Capacitated Production Planning and Inventory Control when Demand is Unpredictable for Most Items: The No B/C Strategy

Transcription

1 SCHOOL OF OPERATIONS RESEARCH AND INDUSTRIAL ENGINEERING COLLEGE OF ENGINEERING CORNELL UNIVERSITY ITHACA, NY TECHNICAL REPORT Jue 200 Capactated Producto Plag ad Ivetory Cotrol whe Demad s Upredctable for Most Items: The No B/C Strategy Joh A. Muckstadt Corell Uversty Davd H. Murray College of Wllam & Mary James A. Rappold Uversty of Wscos COPYRIGHT 200 Ths research was partally fuded by the Natoal Scece Foudato (Grat DMI ) ad by Aspe Techology.

2 Capactated Producto Plag ad Ivetory Cotrol whe Demad s Upredctable for Most Items: The No B/C Strategy Joh A. Muckstadt School of Operatos Research ad Idustral Egeerg Corell Uversty, Ithaca, NY 4853 Davd H. Murray School of Busess Admstrato The College of Wllam & Mary, Wllamsburg, VA 2387 James A. Rappold School of Busess Uversty of Wscos, Madso, WI Jue 200 ABSTRACT I ths paper, we exame a dscrete-tme, perodc-revew producto evromet that assembles several hudred tems ad that possesses lmted, perhaps radom producto capacty. The demad for a large subset of these tems s hghly erratc ad extremely dffcult, f ot mpossble, to predct accurately. Cosequetly, a coordated producto-vetory strategy, such as the No B/C Strategy preseted Carr, et al. (993), s ecessary. I such a strategy, vetory s carred oly hgh demad rate, predctable tems ad producto prorty s gve to ostocked tems. Producto s cotrolled for stocked tems through a modfed base stock polcy. A key feature of our approach s that t does ot rely o tem-level forecasts for each tem. Our objectve s to develop ad test a computatoally effcet ad accurate procedure for establshg base-stock levels that mmzes the expected holdg ad backorder costs per perod over a fte horzo. We cosder two alterate operatg scearos ad compare them. I the frst, we cosder a formato poor evromet ad assume that producto decsos must be made advace of observg demad. I the secod, we assume a formato rch evromet ad observe demad pror to makg producto decsos. We quatfy the value of ths formato terms of operatg performace. We demostrate that as the capacty utlzato creases, the value of obtag advaced demad formato, whle stll valuable, dmshes. I these evromets, there are two types of safety stock eeded to mmze expected costs: demad-drve safety stock ad capacty-drve safety stock. We demostrate that for capacty-drve safety stock, tradtoal equal-fractle allocato polces may lead to substatal vetory mbalaces over tme ad result lower tha expected servce levels. To resolve ths, we preset a alteratve meas for allocatg capacty-drve safety stock that favors storg vetory tems for whch the rsk of ot sellg them s lower. Through a seres of umercal expermets, the model s show to be accurate ad o average performs wth 0.26% of a lower boud. The formato rch case resulted, o average, 35% lower safety stock ad 57% lower expected costs per perod tha the formato poor case. Usg our proposed capacty allocato scheme resulted 23% less uts mbalace tha f the ewsvedor allocato were used. We coclude wth a expermet of the approach a dustral evromet ad demostrate ts effectveess over curret maagemet practces. KEY WORDS: PRODUCTION PLANNING AND CONTROL, MANUFACTURING STRATEGY, NO B/C STRATEGY, VALUE OF INFORMATION, INVENTORY SHORTFALL, COLLABORATIVE PLANNING.

3 Itroducto I ths paper, we explore a alteratve way of makg capacty allocato ad vetory stockg decsos that s desged for lea maufacturg evromets. These evromets have a wde product varety, hghly ucerta demad, lmted producto capacty (perhaps radom), drastcally shorteed flow tmes, reduced setup tmes, ad small batch szes. Ivestmets made ew equpmet ad employee trag have resulted assembly evromets wth very short, predctable, ad repeatable flow tmes. Typcally, there s a effectve maxmum producto capacty per perod. Depedg o how the producto faclty has bee desged, ths producto capacty may be determstc or radom, ad may be scalable through the dyamc addto of equpmet or labor. The tmg of demad formato may also be dfferet dfferet evromets. I some cases, producto decsos must be made pror to the realzato of actual demad. I ths case, a forecast s geerated for each tem over a leadtme ad vetory s held due to forecast errors. The lead-tme s ofte used as a determstc surrogate for lmted producto capacty. We refer to ths vetory as demad-drve safety stock. I other cases, demad formato s kow pror to makg producto decsos. If capacty were ulmted, the o fshed goods vetory would be eeded; however, due to lmted capacty, there may or may ot be suffcet capacty a perod to satsfy all demad. Cosequetly, some vetory s eeded to protect customer servce objectves. We refer to ths type of vetory as capacty-drve safety stock. The more volatle ad upredctable the demad s, the more safety stock s eeded for a gve level of customer servce. Based o our work ad observatos both the hydraulc flud coveyace ad electrcal cotrols dustres, the exstg methodologes used practce for goverg the maagemet, plag, ad cotrol of capacty ad materals maufacturg systems are ot well suted for these ew evromets. Specfcally, the assumpto that demad forecast errors for a tem over a short repleshmet lead-tme ca be accurately represeted usg a probablty dstrbuto s flawed, as we shall demostrate. Thus, producto ad vetory plas based o resultg accurate demad forecasts are themselves flawed ad ca lead to a sgfcat msallocato of capacty over tme. I some cases, ths capacty msallocato has resulted large vetores of low demad rate tems ad vetory wrtedows. 2

4 Examples of the tems cosdered ths research are assembled products, such as computers. I order to receve, process, ad fll these orders, frms must balace assembly capacty lmtatos wth fshed goods vetory vestmet ad customer servce levels. Whle t s typcally ot ecoomcal to stock a wde breadth of products atcpato of demad, there are lkely to be several popular models for whch there s a lower rsk of obsolescece. It s these products that we cosder possble stockg advace of demad order to protect customer servce agast suffcet producto capacty. The value of formato for producto ad vetory decso-makg depeds heavly o the tmg of evets the system. If the maufacturg flow-tme s greater tha the customer order lead-tme, the a forecast must be made ad vetory must be held. If the maufacturg flow-tme s wth the customer order lead-tme, the the frm ca potetally satsfy demad wthout the eed for tem-level forecasts for all tems. If demad s observed before producto decsos are made, ths permts a oe-day reducto order fulfllmet lead-tme for low demad tems. O the other had, f producto decsos must be made before the observato of the perod's demad, ths results a addtoal oe day of order fulfllmet lead-tme. Curretly, vetory systems wth several hudred or thousad tems are maaged fudametally o the oto of ABC vetory, as descrbed Slver ad Peterso (985). Uder ths system, tems are categorzed to ether A, B, or C categores. I ths categorzato, the A-type tems are fast-sellg tems, whch typcally accout for 20% of all the tems, ad whch accout for 80% of all sales dollars. The B-type tems are medum-sellg tems whch accout for 30% of the tems, ad whch accout for 5% of sales dollars. The C-type tems are the slowest-sellg tems, whch accout for the remag 50% of the tems, ad whch accout for the remag 5% of sales dollars. Ths s the classc Pareto aalyss. I our experece, however, we rarely see dstrbutos of demad amog tems that are lke ths classcal oe. The oe show Fgure s more represetatve of the may that we have observed. Furthermore, ths categorzato does ot accout for the dffereces ucertaty that exsts the demad amog tems. For some tems, demad s ucerta but predctable, whle for others the demad s ot predctable. That s, the dstrbuto of lead-tme forecast errors oe case ca be represeted accurately wth a Posso, Negatve Bomal, Normal or Laplace dstrbutos, for example. I the secod case, the coeffcet of varato of forecast errors s so large that a accurate predcto of lead-tme 3

5 demad s ot possble to obta, much less the dstrbuto of demad. Examples of tme seres for some tems are gve Fgure 2. Le Actvty by Part (All Customers) Percet of Total Les 00% 90% 80% A B C 70% 60% 50% 40% 30% 20% 0% 0% A: 4,97 2% B: 9,43 9% C: 8,482 88% 0 20,000 40,000 60,000 80,000 00,000 20,000 40,000 60,000 80, ,000 Part Rak Fgure. A example part categorzato for a processg faclty of automotve spares Example Part (Category A) Example Part 2 (Category A) Quatty 20,000 5,000 0,000 5, Perod Quatty 0,000 8,000 6,000 4,000 2, Perod Example Part 3 (Category C) Example Part 4 (Category C) Quatty 2,000,500, Perod Quatty 3,500 3,000 2,500 2,000,500, Perod Fgure 2. Example Tme seres for some products. The research questos of terest are: What s a approprate producto polcy for cotrollg producto ad vetory? How should capacty be allocated to dvdual tems? How ca the base-stock levels be computed effcetly for large-scale systems? What s the value of obtag advaced demad formato? 4

6 The approach we preset s dfferet from the curret paradgm for maagg producto ad vetory ths type of evromet, sce t cosders hghly varable demad ad lmted producto capacty. As we have stated, the assumpto that the demad process for each tem s accurately represetable va a probablty dstrbuto s approprate may stuatos. Cosequetly, we wll ot assume a forecast s possble for these tems. Nor wll we assume a fxed lead-tme as s commo practce. We propose a approach for maagg producto capacty ad vetory that s called the No B/C Stockg Strategy. It was frst descrbed Carr, et al. (993), ad has the followg operatg characterstcs. Frst, vetory s ot held ay B- or C-type tems, but rather oly A- type tems are produced o a make-to-stock bass. Secod, B- ad C-type tems are gve producto prorty whe demad for them occurs. Thus, they are produced o a make-to-order bass. Ay excess producto capacty remag a perod after producto of the B- ad C- type tems may be used to replesh A-type tem vetory up to a predetermed base stock level. To operate uder ths polcy, A-type tem vetory s requred due to the presece of lmted producto capacty. Although accurate tem-level forecasts may be dffcult for may tems, t s ofte easer to accurately characterze the aggregate demad for capacty usg a probablty dstrbuto. Oe may argue that producg the very low demad tems s ot approprate; however, may staces, the low ad upredctable demad rate tems are the most proftable tems, sce maufacturers of these tems are ofte the suppler of last resort. The decso to categorze a tem as a make-to-stock tem usually volves tem-specfc attrbutes such as demad volume, stablty, maufacturg lead-tme, etc.; however, the categorzato decsos ad the stock level decsos are ot separable lght of capacty lmtatos. Our model determes the categorzato of tems to stocked ad o-stocked types, depedg o the system attrbutes, such as capacty ad demad varablty. Items for whch a probablty dstrbuto for demad s ot avalable or possble to estmate are, by defto, o-stocked tems. Fudametal to our approach s the assumpto that the effect of vetory mbalaces o cost over tme s eglgble. Ivetory mbalace occurs whe there s too much of oe tem ad too lttle of aother tem the system for a gve level of total vetory. I such stuatos, we say that the allocated capacty s a state of mbalace. We make ths assumpto sce t has bee show that mult-echelo vetory systems wth fte producto capacty, vetory 5

7 mbalace does ot have a sgfcat mpact o system cost. See Rappold ad Muckstadt (2000). Uder ths assumpto, we obta a smple lower boud o the expected system cost per perod ad demostrate that ths lower boud s very close to the optmal cost. There are several advatages that we expect from operatg the system uder such a polcy. Frst ad most mportatly, the heret statstcal problems surroudg accurate forecasts for tem-level demads dmsh for B- ad C-type tems. Uder the No B/C strategy, the aggregate demad for producto tme s forecasted across all tems. Because ths aggregato s over a large group of tems, the aggregated demad process possesses a much smaller coeffcet of varato. Ths leads to parameter estmates of demad for capacty wth a hgher level of cofdece, ad ultmately allows for more accurate forecasts of capacty usage. Secod, vetory vestmet s focused fewer tems, each of whch has a hgh turover rato. Thrd, the maagemet of the reduced vetory tems s smplfed ad may of the drect costs assocated wth vetory storage, trackg, ad hadlg are reduced. Fourth, customer servce mproves due to shorter respose tmes for B- ad C-type tems, (sce they are produced o a make-to-order bass) ad due to hgh vetory levels of A-type tems to protect servce from suffcet producto capacty. Ffth, the system ca be maaged as a pull system ad ca be cotrolled usg smple pull sgals for the stocked tems. 2 Lterature Revew There has bee much research accomplshed the area of vetory plag for sgle locato producto ad vetory systems. The research has bee the cotext of cotuous or dscrete tme models for systems that produce ether a sgle or multple tems ad operate wth or wthout setup costs. Our research falls the category of dscrete tme, multple tem models wth o setup costs. Where our problem dffers s where past research has assumed the ablty to characterze tem level demads wth a probablty dstrbuto, we assume that for the majorty of tems (B- ad C-type tems), o such characterzato s possble. Curretly, there are two fudametal approaches for examg ths problem. The frst approach, by Federgrue ad Katala (994), regards tme as a cotuum ad apples queug theory ad pollg system theory to descrbe the behavor of the producto system. Uder ths 6

8 paradgm, A-type tems are produced accordg to a cyclc producto schedule. The methodology s able to cosder both setup tmes ad processg tmes for each tem. The producto of B/C-type tems s trggered oce a order for such a tem s receved. Producto prorty s gve to the B/C-type tem(s) by ether pre-emptg the curret A-type tem producto or by watg utl the completo of the curret A-type tem. Oce the producto of the B/Ctype tem s complete, producto of the A-type tems resumes accordg to the cyclc schedule. Producto of A-type tems s cotrolled va a base-stock polcy whch producto of the A- type tem stops oce ts vetory level reaches a pre-determed target level, or base-stock level. The framework s computatoally effcet ad qute geeral, that t allows setup tmes. As a cosequece of ths polcy, the cycle legth (the tme betwee successve producto rus of the same A-type tem) s radom ad, depedg o the volatlty of demad for all tems, may be hghly varable. Ths cycle tme varato creases safety stock requremets ad may hder the ablty to provde predctable servce to customers whe demad s hghly ucerta. Because of ths varato lead tmes, t s dffcult to mplemet practce. Furthermore, wth ucerta lead tmes, t s dffcult to use ths approach wth a mathematcal programmg based producto ad vetory cotrol model. Our approach lkewse assumes that f the dvdual tem demad for B/C-type tems s upredctable, the o vetory should be stored such tems. To compesate for ths, producto prorty wll be gve to B/C-type tems. Furthermore, to esure that customer servce s protected for A-type tems, addtoal A-type tem vetory wll be held. Both approaches are also computatoally vable for real-szed producto evromets. Where they dffer s the fudametal perspectve of the system - perodc versus cotuous revew. Our model has the operatoal advatage that, because of the repettve ature of the system, lead tmes are relatvely costat, whch s, all real systems we have examed, a requremet. More recetly, Sox, Thomas, ad McCla (997) also cosder a producto-vetory strategy whch vetory s held oly hgh demad tems. Ther approach s based o a cotuous tme aalyss ad uses queueg aalyss to model system behavor uder a varety of producto schedulg dscples. Ther model has the attractve feature that t s smple ad as the capacty utlzato approaches 00%, relatvely more vetory s held the hgh demad tems. They, lke may other researchers, assume that tem-level demad processes are Posso processes. I practce, we have foud that the Posso model s approprate because the 7

9 varato demad over a short lead-tme s much greater tha the Posso model suggests. We are therefore most terested studyg hghly ucerta demad evromets. Hece, we have elected to model A-type tem ad aggregate B/C-type tem demad usg Negatve Bomal dstrbutos. Much of the past research to ths problem has ether assumed ulmted supply or allocato rules that are heretly computatoally complex to hadle mbalaced vetory stuatos. Basc sghts to the problem of vetory mbalaces are made Zpk (984). Other work by Avv ad Federgrue (997) ad by Kapuscsk ad Tayur (996) have made great strdes furtherg the uderstadg of these complex systems, but are computatoally uattractve whe appled to large systems. Approxmatos such as those developed Glasserma (996, 997) are very sghtful explag system dyamcs as the capacty utlzato approaches 00% sgle tem systems. I our observatos of real systems, as the capacty utlzato approaches 00%, chages are made to the system order to complete producto o tme. Examples of such chages clude outsourcg a fracto of the producto requremet ad usg overtme. The choce of a base stock polcy for cotrollg producto ad vetory for each tem s based o the results developed Federgrue ad Zpk (984a) ad DeCrox ad Arreola- Rsa (998). Our model dffers slghtly from thers sce they cosder the stuato whch producto decsos are made pror to the realzato of demad. They argue that for systems whch there s fte capacty ad o fxed setup costs, a modfed base stock polcy s optmal. Thus, whle a base stock polcy may ot be the true optmal polcy, t s a coceptually easy to follow polcy rederg t attractve from a maageral perspectve. Carallo et al. (994) exame the case whe capacty s fte ad radom. As we wll show subsequetly, the case of ucerta capacty ca be modeled as creased ucertaty the demad process. There are several cotrbutos ths paper. Frst, we propose the No B/C strategy as a heurstc ad easy to uderstad polcy, whose performace we explore ad valdate. Secod, we develop a model to effcetly determe tem vetory levels for systems cotag a large umber of tems. Lastly, we descrbe a dustral mplemetato of such a strategy. 8

10 3 Assumptos ad Geeral Notato I ths secto we state our modelg assumptos ad defe the otato that we use throughout the rest of the paper. Model correspods to the formato poor case ad Model 2 correspods to the formato rch case. For x real, let [ x ] + : = m{ x, 0}. System Attrbutes A B P the set of all potetally stocked (A-type) tems; the set of all o-stocked (B/C-type) tems; = A, the total umber of A-type tems; m the dex for tems; the dex for tme perods; the dex for models, m= or 2, for Model or 2, respectvely; h the holdg cost per ut per perod for tem A B ; π the backorder cost per ut per perod for tem A B ; T m the system base-stock level; τ m = [ τ m ] the vector of base-stock levels for dvdual tems, where τ m = T m. Radom Varables ad Probablty Dstrbutos C the dscrete, o-egatve radom varable (r.v.) represetg the per perod producto capacty; c the realzed producto capacty perod ; F C the cumulatve dstrbuto fucto of C; A A the dscrete, o-egatve r.v. represetg the per perod demad for capacty, for tem A ; F the cumulatve dstrbuto fucto of A ; A the cumulatve demad r.v. over perods for A ; () () F the cumulatve dstrbuto fucto of () A ; 9

11 A the dscrete, o-egatve r.v. represetg the per perod aggregate demad for capacty by all A-type tems, where A = ; A A B D the dscrete, o-egatve r.v. represetg the aggregate demad for capacty by all B/C-type tems; the r.v. represetg the aggregate demad for capacty, where D=A+B; V the vetory shortfall perod for Model ; V U Vectors the statoary vetory shortfall r.v. for Model, f t exsts. U the vetory shortfall perod for Model 2; the statoary vetory shortfall r.v. for Model 2, f t exsts. D a ( P + ) vector of all demad r.v.'s, where D = A, A,, A, ]; [ 2 P B d a ( P + ) vector of realzed demads perod, wth d the realzed demad for tem =, 2,, P, perod, ad d, P+ the realzed demad for capacty by all B/C-type tems perod ; x a ( P + ) vector of et vetory levels at the begg of perod ; y a ( P + ) vector of et vetory levels after producto decsos perod for Model ; z a ( P + ) vector of et vetory levels after producto decsos perod for Model 2; We make the followg set of assumptos. Frst, we assume that there are o fxed setup tmes or costs betwee tems whch affect producto decsos. We assume that whle the producto capacty s radom each perod, t s kow the perod before makg producto decsos. We assume that the system s stable, E(D) < E(C). () We assume that we always have adequate capacty to satsfy B/C-type tem demad a perod, Pr{B < C} =. (2) Sce B/C-type tems receve producto prorty over A-type tems ad Pr{B < C}=, B/C-type tems wll ever be backordered. Ths assumpto s prmarly for coveece ad ease of 0

12 exposto. As we wll dscuss subsequetly, ths assumpto may be relaxed, as t does ot alter the valdty of the problem or soluto procedure. We assume that customer order lead tmes for B/C-type tems are oe day. Wthout loss of geeralty, we assume that the per ut processg tmes across the tems are detcal. Thus, we may express demad for each tem terms of requred tme whe allocatg producto capacty. We assume that the demad r.v. ad the capacty r.v. are statoary ad depedet betwee tems ad perods. Ivetory s pulled from fshed goods vetory to satsfy exteral demad. Whe stock s adequate to meet demad, excess demad s backordered. A modfed base stock polcy s followed to cotrol producto ad vetory for each A-type tem. For Model m, accordg to a modfed base stock polcy, there s a prescrbed target vetory level τ m for each A-type tem, whch s also called a order-up-to level, or base stock level. We deote A T = τ as the system target vetory level. The producto faclty attempts to restore all A-type tem vetory levels to τ m, but may ot be able to do so due to suffcet capacty. We assume that tems produced a perod are avalable for shpmet at the ed of that perod. m m Producto Capacty Item-Level Demad Aggregate Demad Iformato Avalablty Iformato Poor Iformato Rch Step System Stock Level, T System Stock Level, T 2 Step 2 Newsvedor Allocato Q-Fucto Allocato Item Stock Levels, τ Item Stock Levels, τ 2 Fgure 3. Overvew of modelg approach Our approach for determg these vetory levels s a two-step process show Fgure 3. Frst, we determe how much vetory should be held the system. Secod, we allocate ths aggregate vetory level to dvdual A-type tems usg oe of two allocato schemes, depedg o the formato avalablty.

13 4 Model Iformato Poor Evromet Uder ths frst scearo, producto decsos are made pror to observg the perod's demad. The sequece of daly evets s as follows. At the begg of perod, x s the vector of et vetory levels (o-had mus backorders) for the stocked tems. Producto capacty c becomes kow. Producto decsos are made to satsfy B/C-type demad occurrg the prevous perod ad to rase x to τ, for each A, f possble, resultg post-producto et vetory levels of y. The system may be uable to restore the et vetory levels to ther targets due to suffcet capacty. We defe the vetory shortfall perod as V, measured after producto but before demad occurs as descrbed Tayur (992). The sgle perod expected cost fucto s, where h ad G P + ( y ) = h E[ y A ] + E[ A y ] + = π, (3) π are the holdg ad backorder costs, respectvely, for tem A. Ths s the famlar ewsvedor fucto. It s the mechasm by whch capacty allocato decsos wll be made Model. Sce B/C-type tems are gve producto prorty ad Pr{ B < C} =, o holdg or backorder costs wll be curred for these tems. The perod s demads d are the realzed ad are ether satsfed or backordered. Fally, the perod s costs are computed. Ths sequece of evets s depcted Fgure 4. Perod Demad occurs for A-tems (due perod ) Demad occurs for B/C-tems (due perod ) Perod Demad occurs for A-tems (due perod ) Demad occurs for B/C-tems (due perod +) Evets: Pla ad execute producto for perod, kowg oly demad occurrg perod 2 Observe demad occurrg perod Shp A-tem demad occurrg perod, due perod Shp B/C-tem demad occurrg perod 2, due perod Pla ad execute producto for perod, kowg oly demad occurrg perod Observe demad occurrg perod Shp A-tem demad occurrg perod, due perod Shp B/C-tem demad occurrg perod, due perod Vectors: Shortfall Dstrbuto: c D x y V x c y D [ V + d c ] + V = tme Fgure 4. Perodc sequece of evets for Model - Producto decsos pror to demad observatos. 2

14 I ths case, f the producto capacty were fte, vetores would be held A-type tems to mmze the expected holdg ad backorder costs the curret perod. Our objectve s to determe the value of the target aggregate vetory level, T, ad dvdual tem target levels τ that mmzes the expected holdg ad backorder costs per perod over a fte horzo. For Model, the target vetory level, T, s defed as the deal amout of vetory after producto but before demad a perod. Ay tem for whch τ 0 s a make-to-order tem. Whle statg the precse problem s coceptually smple, due to the eormous umber of possble states that may exst, solvg the assocated stochastc dyamc program explctly s ot a practcal opto. Thus, our strategy wll be to costruct a very close approxmato that s computatoally effcet. = 4. A Exact Formulato We may state the precse decso problem as, f { m G( y ) + Ef ( y D) } ( x ) = EC + y R( x, C), (4) where R( x, C) { y : y x, = (5) P + = ( y x ) C}, (6) s the set of feasble decsos perod. Costrats (5) prevet the tetoal removal of vetory, sce vetory may be lowered oly through demad. We call these the vetory mbalace costrats. Costrat (6) lmts producto to the avalable capacty the perod. The results preseted Federgrue ad Zpk (984a) ad DeCrox ad Arreola-Rsa (998) show that a modfed base stock polcy s optmal whe there s capacty costrat ad there are o fxed orderg or setup costs. Uder ths polcy, each stocked tem has a prescrbed target vetory level. Each perod, producto occurs to rase each tem up to ts target level, or to get as close as s possble. We wll assume that ths producto polcy s used our approach. 3

15 4.2 A Approxmato usg the Ivetory Shortfall Process Solvg (4) each perod s ot possble due to the eormous umber of states ad the resultg computatoal complexty. Istead, we propose usg the vetory shortfall process to descrbe the evoluto of the system ad solvg the followg approxmato each perod. Ths method s a approxmato; t does ot ecessarly fd the optmal producto ad vetory pla that would be foud by solvg (4). However, the method s computatoally tractable ad, as we wll demostrate, s effectve fdg good solutos The Ivetory Shortfall Process Sce capacty s lmted, the vetory level may ot be restored T after producto. We defe the devato from T after producto a perod as the vetory shortfall, ad deote ths quatty as 0 = 0 V = T. The vetory shortfall process satsfes the Ldley recurso y A V, V = [ V + D C] +. Recall that both D ad C are depedet, detcally dstrbuted radom varables ad observe that { } > 0 V s a Markov cha. Let V be ts statoary r.v., f t exsts. Assumpto () provdes ecessary ad suffcet codtos for the exstece of V. From Cha, et al. (999), the trasto matrx for the shortfall process ca be descrbed as follows. Let p = Pr{ V = j V }. The, j = p Pr{ D k C = k} Pr{ C = k}, k = Pr{ D = k + j C = k} Pr{ C = k}, k j 0, for for j = 0, j > 0, otherwse, 0, 0, j (7) where D represets the radom varable for the aggregate demad for capacty over all tems ~ a perod, ad C s the radom varable for capacty. Defe C = C B, where B s the aggregate demad for the B/C-type tems. Sce the demad for all tems all perods are depedet of each other ad that producto wll frst be allocated to B/C-type tems, we ca descrbe a vetory shortfall process for A-type tems usg the trasto probabltes gve (7). Thus, 4

16 C ~ s the radom capacty avalable for the producto of A-type tems ad ~ ~ Pr{ C = k} = Pr{ B = C k}. Sce E ( D) < E( C), we have that E ( A) < E( C) E( B) = E( C). A key fact s that the vetory shortfall process { } > 0 V s depedet of the aggregate target vetory level T. There are two prcple purposes for safety stock ths system. Frst, safety stock exsts to satsfy demad varato the curret perod. Sce demad s ot observed pror to the producto decso, a demad forecast s made for each A-type tem. We refer to ths type of safety stock as demad-drve safety stock. It exsts due to formato delays or whe customer order lead tmes are less tha maufacturg lead tmes. Secod, safety stock exsts to protect the system agast suffcet producto capacty prevous perods. We refer to ths secod type of safety stock as capacty-drve safety stock. Istead of solvg (4) every perod, we use { } > 0 V ad propose solvg, where m y R ( x, T V ) ' G( y ) (8) R' ( x, T V ) = { y : y x, (9) A } y = T. (0) V From the vetory mbalace costrats (9), the state of the system, x, at the begg of perod costras the soluto to the producto problem. I partcular, the set of feasble solutos R s a result of the assumpto that a modfed base stock polcy s followed. The et vetory at the begg of a perod plus the producto a perod wll ot exceed T. 4.3 Determg the Target System Ivetory Level, T Whle solvg (8) s very effectve for makg perodc allocato decsos, t caot be easly used to determe the optmal target vetory level * T, sce all possble combatos of begg vetory levels x must stll be cosdered. To fd T *, we costruct a approxmato that gores potetal vetory mbalaces. We wll gore the allocato of curret stock amog the tems by relaxg the costrats (9). We assume that the vetory s 5

17 properly allocated amog the tems so that vetory mbalace s ot lkely to occur. Imbalace wll occur, of course; however, as we wll demostrate, ts effect o expected per perod cost s ot substatal. We assume that state of the system at the begg of a perod s o loger defed by x ad V, but rather oly by V. Ths results a drastc smplfcato the descrpto of the system evoluto. Let, where { G( y) : y R''( T )} * y ( T V ) = arg m, () V R' '( T V ) = y : y = T V. A Equato () s smply the vector of optmal producto decsos, gve a total supply of T V uts perod, assumg that vetory mbalaces do ot occur over tme. The resultg expected cost perod s, * ( y ( T V )) J ( T V ) = G. (2) Ths s a lower boud o the true expected cost per perod, as ay msallocato of vetory amog tems ca oly crease ths cost. We explore the accuracy of ths approxmato through a computatoal expermet secto 6. We appeal to the strog law of large umbers for Markov Chas to establsh that the log ru average cost per perod coverges almost surely to ts expected value. To determe m T E V * T we solve, [ J ( T V )] = m J ( T k) Pr{ V = k} T k 0, (3) where V s the statoary shortfall radom varable. As show Rappold ad Muckstadt (2000), the objectve fucto (3) s covex ad ca be solved very quckly. 5 Model 2 Iformato Rch Evromet I ths secto, we alter the perodc sequece of evets. We assume that the perod s demad s realzed pror to producto decsos. The sequece of perodc evets s show Fgure 5. At the begg of a perod, x s the amout of o-had vetory (or backorders). As before, = 0, B, sce B/C-type tems receve producto prorty ad Pr{ B < C} =. x 6

18 Next, demad s realzed ad a producto decso s made. Producto occurs frst to satsfy B/C-type tem demad ad the to restore A-type tem vetory levels to ther prescrbed target vetory levels, τ 2. Perod Demad occurs for A-tems (due perod ) Demad occurs for B/C-tems (due perod ) Perod Demad occurs for A-tems (due perod ) Demad occurs for B/C-tems (due perod +) Evets: Observe demad occurrg perod Pla ad execute producto for perod, kowg demad due perod Shp A-tem demad occurrg perod, due perod Shp B/C-tem demad occurrg perod 2, due perod Observe demad occurrg perod Pla ad execute producto for perod, kowg demad due perod Shp A-tem demad occurrg perod, due perod Shp B/C-tem demad occurrg perod, due perod Vectors: Shortfall Dstrbuto: x d c z U x d c z [ U + d c ] + U = U tme Fgure 5. Perodc sequece of evets for Model 2 - Demad observed pror to producto decsos. If capacty were ulmted, the the target vetory levels would be zero for all tems, sce demad could be satsfed the same perod. Equvaletly, all tems would be make-toorder tems, sce ay tem for whch τ 0 s a make-to-order tem. Therefore, the system 2 = target vetory level, T 2, represets solely the capacty-drve safety stock the system. There s o demad-drve safety stock ths case. We aga determe τ, A, two steps. Frst, we determe how much total 2 vetory should be held the system, across all tems. Secod, we allocate ths aggregate quatty to dvdual A-type tems usg a ovel allocato fucto. Let x ad z be vectors represetg et vetory levels (o-had mus backorders) at the begg ad at the ed of perod, respectvely. Let T 2 deote the aggregate base stock level for Model 2 ad τ be the base stock level of tem, where τ = 2 oe perod cost fucto as, sce the demad for the perod s kow. + + ( h ( z ) + ( z ) A A 2 T 2. We defe the H ( z ) = π ), (4) 7

19 The decso problem s, where g ( x, d ) = EC m H ( z ) + Eg + ( z, D) z S ( x, d, C), (5) S( x, d, C) = { z : z x d, (6) A B z x + d ) C } (. (7) The set of feasble decsos S( x, d, C) s a fucto of the begg vetory levels, the realzed demad the perod, ad the producto capacty. Costrats (6) are the vetory mbalace costrats. Costrat (7) lmts producto to the avalable capacty the perod. 5. A New Capacty Allocato Fucto, Q () As before, the state space becomes exceedgly large for reasoable szed problems, rederg (5) usolvable practcal stuatos. Rather tha solvg (5), we wll solve a approxmato to t. The goal of the dyamc program s to mmze the curret perod s costs ad also produce a mx of tems the proper quattes so that future perod s demads ca be satsfed whle ot carryg excessve stock. Thus, a approxmato should mmze the sum of the curret perod s shortage ad holdg cost plus expected future costs that result from the producto decso ths perod. Whe the holdg costs amog the tems are the same, the curret producto capacty should be allocated to tems for whch there s lkely to be demad that wll cosume the stock the very ear future. Lttle, f ay, of the curret perod's capacty should be used to produce tems that wll ot lkely be used for may perods the future. Let A be the radom varable represetg the cumulatve demad for tem A over () perods ad defe the followg fucto, ( ) [ A ] = = k= 0 + w ( ) Q ( w) = E w = F ( k), (8) as the expected umber of vetory-perods assocated wth stockg w uts of tem the system. Ths fucto was frst troduced Cha, Muckstadt, ad Rappold (999). Ths fucto s covex ad has a hgher value for tems that possess a hghly varable demad 8

20 process. Whe weghted by h, the quatty h Q (w) s a lower boud o all future udscouted holdg costs assocated wth stockg w uts of tem. 5.2 A Approxmato usg the Ivetory Shortfall Process Uder the defto of the vetory shortfall process descrbed Fgure 5, the shortfall process for Model, { V } > 0, s detcal to the shortfall process for Model 2, { U } > 0. Thus, the respectve statoary dstrbutos are detcal. That s Pr{ V = k} = Pr{ U = k}, for k = 0,,2,... Usg { } > 0 U, we propose the followg approxmato to (5), m z S '( x, d, T2 U ) where the set of feasble solutos s, H ( z ) + A h Q ( z ), (9) S' ( x, d, T2 U ) = { z : z x d, (20) U A B } z = T2. The approxmato (9) recogzes the vetory mbalace costrats (20) ad makes allocato decsos that mmze the curret perod s costs ad expected future holdg costs. It s a approxmato sce t gores future expected shortage costs ad lmts the amout of vetory the system to T 2, regardless of the precse state of the system. As we shall demostrate subsequetly, ths assumpto does ot have a sgfcat mpact o system costs. 5.3 Determg the Target System Ivetory Level, T 2 Usg (9) to determe a sutable value of T 2 s ot possble due to the eormous state space. We aga develop a approxmato by relaxg the mbalace costrats. We assume the state of the system at the begg of a perod s o loger defed by x but rather by the value of the vetory shortfall radom varable, allocated amog the tems so that mbalace s ot lkely to occur. U. We assume that the vetory s properly 9

21 Let, where, * z ( T2 U ) = arg m H ( z ) + hq ( z ) : z S''( T2 U A ) (2) S' '( T2 U ) = z : z = T2 U. A Equato (2) s the vector of optmal producto decsos, gve a total supply of T2 U uts perod, assumg that vetory mbalaces do ot occur over tme. The resultg expected cost perod s, L( T U ) = H 2 To determe a sutable value of T 2 we solve, m * ( z ( T U )). (22) 2 [ L( T U )] = m U 2 L( T2 k) Pr{ U = k} T2 T2 k 0 E, (23) where U s the statoary shortfall radom varable. The objectve fucto (23) s covex ad ca be solved very quckly. Oce T * 2 s computed, the dvdual tem target levels may be * * determed as τ = z ( ). 2 T2 I summary, for ay level of system vetory the desred allocato amogst tems. Sce the term A * T2 U, the vector z ( T 2 U ) provdes h Q ( z ) s ot a curred cost but rather a mechasm for allocatg capacty-drve safety stock, the cost assocated wth havg T U * 2 uts of vetory the system s gve as L( T2 U ) = H ( z ( T2 U )) Sce. * T 2 s oly capacty-drve safety stock ad the shortfall dstrbutos for both Model ad Model 2 are detcal, T s the vetory reducto due to creased * * T2 formato avalablty. I Model, * edg vetory a perod s E( ) D * T s the target vetory pror to demad. The desred * * T. Cosequetly, ( T E( D ) T reducto assocated wth creased formato avalablty. s the safety stock ) 2 20

22 6 Numercal Study A extesve smulato study was coducted to measure the accuracy of the proposed approach. Sce we assume that vetory mbalace amog tems ca effectvely be gored, oe objectve these expermets s to measure the degree to whch ths assumpto caused operatg costs to dffer from the computed lower boud. Aother objectve s to provde sght to the value of formato ths evromet. We also demostrate the performace of the Q-fucto over the ewsvedor fucto. I addto to these umercal expermets, we llustrate the effectveess of the Q-fucto a dustral expermet descrbed secto The Desg of the Expermet We cosdered 72 operatg evromets. I all cases, holdg costs were set at $/ut/perod ad the backorder costs at $9/ut/perod. The operatg evromets dffered terms of formato avalablty, capacty utlzato, demad process varato, umber of A- type tems, ad how total A-type tem demad was dstrbuted amog the A-type tems. The latter two factors cotrbuted o addtoal sght to the operatoal dyamcs of the system, ad so the results that follow do ot report these factors explctly. The crtcal factors are summarzed Table. The formato evromets dffered terms of the pot the tme perod whe producto plaers kew the perod s demad. I Model, producto plaers dd ot have access to the curret perod's demad ad thus had to atcpate demad. We assume that the true demad probablty dstrbutos ad parameters were kow. I Model 2, decso makers had access to demad formato before makg producto decsos that perod. I Model 2, we performed each expermet twce, oce usg the ewsvedor allocato fucto (3) to allocate the avalable vetores amog tems, ad oce usg the Q-fucto (8) for the allocato. System Factor Abbrevato Factor Levels Iformato Avalablty Iformato Model, Model 2 Capacty Utlzato Utlzato 83%, 9%, 95% Demad Process Varace-to-Mea Rato VTMR.0, 2, 5 Allocato Method (Model 2 oly) Allocato Newsvedor, Q-fucto Table. Factors uder study. 2

23 I each of the expermets, we frst computed the statoary vetory shortfall dstrbuto by solvg the assocated Markov cha. Usg V, we computed T *, the optmal target system vetory level, ad the assocated lower boud o the expected cost per perod. We the determed the dvdual tem target vetory levels τ by solvg () ad (2). For each of 0 radom seeds, we smulated 2 mllo cycles of actvty, where a cycle s defed to be the tme betwee successve restoratos to T *. Of these cycles, mllo were geerated usg the base radom umber stream, ad the remag mllo were geerated usg ts atthetc stream. Radom demads were geerated from egatve bomal dstrbutos whose meas totaled 00 uts per perod ad varace-to-mea ratos were as dcated Table. The capacty per perod was vared to acheve the levels of utlzato dcated Table. I perods where there was suffcet capacty to rase the system vetory to T *, capacty was allocated accordace wth (8) for Model, ad both (8) ad (9) for Model 2. Ivetory mbalaces arse the smulato ad costs were curred accordgly. The performace measures of terest are lsted Table 2. Measure Descrpto Ivetory Levels The total system vetory level, T. Expected Cost The expected cost of operatg wth T uts of vetory, assumg that o vetory mbalaces occur. Percet Cost Error The value of ActualCost ExpectedCost ExpectedCost Average Imbalace The average umber of uts mbalace per perod. Table 2. System performace measures. The followg three subsectos descrbe the performace of our approach relato to these performace measures. 6.2 Accuracy of the Model Across all expermets, the actual observed cost dffered from the lower boud o expected cost by a average of 0.26%. Ideed, as show Fgure 6, all cases the actual operatg costs were wth 0.53% of the expected cost lower boud. Observe that cost errors 22

24 ted to crease wth VTMR, sce hgher levels of vetory mbalace are lkely to occur as the predctablty of tem-level demad decreases. I the formato-rch evromet (Model 2), expermetal costs dffered from theoretcal lower boud costs by 0.26% ad the formatopoor evromet (Model ) by 0.4%. Ths occurred due to the fact that the expected cost per perod s sgfcatly lower for Model 2. Avg Cost Error % VTMR Iformato Utlzato Grad Total Model 83.33% 0.03% 0.04% 0.09% 0.05% 90.9% 0.02% 0.2% 0.22% 0.2% 95.24% 0.06% 0.26% 0.44% 0.25% Model Total 0.03% 0.4% 0.25% 0.4% Model % 0.36% 0.4% 0.39% 0.39% 90.9% 0.7% 0.42% 0.5% 0.36% 95.24% 0.9% 0.53% 0.50% 0.4% Model 2 Total 0.24% 0.46% 0.47% 0.39% Grad Total 0.4% 0.30% 0.36% 0.26% Fgure 6. Average cost error uder dfferet system scearos. From these results, we coclude that the vetory mbalace assumpto s reasoable ad that the approach s qute accurate. We defe vetory mbalace as the umber of uts at the ed of a perod that have bee stocked correctly. These are uts that are ether above or below the optmal tem-level targets for a gve amout of vetory the system. The tet of the Q-fucto s to allocate vetory oly to those tems that have low coeffcets of varato, thereby decreasg the umber of uts mbalace. Fgure 7 shows the average umber of uts mbalace all expermets. Observe that mbalace creases wth each model as utlzato ad demad varato crease. Notce that whe mbalace does occur, t s lower for Model 2 tha for Model at each level of utlzato ad demad varato. Average Uts I Ibalace VTMR Iformato Allocato Utlzato Grad Total Model Newsboy 83.33% % % Model Total Model 2 Q-Fucto 83.33% % % Model 2 Total Grad Total Fgure 7. Average umber of vetory uts mbalace each perod. 23

25 Avg Cost Error % VTMR Utlzato Allocato Grad Total 83.33% Q-fucto vs. Newsvedor 0.00% 0.00% 0.00% 0.00% 90.9% Q-fucto vs. Newsvedor 0.00% 0.00% -.43% -0.67% 95.24% Q-fucto vs. Newsvedor 0.00% 0.00% -2.50% -.04% Fgure 8. Chage Avg. Cost Error % whe usg Q-fucto stead of Newsvedor allocato for Model 2. The allocato scheme used the Model case was the famlar ewsvedor method, whch, because t cosders oly a sgle perod, s drve sgfcatly by backorder costs whe decdg how much of each tem to stock. It s sestve to the relatve amout of varato the demad patters of the dvdual tems beg stocked over loger perods of tme. Model 2 used the Q-fucto to allocate avalable vetory, whch cosders multple perods to the future. The Q-fucto stores capacty the form of vetory that has a hgher probablty of beg cosumed the mmedate future. Storg capacty hghly varable tems creases the rsk of vetory mbalace, ad cosequetly, creases holdg ad backorder costs. The Q- fucto therefore creates fewer mbalaces tha the ewsvedor method. For Model 2, Fgure 8 shows the chage the cost error whe usg the Q-fucto stead of the ewsvedor fucto for allocato decsos. I the dustral expermet dscussed secto 7, the demad processes are substatally more varable ad the mproved performace of the Q-fucto over the ewsvedor fucto s eve more sgfcat. 6.3 Model vs. Model 2: The Value of Iformato I ths subsecto, we exame the performace dffereces betwee Models ad 2. Fgure 9 ad Fgure 0 cota the values ad relatve dffereces, respectvely, of the desred perod-edg vetory for Models ad 2. These levels of safety stock are held to protect the system agast ucertaty of two types demad ucertaty ad supply ucertaty. Wth each model, vetory creases as VTMR (demad ucertaty) ad capacty utlzato (supply ucertaty) crease, as oe would expect. Betwee models ad 2, the levels of safety stock held are uformly hgher Model tha Model 2 for a gve level of VTMR ad capacty utlzato, sce, Model 2, demad s kow the curret perod. I Model 2, there s oe perod s less demad ucertaty to accout for tha Model. 24

26 Desred Perod-Edg Ivetory VTMR Iformato Utlzato Grad Total Model 83.33% % % Model Total Model % % % Model 2 Total Fgure 9. Desred perod-edg vetory levels (safety stock) for dfferet system scearos. Relatve Dfferece Perod-Edg Ivetory VTMR Iformato Utlzato Grad Total Model vs % -00% -00% -69% -8% 90.9% -62% -48% -33% -40% 95.24% -5% -2% -9% -9% Fgure 0. Relatve chage vetory levels depedg o system scearo. Fgure ad Fgure 2 cota the values ad relatve dffereces, respectvely, of the expected costs per perod for Models ad 2. Wth each model, the expected cost creases as demad ucertaty ad supply ucertaty crease ad, betwee models, the expected costs are uformly hgher Model tha Model 2 for a gve level of VTMR ad capacty utlzato. Average of Exp Cost VTMR Iformato Utlzato Grad Total Model 83.33% $ $ $ 4.34 $ % $ $ $ $ % $ $ 77.6 $ $ Model Total $ $ $ 3.7 $ Model % $.03 $ 6.8 $ $ % $ 9.83 $ $ 6.4 $ % $ $ $ $ Model 2 Total $.23 $ 25.5 $ $ 35.7 Grad Total $ $ 48.0 $ 0.06 $ 59.3 Fgure. The expected holdg ad backorder costs per perod. Relatve Dfferece Expected Costs VTMR Iformato Utlzato Grad Total Model vs % -98% -90% -74% 83% 90.9% -78% -67% -50% -60% 95.24% -52% -39% -23% -33% Fgure 2. The value of formato: relatve chage the expected cost per perod betwee Models ad 2. 25

27 Fgure 3 ad Fgure 4 cota the rates of chage expected costs for each model as supply ad demad ucertaty crease, respectvely. Observe that the rate of chage expected cost as both VTMR ad capacty utlzato crease s greater for Model 2 tha for Model. That s, the expected cost per perod s more sestve to VTMR ad capacty utlzato whe operatg the leaer, formato rch evromet. Chage Cost as Utlzato Icreases VTMR Iformato Utlzato Model 83.33% Base Base Base 90.9% % 3% 8% 95.24% 0% 8% 38% Model % Base Base Base 90.9% 859% 234% 09% 95.24% 226% 590% 3% Fgure 3. Rates of chage expected cost as supply ucertaty creases Chage Cost as VTMR Icreases VTMR Iformato Utlzato Model 83.33% Base 50% 6% 90.9% Base 53% 78% 95.24% Base 62% 228% Model % Base 564% 276% 90.9% Base 3% 525% 95.24% Base 06% 428% Fgure 4. Rates of chage expected cost as demad ucertaty creases Ths observato has mportat cosequeces for supply cha desgers attemptg to reduce costs by mprovg the dowstream vsblty of demad formato. Improvg the rchess of the avalable formato decreases costs, but also creases the system s sestvty to capacty utlzato ad aggregate demad ucertaty. Ths s relevat sce the curret stateof-practce producto plag ad cotrol systems s, at worst, to gore the effects of capacty costrats etrely ad, at best, to capture capacty lmts by smple heurstcs such as fxed maufacturg lead tmes. We explore the mpact of such practces the ext subsecto. 6.4 The Impact of Igorg Capacty Costrats Much has bee sad ad wrtte recetly about the mportace of desgg supply chas to provde the kd of formato vsblty preset Model 2. For example, see Muckstadt, et 26

28 al. (200). The results the prevous subsecto cofrm ad quatfy these deas. I ths subsecto, we reforce the mportace of cosderg lmted capacty producto plag decso-makg, ad observe that the advatages of creased formato avalablty ca deterorate rapdly f fte capacty s gored. Fgure 5 shows the expected cost per perod whe gorg capacty costrats. To costruct the Exp Cost Igorg Capacty values, capacty was assumed to be ulmted, thereby gorg ay supply ucertaty. Fgure 6 cotas the relatve dffereces expected costs. VTMR Iformato Utlzato Data Grad Total Model 83.33% Average of Exp Cost $ $ $ 4.34 $ Exp Cost Igorg Capacty $ $ 66.0 $ 5.00 $ % Average of Exp Cost $ $ $ $ Exp Cost Igorg Capacty $ $ $ 3.5 $ % Average of Exp Cost $ $ 77.6 $ $ Exp Cost Igorg Capacty $ $ 88.6 $ $ 7.23 Model Average of Exp Cost $ $ $ 3.7 $ Model Exp Cost Igorg Capacty $ $ $ $ 9.29 Model % Average of Exp Cost $.03 $ 6.8 $ $ 2.39 Exp Cost Igorg Capacty $.03 $ 6.8 $ $ % Average of Exp Cost $ 9.83 $ $ 6.4 $ 3.33 Exp Cost Igorg Capacty $ 2.25 $ $ 38.3 $ % Average of Exp Cost $ $ $ $ Exp Cost Igorg Capacty $ $ 9.90 $ $ Model 2 Average of Exp Cost $.23 $ 25.5 $ $ 35.7 Model 2 Exp Cost Igorg Capacty $ $ $ 77.7 $ Total Average of Exp Cost $ $ 48.0 $ 0.06 $ 59.3 Total Exp Cost Igorg Capacty $ $ $ $ Fgure 5. Expected costs of cosderg fte capacty vs. gorg fte capacty. VTMR Iformato Utlzato Grad Total Model 83.33% Chage Exp Cost 0.002% 0.05% 0.574% 0.324% 90.9% Chage Exp Cost 0.260%.897% 6.235% 3.858% 95.24% Chage Exp Cost 4.599% 4.86% 35.28% % Model Chage Exp Cost.706% 5.846% 6.63% 0.573% Model % Chage Exp Cost 0.000% 0.000% % % 90.9% Chage Exp Cost % % % % 95.24% Chage Exp Cost 7.202% 55.59% % 75.54% Model 2 Chage Exp Cost % 5.666% % % Total Chage Exp Cost 8.528% % % % Fgure 6. Relatve chage the expected cost per perod whe gorg fte capacty. 27