A Hybrid Method for Forecasting with an Introduction of a Day of the Week Index to the Daily Shipping Data of Sanitary Materials

Transcription

1 Journal of Communcaon and Compuer (05) 0-07 do: 0.765/ / D DAVID PUBLISHING A Hyrd Mehod for Forecasng wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals Dasuke Takeyasu, Hroake Yamasha and Kazuhro Takeyasu. Graduae School of Culure and Scence, he Open Unversy of Japan, - Wakaa, Mhama-Dsrc, Cha Cy , Japan. College of Busness Admnsraon and Informaon Scence, Chuu Unversy, 00 Masumoo-cho Kasuga, Ach , Japan. College of Busness Admnsraon, Tokoha Unversy 5 Oouch, Fuj Cy, Shzuoka , Japan Asrac: Correc sales forecasng s nevale n ndusres. In ndusres, how o mprove forecasng accuracy such as sales, shppng s an mporan ssue. There are many researches made on hs. In hs paper, we propose a new mehod o mprove forecasng accuracy and confrm hem y he numercal eample. Focusng ha he equaon of ESM (eponenal smoohng mehod) s equvalen o (,) order ARMA model equaon, a new mehod of esmaon of smoohng consan n eponenal smoohng mehod s proposed efore y us whch sasfes mnmum varance of forecasng error. Generally, smoohng consan s seleced arrarly. Bu n hs paper, we ulze aove saed heorecal soluon. Frsly, we make esmaon of ARMA model parameer and hen esmae smoohng consans. Thus heorecal soluon s derved n a smple way and may e ulzed n varous felds. Comnng he rend removng mehod wh hs mehod, we am o mprove forecasng accuracy. Furhermore, a day of he week nde s newly nroduced for he daly daa and he forecasng s eecued o he manufacurer s daa of sanary maerals. We have oaned good resul. The effecveness of hs mehod should e eamned n varous cases. Key word: Mnmum varance, eponenal smoohng mehod, forecasng, rend, sanary maerals.. Inroducon Correc sales forecasng s nevale n ndusres. Poor sales forecasng accuracy leads o ncreased nvenory and prolonged dwell me of produc. In order o mprove forecasng accuracy, we have devsed rend removal mehods as well as searchng opmal parameers and oaned good resuls. We creaed a new mehod and appled o varous me seres and eamned he effecveness of he mehod. Appled daa are sales daa, producon daa, shppng daa, sock marke prce daa, flgh passenger daa ec. Many mehods for me seres analyss have een Correspondng auhor: Hroake Yamasha, professor/mba, research felds: supply chan managemen. E-mal: hr-yama@sc.chuu.ac.jp. presened such as AR Model (auoregressve model), ARMA Model (auoregressve movng average Model) and ESM (eponenal smoohng mehod) [-4]. Among hese, ESM s sad o e a praccal smple mehod. For hs mehod, varous mprovng mehod such as addng compensang em for me lag, copng wh he me seres wh rend [5], ulzng Kalman Fler [6], Bayes Forecasng [7], adapve ESM [8], eponenally weghed Movng Averages wh rregular updang perods [9], makng averages of forecass usng plural mehod [0] are presened. For eample, Ref. [6] calculaed smoohng consan n relaonshp wh S/N rao under he assumpon ha he oservaon nose was added o he sysem. Bu he had o calculae under supposed nose ecause he

2 0 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals couldn grasp oservaon nose. I can e sad ha does no pursue opmum soluon from he very daa hemselves whch should e derved y hose esmaon. Ref. [] poned ou ha he opmal smoohng consan was he soluon of nfne order equaon, u he dd no show analycal soluon. Based on hese facs, we proposed a new mehod of esmaon of smoohng consan n ESM efore Ref. []. Focusng ha he equaon of ESM s equvalen o (,) order ARMA model equaon, a new mehod of esmaon of smoohng consan n ESM was derved. l0 l l ˆ () In hs paper, ulzng aove saed mehod, revsed forecasng mehod s proposed. In makng forecas such as shppng daa, rend removng mehod s devsed. Trend removng y he comnaon of lnear and nd order non-lnear funcon and rd order non-lnear funcon s eecued o he manufacurer s daa of sanary maerals. The weghs for hese funcons are vared y 0.0 ncremen and opmal weghs are searched. A DWI (day of he week nde) s newly nroduced for he daly daa and a day of he week rend s removed. Theorecal soluon of smoohng consan of ESM s calculaed for oh of he DWI rend removng daa and he non DWI rend removng daa. Then forecasng s eecued on hese daa. Ths s a revsed forecasng mehod. Varance of forecasng error of hs newly proposed mehod s assumed o e less han hose of he prevously proposed mehod. The res of he paper s organzed as follows: In Secon, ESM s saed y ARMA model and esmaon mehod of smoohng consan s derved usng ARMA model denfcaon; The comnaon of lnear and non-lnear funcon s nroduced for rend removng n Secon ; DWI s newly nroduced n Secon 4; Forecasng s eecued n Secon 5, and esmaon accuracy s eamned.. Descrpon of ESM Usng ARMA Model In ESM, forecasng a me + s saed n he followng equaon. ˆ ˆ ˆ () ˆ Here, ˆ : forecasng a ; : realzed value a ; : smoohng consan 0 ; Eq. () s re-saed as: By he way, we consder he followng (,) order ARMA model. e e () Generally, p, q order ARMA model s saed as: p q a e j e j j (4) Here, : Sample process of Saonary Ergodc Gaussan Process,,, N, ; : Gaussan Whe Nose wh 0 mean e varance. e MA process n Eq. (4) s supposed o sasfy converly condon. Ulzng he relaon ha: E e e, e, 0 We ge he followng equaon from Eq. (). ˆ e (5) Operang hs scheme on +, we fnally ge: ˆ ˆ ˆ e ˆ (6) If we se, he aove equaon s he same wh (),.e., equaon of ESM s equvalen o (,) order ARMA model, or s sad o e (0,,) order ARIMA model ecause s order AR parameer s ( [, ]).

3 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals 0 Comparng wh Eq. () and Eq. (4), we oan: a From Eqs. () and (6), Therefore, we ge: a (7) From aove, we can ge esmaon of smoohng consan afer we denfy he parameer of MA par of ARMA model. Bu, generally MA par of ARMA model ecome non-lnear equaons whch are descred elow. Le Eq. (4) e: p ~ a (8) ~ e q j e j j (9) We epress he auocorrelaon funcon of ~ as ~ r k and from Eqs. (8) and (9), we ge he followng non-lnear equaons whch are well known []. ~ r k ~ r 0 0 qk e j0 q e j0 j k j j ( k q) ( k q ) (0) For hese equaons, recursve algorhm has een developed. In hs paper, parameer o e esmaed s only, so can e solved n he followng way. From Eqs. (), (4), (7) and (0), we ge: If we se: q a ~ r0 e ~ r e 0 () ~ r ~ k k r () he followng equaon s derved. () We can ge as follows. 4 (4) In order o have real roos, mus sasfy: (5) From nverly condon, mus sasfy: From Eq. (), usng he ne relaon, 0 0 Eq. (5) always holds. As s whn he range of: Fnally we ge: (6) whch sasfy aove condon. Thus we can oan a heorecal soluon y a smple way. Here mus sasfy: 0 (7) n order o sasfy 0. Focusng on he dea ha he equaon of ESM s equvalen o (, ) order ARMA model equaon, we can esmae smoohng consan afer esmang ARMA model parameer. I can e esmaed only y calculang 0h and s order auocorrelaon funcon.

4 04 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals. Trend Removal Mehod As rend removal mehod, we descre he comnaon of lnear and non-lnear funcon. () Lnear funcon We se: y a (8) as a lnear funcon. () Non-lnear funcon We se: y y a c (9) a c d (0) as a nd and a rd order non-lnear funcon. () The comnaon of lnear and non-lnear funcon We se: y a a c a c d () 0, 0, 0 () as he comnaon of lnear and nd order non-lnear and rd order non-lnear funcon. Trend s removed y dvdng he daa y Eq. (). ( -h week), j means he order n a week ( j -h order n a week; for eample j : Monday, j 7 : Sunday) and j s daly shppng daa of sanary maerals. Then, DWI j s calculaed as follows: DWI j L L 7 L L j 7 j j () DWI rend removal s eecued y dvdng he daa y Eq. (). Numercal eamples oh of DWI removal case and non-removal case are dscussed n Secon Forecasng he Sanary Maerals Daa 5.. Analyss Procedure The shppng daa of cases from January, 0 o Aprl 7, 0 are analyzed. Frs of all, graphcal chars of hese me seres daa are ehed n Fgs. and. 4. A Day of he Week Inde DWI s newly nroduced for he daly daa of sanary maerals. The forecasng accuracy would e mproved afer we denfy he a day of he week nde and ulze hem. Ths me n hs paper, he daa we handle conss y Monday hrough Sunday, we calculae DWI j j,,7 for Monday hrough Sunday. For eample, f here s he daly daa of L weeks as saed ellow:,, L j,,7 j Where, j R n whch L means he numer of weeks (Here L 0 ), means he order of weeks Fg. Daly shppng daa of produc A. Fg. Daly shppng daa of produc B.

5 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals 05 Analyss procedure s as follows. There are 6 daly daa for each case. We use 49 daa ( o 49) and remove rend y he mehod saed n Secon. Then we calculae a DWI y he mehod saed n Secon 4. Afer removng DWI rend, he mehod saed n Secon s appled and Eponenal Smoohng Consan wh mnmum varance of forecasng error s esmaed. Then sep forecas s eecued. Thus, daa s shfed o nd o 50h and he forecas for 5s daa s eecued consecuvely, whch fnally reaches forecas of 6rd daa. To eamne he accuracy of forecasng, varance of forecasng error s calculaed for he daa of 50h o 6rd daa. Fnal forecasng daa s oaned y mulplyng DWI and rend. Forecasng error s epressed as: ˆ N N Varance of forecasng error s calculaed y: (4) (5) N N (6) In hs paper, we eamne he wo cases saed n Tale. 5.. Trend Removng Trend s removed y dvdng orgnal daa y Eq. (). Here, he wegh of and are shfed y 0.0 ncremen n Eq. () whch sasfy he equaon Eq. (). The es soluon s seleced whch mnmzes he varance of forecasng error. Esmaon resuls of coeffcen of Eqs. (8)-(0) are ehed n Tale. Daa are fed o Eqs. (8)-(0), and usng he leas square mehod, parameers of Eqs. (8)-(0) are esmaed. Esmaon resuls of weghs of Eq. () are ehed n Tale. The weghng parameer As a resul, we can oserve he followng hree paerns. () Seleced lner model: Produc A Case, Produc B Case () Seleced nd order model: Produc B Case, () Seleced rd order model: Produc A Case Graphcal chars of rend are ehed n Fgs. and Removng Trend y DWI Afer removng rend, a day of he week nde s calculaed y he mehod saed n Secon 4. Calculaon resul for s o 49h daa s ehed n Tale 4. Tale The comnaon of he case of rend removal and DWI rend removal. Case Trend DWI rend Case Removal Removal Case Removal Non removal Tale Coeffcen of Eqs. (8)-(0). a s nd rd a c a a c Produc A Produc B d Tale Weghs of Eq. (). Case Case Produc A Case Produc B Case Case

6 06 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals s o 49h daa s ehed n Tale Forecasng and Varance of Forecasng Error Fg. Daly shppng daa of produc A. Ulzng smoohng consan esmaed n he prevous secon, forecasng s eecued for he daa of 50h o 6rd daa. Fnal forecasng daa s oaned y mulplyng DWI and rend. Varance of forecasng error s calculaed y Eq. (6). Forecasng resuls are ehed n Fgs. 5 and 6 Varance of forecasng error s ehed n Tale 6. Fg. 4 Daly shppng daa of produc B. 5.4 Esmaon of Smoohng Consan wh Mnmum Varance of Forecasng Error Afer removng DWI rend, Smoohng Consan wh mnmum varance of forecasng error s esmaed ulzng Eq. (6). There are cases ha we canno oan a heorecal soluon ecause hey do no sasfy he condon of Eq. (7). In hose cases, Smoohng Consan wh mnmum varance of forecasng error s derved y shfng varale from 0.0 o 0.99 wh 0.0 nerval. Calculaon resul for Tale 4 Arlne A day of he week nde. Case 6. Conclusons Correc sales forecasng s nevale n ndusres. Focusng on he dea ha he equaon of ESM was equvalen o (,) order ARMA model equaon, a new mehod of esmaon of smoohng consan n eponenal smoohng mehod was proposed efore y us whch sasfed mnmum varance of forecasng error. Generally, smoohng consan was seleced arrarly. Bu n hs paper, we ulzed aove saed heorecal soluon. Frsly, we made esmaon of ARMA model parameer and hen esmaed smoohng consans. Thus heorecal soluon was derved n a smple way and mgh e ulzed n varous felds. Furhermore, comnng he rend removal mehod wh hs mehod, we amed o ncrease forecasng accuracy. An approach o hs mehod was eecued n he followng mehod. Trend removal y a lnear funcon was appled o he daly shppng daa of sanary maerals. The comnaon of lnear and A day of he week nde Thu. Fr. Sa. Sun. Mon. Tue. Wed. A Case B Case Tale 5 Produc A Produc B Esmaed smoohng consan wh mnmum varance. Case Case Case Case Case

7 A Hyrd Mehod for Forecasng Wh an Inroducon of a Day of he Week Inde o he Daly Shppng Daa of Sanary Maerals 07 case (DWI s no medded). I can e sad ha he nroducon of DWI has worked well. I s our fuure works o asceran our newly proposed mehod n many oher cases. The effecveness of hs mehod should e eamned n varous cases. In he end, we apprecae Mr. Noro Funao for hs helpful suppor of our sudy. References Fg. 5 Forecasng resuls of produc A. Fg. 6 Forecasng resuls of produc B. Tale 6 Produc A Produc B Varance of forecasng error. Case Varance of Forecasng Error Case 9, * Case 5,9. Case 49,54.45 * Case 755,7.7 non-lnear funcon was also nroduced n rend removng. DWI s newly nroduced for he daly daa and a day of he week rend s removed. Theorecal soluon of smoohng consan of ESM was calculaed for oh of he DWI rend removng daa and he non DWI rend removng daa. Then forecasng was eecued on hese daa. Regardng oh daa, he forecasng accuracy of case (DWI s medded) was eer han hose of [] Jenkns, B Tme Seres Analyss Thrd Edon. Prence Hall. [] Brown, R. G. 96. Smoohng, Forecasng and Predcon of Dscree -Tme Seres. Prence Hall. [] Tokumaru, H. 98. Analyss and Measuremen-Theory and Applcaon of Random Daa Handlng. Bafukan Pulshng. [4] Koayash, K. 99. Sales Forecasng for Budgeng. Chuokeza-Sha Pulshng. [5] Wners, P. R Forecasng Sales y Eponenally Weghed Movng Averages. Managemen Scence 6: 4-4. [6] Maeda, K Smoohng Consan of Eponenal Smoohng Mehod. Seke Unversy Repor Faculy of Engneerng: [7] Wes, M., and Harrson, P. J Baysan Forecasng and Dynamc Models. New York: Sprnger-Verlag. [8] Ekern, S. 98. Adapve Eponenal Smoohng Revsed. Journal of he Operaonal Research Socey : [9] Johnson, F. R. 99. Eponenally Weghed Movng Average (EWMA) wh Irregular Updang Perods. Journal of he Operaonal Research Socey 44: 7-6. [0] Makrdaks, S., and Wnkler, R. L. 98. Averages of Forecass; Some Emprcal Resuls. Managemen Scence 9: [] Ish, N. 99. Blaeral Eponenal Smoohng of Tme Seres. In.J.Sysem Sc. : [] Takeyasu, K., and Nagao, K Esmaon of Smoohng Consan of Mnmum Varance and Is Applcaon o Indusral Daa. Indusral Engneerng and Managemen Sysems 7: