Simulation-based Analysis of Service Levels in Stable Production- Inventory Systems

Transcription

1 Simulatio-based Aalysis of Service Levels i Stable Productio- Ivetory Systems Jayedra Vekateswara, Kaushik Margabadu#, D. Bijulal*, N. Hemachadra, Idustrial Egieerig ad Operatios Research, Idia Istitute of Techology Bombay, Powai, Mumbai, MH , Idia jayedra@iitb.c.i, h@iitb.ac.i #School of Mechaical Egieerig, SASTRA Uiversity, Thajavur, TN 6340, Idia kaushikmargabadu@gmail.com *Departmet of Mechaical Egieerig College of Egieerig, Trivadrum, KL 69506, Idia bijulal@iitb.ac.i, Abstract The performace aalysis of a geeral productio ivetory cotrol system uder ucertai demad is preseted. I the model, the productio order releases are determied based o the iformatio feedback o the forecasted demad, work-i-process discrepacy ad ivetory discrepacy. Stability coditios are obtaied i terms of the cotrol parameters that maage the rate at which the above discrepacies are corrected. The service ad cost performaces of the system i terms of order fill rate, item fill rate ad average system cost are aalyzed for various values of the cotrol parameters withi the stability regio. Additioal safety stock is cosidered to help achieve a desired level of service (desired order fill rate). Results based o umerical simulatios are preseted ad their implicatios are discussed. Keywords: productio orderig; stability; order fill rate; ivetory discrepacy; safety stock.. Itroductio Productio ivetory systems are itegral to ay maufacturig eterprise. A efficiet ad effective cotrol scheme for such systems becomes crucial for idustries to maitai their competitive edge, especially i the face of ucertai market coditios. The foudatios for a cotrol system view of productio ivetory systems were laid dow by Forrester (96) i his book Idustrial Dyamics, which is based o system dyamics methodology. The system dyamics models have sice bee used to capture the productioivetory order behavior at a aggregate level usig iformatio feedback structures, with the model represeted by differetial/differece equatios (Forrester 96, Towill 996, Sterma 000, Vekateswara ad So 007). Past works i literature have maily focused o the stability ad cotrollability of geeric productio ivetory cotrol system models, ad their implicatios (Ortega ad Li 004, Disey ad Towill 00, Riddalls ad Beett 00, Disey et al. 006, Vekateswara ad So 007, Bijulal et al. 0). A popular productio ivetory cotrol model i literature, which is comparable to the Forrester s model, is the Automatic Pipelie ad Variable Ivetory ad Order based Productio Cotrol System (APVIOBPCS), well studied by Disey et al. (006), Dejockheere et al. (003) ad others. APVIOBPCS models a productio system with The work was carried out durig the author s stay at Idia Istitute of Techology Bombay as research iter.

2 pipelie delay, ad cosiders the discrepacies i work-i-progress (WIP) ad i edivetory to determie the productio order releases. These policies have bee defied i cotiuous domai, ad the equivalet discrete-time models come uder the family of orderup-to policies (Axsäter 000). Ivetory cotrol ad productio plaig have also bee well researched over the years from the Operatios Research perspective, where the focus has bee o the developmet of prescriptive models primarily aimed at miimizig ivetory related costs (Hax ad Cadea 984, Axsäter 000). Some key performace measures idetified i literature are the ivetory holdig costs, backlog costs, lost sales costs ad service level measures of order fill rate ad item fill rate. To achieve the desired level of service i face of radom demad, the system typically stocks additioal ivetory, i.e. safety stock, which teds to icrease the system cost. A few works ca be foud o the cost performace of differet types of productio ivetory cotrol systems (Disey ad Grubbström 004, Disey et al. 006, Che ad Disey 007, Caella ad Ciacimio 00). All these works have aalyzed the ivetory ad orderig cost performace of the APVIOBPCS model uder differet demad patters. They have, however, focused their fidig oly alog the Deziel ad Eilo (D-E) settig (Deziel ad Eilo 967). The D-E settig correspods to the sceario where the rate of adjustmet for WIP discrepacy equals the rate of adjustmet of ivetory discrepacy. Bijulal et al. (0) have attempted to establish the variatios i order fill rates obtaied throughout the stability regio of a slightly differet productio ivetory cotrol system. However, the applicability of their model is restricted by the defiitio that a period s demad is cosidered fulfilled oly if there is sufficiet ivetory at the start of the period to satisfy the demad. I this paper some results from ogoig research work o the performace aalysis of a geeral productio ivetory cotrol system model uder ucertai demad is preseted. I the model, the productio order releases are computed based o the forecasted demad, adjustmets for WIP discrepacy ad adjustmets for ivetory discrepacy. The rates of adjustmets of these discrepacies are idetified as the system cotrol parameters. Stability coditios based o these parameters are derived. The service ad cost performaces of the system i terms of order fill rate, item fill rate ad average system cost are aalyzed for various values of the cotrol parameters withi the stability regio. Additioal safety stock is hadled i the system to help achieve higher levels of service. Results based o umerical simulatios are preseted ad their implicatios are discussed.. Productio Ivetory Model. Notatios Used Symbols, otatios ad abbreviatios used i this article are summarized i Table.. Model Descriptio The productio ivetory cotrol structure model discussed i this paper is similar to the classical idustrial dyamics model by Forrester (96), ad those studied by Bijulal et al. (0) ad is also comparable to the APVIOBPCS family of models. The equatios uderlyig the model are as follows: FD FD ρ ( CD FD ) () = INV () = INV PCR CD WIP (3) = WIP PREL PCR

3 PCR = PREL L (4) PREL = FD WIPADJ INVADJ (5) WIPADJ = α DWIP WIP ) = α ( L FD WIP ) (6) ( INVADJ = β ( DINV INV ) (7) Table : Notatios Used Symbol Descriptio Uits α Fractioal rate of adjustmet of WIP discrepacy (rate) /time β Fractioal rate of adjustmet of ivetory discrepacy (rate) /time ρ Smoothig factor (costat) for forecast L Productio delay time period T w Time to adjust WIP discrepacy time period T i Time to adjust ivetory discrepacy time period CD Demad i period uits/ time period DINV Desired Ivetory i period uits DWIP Desired work-i-process i period uits FD Demad forecast for period uits/ time period INV Ivetory at the start of period uits INVADJ Adjustmets for ivetory discrepacy i period uits/ time period PCR Productio completio rate i period uits/ time period PREL Productio order release i period uits/ time period WIP Work-i-process at the start of period uits WIPADJ Adjustmets for WIP discrepacy i period uits/ time period b Cost of backorderig oe uit per period Rupees/uit/time h Cost of o holdig uit of ivetory per period Rupees/uit/time Scalig factor for safety ivetory k DOFR s D Estimated stadard deviatio of demad ASC Average system cost util period Rupees. ABC Average backorder cost util period Rupees. AHC Average ivetory holdig cost util period Rupees. DOFR IFR OF OFR Desired order fill rate Item Fill Rate util period 0- variable. idicates demad is fulfilled i period Order Fill rate util period The forecasted demad (FD ) for period is based o the first order expoetial smoothig of the previous period s demad (CD ), with smoothig costat ρ (Equatio ). The fiished goods ivetory INV at the start of ay period is the previous period s startig ivetory (INV ) plus the differece of the previous period s productio PCR ad demad CD, as show i Equatio (). Negative ivetory represets backordered quatities. Similarly, system WIP at the start of a period is the previous period s startig WIP (WIP ) plus the differece of the previous period s productio orders PREL ad productio, as show i Equatio (3). Equatio (4) represets the productio completio rate PCR i the system as a pipelie material delay process to the productio release, with fixed delay L. The productio order release for period, PREL, is the sum of demad forecast, FD, the adjustmet for WIP discrepacy, ad the adjustmet for ivetory discrepacy, as 3

4 show i Equatio (5). The WIP discrepacy is adjusted by a fractioal rate α (Equatio 6). The ivetory discrepacy is adjusted by a fractioal rate β (Equatio 7). It is oted that feedback gais modeled here as α ad β are modeled as /T w ad /T i, respectively, i the past literature. Typically, the desired WIP ad desired ivetory levels are computed based o the expected system performace i steady or equilibrium state. I steady state, the system iflows balace the outflows such that the stock levels (i.e. system state) remai the same (stable). Thus, i steady state the forecasted (expected) demad (FD ) will equal the mea ed customer demad, ad productio order release (ad productio rate) will ted towards the expected demad. Little s Law states that i steady state the expected WIP i the system is the product of the expected throughput rate ad the lead time (L). Hece the desired WIP is set as the product of forecasted demad ad the productio lead time (Equatio 6). Now, the desired ivetory (DINV ) ca be set to 0, i a bid to operate lea ad miimize ivetory holdig costs. However this settigs (DINV =0) may ot provide adequate ivetory coverage i the face of radom demad. I order to achieve higher ivetory coverage (ad hece higher service level) the desired ivetory i the system has to bee modified by addig a safety stock..3 Calculatio of Safety Stock Suppose that the demad (CD ) i each period is idepedet ad Normally distributed with mea μ D ad stadard deviatio σ D. The eterprise may carry some safety stock to cover the radom variatios i demad. The amout of variatios covered by the safety stock will deped o the desired probability of stock out or the customer service level. Desired order fill rate (DOFR), a service level measure, is defied as the proportio of periods i which demad is fulfilled etirely. Sice there is some lead time ivolved to produce the orders ad repleish the ed ivetory, the DOFR ca also be viewed as the probability of ot havig a stock out situatio durig the repleishmet lead time (stock-out occurs whe demad exceeds available stock). Thus safety stock is the amout of additioal ivetory to be stocked to achieve the desired fill rate, ad is give as SafetyStock= k DOFR s R. I the above equatio, safety factor, k DOFR = Φ (DOFR), where Φ( ) is the stadard ormal cumulative distributio fuctio, ad s R represets the stadard deviatio of demad durig the repleishmet lead time. I the system dyamics model uder study, there is a oe period orderig delay ad L periods of productio delay, resultig i a total of L periods of repleishmet delay. Sice demad i each period is idepedet, the safety stock ca be writte as show i Equatio (8), where s D represets the estimated stadard deviatio of demad. SafetyStock = k s L DOFR D (8).3 Calculatio of Desired Ivetory Based o how the demad is beig fulfilled, differet settigs of desired ivetory (DINV) becomes ecessary: (i) A period s demad is fulfilled from the ivetory available at the start of that period. That is, the productio i a period is ot used to satisfy the demad i that period. I this case, there may ot be sufficiet ivetory to meet the demad i that period, but still there could be a positive ivetory holdig at the start of the ext period whe the latest productio quatity is icluded to ivetory. The desired ivetory required would be the mea demad plus safety stock (a detailed aalysis of this sceario has bee preseted i Bijulal et al. 0). (ii) A period s demad is fulfilled (say, at the ed of the week) from the ivetory available at the start of that period plus the productio i that period. I this case, whe the demad i a period is ot fulfilled, the excess demad is 4

5 backordered (captured as egative ivetory). This latter sceario is modeled ad aalyzed i this paper. Sice the demad is fulfilled (say, at the ed of the week) from the ivetory available at the start of that period plus the productio i that period, desired ivetory (DINV) ca be set to cover oly the variability i the demad, as follows: DINV = SafetyStock = kdofr sd L (9).3 Measurig System Performace Order fill rate (OFR), Item fill rate (IFR) ad average system cost per period (ASC) are the system performace measures selected to aalyze the system behavior uder stable parameter settigs. OFR is the proportio of periods that the demad was satisfied i etirety, ad is modeled usig Equatio (0) ad (). OFR = OF i (0) PCRi INVi CDi OF i = 0 otherwise () IFR is the average fractio of demad uits fulfilled durig ay period of time, ad is modeled usig Equatio (). IFR = The system cost has bee assumed to have two compoets: the holdig costs ad the backorder costs. Every uit stocked i ivetory ad carried over to the ext period icurs cost h per period ad every demad uit backordered icurs cost b per period. The average system cost (ASC) has bee estimated as: ASC = ( h INVi b INVi ) (3) i= where INV i ad INV i represets the excess ivetory ad backorder quatities, respectively. INVi INVi 0 INVi INVi < 0 INV i = 0 otherwise ad INV i = 0 otherwise As metioed earlier, the demad (CD ) i each period is idepedet ad Normally distributed with mea μ D ad stadard deviatio σ D. A stock out is said to occur if the sum of the ivetory at the start of a period ad productio i that period is less tha the demad i that period. Suppose the desired ivetory (DINV) is set as 0, the the demad i the period is almost etirely met by that period s productio. Now, the expected productio rate teds to the mea demad at steady state. Therefore the probability of stock out will be ~0.5. The probability that the order is fully met from stock, the order fill rate (OFR), the becomes Pr{stock out} 0.5. Hece the productio ivetory system preseted i Sectio ca be expected to give OFR of about 50% whe DINV =0. i= i= AHC i= INV CD i i ABC () 5

6 Suppose the desired ivetory is set as per Equatio (9); still, the system may ot achieve the desired order fill rate as the system performace is iflueced by parameters α, β ad ρ. Also, the DINV is a fuctio of s D, the estimated stadard deviatio of demad (see Equatio 9). Hece the accuracy of this estimate will also ifluece the system performace. The productio ivetory model, as described i this sectio, is represeted i Figure as a stock ad flow diagram. Productio Order Release (PREL) Work i Progress (WIP) Productio Completio (PCR) - - L WIPADJ α Ivetroy (INV) - INVADJ DWIP DINV Customer Demad (CD) β Forecasted Demad (FD) Chage i forecast - ρ (a) Sub-model of the productio-ivetory cotrol system <Productio Completio (PCR)> <Ivetroy (INV)> <Customer Demad (CD)> Are orders fulfilled? (OF) - Total OF Order Fill Rate (OFR) - <Time> <Ivetroy (INV)> Cumulative Demad Item Fill Rate (IFR) - INV held per period Total Ivetory Held Backlog per period Total backlog <Time> Cum. Holdig Cost - Backlog cost per Holdig Cost per Average Holdig period period Cost (AHC) Cum. Backlog Cost <Time> - Average Backlog Cost (ABC) Holdig Cost per uit Backorder Cost per uit Average System Cost (ASC) (b) Sub-model of performace measuremet Figure : Stock Flow Diagram of the Productio-Ivetory Cotrol System 6

7 3. Regio of Stability of the Productio-Ivetory System The productio ivetory system is represeted by equatios () (3). The coditios for stability are obtaied from the trasfer fuctio derived from the z-trasforms of Equatios () (7), (9). Equatios (0) (3) models the performace measures that do ot ifluece the model dyamics, ad hece are ot required for stability aalysis. To derive the system trasfer fuctio, the differece equatios i time preseted i Sectio are trasformed oto the z-domai, preseted i Equatio (4) (0). Equatio (0) is the trasformed equatio after combiig Equatios (7) ad (9). ρ FD( = CD( (4) z ρ INV ( = ( PCR( CD( ) (5) z WIP( = ( PREL( PCR( ) (6) z L PCR ( = PREL( / z (7) PREL ( = FD( WIPADJ ( INVADJ ( (8) ( L FD( WIP( )) WIPADJ ( = α z (9) INVADJ ( = β ( k s L INV ( ) (0) DOFR The above simultaeous equatios i z are solved to obtai the system trasfer fuctio betwee output variable PREL ad iput variable CD, as show i Equatio (). L PREL( z ( z ){( z )( Lα ) ρ ( k ( ))( ρ ) β} = DOFRsD L z z CD( L L () ( z ρ )( z ){ z z ( α ) α β} Equatio (0) shows the geeral expressio of the system with a fixed productio delay L, the smoothig factor of demad forecast ρ, fractioal adjustmet rates α ad β, the scalig factor of safety stock (k DOFR ) ad the estimated stadard deviatio of demad s D. It is observed that the deomiator polyomial is idepedet of k DOFR ad s D, which implies that these parameters (ad hece desired ivetory value) do ot affect the stability of the system. However, k DOFR ad s D appear i the umerator polyomial ad hece ca be expected to affect the dyamics of the system respose. To determie the stability bouds, the deomiator polyomial eeds to be solved. Sice it is ot possible to solve for geeral L (due to presece of z L ), a fixed pipelie delay of L = 3 periods has bee assumed. All further discussios i this article pertai to a model with pipelie delay of 3 periods. Equatio () represets the system trasfer fuctio with L = 3. 3 PREL( z ( z ){( z )( 3α ) ρ ( k ( ))( ρ ) β} = DOFRsD z z CD( 4 3 () ( z ρ )( z ){ z z ( α ) α β} The trasfer fuctio system preseted i Equatio () is stable, i the boudediput bouded-output (BIBO) sese, if the roots of the deomiator polyomial are iside the uit circle i the complex plae (Vekateswara ad So 007). The coditios for stability i terms of the parameters α, β ad ρ have bee obtaied usig Jury s test (964). Equatios (3) (6) show the bidig coditios for stability. Equatio (3) represets the expressio for the right boudary, Equatio (4) represets the expressio for the lower boudary D 7

8 ad Equatio (5) represets the upper boudary. The regio of stability defied by these equatios i the (α, β) plae is show i Figure. It is further assumed that (α, β ad ρ) are o-egative. 3β 4β ( β ) 4 β α (3) ( 3 β ) 3β 4β ( β ) 4 β α ( 3 β ) (4) α β (5) ρ (6) The system guaratees to be stable whe the values of α ad β are iside the stable regio. Parameter selectio o the boudary makes the system variables to cotiue sustaied oscillatios; while parameter selectio outside the boudary causes the system variables to cotiue oscillatios with expoetially icreasig amplitude. Prelimiary simulatio rus with i.i.d. Normal demad revealed that time varyig stochastic iput to the system does ot affect the system s BIBO stability. Also, it has bee foud that the system performace measures OFR, IFR ad ASC deteriorates for ustable parameter selectios. Hece, the ivestigatios of system performaces preseted i this article are limited withi the stable regio of the parameter settig. Figure : Stability boudary of the productio-ivetory system with L = 3 (Source: Bijulal et al. 0) 4. Experimetal Settigs System performace measures OFR, IFR ad ASC (Equatio (0) (3)) for various settigs of the parameters α, β ad ρ ad the methods of computig s D have bee aalyzed. I order to uderstad the impact of s D o the system performace, three differet settigs are take. I the first case, the estimated stadard deviatio of demad equals the actual stadard deviatio of demad (s D = σ D ). This captures the sceario where the demad variace is kow ad hece used i decisio makig. I the secod case, s D = ν FD where the coefficiet of demad variatio ν = μ D /σ D. This captures the sceario where ν of demad process is kow ad the demad forecast FD is a true estimate of the mea demad 8

9 (Makridakis et al. 005, Axsäter 000). I the third case, s D is computed dyamically as the sample stadard deviatio of the observed demad process, i.e. s D = ( CD i FD i ). i= This captures the sceario where the demad process is ukow. To observe the effect of smoothig factor, ρ, o the system performace, three settigs of ρ, have bee selected as 0, 0. ad. Whe ρ = 0, the forecasted demad remais uchaged from its iitial value throughout the time horizo. A total of about 335 (α, β) pairs spread throughout the stable regio (see Figure ) have bee take for the simulatio study. The poits are listed i Table A i the Appedix. Now, it ca be expected that the settig ρ =0, α =, β = achieve the desired OFR sice at this settig the WIP ad ivetory discrepacies are always fully accouted for ad o oise due to forecastig is part of the orderig process. 4. Geeral Simulatio Settigs The productio lead time L has bee fixed as 3 time periods. The values of backorder cost b ad holdig cost h are both take as Rs. per item per period. DOFR has bee selected as 80% which results i k DOFR = Φ (DOFR) = The iitial values of the stocks, FD 0, WIP 0 ad INV 0 are set equal to the mea demad, desired WIP ad desired ivetory values respectively. The demad i each period has bee assumed to be Normal(000, 0). It has bee esured that the same radom demad patter is used across multiple simulatio rus, which allows for a valid compariso of the results. The simulatio model as described by the Equatios () (3) is modeled ad aalyzed usig Powersim.5. The simulatio ru legth is kept as 3650 time periods, with update iterval as time period. As it was observed that the system reached steady state i the early part of the ru, the replicatio legth of 3650 has bee foud to be adequate for this study. A roud-off error of up to 0.0 was assumed to be acceptable i evaluatig the coditioal statemet i Equatio (). 5. Results ad Observatios A prelimiary set of experimets were coducted with DINV = 0 (o safety stock) to determie the impact of radom demad o system performace. All other system parameters are as described i Sectio 4. The results are summarized i Table. For both settigs ρ=0. ad ρ=, the best performace (max OFR, max IFR ad mi ASC) was obtaied at (α, β) = (0, 0). Apart from this particular poit, the OFR respose of all other poits was quite close to 0.5, validatig our earlier claim. The high values of IFR (99%) are attributed to the low stadard deviatio of demad as compared to the mea demad. Also, it is see that with ρ=, the service levels have improved, but the ASC has worseed. Table : System performace with o safety stock ρ=0. ρ= Max mi mea media max mi mea Media OFR IFR ASC Next, experimets have bee coducted by cosiderig safety stock, i.e., DINV is set as per Equatio (9), with desired order fill rate (DOFR) as 80%. The variatio of the performace parameters, i respose to the differet settigs of s D, ad ρ over the (α, β) plae is aalyzed. A total of =305 experimets have bee coducted. 9

10 Recall that the estimated stadard deviatio of demad, s D = σ D = 0 i the first case, for all settigs of ρ. I the secod case, s D = ν FD = 0.0FD = 0 whe ρ =0; the s D varied betwee 9.95 ad 0.05 whe ρ =0.; ad the s D varied betwee 9.8 ad 0. whe ρ =. I the third case, s D is the sample stadard deviatio, which stabilized at: 0.03 whe ρ =0; 0.55 whe ρ =0.; ad 4.00 whe ρ =. Based o the results, it is observed that the order fill rates obtaied are higher ad closer to the DOFR whe safety stock is cosidered, as expected. For (α, β) poits o the stability boudary, the OFR obtaied were betwee across all settigs of s D, ad ρ. This is agai as expected sice these critically stable poits will cause sustaied oscillatio, with costat amplitude, of the system output PREL OFR α=β ρ =0. ρ =..4.6 ρ = OFR ρ =0. ρ = α=β..4.6 ρ =0.8 (a) OFR for s D = 0 (b) OFR for s D = ν.fd OFR ρ =0. ρ = ρ = α=β (c) OFR for s D = sample stadard deviatio of demad process Figure 3: OFR obtaied uder differet settig s D ad ρ across alog D-E lie Figure 3(a) (c) illustrates the achieved OFR for differet combiatios of s D ad ρ across various α=β settigs. The α=β settig idicates equal weightage to both WIP ad ivetory discrepacy i the orderig scheme, ad is commoly referred to as the Deziel- Eilo (D-E) lie (Deziel ad Eilo 967). The (α, β) poits used to geerate the above curves are give i Table A i the Appedix. It ca be see from Figure that (α, β) values greater tha alog the D-E lie will result i ustable system respose. Some commo patters of behavior are observed for all three settigs of s D (see Figure 3). Whe ρ =0, three distict regios of OFR ca be observed as oe moves alog the D-E lie: (i) Iitially, there is a steep icrease i the OFR as the (α, β) value is icreased from 0 alog the D-E lie, with the OFR reachig about 0.70 at value 0. o the D-E lie; (ii) The OFR the stabilizes aroud 0.78 for (α, β) values betwee 0.7 to.6 o the D-E lie; (iii) Fially, a steep drop i the OFR is observed for (α, β) values beyod.8 o the D-E lie, 0

11 with the OFR reachig about 0.70 at value.0 o the D-E lie. Iterestigly, whe ρ =0., low settigs o the D-E lie (< 0.6) results i high OFR of about 0.77; mid settigs o D-E lie (. to.8) achieves a lower OFR of about 0.7; ad high settigs o the D-E lie (>.9) results i a steep drop i OFR. Now, whe ρ =, it ca be see that OFR steadily deteriorates from a high OFR of 0.8 (at α=β =0.05) to a very low OFR of 0.5 (at α=β =.0) as oe moves upwards alog the D-E lie. It is see that whe s D is computed dyamically as the sample stadard deviatio of the observed demad process, a OFR of about 0.85 is achieved at α=β =0.05 settig. This apparet aomaly could be attributed to the higher value of s D, ad hece DINV, i these settigs. I geeral, there seem to be some critical poit alog the D-E lie, whe ρ > 0, beyod which the OFR measures deteriorates rapidly. Also, it is oted that the settig α = β = ad ρ =0 achieves high OFR, close to the desired order fill rate, for all s D settigs. The empirical cotour graphs for OFR obtaied from the simulatio results are preseted i Figures 4(a) to 4(i). Overall, it is see that amog the OFR cotours, the 0.75 cotour (the cotour markig the regio with high OFR) decreases i area as ρ icreases from 0 to, for all settigs of s D. Also, there is a marked shift i the regio withi the (α, β) plae where high OFR is obtaied for differet values of ρ. Figures 4(b), 4(e) ad 4(h) clearly show that OFR improves above the α=β lie, i.e. whe α > β while it reduces below it, i.e. whe α < β. Also, better OFR performace is observed for α ad β regio across ay settig of s D ad ρ. The results thus idicate that there exists a well defied operatig regio withi the stability regio which ca help achieve better system performace, i.e., high OFR, uder statioary radom demad. It is further observed that the sceario α=β= (represetig the case whe the discrepacy i WIP ad discrepacy i ivetory are completely accouted for i every order) does ot result i the highest OFR values, except whe ρ = 0. Eve the, it is ot the uique best poit. Figures 4(a) to 4(f) shows that the DOFR of 80% is achieved either with s D = σ, or with s D = ν FD though the achieved OFR comes close (~78%) for some parameter settigs. However for the case of s D = sample stadard deviatio a OFR greater tha 80% is achieved for ρ > 0. (see Figures 4(h) ad 4(i)). This may be attributed to the higher safety stock carried i the system uder this s D settig. The average system costs (ASC) appears to be the lowest at the regios of maximal OFR, for a give settig of s D, ad ρ. The media value of ASC was foud to be about Rs.47 whe ρ =0, about Rs.53 whe ρ =0., ad about Rs.89 whe ρ =, across all settigs of s D. Also, for similar values of OFR, the ASC icreases as ρ goes from 0 to, for a give s D. The domiat costs for ASC were foud to be the holdig costs. The holdig costs were geerally higher whe s D was set as the sample stadard deviatio of the demad, sice i this settig, the DINV levels were higher. The itersectio regios of high OFR ad low ASC are illustrated i Figures 5(a) to 5(i). I the plots, the thick black lies deote the ASC cotours ad the thick grey lies deote the OFR cotours. The desired order fill rate (DOFR) is take as 80%. Now, as with OFR cotours, it is observed that the ASC cotours also decreases i area as ρ icreases from 0 to, for all settigs of s D. Also, it is see that the sceario α=β= does ot result i the lowest ASC values, except whe ρ = 0. A iterestig observatio is that i geeral the regio of low ASC is much larger tha ad completely ecompasses the regio of high OFR for all settigs of s D ad ρ. That is, for the same system cost ASC values, the OFR service levels of 75% or 78% ca be obtaied. This implies that the system parameters ca be fie-tued to obtai higher service levels without icreasig system costs.

12 (a) s D = 0, ρ =0 (b) s D = 0, ρ =0. (c) s D = 0, ρ = (d) s D = ν.fd, ρ =0 (e) s D = ν.fd, ρ =0. (f) s D = ν.fd, ρ = (g) s D = sample std.dev, ρ =0 (h) s D = sample std.dev, ρ =0. (i) s D = sample std.dev, ρ = Figure 4: OFR cotours obtaied uder differet settig s D ad ρ for stable (α, β) poits

13 (a) s D = 0, ρ =0 (b) s D = 0, ρ =0. (c) s D = 0, ρ = (d) s D = ν.fd, ρ =0 (e) s D = ν.fd, ρ =0. (f) s D = ν.fd, ρ = (g) s D = sample std.dev, ρ =0 (h) s D = sample std.dev, ρ =0. (i) s D = sample std.dev, ρ = Figure 5: Superimposed OFR ad ASC cotours obtaied uder differet settig s D ad ρ for stable (α, β) poits 3

14 6. Discussios ad Future Work The productio-ivetory cotrol system based o the classical idustrial dyamics model has bee modeled ad aalyzed for its service level ad cost performace. The cotrol parameters for the system are: α, the fractioal rate of adjustmet for WIP, β, the fractioal rate of adjustmet for ivetory, ρ, the smoothig costat used i forecastig, ad s D, the stadard deviatio of demad used to determie the safety stock levels. Experimets are coducted with various cotrol parameter settigs i order to quatify the system performaces, ad the results are preseted i Sectio 5. The results obtaied do offer some isights, which could be beeficial to eterprise maagemet. Firstly, holdig of additioal safety stock is essetial for achievig higher service levels. Uder the base case of ot havig ay safety stock, the system was able to achieve a order fill rate of about 50% oly. Secodly, settig of cotrol parameters iside the stability regio is required, eve with safety stock, to achieve higher order fill rates. (α, β) poits o the stability boudary result i OFR of about 50% oly, while OFR rapidly deteriorates for poits outside the stability regio. Ufortuately, the DOFR of 80% is ever achieved uder most cofiguratios though the OFR does come close (~78%) for some parameter settigs. Thirdly, it is observed that the cotour markig the regio with high OFR decreases i area as ρ icreases from 0 to, for all settigs of s D. Fourthly, the results seem to idicate that uder statioary demad coditios, the WIP discrepacies are to be adjusted at a higher rate tha ivetory discrepacies, that is, α β is desired. These make ituitive sese sice WIP is purely iteral to the productio system while the ivetory discrepacies are triggered by exteral (ukow) demad. Uder all combiatios of s D, ad ρ, the results idicate that the regio eclosed by α, β ad α β has comparatively better performaces tha other regios. Hece it is desirable for the productio system to adjust WIP discrepacies at the same or higher rate tha ivetory discrepacies. Fifthly, completely accoutig for the discrepacies, i.e. α =β =, may ot always be the best optio. α =β = meas that the system tries to adjust the discrepacies fully i each period which may ot be required uder statioary demad coditios. Sixthly, the system parameters ca be fie-tued to obtai higher service levels without icreasig the system costs. That is, for the same system cost ASC values, higher OFR service levels ca be obtaied by fie tuig the cotrol parameters. As future work, the trasfer fuctio i z-domai is to be explored to seek aalytical isights ito the amplitudes ad settlig times of the system respose. Further, appropriate procedures to determie the optimal safety stock required to achieve the desired order fill rate based o the cotrol parameter settigs ca be ivestigated. Refereces Axsäter, S Ivetory cotrol. Kluwer Academic Publishers: Dordrecht. Bijulal, D., Vekateswara, J. ad Hemachadra, N. 0. Service levels, system cost ad stability of productio-ivetory cotrol systems. Iteratioal Joural of Productio Research.. doi: 0.080/ Caella, S. ad Ciacimio, E., 00. O the bullwhip avoidace phase: supply chai collaboratio ad order smoothig. Iteratioal Joural of Productio Research, 48 (), Che, Y.F. ad Disey, S.M., 007. The myopic Order-Up-To policy with a proportioal feedback cotroller. Iteratioal Joural of Productio Research, 45 (), Dejockheere, J., Disey, S.M., Lambrecht, M.R. ad Towill, D.R., 003. Measurig ad avoidig the bullwhip effect: A cotrol theoretic approach. Europea Joural of Operatioal Research, 47 (3),

15 Deziel, D. ad Eilo, S., 967. A liear productio ivetory cotrol rule. The Productio Egieer, 43, Disey, S.M., Farasy, I., Lambrecht, M., Towill, D. ad Va de Velde, W., 006. Tamig the bull-whip effect whilst watchig customer service i a sigle supply chai echelo. Europea Joural of Operatioal Research, 73 (), 5 7. Disey, S.M. ad Grubbström, R.W., 004. Ecoomic cosequeces of a productio ad ivetory cotrol policy. Iteratioal Joural of Productio Research, 4 (7), Disey, S.M. ad Towill, D.R. 00. A discrete trasfer fuctio model to determie the dyamic stability of a vedor maaged ivetory supply chai. Iteratioal Joural of Productio Research. 40 (), Forrester, J.W. 96. Idustrial dyamics. MIT Press: Cambridge, MA. Hax, A.C. ad Cadea, D. 984, Productio ad Ivetory Maagemet. Pretice Hall: Eglewood cliffs, NJ. Jury, E.I., 964. Theory ad applicatio of the z-trasform method. Robert E. Krieger: NY. Makridakis, S., Wheelwright, S.C., ad Hydma, R.J Forecastig: methods ad Applicatios. 3 rd ed. Joh Wiley & Sos (ASIA) Pvt Ltd: Sigapore. Ortega, M. ad Li, L Cotrol theory applicatios to the productio-ivetory problem: a review. Iteratioal Joural of Productio Research. 4 (), Riddalls, C.E. ad Beett, S., 00. The stability of supply chais. Iteratioal Joural of Productio Research. 40 (), Sterma, J.D Busiess Dyamics: Systems Thikig ad Modelig for a Complex World. McGraw-Hill: Bosto, MA. Vekateswara, J. ad So, Y.J Effect of iformatio update frequecy o the stability of productio-ivetory cotrol systems. Iteratioal Joural of Productio Ecoomics 06 (), Towill, D.R Idustrial dyamics modelig of supply chais. Iteratioal Joural of Physical Distributio & Logistics Maagemet. 6 (),

16 Appedix Table A: (α, β) poits used i the simulatio rus No. α β No. α β No. α β No. α β No. α β

17 No. α β No. α β No. α β No. α β No. α β Table A: Poits used alog the D-E lie, i.e. (α = β) S.No α = β S.No α = β S.No α = β