The Dynamc Programmng Models for Invenory Conrol Sysem wh Tme-varyng Demand Truong Hong Trnh (Correspondng auhor) The Unversy of Danang, Unversy of Economcs, Venam Tel: 84-236-352-5459 E-mal: rnh.h@due.edu.vn Receved: February 21, 2017 Acceped: March 16, 2017 do:10.5296/ber.v71.10965 URL: hps://do.org/10.5296/ber.v71.10965 Absrac The concep of dependen and ndependen demand s mporan n nvenory plannng and replenshmen ha also requres dfferen nvenory conrol soluons. Ths paper employs he dynamc programmng echnque for nvenory conrol sysem wh me-varyng demand o propose he replenshmen polcy n erms of he economc order quany, number of replenshmen, and reorder pon where oal nvenory cos s mnmzed. The sudy resul ndcaes ha he dynamc programmng models ouperform he radonal lo szng models n erms of oal nvenory cos. Moreover, he paper creaes opporunes for exendng furher researches on dynamc nvenory relaed o capacy consrans and uncerany condons of demand, yeld, lead me. Keywords: Dynamc nvenory, Lo szng, Order pon sysem, Maeral requremen plannng, Dynamc programmng model, Invenory conrol sysem 1. Background In busness managemen, nvenory consss of a ls of goods and maerals held avalable n sock. The key quesons of wha and how much nvenory are relaed. Plannng s underaken o deermne he level of nvenory ha wll be needed for operaons, and replenshmen s he process of mananng hs level hrough some combnaons of reorder and oher echnques. To deermne he level of nvenory needed for operaons, s useful o denfy he source of he demand. The concep of dependen and ndependen demand s mporan n nvenory plannng and replenshmen. An em has ndependen demand when we can no conrol or e drecly o anoher em s demand. Whle an em has dependen demand when he demand for an em s conrolled drecly, or ed o he producon of somehng else. Therefore, nvenory sysems wh ndependen demand and dependen demand also requre very dfferen soluons. 128 hp://ber.macrohnk.org
There s an abundan leraure on nvenory conrol polces whch exend snce he 30 s. The deals on hese polces may refer o research of Peerson and Slver (1979), and research of Zpkn (2000). In nvenory plannng and replenshmen, he radonal lo szng models are mosly used for nvenory conrol sysems. Each lo szng mehod ouperforms under some assumpons and demand condons n whch he demand does no presen a monoonous behavor and vares from perod o perod. There s also some leraure ha sudes dynamc nvenory conrol polces based on he nvesgaons of Karln (1960) and Scarf (1959). Wagner and Whn (1958) nroduced a dynamc programmng model n whch demand s a funcon of me. Slver and Meal (1973) proposed a heursc mehod ha fnds he opmal order quany, mnmzng he sorage and delvery coss. These deermnsc and sochasc models srongly reled on mahemacal background ha s no easy o undersand and mplemen he opmal nvenory conrol polces n realy. Ths paper aemps o develop he dynamc programmng models for boh ndependen nvenory sysem and dependen nvenory sysem wh me-varyng demand. These models are evaluaed wh radonal lo szng models such as Lo for Lo (LFL), Economc Order Quany (EOQ), Perod Order Quany (POQ), and Mnmum Cos per Perod (MCP). The paper provdes a basc framework for exendng dynamc nvenory researches wh capacy consrans and uncerany condons of demand, yeld, lead me. 2. Order Pon Sysem (OPS) Order Pon Sysem (OPS) s he nvenory conrol sysem for he ndependen demand. The mul-perod nvenory model wh me-varyng demand s developed o propose he replenshmen polcy n erms of order quany, number of replenshmen, and reorder pon. Fgure 1 llusraes he ypcal ndependen nvenory sysem ha has several end-ems wh ndependen demand. A mul-perod nvenory model wh me-varyng demand for ousourcng maerals s modfed on he bass of Wagner s model (Wagner, 1969) under he followng assumpons. 1. Backorder s no allowed. 2. Lead me s known wh cerany, and assumed consan durng he plannng horzon. 3. All relevan coss are assumed consan a each perod durng he plannng horzon. 4. No safey sock s assumed. 5. Orderng and holdng coss per perod are known. 6. Purchase cos s neglgble snce prces are assumed consan a each perod durng he plannng horzon. 7. Invenory level a each perod s assumed consan n each perod. 8. No quany dscoun s allowed. 9. Cos of capal s no consdered. 129 hp://ber.macrohnk.org
Order quany Sock Sock Sysem Fgure 1. The ndependen nvenory sysem The dynamc programmng model for he ndependen nvenory sysem (OPS) can be expressed as follows. Mnmze: Lead me Subjec o: X m T m T O N, H X, 1 1 1 1 (1) X D Q ; 1..m, (2),, 1,, 1 R Q 0 ; 1..m, 1 LT,, LT (3) Q N LS ; 1..m, (4),, 1 Q N M ; 1..m, (5),, 1 0, 1 ; 1..m, X, 0, Q, 0, and N, 1 (6) Where, O = Orderng cos per replenshmen for em a perod h H = Holdng cos per un for em a perod h N, = Number of orders akng place for em a perod h D, = Demand for em a perod h Q, = Order quany each me an order akes place for em a perod h X, = Invenory level for em a he end of perod h LT = Lead me for each replenshmen for em a perod h 130 hp://ber.macrohnk.org
R, = Reorder pon for em a perod h = Perod of me durng he plannng horzon m = Toal number of ems T = Toal number of perod of me durng he plannng horzon The mul-perod nvenory sysem under sudy has hree producs (end-ems) wh ndependen demand. These ems have lumpy demand due o seasonaly, rend, and economc condons. Informaon abou demand and properes of he sysem are gven as npus of he nvenory conrol models. Table 1 gves he demand of hree end-ems n nex egh perods. Table 1. The demand of ems n he plannng horzon Iem 1 2 3 4 5 6 7 8 A 200 200 300 300 350 350 400 400 B 300 300 300 300 300 300 300 300 C 200 250 300 350 350 300 250 200 The properes of he nvenory sysem provde nformaon relaed o nvenory coss, nal nvenory, lead me and lo sze. Ths nformaon s gven n Table 2. Table 2. The properes of he ndependen nvenory sysem Iem () Orderng cos Holdng cos Inal Invenory Lead me Lo sze (O ) (H ) (X0 ) (LT ) (LS ) A 1000 2 200 1 100 B 1500 3 600 2 100 C 2000 5 400 1 100 There are many dfferen mehods for deermnng replenshmen polcy such as Lo for Lo (LFL), Economc Order Quany (EOQ), Perod Order Quany (POQ), and Mnmum Cos per Perod (MCP). These lo szng mehods are used o compare wh he dynamc programmng model ha s called he OPS model. Table 3 shows he oal cos of models under he sudy. The resul ndcaes ha he OPS model s beer han oher lo szng models n erms of oal nvenory cos. Table 3. Toal nvenory cos of he models under he sudy Mehod LFL EOQ POQ MCP OPS Iem A 7000 9580 6100 6100 6100 Iem B 9900 12660 8100 8100 8100 Iem C 15000 16800 17500 13500 13000 Toal cos 31900 39040 31700 27700 27200 3. Maeral Requremen Plannng (MRP) Maeral Requremen Plannng (MRP) s used for he dependen nvenory sysem. The MRP 131 hp://ber.macrohnk.org
model uses a lo of daa abou ems and componens. The erm of em s used o refer o fnal produc, componens and componens of componens. For each em, needs o know: The lead me, he me leg beween he release of an order o he shop floor or o a suppler and he recep of he ems. The lo sze, a mnmum producon quany (referred o as a mnmum lo sze for ems ha are manufacured n-house) or a mnmum order quany for purchased ems. The nvenory saus (sock on hand ha calculaes based on nal nvenory, scheduled recep and demand requremen n each perod). Componens needed, whch s ofen referred o as a bll of maerals (BOM). Order Quany M A T E R I A L Sock C Sub-Assembles Fnal-Assembly A Sock Fgure 2. The dependen nvenory sysem An opmzaon nvenory model s no needed o use MRP calculaon, he purpose of he sudy s o creae an opmzaon problem ha maches MRP no for s own sake bu o ge sared wh models ha mach classc plannng sysems. Usng hs model as a sarng pon, s easy o go on o more sophscaed models (Voßand Woodruff, 2006). The dynamc programmng model for he dependen nvenory sysem (MRP) can be expressed as follows. Mnmze: Lead me m T m T O N, H X, 1 1 1 1 (7) Subjec o: m X, X D P R, 1,, j j, Q, ; 1..m, 1 (8) j1 132 hp://ber.macrohnk.org
R Q 0 ; 1..m, 1 LT,, LT (9) Q N LS ; 1..m, (10),, 1 Q N M ; 1..m, (11),, 1 0, 1 ; 1..m, X, 0, Q, 0, and N, 1 (12) Where, O = Orderng cos per replenshmen for em a perod h H = Holdng cos per un for em a perod h N, = Number of orders akng place for em a perod h Q, = Order quany each me an order akes place for em a perod h X, = Invenory level for em a he end of perod h D, = Exernal demand for em a perod h P,j = Number of em need o make one j R, = Reorder pon for em a perod h LT = Lead me for each replenshmen for em LS = Mnmum lo sze for em M = A large number = Perod of me durng he plannng horzon m = Toal number of ems T = Toal number of perod of me durng he plannng horzon The mul-sage nvenory sysem s consdered o llusrae he nvenory conrol polcy wh me-varyng demand. Suppose ha here s a sngle end-em A ha has a bll of maerals as shown n Fgure 3. 133 hp://ber.macrohnk.org
A (1) B (1) C (1) D (2) E (3) Fgure 3. BOM srucure for he sysem In hs sysem, here are wo ems wh ndependen demand, A and B. Iem B s also a componen o make end-em A, so em B has boh dependen and ndependen demand. The assembly of A requres 1 em B and 1 em C. In order o assembly 1 em C, requres 2 ems D and 3 ems E. Iems B, D, E are raw maerals ha are ordered from ousde supplers. Table 4 descrbes he properes of he nvenory sysem. Table 4. The properes of he dependen nvenory sysem Iem () Orderng cos Holdng cos Inal Invenory Lead me Lo sze (O ) (H ) (X0 ) (LT ) (LS ) A 1000 8 300 1 100 B 1500 6 600 2 100 C 1800 5 400 1 100 D 1300 4 500 1 100 E 2000 7 600 2 100 The demand for em A and em B n he nex egh perods s gven n Table 5. Table 5. The demand for em A and em B n he plannng horzon Iem 1 2 3 4 5 6 7 8 A 200 0 300 300 350 350 0 400 B 0 300 0 300 300 0 300 0 Based on nformaon abou bll of maerals (BOM) and daa from Table 4 and Table 5, he MRP model provdes opmal soluon for scheduled receps, sock on hand and planned order release n each perod. The opmal nvenory model s developed for such sysem based on prevous dynamc programmng model. The objecve of he model s o mach demand requremens and mnmze oal nvenory cos. The effecveness of MRP model s depended on lo szng mehods. Some radonal lo szng mehods are used o compare lo szng generang n he MRP model under oal nvenory cos. Lo szng mehods are employed n he sudy ncludng Lo for Lo (LFL), Perod Order Quany (POQ), and Mnmum Cos per Perod (MCP). Table 6 shows oal nvenory cos of he models under he sudy. 134 hp://ber.macrohnk.org
Table 6. Toal nvenory cos of he models under he sudy Mehod LFL EOQ POQ MCP MRP Iem A 6600 6952 6600 6600 9800 Iem B 10200 11244 10200 10200 7800 Iem C 10200 19622 12350 10150 9650 Iem D 7100 6624 4600 5900 3400 Iem E 12300 10200 8200 10200 4000 Toal cos 46400 54642 41950 43050 34650 The above resul ndcaes ha he MRP model has he leas oal nvenory cos. I reveals ha he MRP model s beer han oher models wh proposed lo szng mehods n erms of oal nvenory cos. Moreover, s neresed n exendng he MRP model o more sophscaed models wh capacy consrans. 4. Conclusons The paper has developed he dynamc programmng models for boh he ndependen and ndependen nvenory sysems. These dynamc programmng models are very basc for exendng o sophscaed nvenory conrol models. Some fndngs are summarzed as follows. For he ndependen demand, he dynamc programmng model for he ndependen nvenory sysem (OPS) s developed for he mul-perod nvenory model wh me-varyng demand. The resul ndcaes ha he OPS model provdes he opmal nvenory soluon n erms of oal nvenory cos. Moreover, he model may exend o nvenory conrol polcy wh unceranes n demand, yeld and lead me. (Baba and Dallery, 2006). Accordng o he assumpons of he model, s applcaon s lmed o some cases n pracce, especally he accuracy of he forecased demand n each perod. However, s found ha he model can provde an alernave replenshmen polcy wh sgnfcan savng o he decson maker n managng her sysem effcenly. For he dependen demand, here s no perfec model for Maeral Requremen Plannng. In fac, he MRP model has a number of well-known and very severe problems. Perhaps he wo mos serous problems are ha lo szng can cause nervousness and here are no capacy consrans. Even havng serous problems, opmal MRP model can sll be very useful for solvng he lo szng problems. For one hng, s usually much beer han non-plannng model a all. Ths s parcularly rue n ndusres wh changng demand paerns where sandard orders canno be used. The MRP model provdes a good sarng pon for plannng and for he orderng of raw maerals. The sudy resul ndcaes ha he MRP model s beer han oher radonal lo szng models as a whole. In addon, usng hs model as a sarng pon, s easy o go on o more sophscaed models, especally capacy consrans. References Baba, M. Z., & Dallery, Y. (2006). A dynamc nvenory conrol polcy under demand, yeld and lead me unceranes. In Servce Sysems and Servce Managemen, 2006 Inernaonal 135 hp://ber.macrohnk.org
Conference on (Vol. 2, pp. 1026-1031). IEEE. hps://do.org/10.1109/csssm.2006.320649 Karln, S. (1960). Dynamc nvenory polcy wh varyng sochasc demands. Managemen Scence, 6(3), 231-258. hps://do.org/10.1287/mnsc.6.3.231 Peerson, R., & Slver, E. A. (1979). Decson sysems for nvenory managemen and producon plannng. New York: Wley. Scarf, H. (1959). The opmaly of (S, s) polces n he Dynamc Invenory Problem. Mahemacal Mehods n he Socal Scences, Sanford Unversy Press, Sanford, Calforna. Slver, E. A., & Meal, H. C. (1973). A heursc for selecng lo sze quanes for he case of a deermnsc me-varyng demand rae and dscree opporunes for replenshmen. Producon and Invenory Managemen, 14(2), 64-74. Voß, S., & Woodruff, D. L. (2006). Inroducon o compuaonal opmzaon models for producon plannng n a supply chan (Vol. 240). Sprnger Scence & Busness Meda. Wagner, H. M. (1969). Prncples of operaons research: wh applcaons o manageral decsons. In Prncples of operaons research: wh applcaons o manageral decsons. Prence-Hall. Wagner, H. M., & Whn, T. M. (1958). Dynamc verson of he economc lo sze model. Managemen scence, 5(1), 89-96. hps://do.org/10.1287/mnsc.5.1.89 Zpkn, P. H. (2000). Foundaons of nvenory managemen (Vol. 2). New York: McGraw-Hll. Appendx Appendx 1. The OPS Model n Lngo MODEL:! The dynamc programmng model of ndependen nvenory sysem;! Keywords: Order Pon Sysem (OPS); SETS:! Index of ems; ITEM/1..3/:O,H,X0,LT,LS;! The plannng horzon; TIME/1..8/;! Se of em & me, npu & oupu; LINK(ITEM,TIME):D,N,Q,R,X; ENDSETS DATA:! The properes of he sysem; O,H,X0,LT,LS = 1000 2 200 1 100 1500 3 600 2 100 136 hp://ber.macrohnk.org
2000 5 400 1 100;! The demand requremens; D = 200 200 300 300 350 350 400 400 300 300 300 300 300 300 300 300 200 250 300 350 350 300 250 200; ENDDATA! A large number; M=10000;! The objecve funcon; MIN=@SUM(LINK(I,T):O(I)*N(I,T))+@SUM(LINK(I,T):H(I)*X(I,T));! The consrans for nvenory saus; @FOR(ITEM(I):X(I,1)=X0(I)-D(I,1)+Q(I,1)); @FOR(LINK(I,T) T #LT# @SIZE(TIME):X(I,T+1)=X(I,T)-D(I,T+1)+Q(I,T+1));! The consrans for planned order release; @FOR(ITEM(I):R(I,1)=@SUM(TIME(T) T #LE# LT(I)+1:Q(I,T))); @FOR(LINK(I,T) T #GT# LT(I)+1:R(I,T-LT(I))=Q(I,T));! The consrans for scheduled receps; @FOR(LINK(I,T):Q(I,T)>=N(I,T)*LS(I);Q(I,T)<=N(I,T)*M);! The decson varables; @FOR(LINK(I,T):@GIN(X);@GIN(Q)); @FOR(LINK(I,T):@BIN(N)); END Source: Auhor s work Appendx 2. The MRP Model n Lngo MODEL:! The dynamc programmng model of dependen nvenory sysem! Keywords: Maeral Requremen Plannng (MRP); SETS:! Index of ems; ITEM/1..5/:O,H,X0,LT,LS;! The plannng horzon; TIME/1..8/;! Se of em & me, npu & oupu; LINK(ITEM,TIME):D,N,Q,R,X;! Bll of maeral srucure; PART(ITEM,ITEM):P; ENDSETS DATA:! The properes of he sysem; O,H,X0,LT,LS = 1000 8 300 1 100 Busness and Economc Research 137 hp://ber.macrohnk.org
1500 6 600 2 100 1800 5 400 1 100 1300 4 500 1 100 2000 7 600 2 100;! The demand requremens; D = 200 0 300 300 350 350 0 400 0 300 0 300 300 0 300 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;! Bll of maeral srucure; P = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 3 0 0; ENDDATA! A large number; M=10000;! The objecve funcon; MIN=@SUM(LINK(I,T):O(I)*N(I,T))+@SUM(LINK(I,T):H(I)*X(I,T));! The consrans for nvenory saus; @FOR(ITEM(I):X(I,1)=X0(I)-D(I,1)-@SUM(ITEM(J):P(I,J)*R(J,1))+ Q(I,1)); @FOR(LINK(I,T) T #GT# 1:X(I,T)=X(I,T-1)-D(I,T)- @SUM(ITEM(J):P(I,J)*R(J,T))+Q(I,T));! The consrans for planned order release; @FOR(ITEM(I):R(I,1)=@SUM(TIME(T) T #LE# LT(I)+1:Q(I,T))); @FOR(LINK(I,T) T #GT# LT(I)+1:R(I,T-LT(I))=Q(I,T));! The consrans for scheduled receps; @FOR(LINK(I,T):Q(I,T)>=N(I,T)*LS(I);Q(I,T)<=N(I,T)*M);! The decson varables; @FOR(LINK(I,T):@GIN(X);@GIN(Q)); @FOR(LINK(I,T):@BIN(N)); END Source: Auhor s work Copyrgh Dsclamer Busness and Economc Research Copyrgh for hs arcle s reaned by he auhor(s), wh frs publcaon rghs graned o he journal. Ths s an open-access arcle dsrbued under he erms and condons of he Creave Commons Arbuon lcense (hp://creavecommons.org/lcenses/by/3.0/). 138 hp://ber.macrohnk.org