A Single-Producer Multi-Retailer Integrated Inventory System with Scrap in Production

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1 Research Joural of Appled Sceces, Egeerg ad Techology 5(4): 54-59, 03 ISSN: ; e-issn: Maxwell Scetfc Orgazato, 03 Subtted: July 09, 0 Accepted: August 08, 0 Publshed: February 0, 03 A Sgle-Producer Mult-Retaler Itegrated Ivetory Syste wth Scrap Producto Sga Wag Chu, Kuo-Tsu Lu, Che-Hua Lee ad Yua-Shy Peter Chu Departet of Busess Adstrato, Departet of Idustral Egeerg ad Maageet, Chaoyag Uversty of Techology, Tachug 43, Tawa Abstract: Ths study deteres the repleshet-dstrbuto polcy for a sgle-producer ult-retaler tegrated vetory syste wth scrap rate producto. The objectve s to fd the optal lot-sze ad uber of shpets that zes the total expected costs for such a specfc syste. It s assued that a te s aufactured by a producer ad the delvered to ts dfferet retalers for sale. Each retaler has ts ow aual product dead ad the dead wll be satsfed by sychrozed ultple shpets durg each producto cycle. Ulke the classc Fte Producto Rate (FPR) odel assug perfect producto ad a cotuous vetory ssug polces, the proposed syste assues that there s a evtable rado defectve rate producto ad all ocoforg tes are scrap ad delvery of product s uder a practcal ultple shpet polcy. Matheatcal odelg s eployed ad the reewal reward theore s used to cope wth the varable cycle legth. The the expected syste cost fucto s derved ad the covexty of ths fucto s proved. Fally, a closed-for optal repleshetdstrbuto polcy for such a specfc sgle-producer ult-retaler tegrated syste s obtaed. A uercal exaple s provded to deostrate the practcal usage of the obtaed results. Keywords: FPR odel, vetory, aufacturg, ultple retalers, ultple shpets, optal batch sze, scrap INTRODUCTION The purpose of ths study s to detere the optal lot-sze ad uber of shpets that zes the total expected cost for a sgle-producer ult-retaler tegrated vetory syste wth scrap producto. The classc FPR odel (also kow as the Ecooc Producto uatty (EP) odel) assues a perfect producto (Taft, 98; Nahas, 009). However, a real lfe aufacturg evroet, due to dfferet upredctable factors t sees to be evtable to have perfect producto (Hadley ad Wht, 963; Scheder, 979; Beleck ad Kuar, 988; Cher ad Yag, 999; Booe et al., 000; Teuter ad Flapper, 003; Ojha et al., 007; L et al., 008; Chu et al., 009a, 0a; Lee et al., 0; Che, 0). Aother urealstc assupto the classc FPR odel s the cotuous vetory ssug polcy to satsfy product dead. I the real world vedor-buyer tegrated syste, however, the ultple or perodc delveres of fshed products are ofte used. Schwarz (973) frst exaed a oe-warehouse N-retaler deterstc vetory syste wth the objectve of deterg the stockg polcy that zes the log-ru average syste cost per ut te. The optal solutos alog wth a few ecessary propertes are derved for such a oe-retaler ad N detcal retaler probles. Heurstc solutos for the geeral proble were also suggested. Goyal (977) studed the tegrated vetory odel for the sgle supplersgle custoer proble. He proposed a ethod that s typcally applcable to those vetory probles where a product s procured by a sgle custoer fro a sgle suppler ad a exaple was provded to llustrate hs proposed ethod. Schwarz et al. (985) cosdered the syste fll-rate of a oe-warehouse N-detcal retaler dstrbuto syste as a fucto of warehouse ad retaler safety stock. They used a approxato odel fro a pror study to axze syste fll-rate subject to a costrat o syste safety stock. As results, propertes of fll-rate polcy les are suggested. They ay be used to provde aageral sght to syste optzato ad as the bass for heurstcs. Baerjee (986) studed a jot ecooc lot-sze odel for purchaser ad vedor, wth the focus o zg the jot total relevat cost. He cocluded that a jotly optal orderg polcy, together wth a approprate prce adjustet, could be ecoocally beefcal for both partes, but deftely ot dsadvatageous to ether party. Studes have sce bee carred out to address varous aspects of vedor-buyer supply cha optzato ssues (K ad Hwag, 988; Baerjee ad Burto, 994; Parja ad Sarker, 999; Cetkaya ad Lee, 000; Yao ad Chou, 004; Hoque, 008; Sarker ad Dpoegoro, 009; Chu et al., 0, Correspodg Author: Yua-Shy Peter Chu, Departet of Idustral Egeerg ad Maageet, Chaoyag Uversty of Techology, Tachug 43, Tawa 54

2 Res. J. Appl. Sc. Eg. Techol., 5(4): 54-59, 03 Fg. : O-had vetory of perfect qualty tes the producer sde 0b). Because lttle atteto has bee pad to the area of dervg the jot repleshet-dstrbuto polcy for a sgle-producer ult-retaler FPR odel wth scrap producto, ths study teds to brdge the gap. MODELING AND ANALYSIS Ths study cosders a sgle-producer N-retaler tegrated vetory syste wth scrap. It s assued that the aufacturg process producer s sde ay radoly produce a x porto of defectve tes at a producto rate d. All tes produced are screeed ad ut specto cost s cluded ut producto cost C. All ocoforg tes are assued to be scrap; they are dscarded at the ed of producto. Uder the oral operato, to prevet shortages fro occurrg, the costat producto rate P satsfes (P-d-λ)>0 or (- x-λ/p)>0 ad d ca be expressed as d = Px. Ulke the classc FPR odel assug a cotuous ssug polcy for eetg dead, ths research cosders a ult-shpet polcy. It s also assued that the fshed tes ca oly be delvered to retalers f the whole lot s qualty assured at the ed of producto. Each retaler has ts ow aual dead rate λ. Fxed quatty stallets of the fshed batch are delvered to ultple retalers sychroously at a fxed terval of te durg the dowte t (Fg. ). Cost paraeters used ths study are as follows: the ut producto cost C, ut dsposal cost C S, the producto setup cost K, ut holdg cost h, the fxed delvery cost K per shpet delvered to retaler, ut holdg cost h for te kept by retaler ad ut shppg cost C for te shpped to retaler. Addtoal otato s lsted as follows: = Nuber of fxed quatty stallets of the fshed batch to be delvered to retalers for each cycle, a decso varable (to be detered) = Nuber of retalers t = The producto upte for the proposed syste H = Maxu level of o-had vetory uts whe regular producto process eds t = Te requred for delverg all qualty assured fshed products to retalers t = A fxed terval of te betwee each stallet of fshed products delvered durg producto dowte t T = Producto cycle legth I (t) = O-had vetory of perfect qualty tes at te t I c (t) = O-had vetory at the retalers at te t TC (, ) = Total producto-vetory-delvery costs per cycle for the proposed syste E [TCU (, )] = Total expected producto-vetorydelvery costs per ut te for the proposed syste Fro Fg., the followg equatos ca be drectly obtaed: T t t () H t P P d () T x (3) x t t P (4) H Pdt Pd x (5) P (6) The o-had vetory of scrap tes durg producto upte t s: dt Pxt x. (7) Cost for each delvery s: = Producto lot sze per cycle, a decso varable (to be detered) 55 K C T (8)

3 Res. J. Appl. Sc. Eg. Techol., 5(4): 54-59, 03 Total producto-vetory-delvery cost per cycle TC (, ) cossts of the setup cost, varable producto cost, dsposal cost, fxed ad varable delvery cost, holdg cost durg producto upte t ad holdg cost for fshed goods kept by both the aufacturer ad the custoer durg the delvery te t. Therefore, TC (, ) s:, S TC C K C x K C T H dt Tt h t Ht h Tt () Because the scrap rate x durg producto s assued to be a rado varable wth a kow probablty desty fucto. I order to take the radoess of scrap rate to accout, the expected values of x ca be used the cost aalyss. Substtutg all paraeters fro Eq. () to () TC (, ) ad wth further dervatos the expected cost E [TCU (, )] ca be obtaed as follows: Fg. : O-had vetory retaler sdes Total delvery costs for shpets a cycle are: K C T (9) The varable holdg costs for fshed products kept by the aufacturer, durg the delvery te t where fxed-quatty stallets of the fshed batch are delvered to custoers at a fxed terval of te, are as follows (Chu et al., 009b): h Ht (0) ETC ETCU,, C K K CSEx ET Ex Ex Ex h h E x P Ex P C h h E x P DERIVATION OF THE OPTIMAL SOLUTIONS (3) Proof of covexty: For the proof of covexty of E [TCU (, )], oe ca use the Hessa atrx equatos (Rard, 998) ad verfy whether the followg codto (Eq. (4) holds:, E TCU, ETCU 0 E TCU, E TCU, (4) Fro Eq. (3), oe obtas: The varable holdg costs for fshed products kept by the custoer durg the delvery te t, are as follows (Fg. ad Appedx for detals): h Tt Tt () 56 K K h, h ETCU E x Ex PEx P h h Ex P (5)

4 K K ETCU, E x Res. J. Appl. Sc. Eg. Techol., 5(4): 54-59, 03 3 (6) K, ETCU E x h h Ex P (7) Ex h h (8) ETCU, 3 P K, ETCU E x h h Ex P (9) Substtutg Eq. (6), (8) ad (9) Eq. (4) oe has:, E TCU, K, E TCU, Ex ETCU 0 ETCU (0) Equato (0) s resultg postve, because K, λ, (-E [x]) ad are all postve. Hece, E [TCU (, )] s a strctly covex fucto for all ad dfferet fro zero. Therefore, the covexty of E [TCU (, )] s proved ad there exsts a u of E [TCU (, )]. Dervato of the optal polcy: To detere jotly the repleshet-dstrbuto polcy for the proposed sgle-producer ult-retaler tegrated vetory syste wth scrap, oe ca solve the lear syste of Eq. (5) ad (7) by settg these partal dervatves equal to zero. Further dervatos oe obtas: K K * h hex h h Ex P P h Ex ad () 57 * K h h Ex P K h h Ex P () E x h P It s oted that although real-world stuato the uber of delveres takes teger values oly, Eq. () results a real uber. I order to fd the teger value of * that zes the expected syste cost, two adjacet tegers to ust be exaed respectvely (Chu et al., 009b). Let + deote the sallest teger greater tha or equal to (derved fro Eq. ()) ad - deote the largest teger less tha or equal to. Substtute + ad - respectvely Eq. (), the applyg the resultg (, + ) ad (, - ) Eq. (3) respectvely. By selectg the oe that gves the u log-ru average cost as the optal repleshet-dstrbuto polcy (*, *). A uercal exaple deostratg ther practcal usages s provded ext secto. NUMERICAL EXAMPLE Assue a producer ca aufacture a product at a aual rate of 60,000 uts. Aual deads λ of ths te fro 5 dfferet retalers are 400, 500, 600, 700 ad 800 respectvely (total dead s 3000/year). There s a rado scrap rate durg producto upte whch follows a ufor dstrbuto over the terval (0, 0.3). Values for addtoal paraeters are: h = Ut holdg cost per te at the producer sde, $0 per te per year K = $35000 per producto ru C = $00 per te C S = $0, dsposal cost per scrap te K = The fxed delvery cost per shpet for retaler, they are $ 00, 00, 300, 400 ad 500, respectvely E [TCU ( )] *, * $480,000 $460,000 $440,000 $40,000 $400,000 $380,000 ( *, * ) = (85,5) ( *, * ) = (96,5) E [TCU (3,5) = $460,408] $360, Fg. 3: Varato of rado scrap rate effects o the (*, *) polcy ad E [TCU (*, *)] x

5 h = ut holdg cost for te kept by retaler, they are $75, 70, 65, 60 ad 55 per te, respectvely C = ut trasportato cost for te delvered to retaler, they are $0.5, 0.4, 0.3, 0. ad 0., respectvely Fro Eq. (), () ad (3), oe obtas the optal uber of delvery * = 5, the optal repleshet lot sze * = 3 ad total expected cost E [TCU (*, *)] = $ 460,408. It s oted that uber of delvery should practcally be a teger uber whle Eq. () results 5.39, by applyg the aforeetoed (, + ) ad (, - ) Eq. (3) respectvely, oe fds the optal uber of delvery * = 5. Varato of rado scrap rate effects o the optal (*, *) polcy ad o the syste cost E [TCU (*, *)] are depcted Fg. 3. It s oted that as the rado scrap rate x creases, the value of the syste cost E [TCU (*, *)] creases sgfcatly. CONCLUDING REMARKS I real world busess evroets, t s coo to have a producer suppled a product to ts ultple retalers. I such a specfc teral supply chas, aageet would lke to fgure out a best repleshet-dstrbuto polcy order to ze total syste costs. Ths study vestgates the optal repleshet-dstrbuto polcy for such a sgleproducer ult-retaler tegrated vetory syste wth scrap rate producto. Matheatcal odelg ad aalyss are used. The reewal reward theore s eployed to cope wth the varable cycle legth (due to the radoess of scrap rate producto). The log-ru expected producto-vetory-delvery cost per ut te for the proposed syste odel s derved ad ts covexty s proved. The, a closed-for soluto of the optal repleshet-dstrbuto polcy for such a sgle-producer ult-retaler tegrated syste s obtaed. Effects of varous scrap rates o the optal soluto are also exaed. Oe terestg topc for future research wll be to vestgate the effect of rework o the optal repleshet-dstrbuto polcy such a specfc supply chas. ACKNOWLEDGMENT The authors apprecate Natoal Scece Coucl of Tawa for supportg ths research uder grat uber: NSC H MY3 (3/3). APPENDIX Fro Fg., stallets (fxed quatty D) of the fshed lot are delvered to custoer at a fxed terval of te t, oe has: H D Res. J. Appl. Sc. Eg. Techol., 5(4): 54-59, 03 (A) 58 t t (A) Fro Fg., the coputatos of retalers holdg cost durg t (Eq. ()) are as follows: D I t I t h I t D It It h I t D It It h I t D I t I t h I t h D It It It (A3) Because stallets (fxed quatty D) of the fshed lot are delvered to custoer at a fxed terval of te t, oe has the followg: I t (A4) D t I (A5) where, I deotes uber of left over tes for each retaler after dead durg each fxed terval of te t has bee satsfed (Fg. ). Eq. (A3), the retalers holdg cost durg t becoes: h D It It t h t It tt t Tt h tt REFERENCES (A6) Baerjee, A., 986. A jot ecooc-lot-sze odel for purchaser ad vedor. Decso Sc., 7(3): 9-3. Baerjee, A. ad J.S. Burto, 994. Coordated vs. depedet vetory repleshet polces for a vedor ad ultple buyers. It. J. Prod. Eco., 35: 5-. Beleck, T. ad P.R. Kuar, 988. Optalty of zerovetory polces for urelable producto faclty. Oper. Res., 36: Booe, T., R. Gaesha, Y. Guo ad J.K. Ord, 000. The pact of perfect processes o producto ru tes. Decs. Sc., 3(4): Cetkaya, S. ad C.Y. Lee, 000. Stock repleshet ad shpet schedulg for vedor aaged vetory systes. Maage. Sc., 46(): 7-3. Che, C.T., 0. Dyac prevetve ateace strategy for a agg ad deteroratg producto syste. Exp. Syst. Appl., 38(5):

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