Demand cae Linear Linear 2 Exponenial Exponenial 2 Opimal Heuriic Heuriic Heuriic Heuriic Heuriic Concluion In hi paper, we found he near opimal oluion for an invenory replenihmen model wih ime-varying demand, conan deerioraion and horage uing GA. Baed on our numerical reul, we have hown ha our procedure perform beer han ome heuriical procedure. he be oluion found opimal value in all he daa e. Inernaional Journal of he Compuer, he Inerne and anagemen Vol.6. N.o.2 (ay-augu, 28 pp

6 ohd Omar, Noor Hanah oin, and Ivan Yeo Appendix A: Opimal procedure Referring o Secion 2, he e of equaion ( give u c 3 d c a exp( θ + exp( θ d =, =,2, K, n 2. (3 From hi equaion, oberve ha >. + If we define = and n = H, i i eay o ee ha once = x i fixed, he e of equaion ( and (3 will deermine all oher and recurively a funcion of x. o ee hi, le = F(,, (4 + = G(,. (5 where (4 and (5 are he oluion o equaion ( and (3 repecively. Now we have = x = F(, x 2 = G(, 2 = F(, 2 n = G( n 2, n 2 = F(, n n 2 n o ummarize i, our procedure for finding he opimal replenihmen policy for a given n i o fix = x, obain all he oher and uing (4 and olving (5 repecively, and finally check if n = H. Since we have aumed n o be fixed, he opimal value of n can be obained by repeaing our procedure for differen value of n ( n = 2,3, K unil he oal invenory co ar o increae. We noe ha he econd order condiion canno be rivially checked, o we hall aume he e of equaion (9 and ( are enough for deermining he minimum of (8. Appendix B: Heuriic procedure he heuriic procedure ued for comparion in Secion 4 are baed on hoe in Hariga and Benkherouf (994. he variable n ued in he following ex will refer o he number of replenihmen excluding he final back-order, ince he final back-order only add a conan value o he oal invenory co. Heuriic : Conan Demand Approximaion In hi heuriic, i aumed a conan demand rae over an infinie planning horizon. Le D be he average demand per uni of ime, ha i, H D = udu. (6 H For each replenihmen cycle on he planning horizon, le be he lengh of ha cycle and I be he lengh of ime during which invenory i carried (I <. hen, he amoun of invenory carried during a cycle i I I(, I = ( exp( θ Dd, (7 θ and he amoun of horage incurred during a cycle i (, = { d } S I D u d. (8 I I Finally, uing (7 and (8, he oal invenory co per uni of ime i CU ( I, = { c + ca I(, I + c3s( I, }. hen, he opimal I and are he oluion of he following wo equaion: CU = F ( I, =, and (9 I CU = G ( I, =. (2 he number of replenihmen order placed during he planning horizon i hen n = [ H / ], where [x] refer o he large ineger number which i maller or equal o x. 22

7 An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm Nex, we preen he differen ep of he heuriric mehod.. Compue c a = c 2 + c 4 θ and D from (6.. For I (, H, compue from (9 and ue (I, on (2. 2. If (2 i olved, go o If (2 i no olved, go o. 4. If H, n = [ H / ]. 5. If > H, = H, and n =. 6. Find he replenihmen chedule a follow: = for =,, K, n and H. n = 7. Compue he oal invenory co of he replenihmen chedule. Heuriic 2: Equal Replenihmen Cycle In hi heuriic, i aumed ha all replenihmen are made a equal cycle of lengh. For n replenihmen and = =,2, HK/ n, n, we have =, and = Leing. = I, ( give u ca H I + [exp( θ I ] =, c θ n (2 3 which ha he oluion I = I for all replenihmen cycle. oreover, oberve ha I = I(n. Now, for he cae of linear demand, (8 can be wrien a ca bhn ( φ φ I W( n = nc + an+ 2 bn I I θ θ θ θ θ 2 H an bhn ( + bn H + Iφ + I, n n (22 where φ = φ( n = exp( θ I, and for he cae of exponenial demand a (exp{( θ + b I} exp( θ + b bh ac a W( n = nc + exp{( bh/ n} θ (exp( bi b ac3 bh exp exp( + bψ + ψ, b n b b (23 where ψ = ψ ( n = ( caφ /( c3θ. Auming ha he funcion W (n i convex, he opimal number of order under he rericion of equal replenihmen cycle i he fir ineger which aifie W ( n < W ( n and W ( n < W ( n +. Nex, we preen he differen ep of he heuriic mehod.. n =.. Compue W (n and W ( n If W ( n < W ( n +, op, and go o If W ( n > W ( n +, n = n +, and go o. 4. Find he replenihmen chedule a follow: = for =,2, K, n. 5. Compue he oal invenory co of he replenihmen chedule. Heuriic 3: Exended Silver-eal Heuriic In hi heuriic, i deermine each lo ize equenially, one a a ime, by minimizing he oal invenory co per uni of ime, raher han minimizing he oal invenory co up o he ime horizon. he oal invenory co per uni of ime i CU ( I, = { c + ca I (, I + c3s( I, }, where I I (, I = [exp( θ ] d, (24 θ and { u du} d. I I S( I, = (25 Inernaional Journal of he Compuer, he Inerne and anagemen Vol.6. N.o.2 (ay-augu, 28 pp

8 ohd Omar, Noor Hanah oin, and Ivan Yeo he neceary condiion for he opimal. Le = and =. I and are. For I (, H, compue from (27 and ue ( I, on (28. CU 2. If (28 i olved, go o 4. = F2 ( I, =, and (26 I 3. If (28 i no olved, go o. CU Sep 4 o 9 are he ame a in he = G2 ( I, =. (27 Silver-eal heuriic. Nex, we preen he differen ep of he heuriic mehod.. Le = and =.. For I (, H, compue from (26 and ue ( I, on ( If (27 i olved, go o If (27 i no olved, go o. 4. If = H, go o If > H, go o If < H, le = +, a = a + b (linear demand or a = a exp( b (exponenial demand, H = H, = +, and go o. 7. Le = +, n =, and go o Le = H, = +, n =, and go o Compue he oal invenory co of he,, K, chedule. ( n Heuriic 4: he Exended Lea Co Heuriic In hi heuriic, i equae he ordering co wih he um of he holding, horage, and deerioraion co o find each lo ize equenially and one a a ime. By uch equalizaion, we have c ca I(, I c3s( I, =, (28 where I (, I and S( I, are from (24 and (25 repecively. Nex, we preen he differen ep of he heuriic mehod. Heuriic 5: he Exended Lea-uni Co Heuriic he procedure of hi heuriic i he ame a ha of he Silver-eal heuriic excep for i co funcion, which i he oal co per uni demand. he mahemaical expreion of hi oal co per uni demand i given by c + ca I(, I + c3s( I, CU I, =, (29 d where I (, I and S( I, are from (24 and (25 repecively. hen, he opimal I and are he oluion of he follwing wo equaion: CUD I CUD = F3 ( I, = = G3 ( I, =., and (3 (3 Nex, we preen he differen ep in he heuriic mehod.. Le = and =.. For I (, H, compue from (3 and ue ( I, on (3. 2. If (3 i olved, go o If (3 i no olved, go o. Sep 4 o 9 are he ame a in he Silver-eal heuriic. 24

9 An Invenory Replenihmen odel for Deerioraing Iem wih ime-varying Demand and Shorage uing Geneic Algorihm Reference Baker, J.E. (987. Reducing bia and inefficiency in he elecion algorihm. Proceeding of he Second Inernaional Conference on Geneic Algorihm and heir Applicaion, Hilldale, New Jerey, USA: Lawrence Erlbaum Aociae, 4 2. Benkherouf, L. (995. On an invenory model wih deerioraing iem and decreaing ime-varying demand and horage. European Journal of Operaional Reearch, 86, Benkherouf, L (998. Noe on a deerminiic lo-ize invenory model for deerioraing iem wih horage and a declining marke. Compuer & Operaion Reearch, 2(, Chipperfiled, A., Fleming, P., Pohlheim, H. and Foneca, C. (993. Geneic Algorihm OOLBOX for Ue wih ALAB. Deparmen of Auomaic Conrol and Syem Engineering, Univeriy of Sheffield, U.K. Chu, P. and Chen, P.S. (22. A noe on invenory replenihmen policie for deerioraing iem in an exponenially declining marke. Compuer & Operaion Reearch, 29, Giri, B.C., Chakrabary,. and Chaudhuri, K.S. (2. A noe on a lo-izing heuriic for deerioraing iem wih ime-varying demand and horage. Compuer & Operaion Reearch, 27, Hariga,. (994. he invenory lo-izing problem wih coninuou ime-varying demand and horage. Journal of he Operaional Reearch Sociey, 45, Hariga,. and Al-Alyan, A. (997. A loizing heuriic for deerioraing iem wih horage in growing and declining marke. Compuer & Operaion Reearch, 24, Hariga,. and Benkherouf, L. (994. Opimal and heuriic invenory replenihmen model for deerioraing iem wih exponenial ime-varying demand. European Journal of Operaional Reearch, 79, ühlenbein, H. and Schierkamp-Vooen, D. (993. Predicive odel for he Breeder Geneic Algorihm I - Coninuou Parameer Opimizaion. Evoluionary Compuaion, (, Wee, H.. (995. A deerminiic lo-ize invenory model for deerioraing iem wih horage and a declining marke. Compuer & Operaion Reearch, 22, Inernaional Journal of he Compuer, he Inerne and anagemen Vol.6. N.o.2 (ay-augu, 28 pp