Heuristics for Production Allocation and Ordering Policies in Multi-Plants with Capacity

Transcription

1 J. Servce Scece & Maageet, 00, 3: 6- do:0.436ss Publshed Ole March 00 ( eurstcs for Producto llocato ad Orderg Polces Mult-Plats th apacty Je Zag,, Jafu Tag School of Iforato Scece ad Egeerg, Northeaster Uversty, Sheyag, ha; School of Iforato, Laog Uversty, Sheyag, ha. Eal: eceved October 9 th, 009; revsed Noveber 7 th, 009; accepted Jauary 5 th, 00. BSTT Jot decsos producto allocato ad orderg polces for sgle ad ultple products a producto-dstrbuto etork syste cosstg of ultple plats are dscussed, producto capacty costrats of ult-plats ad ut producto capacty for producg a product are cosdered. Based o the average total cost ut te, the decsve odel s establshed. It tres to detere the producto cycle legth, delvery frequecy a cycle fro the arehouse to the retaler ad the ecooc producto allocato. The approach hges o provdg a optzed soluto to the ot decso odel through the heurstcs ethods. The heurstc algorths are proposed to solve the sgle-product ot decso odel ad the ult-products decso proble. Sulatos o dfferet szes of probles have sho that the heurstcs s effectve, ad geeral ore effectve tha Quas-Neto ethod (QNM). Keyords: Jot Decsos, Producto-Dstrbuto; Multple Plats, apactated, eurstc lgorth. Itroducto I the past, logstc decso aog ateral procureet aageet, producto ad dstrbuto ere ade solato. Prevous studes have exaed producto, trasportato ad vetory separately. These aor actvtes are closely related th each other ad should be coordated effectvely to ehace ts proft today s copettve arket. Ucoordated ad solated decso-akg aog fuctoal related actvtes supply cha syste ay eake ts syste-de copettveess. ece, ore efforts are o beg ade to tegrate coordate producto ad dstrbuto, producto ad trasportato, producto ad vetory, as ell as trasportato ad vetory the for of supply cha aageet. Kg [ descrbed the pleetato of a coordated producto-dstrbuto syste, a aor tre aufacturer th four factores ad e aor dstrbuto ceters. llas [ cosdered the proble of ot schedulg of producto ad dstrbuto a coplex etork, the obectve of the proble as to ze average producto ad dstrbuto cost per perod. ll [3 dscussed producto-delvery polces a sgle aufacturer ad a sgle retaler. Davd [4 attepted to detfy lot szg ad delvery schedulg a sgle aufacturer ad a sgle retaler syste. K [5 dscussed the producto ad orderg polces a supply cha cosstg of a sgle aufacturer ad a sgle retaler. e proposes a effcet heurstc algorth to detere the ear optal producto allocato ratos. K [6 exteded ther paper ad develops ot ecooc producto allocato, lot-szg, ad shpet polces a supply cha here a aufacturer produces ultple tes ultple producto les ad shps the tes to the respectve retalers. Ther forulatos are ofte based o ecooc order quatty (EOQ) ad athe- atcal prograg. ccordgly, the correspodg soluto ethods are EOQ [7,8, heurstcs [5,6,9 ad decoposto [0,. I recet studes, odel for coordatg productodstrbuto etork systes have teded to focus o ot decsos o all actvtes. More coplcated tegrated decsos o producto, trasportato, ad vetory have receved relatvely lttle atteto, as [ ad [3. Tag [ dscussed a tegrated decso o producto assget, lot-szg, trasportato, ad order quatty for a ultple-supplerultple-destatos logstcs etork a global aufacturg syste ad proposed a heurstcs to solve edu ad largescale tegrated decso probles. Yug [3 attepted opyrght 00 Sces

2 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty 7 to tackle ot decsos assgg producto, lot-sze, trasportato, ad order quatty for sg ad ultple products a producto-dstrbuto etork syste th ultple supplers ad ultple destatos. e provded a optzed soluto to solve the ot decso odel through a to-layer decoposto ethod that cobes several heurstcs. Ths paper addresses the ssue of ho to effectvely allocate producto requreet to ultple plats supply cha syste. K [5,6 dscussed the producto ad orderg polces a supply cha cosstg of a sgle aufacturer th ultple plats ad a sgle retaler or ultple retalers. The retalers place orders based o the EOQ-lke polcy, ad the ultple plats produce dead requreet fro the retalers. Each of ultple plats has ts producto ad trasfer rates. I real lfe, all the plats the aufacturer have producto capacty costrats. ll the plats should produce th ts capacty to eet the deads of the retalers. The proble dscussed ths paper exteds the odel proposed by [5,6, ad producto capacty costrats of ult-plats ad ut producto capacty for producg a product are cosdered the odel. The heurstcs ethods have bee developed to solve the proble th sgle product ad ultple products, respectvely. I ths paper, the odel for a sgle product ll be dscussed Secto, folloed by detaled dscusso to solve ultple products Secto 3. Oe llustrated exaple th several testg probles ad ther respectve sulato results ad aalyses are preseted Secto 4.. Forulatos ad eurstcs th Sgle Product. Proble Forulatos I a global aufacturg eterprse, there are plats each producg ultple parts ad ultple assebles that serve ultple assebly plats a year, or alteratvely, each assebly plat deads ultple parts fro ay dfferet supplers. ece, such a global aufacturg eterprse ca be forulated as a cobed producto-dstrbuto etork cosstg of ultple supplers ad ultple destatos. I ths paper, e cosder a producto-dstrbuto etork coposed of a sgle aufacturer th ultple plats ad ultple retalers. The retalers are gve aual dead of the product. To eet the aual deads of the product, the aufacturer procures the aterals ad ult-plats produce th ther capacty the aufacturer. The ult-plats of the aufacturer have ther producto rate. The fshed products are trasferred to the coo arehouse at the plats trasfer rate. Fally, the arehouse delvers the ordered lots of a fxed sze to the retaler perodcally. The etork s sho Fgure. The cost copoets cosdered clude to parts, the frst part s the orderg cost fro ra aterals, the pro- Procuree Suppler vetory Plat Plat Plat arehouse Fgure. Producto-dstrbuto etork etaler etaler etaler ducto setup cost, the orderg cost at the arehouse, ad the orderg cost of the retaler; the secod part s the holdg costs for ra aterals, ork--process vetores, fshed tes at the arehouse ad the retaler. ssue that there are plats a aufacturer, here each of the plats s dcated by the subscrpts. The follog otatos ad decso varables are appled. P = aual producto rate at plat (utyear) Q = aual producto capacty at plat (year) d = aual trasfer rate fro plat to the arehouse (ut) u = producto capacty eeded to produce ut product at plat (year) h = holdg cost for ork--processes at plat S p = producto setup cost at the aufacturer =orderg cost for ra aterals at the aufacturer = order hadlg cost for fshed products at the arehouse r = orderg cost at the retaler = holdg cost for ra aterals at the aufacturer = holdg cost for fshed products at the arehouse r = holdg cost for fshed products at the retaler D =dead rate uts at the retaler (utyear) T = decso varable, producto cycle legth at the aufacturer (year) = decso varable, delvery frequecy a producto cycle fro the arehouse to the retaler (,..., ) decso varable, producto allocato for ultple plats These otatos ll be exteded Secto 3 to clude ultple products. ccordgly, fro the above paraeters ad decso varables, d D ad P d should be satsfed for the relevace of the pro- opyrght 00 Sces

3 8 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty posed odel.. Jot Decso Model for a Sgle Product The average cost copoets cosdered ths proble clude to parts, the frst part s the orderg cost; the secod part s the holdg costs these to parts of the costs are deoted by F (T, ) ad F (, ) respectvely. I a producto cycle has delvery fro the arehouse to the retaler, so the orderg cost F (T, ) are gve as F ( T, ) [( S) ( ) T () p r For the secod part of the costs, the average vetory levels for ra aterals, ork--process plat ( ), ad fshed products at the arehouse ad the retaler over the producto cycle are deoted by I, I, I ad I r, respectvely. I ad I ca be derved by the appedx of eferece [5. Fro the decso varables, e ca derved the producto lot sze s DT, ad the apportoed producto lot sze for plat s DT. Durg a producto cycle, the producto te s DT P, the delvery te s DT d, as llustrated Fgure. It ca be sho that, the average vetory for ork--process I s I [ ( DT d ) DT ( DT P) DT T DT d d P ( )[( )( ) ece, I, I, I ad I r [5 are gve as ( ) I DT P () I DT d d P (3) ( )[( )( ) ( )( ) ( ) I DT D T d (4) Ir DT (5) ece, the holdg cost F (, ) are gve as F (, ) I h I I I (6) r r Substtutg () (5) to (6), e ca obta Fgure. Ivetroy traectory for ork--process plat (, ) ( )[ ( r) F DT D here h h P d The tegrated decsos of the ecooc producto allocato ad delvery polces are expressed as the follog odel: M = F (T, ) +F (, ) (8) [( S ) ( ) T p r ( DT )[ ( r) D (7) s.t. (9) 0 d D,,..., (0) 0 Q Du,,..., () I ths odel, (8) s the obectve of zg the average orderg ad holdg cost for ra aterals, ork--process, fshed products at the arehouse ad the retaler. The costrat (9) s the allocato vector for ultple plats. The costrats (0) ad () should be satsfed by defto, respectvely..3 eurstcs Soluto Procedures The odel s a fractoal olear prograg odel that s ether covex or cocave ad s dffcult to be solved. So e trasfor ths odel th the decso varables (T,, ) to a ore splfed ad equvalet proble th a decso varable, the last trasfored proble s coputed usg a heurstc procedure. Frst, the proble s strctly covex th respect to T, thus the optal cycle legth T*(, ) for a fxed par of ad ca be uquely derved by solvg ddt=0: [( Sp) ( r) T* [ ( r) ( D ) D Substtutg T* to (8), e ca derve E(, ): E (, ) T ( *,, ) {[( S ) ( ) p r r [ ( ) ( D ) D} () (3) For (3), e ca derve: S ( ) [( SP) ( r) (4) [ ( r) ( D ) D opyrght 00 Sces

4 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty 9 e ca obta (5) for fxed : ds( ) ( r)[ D d ( Sp)( r) d S( ) ( Sp)( r) 3 d (5) Sce d S( ) d >0, e ca obta fro dsd=0 ad s gve by ( ) {( Sp)( r)( r) (6) [ ( D )} Sce other ters (7) are costat regardless of except D, e reforulate the ext proble equvaletly as follos: MaxG( ) s.t. (9),(0),() Ths proble belogs to the class of quadratc axzato probles subect to lear costrats th a postve defte quadratc ter. eferece [4 has proved t s a NP-hard proble. Sce ths proble as to assg producto allocato, a heurstc procedure s proposed as follos to solve t. The heurstc algorth steps Step. esequece the descedg order, such that 3 ; Step. Let t be the curret dex uber of the plat to t be assged, ad t be the total aout of the producto allocato t=0, t =0; Step3. t=t+ assget to producto to the tth plat pot: If t- < set {, d D, Q Du } t t t t t t t t Else t 0, t t t Ed f Step4. If t<, go to Step 3; else, go to Step5; Step5. alculate the MaxG( ), the stop. fter dervg *, e ca obta * ad T* fro (6) ad (). 3. Jot Decsos for Multple Products 3. Forulato th Multple Products I ay real cases, the aufacture ofte produces ultple products to eet the eed of the retalers. I ths producto-dstrbuto etork of ultple products, the a ssue s ho ot decsos ca be ade aually o producto cycle legth, delvery frequecy ad producto allocato at a al average cost to the etork. To derve the soluto, the otatos are defed as follos: P = aual producto rate for product at plat (utyear) Q = aual producto capacty for product at plat (year) d = aual trasfer rate for product fro plat to the arehouse (ut) u = producto capacty eeded to produce ut product at plat (year) h = holdg cost for product at plat S = producto setup cost for product at the aufacturer = orderg cost of ra aterals for product = order hadlg cost for fshed product at the arehouse = orderg cost for product at the retaler = holdg cost of ra aterals for product = holdg cost for fshed product at the arehouse = holdg cost for fshed product at the retaler D = dead rate for product (utyear) T = decso varable, producto cycle legth at the aufacturer (year) = decso varable, delvery frequecy for product a producto cycle fro the arehouse to the retaler = decso varable, producto allocato for product plat Slar to the average cost structure of a sgle product, the orderg costs ad the holdg costs are represeted as follos, respectvely: F ( T, ) [( S) ( ) T (8) The secod part s the holdg costs for ra aterals, ork--process vetores, fshed tes at the arehouse ad the retaler. They are deoted by I, I, I, I respectvely ( ) I DT P (9) I ( DT ) [( d )( d P ) (0) ( )( ) ( ) I DT D T d () I DT () ece, the holdg cost F (, ) are gve as opyrght 00 Sces

5 0 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty (, ) F I h I I I (3) Substtutg (9) () to (3), e ca obta F T D D (, ) ( ) [ ( ) (4) ( h ) P ( h ) d (5) The tegrated decsos of the ecooc producto allocato ad delvery polces are expressed as the follog odel: F F( T, ) F(, ) [( S) ( ) T (6) ( T ) D[ ( ) D s. t (7) 0 d D, (8) 0 Q Du (9) 3. eurstcs Method for Multple Products The odel s a fractoal olear prograg odel that s the sae as the odel th the sgle product. It ca be solved by tradtoal olear prograg techques, such as GINO, gradet search ethods, here oly the local optal soluto ay be foud. heurstcs s proposed to solve ths proble. Frst, the proble s strctly covex th respect to T, thus the optal cycle legth T*(, ) for a fxed par of ad ca be uquely derved by solvg d F d T =0: [( S) ( ) T*(, ) D [ ( ) D (30) Substtutg T* to (6), e ca derve: F' {[ (( S ) ( )) [ D ( ( ) D )} For (3), e ca derve: (3) SE( ) { [( S) ( )} (3) { D[ ( ) D } e ca obta (33) for fxed : ds ( ) E D D d ( ) D ( S) ( ) ( ) (33) ( ) D ( S ) d SE( ) 3 d (34) Sce d SE ( ) d >0, e ca obta fro ds E d =0 ad s gve by ( ) ( ) 0 D S ( ) ( ) D( D ) Substtutg (35) to (3), e get F( ): (35) [( )( ) D F( ) { ( S )[ D ( D )} (36) Sce other ters (36) are costat regardless of except D, e reforulate the ext proble equvaletly as follos: ( ) Maxze G s.t (7), (8), (9) Ths proble belogs to the class of quadratc axzato probles subect to lear costrats th a postve defte quadratc ter. Sce ths proble as to assg producto allocato, a heurstc procedure s proposed as follos to solve the odel. The heurstcs s Step. esequece the descedg order for product, such that 3 ; Step. =+, Let t be the curret dex uber of the plat to be assged, ad t t t, t 0, 0 Step3. t=t+ assget to producto to the tth plat pot: If t < set {, d D, Q Du } t t t t t go Step 4 t, t t Else t 0, t, t t Ed f Stetp4. f t<, go to Step 3; else, go to Step5; Step5. alculate G( ) ; Step6. f <, go Step 5 opyrght 00 Sces

6 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty Step7. alculate D Step8. alculate *,f * s t terger, the * arg { F' (,..., I), 0 0 { ( *), ( *) }} Else go Step Step9. alculate T* Step0. alculate F, stop 4. Sulatos ad Perforace alyss To test the perforace of the heurstcs, soe coputato experetatos are coducted ad ther sulato results, as ell as aalyss, are preseted ths secto. The coparso of the heurstcs ad tradtoal quas- Neto ethod are reported ad aalyzed. To shorte the legth of the paper ad thout loss of geeralty, ths secto presets a exaple th ultple products to llustrate the applcato of the odel ad the heurstcs. For splcty, ultple plats at the aufacturer are deoted by,, 3, they ca all produce three products, B,. The producto rate, aual producto capacty, Trasfer rate ad the holdg cost are preseted Table. The producto setup cost, the orderg cost ad the holdg cost at the aufacturer, the arehouse ad the retaler ad the dead rate are preseted Table. Plats Table. The paraeters used sulato tests Producto rate(ut) ual apacty(hours) B B Plats Trasfer rate(utyear) oldg cost($ut) B B Table. Basc data of the test costs B Setup cost O of ra ateral Shppg cost O at the retaler of ra ateral 3 at arehouse at the retaler Dead rate(ut) Fro Table 3, the producto cycle legth, delvery frequecy a cycle fro the arehouse to the retaler, the producto allocato of the to solutos th heurstcs ad QNM are copared. ear-optal soluto th relatve devato of 0.98% s obtaed fro the best solutos by QNM th feasble tal soluto. ece, oe ca coclude that the heurstcs ethod s ore effectve tha QNM the case of the above exaple. I partcular, the results have poted out the sgfcace of assgg producto aog the plats. It reveals that busess operatos, cludg producto ad dstrbuto aog the plats, should be cosdered a tegratve aer so as to reduce costs ad ehace the eterprse s copettveess. To llustrate the effectveess of the heurstcs, four radoly geerated exaples th 3*5, 5*5, 5*0, 0*0 (plats*products) are cted to ake the coparso betee the heurstcs ad the QNM. Fro Table 4, oe ca see that the heurstc algorth s better tha the QNM these exaples ters of qualty of the soluto. 5. oclusos Oe of the core probles of supply cha aageet s the coordato of producto ad dstrbuto. Ths paper cosders ot decsos producto cycle legth, delvery frequecy ad producto allocato for a sgle Table 3. oparso of the heurstcs ad QNM Delvery frequecy Producto allocato heurstcs QNM heurstcs QNM B 5 5 -B Producto heurstcs cycle QNM B The total costs heurstcs QNM B dfferece% Table 4. oparso of the heurstcs ad QNM for dfferet sze of exaples Proble sze osts heurstc QNM Dff (e%) 3* * * * opyrght 00 Sces

7 eurstcs for Producto llocato ad Orderg polces Mult-Plats th apacty product ad for ultple products a producto- dstrbuto etork syste th ultple plats ad ultple retalers. ll plats are all capactated. Based o the producto capacty ad the ut producto capacty for producg a product, the atheatcal prograg odel s preseted to dstrbute the dead of the retaler to ult-plats to acheve a obectve of zg the average costs. To effectve heurstc ethods are developed to solve the ot decso proble th sgle product ad ultple products. The sulato results have sho that the heurstcs s easly pleeted ad effectve for the decso probles. Future ork cludes: the ecooc allocato of the coplex product ultple plats. 6. ckoledget The paper s facally supported by the Natural Scece Foudato of ha (NSF , 70700), Natoal 973 progra (009B3060) of ha, ad proect of Mstry of Educato (MOE) ha th uber B0805. EFEENES [.. Kg ad.. Love, oordatg decsos for creased profts, Iterfaces, Vol. 0, pp. 4 9, 980. [ J. F. llas, eurstc techques for sultaeous schedulg of producto ad dstrbuto ult-echelos structures: Theory ad eprcal coparsos, Maageet Scece, Vol. 7, pp , 98. [3. M. ll, The sgle-aufacturer sgle-retaler tegrated producto-vetory odel th a geeralzed polcy, Europea Joural of Operatoal esearch, Vol. 97, pp , 997. [4 I. Davd ad M. Ebe-hae, o far should JIT vedor buyer relatoshps go? Iteratoal Joural of Producto Ecooc, Vol. 8, pp , 003. [5 T. K, Y. og, ad J. Lee, Jot ecooc producto allocato ad orderg polces a supply cha cosstg of ultple plats ad a sgle retaler, Iteratoal Joural of Producto esearch, Vol. 43, pp , 005. [6 T. K ad Y. og, Techcal ote: Producto allocato, lot-szg, ad shpet polces for ultple tes ultple producto les, Iteratoal Joural of Producto esearch, Vol. 46, pp , 008. [7.. all, O the tegrato of producto ad dstrbuto: Ecooc order ad producto quatty plcatos, Trasportato esearch, Part B, Vol. 30, pp , 996. [8 J. ah ad.. Yao, The ecooc lot ad delvery schedulg proble: The sgle te case, Iteratoal Joural Producto Ecoo, Vol. 8, pp. 38 5, 99. [9 J. Bea, aalyss of vetory ad trasportato costs a costraed etork, Trasportato Scece, Vol. 3, pp , 989. [0 D. E. Bluefeld, L. D. Burs, ad. F. Dagazo, Sychrozg producto ad trasportato schedules, Trasportato esearch, Vol. 5, pp. 3 37, 99. [ F. Fuero ad. Vercells, Sychrozed developet of producto, vetory ad dstrbuto schedule, Trasportato Scece, Vol. 33, pp , 999. [ J. F. Tag, K. L. Yug, ad... Ip., eurstcs based tegrated decsos for logstcs etork systes, Joural Maufacturg Systes, Vol. 3, pp. 3, 004. [3 K. L. Yug, J. F. Tag,... Ip., ad D.. ag, eurstcs for ot decsos producto, trasportato, ad order quatty, Trasportato Scece, Vol. 40, pp. 99 6, 006. [4 S. Sah, oputatoally related probles, SIM Joural o oputg, Vol. 3, pp. 6 79, 974. opyrght 00 Sces