Damped Trend forecasting and the Order-Up-To replenishment policy

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1 Li, Q. and Disney, S.M., (01), Damed Trend forecasing and he Order-U-To relenishmen olicy, Pre-rins of he 17 h Inernaional Working Seminar of Producion Economics, Innsbruck, Ausria, February 0 h 4 h, Vol. 3, Damed Trend forecasing and he Order-U-To relenishmen olicy Qinyun Li 1, Sehen M. Disney Cardiff Business School, Aberconway Building, Colum Drive, Cardiff, CF10 3EU, UK. 1 LiQY@cardiff.ac.uk, DisneySM@cardiff.ac.uk Absrac We develo a z-ransform ransfer funcion model of he Damed Trend forecasing mechanism from which we deermine is sabiliy boundary. We show ha he Damed Trend forecasing mechanism is sable for a much larger roorion of he aramerical sace han is currenly acknowledged in he lieraure. We incororae he Damed Trend forecasing mechanism ino an Order-U-To (OUT) relenishmen olicy and invesigae he frequency resonse of his sysem. We rove ha Naïve, Exonenial Smoohing and Hols forecass, when used wihin he OUT olicy, will always generae bullwhi, for every ossible demand rocess, for any lead-ime. However, he Damed Trend forecasing mechanism, when used wihin he OUT olicy, behaves differenly. Someimes i will generae bullwhi and someimes i will no. Bullwhi avoidance occurs when demand is dominaed by low frequencies in some insances. In oher insances bullwhi avoidance haens a high frequencies. We are also able o demonsrae a comlex odd-even lead-ime effec exiss. Bullwhi may be avoided when he lead-ime is odd for a aricular demand aern, bu re-aears when he lead-ime changes o an even number. Keywords: Damed Trend, Forecasing, Order-U-To Relenishmen Policy, Bullwhi, Sabiliy. 1. Inroducion The Damed Trend (DT) forecasing mehod develoed by [1] has ofen been romoed as he mos accurae forecasing echnique in he so-called M-comeiions []. [3] find he DT mehod is he bes mehod for 84% of he 3003 ime series in he M3 forecasing comeiion when using local iniial values. I was he bes mehod 70% of he ime when using global iniial values. [4] also conclude ha DT forecasing can reasonably claim o be a benchmark forecasing mehod for all ohers o bea. The grea virual of DT is ha fuure forecass are no simly fla line exensions of he curren, nex eriod forecas. I is able o deec and forecas rends and fuure forecass change wih hese rends. The DT forecasing mehodology also conains a leas eleven differen forecasing mehods when all arameers are seleced from he real 0,1 inerval, [3]. This makes i a owerful and very general forecasing aroach as uning he DT arameers effecively auomaes model selecion. The frequency resonse aroach ha we ake is aricularly owerful as we are able o generae resuls ha are alicable for any demand rocesses. This is because all demand rocesses can be decomosed ino a se of harmonic frequencies via he Fourier Transform. By undersanding how he sysem reacs o he comlee se of harmonic frequencies (via he Amliude Raio wihin he frequency resonse grah) we are han able o gain insighs and draw conclusions ha are valid for all ossible demand aern. Many of he resuls ha we obain are also valid for any lead-ime. Our findings srenghen, sharen and refine he argumens of [6]. In his aer we derive a discree-ime ransfer funcion of he DT mechanism in secion. In secion 3 we idenify he sabiliy boundaries of DT forecasing mechanism via Jury s Inners Aroach [5]. Secion 4 describes he relenishmen olicy used. We incororae he DT forecasing mehodology ino he Order-U-To (OUT) relenishmen olicy, develo a discree-ime z-ransform ransfer funcion reresenaion of he combined forecasing and relenishmen sysem and analyze is frequency resonse lo. Secion 5 rovides summary numerical resuls, confirming our heoreical findings. Secion 6 concludes.

2 . The Damed Trend forecasing mehod DT forecass [1] are generaed by aˆ ˆ ˆ 1 a 1b1d bˆ 1 ˆ 1 ˆ ˆ b a a 1. (1) k ˆ ˆ ˆ i d, k ab i1 Here d is he ime series being forecased. a is he curren esimae of he level, exonenially smoohed by he consan. b is he curren esimae of he rend, exonenially smoohed by he consan. b 0 is he iniial value of he rend, assumed o be zero, b0 0. is he daming arameer ha can be inerreed as a measure of he ersisence of he rend. k is he number of eriods ahead ha he forecas is required o redic. dˆ, k, is he forecas, made a ime of demand in he eriod k. Several well-known forecasing aroaches are encasulaed wihin he DT model. These include Hols mehod where here is no daming of he rend comonen when 1, Simle Exonenial Smoohing (SES) when 0 and Naïve forecasing when 1 and 0, see Table 1. Forecasing mehod Hols mehod Simle exonenial smoohing, SES Naïve forecasing Parameer seings 1 0 0, 1 Noes By seing 1, dˆ k aˆ kbˆ, resuls. The fuure forecass hen becomes a linear exraolaion of he curren esimae of he rend. 0 imlies ha b 0 0. I hen follows ha aˆ (1 ) aˆ 1 d. This in urn means dˆ, k aˆ. Here we have made exlici he fac ha he SES forecas of all fuure forecass ( k eriods ahead) is simly he forecas of he nex eriods demand. Ignoring he subscri ha gives informaion on which eriod we are forecasing yields he common SES formula, d ˆ (1 ) dˆ d. 1 This is easy o see from he exonenial smoohing formula as hese arameers yields d ˆ d., k Table 1. Three oular forecasing mehods encasulaed wih he Damed Trend mehod Transfer funcions are useful ools for sudying linear sysems, as hey allow convoluion in he ime domain o be relaced by simly algebra in he comlex frequency domain. In he frequency domain here is also a wide range of ools develoed by conrol engineering heoriss for undersanding he dynamic behaviour of such sysems. I is a relaively simle ask o develo a block diagram of (1) and maniulae i o obain he ransfer funcion of he DT forecasing mechanism (Figure 1). We refer ineresed readers o [7] for informaion on how o achieve his.

3 is he z-ransform of, is he z-ransform of Figure 1. Block diagram of Damed Trend forecasing The discree ime ransfer funcion of (1) is given by k1 k1 z Dˆ ( ) k z z ( z) 1 1 z 1 1 z 1, () n which, is in sandard form as coefficiens of he z-ransform oeraor, z xnz. We n0 noe from () ha he co-efficien of he highes ower of z in he denominaor is osiive when 1 and negaive when 1. This is imoran as Jury s Inners aroach, which will be exloied in he nex secion, requires his o be osiive [5]. 3. Sabiliy of Damed Trend forecass via Jury s Inners Aroach The quesion of sabiliy is a fundamenal asec of dynamic sysems. A sable sysem will reac o a finie inu and reurn o seady sae condiions in a finie ime. An unsable sysem will eiher diverge exonenially o osiive or negaive infiniy or oscillae wih ever increasing amliude. A criically sable sysem will fall ino a limi cycle of consan amliude o any finie inu. Oscillaions in he forecass and order raes in suly chains are cosly. So, as a firs se o dynamically designing a suly chain relenishmen rule, we mus ensure ha a relenishmen rule and all he comonens (such as he forecasing sysem) are sable. [5] shows ha he necessary and sufficien condiions for sabiliy of a linear discree sysem are given by: A ( 1) 0, ( 1) n A( 1) 0, and he marices n 1 n1 n are osiive innerwise. 1 For he sysem we are sudying here, A (z) is he denominaor of (), and he co-efficien of he highes ower of z in his case a, see (3) mus be osiive. This can be easily achieved by mulilying boh numeraor and denominaor by 1. However, i is ineresing o noe ha no maer wheher we need o do he mulilicaion or no, he values of he coefficiens in he denominaor A (z) always remain he same. Therefore, before using Jury s aroach, we rewrie he denominaor of () by subsiuing 1 wih is absolue value 1 o simlify fuure analysis. Then, A (z) can be exressed as Az ( ) 1(1 ) z1( 1 ) z 1, a0 a1 a (3) and n 1 are simly scalars as he original ransfer funcion () is only of second order ( n ). Secifically n 1 are 3

4 n 1 a a0 1 1(1 ), n 1 a a0 1 1(1 ). (4) Taking each crieria in urn: A (1) mus be greaer han zero: A ( 1) A( z), ha is A (1) is given by (3) wih he z is z 1 relaced wih 1, (5) slis he A(1) 1 1 ( 1) 0. (5), aramerical lane ino quarers along he lines given by 0 and ( 1). ( 1) n A( 1) 0 mus be greaer han zero. In he same manner as above, ( 1) n A( 1 ) given by (3) wih he z is relaced by 1 and n, is (6) divides he ( 1) n A( 1) 1 0 (6), aramerical lane along he curve ( ), which has an asymoe a 0. n mus be osiive innerwise. A marix is osiive innerwise if is deerminan is osiive and all he deerminans of is Inners are also osiive. Because he order of he ransfer funcion n, hen he n 1 marices only conain one elemen [8]. To ensure ha he elemens are osiive innerwise, i is enough ha 1 n1 n1 n 1 a a n 1 a a (7) The crieria n 1 divides he arameric lane along 1 arameric lane along 1., 1 n divides he Figure rovides a conceual ma of he sabiliy boundary and how i changes for differen. I is common racice in exonenial smoohing models o resric he smoohing arameers o he 0,1 inerval [9],[10]. A series of aers [1],[11] have also roosed he daming arameer is resriced o 0 1. However, i is ineresing o noe ha here are sable DT forecass for a much broader range of arameer values han hose usually recommended in he lieraure. Similar findings were observed for Hols mehod and SES. When 1, when we have he Hols mehod, he sabiliy condiions are 0, 0 (4 ). When 0, he SES sabiliy boundary can be observed, Using Damed Trend Forecasing wihin he Order-U-To Policy A single reailer firs receives goods in each eriod. He observes and saisfies cusomer demand wihin he relenishmen eriod, d. Any unfilled demand is backlogged. The reailer observes his invenory level and laces a relenishmen order, o, a he end of each eriod. There is a fixed ime eriod of T beween lacing an order and receiving ha order in sock. We assume ha he reailer follows a simle Order-U-To invenory olicy. In an OUT olicy, orders are laced o raise he invenory osiion i u o an OUT level or base sock level s, 4

5 Figure. The Damed Trend Sabiliy region o s i. (8) The invenory osiion is he amoun of invenory on-hand invenory on-order backlog. The amoun of invenory on-hand minus he backlog is he ne sock ns level. The invenory on-order is also known as he Work-In-Progress (WIP), wi. The invenory osiion a ime eriod, i is given by i ns wi. (9) The OUT level is ofen esimaed from he observed demand. I can be wrien as 5

6 s ˆ ˆ nsd, T 1 d, i, i1 T dwi (10) where d ˆ, T 1 is he forecased demand in eriod T 1 made in eriod. The Targe Ne Sock, ns, is a safey sock used o ensure a sraegic level of invenory availabiliy. ns is a ime invarian consan. Under he assumions of normally disribued forecas errors and iece-wise linear convex invenory holding (h) and backlog coss (b) hen i is common o 1 b assume ns z ns; z b h. Here ns is he sandard deviaion of he ne sock levels and 1 x is he inverse of he cumulaive normal disribuion funcion evaluaed a x. The ime T varying Desired Work In Progress, dwi ˆ d i1, i is he sum of he forecass, made a ime in he eriods from 1 o T. [1] show he order decision can be rewrien as o ˆ ns d, T 1 dwi wi ns s i T T. (11) ns dˆ dˆ o ns s s d, T 1, i i 1 i1 i1 s wi The z-ransform ransfer funcion for he order rae, exressed in a manner in which he forecasing sysem has been lef unsecified, is given by ˆ Oz ( ) D 1 T 1( z) DWIPz ( ) (1 z ) 1. ( z) ( z) ( z) s (1) (1) is a useful dearure oin for furher analysis as he forecasing comonens can be simly sloed ino Dˆ T 1 z and DWIP z o yield he sysem ransfer funcion. We noice in (11) and ha he OUT olicy requires wo forecass. One of hese forecass is a redicion, made a ime of he demand in he eriod T 1. Adaing he DT forecas o achieve his is done wih T 1 T 1 ˆ ˆ i ˆ ( 1) d ˆ ˆ, T 1 a b a b. (13) 1 i1 The oher forecas required by he OUT olicy is a redicion, made a ime, of demand over he lead-ime. Tha is, he demand in eriods 1,,..., T. In he ime domain his is T T i 1 ˆ ˆ ˆ ( 1) T T dwi ˆ ˆ ˆ d d, 1, T, i at b at b i1 i1 1 1 T 1. (14) Figure 3 shows ha he ransfer funcions of he DT forecas and WIP arge can be buil u from auxiliary variables, b z. These are az and 6

7 1 1 D(Demand) DWIP 1 O(Order) R (Receis) NS (Ne sock) WIP 1 1 TNS Invenory Posiion + + Figure 3. Block diagram of OUT olicy wih Damed Trend forecass a z b z z z 1 1 z 1 1 z1z 1 z 1 1 z z The z-ransforms of he wo DT forecass required by he OUT olicy are 1 z T Dˆ z z1 z 1 1 T z z z. (15) T DWIP z z z T z z z z 1 1 1, (16) The ransfer funcions of he wo required forecass when Hols Mehod is used are 1 z z T T z 1 z z. (17) Dˆ T z z 1z. (18) DWIP( z) z T 1T zt 1T ( z) 1 1 As he rend comonen of he demand rocess is no exlicily forecased in he SES and he Naïve forecasing models, he majoriy of scholars consider he DWIP erm o be simly he roduc of he lead-ime and he mos recen forecas [6], [13]. The ransfer funcions of D ˆ 1 T 7

8 and DWIP for SES are hen 1 Dˆ T z z DWIP( z) zt ;. (19) z z 1 ( z) z 1 The ransfer funcions of Dˆ 1 and DWIP and Naïve forecasing mehods are T 1 z Dˆ T z DWIP( z) 1; T. (0) ( z) Once we have he ransfer funcions of hese wo forecass, we can subsiue hem ino (1) o obain he order rae ransfer funcion. The order rae ransfer funcion is imoran as i conains informaion on he well-known bullwhi effec. The bullwhi effec is resen if he variance of he orders is greaer han he variance of he demand. If cusomer demand is indeendenly and idenically disribued (i.i.d.) hen he following relaionshi holds, Bullwhi o Z O z o 0 0 d 1. (1) 1 In (1) Z x is he inverse z-ransform oeraor. Via Parseval s heorem, we can make he link beween he bullwhi effec and he frequency resonse ( i ) ( i o ) ( i Oe d Oe Oe ) d. () Amliude Raio, AR Here Oz ( ) is he z-ransform of o and reresens he angular frequency of o. If we invesigae he Amliude Raio (AR) of differen frequencies we are able o gain insigh ino how he forecasing and relenishmen sysem behaves o any demand aern. This is because all demand aerns can be decomosed ino a se of harmonic frequencies. i.i.d. demands mean ha all frequencies are resen wih equal densiy in he demand signal. The frequency resonse of discree ime sysems is a funcion wih a eriodiciy of. However, we only need o sudy he AR for frequencies in he eriod 0,, as he frequency resonse lo on,0 is a simle reflecion of 0, abou he origin. Furhermore we noe ha AR 1 0 and d AR d 0 for all sysems. Le s now ake a look a he frequency resonse for he OUT 0 olicy wih differen forecasing mechanisms. 4.1 Frequency Resonse of he OUT Policy wih Naïve Forecass. The AR is sricly dar increasing in as d T 3T sin, AR 1 and AR 0 3 T wihin he inerval 0,, see Figure 4a. 4. Frequency Resonse of he OUT Policy wih SES Forecass. For sable SES forecass (Figure 4b) he AR is a sricly increasing wihin he frequency inerval 0, as dar d T T 3 1 sin cos cos 0. (3) 8

9 Togeher wih AR 1 we can deduce ha he value of AR is always greaer han 1 for all 1 frequencies. In anoher words, he OUT olicy wih SES forecasing will always roduce bullwhi for all demand aerns for all lead-imes. This finding is consisen wih he resuls in [6], bu hey failed o comleely characerise he frequency resonse. 4.3 Frequency Resonse of he OUT Policy wih Hols forecasing. For sable Hols forecass he AR originaes a 1 and ends a 0 AR 4 T T AR 4 1, (4) see Figure 4c. However, in beween hese wo oins here are wo differen AR resonses. Eiher he AR is sricly increasing in or, when 4 ( ), here is a saionary oin wihin he 0, inerval. Then he AR is an increasing funcion in unil arccos a which oin i becomes a decreasing funcion unil =. The saionary oin, if i exiss, will be a maximum. Using hese facs we are able o rove ha he Order-U-To wih Naïve, SES and Hols forecass will, for any demand aerns and all lead-imes, always generae bullwhi. 4.4 Frequency Resonse of he OUT Policy wih Damed Trend Forecasing. The DT frequency resonse is more comlex han hose reviously considered. We firs invesigae wo siuaions: low-frequency resonses ( near 0) and high-frequency resonses ( near ). We hen ay aenion o frequencies beween 0 and. Alhough AR 0 1 and d AR d 1 he second derivae can be osiive, zero or negaive. The sign of he second 0 derivae has geomerical imlicaions. If he second derivae is osiive, he grah of AR will be convex near 0 wih a local minimum a 0. If he second derivae is negaive, he AR curve will be concave near he origin and he oin a 0 is a local maximum. A concave AR will imly ha he DT forecas enabled OUT olicy will be able o avoid generaing bullwhi for low frequency demand. The lowes-order non-zero derivaive is always of even order for any of he DT seings. This means ha a saionary oin a 0 canno be an inflecion oin when he second derivae is zero - i has o be eiher a local maximum or a local minimum. This fac also concurs wih common knowledge of he eriodiciy of he frequency resonse. Figure 4. Frequency resonse of he OUT olicy wih (a) Naïve, (b) SES and (c) Hols Mehod forecasing 9

10 d AR Consider low frequency behaviour when T 1. Wih 1 1, 0. This means d 0 ha hese seings will always generae bullwhi when is near 0. When 1, he AR near 0 is always concave, imlying bullwhi is avoided a low frequencies. However when 1, he second derivae can be osiive, negaive or zero. Figure 5 mas ou he areas of he arameric lane where he bullwhi effec can be avoided when he demand rocess conains low frequency harmonics. The curves which searae ou he differen classes of 1 bullwhi behaviour for when 3 1 are and, 1. When he, and, 1 deermine he differen classes of bullwhi behaviour. d AR These were all obained by seing 0 and solving for he relevan variables. d Consider high frequency bullwhi behaviour near when T 1. DT forecass wih 1 or 3 always generae bullwhi for high-frequency demands as AR 1. If 1, hen AR 1. There are also some circumsances ha he AR 1, see Figure 6. When AR 1 he DT enabled OUT olicy avoids inducing he bullwhi when he demand rocesses conain only high frequency harmonics. This bullwhi avoidance occurs for: when and ; 1 0 when 0 and ; Figure 5. Concaviy of AR near 0 for T 1 10

11 3 1 when 1 and When he lead-ime increases, for low frequencies near 0 and 1 he influence of he arameers seings (Figure 5a) remains exacly he same. Tha is AR < 1 near 0. For high frequency demand near when 0 1 o he area of he aramerical lane ha is able o avoid he bullwhi effec will become smaller. When 1 0, he region wihin he aramerical lane where bullwhi is aenuaed changes in a comlex manner. I has differen shaes when he lead-ime changes from an odd number o an even number. When 1, bullwhi avoiding areas of he aramerical lane for boh low-frequency and high frequency demand will disaear and reaear in sohisicaed manners when he lead-ime swiches beween an odd number and an even number. So far we have no been able o deermine he characerisics of he wo saionary oins wihin he inerval 0,. However he resuls ha we have obained a 0, indicaes ha for some demand aerns he OUT olicy wih DT forecasing mechanism is able o avoid he bullwhi effec. This is a ye of dynamic behaviour ha is no resen when he Naïve, SES and Hols Mehod is used as a forecasing mehod wihin he OUT olicy. 5. Numerical verificaion We consruced an Excel based simulaion of he OUT olicy wih DT forecasing and uni lead-imes. Demand was assumed o be made u of a single sine wave wih a mean of 10, uni amliude and a frequency of 0.0,3.1 radians er eriod. We deermined he bullwhi o ns and ne sock variance amlificaion raio, NSAm = from 4000 eriods afer an d iniialisaion eriod of 1000 eriods. Samle numerical resuls are given below in Table. They verify ha he OUT olicy wih DT forecasing can indeed eliminae he bullwhi effec. I is ineresing o noe ha whils we have no sudied he NSAm measure from a heoreical sandoin in his aer, he numerical resuls in Table hin a he ossibiliy ha no only is bullwhi reduced wih DT forecasing, bu here aears o be good conrol over invenory levels as well. 6. Concluding remarks We have sudied he sabiliy and bullwhi behaviour of an OUT olicy ha incororaes DT forecasing. We have demonsraed ha he DT forecass are sable over a much broader range of arameer values han is usually recommended in he lieraure. We have shown ha he OUT wih hree differen yes of forecass, Naïve, SES and he Hols mehod forecass, will d Figure 6. Possible seings ha resul in AR 1 near when T 1. 11

12 Demand Frequency (radians er eriod) Bullwhi o d ns d NSAm Table. Numerical resuls from a 4000 eriod simulaion verifying our heoreical resuls always generae bullwhi, for any demand aerns and for all lead-imes. For he DT forecasing mehod, we have shown ha for some demand aerns he OUT relenishmen olicy wih DT forecasing mechanism is able o avoid generaing bullwhi. This is a qualiaively differen bullwhi behaviour ha is no resen wih oher, more radiional forecasing olicies. This suggess ha he DT forecasing mehodology deserves much more aenion in he OR/OM lieraure han i currenly receives. 6. References [1] Gardner, E.S. Jr, McKenzie, E., Forecasing rends in ime series. Managemen Science, 31 (10), [] Makridakis, S., Hibon, M., 000. The M3-comeiion: resuls, conclusions and imlicaions. Inernaional Journal of Forecasing, 16 (4), [3] Gardner, E.S. Jr, McKenzie, E., 011. Why he damed rend works. Journal of he Oeraional Research Sociey, 6 (6), [4] Fildes, R., Nikoloous, K., Crone, S., Syneos, A., 008. Forecasing and oeraional research: a review. Journal of he Oeraional Research Sociey, 59 (9), 1-3. [5] Jury, E.I., Inners aroach o some roblems of sysem heory. IEEE Transacions on Auomaic Conrol, 16 (3), [6] Dejonckheere, J., Disney, S.M., Lambrech, M.R., Towill, D.R., 003. Measuring and avoiding he bullwhi effec: a conrol heoreic aroach. Euroean Journal of Oeraional Research, 147 (3), [7] Nise, N.S., 004. Conrol sysems Engineering 4 h ed. Hobeken, New Jersey: John Wiley & Sons. [8] Disney, S.M., 008. Suly chain aeriodiciy, bullwhi and sabiliy analysis wih Jury s inners. IMA Journal of Managemen Mahemaics, 19 (), [9] Winers, P.R., Forecasing sales by weighed moving averages. Managemen Science, 6 (3), [10] Hol, C.C., 004. Forecasing seasonals and rends by exonenial weighed moving averages. Inernaional Journal of Forecasing, 0 (1), [11] Gardner, E.S. Jr, McKenzie, E., Seasonal exonenial smoohing wih damed rends. Managemen Science, 35 (3), [1] Hosoda, T., Disney, S.M., 006. On variance amlificaion in a hree-echelon suly chain wih minimum mean squared error forecasing. OMEGA: The Inernaional Journal of Managemen Science, 34 (4), [13] Chen, F., Ryan, J.K., Simchi-Levi, D., 000. The Imac of exonenial smoohing forecass on he bullwhi effec. Naval Research Logisics, 47 (4),