Improving Forecasting Accuracy in the Case of Intermittent Demand Forecasting

Transcription

1 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 Improvng Forecasng Accuracy n he Case of Inermen Demand Forecasng Dasuke Takeyasu The Open Unversy of Japan, Chba Cy, Japan Asam Shara Tax Corporaon Arkne, Shzuoka Cy, Japan Kazuhro Takeyasu College of Busness Admnsraon, Tokoha Unversy, Shzuoka, Japan Absrac In makng forecasng, here are many knds of daa. Saonary me seres daa are relavely easy o make forecasng bu random daa are very dffcul n s execuon for forecasng. Inermen daa are ofen seen n ndusres. Bu s raher dffcul o make forecasng n general. In recen years, he needs for nermen demand forecasng are ncreasng because of he consrans of src Supply Chan Managemen. How o mprove he forecasng accuracy s an mporan ssue. There are many researches made on hs. Bu here are rooms for mprovemen. In hs paper, a new mehod for cumulave forecasng mehod s proposed. The daa s cumulaed and o hs cumulaed me seres, he followng mehod s appled o mprove he forecasng accuracy. Trend removng by he combnaon of lnear and nd order non-lnear funcon and rd order non-lnear funcon s execued o he producon daa of X- ray mage nensfer ube devce and Dagnosc X-ray mage processng apparaus. The forecasng resul s compared wh hose of he non-cumulave forecasng mehod. The new mehod shows ha s useful for he forecasng of nermen demand daa. The effecveness of hs mehod should be examned n varous cases. Keywords nermen demand forecasng; mnmum varance; exponenal smoohng mehod; rend I. INTRODUCTION Supply chan managemen s nevable n ndusres n recen years. Demand forecasng s he bass n supply chan managemen. In ndusres, how o mprove forecasng accuracy such as sales, shppng s an mporan ssue. There are cases ha nermen demand forecasng s requred. Bu he mere applcaon of he pas mehod does no bear good esmaon of parameers and exquse forecasng. There are many researchers made on hs. Based upon he Croson s model (Box e al.008), Shensone and Hyndma (005) analyzed he nermen demand forecasng. Troung e al. (0) appled Neural Nework o nermen demand forecasng. Tanaka e al. (0) has bul sales forecasng model for book publshng, where hey have devsed cumulave forecasng mehod. In hs paper, we furher develop hs cumulave forecasng mehod n order o mprove he forecasng accuracy for nermen demand. A new mehod for cumulave forecasng mehod s proposed. The daa s cumulaed and o hs cumulaed me seres, he followng mehod s appled o mprove he forecasng accuracy. Focusng ha he equaon of exponenal smoohng mehod(esm) s equvalen o (,) order ARMA model equaon, a new mehod of esmaon of smoohng consan n exponenal smoohng mehod s proposed before by us whch sasfes mnmum varance of forecasng error[7]. Generally, smoohng consan s seleced arbrarly. Bu n hs paper, we ulze above 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 be ulzed n varous felds. Furhermore, combnng he rend removng mehod wh hs mehod, we am o mprove he forecasng accuracy. An approach o hs mehod s execued n he followng mehod. Trend removng by he combnaon of lnear and nd order non-lnear funcon and rd order non-lnear funcon s execued o he daa of X-ray mage nensfer ube devce and Dagnosc X-ray mage processng apparaus. The weghs for hese funcons are se 0.5 for wo paerns a frs and hen vared by 0.0 ncremen for hree paerns and opmal weghs are searched. For he comparson, monhly rend s removed afer ha. Theorecal soluon of smoohng consan of ESM s calculaed for boh of he monhly rend removng daa and he non-monhly rend removng daa. Then forecasng s execued on hese daa. The forecasng resul s compared wh hose of he noncumulave forecasng mehod. The new mehod shows ha s useful for he forecasng of nermen demand daa. The effecveness of hs mehod should be examned n varous cases. The res of he paper s organzed as follows. In secon, he new mehod s descrbed. ESM s saed by ARMA model and esmaon mehod of smoohng consan s derved usng ARMA model denfcaon. The combnaon of lnear and non-lnear funcon s nroduced for rend removng and he Monhly Rao s also referred. Forecasng s execued n secon, and esmaon accuracy s examned, whch s followed by he Dscusson of secon P a g e

2 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 II. DESCRIPTION OF THE NEW METHOD A. Descrpon of ESM Usng ARMA Model [5] In ESM, forecasng a me + s saed n he followng equaon. xˆ xˆ x xˆ xˆ x Here, x ˆ : forecasng a x : realzed value a : smoohng consan 0 () s re-saed as l0 l x l () xˆ () By he way, we consder he followng (,) order ARMA model. x x e e () Generally, p, q order ARMA model s saed as x p q a x e j b e j j (4) Here, x : Sample process of Saonary Ergodc Gaussan x,,, N, Process e :Gaussan Whe Nose wh 0 mean e varance MA process n (4) s supposed o sasfy converbly condon. Ulzng he relaon ha E e e, e, 0 we ge he followng equaon from (). xˆ x e (5) Operang hs scheme on +, we fnally ge e x xˆ xˆ xˆ (6) xˆ If we se, he above equaon s he same wh (),.e., equaon of ESM s equvalen o (,) order ARMA model, or s sad o be (0,,) order ARIMA model because s order AR parameer s []. Comparng wh () and (4), we oban a b From (), (6), Therefore, we ge a (7) b From above, 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 become non-lnear equaons whch are descrbed below. Le (4) be ~ x x a x (8) p q bje j j ~ x e (9) We express he auocorrelaon funcon of x~ as ~ r k and from (8), (9), we ge he followng non-lnear equaons whch are well known. ~ r k ~ r 0 0 qk e j0 q e j0 b b b j k j j ( k q) ( k q ) (0) For hese equaons, a recursve algorhm has been developed. In hs paper, parameer o be esmaed s only b, so can be solved n he followng way. From () (4) (7) (0), we ge q a b ~ r0 b e ~ r b e () If we se ~ r ~ k k () r0 he followng equaon s derved. b () b We can ge b as follows. 4 b (4) 40 P a g e

3 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 In order o have real roos, mus sasfy (5) From nverbly condon, b mus sasfy b From (), usng he nex relaon, (5) always holds. As b 0 b 0 b b s whn he range of Fnally we ge b b (6) whch sasfy above condon. Thus we can oban a heorecal soluon by 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 be esmaed only by calculang 0h and s order auocorrelaon funcon. B. Trend Removal Mehod [5] As rend removal mehod, we descrbe he combnaon of lnear and non-lnear funcon. [] Lnear funcon We se as a lnear funcon. [] Non-lnear funcon y a x (8) b We se y a (9) x b x c y a (0) x b x cx d as a nd and a rd order non-lnear funcon. [] The combnaon of lnear and non-lnear funcon We se a x b a x b x y () c a x b β a x b x c x y () β d y γ a x b γ a x b x c γ a x b x c x d () as he combnaon of lnear and nd order non-lnear and rd order non-lnear funcon. Here,, β β, γ ( γ γ ). Comparave dscusson concernng (), () and () are descrbed n secon 5. C. Monhly Rao [5] For example, f here s he monhly daa of L years as saed bellow: Where,,, L j,, x j x j R n whch j means monh and means year and x j s a shppng daa of -h year, j-h monh. Then, monhly rao III. x~,, j j s calculaed as follows. L xj ~ L x j L (4) xj L j FORECASTING THE PRODUCTION DATA A. Analyss Procedure Sum oal daa of producon daa of X-ray mage nensfer ube devce and Dagnosc X-ray mage processng apparaus from January 00 o December 0 are analyzed. These daa are obaned from he Annual Repor of Sascal Invesgaon on Sascal-Survey-on-Trends-n- Pharmaceucal-Producon by Mnsry of Healh, Labour and Welfare n Japan. The orgnal daa are accumulaed for X-ray mage nensfer ube devce daa and Dagnosc X-ray mage processng apparaus daa. Analyss procedure s as follows. There are 6 monhly daa for each case. We use 4 daa ( o 4) and remove rend by he mehod saed n.. Then we calculae monhly rao 4 P a g e

4 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 by he mehod saed n.. Afer removng monhly rend, he mehod saed n s appled and Exponenal Smoohng Consan wh mnmum varance of forecasng error s esmaed. Then sep forecas s execued. Thus, daa s shfed o nd o 5h and he forecas for 6h daa s execued consecuvely, whch fnally reaches forecas of 6h daa. To examne he accuracy of forecasng, varance of forecasng error s calculaed for he daa of 5h o 6h daa. Fnal forecasng daa s obaned by mulplyng monhly rao and rend. Forecasng error s expressed as: Varance of forecasng error s calculaed by: N (7) N B. Trend Removng Trend s removed by dvdng orgnal daa by,(),(),(). The paerns of rend removal are exhbed n Table. TABLE I. xˆ x (5) THE PATTERNS OF TREND REMOVAL Paern, are se 0.5 n he equaon () Paern, are se 0.5 n he equaon () Paern s shfed by 0.0 ncremen n () Paern4 s shfed by 0.0 ncremen n () N N Paern5 γ and γ are shfed by 0.0 ncremen n () In paern and, he wegh of,,, are se 0.5 n he equaon (),(). In paern, he wegh of s shfed by 0.0 ncremen n () whch sasfy he range In paern4, he wegh of s shfed n he same way whch sasfy he range In paern5, he wegh of and are shfed by 0.0 ncremen n () whch sasfy he range 0. 00, The bes soluon s seleced whch mnmzes he varance of forecasng error. C. Removng rend of monhly rao Afer removng rend, monhly rao s calculaed by he mehod saed n.. D. Esmaon of Smoohng Consan wh Mnmum Varance of Forecasng Error Afer removng monhly rend, Smoohng Consan wh mnmum varance of forecasng error s esmaed ulzng (6). There are cases ha we canno oban a heorecal soluon because hey do no sasfy he condon of (5). (6) In hose cases, Smoohng Consan wh mnmum varance of forecasng error s derved by shfng varable from 0.0 o 0.99 wh 0.0 nerval. The nermen demand daa ofen nclude 0 daa. If here are so many 0 daa, here s a case we canno calculae he heorecal solaon of smoohng consan. In ha case, we add very ny daa whch s no 0 bu close o 0 ha does no affec anyhng n calculang parameers (.e. neglgble small). E. Forecasng AND Varance of Forecasng Error Ulzng smoohng consan esmaed n he prevous secon, forecasng s execued for he daa of 5h o 6h daa. Fnal forecasng daa s obaned by mulplyng monhly rao and rend. Varance of forecasng error s calculaed by (7). As we have made accumulaed daa case and ny daa close o 0 added case, we have he followng cases alogeher.. Non Monhly Trend Removal () Accumulaed Daa () Non Accumulaed Daa (-) Forecasng from he Accumulaed daa (Accumulaed forecasng daa a me n-accumulaed daa (a me n-) ) A. Paern B. Paern C. Paern D. Paern4 E. Paern5 (-) Forecasng from he ny daa close o 0 added case A. Paern B. Paern C. Paern D. Paern4 E. Paern5. Monhly Trend Removal () Accumulaed Daa () Non Accumulaed Daa (-) Forecasng from he Accumulaed daa (Accumulaed forecasng daa a me n-accumulaed daa (a me n-) ) A. Paern B. Paern C. Paern D. Paern4 E. Paern5 (-) Forecasng from he ny daa close o 0 added case A. Paern B. Paern C. Paern D. Paern4 E. Paern5 We can make forecasng by reversely makng he daa from he forecasng accumulaed daa,.e., ha s shown a (-). Now, we show hem a Fgure hrough 6. Fgure, and show he Non-monhly Trend Removal Case n X-ray mage nensfer ube devce. I ncludes all cases classfed above. Fgure shows he Accumulaed Daa Case n Non- Monhly Trend Removal. Fgure shows he Forecasng from he Accumulaed Daa Case n Non-Monhly Trend Removal. 4 P a g e

5 Fgure shows he Forecasng from he ny daa close o 0 added case n Non-Monhly Trend Removal. Table, and 4 show he correspondng varance of forecasng error for each Fgure, and. (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 Fg.. Forecasng from he Tny Daa close o 0 Added case n Non- Monhly Trend Removal (-(-)) TABLE IV. VARIANCE OF FORECASTING ERROR (-(-)) Fg.. Forecasng from he Accumulaed Daa Case n Non-Monhly Trend Removal (-()) TABLE II. VARIANCE OF FORECASTING ERROR (-()) Paern Paern Paern Paern4 Paern Paern Paern Paern Paern4 Paern Nex, we see he Monhly Trend Removal case. Fgure 4,5 and 6 show he Monhly Trend Removal Case n X-ray mage nensfer ube devce. I ncludes all cases classfed above. Fgure 4 shows he Accumulaed Daa Case n Monhly Trend Removal. Fgure 5 shows he Forecasng from he Accumulaed Daa Case n Monhly Trend Removal. Fgure 6 shows he Forecasng from he ny daa close o 0 added case n Monhly Trend Removal. Table 5,6 and 7 show he correspondng varance of forecasng error for each Fgure 4,5 and 6. Fg.. Forecasng from he Accumulaed Daa Case n Non-monhly Trend Removal (-(-)) TABLE III. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Fg. 4. Accumulaed Daa case n Monhly Trend Removal (-()) 4 P a g e

6 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 TABLE V. VARIANCE OF FORECASTING ERROR (-()) Paern Paern Paern Paern4 Paern Now, we proceed o he case of Dagnosc X-ray mage processng apparaus. Fgure 7,8 and 9 show he Non-monhly Trend Removal Case n Dagnosc X-ray mage processng apparaus. I ncludes all cases classfed above. Fgure 7 shows he Accumulaed Daa Case n Non- Monhly Trend Removal. Fgure 8 shows he Forecasng from he Accumulaed Daa Case n Non-Monhly Trend Removal. Fgure 9 shows he Forecasng from he ny daa close o 0 added case n Non-Monhly Trend Removal. Table 9,0 and show he correspondng varance of forecasng error for each Fgure 7,8 and 9. Fg. 5. Forecasng from he accumulaed Daa case n Monhly Trend Removal (-(-)) TABLE VI. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Fg. 7. Accumulaed Daa case n Non-Monhly Trend Removal (-()) TABLE IX. VARIANCE OF FORECASTING ERROR (-()) Paern Paern Paern Paern4 Paern Fg. 6. Forecasng from he Tny Daa close o 0 Added case n Monhly Trend Removal (-(-)) TABLE VII. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Table 8 shows he summary for X-ray mage nensfer ube devc by he Varance of forecasng error. Name TABLE VIII. Mnmum varance of Forecasng Error X-ray mage Accumulae nensfer d Daa ube devce SUMMARY FOR X-RAY IMAGE INTENSIFIER TUBE DEVICE Monhly Trend Removal Forecasng Tny daa Valueclose o 0 Accumulae added case d Value Accumulae d Daa Non Monhly Trend Removal Forecasng Tny daa Valueclose o 0 Accumulae added case d Value Fg. 8. Forecasng from he accumulaed Daa case n Non-Monhly Trend Removal (-(-)) TABLE X. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern P a g e

7 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 Fg. 9. Forecasng from he Tny Daa close o 0 Added case n Non- Monhly Trend Removal (-(-)) TABLE XI. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Nex, we see he Monhly Trend Removal case. Fgure 0, and show he Monhly Trend Removal Case n Dagnosc X-ray mage processng apparaus.. I ncludes all cases classfed above. Fgure 0 shows he Accumulaed Daa Case n Monhly Trend Removal. Fgure shows he Forecasng from he Accumulaed Daa Case n Monhly Trend Removal. Fgure shows he Forecasng from he ny daa close o 0 added case n Monhly Trend Removal. Table, and 4 show he correspondng varance of forecasng error for each Fgure 0, and. Fg.. Forecasng from he accumulaed Daa case n Monhly Trend Removal (-(-)) TABLE XIII. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Fg.. Forecasng from he Tny Daa close o 0 Added case n Monhly Trend Removal (-(-)) TABLE XIV. VARIANCE OF FORECASTING ERROR (-(-)) Paern Paern Paern Paern4 Paern Table 7 shows he summary for Dagnosc X-ray mage processng apparaus by he Varance of forecasng error. TABLE XV. SUMMARY FOR DIAGNOSTIC X-RAY IMAGE PROCESSING APPARATUS Fg. 0. Accumulaed Daa case n Monhly Trend Removal (-()) TABLE XII. VARIANCE OF FORECASTING ERROR (-()) Paern Paern Paern Paern4 Paern Name Monhly Trend Removal Dagnosc X-ray mage Accumulae processng d Daa apparaus of oher Mnmum varance of Forecasng Error Forecasng Tny daa Valueclose o 0 Accumulae added case d Value IV. DISCUSSION Accumulae d Daa Non Monhly Trend Removal Forecasng Tny daa Valueclose o 0 Accumulae added case d Value In he case of X-ray mage nensfer ube devce, Monhly Trend Removal case was beer han Non-Monhly Trend 45 P a g e

8 (IJACSA) Inernaonal Journal of Advanced Compuer Scence and Applcaons, Vol. 5, No. 5, 04 Removal case. Ths me seres had a raher clear monhly rend and he resul had refleced hem. Forecasng from he accumulaed daa case (-(-)) was beer han hose of he ny daa close o 0 added case (-(-)) n hs Monhly Trend Removal case for X-ray mage nensfer ube devce. On he oher hand, n he case of Dagnosc X-ray mage processng apparaus, Non-Monhly Trend Removal case was beer han Monhly Trend Removal case. The me seres of Dagnosc X-ray mage processng apparaus does no have clear monhly rend. Forecasng from he ny daa close o 0 added case (- (-)) was beer han hose of Accumulaed daa case (-(- )). By he way, forecasng of accumulaed daa (-(), -()) shows raher good resul. I can be used as one of he ool o decde when and how much volume o procure he maerals ec.. I can be ulzed as a new mehod o procure n supply chan managemen. V. CONCLUSION The needs for nermen demand forecasng are ncreasng. In hs paper, a new mehod for cumulave forecasng mehod was proposed. The daa was cumulaed and o hs cumulaed me seres, he new mehod was appled o mprove he forecasng accuracy. The forecasng resul was compared wh hose of he non-cumulave forecasng mehod. The new mehod shows ha s useful for he forecasng of nermen demand daa. Forecasng from he accumulaed daa case (-(-)) was beer han hose of he ny daa close o 0 added case (-(-)) n hs Monhly Trend Removal case for X-ray mage nensfer ube devce. On he oher hand, n he case of Dagnosc X-ray mage processng apparaus, forecasng from he ny daa close o 0 added case (-(-)) was beer han hose of Accumulaed daa case (- (-)). Among hem, forecasng of accumulaed daa (-(), -()) shows raher good resul. I can be used as one of he ool o decde when and how much volume o procure he maerals ec.. I can be ulzed as a new mehod o procure n supply chan managemen. VI. FUTURE WORKS I s our fuure works o nvesgae much furher cases o confrm he effecveness of our new mehod. The effecveness of hs mehod should be examned n varous cases. REFERENCES [] Box, G..E.P., Jenkns, G.M.& Rensel, G.C.,Tme Seres analyss: forecasng and conrol, Wley, 4 h edn.,008. [] Lyda Shensone and Rob J. Hyndma, Sochasc models underlyng Croson s mehod for nermen demand forecasng, Journal of Forecasng, 4:89-40,005. [] Kenj Tanaka, Yukhro Myamura and Jng Zhang, The Cluser Groupng Approach of Sales Forecasng Model for Book Publshng, Inernaonal Journal of Japan Assocaon for Managemen Sysems, Vol.4, No.,pp.-5,0. [4] Nguyen Khoa Ve Froung, Shn Sangmun, Vo Thanh Nha, Kwon Ichon, Inermen Demand forecasng by usng Neural Nework wh smulaed daa, Proceedngs of he 0 Inernaonal Engneerng and Operaons Managemen Kuala Lumpur, Malasa, pp.7-78,0. [5] Kazuhro Takeyasu and Keko Nagaa, Esmaon of Smoohng Consan of Mnmum Varance wh Opmal Parameers of Wegh, Inernaonal Journal of Compuaonal Scence Vol.4,No.5, pp. 4-45, P a g e