Incorporating Delay Mechanism in Ordering Policies in Multi- Echelon Distribution Systems

Transcription

1 Incorporaing Delay Mechanism in Ordering Policies in Muli- Echelon Disribuion Sysems Kamran Moinzadeh and Yong-Pin Zhou Universiy of Washingon Business School {amran, June, 2005; revised July, 2006; December 2006 Absrac In his paper, we devise a framewor for obaining he opimal ordering policy in a single locaion, coninuous review invenory sysem wih arbirary iner-demand imes. We show ha i is opimal o order a demand arrival epochs only if he iner-demand ime has consan or decreasing failure rae (CFR or DFR). When he iner-demand ime has increasing failure rae (IFR), we show ha he opimal policy is o delay he order. We hen exend his policy o muli-echelon disribuion sysems consising of one supplier and many reailers. Boh decenralized and cenralized sysems are considered. We derive expressions and procedures for he evaluaion of oal cos and he compuaion of opimal delay in all seings considered. More imporanly, we sudy he impac of our delay policy in all he seings. The numerical resuls indicae ha for he single locaion model, he opimal delay can reduce he oal cos significanly. Resuls from he single locaion model can be applied o he decenralized muli-echelon sysem, where he upsream supplier acs as a single locaion sysem. The supplier order delay can also have a significan impac (eiher posiive or negaive) on he reailers oal cos as well as he sysem s oal coss. Finally, he impac of supplier order delay is minimal in he cenralized muli-echelon seing. We offer an inuiive explanaion for his observaion. - -

2 . Inroducion The lieraure on single locaion coninuous review invenory models is rich and many sudies have developed framewors for analyzing and compuing opimal or near opimal parameers of he well nown (s,s) and (,R) policies under differen seings (see Hadley and Whiin 963, Tijms 97, Sivlazian 974, Nahmias 976, Sahin 979 and 982, Zipin 986a &b, Zheng 992, and Federgruen and Zheng 992, ec.). I is well nown ha in single locaion sysems under coninuous review and Poisson demand, he (,R) policy, where orders are placed only a demand epochs, is opimal. However, when demand follows an arbirary renewal disribuion, he (,R) policy is no longer opimal as order epochs may no coincide wih demand epochs. In his paper, we consider coninuous review invenory sysems where demand forms a renewal disribuion wih arbirary iner-demand disribuion, and inroduce an order delay policy referred o as he (,R,T) policy. Under his policy, a replenishmen order will be placed T ime unis afer invenory posiion reaches R, or when he nex demand occurs (and he invenory posiion decreases o R-), whichever occurs firs. I can be easily verified ha he classical (,R) policy is a special case of he (,R,T) policy for T=0. The quesion we as in his paper is no wheher he class of (,R,T) policies will ouperform he class of (,R) policies, bu raher by how much. More imporanly, we analyze his policy in a muli-echelon seing as well. Finally, we invesigae he effec of sysem parameers (cos, leadime, demand, ec.) and sysem configuraion (single locaion vs. muli-echelon, cenralized vs. decenralized, ec.) on he magniude of cos savings. We will show ha in he single echelon seing, when he iner-demand ime exhibis decreasing or consan (i.e., Poisson demand) failure rae (DFR or CFR), i is indeed opimal never o delay he order. However, when iner-demand ime has increasing failure rae (IFR), hen by delaying and observing he elapsed ime since las demand, he sysem can ge more up-o-dae informaion on he ime unil nex demand and improve is performance by delaying he order placemen. We show ha he magniude of such improvemens can be quie significan. An - 2 -

3 imporan applicaion of his model is o he upsream sie (supplier or warehouse) in a disribuion sysem, where due o downsream (reailers) bach-ordering, (see Duermeyer and Schwarz 98, Nahmias 98, Lee and Moinzadeh 987, Moinzadeh and Lee 986, Svoronos and Zipin 988 and Axsaer 990 for examples), iner-order imes from each of he downsream sies is IFR. For insance, when exernal demand a he downsream locaions is Poisson, he iner-order imes from each sie is Erlang, which is IFR. Clearly, in such seings, i may be opimal for he up-sream insallaion o delay is order placemen. In his paper, we also sudy order delay in a muli-echelon seing, boh decenralized and cenralized. In a decenralized disribuion sysem where reailers order in baches larger han one, he supplier operaes lie a single locaion model facing IFR demand and uses an order delay policy. Clearly, he single locaion resuls sugges ha he supplier will be beer off. However, he impac of he supplier order delay on he reailers is more complex. For insance, if he supplier eeps he same R as ha in he classical (,R) policy and uses a delay T, hen he reailers will be worse off as heir leadimes will be sochasically larger. Bu, if he possibiliy of an order delay causes he supplier o increase is R o R and hen use a delay T, hen he delay policy used by he supplier can acually benefi he reailers. In a numerical experimen, we observe ha he impac on he reailer coss can be significan. In a cenralized serial invenory sysem, i is well nown ha echelon-based invenory policy is opimal (see Clar and Scarf 960, Axsaer and Rosling 993 and Chen 998). For cenralized disribuion sysems wih muliple reailers and one supplier (i.e., warehouse), he general form of he opimal policy is no nown. In all previous wors, (,R)-ype policies have been employed as he order policy of choice in all locaions and he opimal parameer values are found for boh he reailers and he supplier. For examples, see Duermeyer and Schwarz (984), Lee and Moinzadeh (987), Moinzadeh and Lee (986), Svoronos and Zipin (988) and Axsaer (990). In his paper, we will consider a similar sysem o hose menioned above, employ he (,R,T) order policy a he supplier and sudy is impac on such sysems. Numerical resuls sugges ha he order delay has - 3 -

4 minimal impac on he sysem cos. We provide an inuiive explanaion based on he resuls in he decenralized seing. To summarize, he conribuion of his paper is wo fold: Firs, in a single locaion seing, we use a delay policy and idenify he condiions when i is opimal. Second and more imporan, we apply he order delay policy o boh he cenralized and decenralized disribuion sysems, and compare hem wih sysems ha do no use order delay. Furhermore, we idenify he seings where employmen of delay policies a he supplier is mos criical in such disribuion sysems. The only papers nown o he auhors ha deal wih general, non-poisson, demand process are Schulz (989), Moinzadeh (200) and Kaircioglu (996). Schulz (989) sudies he concep of order delay in an (S-,S) invenory seing where S, he maximum socing level, is se a one. Moinzadeh (200) considers a single locaion invenory sysem in he (S-,S) seing where demand follows an arbirary renewal disribuion. He proposes an improved order policy, which is based on delaying he order placemen in such sysems; ha is, an order will be placed T ime unis afer a demand occurs. The opimal delay, T, however, is fixed and no deermined in real ime. This policy is subopimal as i does no ae ino accoun he demand aciviies during he delay period, T. Even so, i is shown ha his order policy wih delay can resul in significan cos savings compared wih he radiional (S-,S) policy. The mehod of sudy in his paper is based on Moinzadeh (200), and provides a significan improvemen. Kaircioglu (996) also considers he single locaion version of our problem. Alhough he resuls are similar, our mehod of analysis is differen. Furhermore, unlie Moinzadeh (200) and Kaircioglu (996), we consider in his paper he order delay policy in disribuion sysems consising of one supplier and muliple reailers, boh cenralized and decenralized, and sudy he effec of such a delay on sysem coss. We idenify he seings when employmen of delay policies a he supplier is mos criical in such disribuion sysems. The res of he paper is organized as follows: In Secion 2, we sudy he delay policy in a single-echelon seing and idenify he condiions when i is opimal. Then in Secion 3, we expand - 4 -

5 our analysis o a muli-echelon disribuion sysem. In Secion 4, we numerically sudy he impac of he delay policies on coss for single locaion, boh cenralized and decenralized muli-echelon disribuion sysems. We close in Secion 5 by summarizing he conribuion of his sudy and suggesing avenues for fuure research. All he proofs can be found in he appendix. 2. A Single Locaion Model We sudy a coninuous review invenory sysem where demand is of uni size and follows a renewal process wih a mean rae λ. Lead ime is a consan, L, and any excess demand is bacordered. We propose he following dynamic delayed ordering policy, which we call he delayed-order policy: When he curren sysem invenory posiion is r and i has been ime unis since las demand occurrence, an order of will be placed τ r () ime unis from now. Noe ha he delayed-order-policy is he opimal class of policies in such sysems as a any invenory level, he ordering decision is examined a any poin in ime. We will show ha he opimal policy parameers under his policy will be such ha he resuling policy operaes lie a (,R,T) policy. For fixed, he opimal value of τ r () can be found by minimizing he average oal cos rae when invenory posiion is r and an order will be placed. We now derive an expression for his average cos rae. Consider a uni in an order, say, he h uni (=,,). This uni will be assigned o he (r) h fuure demand. The expeced cos associaed wih his uni, consising of holding and bacorder coss, will be: Yr { r ( τr() ) π ( τr() ) r } E h Y L L Y, where Y r denoes he ime unil he (r) h fuure demand when he las demand occurred ime unis ago (all necessary noaions are defined in Table )

6 Table : Noaion for he single locaion model R τ r () fixed order quaniy maximum invenory posiion level ha riggers an order order delay when invenory posiion is r and ime have elapsed since las demand (o simplify noaion, he subscrip will someimes be suppressed) L h π X consan order leadime uni holding cos per uni ime uni bacorder cos per ime uni random variable represening he i.i.d. iner-demand imes f ( ) / F( ) PDF/CDF of X Z = i = X Y i -fold convoluion of X for > 0, Z = 0 0, Z = Z for < 0 random residual life of X when ime unis have elapsed; i.e., ime unil g ) / G ( ) PDF/CDF of ( he nex demand when he las demand occurred ime unis ago Y Y = Y Z, ime unil he h fuure arrival, when ime unis have elapsed since las G (x) CDF of j demand epoch for > 0, Y = Y Y = Z for 0, Hi, j( x) = G( x) The sum of he G funcions for = i o j = i We will fix hroughou he paper; hen for each r and, we find he opimal delay τ r () which minimizes he average oal cos rae when invenory posiion is r and an order will be placed as follows: = r { ( τr ) π ( τr ) } r λ min E h Y L ( ) L ( ) Y. () τ r () Y - 6 -

7 Finally, once he opimal delay τ r () is fully characerized, we find he opimal value of r for he delayed order policy. Noe ha since τ r () is updaed coninuously for all r and, an order will be placed only when τ r ()=0 (when τ r ()>0, is exac value is no imporan; an order will no be placed). The objecive funcion in () is convex in τ r (), so i suffices o analyze he firs order derivaive of he objecive funcion (where H, ( x) = G ( x) ): i j j = i r λ λ FO = h Pr ( Y > L τr() ) π Pr ( Y L τr() ) = h ( h π) H r, r ( L τr() ). = r (2) All he proofs of he Lemmas and Theorems o follow can be found in he Appendix. Lemma () When r 0, i is opimal no o delay orders. Tha is, τ r () 0. h τ r() = Hr, r L π h. (2) When r>0, he opimal delay saisfies ( ) Lemma 2 () When X has DFR, τ r () is non-decreasing in. (2) When X has CFR, τ r () is consan in. (3) When X has IFR, τ r () is non-increasing in. The firs wo saemens in Lemma 2 imply ha if i is opimal no o order a a demand epoch, i is opimal no o order as long as he invenory posiion remains he same (i.e., τ (0) > 0 τ ( ) > 0, ). Therefore, we have: r r Theorem If he iner-demand ime disribuions have DFR or CFR, hen i is opimal o place orders only a demand epochs. Corollary When he demand process follows a Poisson process, i is opimal o order only a demand epochs

8 Now we focus on he more ineresing case of IFR. Examples of IFR demand abound in pracice. For example, uniform and Erlang disribuions have IFR, and for some parameer combinaions, he Weibull disribuion also has IFR. Lemma 3 Suppose he iner-demand ime has IFR. For any r, τ () is decreasing in, and lim τ ( ) τ (0). r r r In Theorem 2, we characerize he opimal policy parameers under he proposed delayed policy. Theorem 2 Le r = min{r: τ ( 0) > 0 }. r (a) If τ ( ) = 0 for some finie, hen le R= r and T = inf{: τ ( ) = 0 }. r (b) If τ ( ) > 0 for all finie, hen le R= r and T = 0. r The following policy is he opimal delayed-order policy: () When r>r, i is opimal no o order for all. (2) When r=r, i is opimal no o order for <T; and i is opimal o order a T. (3) When r<r, i is opimal o order for =0 (i.e., no delay). The resuls in Lemma 3 and Theorem 2 are illusraed in Figures and 2 where he behavior of τ r () is depiced as a funcion of for differen values of r. In Figures and 2, he wo (a) panels correspond o (2) in Theorem 2, and he wo (b) panels correspond o (3) in Theorem 2. R τ r () r=r3 lim τ R 3 ( ) = τ R 2 (0) r=r2 lim τ R 2 ( ) = τ R (0) r=r lim τ ( ) = τ (0) R R all he r<r r=r curves lie on he horizonal axis (r) T (a) τ r () r=r4 r=r3 r=r2 r=r T=0 (r) all he r R curves lie on he horizonal axis (b) Figure : The τ r () Curves for lim τ r( ) = τ r (0) - 8 -

9 lim ( ) τ R 2 τ R (0) lim ( ) τ R τ R (0) all he r<r curves lie on he horizonal axis τ r () r=r2 r=r r=r (r) T (a) τ r () r=r3 r=r2 r=r T=0 (r) all he r R curves lie on he horizonal axis (b) Figure 2: The τ r () Curves for lim τ r( ) > τ r (0) Numerically, one needs o calculae τ r (0) for he various values of r, and use Theorem 2 o find R. Moreover, for his R, if he opimal T is posiive, he firs order condiion of opimaliy can be h π T T used o find he T such ha τ R (T)=0; ha is, = G ( L) = H ( L) R R, R. The righ hand h = R side of he equaion is decreasing in T, so a numerical mehod could be used o solve his equaion. An imporan implicaion of Theorem 2 is ha if he sysem sars ou wih a very high invenory posiion r, hen he curve would never inersec wih he horizonal axis (case in Theorem 2). Then no orders will be placed unil he nex demand epoch, and so on, unil he invenory posiion finally drops down o R (case 2). So effecively, we are reducing he invenory posiion from r o R. Similarly, when he sysem sars ou wih very low invenory posiion r (case 3), successive orders of size will be placed righ away o bring he invenory posiion o above R. When he invenory posiion reaches R, an order will be placed afer T ime unis or when he nex demand arrives, whichever occurs firs. Therefore, when he sysem is in seady sae, an order will be placed only in wo siuaions: () invenory posiion is R-; or (2) invenory posiion is R, and i is been T ime unis since las demand

10 This clearly demonsraes ha he opimal delayed-order policy is a (,R,T) policy. Now, he expeced oal cos rae of a (,R,T) policy for any fixed, which consiss of he average cos rae if demand occurs before T is expired and he average cos rae if T expires before an occurrence of a demand, can be expressed as: T E h( Z x T L) π ( L T x Z ) πe [ Z T L] f( x) dx Z Z R R R R > 0 0 T E ( ) ( ) ( ) 0 Z h Z ( ) L π L Z EZ π L x Z f x dx R R R R > 0 0 (3) The opimal delay T can be found by applying Lemma and Theorem 2. The firs order derivaive is as follows: FO2 = Pr ( 0) Pr ( 0 ) ( ) T h Z x T L> π L T x Z > π f x dx R R R R > 0 0 E Z h( Z ) ( ) [ ] ( ) L π L Z πez Z T L f T R R R R > 0 0 E h Z L L Z ( ) π ( ) Z R R R R > 0 0 T = FT ( ) π ( h π) Pr( Z 0 ) 0 y L> g ( ydy ) R R > 0 T = FT ( ) π ( h π) G ( L) R R > 0 T = FT ( ) π ( h π) G ( L) R R = FT ( ) h ( h π ) H ( ) T L R, R. EZ π ( L T Z ) ( ) f T (4) - 0 -

11 Comparing equaion (4) wih equaion (2), we find ha for any opimal (,R,T) policy where FO 2 =0 in (4), seing τ ( ) = 0 will also saisfy FO=0 in (2). Therefore he opimal R, T values R T are he same for boh he (,R,T) policy and he delayed-order policy. Remar : The model in Moinzadeh (200) is a special case of our delayed-order policy model for he (S-,S) seing. Moreover, Moinzadeh (200) assumes ha he order delay, τ, is saic, fixed, and no allowed o change in real ime. In he even ha many demand orders occur afer invenory posiion reaches S-, and before τ has expired, which would have riggered an order under our policy before τ, he policy in Moinzadeh (200) would sill place an order afer he fixed τ. Clearly his is less effecive. Remar 2: Kaircioglu (996) sudies he (,R,T) policy using a differen approach. He firs analyzes a single uni in he order, hen uses inducion o exend i o muliple unis, and, laer, infinie ime horizon. Lie Moinzadeh (200), Kaircioglu (996) sudies only he single-echelon problem. Our analysis of he muli-echelon disribuion sysem is presened in he nex secion. 3. A Muli-Echelon Disribuion Sysem Now we will apply our (,R,T) policy o a disribuion sysem. We will follow he classic seing for disribuion sysems (see Svoronos and Zipin 986, Axsaer 990 and Moinzadeh 2002). In such a sysem, here are N idenical reailers and one supplier. Demand a each reailer follows a Poisson process wih rae λ; as a resul, each reailer follows a (,R) policy when placing orders. Excess demand is bacordered a he reailers. Upon placemen of an order, if he supplier has soc, i will fill he order immediaely; oherwise, he reailer s order will be delayed unil he supplier has sufficien soc o fill he order. I is assumed ha he ransi ime from he supplier o he reailer is fixed and consan L. The supplier orders is soc from an ouside source, which is assumed o have - -

12 ample soc. We assume ha he supplier s order quaniy, ˆ, is an ineger muliple of reailer s order quaniy, ; ha is, ˆ =m (m=,2,..). Finally, he order leadime from he ouside source o he supplier is assumed o be a consan ˆL. (Throughou his secion, we will apply he ˆ noaion o all he parameers, variables, and funcions associaed wih he supplier.) In his secion, we firs analyze he demand process a he supplier, and show ha i follows an IFR process. Then, we will examine he supplier s use of he delayed-order policy boh in he cenralized and decenralized seings. Since demand a he reailers is Poisson demand and a (,R) policy is employed by he reailers, each reailer s iner-order ime has an Erlang disribuion wih mean /λ and shape parameer (denoed as Erlang(λ/,)). We sar by examining a random epoch when he supplier receives a reailer order. Wihou loss of generaliy, assume ha Reailer jus placed he order. Thus, he invenory posiion a Reailer will be R jus afer order placemen. I is well nown in he lieraure ha he invenory posiions a he oher reailers are uniformly disribued on {R, R2,, R}. The following hree Lemmas will show ha he random ime unil he nex order a he supplier by any reailer has IFR. Lemma 5 Lemma 6 Lemma 7 Any Erlang disribuion has IFR. A any reailer order epoch, all he oher reailers have IFR ime unil nex order. The minimum of independen IFRs is also IFR. Togeher, he hree Lemmas imply ha he iner-demand ime seen by he supplier is IFR. Proposiion If he reailer order size,, is greaer han, hen he iner-order ime a he supplier has IFR. Since reailers see Poisson demand, hey will order only a heir own demand epochs. I is he supplier who should opimally inroduce delay ino is own ordering

13 Cenralized Seing In he cenralized seing, he reailers and he supplier belong o he same company. Therefore he objecive is o find he opimal invenory policy parameers for he reailers and he supplier joinly, in order o minimize oal supply chain cos, consising of holding and penaly cos a he reailers and holding cos a he supplier. We assume ha he reailers are idenical and hey all use a (,R) policy. We assume ha he supplier follows a ( ˆ, ˆR, ˆ T ) policy. When reailers are idenical, under he sandard assumpions made here which is in line wih previous wor in he lieraure (see Svoronos and Zipin 986, Chen 998 and Axsaer 993), i is reasonable o se he supplier s reorder poin o be a muliple of (see Zipin 2000, pp. 320 for a discussion). Thus, we assume ha ˆR = ˆr. Recall ha a delayed-order policy for he supplier means: When he invenory posiion reaches ˆR, he supplier places an order of quaniy ˆ, ˆ T ime unis in he fuure or when he nex reailer order is received, whichever occurs firs. Le us now follow a supplier order. This shipmen will be used o saisfy he ( r ˆ ) h, ( rˆ m) h reailers fuure orders. There are wo coss associaed wih his order:, ( r ˆ 2) h, C, he supplier s expeced holding cos (when a shipmen arrives a he supplier before is corresponding reailer order) per uni in an order. C 2, he expeced reailer holding (when unis arrive a he reailer before heir corresponding demands arrive a he reailer) and penaly cos (when unis arrive a he reailer afer heir corresponding demands arrive a he reailer) per uni in an order. We will analyze he wo coss one by one: Le Vˆ j ˆ ˆ ˆ ˆ Z j Z T L j > = j ( ) if 0 0 if 0, hen V ˆj is he ime ha he supplier order arrives a he supplier before he ( j rˆ ) h fuure reailer order shows up, where j = ˆr, ˆr 2,, ˆr m. Consequenly, - 3 -

14 rˆ m rˆ m ˆ ˆ ˆ ˆ ˆ ˆ ˆ = [ j] = j ( ) j= rˆ j= rˆ. C he V m he Z Z T L m Moreover, le Wˆ j ˆ ˆ ˆ L ( Z ˆ T) Z j if j > 0 =. (5) Zˆ min { ˆ, ˆ } ˆ j T Z L if j 0 Then W ˆ j is he ime ha he supplier order arrives a he supplier afer he ( j rˆ ) h fuure reailer order shows up, where j = ˆr, ˆr 2,, ˆr m. (W is also called he reard). When a reailer orders from he supplier, if i becomes he j h order in he supplier order, hen he oal leadime for his reailer s order is ˆ j L W. Define he ime i aes for he ( i R) h fuure cusomer order o arrive by Z i, i =R, R2,, R. Then Z i is an Erlang random variable wih parameers λ/i and i. Clearly, he reailer order arrives ahead of he corresponding cusomer order if and only if Z ˆ i > L Wj. Therefore, he holding cos is h Z ˆ i L W j and he penaly cos is ˆ j π L W Z i. Thus, 2 = { i j π j i } C he Z L Wˆ E L Wˆ Z ( m). i= R, R 2,..., R j= rˆ, rˆ 2,..., rˆ m Theorem 3 In he cenralized seing, he expeced oal sysem cos rae can be expressed as: ( ) TC = Nλ C C = 2 { π } ˆ r m ˆ ˆ ˆ ˆ ˆ ˆ Nλ he Z j ( Z T) L m he Zi L W j E L Wj Z i ( m). j= rˆ i= R, R 2,..., R j= rˆ, rˆ 2,..., rˆ m We can hen use Theorem 3 o search for he opimal R and T for he supplier s (,R,T) policy

15 Remar 3: When he reailers are non-idenical, he analysis becomes very complicaed. One may need o eep rac of each sub-bach as Forsberg (996) and Axsaer (2000) did. We will no carry ou ha analysis in his paper, and consider his a challenging bu imporan fuure research opic. Decenralized Seing In he decenralized seing, he reailers and he supplier belong o differen companies. Therefore hey each find he opimal invenory policy parameers o minimize oal invenory cos for hemselves. Specifically he supplier chooses he opimal (,R) policy parameers o minimize is own holding and penaly coss, while he reailers chooses heir own opimal (,R,T) policy parameers o minimize heir own holding and penaly coss. Since demand a he reailers is saionary Poisson, i is opimal for he reailers o use he radiional (,R) ordering policy. However, o derive he opimal parameer, he reailer needs o use LW as he random order leadime, where W is given in (5). No maer wha value of R he reailers use, heir orders o he supplier have Erlang inerorder ime disribuion. According o Proposiion, he supplier sees IFR demand process, so i is opimal for he supplier o use he (,R,T) ordering policy. The derivaion of he operaing characerisics for he supplier is similar o ha of he cenralized case, so he deailed are omied. Remar 4: When he reailers are non-idenical bu independen, Lemmas 5-6 coninue o hold. Thus i is sill opimal for he supplier o delay is ordering. 4. Numerical Analysis In his secion we es he effeciveness of (,R,T) policy by comparing i wih he classical (,R) policy via a numerical experimen. We presen resuls for he single locaion seing firs in Secion 4.; hen we focus on he muli-echelon disribuion seing in Secion 4.2. In doing so, we - 5 -

16 sudy he decenralized and he cenralized muli-echelon disribuion sysems in Secions 4.2. and 4.2.2, respecively. 4. Single Locaion Model We now sudy he effeciveness of he order delay policy compared wih he classical (,R) policies in a single locaion invenory sysem discussed in Secion 2. To compare he effeciveness of he (,R,T) and he classical (,R) policy, we compue he percen deviaion as: % deviaion = 00 * Average Toal Cos(, R,T ) Average Toal Cos(, R) Average Toal Cos(, R) In our numerical experimen, we only focus on he non-rivial IFR case, and assume ha he inerdemand imes follow an Erlang(λ/, ). The choice of Erlang disribuion serves as he building bloc for he muli-echelon problem, which is analyzed in more deails in he secions ha follow. Noe ha he squared coefficien of variance (CV 2 ) of Erlang is /. In order o see he impac of he variance, no he mean, of he iner-order ime, we normalize he mean of he Erlang disribuions considered o uniy (i.e., λ= ), and vary. We vary he leadime L, he holding and penaly cos π parameers (h and π), and he resulan criical raio, which approximaes he service level. π h π Specifically, we le λ= {2, 4, 8}, L {0., 0.5,, 2, 5}, h=, {0.6, 0.8, 0.9, 0.95}, and π h {,2,4,8}. Resuls of he numerical ess are shown in Figure

17 Average Cos Reducion (%) Average Cos Reducion (%) (a) (c) Figure 3: Comparison of delayed and no-delay policies: single locaion seing Average Reorder Poin Average Cos Reducion (%) Lead Time Lead Time Service Level (b) (d) Noe ha he iner-demand ime has an Erlang disribuion of mean. The squared coefficien of variaion (CV 2 ), however, is /. I is quie clear ha as increases, he CV of he iner-demand ime decreases, and he demand arrivals become more predicable. In such cases, he order delay policy will resul in larger savings over he (,R) policy (see Figure 3(a)). In one exreme, when =, he iner-demand ime is Poisson, and here will be no cos savings whasoever by delaying he order. In he oher exreme, when he iner-demand ime is consan (CV=0 or = ), he opimal delay will perfecly mach he supply wih demand resuling in no holding and penaly coss. This observaion was made by boh Kaircioglu (996) and Moinzadeh (200). This resuls in he mos cos savings. From Figure 3(b), we observe ha as he order leadime increases, he opimal reorder poin also increases. Thus, for each replenishmen order cycle, he ime unil corresponding demand arrival is composed of more iner-demand imes. The convoluion of more iner-demand imes resuls in less - 7 -

18 variable ime unil corresponding demand which resuls in lower % cos difference beween he order delay policy and he (,R) policy. This observaion can be verified in Figure 3(c). From Figure 3(c), i is worh noing ha when leadimes are very small (one enh of he average inerdemand ime), he average cos savings by he ordering delay is subsanial. In Figure 3(d), service level (represened by π /( π h) ) is higher and he cos reducion is increasing in service level. Theoreically, however, cos reducion may decrease in service level when service level is low. Since hey are unusual in pracice, we do no perform any analysis for low service levels. Remar 5: The cos savings in Figure 3 are all average numbers for a fixed parameer value. For individual cases, he cos saving can be large. Since he cos difference is larger when leadime is smaller and is larger, he highes cos saving, 48.0%, occurs, no surprisingly, a =8, L=0., and π=9. The magniude of cos savings is consisen wih hose presened in Kaircioglu (996), indicaing ha he resuls are indeed very general, no specific o a paricular iner-demand ime disribuion. 4.2 A Muli-Echelon Disribuion Sysem 4.2. Decenralized Seing In his secion, we sudy a decenralized disribuion sysem discussed in Secion 3. In such a sysem, wihou having o be concerned wih he reailer coss, he supplier can choose is delay order o significanly reduce is average oal cos (as shown in Secion 4.). However, we aim o invesigae he effec of he supplier s decision on he reailers coss as well as he oal sysem cos. In our numerical experimen, we vary he following parameers: for he reailers, N {2, 4, 8, 6}, h=, π ( π h) =0.9, L=, {2,4,6}; for he supplier, we considered, ˆL {0., 0.5}, ĥ {0.2, - 8 -

19 0.5}, ˆ π ( ˆ π hˆ ) {0.8, 0.9}, and m {, 2, 4, 8}. We also normalize he mean of he reailers inerdemand ime o uniy (i.e., λ=). In he decenralized seing, when he supplier uses order delays o reduce is cos, i is naural o hin ha his will increase he reailers coss. However, his is no necessarily rue. Suppose ha wihou using he order delay, i is opimal for he supplier o reorder a R; hen wih he delay he supplier could do one of wo hings: use he same R and find T, or increase R by and hen find T. Therefore, he impac of he order delay on he reailers can be eiher posiive (when R is increased by one) or negaive (when R is ep he same). Thus, even in he decenralized disribuion seings, he delayed order policy can be used o lower coss for boh he supplier and he reailers, and hence he sysem. In all, we considered a oal of 384 cases. As can be seen in Figure 4 and Table 2, when averaging over all he cases, boh he reailers and he sysem see heir oal cos go up. The small values of he overall averages do no ell he whole sory, however. In mos cases here is no difference beween he delayed and non-delayed policies, bu when here is a difference, he % cos differences can be quie large in eiher direcion. A furher loo a he breadown of he cos difference reveals more informaion. When parameers are such ha he order delay by he supplier causes he reailers (sysem s) oal cos o go up, he average increase is quie significan, 9.799% (7.33%). On he oher hand, when he delay causes he reailers (sysem s) coss o decrease, on average he decrease is relaively small, 0.345% (0.375%). Furhermore, he maximum and minimum % cos differences are significan for he reailers, he supplier, as well as he sysem as a whole. Given ha he supplier order delay policy can have a significan impac for all he paries involved, we nex sudy he impac of sysem parameers in hese sysems

20 Frequency of Cos difference % reailer supplier sysem Frequency <-5-5~-0-0~- -5~0 0~5 5~0 0~5 5~20 20~25 25~30 30~35 35~40 40~45 Cos difference (%) Figure 4: Frequency of cos improvemen (delayed vs no delay) for he muli-echelon decenralized seing Table 2: Percen cos difference from delayed o no-delay policy: muli-echelon decenralized seing Reailer Supplier Sysem Average of all cases.353% -.02% 0.99% Average of posiive 9.799% N/A 7.33% Average of negaive % % % Max % % 0% 3.595% Min % % % Figure 5 shows how he cos differences for he reailers, he supplier, and he sysem change wih respec o, ˆL, and ˆ π ( ˆ π hˆ ), respecively. No surprisingly, similar o he single locaion seing, he supplier s savings increases as he reailer s order quaniy ges larger maing supplier s iner-demand imes more predicable. The behavior of reailer s cos, as well as he sysem s, seems o go in he opposie direcion. And he higher he cos savings for he supplier, he higher he cos increases for he reailers and he sysem

21 2.5 2 reailer 2.5 supplier sysem Cos Difference % Cos Difference % reailer supplier sysem -.5 Reailer Bach Size Supplier Leadime reailer supplier sysem Cos Difference % Service Level Figure 5: Average cos differences beween delay policy and no-delay policy: muli-echelon decenralized seing Nex we analyze he impac of wo oher parameers: N, he number of reailers, and m, he supplier s bach size expressed in erms of he number of reailer s bach size. As we only consider idenical reailers, when N ges large, he oal order sream o he supplier becomes more regular. In he limi when N, he supplier s demand sream becomes Poisson, and he opimal acions are no o delay any orders (i.e., T 0). Therefore, i is reasonable o expec ha he cos difference is larger when N is smaller. This observaion is verified in Figure 6(a). I is ineresing o noe again ha i seems he higher he cos savings for he supplier, he higher he cos increases for he reailers and he sysem. 5 4 reailer supplier 6 5 reailer supplier 3 sysem 4 sysem Cos Difference % Cos Difference % Number of Reailers Supplier Bach Size (a) (b) Figure 6: Average cos differences beween delay policy and no-delay policy: muli-echelon decenralized seing - 2 -

22 The impac of supplier order delay as m is varied is less clear. As can be seen in Figure 6(b), as m increases, he supplier s savings will firs decrease and hen increase. This can be explained as follows: As he number of sub-baches ges large (e.g., m=8), supplier s bach size will ge large. Thus, he supplier s cos will be dominaed by he holding cos due o his large order quaniy. Therefore, beer iming of when an order is placed becomes more criical and hence, he supplier s savings goes up. Similarly, when he supplier s bach size is small (e.g., m=), a more accurae iming of order placemen is criical in balancing he holding and he shorage coss Cenralized Seing We now consider he impac of he delayed order policy in cenralized disribuion sysems. In he muli-echelon disribuion sysem, all he reailers face Poisson demand, so i is opimal for hem no o delay orders. However, he order delay is opimal for he supplier, who faces an IFR iner-demand ime (see Proposiion and Theorem 4). In his secion, we analyze how he supplier delay as well as various sysem parameers affecs he oal sysem coss. Similar o he decenralized seing, we sudy a comprehensive se of parameers by fixing h=, L=, and varying he values of λ,, and π for he reailers, and he values of N, m, ˆL, and ĥ for he supplier. We choose no o presen he deailed resuls here, however, as he percenage difference of oal coss is minimal, and he overall average is small. We offer wo explanaions for his: Firs, in he cenralized disribuion sysem, reailers oal holding cos is a dominan componen of he oal cos, especially for large N and, and opimal delay order a he supplier has minimal impac on his holding cos. Second, as was shown in he previous secion, whenever he supplier inroduces a delay in he order, i may have a negaive impac on he reailers (sochasically increasing he reailer leadime) if no accompanied by an increase in he supplier reorder poin R. In many of he cases we considered, he delay is no accompanied by he increase in R. Therefore, he cos decrease a he supplier is offse by he cos increase a he reailers. As a whole, sysem cos

23 improves only very slighly. Noe ha his is also consisen wih our observaion in decenralized sysems where he oal sysem cos difference beween delayed order and (,R) policies was minimal. 5. Conclusion Tradiional invenory models assume eiher Poisson demands or ha orders can only be placed a demand arrival epochs. In his paper we sudy a general demand process for a single locaion invenory sysem, and show ha i is opimal o order a demand arrival epochs only if he inerdemand ime has consan or decreasing failure rae (CFR or DFR). We also show ha when he iner-demand ime has increasing failure rae (IFR), he opimal policy is o delay he order. This improves he classical (,R) policy o he (,R,T) policy where T represens he delay in ordering. We hen exend his policy o he decenralized muli-echelon disribuion seing. When he reailers face i.i.d. Poisson demands and use he (,R) ordering policy, heir orders o he supplier follow an Erlang process. We show ha he aggregaion of all hese order sreams also has IFR, herefore i is opimal for he supplier o use he (,R,T) policy for ordering. In he case of a cenralized muli-echelon disribuion sysem, we show ha when he wo echelons have idenical uni holding cos, i is again opimal for he supplier o delay is order in he (,R) framewor. In all hree seings we give explici expressions and procedures for he evaluaion of oal cos and he compuaion of opimal delay T. Using hese we carry ou comprehensive numerical ess and find ha for he single locaion model he opimal delay can reduce he oal cos significanly. The cos savings are he highes when leadime is shor and iner-demand ime is more variable (high CV). The cos savings, as a funcion of he desired service level, follow a cyclic paern. We provide an inuiive explanaion for such behavior. In he decenralized muli-echelon disribuion sysem, we also find ha he order delay can have significan impac on he reailers and he supplier separaely. We find parameer combinaions ha have he mos impac and provide inuiive explanaions. Given he decenralized seing, i is

24 obvious ha he supplier always benefis from he order delay. The impac on he reailer, however, is no immediaely clear. We show ha he supplier order delay can eiher benefi or hur he reailers, bu mosly he negaive impac is larger in magniude han he posiive ones. In he cenralized muli-echelon disribuion sysem, however, we find ha he order delay adoped by he supplier has minimal impac on he oal sysem cos. This has o do wih he fac ha holding cos a all he reailers end o dominae he oal cos and, for given, hey are no affeced by he order delay. Moreover, someimes he order delay ha benefis he supplier may have negaive impac on he reailers (his is made clearer in he decenralized cases), offseing he impac on he oal sysem cos. This (negaive) resul suggess ha he (,R) policies commonly used in disribuion invenory sysems, while heoreically subopimal, are very good numerically. To summarize, based on our analysis, he proposed delay policy has more impac on he average cos savings in single locaion seings wih more predicable iner-demand imes, shor leadimes and high service levels. In cenralized disribuion sysems, our findings indicae ha he policy has lile impac on oal sysem s cos. However, when he sysem is decenralized, hen he supplier can benefi by adoping he delay policy; his is usually achieved a reailer s expense. In addiion o condiions saed for he single locaion siuaion, his benefi is more significan in disribuion sysems wih few reailers. I is possible o analyze how he decenralized muli-echelon disribuion sysem can be coordinaed o perform lie he cenralized one. One possible mechanism for achieving he coordinaion is o impose an appropriae bacorder penaly cos on he supplier. Given ha he supplier s order delay resuls in a coninuous specrum of he service level on [0,), i is always possible o find he appropriae penaly cos. We do no sudy his issue in his paper as he coordinaion issue generally applies o he disribuion sysem, wih or wihou order delays, and is analysis exceeds he scope of his paper. I is cerainly a very ineresing and promising opic for fuure research. Anoher ineresing exension o his wor will be o consider he case when reailers are no idenical. As a resul, in such siuaions, reailers order quaniies will be differen. This

25 maes he analysis of he operaing characerisics of he sysem complicaed. In fac, here are only a few sudies ha deal wih such disribuion sysems even under he radiional (,R) policies (e.g., Forsberg 997, Axsaer 2000). References. Axsaer, S., Simple soluion procedures for a class of wo-echelon invenory problems, Operaions Research, 38 (990), Axsaer, S., Exac analysis of coninuous review (R,) policies in wo-echelon invenory sysems wih compound Poisson demand, Operaions Research, 48 (2000), Axsaer, S. and K. Rosling, Insallaion and echelon soc policies for mulilevel invenory conrol, Managemen Science, 39 (993), Barlow, R. and F. Proschan, Saisical heory of reliabiliy and life esing. Hol, Rinehar and Winson, Inc. (976). 5. Chen, F., Echelon Reorder Poins, Insallaion Reorder Poins, and he Value of Cenralized Demand Informaion, Managemen Science 44 (998), S22-S Clar, A. and H. Scarf, Opimal Policies for a Muli-Echelon Invenory Problem, Managemen Science, 6, (960). 7. Deurmeyer, B. and L. Schwarz, A model for he analysis of sysem service level in warehouse/reailer disribuion sysems: he idenical reailer case, L. Schwarz, ed. Mulilevel Producion/Invenory Conrol. Chaper 3 (TIMS Sudies in Managemen Science 6). Elsevier, New Yor (98). 8. Federgruen, A. and Y.S. Zheng, An Efficien Algorihm for Compuing an Opimal (r,) Policy for Coninuous Review Sochasic Invenory Sysems, Operaions Research, 40 (992), Forsberg, R. Exac evaluaion of (R,)-policies for wo-level invenory sysems wih Poisson demand, European Journal of Operaional Research, 96 (996),

26 0. Hadley, G. and T.M. Whiin, Analysis of invenory sysems. Prenice Hall (963).. Lee, H. and K. Moinzadeh, Two-parameer approximaions for muliechelon repairable invenory models wih bach ordering policy, IIE Transacions, 9 (987), Kaircioglu, K., Essays in invenory conrol. Ph.D. disseraion. The Universiy of Briish Columbia (996). 3. Moinzadeh, K and H. Lee, Bach size and socing levels in muliechelon repairable sysems, Managemen Science, 32 (986), Moinzadeh, K., An improved ordering policy for coninuous review invenory sysems wih arbirary iner-demand ime disribuions, IIE Transacions, 33 (200), Moinzadeh, K., A muli-echelon invenory sysem wih informaion exchange, Managemen Science, 48 (2002), Nahmias, S., On he Equivalence of Three Approximae Coninuous Review Invenory Model, Naval Research Logisics uarerly, 23 (976), Nahmias, S., Managing reparable Iem Invenory Sysems: A Review, in L.B. Schwarz (Ed.), Muli-Level Producion/Invenory Conrol Sysems: Theory and Pracice, TIMS Sudies in Managemen Sciences, 6, Norh Holland, Amserdam (98). 8. Sahin, I., On he Saionary Analysis of Coninuous Review (s,s) Invenory Sysems wih Consan Leadimes, Operaions Research, 27 (979), Sahin, I., On he Objecive Funcion Behavior in (s,s) Invenory Models, Operaions Research, 30 (982), Schulz, C.R., Replenishmen Delays For Expensive Slow-Moving Iems, Managemen Science, 35 (989), Sivlazian, B.D., A Coninuous Review (s,s) Invenory Sysem wih Arbirary Inerarrival Disribuion beween Uni Demand, Operaions Research, 22 (974), Svoronos, A. and P. Zipin, Esimaing he performance of muli-echelon invenory sysems, Operaions Research, 37 (988),

27 23. Tijms, H.C., Analysis of (s,s) Invenory Models, Mahemaical Cenre Trachs, 40, Mahemaical Cenrum, Amserdam (972). 24. Zheng, Y.S., On Properies of Sochasic Invenory Sysems, Managemen Science, 38 (992), Zipin, P., Sochasic Leadimes in Coninuous Review Invenory Sysems, Naval Research Logisics uarerly, 35 (986 a), Zipin, P., Invenory Service Level Measures: Convexiy and Approximaions, Managemen Science, 32 (986 b), Zipin. P., Foundaions of Invenory Managemen. The McGraw-Hill Companies (2000)