Stability analysis of constrained inventory systems with transportation delay

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1 Sabiliy analysis of consrained invenory sysems wih ransoraion delay Xun Wang a,*, Sehen M. Disney b, Jing Wang a a School of Economics and Managemen, BeiHang Universiy, Beijing, 009, China b Logisic Sysems Dynamics Grou, Cardiff Business School, Cardiff Universiy, Aberconway Building, Colum Drive, Cardiff, CF0 3EU,UK Absrac Sabiliy is a fundamenal design roery of invenory sysems. However, he ofen exloied lineariy assumions in he curren lieraure creae a major ga beween heory and racice. In his aer he sabiliy of a consrained roducion and invenory sysem wih a Forbidden Reurns consrain (ha is, a non-negaive order rae) is sudied via a iecewise linear model, an eigenvalue analysis and a simulaion invesigaion. The APVIOBPCS (Auomaic Pieline, Variable Invenory and Order Based Producion Conrol Sysem) and EPVIOBPCS (Esimaed Pieline, Variable Invenory and Order Based Producion Conrol Sysem) relenishmen olicies are adoed. Surrisingly, all kinds of non-linear dynamical behaviours of sysems can be observed in hese simle models. Exac exressions of he asymoic sabiliy boundaries and Lyaunovian sabiliy boundaries are derived when acual and erceived ransoraion lead-ime is and eriods long resecively. Asymoically sable regions in he non-linear Forbidden Reurn sysems are idenical o he sable regions in is unconsrained counerar. However, regions of bounded flucuaions ha coninue forever, including boh eriodiciy and chaos, exis in he aramerical lane ouside he asymoically sable region. Simulaion shows a comlex and delicae srucure in hese regions. The resuls sugges ha accurae lead-ime informaion is essenial o eliminae invenory drif and insabiliy and ha ordering olicies have o be designed roerly in accordance wih he acual lead-ime o avoid hese flucuaions and divergence. Keywords: Invenory; logisics; sysem dynamics; comlexiy heory. The inroducion and moivaion One of he main objecives when designing an invenory sysem is o mainain is sabiliy and robusness in he face of exerior disurbances. Since he inroducion of conrol heory and sysem dynamics aroaches o he field of invenory conrol (Simon, 95; Forreser, 96), many works have sudied his roblem. However, he significance of resuls obained are frequenly limied by boh he uncerainy and comlexiy of he sysem srucure. Ofen omied facors in invenory sysem design include sauraion (logisics consrains) and mis-secified delays (lead-imes). In revious suly chain sabiliy sudies (Riddalls and Benne, 00; Nagaani and Helbing, 004; Warburon e al., 004; Disney, 008), linear invenory sysem models are usually adoed. Lineariy assumions include infinie caaciy, ignoring invenory limiaions and reurn resricions. This has grealy limied he alicabiliy of ublished resuls, and has failed o exlain many business henomena (Riddalls e al., 000). For insance, o mainain lineariy of he commonly invesigaed IOBPCS (Invenory and Order Based Producion Conrol Sysem) models, order raes are ermied o ake negaive values. This means ha all aricians in a suly chain are allowed o reurn excess roduc freely. Secifically, a negaive order rae value leads o a decrease in he invenory level a he consuming echelon and an immediae increase in he invenory level a he sulying echelon. Pracically his may mean ha he excess invenory is no moved from one locaion o anoher bu insead will be considered o be in he ossession of he usream member unil being used as ar of a fuure relenishmen (Hosoda and Disney, 009). This assumion may be difficul o realize in realiy. Non-linear effecs also lay an imoran role in invenory sysems, someimes even a dominan role (Nagaani and Helbing, 004). When lineariy assumions are removed comlex dynamic behaviours are resen. They may even be chaoic or suer-chaoic. Mosekilde and Larsen (988) found ha he oeraing cos of a consrained sysem could be 500 imes higher han is linear counerar. Larsen e al. (999) furher demonsraed his fac. Through heir simulaion exerimens, Thomson e al. (99) concluded ha economic and business sysems scarcely oerae nearby heir seady sae. More recenly, Wang e al. (005) used he Lyaunov Exonen o idenify wheher an invenory sysem is in a chaoic sae. Wu and Zhang (007) found ha he aracors of an invenory sysem model move wih he assumed iniial saes, rendering i imossible o rovide guidelines o avoid chaos by bifurcaion analysis. Hwarng and Xie (008) invesigaed several sysem facors ha affec chaoic behaviour and discovered a chaosamlificaion henomenon beween suly chain echelons. Liu (005) and Rodrigues and Boukas (006) analyzed he sabiliy of suly chain invenory sysems wih iecewise linear echniques. Laugessen and Mosekilde (006) and Mosekilde and Laugessen (007) sudied he border-collision behaviour in iecewise linear suly chain sysems. Mahemaical roeries of such sysems, such as local and global sabiliy condiions and bifurcaions, are very hard * Corresonding auhor. Tel.: addresses: aul.wong.buaa@gmail.com (X. Wang), disneysm@cardiff.ac.uk (S.M. Disney), ekwjing@ublic.ba.ne.cn (J. Wang)

2 Fig. : Lead-ime erformance in an indusrial seing, Coleman (994). o invesigae and nooriously challenging (Sun, 00). In an ideal siuaion, roducion lead-ime would no vary and suly chain aricians would have erfec knowledge of lead-imes (Cheema, 994). However we should no necessarily assume his is so. Time varying and miserceived lead-imes are commonly observed in indusry (Fig., adoed from Coleman, 994). In aricular, varying lead-ime increases suly chain cos (Chaharsooghi and Heydari, 00), exacerbaes he bullwhi effec (Kim e al., 006) and significanly affecs olicy making (Rossi e al., 00). Miserceived lead-ime creaes a so-called henomenon of invenory drif and aenuaes he efficacy of meeing safey sock requiremens (Disney and Towill, 005). From his ersecive, he main concern of his aer is o invesigae he effec of lead-ime on consrained suly chain sabiliy, or, more secifically, a suly chain relenishmen olicy wih Forbidden Reurns. This aer is organised as follows: Secion models he consrained one echelon suly chain sysem iecewise-linearly and conducs an eigenvalue analysis; Secion 3 invesigaes he sabiliy of he invenory sysem wih erfec lead-ime informaion under uni lead-ime; Secion 4 focuses on he effec of lead-ime change and lead-ime miserceion on sysem sabiliy; Secion 5 concludes.. Model of consrained suly chain sysems This aer ados he APVIOBPCS ordering olicy. A brief review of his olicy will now be rovided. Towill (98) resened a model named Invenory and Order Based Producion Conrol Sysem (IOBPCS) in he form of a coninuous ime, Lalace Transform block diagram. Edghill and Towill (989) inroduced a variable invenory arge o he basic IOBPCS model, creaing he VIOBPCS (Variable Invenory and Order Based Producion Conrol Sysem) model. John e al. (994) roosed an Auomaic Pieline, Invenory and Order Based Producion Conrol Sysem, or APIOBPCS by adding work-in-rocess (ieline invenory or suly line) informaion feedback ino he roducion arge decision rocess. This model has been frequenly adoed and researched for is many disinguished advanages (Disney and Towill, 003). Furher, by combining VIOBPCS and APIOBPCS, a more general model, Auomaic Pieline, Variable Invenory and Order Based Producion Conrol Sysem (APVIOBPCS), will be derived. The full IOBPCS family is comrehensively reviewed by Sarimveis e al. (008). We sudy he discree ime APVIOPBCS relenishmen olicy. This olicy has been frequenly adoed and researched as i is of a very general naure. The oular order-u-o (OUT) olicy is a secial case of his APVIOBPCS model, Dejonckheere e al. (003). We refer ineresed readers o Disney e al. (003) for more informaion on he APVIOBPCS model, bu we also give a definiion here. When T =, he difference equaions for he APVIOBPCS invenory sysem are given by () o (6) Ta Forecasing AVCON CONS AVCON, () T T a a

3 Invenory AINV AINV COMRATE CONS, () Work-in-rocess WIP WIP ORATE COMRATE, (3) Transoraion delay TRANS ORATE, (4) Comleions COMRATE TRANS, (5) ORATE Ordering decision AVCON S AVCON AINVAVCON WIP Invenory Discreancy ( ) AVCON AINV WIP S S WIP Discreancy. (6) Equaion () deails AVCON, an esimae of he AVerage CONsumion, used a forecas of fuure demand. The subscri is used o index ime. This forecas can be generaed using any forecasing mehod, bu here we use he exonenial smoohing forecas mehod. T is he average age of he daa in he exonenial smoohing forecas and is a linked o he so-called smoohing consan,, ofen used in he lieraure via T. () gives he invenory a balance equaion, where he new realisaion of he Acual INVenory, AINV is he sum of he revious acual invenory level, AINV lus COMRATE, he COMleions from he roducion faciliies or (deliveries from he suliers), minus he curren demand or CONSumion, CONS. (3) is he equivalen Work In Progress (WIP) balance equaion, where ORATE is he Order RATE a ime. TRANS in (4) is an assisan variable used o describe he ransoraion delay in marix form. In (5) COMRATE is he COMleion RATE. (6) describes he roducion / disribuion / relenishmen Order RATE decision. I is made u of a single feed-forward loo based on he forecas, AVCON and wo feedback loos, AINV and WIP. We also have wo addiional conrollable feedback arameers,,, ha are used o regulae how he feedback on he invenory levels, AVCON AINV, and S S he WIP levels AVCON WIP is incororaed ino he roducion ordering decision. Forbidden Reurns (nonnegaive orders) are enforced wih he maximum oeraor, [x] + = max(0,x), in (6). Since here is no non-negaive consrain on acual invenory, he following underlying assumions of his model are necessary: ousourcing is available (downsream demand can sill be fulfilled even if sulier s invenory is insufficien); and shorage backorder is allowed (negaive invenory can be accumulaed ino he nex eriod). The above difference equaions can be used o develo a dynamic simulaion of he olicy in sofware such as Excel, or more efficienly MATLAB, where evoluion of each variable can be calculaed ieraively. They are also easily convered ino marices ha describe he sysem of equaions: Ta Ta Ta( S ) S S A Ta , b Ta S S Ta, 0 0 A Ta Ta , b T a 0 0 0, x [ AVCON ORATE AINV WIP TRANS] T. 3

4 The iecewise affine model for his sysem is given by x Ax bcons, x Ax bcons, x S S (7) where S { x ORATE 0} and S { x ORATE 0} are boh non-degenerae olyhedral ariions of he sae sace. Tha is, each region S i is a (convex) olyhedron wih a non-emy inerior. The (n ) dimensional hyer-lane n ORATE = 0 is he boundary of he ariions. SS, S S S S. S is he inerior of S and n is he dimension of x. I should be noed ha he boundaries are coninuous, i.e., Ax Ax when x S S. Common conces, such as region and boundary, will be used in eiher he hase sace or he aramerical sace. This model can be furher simlified. When T =, he work-in-rocess is a flow rae raher han a sock level, i.e., TRANS = WIP. Moreover, since he forecass are solely a feed-forward loo in he sysem, hey do no affec sysem sabiliy. This allows us o decrease he dimension of he invenory sysem model. If we le T a = 0, indicaing ha he demand forecas is, a all imes, equal o he las observed demand we can exress he invenory sysem in hree dimensions wih he following marices: S S A 0 0 0, b S 0, A , b 0 ORATE, x AINV. 0 WIP Furher noicing ha A and A are boh linear deenden, he dimension of he sysem can be furher reduced. Denoing he sum of on-hand invenory and work-in-rocess as he invenory osiion, ha is IP = AINV + WIP, we have A S, b S, A 0 0, b 0, ORATE x IP (8) In he following secions, we will show ha sabiliy and eriodiciy of he invenory sysem described by Eq. (~6) are deermined by he wo-dimensional dynamical sysem exressed in (7) and (8), and more secifically, by he eigenvalues for A, A and A A. These eigenvalues are: λ A (9) λ 0, (0), ( ) 4 S ( ) 4 S A λ AA (0, ) () S Noice A has wo eigenvalues and A A has one eigenvalue associaed wih he feedback loos which need o be invesigaed. The eigenvalues of hese marices yield regions in he aramerical lane {, S } in which he invenory sysem behaves differenly and hese will now be derived and invesigaed. 3. Sabiliy of he consrained APVIOBPCS model The sabiliy of linear sysems is much easier o invesigae comared o non-linear sysems. There are only wo aerns of dynamic behaviours which are hysically ossible from a sabiliy ersecive. The sysem could be sable, which means he rajecory will evenually reurn o an equilibrium oin (node), no maer where i is sared (insensiiviy o iniial value). I could also be unsable, which means he rajecory will escae o infiniy. There is also a hird aern, aearing when he sysem is on he very edge of sabiliy boundary called criical sabiliy. The sysem will oscillae a a regular inerval. However, since he boundary has a measure of zero in he aramerical sace, criical sabiliy is only available mahemaically, and is no resen in hysical sysems. Wheher i exiss in a suly chain sysem ha is driven by a comuer algorihm (i.e. mahemaically) is a maer for debae. 4

5 In non-linear sysems however, he range of dynamic behaviours he sysem could exhibi is much larger. Firs, here could be a saddle oin. A saddle oin aracs rajecories from some direcions and reels rajecories from oher direcions. Second, here are oher yes of aracors such as limi cycles and srange aracors ha may be resen. Trajecories of non-linear sysems could be eiher convergen or divergen and can even oscillae in a bounded fashion. I could oscillae in a regular reeaing aern or in a seemingly random one. The dynamic behaviour of he sysem could also be highly sensiive o iniial values. Dynamic behaviours of he invenory levels in non-linear sysems are caegorized in Fig.. Here we have also ranked he differen dynamic behaviours from an inuiive suly chain cos viewoin. Deailed invesigaions on he sabiliy of he consrained suly chain will now be conduced. Fig. 3 shows yical bounded dynamic aerns in he ime domain. Fig. : Caegorizaion of dynamic behaviours of a non-linear sysem. Fig. 3: Four yical dynamic order rae aerns (for he se resonse) generaed by he consrained invenory sysem. (a) asymoic sable; (b) eriodic; (c) quasi-eriodic; (d) chaoic 3. Lyaunovian sabiliy and divergence If all soluions of a dynamical sysem ha sar ou near an equilibrium oin x e, say near x e forever, hen he sysem is Lyaunovian sable. Noe ha, even when he sysem flucuaes bu never aroaches he equilibrium, as long as he flucuaion is bounded, he sysem is Lyaunovian sable. Lyaunovian insabiliy indicaes an unbounded oscillaion and a rajecory ha ends o infiniy. Tha is, i diverges. There are wo facors in he invenory sysem wih non-negaive order raes ha cause divergence. One is he marix coefficien (A ) which creaes an exonenial (mulilicaive) monoonic divergence. The oher is he affine erm (b ) which creaes a linear (addiive) divergence. Someimes hese wo facors may combine and lead o an exonenial and 5

6 oscillaing divergence (he characerisaion of which is beyond he scoe of his aer). The crieria for exonenial monoonic divergence is im( λ A ) 0 and λ A, where im(z) is he imaginary ar of he comlex number z. These relaions give he Lyaunovian sabiliy regions. In linear sysems, if he eigenvalues of sysem marices are comlex and he absolue value of eigenvalues are bigger han, he sysem will oscillae wih exonenial divergence. However, in iecewise linear Forbidden Reurns sysems, such rajecories will evenually hi he boundary. In oher words, he boundary consrains or limis such rajecories from divergence. Thus he exonenial divergence is always monoonic and away from he boundary. We can derive he aramerical boundaries ha searae exonenial divergence from bounded resonses. For T =, when A has wo real eigenvalues greaer han, exonenial divergence can be observed. Thus, he Lyaunovian sabiliy boundary for T = is given by S = ( + ) / 4. () 3. Asymoic sabiliy If x e is Lyaunovian sable and all soluions ha sar near x e converge o x e, hen more srongly, x e is asymoically sable. This means ha he rajecory aroaches an equilibrium oin over ime (Fig. 3a). This conce has similar meaning wih sabiliy in a classical linear conrol heory sense. In he asymoically sable region, he sysem will evenually reurn o equilibrium. The condiion for asymoic sabiliy is abs( λ ). When T = his amouns o and A 0 < S <, 0 < < (3) + < S <, < < 3. (4) 3.3 Periodiciy Periodiciy of a sysem is a oin which he sysem reurns o afer a cerain number of funcion ieraions or a cerain amoun of ime (Fig. 3b). Periodic behavior is defined as recurring a regular inervals, such as every 4 hours, or every 4 weeks. In his iecewise linear discree sysem, boundaries of eriodic movemens can be obained by sudying he asymoic sabiliy of he corresonding marix reresening he eriodic aern. Tha is o say, if eriod m n S S is discovered under cerain arameer seings, where mn, and ower is used o exress he rajecory m n saying in one region, hen he marix A A is asymoically sable. We noice ha he value of n does no affec m n eigenvalues of marix A A as long as n > 0 since he wo dimensional A (Eq. 8) is idemoen. n For insance, condiions of eriodic movemen SS in T = sysem can be derived by solving abs( λ ) n, which AA leads o 0 < S <. Likewise, he boundary for eriodiciy of n SS can be obained from abs( λ ), i.e.,, where we can obain S S n AA ands. (5) Fig. 4 shows Lyaunovian sable (whie), asymoically sable (black) and eriodic (dark grey) regions for he invenory sysem when T =. The marices used o derive he boundaries are also labeled. Noice ha here is an infinie number of such dark grey branches above he asymoic sabiliy boundary. 3.4 Quasi-eriodiciy and chaos Quasi-eriodiciy is he roery of a sysem ha dislays irregular eriodiciy. Quasi-eriodic behaviour is a aern of recurrence wih a comonen of unredicabiliy ha does no lead iself o recise measuremen (Fig. 3c). Quasieriodic moion is, in rough erms, he ye of moion execued by a dynamical sysem conaining a finie number (wo or more) of incommensurable frequencies. Values of quasi-eriodic oins are dense everywhere. Mahemaically, chaos refers o a very secific kind of unredicabiliy: deerminisic behaviour ha is very sensiive o is iniial condiions. In oher words, infiniesimal variaions in iniial condiions for a chaoic dynamic sysem lead o large variaions in behaviour over ime. Chaoic sysems consequenly aear disordered and random (Fig. 3d). 6

7 When chaoic dynamics occurs, he sysem moves in a region of hase sace ha is densely filled wih unsable eriodic orbis. The rajecory is araced and reelled in differen direcions by hese orbis, and he resul is an irregular behaviour ha is sensiive o small erurbaions and arameer changes (Mosekilde and Laugessen. 007). Quasi-eriodiciy and chaoic moion are boh characerized by he fac ha, alhough bounded in hase sace (Lyaunovian sable), he rajecory never recisely reeas iself. This research shows ha, comared o he fourechelon beer-game model sudied in Mosekilde and Laugessen (007), such comlex behaviours can be found even in his simle model of a consrained invenory sysem. When he sysem behaves quasi-eriodically or chaoically, A has wo comlex eigenvalues ouside he uni circle, and no sable eriodic movemen can be found. In Fig. 4, Quasi-eriodiciy and chaos are reresened by ligh grey. Fig. 4. Sabiliy diagram of he Forbidden Reurns invenory sysem when T =. To summarise he aforemenioned boundaries divide he aramerical lane ino several regions in which he invenory sysem behaves differenly: asymoically sable, eriodic, quasi-eriodic, chaoic and divergen. As inuiion would dicae, a osiive invenory feedback arameer is essenial o mainain sabiliy of he sysem. Furhermore, when he absolue value of is small, he sysem will be asymoically sable. This region is limied and shaed as he black riangle. Luckily he Order-U-To olicy ( S =, = ) lies wihin his region. When is negaive, exonenial divergence can be observed. Regular eriodiciy can be discovered when S is small and is osiively large. Oher areas are filled by eriodical branches, quasi-eriodic and chaoic arameer seings. Wha we can infer from he above analysis is ha high S and values lead o chaos. 4. Sabiliy analysis on he effec of lead-imes In his secion we will analyze he effec of lead-ime change and lead-ime miserceion on suly chain sabiliy. We focus on acual lead-ime change in he firs subsecion and lead-ime miserceion in he second. For he sake of simliciy, in hese wo subsecions, we resrain ourselves o cases where boh acual lead-ime and erceived leadime are eiher one or wo imes of he lengh of he ordering cycle, ha is,t, T {,}. In he las subsecion, we invesigae how he lead-ime affecs he size of asymoically sable region and Lyaunovian sabiliy boundaries. 7

8 Mehods o deermine sabiliy boundaries which are roosed in he revious secion are aricularly useful in conducing his ar of analysis. 4. Acual lead-ime changes wih erfec knowledge For T =, using he same echniques as in he revious secion, i is easy o derive sabiliy and eriodiciy condiions. For he sake of breviy we omi he redicable modeling and analysis rocedures. The Lyaunovian sabiliy boundary is given by, (6) S 0 and is asymoic sabiliy condiions are: 0 S ( ) 3 5 for -0.5 < < ; (7) ( ) 3 5 for < <.5. (8) S If we draw boundaries of T = and T = ogeher as shown in Fig. 5, we are able o visualise he differen dynamic behaviours ha exis when he lead-ime increases from o and we know of his fac. Table summarizes he dynamic behaviour under each lead-ime value, and highlighs he regions where a srucurally differen behaviour can be observed. Since he eriodic boundaries are fracal as was discovered in he revious secion, we decided no o disinguish eriodic, quasi-eriodic and chaoic movemens hereafer bu idenify hem aggregaely as bounded oscillaion. Alhough we have assumed he lead-ime can be accuraely measured and he relenishmen olicy can be correcly udaed wih his lead-ime informaion, a sudden increase of lead-ime could sill jeoardize sysem sabiliy. Tha is, a sysem could be asymoically sable wih one lead-ime bu wih a higher lead-ime i could exhibi bounded oscillaion (region I and II in Fig. 5) or even divergen behaviour (IV). In region III, a lead-ime increase will acually sabilize he sysem. Highly volaile lead-imes will increase he difficuly of mainaining sysem sabiliy. Consider he following examle: he decision maker ignores WIP feedback and uses 80% of invenory feedback in ordering decision ( S = 0.8, = 0, IOBPCS sysem). The sysem is asymoically sable under uni lead-ime. However a limied vibraing dynamic behaviour will be observed when lead-ime is (Fig. 6b). Fig. 5 and Table are verified via simulaion as shown in Fig. 6, where he sysem is driven by a uni se demand rocess, and a doed line reresens he ime when lead-ime changes. The lengh of he MATLAB simulaion and he iming of he lead-ime changes have been adjused o allow he figures o be resened clearly. 4. Unobserved lead-ime changes In his secion, he effec of lead-ime on sysem sabiliy in he Esimaed Pieline Variable Invenory and Order Based Producion Conrol Sysem (EPVIOBPCS) will be analyzed. This olicy was inroduced in Disney and Towill (005) o eliminae a henomenon known as invenory drif (when he sysem falls ino a seady sae wih a ermanen difference beween he arge and acual invenory levels), by calculaing work-in-rocess level using erceived lead-ime as WIP WIP ORATE ORATE (9) T or, afer z-ransform, T WIP z ORATE z. (0) where T is he erceived lead-ime. When T T he EPVIOBPCS roduces he same dynamic resonse as he APVIOBPCS. However, when he erceived lead-ime is incorrec, i.e., T T, his mehod generaes arificial WIP values ha are mached o he DWIP arge levels and his allows he invenory o reurn o arge levels. I is already known ha when work-in-rocess is calculaed in he convenional way (via Eq. 5), he value of T does no affec he sabiliy of he sysem, since i only aears in a feed-forward ah (calculaing he desired work-in-rocess). However, 8

9 Fig. 5: Sabiliy comarison beween T = and T =. Table : Change of dynamic behaviour in each region wih known T changes observed in Fig. 5. T = T = I Asymoically sable Bounded Oscillaion II Asymoically sable Bounded Oscillaion III Bounded Oscillaion Asymoically sable IV Asymoically sable Divergen V Bounded Oscillaion Divergen when work-in-rocess is calculaed based on he erceived lead-ime, T, (wih Eq. 9), hen T changes he dimension of he invenory sysem and hus i has a dramaic effec on sabiliy and he dynamic behaviour. The characerisic olynomial for EPVIOBPCS, in he z-domain, i s T T TT z ( ) z z S 0 for T T T T TT z ( ) z z 0for T T () S When erceived lead-ime is correc ( T T), his olynomial becomes T T S z ( ) z 0 () 9

10 Fig. 6: Dynamic resonses of Forbidden Reurns APVIOBPCS wih a sudden increase of lead-ime under mulile arameer seings (verificaion of Fig. 5 and Table ). (a) Region I: α S =.5, α = ; (b) Region II: α S = 0.8, α = 0; (c) Region III: α S =.8, α = ; (d) Region IV: α S = 0.05, α = 0.8; (e) Region V: α S =, α = which characerises he APVIOBPCS relenishmen olicy. To mainain asymoic sabiliy of he sysem, (comlex) soluions mus lie wihin he uni circle on he comlex lane. For Lyaunovian sabiliy, a leas one soluion mus be real and larger han one. The longer lead-ime beween he acual and he erceived ones deermines he order of he olynomial. For secific lead-ime combinaions i is ossible o derive he crieria analyically wih he Inners aroach of Jury (974). For he general case unsecified lead-ime combinaions, we do no know of a soluion. Le s examine he following cases: Case (-), an overesimaion of he acual lead-ime; Case (-), an under 0

11 esimaion of he acual lead-ime. Noe ha we are using wo hyhenaed numbers in brackes o reresen lead-ime mis-secificaion scenarios, he firs number reresening he acual lead-ime, T, and he second one erceived leadime, T. Paramerical boundaries for he above wo cases can be derived. For he (-) case, he asymoic sabiliy region is: 0 S, 0.5. (3) The Lyaunovian sabiliy boundary is: (4) S 54S 45 6S 54S 3S S 0 For he (-) case, he asymoic sabiliy boundary is Is Lyaunovian sabiliy boundary is: 5 S, 0.5 (5), 0.5. (6) S (7) S 6 8S 3 8S 8S S 0 Similar o he analysis in he revious secion, we obain srucurally differen behaviours when lead-ime missecificaion occurs. Firs, by overlaing he sabiliy diagram of case (-) wih ha of case (-) (Fig. 7), he effec of incorrecly assuming he lead-ime is raher han correcly assuming ha i is can be analyzed. I can be seen ha he lead-ime mis-secificaion leads o a major reducion in he size of asymoically sable region and an increase in he size of he exonenially divergen region. Moreover, in mos regions, he dynamic behaviour of he invenory sysem deerioraes (Table ). Fig. 7 and Table highligh a raher worrying siuaion ha could exis for he classical OUT olicy. Using S = = when boh he acual and he erceived lead-ime is resuls in an asymoically sable sysem. However, if he erceived lead-ime is, bu he acual lead-ime is, hen an oscillaory sysem is resen. This is a raher alarming resul given he revalence of he classical OUT in indusrial seings and highlighs he need for knowing and using correc lead-ime informaion in relenishmen olicies. By overlaing he sabiliy ma of case (-) wih ha of case (-) (Fig. 8), we can analyze he effec of incorrecly using a lead-ime of one when we should be using a lead-ime of wo in he relenishmen sysem. Dynamic behaviour comarisons are shown in Table 3. This lead-ime mis-secificaion decreases he size of asymoically sable region and increases he size of he Lyaunovian sable region. Again, alarmingly, he indusrially revalen seing of S = = in he OUT olicy exhibis eriodic behaviour when lead-ime mis-secificaion occurs, alhough i is asymoically sable when he relenishmen sysem is se u wih correc lead-ime informaion. Table 4 shows he effec of acual lead-ime and erceived lead-ime on he size of asymoically sable region. I can be seen ha when lead-ime is erceived correcly (numbers on he main diagonal), he size of asymoic sabiliy is much bigger han hose when lead-ime informaion is incorrec (numbers above or below he diagonal). The imorance of accurae lead-ime informaion on asymoic sabiliy region is obvious. 5. Conclusion and discussion This aer highlighs he range of dynamic behaviours ha are resen in a consrained invenory sysem wih only one consrain. I is ineresing ha even a simle deerminisic model wih a shor lead-ime is sufficien o generae such comlex henomena. We wonder wha sor of imac sochasic models, longer lead-imes, more consrains and mulile echelons will have? Comared o he characer of he sable regions, he unsable regions are relaively unknown. We have shown ha a comlex and diversified se of behaviours and aerns exis in he unsable region. Managerially, in a suly chain or roducion seing, inuiively we may rank hese classes of dynamic behaviours from good o bad as follows; asymoic sabiliy, eriodiciy, quasi-eriodiciy, chaos and divergence. The mos surrising resul we have revealed here is he fac ha using wrong lead-ime informaion in he EPVIOBPCS can

12 Fig. 7: Sabiliy Comarison beween (-) and (-). Table : Change of dynamic behaviour in each region wih T and T changes observed in Fig. 7. T =, T = T =, T = I Asymoically sable Bounded Oscillaion II Asymoically sable Divergen III Bounded Oscillaion Divergen Fig. 8: Dynamic resonses of Forbidden Reurns EPVIOBPCS wih a sudden increase of erceived lead-ime under mulile arameer seings (verificaion of Fig. 7 and Table ). (a) Region I: α S =.3, α = ; (b) Region II: α S = 0.05, α = 0.9; (c) Region III: α S = 0.4, α =.5.

13 Fig. 9: Sabiliy Comarison beween (-) and (-). Table 3: Change of dynamic behaviour in each region wih T and T changes observed in Fig. 9. T =, T = T =, T = I Asymoically sable Bounded Oscillaion II Bounded Oscillaion Asymoically sable III Divergen Asymoically sable IV Divergen Bounded Oscillaion 3

14 Fig. 0: Dynamic resonses of Forbidden Reurns EPVIOBPCS wih a sudden decrease of erceived leadime under mulile arameer seings (verificaion of Fig. 9 and Table 3). (a) Region I: α S =.5, α =.; (b) Region II: α S = 0., α = -0.5; (c) Region III: α S = 0.05, α = -0.7; (d) Region IV: α S = 0., α = -. Table 4: Size of he asymoic sabiliy region under differen acual and erceived lead-imes. T T resul in a eriodic or a chaoic sysem. As he EPVIOBPCS is a general case of he Order-U-To (OUT) olicy his is a worrying resul as he OUT olicy is robably one of he mos oular relenishmen algorihms in indusry. Fig. akes anoher look a his issue by resening four bifurcaion diagrams, wo wih correc lead-imes, wo wih missecified lead-imes. Fig. was roduced wih MATLAB. For each bifurcaion diagram, we run 500 simulaion exerimens wih he conrol arameer (α ) changing. In each simulaion he se resonse of ORATE for 000 eriods is generaed and ransien daa in he firs 800 eriods is discarded. The remaining seady sae daa is hen used o derive he bifurcaion diagram. We see ha he bifurcaion crieria analyically obained in Secion 3 can be verified by hese diagrams, and also he rediced comlex and rich dynamic aerns can be observed, esecially in (b) and (c). Wih incorrec lead-ime informaion, he indusrially revalen OUT olicy is no asymoically sable. This 4

15 means ha i is exremely imoran o obain and use accurae lead-ime informaion in a relenishmen sysem. We also noe ha he esimaed lead-ime will affec sysem sabiliy only in he EPVIOBPCS version of he OUT olicy, which is designed o eliminae invenory drif. Hence, here migh be a rade-off beween effecive invenory conrol and sysem sabiliy. This requires more deailed invesigaion in he fuure. There is also a erformance measuremen / behaviour asec o consider here. We have exerience of a comany where bonuses were awarded based on reducions in sulier lead-imes. The informaion used o deermine wheher bonuses were awarded was gahered from he comanies ERP sysem ha was also used o schedule he roducion and coordinae he suly chain. Emloyees knew of his and coverly changed lead-ime informaion in he ERP sysem wihou he subsequen hysical reengineering effors. The comany was exeriencing exreme dynamic behaviour in boh roducion and invenory. We believe a leas ar of his dramaic dynamic behaviour can be exlained by he henomena described by his aer. In his aer, we have adoed he assumion ha he las observed demand is used as he forecas for he curren eriod, i.e., T or α =, also known as he naïve forecasing mehod. This is acceable in a se demand scenario a as he forecas generaed by he simle smoohing mehod (wih T 0.5 o ensure sabiliy) will evenually be equal a o he demand. Using he naïve forecasing mehod only removes he ransien resonse and does no affec he seady sae. Furhermore, o obain closed-form exressions of he sabiliy crieria, we concenraed rimarily on shor leadime scenarios, such as one or wo eriods. When longer lead-imes are resen, he dynamic behaviour of he invenory sysem can be highly jeoardised; mos imoranly he asymoic sabiliy region will shrink in size as we demonsrae in Table 4. Ordering olicies wih high invenory feedback and low WIP feedback are esecially vulnerable o lead-ime increases. The Lyaunovian sabiliy boundary moves o he righ when lead-ime increases, increasing he ossibiliy of a divergen resonse. Meanwhile longer lead-imes may increase he risk of lead-ime errors being inroduced, harming dynamic erformance. In he field of dynamical sysems, he characerisaion of high dimensional iecewise linear sysems are far from being solved. Hence, o exlore he dynamical behaviours of he consrained invenory sysem, a simulaion-based echnique has been incororaed wih an eigenvalue analysis and knowledge of dynamical sysems. Due o he unique naure of he Forbidden Reurns sysem, a linear sysem sabiliy analysis was sufficien o derive he asymoic sabiliy boundaries. However, in general his may no always be he case. Acknowledgmens Xun Wang would like o hank he Chinese Scholarshi Council for roviding financial suor (No ) ha enabled him o visi Cardiff Universiy for year. The suor of Naional Naural Science Foundaion of China (Nos ; ; 7706) is also acknowledged. 5

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