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2 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 104 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy MEI-FANG WU*, YUAN-SHYI PETER CHIU** AND PENG-CHENG SUNG** *Department of Industral Engneerng and Systems Management, Feng Cha Unversty, Tachung 407, Tawan **Department of Industral Engneerng & Management, Chaoyang Unversty of Technology, Wufong Dstrct, Tachung 413, Tawan Emal: **ypchu@cyut.edu.tw ABSTRACT Ths paper consders optmzaton of a mult-product economc producton quantty (EPQ) model wth scrap tems and an mproved mult-delvery polcy. We extend the work of Chu et al. (2013a) by ncorporatng an mproved n + 1 mult-delvery plan nto ther model wth an am at reducng vendor s nventory holdng cost. Under such a specfc shpment polcy, for each product an extra delvery of fnshed goods s made durng producton uptme to satsfy the product demand for the perod of vendor s uptme. Then, when rework process comes to an end (.e., the rest of the producton lot s qualty assured) n fxed quantty nstallments of fnshed goods are delvered to buyers at a fxed tme nterval. The objectves of ths study are to determne the optmal rotaton producton cycle tme that mnmzes the long-run average system cost per unt tme and reveals the effects of random scrap rate and the mproved delvery polcy on the optmal producton cycle tme. Wth the help of a mathematcal model a closedform optmal common cycle tme s derved. Through a numercal example, practcal usages of obtaned results and sgnfcant savngs n vendor s stock holdng cost are demonstrated. Keywords: Mult-product EPQ model; mult-delvery plan; optmzaton; rotaton cycle tme; scrap. INTRODUCTION A mult-product economc producton quantty (EPQ) model wth scrap tems and an mproved mult-delvery plan s studed n ths paper. The conventonal EPQ model (Taft, 1918) consders a perfect manufacturng process for a sngle product and a contnuous ssung polcy of fnshed tems. EPQ model employed a mathematcal technque and optmzaton process, balanced producton setup and nventory holdng

3 105 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung costs, and derved a most economc producton lot that mnmzes total producton cost for proposed manufacturng system. Despte ts smplcty n assumpton, the concept of tradtonal EPQ model has snce been adopted and appled ndustry-wde (Wagner & Whtn, 1958; Hadley & Whtn, 1963; Osteryoung et al., 1986; Tersne, 1994). In real-lfe manufacturng envronments, due to dfferent unpredctable factors, producton of defectve tems seems to be nevtable. Mak (1985) employed mathematcal modelng to examne an nventory system, where the number of unts of acceptable qualty n a replenshment lot s uncertan and the demand s partally captve. He assumed that the fracton of the demand durng the stock-out perod whch can be backordered s a random varable whose probablty dstrbuton s known. The optmal replenshment polcy was syntheszed for such a system. A numercal example was used to llustrate the theory. The results ndcated that the optmal replenshment polcy s senstve to the nature of the demand durng the stock-out perod. Schwaller (1988) studed an EOQ model by ncludng both fxed and varable nspecton costs, for fndng and removng a known proporton of defectve tems n ncomng lots. Harga and Ben-Daya (1998) consdered the economc producton quantty problem n the presence of mperfect processes. The tme to shft from the n-control state to the out-of-control state was assumed to be flexble, and they provded dstrbutonbased and dstrbuton-free bounds on the optmal cost. For the exponental case, they compared the optmal solutons to approxmate solutons proposed n the lterature. Inderfurth et al. (2006) nvestgated a determnstc problem of plannng the producton of new and recoverng defectve tems of the same product manufactured on the same faclty. The processng of a batch ncludes two stages: the regular producton and the rework process. Whle watng for rework, defectve tems deterorate and there s a gven deteroraton tme lmt. Deteroraton results n an ncrease n tme and cost for performng rework processes. The objectve of ther study was to fnd batch szes and postons of tems to be reworked such that overall producton-nventory costs are mnmzed. A polynomal dynamc programmng algorthm was presented to solve ths problem. Other studes that addressed dfferent aspects of mperfect producton systems can also be found n Chu et al. (2013b); Kamsu-Foguem et al. (2013); Sarkar & Sarkar (2013); n et al. (2013). The contnuous fnshed tems dstrbuton polcy n conventonal EPQ model s unrealstc when fnshed goods need to be dstrbuted to outsde customers n supply chans envronment. Multple or perodc delvery polcy s commonly adopted n such a vendor-buyer system. Goyal (1977) nvestgated an ntegrated nventory model for a sngle suppler - sngle customer problem. A method was proposed for solvng those nventory problems, wheren a product made by a sngle suppler s procured by a sngle customer. A numercal example was provded to verfy ths soluton process. Schwarz et al. (1985) determned the fll-rate of a one-warehouse N-dentcal retaler

4 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 106 dstrbuton system. An approxmaton model was adopted from a pror study to maxmze system fll-rate subject to a constrant on system safety stock. As a result, propertes of fll-rate, polcy were suggested to provde management, when lookng nto system optmzaton. Aderohunmu et al. (1995) demonstrated that a co-operatve batchng polcy, based on cost nformaton exchange between the vendor and the buyer, can sgnfcantly reduce total cost n the just-n-tme (JIT) envronment. The mpact of such co-operaton on total costs (ncludng orderng, set-up, transportaton and nventory holdng costs) was examned for a long-term supply relatonshp. They ndcated that jont optmzaton of both the vendor and the buyer s operatons do not necessarly result n a common lot sze. Senstvty of the resultng cost savngs due to the exchange of cost nformaton to changes n the relevant operatng parameters was also analyzed. Swenseth & Godfrey (2002) demonstrated that the freght rate functons presented n the lterature can be ncorporated nto nventory replenshment decsons wthout compromsng the accuracy of the decson. They concluded that these functons can be ncorporated wthout addng undue complexty to the decson process. Abdul-Jalbar et al. (2008) consdered a mult-echelon nventory system n whch one vendor supples an tem to multple buyers. They assumed that the vendor produces the tem at a fnte rate and customer demand occurs at each buyer at a constant rate. The objectve s to determne the order quanttes at the buyers and the producton and shpment schedule at the vendor n order to mnmze the average total cost per unt tme. The problem was formulated n terms of nteger-rato polces and a heurstc procedure was developed. Both soluton procedures were llustrated wth a numercal example. Performance of the heurstc for computng nteger-rato polces was demonstrated. Addtonal studes that addressed varous aspects of perodc or mult-delvery ssues n the vendor-buyer ntegrated systems can also be found n Chu et al. (2013c); Cedllo-Campos & Sánchez-Ramírez (2013). In manufacturng sector, for the purpose of maxmzng machne utlzaton, vendors often manufacture multple products n turn on a sngle machne. Gordon & Surks (1975) presented an approach to determne control polces for a mult-tem nventory envronment, where the tems are ordered from a sngle suppler and the demand for tems are subject to severe fluctuatons. The tme between orders can ether be fxed or based on accumulatng a fxed order quantty for all products. Ther model balanced carryng and stock-out costs. An operatonal system structure was developed and a smulaton procedure was used to determne the approprate value of ther nventory factor n the model. Zahork et al. (1984) studed a mult-tem, mult-level producton schedulng problem wth lnear costs and producton and nventory constrants at one key faclty. Two mult-tem problems were consdered, one n whch the constrant was on shppng capablty and the other n whch there was a fnal stage bottleneck machne. A mult-tem facltes-n-seres problem was formulated as a lnear program. A 3-perod result was used as the bass for a rollng

5 107 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung heurstc for T-perod problems. Condtons under whch ths heurstc fals to fnd optmal solutons were dscussed, and computatonal comparsons to standard lnear programmng were provded. Byrne (1990) presented an approach to the mult-tem producton lot szng problem based on the use of smulaton, to model the nteractons occurrng n the system. A search algorthm was used to adjust the lot szes on the bass of the results of prevous smulaton runs, wth the objectve of achevng a mnmum total cost soluton. Zpkn (1995) explored the performance of a mult-tem productonnventory system. He compared two alternatve polces, representng dfferent modes of collectng and utlzng nformaton, then derved a closed-form measure of performance for one of them, the famlar frst-come-frst-served (FCFS) polcy, and proposed a comparable approxmaton for the other, the longest-queue (Q) polcy. These results were llustrated, tested through smulatons, and used to address several basc manageral ssues. Khoury et al. (2001) consdered a producton faclty that s used to produce more than a sngle tem n order to maxmze ts utlzaton. They assumed the exstence of capacty constrant and analyzed the lot schedulng problem under such an nsuffcent capacty. A two-product problem usng the common cycle approach was examned. Extenson to any number of products s dscussed. Clausen & Ju (2006) studed an economc lot and delvery schedulng problem (EDSP), where a suppler produces and delvers components of dfferent types to a consumer n batches. The objectve s to determne the cycle tme, whch mnmzes the total cost per tme unt. The purpose of ther study ncludes the determnaton of the producton sequence of the component types wthn each cycle. Computatonal behavor of two publshed algorthms; a heurstc and an optmal algorthm were nvestgated. Wth large number of component types, the optmal algorthm has long runnng tmes, so they developed a hybrd algorthm to assst them to fnd optmal soluton n a more effcent manner. Guchhat et al. (2010) consdered a mult-tem nventory model of breakable tems where demands of the tems are stock dependent, and breakablty rates ncrease lnearly wth stock and nonlnearly wth tme. Due to non-lnearty and complexty of the problem, ther model was solved numercally and fnal decsons were made usng genetc algorthm (GA). They consdered both fuzzy and stochastc uncertan nventory costs, and a chance constraned approach was used to deal wth smultaneous presence of stochastc and fuzzy parameters. Dfferent numercal examples were used to llustrate dfferent cases of the problem. Björk (2012) developed a fuzzy multtem economc producton quantty (EPQ) model, where a company has to decde the sze of some producton batches under uncertan cycle tmes. The uncertanty was handled wth trangular fuzzy numbers and an analytcal soluton was found to the optmzaton problem. Chu et al. (2013a) derved the optmal common producton cycle tme for a mult-tem fnte producton rate (FPR) model wth random scrap rate and n fxed quantty multple delvery polcy. Mathematcal modelng along wth optmzaton technque s used to solve the problem. A closed-form optmal common

6 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 108 producton cycle tme that mnmzes the expected system costs s obtaned. Effect of scrap rate on the optmal common cycle tme was nvestgated and dscussed through a numercal example. Studes related to dfferent aspects of the mult-tem producton plannng and optmzaton ssues can also be found n Jodlbauer & Retner (2012); Chu et al. (2013d). Amng at lowerng vendor s nventory holdng cost, ths paper extends Chu et al. s work (2013a) by ncorporatng an mproved n+1 shpment polcy nto ther model. Under the proposed polcy, one extra delvery of fnshed tems s made n vendor s producton uptme to satsfy buyer s product demands durng the uptme. Then, upon completon of producton, n fxed quantty nstallments of fnshed tems are delvered to buyers at a fxed tme nterval. The objectves are to determne the optmal common producton cycle tme that mnmzes the long-run average system cost per unt tme, and study the effects of random scrap rate and mproved delvery polcy on the optmal cycle tme and on the system costs. MODEING AND FORMUATIONS In ths paper a mult-product economc producton quantty model wth random scrap rate and an mproved mult-delvery plan under the rotaton cycle polcy s studed. Descrpton of the proposed model s as follows: consder products are manufactured n turn on a sngle machne and all tems are screened and nspecton cost s ncluded n unt producton cost C (where = 1, 2,, ). Durng producton uptme of product there s a x porton of random scrap tems produced at a rate d. Under the regular operatng polcy, shortages are not allowed, so the constant producton rate for product, P must satsfy (P d λ ) > 0, where λ ι s the annual demand rate for product, and d can be expressed as d = x P. For the purpose of lowerng producer s nventory holdng cost, ths study extends the work of Chu et al. (2013a) by adoptng a specfc n+1 mult-shpment polcy, and under the proposed n+1 delvery polcy an ntal shpment of fnshed goods s to meet customer s product demands durng producer s uptme; then upon completon of regular producton n fxed quantty nstallments of the fnshed products are transported to customers, at a fxed tme nterval t n (see Fgure 1). Fgure 2 depcts producer s on-hand nventory level of perfect qualty tems of product n the proposed n+1 delvery model (n blue lnes) and the expected reducton n producer s stock holdng costs (green shaded area) n comparson wth that n Chu et al. (2013a) (n black lnes). Addtonal cost varables used n ths study nclude: producton setup cost K (where = 1, 2,, ), unt holdng cost h, dsposal cost C S per scrapped tem, the fxed delvery cost K 1 for product per delvery, and unt shppng cost C T for each product. Other parameters used n ths paper are lsted as follows:

7 109 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung Fg. 1. On-hand nventory of perfect qualty tems n the proposed mult-product EPQ model under rotaton cycle tme polcy Fg. 2. Expected reducton n producer s stock holdng costs (n green) for each product n the proposed model n comparson wth Chu et al. s model (2013a) T = rotaton cycle tme - the decson varable, t = tme needed to produce enough tems of product to meet customer s demand durng uptme t 1, t 1 = producton uptme for product, t 2 = the delvery tme for product, H 1 = level of on-hand nventory n unts of product for meetng customer s demand durng uptme t 1,

8 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 110 H = maxmum level of on-hand nventory of product when the regular producton ends, n = number of fxed quantty nstallments of the fnshed lot to be delvered to customers n each cycle, Q = producton lot sze per cycle for product, t n = fxed nterval of tme between each nstallment of fnshed product beng delvered durng t 2, I(t) = on-hand nventory of perfect qualty tems at tme t, I S (t) = on-hand nventory of scrap tems for product at tme t, TC(Q ) = total producton-nventory-delvery cost per cycle for product, E[TCU(Q )] = total expected producton-nventory-delvery cost per unt tme for products n the proposed model under the rotaton cycle tme polcy, E[TCU(T)] = total expected producton-nventory-delvery cost per unt tme for products n the proposed system usng the rotaton cycle tme T as decson varable. By observng from Fgure 1, the followng formulas can be drectly obtaned for any = 1, 2,, : T t t Q = = (1) λ t H1 λt1 = = P d P d (2) t 1 Q H + H = = P P d 1 (3) t = T t = nt (4) 2 1 n H = λ t (5) 1 1 ( )( ) H = P d t t (6) 1

9 111 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung λ = λ (7) = 1 Fgure 3 llustrates on-hand nventory of scrap tems n the proposed mult-product EPQ model under rotaton cycle tme polcy. From Fgure 2, the total scrap tems of product can be obtaned as follows: dt = xq (8) 1 Fg. 3. On-hand nventory of scrap tems n the proposed mult-product EPQ model under rotaton cycle tme polcy Total delvery cost for product (.e., n+1 shpments) n a gven producton cycle s ( ) 1 T n+ K + C Q (9) 1 The varable holdng costs for fnshed tems of product durng delvery tme t 2 are as follows (Chu et al., 2009): n 1 h H t 2n 2 (10) Total producton-nventory-delvery cost per cycle for products, conssts of producton setup cost, varable producton cost, varable dsposal cost, fxed and varable delvery cost, holdng cost durng producton uptme t 1, and holdng cost for fnshed goods kept durng the delvery tme t 2. Hence, total TC(Q ) for products are

10 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 112 ( ) ( 1) K + CQ + CS xq + n+ K1 + C TQ TC( Q ) = H H dt n 1 + () + ( ) + ( ) n 1 1 = 1 = 1 h t t1 t t1 Ht2 (11) The defectve rate x s assumed to be a random varable wth a known probablty densty functon; hence, to take the randomness of x nto account the expected value of x s used n ths study. Substtutng all varables from equatons (1) to (10) n Eq. (11) and wth further dervatons the expected E[TCU(Q )] can be obtaned as K Cλ + + C λe x + C λ + [ ] ( n+ 1) 1 S T T T () 2 = 2 1 ht λ 1 2λ = + λ P P(1 E[ x] ) P n λ P E TCU T K (12) Dervaton of the optmal rotaton cycle tme As stated earler, the objectve of ths study s to determne the optmal rotaton producton cycle tme to mnmze the long-run average system cost per unt tme (.e., Eq. (12)). In order to obtan the optmal rotaton cycle tme T* one must frst prove the exstence of the mnmum of the expected cost functon E[TCU(T)]. Because the system cost has one decson varable, T, dfferentatng E[TCU(T)] wth respect to T gves the frst and the second dervatve of E[TCU(T)] as [ ( )] de TCU T dt K ( n+ 1) K T T = hλ 1 2λ = + λ P P(1 E[ x] ) P n λ P (13) [ ( )] 2 K + ( n+ 1) K 1 = 2 3 (14) 2 d E TCU T dt = 1 The second-order dervatve s postve because K, n, K 1, and T are all postve. Thus, E[TCU(T)] s a convex functon for all T dfferent from zero. It follows that the optmal rotaton producton cycle tme T* can be obtaned by settng the frst dervatve of E[TCU(T)] as equal to zero: T

11 113 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung [ ( )] de TCU T dt K ( n+ 1) K T T = hλ 1 2λ = = + λ P P(1 E[ x] ) P n λ P 0 (15) or hλ 1 2λ K ( n 1) K1 1 T + + = λ + + = 1 = 1 2 P P(1 E[ x] ) P1 n λ P (16) Wth further dervatons one obtans T ( ) 2 K + n+ 1 K 1 * = 1 = 2 2 hλ 1 2λ λ = 1 2 P P(1 E[ x] ) P n λ P (17) Effect of producton setup tme on the rotaton cycle tme The producton setup tme s relatvely short n general, when compared to the producton uptme. However, f setup tme becomes a factor, one has to check whether the producton cycle length s long enough to account for the setup and producton tmes of products (Chu et al., 2009). et S denote producton setup tme for product ; Eq. (18) must hold n order to ensure that each cycle has suffcent tme for the setup and producton of products. S + ( Q / P) < T (18) = 1 Substtutng Eq. (1) n Eq. (18) gves S = 1 ( λ P) T > = T 1 / = 1 Hence, f the setup tme becomes a sgnfcant factor n producton plannng, one should choose the optmal rotaton cycle tme from max[t*, T mn ] (Chu et al., 2009). mn. NUMERICA EXAMPES In order to lessen comparson efforts for readers the same numercal example s used n ths secton as n Chu et al. (2013a). Consder a producton plan for makng fve (19)

12 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 114 dfferent products n turn on a sngle machne under a rotaton cycle tme polcy. Assume that for product = 1, 2,, 5, respectvely, the annual producton rate P s 58000, 59000, 60000, 61000, and 62000; the demand rate λ s 3000, 3200, 3400, 3600, and 3800 per year; the random defectve rate x durng the producton processes follows the unform dstrbuton over the ntervals of [0, 0.05], [0, 010], [0, 0.15], [0, 0.20], and [0, 0.25]; all defectve tems are not reparable; and the dsposal cost C S s $20, $25, $30, $35, and $40. Other system varables used for product = 1, 2,, 5 nclude the followng: K = the producton setup costs are $3800, $3900, $4000, $4100, and $4200, respectvely. h = unt holdng costs are $10, $15, $20, $25, and $30, respectvely. C = unt producton costs are $80, $90, $100, $110, and $120, respectvely. K 1 = fxed costs per delvery are $1800, $1900, $2000, $2100, and $2200, respectvely. C T = unt transportaton costs are $0.10, $0.20, $0.30, $0.40, and $0.5, respectvely. n = number of shpments per cycle; n ths study, t s assumed to be a constant 3 (.e., n+1 = 4). Applyng Eq. (17) the optmal rotaton cycle tme T* = (years) can be obtaned, and from computaton of Eq. (12), one fnds that the expected system cost for makng products E[TCU(T*)] = $2,097,903. The varaton of the average defectve rate of x effects on the expected system cost E[TCU(T)] and on varous components of E[TCU(T*)] s llustrated n Fgure 4. It s noted that, as average defectve rate ncreases, the dsposal cost (for the scrap tems) and the expected system cost E[TCU(T*)] ncreases sgnfcantly. Fg. 4. Varaton of average defectve rate effects on the expected system cost E[TCU(T)] and on varous components of E[TCU(T)]

13 115 Me-Fang Wu, Yuan-Shy Peter Chu and Peng-Cheng Sung Fg. 5. Varaton of rotaton producton cycle tme T effects on the expected system cost E[TCU(T)] Varaton of rotaton producton cycle tme T effects on the expected system cost E[TCU(T)] s depcted n Fgure 5. The proposed model s amed at reducng the producer s nventory holdng cost for each product durng the producton cycle (see Fgure 2). In ths example, the total system cost savngs are $15,291 or 3.89% of total other related costs (.e., total system cost E[TCU(T)] excludes varable producton cost λ C ). A further analyss shows that the percentage of reducton of total holdng cost s 16.7% (.e., reduced from $98,906 [Chu et al., 2013a] to $82,424; see Fg. 6). Further analyss ndcates that the relatve reductons of nventory holdng cost for each product are $429, $1,351, $2,776, $4,712, and $7,251; or 5.72%, 10.68%, 14.78%, 18.19%, and 21.17%, respectvely. In summary, through the aforementoned numercal example, sgnfcant savngs n the vendor s stock holdng cost are demonstrated n our research result. Fg. 6. Comparson of total nventory holdng costs of the proposed n+1 delvery polcy wth that n Chu et al. s model (2013a)

14 Optmzaton of a mult-product EPQ model wth scrap and an mproved mult-delvery polcy 116 An addtonal example to demonstrate the applcablty of the proposed model s as follows. Suppose management of a manufacturng frm has dfferent manufacturng process optons for a mult-product producton system, and assocated wth these optons are dfferent mean scrap rates over the nterval of [0.03, 0.30]. Applyng the research results obtaned from the present study, the effects of varous scrap rates on the optmal rotaton cycle tme T and on the expected system cost E[TCU(T)] can be determned and llustrated as n Fgure 7. It s noted that as mean scrap rate ncreases, the optmal rotaton cycle tme T* decreases slghtly, but the expected system cost E[TCU(T)] ncreases sgnfcantly. In summary, through the aforementoned numercal example, effects of mean scrap rates on the optmal rotaton cycle tme and on the expected system cost can all be obtaned by the use of results of the proposed study. CONCUDING REMARKS Wth a prmary ntenton of reducng the vendor s nventory holdng cost, ths study extends the work of Chu et al. (2013a) by ncorporatng an mproved n + 1 multdelvery plan nto ther model. Mathematcal modelng and optmzaton technques are used to solve the proposed problem. As a result, the optmal rotaton producton cycle tme that mnmzes the long-run average system cost per unt tme s derved. Through a numercal example, the practcal usage of the obtaned result s demonstrated (see Fgures 4 and 5) and the sgnfcant savngs n the vendor s stock holdng cost as compared to pror work (Chu et al., 2013a) s confrmed (see Fgures 2 and 6). In concluson, the objectve of lowerng the vendor s holdng cost has been successfully accomplshed. For future study, one nterestng drecton wll be to consder the effect of stochastc demand rates on the optmal common cycle tme. Fg. 7. Effects of varous scrap rates on the optmal rotaton cycle tme T* and on the expected system cost E[TCU(T)]

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