Forecasting method under the introduction. of a day of the week index to the daily. shipping data of sanitary materials

Transcription

1 Journal of Compuaions & Modelling, vol.7, no., 07, 5-68 ISSN: (prin), (online) Scienpress Ld, 07 Forecasing mehod under he inroducion of a day of he week index o he daily shipping daa of saniary maerials Komei Suzuki, Hiroake Yamashia and Kazuhiro Takeyasu Asrac Correc sales forecasing is indispensale o indusries. In indusries, how o improve forecasing accuracy such as sales, shipping is an imporan issue. There are many researches made on his. In his paper, we propose a new mehod o improve forecasing accuracy and confirm hem y he numerical example. Focusing ha he equaion of exponenial smoohing mehod(esm) is equivalen Graduae School of Humaniies and Social Sciences, Shizuoka Universiy. kome-komei@wind.ocn.ne.p College of Business Adminisraion and Informaion Science, Chuu Universiy. he-yama@isc.chuu.ac.p College of Business Adminisraion, Tokoha Universiy. akeyasu@f.okoha-u.ac.p Aricle Info: Received : Ocoer 7, 06. Revised : Decemer, 06. Pulished online : April 0, 07.

2 5 Forecasing mehod under he inroducion of a day of he week index o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod is proposed efore y us which saisfies minimum variance of forecasing error. Generally, smoohing consan is seleced arirarily. Bu in his paper, we uilize aove saed heoreical soluion. Firs, we make esimaion of ARMA model parameer and hen esimae smoohing consans, which is he heoreical soluion. Comining he rend removing mehod wih his mehod, we aim o improve forecasing accuracy. Furhermore, a day of he week index is newly inroduced for he daily daa and he forecasing is execued o he manufacurer s daa of saniary maerials. We have oained good resul. The effeciveness of his mehod should e examined in various cases. Mahemaics Suec Classificaion: 6J0 Keyword: Minimum Variance; Exponenial Smoohing Mehod; Forecasing, Trend; Saniary Maerials Inroducion Correc sales forecasing is indispensale o indusries. Poor sales forecasing accuracy leads o increased invenory and prolonged dwell ime of produc. In order o improve forecasing accuracy, we have devised rend removal mehods as well as searching opimal parameers and oained good resuls. We creaed a new mehod and applied i o various ime series and examined he effeciveness of he

3 K. Suzuki, H. Yamashia and K. Takeyasu 5 mehod. Applied daa are sales daa, producion daa, shipping daa, sock marke price daa, fligh passenger daa ec. Many mehods for ime series analysis have een presened such as Auoregressive model (AR Model), Auoregressive Moving Average Model (ARMA Model) and Exponenial Smoohing Mehod (ESM) (Box Jenkins []), (R.G.Brown[]), (Tokumaru e al.[]), (Koayashi[4]). Among hese, ESM is said o e a pracical simple mehod. For his mehod, various improving mehod such as adding compensaing iem for ime lag, coping wih he ime series wih rend (Peer [5]), uilizing Kalman Filer (Maeda [6]), Bayes Forecasing (M.Wes e al. [7]), adapive ESM (Seinar [8]), exponenially weighed Moving Averages wih irregular updaing periods (F.R.Johnson [9]), making averages of forecass using plural mehod (Spyros [0]) are presened. For example, Maeda[6] calculaed smoohing consan in relaionship wih S/N raio under he assumpion ha he oservaion noise was added o he sysem. Bu he had o calculae under supposed noise ecause he couldn grasp oservaion noise. I can e said ha i doesn pursue opimum soluion from he very daa hemselves which should e derived y hose esimaion. Ishii[] poined ou ha he opimal smoohing consan was he soluion of infinie order equaion, u he didn show analyical soluion. Based on hese facs, we proposed a new mehod of esimaion of smoohing consan in ESM efore (Takeyasu e al. []). Focusing ha he equaion of ESM is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in ESM was derived. In his paper, uilizing aove saed mehod, a revised forecasing mehod is proposed. a day of he week index (DWI) is newly inroduced for he daily daa and a day of he week rend is removed. Theoreical soluion of smoohing consan of ESM is calculaed for oh of he DWI rend removing daa and he non DWI rend removing daa. Then forecasing is execued o he manufacurer s daa of saniary maerials. This is a revised forecasing mehod. Variance of

4 54 Forecasing mehod under he inroducion of a day of he week index forecasing error of his newly proposed mehod is assumed o e less han hose of he previously proposed mehod. The res of he paper is organized as follows. In secion, ESM is saed y ARMA model and esimaion mehod of smoohing consan is derived using ARMA model idenificaion. The cominaion of linear and non-linear funcion is inroduced for rend removing in secion. a day of he week index (DWI) is newly inroduced in secion 4. Forecasing is execued in secion 5, and esimaion accuracy is examined. Descripion of ESM using ARMA model In ESM, forecasing a ime + is saed in he following equaion. Here, ˆ + x : forecasing a + xˆ + xˆ + α αx + ( x xˆ ) ( α ) xˆ () x : realized value a α : smoohing consan ( 0 < α < ) () is re-saed as: l 0 l ( α ) x l By he way, we consider he following (,) order ARMA model. Generally, ( p, q) order ARMA model is saed as: xˆ + α () x x e β e () Here, x + p q aix i e + i e (4)

5 K. Suzuki, H. Yamashia and K. Takeyasu 55 { }: x Sample process of Saionary Ergodic Gaussian Process ( ) { } e :Gaussian Whie Noise wih 0 mean σ variance MA process in (4) is supposed o saisfy converiiliy condiion. Uilizing he relaion ha: e x,,, N, we ge he following equaion from (). Operaing his scheme on +, we finally ge: If we se β α, he aove equaion is he same wih (), i.e., equaion of ESM is equivalen o (,) order ARMA model, or is said o e (0,,) order ARIMA model ecause s order AR parameer is (Box Jenkins []), (Tokumaru e al.[]). xˆ + [ e e, e, ] 0 Comparing wih () and (4), we oain: E xˆ β (5) xˆ xˆ x e + + ( β ) e ( β )( x xˆ ) (6) From (), (6), a β α β Therefore, we ge: a β α From aove, we can ge esimaion of smoohing consan afer we idenify he parameer of MA par of ARMA model. Bu, generally MA par of ARMA model ecome non-linear equaions which are descried elow. Le (4) e: (7)

6 56 Forecasing mehod under he inroducion of a day of he week index We express he auocorrelaion funcion of x~ as r k ~ and from (8), (9), we ge he following non-linear equaions which are well known []. For hese equaions, recursive algorihm has een developed. In his paper, parameer o e esimaed is only, so i can e solved in he following way. From () (4) (7) (0), we ge: If we se: he following equaion is derived. We can ge as follows. i p i i x a x x + ~ (8) q e e x + ~ (9) + + q e k k q e k r q k q k r ~ ) ( 0 ) ( ~ σ σ (0) ( ) 0 ~ ~ e e r r a q σ σ α β + () 0 ~ ~ r r k k ρ () + ρ () 4 ρ ρ ± (4)

7 K. Suzuki, H. Yamashia and K. Takeyasu 57 In order o have real roos, ρmus saisfy: From inveriiliy condiion, mus saisfy: ρ (5) From (), using he nex relaion, (5) always holds. As is wihin he range of: < ( ) 0 ( + ) 0 α + Finally we ge: < < 0 which saisfy aove condiion. Thus we can oain a heoreical soluion y a simple way. Here ρ mus saisfy: in order o saisfy 0 < α <. 4ρ ρ + ρ 4ρ α ρ Focusing on he idea ha he equaion of ESM is equivalen o (,) order ARMA model equaion, we can esimae smoohing consan afer esimaing ARMA model parameer. I can e esimaed only y calculaing 0h and s order auocorrelaion funcion. (6) 0 < ρ < (7)

8 58 Forecasing mehod under he inroducion of a day of he week index Trend removal mehod As rend removal mehod, we descrie he cominaion of linear and non-linear funcion. [] Linear funcion We se: as a linear funcion. [] Non-linear funcion We se: y a x + (8) y a x + + (9) x c y a x + x + + (0) cx d as a nd and a rd order non-linear funcion. [] The cominaion of linear and non-linear funcion We se: y α + α ( ax + ) + α ( ax + x + c ) ( a x + x + c x + d ) () 0 α, 0 α α + α + α α as he cominaion of linear and nd order non-linear and rd order non-linear funcion. Trend is removed y dividing he daa y ()., 0 () 4 A day of he week index a day of he week index (DWI) is newly inroduced for he daily daa of

9 K. Suzuki, H. Yamashia and K. Takeyasu 59 saniary maerials. The forecasing accuracy would e improved afer we idenify he a day of he week index and uilize hem. This ime in his paper, he daa we handle consis y Monday hrough Sunday, we calculae DWI (,,7) for Monday hrough Sunday. For example, if here is he daily daa of L weeks as saed ellow: { } ( i,, L) (,,7) x i where x i R in which L means he numer of weeks (Here L 0 ), i means he order of weeks ( i -h week), means he order in a week ( -h order in a week; for example : Monday, 7 : Sunday) and x i is Daily shipping daa of saniary maerials. Then, DWI is calculaed as follows. DWI L L 7 L i L x i 7 i x i () DWI rend removal is execued y dividing he daa y (). Numerical examples oh of DWI removal case and non-removal case are discussed in Secion 5. 5 Forecasing he saniary maerials daa 5. Analysis Procedure The shipping daa of cases from January, 0 o April, 0 are analyzed. Firs of all, graphical chars of hese ime series daa are exhiied in Figure 5- o 5-.

10 60 Forecasing mehod under he inroducion of a day of he week index Figure 5-: Daily Shipping Daa of Produc A Figure 5-: Daily Shipping Daa of Produc B Analysis procedure is as follows. There are 68 daily daa for each case. We use 49 daa( o 49) and remove rend y he mehod saed in Secion. Then we calculae a day of he week index (DWI) y he mehod saed in Secion 4. Afer removing DWI rend, he mehod saed in Secion is applied and Exponenial Smoohing Consan wih minimum variance of forecasing error is esimaed.

11 K. Suzuki, H. Yamashia and K. Takeyasu 6 Then sep forecas is execued. Thus, daa is shifed o nd o 50h and he forecas for 5s daa is execued consecuively, which finally reaches forecas of 6rd daa. To examine he accuracy of forecasing, variance of forecasing error is calculaed for he daa of 50h o 6rd daa. Final forecasing daa is oained y muliplying DWI and rend. Forecasing error is expressed as: ε xˆ x (4) i i i N i Variance of forecasing error is calculaed y: ε ε (5) In his paper, we examine he wo cases saed in Tale 5-. N i N σ ε i (6) N i ( ) ε ε Tale 5-: The cominaion of he case of rend removal and DWI rend removal Case Trend DWI rend Case Removal Removal Case Removal Non removal 5. Trend Removing Trend is removed y dividing original daa y (). Here, he weigh of α and α are shifed y 0.0 incremen in () which saisfy he equaion (). The es soluion is seleced which minimizes he variance of forecasing error. Esimaion resuls of coefficien of (8), (9) and (0) are exhiied in Tale 5-.

12 6 Forecasing mehod under he inroducion of a day of he week index Daa are fied o (8), (9) and (0), and using he leas square mehod, parameers of (8), (9) and (0) are esimaed. Esimaion resuls of weighs of () are exhiied in Tale 5-. The weighing parameers are seleced so as o minimize he variance of forecasing error. Tale 5-: Coefficien of (8),(9) and (0) s nd rd a a c a c d Produc A Produc B Tale 5-: Weighs of () Case α α α Produc A Produc B Case Case Case Case As a resul, we can oserve he following wo paerns. Seleced liner model: Produc A Case, Produc B Case, Produc B Case Seleced nd order model: Produc A Case

13 K. Suzuki, H. Yamashia and K. Takeyasu 6 Graphical chars of rend are exhiied in Figure 5- o 5-4. Figure 5-: Daily Shipping Daa of Produc A Figure 5-4: Daily Shipping Daa of Produc B 5. Removing Trend y DWI Afer removing rend, a day of he week index is calculaed y he mehod saed in 4. Calculaion resul for s o 49h daa is exhiied in Tale 5-4.

14 64 Forecasing mehod under he inroducion of a day of he week index Tale 5-4: a day of he week index Case a day of he week index Thu. Fri. Sa. Sun. Mon. Tue. Wed. Produc A Produc B Case Case Esimaion of Smoohing Consan wih Minimum Variance of Forecasing Error Afer removing DWI rend, Smoohing Consan wih minimum variance of forecasing error is esimaed uilizing (6). There are cases ha we canno oain a heoreical soluion ecause hey do no saisfy he condiion of (7). In hose cases, Smoohing Consan wih minimum variance of forecasing error is derived y shifing variale from 0.0 o 0.99 wih 0.0 inerval. Calculaion resul for s o 49h daa is exhiied in Tale 5-5. Tale 5-5: Esimaed Smoohing Consan wih Minimum Variance Case ρ α Produc A Produc B Case Case Case Case

15 K. Suzuki, H. Yamashia and K. Takeyasu Forecasing and Variance of Forecasing Error Uilizing smoohing consan esimaed in he previous secion, forecasing is execued for he daa of 50h o 6rd daa. Final forecasing daa is oained y muliplying DWI and rend. Variance of forecasing error is calculaed y (6). Forecasing resuls are exhiied in Figure 5-5 o 5-6. Figure 5-5: Forecasing Resuls of Produc 8 Figure 5-6: Forecasing Resuls of Produc B

16 66 Forecasing mehod under he inroducion of a day of he week index Variance of forecasing error is exhiied in Tale 5-6. Tale 5-6: Variance of Forecasing Error Case Variance of Forecasing Error Produc A Case 09 * Case 78 Produc B Case 98 * Case Conclusions Correc sales forecasing is indispensale o indusries. Focusing on he idea ha he equaion of exponenial smoohing mehod(esm) was equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod was proposed efore y us which saisfied minimum variance of forecasing error. Generally, smoohing consan was seleced arirarily. Bu in his paper, we uilized aove saed heoreical soluion. Firs, we made esimaion of ARMA model parameer and hen esimaed smoohing consans, which was he heoreical soluion. Furhermore, comining he rend removal mehod wih his mehod, we aimed o increase forecasing accuracy. An approach o his mehod was execued in he following mehod. Trend removal y a linear funcion was applied o he daily shipping daa of saniary maerials. The cominaion of linear and non-linear funcion was also inroduced in rend removing. a day of he week index (DWI) is newly inroduced for he daily daa and a day of he week rend is removed. Theoreical soluion of smoohing consan of ESM was calculaed for oh of he DWI rend removing daa and he non DWI rend removing daa. Then forecasing

17 K. Suzuki, H. Yamashia and K. Takeyasu 67 was execued on hese daa. Regarding oh daa, he forecasing accuracy of case (DWI is imedded) was eer han hose of case (DWI is no imedded). I can e said ha he inroducion of DWI has worked well. I is our fuure works o ascerain our newly proposed mehod in many oher cases. The effeciveness of his mehod should e examined in various cases. In he end, we appreciae Mr. Norio Funao for his helpful suppor of our sudy. References [] Box Jenkins, Time Series Analysis, Third Ediion, Prenice Hall, 994. [] R.G. Brown, Smoohing, Forecasing and Predicion of Discree Time Series, Prenice Hall, 96. [] Hidekasu Tokumaru e al., Analysis and Measuremen Theory and Applicaion of Random daa Handling, Baifukan Pulishing, 98. [4] Kengo Koayashi, Sales Forecasing for Budgeing, Chuokeizai-Sha Pulishing, 99. [5] Peer R.Winers, Forecasing Sales y Exponenially Weighed Moving Averages, Managemen Science, 6(), (984), 4-4. [6] Kasuro Maeda, Smoohing Consan of Exponenial Smoohing Mehod, Seikei Universiy Repor Faculy of Engineering, 8, (984), [7] M. Wes and P.J. Harrison, Baysian Forecasing and Dynamic Models, Springer-Verlag, New York, 989. [8] Seinar Ekern, Adapive Exponenial Smoohing Revisied, Journal of he Operaional Research Sociey,, (98), [9] F.R. Johnson, Exponenially Weighed Moving Average (EWMA) wih Irregular Updaing Periods, Journal of he Operaional Research Sociey, 44(7), (99), 7-76.

18 68 Forecasing mehod under he inroducion of a day of he week index [0] Spyros Makridakis and Roea L.Winkler, Averages of Forecass; Some Empirical Resuls, Managemen Science, 9(9), (98), [] Naohiro Ishii e al., Bilaeral Exponenial Smoohing of Time Series, In. J. Sysem Sci., (8), (99), [] Kazuhiro Takeyasu and Kazuko Nagao, Esimaion of Smoohing Consan of Minimum Variance and is Applicaion o Indusrial Daa, Indusrial Engineering and Managemen Sysems, 7(), (008),