An Integrated Inventory Model with Controllable Lead time and Setup Cost Reduction for Defective and Non-Defective Items

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1 Internatonal Journal of Supply and Operatons Management IJSOM August 14 Volume 1 Issue pp ISSN-Prnt: ISSN-Onlne: An Integrated Inventory Model wth Controllable Lead tme and Setup Cost Reducton for Defectve and Non-Defectve Items M. Vjayashree a R. Uthayakumar a a Department of Mathematcs The Gandhgram Rural Insttute Deemed Unversty Gandhgram Dndgul Taml Nadu Inda Abstract In ths paper the study deals wth the lead tme and setup reducton problem n the vendorpurchaser ntegrated nventory model. The cost of captal (.e. opportunty cost) s one of the key factors n makng the nventory and nvestment decsons. Lead tme s an mportant element n any nventory system. The proposed model s presents an ntegrated nventory model wth controllable lead tme wth setup cost reducton for defectve and non defectve tems under nvestment for qualty mprovement. In ths analyss the proposed model we assumed that the setup cost and process qualty s logarthmc functon. Setup cost reducton for defectve and non defectve tems s the man focus for the proposed model. The objectve of the proposed model s to mnmze the total cost of both the vendor-purchaser. The mathematcal model s derved to nvestgate the effects to the optmal decsons when nvestment strateges n setup cost reductons are adopted. Ths paper attempts to determne optmal order quantty lead tme process qualty and setup cost reducton for producton system such that the total cost s mnmzed. A soluton procedure s developed to fnd the optmal soluton and numercal examples are presented to llustrate the results of the proposed models. correspondng emal address: vjayashreemay3@gmal.com 19

2 Int J Supply Oper Manage (IJSOM) Keywords: Integrated nventory model Vendor-purchaser coordnaton Lead tme crashng cost Setup cost reducton for defectve and non defectve tems. 1. Introducton In modern producton management controllable lead tme and setup cost reducton are keys to busness success and have attracted consderable research attenton. Setup cost reducton s one of the mportant producton actvtes n an ntegrated nventory control. Setup tme reducton n manufacturng operatons s wdely recognzed to provde sgnfcant benefts n areas such as cost aglty and qualty. Many technques are avalable to mprove setup tme such as a revsng setup procedure modfyng toolng for standardzed locatng and clampng or ntroducng robotc change over equpment to name a few. Each technque wll provde a certan level of beneft and has assocated costs. Gven that the goal of the frm s to select the setup cost reducton technques that wll mnmze ther overall costs. In tradtonal economc order quantty (EOQ) and economc producton quantty (EPQ) models most of the lterature treatng nventory proble ether n determnstc or probablty models the stock out or setup cost are regarded as prescrbed constants and equal at the optmum. In modern producton management controllable lead tme and orderng cost reducton and setup cost reducton are keys to busness success and have attracted consderable research attenton. Orderng quantty servce level and busness compettveness can be shown to possbly be nfluenced drectly or ndrectly va lead-tme and or orderng cost control. The ntegrated nventory models treat the orderng cost and setup cost /or lead tme as constants. However n some practcal stuatons lead tme and orderng cost can be controlled and reduced n varous ways. However n practce setup cost can be controlled and reduced through varous efforts such as worker tranng procedural change and specalzed equpment acquston. Through the Japanese experence of Just-n-Tme (JIT) producton the advantages and benefts assocated wth efforts to reduce the setup cost can be clearly supposed. The Just-n-Tme (JIT) system plays an mportant role n present supply chan management. One of the major tasks of mantanng the compettve advantages of JIT producton s to compress the lead tme needed to perform actvtes assocated wth delverng hgh-qualty products to customers. In the dynamc compettve envronment successful companes have devoted consderable attenton to reducng nventory cost and lead tme and mprovng qualty smultaneously. In recent years ndustres have devoted consderable attenton to reducng 191

3 Vjayashree and Uthayakumar nventory cost. We have developed an ntegrated nventory model wth controllable lead tme wth setup cost reducton for defectve and non-defectve tems.. Lterature Revew In recent years most nventory problems have ther focus on the ntegraton between the vendor and the buyer. For the supply chan management establshment of long term strategc partnershps between the buyer and the vendor s advantageous for the two partes regardng costs and therefore profts snce both partes cooperate and share nformaton wth each other to acheve mproved benefts. In Several researchers have shown that the buyer and the vendor can acheve ther own mnmum total cost or ncrease ther mutual beneft through strategc cooperaton wth each other. In the current supply chan management (SCM) envronment companes are usng JIT producton to gan and mantan a completve advantage. JIT requres a sprt of cooperaton between the buyer and the vendor and t has been shown that formng a partnershp between the buyer and the vendor s helpful n achevng substantal benefts for both [Goyal (199)]. In ths complex envronment successful companes have devoted consderable attenton to reducng nventory cost and mprovng qualty smultaneously. The Japanese have successfully ntroduced the just-ntme (JIT) manufacturng phlosophy whch calls for small producton quanttes and low nventores. Small producton quanttes reduce rework and scrap and provde flexblty wth frequent producton runs. It s well known that setup costs/tmes must be reduced n order to reduce producton quanttes. Setup costs/tmes are costs/tmes that t takes to go from the producton of the last good pece of a pror run to the frst good pece of a new producton run. The reduced setup cost can hence result n benefts such as less nvestment n nventory mproved qualty and flexblty [Hall (1983) and Schonberger (198)]. Ths research focused on only one aspect of the advantage of reducng setup costs/tmes namely reduced nventory related costs. In the classcal economc order quantty (EOQ) models the vendor s and buyer s nventory problems are treated n separately and the EOQ formula can gve an optmal soluton for them respectvely. Ths ndependent decson behavor usually cannot assure that the two partes as a whole reach the optmal status. Therefore durng the past four decades many researchers pay more attenton on the problem of jont replenshment that mnmzes the total relevant costs for the vendor and the buyer. 19

4 Int J Supply Oper Manage (IJSOM) The ntegraton between vendor and buyer for mprovng the performance of nventory control has receved a great deal of attenton to the ntegrated approach has been examned for years. Goyal (1976) s among the frst researchers who analyzed an ntegrated nventory model for sngle vendor and sngle buyer system. The framework he proposed has encouraged many researchers to present varous types of ntegrated nventory system. For nstance Banerjee (1986) assumed that the vendor manufactures at a fnte rate and consdered a jont economc lot-sze model n whch a vendor produces to order for a buyer on lot-for-lot bass. Goyal (1988) relaxed the lot-for-lot polcy and suggested that vendor s economc producton quantty should be an nteger multple of buyer purchase quantty. As a result of usng the approach suggested n Goyal s (1988) model sgnfcant reducton n nventory cost can be acheved. Later numerous researchers [see Goyal () Hoque (6) Pandey (7) Tang (4) Teng (11) and Tsou (9)] addressed ntegrated producton-dstrbuton nventory models extendng the deal of Goyal (1988) under varous assumptons. Lead tme reducton s another mportant producton actvty n an ntegrated nventory control. Lead tme can be reduced by an addtonal crashng cost; In other words lead tme s controllable. Accordng to Hsu et al. (9) crashng cost could be expendtures on equpment mprovement nformaton technology order expedte or specal shppng and handlng. By shortenng lead tme buyers can lower the safety stock reduce the out-of-stock loss mprove the customer servce level and ncrease the compettve advantage of busness. Therefore lead tme reducton has been one of the most offered themes for both researchers and practtoners. Lead tme s another essental factor n any supply chan and nventory management system. It generally conssts of numerous consttuents such as order preparaton suppler lead tme delvery and setup tme. Controllng lead tme properly and takng the optmal order quantty are very mportant n attanng the mnmum total nventory cost. The ssue of lead tme reducton has receved a lot of nterest n recent years. Lead tme reducton has numerous benefts ncludng quckly flled customer orders and reducng the fnshed goods nventory level. Lead tme management s a sgnfcant ssue n producton and operaton management. In fact lead tme usually conssts of the followng components [Tersne (1994)]: order preparaton order transt suppler lead tme delvery tme and setup tme. In many practcal stuatons lead tme can be reduced at an added crashng cost; n other words t s controllable. By shortenng the lead tme we can lower the safety stock reduce 193

5 Vjayashree and Uthayakumar the stockout loss and mprove the customer servce level so as to gan compettve edges n busness. The Japanese experence of usng Just-n-Tme (JIT) producton showed that the benefts assocated wth lead tme control are clear. Therefore reducng lead tme s both necessary and benefcal. Inventory models ncorporatng lead tme as a decson varable were developed by several researchers. Lao et al. (1991) frst devsed a probablty nventory model n whch lead tme was the unque decson varable. Ben-Daya et al. (1994) extended Lao et al. (1991) model by consderng both lead tme and order quantty as decson varables. Later some studes [Moon (1998) Ouyang (1999 b ) Ouyang (1999 c ) Ouyang () and Ouyang (1999 d )] n the feld of lead tme reducton generalzed Ben-Daya et al. (1994) model by allowng reorder pont as one of the decson varables. Later many researchers (see Glock (1) Ho (9) and Ouyang (7)) nvestgated varous ntegrated producton-nventory models for lead tme reducton n sngle-vendor sngle-buyer supply chan. In the recent year Glock (1 a ) developed the jont economc lot sze problem. And also Glock (1) developed lead tme reducton strateges n a sngle-vendor- sngle buyer ntegrated nventory model wth lot sze-dependent lead tme and stochastc demand. Hoque (13) developed a vendor buyer ntegrated producton nventory model wth normal dstrbuton of lead tme. The Just-n-Tme (JIT) system has receved a great deal of attenton n the feld of producton /nventory management. From an nventory standpont the ultmate goal of JIT s to produce/order small lot-sze and to reduce nventory level whch can be acheved by reducng setup cost. In the lterature many authors have presented the analytcal models to dscuss the effects of nvestng n reduce setup cost. The ntal result n the development of the setup cost reducton model s that of Porteus (1985) who ntroduced the concept and developed a framework of nvestng n reducng setup cost on EOQ model. The framework has encourage many researchers such as Keller et al. (1988) Nars et al. (199) Km et al. (199) and Paknejad et al. (1987) to examne setup cost reducton. Then Ouyang et al. (1999 a ) nvestgated the nfluence of orderng cost reducton on modfed contnuous revew nventory systems nvolvng varable lead tme wth partal backorders. Later Woo et al. (1) developed an ntegrated nventory model for a sngle vendor and multple buyers wth orderng cost/setup cost reducton. Zhang (7) extended Woo et al. s (1) model by relaxng the assumpton that the cycle tmes for all buyers and the vendor are the same. Later 194

6 Int J Supply Oper Manage (IJSOM) some researchers [see Chang (6) Coates (1996) and Km et al. (199)] addressed setup cost/orderng cost reducton nventory models under varous assumptons. Qualty has been hghly emphaszed n modern producton/nventory management systems. Also t has been evdenced that the success of Just-In-Tme (JIT) producton s partly based on the belef that qualty s a controllable factor whch can be mproved through varous efforts such as worker tranng and specalzed equpment acquston. In the classcal Economc Order Quantty (EOQ) model the qualty-related ssue s often neglected; t mplctly assumes that qualty s fxed at an optmal level (.e. all tems are assumed wth perfect qualty) and not controllable. However ths may not be true. In a real producton envronment we can often observe that there are defectve tems beng produced. These defectve tems must be rejected repared reworked or f they have reached the customer refunded; and n all cases substantal costs are ncurred. Porteus (1986) and Rosenblatt et al. (1986) were the frst to explctly elaborate on the sgnfcant relatonshp between qualty mperfecton and lot sze. Specfcally Porteus (1986) extended the EOQ model to nclude a stuaton where the producton process s mperfect and based on ths model he further studed the effects of nvestment n qualty mprovement by ntroducng the addtonal nvestng optons. Snce Porteus (1987) several authors proposed the qualty mprovement models under varous settngs see e.g. Keller et al. (1988) Hwang et al. (1993) Moon (1994) Hong et al. (1995) and Ouyang et al. (1999). Pan et al. () have developed an ntegrated suppler-purchaser model focused on the beneft from lead tme reducton. Pan et al. (4) have developed an ntegrated nventory model nvolvng determnstc varable lead tme and qualty mprovement nvestment. The objectve of ths paper s to present the vendor-purchaser ntegrated producton nventory model to reduce the setup cost for defectve and non defectve tems under nvestment for qualty mprovement. The remander of ths paper s organzed as follows. Secton 3 descrbes the notatons and assumptons used throughout ths paper. In secton 4 Model formulatons for two cases. Case 1 for non defectve tems wth setup cost reducton n secton 4.1. An effcent algorthm s developed to obtan the optmal soluton for non defectve tems n secton 4.. Case for defectve tems under nvestment for qualty mprovement wth setup cost reducton n secton 4.3. An effcent algorthm s developed to obtan the optmal soluton for defectve tems under nvestment for qualty mprovement n secton 4.4. An llustratve numercal example 195

7 Vjayashree and Uthayakumar has provded both defectve and non defectve tems n secton 5 to llustrate the results. Fnally we draw some concluson n secton Notatons and Assumptons To establsh the mathematcal model the followng notatons and assumptons of the model are as follows 3.1. Notatons To develop the proposed model the followng notatons are used D P Q A S S L C v p Average demand per unt tme on the purchaser Producton rate of the vendor Order quantty of the purchaser (Decson Varable) Purchaser s orderng cost per order Vendor s setup cost per set-up (Decson Varable) Orgnal vendor s setup cost for each producton run. Length of lead tme n weeks Unt producton cost pad by the vendor C Unt purchase cost pad by the purchaser C < C v p m α g The number of delveres of the product delvered from the vendor to the purchaser n one producton cycle a postve nteger (Decson Varable) Vendor s fractonal opportunty cost of captal per unt tme. (e. g. nterest rate) Vendor s unt rework cost per defectve tem I (S) Captal nvestment requred to acheve setup cost < S S S I(θ ) Captal nvestment requred to reduce the out-of-control probablty from θ to θ δ Percentage decrease S per dollar ncrease n nvestment I (S) λ Percentage decrease θ per dollar ncrease n nvestment I (θ ) r θ θ Annual nventory holdng cost per dollar nvested n stocks. Probablty of the vendor s producton process that can go out-of-control. (Decson Varable) Orgnal probablty of the vendor s producton process that can go out-of-control. 196

8 Int J Supply Oper Manage (IJSOM) 3.. Assumptons The assumptons made n the paper are as follows 1. The product s manufactured wth a fnte producton rate P and P > D.. The system conssts of a sngle vendor and a sngle purchaser for a sngle commodty has been consdered. 3. Inventory s contnuously revewed. The purchaser places the order when the nventory poston reaches the reorder pont R. 4. The reorder pont R = the expected demand durng lead tme+ safety stock ( SS ) and ( standard devatonof lead tme demand) SS = k that s R = DL + kσ L where k s a safety factor. 5. The demand X durng lead tme L follows a normal dstrbuton wth mean µ L and standard devatonσ L. 6. The lead tme L conssts of n mutually ndependent components. The th component has a normal duraton b mnmum duraton a and crashng cost per unt tme c. For convenence we rearrange c such that c 1 < c < c3 <... < cn. The components of lead tme are crashed one at a tme startng from the frst component because t has the mnmum unt crashng cost and then the second component and so on. 7. Let L o = b and n = 1 L be the length of lead tme wth components crashed to ther mnmum duraton and then L can be expressed as ( b a ) = 1 n L = Lo = j j... and the lead tme crashng cost per cycle R j 1 1 ( L) c ( L L) + c ( b a ) L [ L L ] 1 j= 1 j j j 1 = 197. In addton the length of lead tme s equal for all shppng cycles and the lead tme crashng costs occur n each shppng cycle. See Lao et al. (1991). 8. The relatonshp between lot sze and qualty s formulated as follows: whle vendor s producng a lot the process can go out of control wth a gven probablty θ each tme another unt s produced. The process s assumed to be n control n the begnnng of the producton process. Once out of control the process produces defectve tems and contnues to do so untl the entre lot s produced. (Ths assumpton s n lne wth Porteus (1986)).

9 Vjayashree and Uthayakumar 9. The extra costs ncurred by the vendor wll be fully transferred to the purchaser f shortened lead tme s requested. 1. Once the producton process shfts to an out-of-control state the shft cannot be detected untl the end of the producton cycle and the process contnuous producton and a fxed proporton of the produced tems are defectve. 11. All defectve tems produced are detected after the producton cycle s over and rework cost for defectve tems wll be ncurred. 1. We assume that the captal nvestment I ( S ) n reducng setup cost s gven by a logarthmc functon I( S) decrease S per dollar ncrease n nvestment I (S). S 1 = q ln for < S S where q = and δ s percentage S δ 13. We assume that captal nvestment I (θ ) n mprovng process qualty (reducng out-ofcontrol probablty for θ I θ = q 1 ln for θ θ toθ ) s gven by a logarthmc functon ( ) 1 < θ θ where q 1 = and λ s percentage decreaseθ per dollar ncrease n λ nvestment I (θ ).(Ths nvestment functon has also been used n [Porteus (1986) Keller(1988) and Hong (1993)]. 14. Defectve tem rework cost per unt tme: the expected number of defectve tems n a run of sze mq wth a gven probabltyθ that the process can go out of control s m Q θ (see Porteus (1986) for detal dervaton). Thus the defectve cost per unt tme s gven by gmqd θ. 4. Model formulaton for both defectve and non defectve tems The purchaser places an order after every Q demand; therefore for average cycle tme of D Q the expected orderng and lead tme crashng costs per unt can be gven by AD and ( ) DR L Q Q respectvely. Purchaser orderng cost per year = Q DA 198

10 Int J Supply Oper Manage (IJSOM) D Q Purchaser lead tme crashng cost per year = R( L) The expected net nventory level just before arrval of a procurement s the safety stock s = R DL. The expected net nventory level mmedately after arrval of procurement s Q Q + s. Hence the average nventory over the cycle can be approxmated by + s.e. Q + kσ Q rcp + kσ L. L (Assumpton 3) so the purchaser s holdng cost per unt tme s Q Purchaser holdng cost per year = + kσ L rc The vendor-purchaser ntegrated system s desgned for a vendor s producton stuaton n whch once the purchaser orders a lot sze Q the purchaser begns producton wth a constant producton rate P and a fnte number of unts are added to nventory untl the producton run has been completed. The vendor produces the tem n lot of sze mq n each producton p cycle of length mq and the purchaser wll receve the supply n m lots each of szeq. The D frst lot of sze Q s ready for delveres after tme P Q just after the start of the producton and then the vendor contnues makng the delvery on average every D Q unts of tme untl the SD nventory level falls to zero ( see Fg 1). The total cost per unt tme s gven by. ( mq) Vendor setup cost per year = D mq S Vendor s average nventory s evaluated as the dfference of the vendor s accumulated nventory and the purchase s accumulated nventory. That s Q Q m Q Q mq + ( m 1) ( ( m 1)) P D P D mq D 199

11 Vjayashree and Uthayakumar Q D D = 1 1 m + P P So the vendor s holdng cost per unt tme s Q D D m = 1 1+ rc v P P. Fgure 1. The nventory pattern for the vendor and the purchaser S Opportunty cost for setup cost reducton = q ln S Opportunty cost for process qualty = q ln 1 θ Defectve cost per year = gmqdθ 4.1. Case 1 - Non defectve tems wth setup cost reducton θ In ths secton a model s developed for the vendor-purchaser ntegrated system to mnmze the total cost per unt tme of the vendor-purchaser ntegrated syste s the sum of the

12 Int J Supply Oper Manage (IJSOM) orderng cost holdng cost and lead tme crashng costs per unt tme for the purchaser nvestment cost requred for setup cost reducton and holdng costs per unt tme for the vendor. In addton the target value of S s constraned on < S S As a result n mathematcal 1 symbolzaton. Therefore the problem under study can be formulated as the followng non lnear programmng model ( Q L m S) = TC Orderng cost + vendor s holdng cost+ purchaser s holdng cost + TC opportunty cost for setup cost+ purchaser s lead tme crashng cost DA D D Q D D ( Q L S) = + R( L) + S + m 1 1+ rc + v rc p Q Q mq P P + + kσ S L rcp + αq ln S Q D S Q D D A + + R Q m ( L) + r m 1 1+ C + v C P P = p + S kσ L + αq ln (1) S rc p Subject to < S S where α s the annual fractonal cost of captal nvestment (e.g. nterest rate). To solve the above nonlnear programmng proble ths study temporarly gnores the constrant < S S and relaxes the nteger requrement on m (the number of shpments from the vendor to the purchaser durng one producton cycle). For fxed ( Q L m S ) Q S and L [ L L ] TC can be proved to be a convex functon of m. Consequently the search for the optmal delvery m s reduced to fnd a local mnmum. Property 1. For fxed Q S and L [ L L ] TC( Q L ) s convex n m. 1 Takng the frst and second partal dervatves of TC ( Q L S) wth respect to m we have TC and TC ( Q L m) m ( Q L m) m DA Q = + C Qm DA = 3 Qm > D p v 1 1

13 Vjayashree and Uthayakumar Therefore ( Q L ) TC s convex n m for fxed S Q and L [ L L ] 1. Ths completes the proof of Property 1. Next the frst partal dervatves of TC( Q S) wth respect to Q S and [ L L ] L are taken for fxed m respectvely. Ths process yelds 1 ( Q L m) TC D S r D D = A + + R () Q Q m ( Q L S m) TC TC S = D Qm αq S 1 ( Q L m) D 1 L ( L) + m 1 1+ C + v C p P P = c + rc pkσ L (4) Q Furthermore for fxed ( Q m) L TC [ L L ] 1 ( Q L m) L because Hence for fxed ( Q m) nterval [ L L ] Q = 1 TC ( Q L S ) (3) s noted to be a concave functon n 3 r = C pkσ L < (5) 4 the mnmum total cost per unt tme occurs at the end ponts of the. On the other hand by settng Equatons. () - (3) equal to zero we obtan S D A + + R( L) m D D r m 1 1+ C v + C p P P αqqm S = (7) D For fxed m and L [ L L ] 1 by solvng Equatons (6)-(7) we can obtan the values of Q S denote these value by ( Q S ). The subsequent algorthm s proposed to fnd the optmal value of order quantty Q Lead tme L Opportunty cost of setup cost reducton S and number of delveres m. (6) 4.. Algorthm n non defectve tems wth setup cost reducton Step 1 set m = 1 snce m s nteger. Step For each L = n. Perform (.1)-(.6) (.1) Start wth S 1 = S.

14 Int J Supply Oper Manage (IJSOM) (.) Substtute S 1 nto Eq. (6) evaluates Q 1. (.3) Utlzng Q 1 determne S s from Eq. (7) (.4) Repeat (.)-(.3) untl no change occurs n the values of Q S (.5) Compare S and S.5.1. If S < S then the soluton found n step 1 s optmal for the gven L. We denote the optmal soluton by ( Q ). then go to step (.6). S.5.. If S S o then take S = S and utlze Eq. (6) to determne the new procedure smlar to the one n (.). The result s denoted by ( ) Q Utlze Eq. (1) to calculate the correspondng TC( Q S L m) step (3). Step 3 Fnd TC( Q S L m). mn= n Let TC( Q ( m ) S L m ) TC( Q S L m) optmal soluton for fxed m. mn= n s S Q by. Then go to = then TC( Q S L ). s the Step 4 Set m = m + 1 repeat Step () step (3) to get TC( Q S L ) Step 5 If TC( Q S L m). TC( Q S L m 1) to step 6. ( m ) + ( m ) m. then go to step 4 otherwse go Step 6 Set TC( Q S L m ) = TC( Q( m + 1) S ( m+ 1) L ( m+ 1) m + 1) then ( Q S L m ) optmal soluton. TC s s the 4.3. Case - Defectve tems under nvestment for qualty mprovement wth setup cost reducton In ths secton a model s developed for the vendor-purchaser ntegrated system to mnmze the total cost per unt tme of the vendor-purchaser ntegrated syste t s the sum of the orderng cost holdng cost and lead tme crashng costs per unt tme for the purchaser nvestment cost requred for setup cost reducton nvestment cost requred for qualty mprovement and holdng costs per unt tme for the vendor. 3

15 Vjayashree and Uthayakumar In addton the target value of S θ s constraned on < S S < θ θ. As a result n mathematcal symbolzaton. Therefore the problem under study can be formulated as the followng non lnear programmng model ( Q L θ ) = TC Orderng cost + vendor s holdng cost+ purchaser s holdng cost + TC opportunty cost for setup cost + Purchaser s lead tme crashng cost+ opportunty cost for process qualty +defectve cost DA D D Q D D Q Q Q mq P P S gmqdθ θ + αq ln + + αq1 ln S θ D S Q ( ) D D = A + + R L + m 1 1+ rcv + rc p + gmdθ Q m P P S θ + rc p kσ L + α q ln + α q1 ln (8) S θ ( Q L θ S ) = + R( L) + S + rc v m kσ L rc p Subject to < S S < θ. where α s the annual fractonal cost of captal nvestment (e.g. nterest rate). θ To solve the above nonlnear programmng proble ths study temporarly gnores the constrant < S S < θ θ and relaxes the nteger requrement on m (the number of delveres from the vendor to the purchaser durng one producton cycle). For fxed and L [ L L ] ( Q L θ ) 1 Q θ TC can be proved to be a convex functon of m. Consequently the search for the optmal delveres Property. For fxed Q θ and [ L L ] we have L TC ( Q L m) 1 m s reduced to fnd a local mnmum. s convex n m. Takng the frst and second partal dervatves of TC( Q θ S) wth respect to m TC( Q θ L) m DS = Qm Q D + Cv 1 P + gdθ TC( Q θ S) DS = 3 m Qm > 4

16 Int J Supply Oper Manage (IJSOM) Therefore ( Q L S ) TC θ s convex n m for fxed Q S θ and L [ L L ] 1. Ths completes the proof of Property. Next the frst partal dervatves of TC( Q θ S) wth respect to Q θ and L [ L L ] are taken for fxed m respectvely. Ths process yelds 1 TC( Q θ Q L) D = Q A + S m + R( L) TC( Q θ L) QgmD αq1 = θ θ TC( Q θ L) = S 1 D D + m 1 1+ rc v + rc p + gmdθ P P D Qm αq S (9) (1) (11) TC 1 ( Q θ L) D 1 = c + rc pkσl L Q Furthermore for fxed ( Q θ m) ( Q L ) L TC [ L L ] 1 ( Q L m) L because Hence for fxed ( Q S m) the nterval L [ L L ] we obtan (1) TC θ s noted to be a concave functon n 3 r = C pkσ L < (13) 4 θ the mnmum total cost per unt tme occurs at the end ponts of 1. On the other hand by settng Equatons. (9) - (11) equal to zero Q = S D A + + R( L) m D D rcv m rc p + gmdθ P P (14) αqqm S = (15) D αq θ = 1 (16) gmdq For fxed m and L [ L L ] 1 by solvng Equatons (14)-(16) we can obtan the values of Q θ denote these value by ( Q θ S ). 5

17 Vjayashree and Uthayakumar The subsequent algorthm s proposed to fnd the optmal value of order quantty Q Lead tme L opportunty cost of setup cost reducton S opportunty cost for process qualty θ and number of delveres m Algorthm n defectve tems under nvestment for qualty mprovement and setup cost reducton Step 1 set m = 1 snce m s nteger. Step For each L = n. Perform (.1)-(.5.6).1 Start wth θ 1 = θ S =. 1 S. Substtuteθ 1 v 1 nto Eq. (14) whch evaluates Q 1..3 Utlzng Q 1 determnes S and θ from Eq. (15) and (16)..4 Repeat (.)-(.3) untl no change occurs n the values of Q S θ.5 Compare θ andθ and S and S respectvely..5.1 If S < S andθ < θ then the current soluton s optmal for the gven L.We ' ' ' denote the optmal soluton by ( Q θ S ). If ( Q θ S ). = ( Q θ S ) then go to step (.5.6) otherwse go to step (.5.)..5. If S < S andθ θ go to step (.5.3). If S S andθ < θ go to step (.5.4). If S S andθ θ then go to step (.5.5)..5.3 Let θ = θ and utlze Eq. (14) and (15) to determne the new ( Q S ) by a procedure smlar to the one n Step the result s denoted by Qˆ ˆ θ ). If Sˆ < S then ( the optmal soluton s obtaned.e. f ( Q θ S ). = Qˆ θ Sˆ ). then go to step (.5.6) otherwse go to step (.5.3.1). ( o Let S = S and utlze Eq. (14) to determne the new Q then go to step (.5.6)..5.4 let S = S and utlze Eq. (14) and (16) to determne the new ( Q θ ) by a procedure smlar to the one n Step the result s denoted by Q r S r r ). If θ < θ then ( 6

18 Int J Supply Oper Manage (IJSOM) the optmal soluton s obtaned.e. f ( Q θ S ). = r r θ S ). then go to step (.5.6) otherwse go to step (.5.4.1). ( Q Let θ = θ and utlze Eq. (14) to determne the new Q then go to step (.5.6)..5.5 Let S = S and θ = θ and utlze Eq. (14) to determne the new Q then go to step (.5.6)..5.6 Utlze Eq. (8) to calculate the correspondng TC( Q θ S L m). Then go to step (3). Step 3 Fnd TC( Q S θ L m). mn= n Let TC( Q S L s m m ) TC( Q S L m) m ( ) ( ) θ θ mn= n ( Q S L ) = then TC θ s the optmal soluton for fxed m. Step 4 Set = m + 1 m repeat step () step (3) to get TC( Q ( m ) L S m) Step 5 If TC( Q L θ S m) TC( Q L θ S m 1) (6) otherwse go to step (4). θ. + Step 6 Set ( Q L θ S m ). ( Q ( m+ 1) L ( m+ 1) θ S m 1) ( Q θ S L m ) s the set of optmal soluton. = ( ) ( m+ 1) ( m+ 1) + m 5. Numercal Examples for both defectve and non defectve tems then go to step then () Non defectve tems Consder an nventory system wth followng characterstcs D = 1 unt/ year P = 3 unt/ year A = $ 5 / order S = $4 / set up q = 35 C = $ / unt r =. α =.1 k =.33 σ v three components wth data shown n table 1. C ( ) = q ln p S = $ 5 / unt I S S and the lead tme has = 7 unt/week 7

19 Vjayashree and Uthayakumar Table 1. Lead tme data for the llustratve example Lead tme Normal Mnmum Unt component duraton b ( days) duraton a ( days) crashng cost c ( $/ days) Applyng the soluton procedure of the proposed algorth the computatonal results are demonstrated n table number of delveres m = optmal lead tme L = 4 weeks optmal order quantty Q = 15 unt Opportunty cost for setup cost reducton S = $88 Total Cost TC = $ Pctoral representatons and numercal analyss are presented to show the convexty of TC ( Q L) n fgure () & (3). Table. Summary of the soluton procedure for the llustratve example for non defectve tems L R ( L ) Q m = 1 m = m = 3 S TC Q S TC Q S TC

20 Int J Supply Oper Manage (IJSOM) Total Cos t (TC) Qrder Quantty (Q) Number of delvers (m) 3 Fgure. Pctoral representaton for optmal soluton for TC when L =4 weeks m= Q = T o t a l C o s t (T C ) () reducton Fgure 3. Pctoral representaton for optmal soluton Defectve tems under nvestment for qualty mprovement wth setup cost Consder an nventory system wth followng characterstcs D = 1 unt/ year P = 3 unt/ year q 1 = 4 θ = Number of delvers (m) A = $ 5 / order g = $15 / per defectve θ S = $4 / set up C p = $ 5 / unt I( θ ) = q1 ln θ unt C = $ / unt r =. α =.1 k =.33 v 9

21 Vjayashree and Uthayakumar S I( S ) = q ln q = 35 σ = 7 unt/week and the lead tme has three components wth data S shown on table 1. Applyng the soluton procedure of the proposed algorth the computatonal results are demonstrated n table 3 number of delveres m = optmal lead tme L = 4 weeks optmal order quantty Q = 118unt Opportunty cost for setup cost S = $83 Opportunty cost for process qualty θ =. 49 Total Cost TC = $ Pctoral representatons and numercal analyss are presented to show the convexty of TC( Q θ L) n fgure (4) & (5). Table 3. Summary of the soluton procedure for the llustratve example for defectve tems L R ( L ) Q m = 1 m = m = 3 S θ TC Q S θ TC Q S θ TC T o t a l C o s t (T C ) Qrder Quantty (Q) Number of delvers(m) 3 Fgure 4. Pctoral representaton for optmal soluton for TC when L =4 weeks m = Q =118 1

22 Int J Supply Oper Manage (IJSOM) Total Cos t (TC) Number of delvers(m) Fgure 5. Pctoral representaton for optmal soluton 6. Concluson Reducton n setup cost yeld varablty and lead tme are mportant strateges n manufacturng system. The prmary purpose of ths paper s to present the vendor-purchaser ntegrated producton nventory model under nvestment n qualty mprovements for defectve and non defectve tems. Many companes have recognzed the sgnfcance of lead tme as a compettve weapon and have used lead tme as a means of dfferentatng themselves n the market poston. In the producton envronment lead tme s an mportant element n any nventory management system. The mathematcal model s derved to nvestgate the effects of the best decsons when captal nvestment strateges n setup cost and nvestment for qualty mprovements are adopted. We developed an algorthm to mnmze the total cost of the vendor-purchaser ntegrated system by smultaneously optmzng the order quantty lead tme the number of delveres process qualty and setup cost reducton. In our model the captal nvestment n process qualty and setup cost reducton s assumed to be a logarthmc functon. An teratve algorthm was devsed to determne the optmal soluton for optmal order quantty process qualty lead tme setup cost reducton and number of delveres between the vendor and the purchaser. Furthermore a numercal s gven to llustrate the results for both defectve and non defectve tems. Pctoral representaton s also presented to llustrate the proposed model. 11

23 Vjayashree and Uthayakumar Acknowledgement The authors would lke to acknowledge the Edtor of the Journal and anonymous revewer for ther support and helpful clarfcaton n the revsng the paper. The 1 st author thankful to the DST INSPIRE Senor Research Fellowshp (SRF) Fellowshp Mnstry of Scence and Technology Government of Inda under the grant no. DST/INSPIRE Fellowshp/AORC- IF/UPGRD/14-15 dated References Banerjee A. (1986). A jont economc-lot-sze model for purchaser and vendor Decson Scences Vol.17 pp Ben-Daya M. and Raouf A. (1994). Inventory models nvolvng lead tme as decson varable Journal of the Operatonal Research Socety Vol. 45 pp Chang H.C. Ouyang L.Y. Wu K.S. and Ho C.H. (6). Integrated vendor- buyer cooperatve nventory models wth controllable lead tme and orderng cost reducton European Journal of Operatonal Research Vol. 17 pp Coates E.R. (1996). Manufacturng setup cost reducton Computers and Industral Engneerng Vol. 31 pp Glock C.H. (1). Lead tme reducton strateges n a sngle vendor-sngle buyer ntegrated nventory model wth lot sze-dependent lead tme and stochastc demand. Internatonal Journal of Producton Economcs Vol. 136 pp Glock C.H. (1 a ). The jont economc lot sze problem: A revew. Internatonal Journal of Producton Economcs Vol. 135 pp Goyal S.K. (1976). An ntegrated nventory model for a sngle suppler sngle customer proble Internatonal Journal of Producton Research Vol.15 pp Goyal S.K. (1988). A jont economc-lot-sze model for purchaser and vendor: A comment Decson Scences Vol.19 pp Goyal S.K. and Nebebe F. (). Determnaton of economc producton-shpment polcy for a sngle-vendor sngle-buyer syste European Journal of Operatons Research Vol. 11 pp Goyal S.K. and Srnvasan G. (199). The ndvdually responsble and ratonal decson approach to economc lot szes for one vendor and many purchasers: a comment Decson Scences Vol.3 pp

24 Int J Supply Oper Manage (IJSOM) Hall R.W. zero Inventores Dow Jones Irwn Homewood IL Ho C.H. (9). A mnmax dstrbuton free procedure for an ntegrated nventory model wth defectve goods and stochastc lead tme demand Internatonal Journal of Informaton and Management Scences Vol. pp Hong J.D. and Hayya J.C. (1995). Jont nvestment n qualty mprovement and setup reducton Computers and Operatons Research Vol. pp Hoque M.A. (13). A vendor-buyer ntegrated producton nventory model wth normal dstrbuton of lead tme. Internatonal Journal of Producton Economcs Vol. 144 pp Hoque M.A. and Goyal S.K. (6). A heurstc soluton procedure for an ntegrated nventory system under controllable lead-tme wth equal or unequal sze batch shpments between a vendor and a buyer Internatonal Journal of Producton Economcs Vol. 1 pp Hsu S.L. and Lee C.C. (9). Replenshment and lead tme decsons n manufacturer retaler chans Transportaton Research Part E Vol.45 pp Hwang H. K D.B. and K Y.D. (1993). Multproduct economc lot sze models wth nvestments costs for setup reducton and qualty mprovement Internatonal Journal of Producton Research Vol. 31 pp Keller G. and Noor H. (1988). Impact of nvestng n qualty mprovement on the lot sze model Omega Vol. 15 pp K K.L. Hayya J.C. and Hong J.D. (199). Setup reducton n economc producton quantty model Decson Scence Vol. 3 pp Lao C.J. and Shyu C.H. (1991). An analytcal determnaton of lead tme wth normal demand Internatonal Journal of Operatons & Producton Management Vol.11 pp Moon I. (1994). Multproduct economc lot sze models wth nvestments costs for setup reducton and qualty mprovement: Revew and extensons Internatonal Journal of Producton Research Vol. 3 pp MoonI. and ChoS. (1998). A note on lead tme and dstrbutonal assumptons n contnuous revew nventory models Computers & Operatons Research Vol. 5 pp

25 Vjayashree and Uthayakumar Nasr F. Affsco J.F. and Paknejad M.T. (199). Setup cost reducton n an nventory model wth fnte range stochastc lead tmes Internatonal Journal of Producton Research Vol. 8 pp Ouyang L.Y. and Chang H.C. (1999). Impact of nvestng n qualty mprovement on (Q r L) model nvolvng the mperfect producton process Producton Plannng Control Vol. 11 pp Ouyang L.Y. Chen C.K. and Chang H.C. (1999 a ). Lead tme and orderng cost reductons n contnuous revew nventory systems wth partal backorders Journal of the Operatonal Research Socety Vol. 5 pp Ouyang L.Y. Wu K.S. and Ho C.H. (4). Integrated vendor-buyer cooperatve models wth stochastc demand n controllable lead tme Internatonal Journal of Producton Economcs Vol. 9 pp Ouyang L.Y. Wu K.S. and Ho C.H. (7). An ntegrated vendor buyer nventory model wth qualty mprovement and lead tme reducton Internatonal Journal of Producton Economcs Ouyang L.Y. and Chuang B.R. (1999 b ). (Q R L) nventory model nvolvng quantty dscounts and a stochastc backorder rate Producton Plannng & Control Vol. 1 pp Ouyang L.Y. and Chuang B.R. (1999 c ). A mnmax dstrbuton free procedure for stochastc nventory models wth a random backorder rate Journal of the Operatons Research Socety of Japan Vol. 4 pp Ouyang L.Y. and Chuang B.R. (). Stochastc nventory models nvolvng varable lead tme wth a servce level constrant Yugoslav Journal of Operatons Research Vol. 1 pp Ouyang L.Y. Chuang B.R. and Wu K.S. (1999 d ). Optmal nventory polces nvolvng varable lead tme wth defectve tems Journal of the Operatonal Research Socety of Inda Vol.36 pp Paknejad M. J. and Affsco J. F. (1987). The effect of nvestment n new technology on optmal batch quantty Proceedngs of the Northeast Decson Scences Insttute DSI RI USA

26 Int J Supply Oper Manage (IJSOM) Pan J.C.H. and Yang J.S. (4). Just-n-tme: an ntegrated nventory model nvolvng determnstc varable lead tme and qualty mprovement nvestment Internatonal Journal of Producton Research Vol. 4 pp Pan J.C.H. and Yang J.S.A. (). Study of an ntegrated nventory wth controllable lead tme Internatonal Journal of Producton Research Vol. 4 pp Pandey A. Masn M. and Prabhu V. (7). Adaptve logstc controller for ntegrated desgn of dstrbuted supply chans Journal of Manufacturng Systems Vol.6 pp Porteus E.L. (1985). Investng n reduced setups n the EOQ model Management Scences Vol. 31 pp Porteus E.L. (1986). Optmal lot szng process qualty mprovement and setup cost reducton Operatons Research Vol. 34 pp Rosenblatt M.J. and Lee H.L. (1986). Economc producton cycles wth mperfect producton processes IIE Transactons Vol.18 pp Schonberger R. (198). Japanese Manufacturng Technques the Free New York. Tang J. Yung K.L. and Ip A. W.H. (4). Heurstcs-based ntegrated decsons for logstcs network systems Journal of Manufacturng Systems Vol.3 pp Teng J.T. Crdenas B.L.E. and Lou K.R. (11). The economc lot-sze of the ntegrated vendor-buyer nventory system derved wthout dervatves: a smple dervaton Appled Mathematcs and Computaton Vol. 17 pp Tersne R. (1994). Prncple of Inventory and Materal Management Fourth Edton. Prentce-Hall USA. Tsou C.S. Fang H.H. Lo H.C. Huang C.H. (9). A study of cooperatve advertsngs n a manufacturer-retaler supply chan Internatonal Journal of Informaton and Management Scences Vol. pp Woo Y.Y. Hsu S.L. and Wu S.H. (1). An ntegrated nventory model for a sngle vendor and multple buyers wth orderng cost reducton Internatonal Journal of Producton Economcs Vol. 73 pp Zhang T. Lang L. Yu Y. and Yan Y. (7). An ntegrated vendor-managed nventory model for a two-echelon system wth order cost reducton Internatonal Journal of Producton Economcs Vol. 19 pp

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