Finite Production Rate Model With Quality Assurance, Multi-customer and Discontinuous Deliveries

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1 Fte Producto Rate Model Wth ualty Assurace, Mult-custoer ad Dscotuous Delveres Yua-Shy Peter Chu, L-We L, Fa-Yu Pa 3, Sga Wag Chu * Departet of Idustral Egeerg Chaoyag Uversty of Techology, Tachug 43, Tawa Departet of Busess Adstrato, Chaoyag Uversty of Techology, Tachug 43, Tawa 3 Departet ad Graduate Isttute of Busess Adstrato, Natoal Chaghua Uversty of Educato, Chaghua 500, Tawa ABSTRACT Ths study s cocered wth the repleshet-shpet decso for a ult-custoer fte producto rate (FPR) odel wth qualty assurace ad dscotuous delveres. We cosder that a product s aufactured by a producer ad all tes are screeed for qualty cotrol purpose. Nocoforg tes wll be pcked up ad categorzed as scrap or reparable tes. The reworkg wll be doe rght after the regular producto each repleshet cycle. After the etre lot s qualty assured, ultple shpets wll be delvered sychroously to ult-custoer each cycle. Each custoer has ts ow aual product dead, ut stock holdg cost, as well as fxed ad varable product delvery costs. Matheatcal odelg alog wth Hessa atrx equatos s eployed to solve the proposed odel. A closed-for optal repleshet-shpet polcy for such a specfc tegrated FPR odel s obtaed. A uercal exaple s provded to show the practcal applcablty of the obtaed results. Keywords: optzato, fte producto rate odel, repleshet-shpet polcy, ultple custoers, dscotuous delveres, scrap, rework.. Itroducto Ths paper exaes a ult-custoer fte producto rate (FPR) odel corporatg the qualty assurace ad a dscotuous product delvery polcy. The FRP odel [-] was frst troduced to assst aufacturers deterg the ost ecooc producto lot sze that zes the log-ru average productovetory cost. Fte producto rate odel assues plctly that all tes produced are of perfect qualty. I real lfe producto settgs, however, due to varous upredctable factors t s evtable to produce defectve tes radoly [3-8]. Certa ocoforg tes soetes ca be reworked ad repared to eet the expected stadard of product qualty as well as to reduce overall producto-vetory costs [9-3]. The classc FRP odel cosders a cotuous vetory ssug polcy. However, real world stuatos, the ult-delvery (or dscotuous) polcy s cooly adopted by the vedor to trasport fshed products to ts custoer [4-0]. Moreover, real lfe supply chas evroets, we ofte have a vedor who aufactures a product ad supples t to custoers dfferet locatos. The aageet of such a tegrated supply chas syste would be axous to kow what the best producto-shpet polcy wll be order to ze the total syste costs. Schwarz [] studed 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 propertes are obtaed for such a oe-retaler ad N detcal retaler probles. Heurstc solutos for the geeral proble were also suggested. Baerjee ad Baerjee [] developed a aalytcal odel for a coordated, order-less vetory syste for the sgle product, sgle vedor, ad ultple purchasers. Such a syste was ade practcal electroc data terchage (EDI) at the te, for the exchage of forato betwee tradg parters. O the bass of the potetal beefts of JouralofAppledResearchadTechology 5

2 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 ths techology, they proposed a coo cycle repleshet approach, where the suppler aloe akes all repleshet decsos, wthout orderg o the part of the custoers. Ther odel ad cocepts were deostrated by a sple uercal exaple ad cocluded that EDI-based vetory cotrol ca be attractve fro the ecooc, as well as other stadpots. Lu [3] cosdered a oe-vedor ult-buyer tegrated odel wth the objectve of zg a vedor s total aual cost, subject to the axu costs that buyers ay be prepared to cur. The buyer's aual dead ad prevous frequecy of order are assued to be kow forato ths odel. As a result, a optal soluto for the oevedor oe-buyer case was obtaed; a heurstc approach for the oe-vedor ult-buyer case was also provded. Woo et al. [4] cosdered a tegrated vetory syste where a sgle vedor purchases ad processes raw aterals order to delver fshed tes to ultple buyers. The vedor ad all buyers are wllg to vest reducg the orderg cost (e.g., establshg a electroc data terchage based vetory cotrol syste) order to decrease ther jot total cost. A aalytcal odel s developed to derve the optal vestet aout ad repleshet decsos for both vedor ad buyers. The expoetal orderg cost fucto s the appled to ther geeral odel, ad a uercal aalyss s perfored to provde terestg sghts of the odel. Nuercal results show that the vedor ad all the buyers ca beeft drectly fro substatal cost savgs by ths orderg cost reducto vestet. Yao ad Chou [5], cosdered a tegrated supply cha odel whch oe vedor supples tes for the dead of ultple buyers. The objectve of ther odel was to ze the vedor's total aual cost subject to the axu cost that the buyer ay be prepared to cur. They explored the optalty structure of ths tegrated odel ad asserted that the optal cost curve s pece-wse covex. Theoretcal results o the breakpots of the optal cost curve provde useful sghts that supported the desg of ther search algorth. Nuercal experets deostrated that ther search algorth s very effcet for t out-perfors other heurstcs, ad t secures the global optal soluto for each of ther 0 experetal probles. Yag ad Lo [6] cosdered a proble of deterg a sutable polcy of vetory ad purchasg aageet, wth the objectve of zg total expected vetory costs wth ultple parters uder cotrollable lead te. May studes that addressed varous aspects of supply cha ssues have also bee extesvely carred out recetly [7-35]. Ths paper deteres the jot optal producto lot sze ad uber of delveres that zes the log-ru expected syste cost for such a tegrated ult-custoer FPR odel wth qualty assurace ad dscotuous delvery polcy. Because lttle atteto has bee pad to ths area, ths paper s teded to brdge the gap.. Proble descrpto, odelg, ad forulatos Ths paper studes a ult-custoer fte producto rate odel wth qualty assurace ad dscotuous delveres. Cosder a product that ca be ade by a vedor at a aual producto rate P ad durg the producto process a x porto of ocoforg tes ay radoly be geerated at a rate d. All tes ade are screeed ad specto cost s cluded the ut producto cost C. Nocoforg tes wll be categorzed as scrap or reparable tes. The reworkg wll be doe rght after the regular producto each repleshet cycle at a rate of P. Uder the oral operato assupto, to avod shortages fro occurrg the costat producto rate P ust satsfy (P-d-)>0, where s the su of aual deads of all custoers (dvdual dead rate s each), ad d ca be expressed as d=px. Ulke the FPR odel assues a cotuous vetory ssug polcy for satsfyg dead, ths study cosders a dscotuous delvery polcy. Specfcally speakg, after the etre lot s qualty assured the ed of rework process, fxed quatty ultple shpets of fshed tes are delvered sychroously to ultcustoer at a fxed terval of te durg the dowte t 3 each cycle (refer to Fgures ad ). Cost paraeters used ths study are as follows: The ut producto cost C, ut holdg cost h, producto setup cost K, ut dsposal cost C S, ut cost C R ad ut holdg cost h for each reworked te, the fxed delvery cost K per shpet delvered to custoer, ut holdg cost h for te kept by custoer, ad ut shppg cost C T for te shpped to custoer. Addtoal otato used ths study cludes: 6 Vol.,February04

3 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 = a porto of ocoforg tes that s categorzed as scrap tes, T = producto cycle legth, = producto lot sze per cycle, a decso varable, = uber of fxed quatty stallets of the fshed batch to be delvered to retalers for each cycle, a decso varable, t = the producto upte for the proposed syste, t = te requred for reworkg the ocoforg tes produced each cycle, t 3 = te requred for delverg all qualty assured fshed products to retalers, Fgure. Vedor s o-had vetory of perfect qualty tes t = a fxed terval of te betwee each stallet of fshed products delvered durg producto dowte t 3, H = level of o-had vetory uts whe regular producto process eds, H = axu level of o-had vetory uts whe the rework process eds, = uber of retalers, I(t) = vedor s o-had vetory of perfect qualty tes at te t, I d (t) = vedor s o-had vetory of defectve tes at te t, I c (t) = custoers o-had vetory at te t, TC(,) = total producto-vetory-delvery costs per cycle for the proposed syste, E[TCU(,)] = total expected productovetory-delvery costs per ut te for the proposed syste. Fgure. Vedor s o-had vetory of defectve tes Fro Fgures ad, we drectly obta the followg equatos: () x T tt t3 () H t P P d (3) JouralofAppledResearchadTechology 7

4 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 t x (4) P 3 t T t t t (5) H Pdt Pd ( x) P (6) H HPt x. (7), TC K C C x C x K C T R S T Hdt HH h t t Ht3 P t Tt3 h t h t tt (3) The o-had vetory of defectve tes durg upte t s (see Fgure ) dt Pxt x. (8) Delvery cost of the th shpet to custoers s K C T (9) Total delvery costs for shpets are: (0) K C T T Vedor s varable holdg costs for fshed products kept durg t 3 where fxed- quatty stallets of the fshed batch are delvered to custoers at a fxed terval of te, are as follows [35]. h Ht 3 () Custoers total stock holdg costs durg a cycle are (see Fgure 3 ad Appedx A for detals). 3 h ttt () Tt TC(,) cossts of the producto setup cost, varable producto cost, cost for the reworkg, dsposal cost, the fxed ad varable delvery cost, vedor s holdg cost upte t, rework te t, ad delvery te t 3, ad custoers holdg costs as follows: Fgure 3. Custoers o-had vetory status Takg to accout the radoess of defectve rate, we ca use the expected values of x cost aalyss of ths study. Substtutg all paraeters fro Equatos to TC(,) ad wth further dervatos, E[TCU(,)] ca be obtaed as (please refer to Appedx B for detals). 8 Vol.,February04

5 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 C K K CREx CSEx ETCU, E x Ex Ex Ex Ex h Ex CT ExEx P P Ex E x E x h h P P P Ex h E x Ex h P P 3. Dervato of the optal polcy 3. Proof of covexty (4) Ths study eploys the Hessa atrx equatos [36] here to prove the covexty of E[TCU(,)]. That s to verfy whether the followg equato holds:, E TCU, ETCU 0 E TCU, E TCU, (5) Applyg dfferetato to Equato 4 we obta K K ETCU, h Ex Ex Ex Ex Ex P P h P P Ex Ex Ex h h E x Ex h P Ex P P ETCU, K K 3 (6) (7) K ETCU, E x E TCU Ex Ex h h P P, Ex Ex (8) h 3 h P P K ETCU, E x (9) Ex Ex h h P P (0) Substtutg Equatos 7, 9 ad 0 Equato 5 we have, E TCU, K, E TCU, Ex ETCU 0 ETCU () Equato resulted postve, because K,,, ad (-E[x]) are all postve. Hece, E[TCU(,)] s a strctly covex fucto for all ad dfferet fro zero. Hece, we kow that there s a u of E[TCU(,)]. 3. The optal repleshet-shpet polcy To jotly detere the optal repleshetshpet polcy for the proposed ult- custoer FPR odel, we ca solve the lear syste of Equatos 6 ad 8 by settg these partal dervatves equal to zero. Wth further dervatos we have the optal lot sze * ad optal uber of delvery * as follows: JouralofAppledResearchadTechology 9

6 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 * ad * K K Ex h ExEx h Ex P P P Ex hex h h Ex P P Ex h Ex () Kh hexex P P Ex h ExEx h Ex P P P K Ex hex h h Ex P P (3) As we ca see, a real-lfe stuato the uber of delveres ca oly be a teger value. I order to detere the teger value of * that zes the expected syste cost, two adjacet tegers to ust be exaed respectvely [3]. Let - deote the largest teger less tha or equal to ad + deote the sallest teger greater tha or equal to (as was derved fro Equato 3. Substtute + ad - respectvely Equato ad the applyg the resultg (, + ) ad (, - ) Equato 4 respectvely. By selectg the oe that gves u log-ru average cost as the optal repleshet- dstrbuto polcy (*, *). A uercal exaple s provded ext to show the practcal applcablty of the obtaed results. 4. Nuercal exaple Suppose a aufacturer ca ake a product at a producto rate P=60000 tes/year ad ths product has expereced stable aual dead fro fve dfferet dustral clets, where s 400, 500, 600, 700, ad 800 respectvely (hece, the su of dead s 3000 per year). Durg the producto process, the producer has expereced a rado defectve rate that follows a ufor dstrbuto over the rage of [0, 0.3]. Aog the defectve tes 0% (.e. =0.) s foud to be scrap tes. The other porto of the ca be reworked ad repared at a reworkg rate P =3600 per year. Other paraeters used ths study clude C K h h C S C R K = $00 per te, = $35000 per producto ru, = $5 per te per year (aufacturer s ut holdg cost per year), = $60 per te per year (aufacturer s ut holdg cost for reworked te), = $0, dsposal cost for each scrapped te, = $60, reparg cost for each reworked te, = $00, $00, $300, $400 ad $500 (the fxed delvery cost per shpet to custoer =,,, ad 5, respectvely), C T = $0.5, $0.4, $0.3, $0., ad $0. (trasportato cost per te delvered to custoer =,,, ad 5, respectvely), h = $75, $70, $65, $60, ad $55 (holdg cost per year per te kept by custoer =,,, ad 5, respectvely). Frst, applyg Equato 3 we have =4.47. By exag two adjacet tegers to ad applyg Equato, we obta (, + )=(47,5) ad (, - )=(385,4). The, substtutg (, + ) ad (, - ) Equato 4 respectvely ad selectg the oe that gves the u syste cost, we obta the optal uber of delvery *=4, optal repleshet *=385, ad the expected cost E[TCU(*,*)] =$440,53. Varato of rado defectve rate ad scrap rate effects o E[TCU(*,*)] s depcted Fgure 4. We ote that as rado defectve rate x creases, the expected syste cost E[TCU(*,*)] creases sgfcatly, ad as the scrap rate creases the expected syste cost E[TCU(*,*)] creases slghtly. 5. Coclusos A ult-custoer fte producto rate odel wth qualty assurace ad dscotuous delveres was studed. It cosders that a product s ade by a aufacturer ad a porto of rado ocoforg tes produced s reworked ad repared rght after the ed of regular producto each repleshet cycle. After the etre lot s 0 Vol.,February04

7 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53 qualty assured, ultple shpets are delvered sychroously to ult-custoer each cycle. Each custoer has ts ow aual product dead, ut stock holdg cost, ad fxed ad varable product delvery costs. I real world supply chas evroets, aageet of such a supply cha syste would certaly lke to fgure out the optal repleshet-dstrbuto polcy hece, the logru average syste cost ca be zed. Ths study accoplshed the purpose by developg a soluto procedure usg the atheatcal odelg to deal wth the aforeetoed supply cha syste. As a result, a closed for soluto of the optal repleshet-dstrbuto polcy s obtaed. Effects of varous syste paraeters o the optal soluto are vestgated (as show Fgures 4) order to provde aageet wth depth sghts of such a realstc ult-custoer supply cha syste. D H (A-) The dervatos of custoers holdg cost (Equato ) are as follows (see Fgure 3). D I t I t t h I t D I t I t t h I t D I t I t t h It D I t I t t h It D It It h I t t (A-3) Because stallets of the fshed lot are delvered to custoer at a fxed terval of te t, we have the followg: I t (A-4) t D t I (A-5) Fgure 4. Varato of rado defectve rate ad scrap rate effects o the E[TCU(*,*)] Ackowledgeets Authors scerely apprecate the Natoal Scece Coucl of Tawa for ts support of ths research uder grat uber: NSC H MY. Appedx A Dervatos of custoers stock holdg Durg the delvery te t 3, stallets of fxed quatty D of the fshed lot are delvered to custoers at a fxed terval of te t, where t t 3 (A-) where I deotes uber of left over tes for each custoer after dead durg each fxed terval of te t has bee satsfed (see Fgure 3). Equato (A-3) the custoers holdg cost becoes h D I t It t t 3 3 h t I t t t t t t 3 3 Tt3 h t tt Appedx B (A-6) Dervatos of the log-ru average syste cost E[TCU(,)] Fro Equato 3 JouralofAppledResearchadTechology

8 FteProductoRateModelWthualtyAssurace,MultcustoeradDscotuousDelveres,YuaShyPeterChuetal./53, TC K C C x C x K C T R S T Hdt HH h t t Ht3 P t Tt3 h t h t tt (3) by substtutg all paraeters fro equatos to Equato 3 we have TC, C K CR x C Sx K CT x h h x xx x x x h P P P P h x h x h xx P P hx x xx h x P P P (B-) wth further dervatos we have TC, C K CR x C Sx K CT x h h x x x h x P P h x xx hx x P P P x xx h P P (B-) Takg to accout of radoess of defectve rate x ad wth further dervatos, we obta E[TCU(,)] as C K K CREx CSEx ETCU, E x Ex Ex Ex Ex h Ex CT ExEx P P Ex E x E x h h P P P Ex h E x Ex h P P (4) Refereces [] E.W. Taft, The ost ecoocal producto lot, Iro Age, vol. 0, pp. 40-4, 98. [] G. Hadley ad T. M. Wht, A optal fal vetory odel, Maageet Scece, vol. 7, pp , 96. [3] W. Shh, Optal vetory polces whe stock-outs result fro defectve products, Iteratoal Joural of Producto Research, vol. 8, o. 6, pp , 980. [4] M. Heg ad Y. Gerchak, Structure of perodc revew polces the presece of rado yeld, Operatos Research, vol. 38, o. 4, pp , 990. [5] T. Booe et al., The pact of perfect processes o producto ru tes, Decso Sceces, vol. 3, o. 4, pp , 000. [6] E. Rubo ad J. C. Jáuregu-Correa, A Wavelet Approach to Estate The ualty of Groud Parts, J. Appl. Res. Tech., vol. 0, o., pp. 8-37, 0. [7] G. C. Mahata, A EP-based vetory odel for expoetally deteroratg tes uder retaler partal trade credt polcy supply cha, Expert Systes wth Applcatos, vol. 39, o. 3, pp , 0. [8] S. W. Chu et al., Producto-shpet polcy for EP odel wth qualty assurace ad a proved delvery schedule, Matheatcal ad Coputer Modellg of Dyacal Systes, vol. 9, o. 4, pp , 03. [9] B. J. Yu ad E. D. McDowell, Optal Ispecto Polces a Seral Producto Syste cludg scrap, rework ad repar: A MILP approach, Iteratoal Joural of Producto Research, vol. 5, o. 0, pp , 987. [0] A. M. Zargar, Effect of rework strateges o cycle te, Coputers ad Idustral Egeerg, vol. 9, o. -4, pp , 995. [] R. H. Teuter ad S. D. P. Flapper, Lot-szg for a sgle-stage sgle-product producto syste wth rework of pershable producto defectves, OR Spectru, vol. 5, o., pp , 003. [] S. W. Chu et al., Note o the atheatcal odelg approach used to detere the repleshet polcy for the EM odel wth rework ad ultple shpets, Appled Matheatcs Letters, vol. 5, o., pp , 0. [3] H-D. L ad Y-S. P. Chu, Note o repleshet ru te proble wth ache breakdow ad falure rework", Expert Systes wth Applcatos, vol. 39, o. 7, pp , 0. Vol.,February04

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