A Hybrid Method to Improve Forecasting Accuracy in the Case of Sanitary Materials Data

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1 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 A Hybrid Mehod o Improve Forecasing Accuracy in he Case of Saniary Maerials Daa Daisuke Takeyasu Graduae School of Culure and Science, The Open Universiy of Japan, Wakaba, Mihama Disric, Chiba Ciy, 6886, Japan Hiroake Yamashia College of Business Adminisraion and Informaion Science, Chubu Universiy, 00 Masumoocho Kasugai, Aichi, 48780, Japan Kazuhiro Takeyasu College of Business Adminisraion, Tokoha Universiy, Oobuchi, Fuji Ciy, Shizuoka, 47080, Japan Absrac Sales forecasing is a saring poin of supply chain managemen, and is accuracy influences business managemen significanly. In indusries, how o improve forecasing accuracy such as sales, shipping is an imporan issue. In his paper, a hybrid mehod is inroduced and plural mehods are compared. Focusing ha he equaion of exponenial smoohing mehod (ESM) is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod is proposed before by Takeyasu e.al. which saisfies minimum variance of forecasing error. Firsly, we make esimaion of ARMA model parameer and hen esimae smoohing consans. In his paper, combining he rend removing mehod wih his mehod, we aim o improve forecasing accuracy. Trend removing by he combinaion of linear and nd order nonlinear funcion and rd order nonlinear funcion is carried ou o he manufacurer s daa of saniary maerials. The new mehod shows ha i is useful for he ime series ha has various rend characerisics and has raher srong seasonal rend. The effeciveness of his mehod should be examined in various cases. Keywords componen; minimum variance; exponenial smoohing mehod; forecasing; rend; saniary maerials I. INTRODUCTION The needs for sales forecasing is prevailing among companies, bu he conens of such needs are undergoing significan changes because of he rapid changes in he recen business environmen. Correc forecasing along wih supply chain managemen is required ha leads o he shorened lead ime and less socks. Time series analysis is ofen used in such hemes as sales forecasing, sock marke price forecasing ec. Sales forecasing is ineviable for Supply Chain Managemen. Bu in fac, i is no well uilized in indusries. I is because here are so many irregular incidens herefore i becomes hard o make sales forecasing. A mere applicaion of mehod does no bear good resul. The big reason is ha sales daa or producion daa are no saionary ime series, while linear model requires he ime series as a saionary one. In order o improve forecasing accuracy, we have devised rend removal mehods as well as searching opimal parameers and obained good resuls. We creaed a new mehod and applied i o various ime series and examined he effeciveness of he 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 been presened such as Auoregressive model (AR Model), Auoregressive Moving Average Model (ARMA Model) and Exponenial Smoohing Mehod (ESM) [] [4]. Among hese, ESM is said o be 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 [], uilizing Kalman Filer [6], Bayes Forecasing [7], adapive ESM [8], exponenially weighed Moving Averages wih irregular updaing periods [9], making averages of forecass using plural mehod [0] are presened. For example, Maeda [6] calculaed smoohing consan in relaionship wih S/N raio under he assumpion ha he observaion noise was added o he sysem. Bu he had o calculae under supposed noise because he could no grasp observaion noise. I can be said ha i does no pursue opimum soluion from he very daa hemselves which should be derived by [] hose esimaion. Ishii poined ou ha he opimal smoohing consan was he soluion of infinie order equaion, bu he didn show analyical soluion. Based on hese facs, a new mehod of esimaion of smoohing consan in ESM was proposed before []. 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. Furhermore, combining he rend removal mehod, forecasing accuracy was improved, where shipping daa, sock marke price daa ec. were examined [] [9]. In his paper, uilizing above saed mehod, a revised forecasing mehod is proposed. A mere applicaion of ESM does no make good forecasing accuracy for he ime series which has nonlinear rend and/or rend by monh. A new mehod o cope wih his issue is required. Therefore, uilizing above saed mehod, a revised forecasing mehod is proposed in his paper o improve forecasing accuracy. In making forecas such as producion daa, rend removing mehod is 9 P a g e

2 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 devised. Trend removing by he combinaion of linear and nd order nonlinear funcion and rd order nonlinear funcion is execued o he manufacurer s daa of saniary maerials. The weighs for hese funcions are se 0. for wo paerns a firs and hen varied by 0.0 incremen for hree paerns and opimal weighs are searched. For he comparison, monhly rend is removed afer ha. Theoreical soluion of smoohing consan of ESM is calculaed for boh of he monhly rend removing daa and he nonmonhly rend removing daa. Then forecasing is execued on hese daa. This is a revised forecasing mehod. Variance of forecasing error of his newly proposed mehod is assumed o be less han hose of previously proposed mehod. The new mehod shows ha i is useful especially for he ime series ha has sable characerisics and has raher srong seasonal rend and also he case ha has nonlinear rend. The res of he paper is organized as follows. In secion, he new mehod is described. ESM is saed by ARMA model and esimaion mehod of smoohing consan is derived using ARMA model idenificaion. The combinaion of linear and nonlinear funcion is inroduced for rend removing and he Monhly Raio is also referred. Forecasing is execued in secion, and esimaion accuracy is examined, which is followed by he Discussion of secion 4 II. DESCRIPTION OF THE NEW METHOD A. Descripion of ESM Using ARMA Model [] In ESM, forecasing a ime + is saed in he following equaion. Here, ˆ xˆ x xˆ x xˆ x : forecasing a x : realized value a xˆ : smoohing consan 0 () is resaed as l0 l x l () xˆ () By he way, we consider he following (,) order ARMA model. x x e e () Generally, p, q order ARMA model is saed as Here, x p q ai x i e i j b e j j (4) x : Sample process of Saionary Ergodic Gaussian Process x,,, N, e :Gaussian Whie Noise wih 0 mean e variance MA process in (4) is supposed o saisfy converibiliy condiion. Uilizing he relaion ha E e e, e, 0 we ge he following equaion from (). xˆ x e () Operaing his scheme on +, we finally ge xˆ xˆ xˆ e x xˆ If we se, he above equaion is he same wih (), i.e., equaion of ESM is equivalen o (,) order ARMA model, or is said o be (0,,) order ARIMA model because s order AR parameer is [][]. Focusing ha he equaion of exponenial smoohing mehod (ESM) is equivalen o (,) order ARMA model equaion, a new mehod of esimaion of smoohing consan in exponenial smoohing mehod is derived. Finally we ge: b 4 4 (6) (7) Thus we can obain a heoreical soluion by a simple way. Here mus saisfy 0 (8) in order o saisfy 0. 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 be esimaed only by calculaing 0h and s order auocorrelaion funcion. B. Trend Removal Mehod [] As ESM is a one of a linear model, forecasing accuracy for he ime series wih nonlinear rend is no necessarily good. How o remove rend for he ime series wih nonlinear 60 P a g e

3 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 rend is a big issue in improving forecasing accuracy. In his paper, we devise o remove his nonlinear rend by uilizing nonlinear funcion. As rend removal mehod, we describe he combinaion of linear and nonlinear funcion. [] Linear funcion We se III. FORECASTING THE SHIPPING DATA OF MANUFACTURER A. Analysis Procedure Manufacurer s daa of saniary maerials from Sepember 009 o Augus 0 are analyzed. Firs of all, graphical chars of hese ime series daa are exhibied in Fig.,,. as a linear funcion. [] Nonlinear funcion We se y a x (9) b y a (0) x b x c y a () x b x cx d as a nd and a rd order nonlinear funcion. [] The combinaion of linear and nonlinear funcion 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 (4) as he combinaion of linear and nd order nonlinear and rd order nonlinear funcion. Here,, β β, γ ( γ γ ). Comparaive discussion concerning (), () and (4) are described in secion. C. Monhly Raio [] For example, if here is he monhly daa of L years as saed bellow: Where, i,, L j,, x ij x ij R in which j means monh and i means year and x ij is a shipping daa of ih year, jh monh. Then, monhly raio x~,, j j is calculaed as follows. L xij ~ L i x j L () xij L i j Monhly rend is removed by dividing he daa by (). Numerical examples boh of monhly rend removal case and nonremoval case are discussed in. Fig.. Produc A Fig.. Produc B Fig.. Produc C Analysis procedure is as follows. There are 6 monhly daa for each case. We use 4 daa( o 4) and remove rend by he mehod saed in.. Then we calculae monhly raio by he mehod saed in.. Afer removing monhly rend, he mehod saed in. is applied and Exponenial Smoohing Consan wih minimum variance of forecasing error is esimaed. 6 P a g e

4 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 Then sep forecas is execued. Thus, daa is shifed o nd o h and he forecas for 6h daa is execued consecuively, which finally reaches forecas of 6h daa. To examine he accuracy of forecasing, variance of forecasing error is calculaed for he daa of h o 6h daa. Final forecasing daa is obained by muliplying monhly raio and rend. Forecasing error is expressed as: xˆ x (6) i i N N i i (7) B. Trend Removing Trend is removed by dividing original daa by (),(),(4). The paerns of rend removal are exhibied in Table. i In paern and, he weigh of,,, are se 0. in he equaion (),(). In paern, he weigh of is shifed by 0.0 incremen in () which saisfy he range In paern4, he weigh of is shifed in he same way which saisfy he range In paern, he weigh of and are shifed by 0.0 incremen in (4) which saisfy he range 0. 00, The bes soluion is seleced which minimizes he variance of forecasing error. Esimaion resuls of coefficien of (9), (0) and () are exhibied in Table. Esimaion resuls of weighs of (), () and (4) are exhibied in Table. Graphical char of rend is exhibied in Fig. 4,, 6 for he cases ha monhly raio is used. TABLE I. COEFFICIENT OF (9),(0) AND () Pro duc A Pro duc B Pro duc C a s nd rd b a b c a b c d TABLE II. THE PATTERNS OF TREND REMOVAL Paern, are se 0. in he equaion () Paern, are se 0. in he equaion () Paern is shifed by 0.0 incremen in () Paern4 is shifed by 0.0 incremen in () Paern γ and γ are shifed by 0.0 incremen in (4) Produc A Produc B Produc C Monhly raio TABLE III. WEIGHTS OF (), () AND (4) Paern Paern Paern Paern4 Paern Used No used Used No used Used No used P a g e

5 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 Fig. 4. Trend of Produc A Fig.. Trend of Produc B Fig. 6. Trend of Produc C 6 P a g e

6 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 C. Removing rend of monhly raio Afer removing rend, monhly raio is calculaed by he mehod saed in.. Calculaion resul for s o 4h daa is exhibied in Table 4 hrough 8. D. Esimaion of Smoohing Consan wih Minimum Variance of Forecasing Error Afer removing monhly rend, Smoohing Consan wih minimum variance of forecasing error is esimaed uilizing (7). There are cases ha we canno obain a heoreical soluion because hey do no saisfy he condiion of (A9). In hose cases, Smoohing Consan wih minimum variance of forecasing error is derived by shifing variable from 0.0 o 0.99 wih 0.0 inerval. Calculaion resul for s o 4h daa is exhibied in Table 9. E. Forecasing and Variance of Forecasing Error Uilizing smoohing consan esimaed in he previous secion, forecasing is execued for he daa of h o 6h daa. Final forecasing daa is obained by muliplying monhly raio and rend. Variance of forecasing error is calculaed by (8). Forecasing resuls are exhibied in Fig. 7, 8, 9 for he cases ha monhly raio is used. TABLE IV. MONTHLY RATIO (PATTERN) Monh Produc A Produc B Produc C TABLE V. MONTHLY RATIO (PATTERN) Monh Produc A Produc B Produc C TABLE VI. MONTHLY RATIO (PATTERN) Monh Produc A Produc B Produc C TABLE VII. MONTHLY RATIO (PATTERN4) Monh Produc A Produc B Produc C TABLE VIII. MONTHLY RATIO (PATTERN) Monh Produc A Produc B Produc C P a g e

7 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 TABLE IX. ESTIMATED SMOOTHING CONSTANT WITH MINIMUM VARIANCE Produc A Produc B Produc C Mon hly raio Paern Paern Paern Paern4 Paern Used No used Used No used Used No used Fig. 7. Forecasing Resuls of Produc A Fig. 8. Forecasing Resuls of Produc B 6 P a g e

8 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 Fig. 9. Forecasing Resuls of Produc C Variance of forecasing error is exhibied in Table 0 TABLE X. VARIANCE OF FORECASTING ERROR Monhly raio Paern Paern Paern Paern4 Paern Produc A Produc B Produc C Used 84,8,64, ,,890,8, No used,,6,67,0.600,476,09,69, Used 7,,, ,6,46, No used 0,76,9, ,44,6, Used,44,99,09,8.00,6,6,9, No used,04,97,068,44.400,7,66,9, F. Remarks These ime series have nonlinear rend and rend by monh. Applying only an ESM does no make good forecasing accuracy. All cases had a good resul in s + nd order wih he case ha monhly raio is used. We can observe ha monhly rend is raher apparen in hese cases. Therefore he mehod has seleced he monhly rend removing case. IV. DISCUSSION Correc sales forecasing is ineviable in indusries. Poor sales forecasing accuracy leads o a gap beween he sales plan and resul, which in urn generaes a gap beween he sales plan and he producion plan. The condiion in which he quaniy in a producion plan exceeds ha in a sales plan (excess producion) pushes up cos caused by increased finished and inermediae produc invenory. Increased invenory and prolonged dwell ime of produc in invenory will lead o increased wase loss as well as exended leadime, affecing cusomer saisfacion. In order o improve forecasing accuracy, we have devised rend removal mehods as well as searching opimal parameers and obained good resuls. We creaed a new mehod. V. CONCLUSION 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 before by Takeyasu e.al.[] which saisfied minimum variance of forecasing error. Combining he rend removal mehod wih his mehod, we aimed o improve forecasing accuracy. A mere applicaion of ESM does no make good forecasing accuracy for he ime series which has nonlinear rend and/or rend by monh. A new mehod o cope wih his issue is required. Therefore, uilizing above saed mehod, a revised forecasing mehod is proposed in his paper o improve forecasing accuracy. An approach o his mehod was execued in he following mehod. Trend removal by a linear funcion was applied o he manufacurer s daa of saniary maerials. The combinaion of linear and nonlinear funcion was also inroduced in rend removing. For he comparison, monhly rend was removed afer ha. Theoreical soluion of smoohing consan of ESM was calculaed for boh of he monhly rend removing daa and he nonmonhly rend removing daa P a g e

9 (IJACSA) Inernaional Journal of Advanced Compuer Science and Applicaions, Vol., No., 04 Then forecasing was execued on hese daa. Produc Ⅰ and Produc Ⅱ had a good resul in s + nd order wih he case ha monhly raio is used, while Produc Ⅲhad a good resul in s + nd order wih he case ha monhly raio is no used. VI. FUTURE WORKS I is our fuure works o invesigae much furher cases o confirm he effeciveness of our new mehod. Various cases should be examined hereafer. In he end, we appreciae Mr. Norio Funao for his helpful suppor of our sudy. REFERENCES [] Box Jenkins, Time Series Analysis Fourh Ediion,Wiley,0. [] Kazuhiro Takeyasu and Yuki Higuchi, Time Series Analysis and is Applicaions, Osaka Municipals Universiies Press,007. [] Toru Kaayama, Inroducion o Sysem Idenificaion, Asakura Shoen,998. [4] Kazuhiro Takeyasu, Yasuo Ishii and Yuki Higuchi, Time Series Analysis and is Applicaions Ⅲ, Osaka Municipals Universiies Press,009. [] Peer R.Winers, Forecasing Sales by Exponenially Weighed Moving Averages, Managemen Science,Vol6,No., pp. 44,984. [6] Kasuro Maeda, Smoohing Consan of Exponenial Smoohing Mehod, Seikei Universiy Repor Faculy of Engineering, No.8, pp ,984. [7] M.Wes and P.J.Harrison, Baysian Forecasingand Dynamic Models, SpringerVerlag,New York,989. [8] Seinar Ekern, Adapive Exponenial Smoohing Revisied,Journal of he Operaional Research Sociey, Vol. pp.7778,98. [9] F.R.Johnson, Exponenially Weighed Moving Average (EWMA) wih Irregular Updaing Periods, Journal of he Operaional Research Sociey,Vol.44,No.7 pp.776,99. [0] Spyros Makridakis and Robea L.Winkler, Averages of Forecass; Some Empirical Resuls, Managemen Science,Vol.9, No.9, pp ,98. [] Naohiro Ishii e al., Bilaeral Exponenial Smoohing of Time Series, In.J.Sysem Sci., Vol., No.8, pp ,99. [] Kazuhiro Takeyasu and Keiko Nagaa, Esimaion of Smoohing Consan of Minimum Variance wih Opimal Parameers of Weigh, Inernaional Journal of Compuaional Science Vol.4,No., pp. 4 4,00. [] Kazuhiro Takeyasu, Keiko Nagaa, Yuki Higuchi, Esimaion of Smoohing Consan of Minimum Variance And Is Applicaion o Shipping Daa Wih Trend Removal Mehod, Indusrial Engineering & Managemen Sysems (IEMS),Vol.8,No.4, pp.76,009, [4] Kazuhiro Takeyasu, Keiko Nagaa, Yui Nishisako, A Hybrid Mehod o Improve Forecasing Accuracy Uilizing Geneic Algorihm And Is Applicaion o Indusrial Daa, NCSP'0, Honolulu,Hawaii,USA,00. [] Kazuhiro Takeyasu, Keiko Nagaa, Kana Takagi, Esimaion of Smoohing Consan of Minimum Variance wih Opimal Parameers of Weigh, NCSP'0, Honolulu,Hawaii,USA,00. [6] Kazuhiro Takeyasu, Keiko Nagaa, Tomoka Kuwahara, Esimaion of Smoohing Consan of Minimum Variance Searching Opimal Parameers of Weigh, NCSP'0, Honolulu,Hawaii,USA,00. [7] Kazuhiro Takeyasu, Keiko Nagaa, Mai Io, Yuki Higuchi, A Hybrid Mehod o Improve Forecasing Accuracy Uilizing Geneic Algorihm, The h APEIMS, Melaka, Malaysia,00. [8] Kazuhiro Takeyasu, Keiko Nagaa, Kaori Masumura, Esimaion of Smoohing Consan of Minimum Variance and Is Applicaion o Sales Daa, JAIMS, Honolulu, Hawaii, USA.0. [9] Hiromasa Takeyasu, Yuki Higuchi, Kazuhiro Takeyasu, A Hybrid Mehod o Improve Forecasing Accuracy in he Case of Bread, Inernaional Journal of Informaion and Communicaion Technology Research, Vol., No., pp.804~8, P a g e