Improvement of Forecasting Accuracy by the Utilization of Genetic Algorithm with an Application to the Sanitary Materials Data

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1 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Improvemen of Forecasing Accuracy by he Uilizaion of Geneic Algorihm wih an Applicaion o he Saniary Maerials Daa Daisuke Takeyasu, Hiroake Yamashia, Kazuhiro Takeyasu Graduae School of Culure and Science, The Open Universiy of Japan, Wakaba, MihamaDisric, Chiba Ciy, 68586, Japan College of Business Adminisraion and Informaion Science, Chubu Universiy, Masumoocho Kasugai, Aichi,48785, Japan College of Business Adminisraion, Tokoha Universiy,5 Oobuchi, Fuji Ciy, Shizuoka, 478, Japan ABSTRACT How o improve forecasing accuracy such as sales, shipping is one of he criical success facor in supply chain managemen. There are many researches made on his. 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, new mehod of esimaion of smoohing consan in exponenial smoohing mehod is proposed before by us which saisfies minimum variance of forecasing error. Generally, smoohing consan is seleced arbirarily. Bu in his paper, we uilize above saed heoreical soluion. Firsly, we make esimaion of ARMA model parameer and hen esimae smoohing consans. Thus heoreical soluion is derived in a simple way and i may be uilized in various fields. Furhermore, combining he rend removing mehod wih his mehod, we aim o improve forecasing accuracy. An approach o his mehod is execued in he following mehod. 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.5 for wo paerns a firs and hen varied by. incremen for hree paerns and opimal weighs are searched. Geneic Algorihm is uilized o search he opimal weigh for he weighing parameers of linear and nonlinear funcion. 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. 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: minimum variance, exponenial smoohing mehod, forecasing, rend, saniary maerials. INTRODUCTION Supply Chain Managemen is ineviable in shorening lead ime and decreasing socks. In paricular, high accuracy demand forecasing plays an imporan role in supply chain managemen. 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 [5], 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 [] 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 doesn 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, we proposed a new mehod of esimaion of smoohing consan in ESM 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. In his paper, uilizing above saed mehod, a revised forecasing mehod is proposed. In making forecas such as pr oducion daa, rend removing mehod is devised. Trend removing by he combinaion of linear and nd order nonlinear funcion and rd order nonlinear funcion is execued o he m anufacurer s daa of saniary maerials. These are used for medical use. The weighs for hese funcions are se.5 for w o paerns a firs and hen varied by. incremen for hree p aerns and opimal weighs are searched. Geneic Algorihm is uilized o search he opimal weigh for he weighing parameers of linear and nonlinear funcion. 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 res of he paper is organized as follows. In secion, 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 in secion. The Monhly Raio is referred in secion 4. Forecasing Accuracy is defined in secion 5. Opimal weighs are searched in secion 6. Forecasing is carried ou in secion 9

2 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// 7, and esimaion accuracy is examined. Therefore, we ge. DESCRIPTION OF ESM USING ARMA MODEL a b (8) In ESM, forecasing a ime + is saed in he following equaion. Here, ˆ x x : : xˆ xˆ x xˆ () forecasing a xˆ x () realized value a : smoohing consan () is resaed as l l x l xˆ () By he way, we consider he following (,) order ARMA model. x x e e (4) Generally, p, order ARMA model is saed as x q p q aix i e i j b e j j Here, MA process in (5) is supposed o saisfy converibiliy condiion. Uilizing he relaion ha E e e, e, we ge he following equaion from (4). Operaing his scheme on (5) xˆ x e (6) +, we finally ge From above, 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 become nonlinear equaions which are described below. Le (5) be p i ~ x x a x (9) q j i i ~ x e b e () j j We express he auocorrelaion funcion of as ~ r k and from (9), (), we ge he following nonlinear equaions which are well known. : x Process Sample process of Saionary Ergodic Gaussian x ~ r k ~ r,,, N, e :Gaussian Whie Noise wih mean qk e j q e j b b b j k j j e x~ ( k q) ( k q ) variance () 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 (,,) order ARIMA model because s order AR parameer is. Comparing wih (4) and (5), we obain From (), (7), a b (7) For hese equaions, recursive algorihm has been developed. In his paper, parameer o be esimaed is only b, so i can be solved in he following way. From (4) (5) (8) (), we ge q a b ~ r b e ~ r b e ()

3 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// If we se ~ r ~ k k r he following equaion is derived. We can ge b as follows. In order o have real roos, () b (4) b 4 b (5) mus saisfy From inveribiliy condiion, b b From (4), using he nex relaion, (6) always holds. As b is wihin he range of Finally we ge b b b b b 4 mus saisfy 4 (6) (7) which saisfies above condiion. Thus we can obain a heoreical soluion by a simple way. 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 h and s order auocorrelaion funcion.. TREND REMOVAL METHOD As rend removal mehod, we describe he combinaion of linear and nonlinear funcion. [] Linear funcion We se as a linear funcion. [] Nonlinear funcion We se y a x (8) b y a x b x (9) c y a () x b x cx d as a nd and a rd order nonlinear funcion. ( d ( a, b, c) and a, b, c, ) are also parameers for a nd and a rd order nonlinear funcions which are esimaed by using leas square mehod. [] The combinaion of linear and nonlinear funcion. We se y ax b a x b x c a x b x c x d (),,, () as he combinaion linear and nd order nonlinear and rd order nonlinear funcion. Trend is removed by dividing he original daa by (). The opimal weighing parameer,,are deermined by uilizing GA. GA mehod is, precisely described in secion MONTHLY RATIO For example, if here is he monhly daa of L years as saed bellow: i,, L j,, x ij Where, x ij R in which j means monh and i means year and is a shipping daa of i h year, j h monh. Then, monhly raio j,, is calculaed as follows. x ij ~ x j 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 FORECASTING ACCURACY Forecasing accuracy is measured by calculaing he variance of he forecasing error. Variance of forecasing error is

4 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// calculaed by: N (4) N i i Where, forecasing error is expressed as: xˆ x i i N N i i (5) (6) 6. SEARCHING OPTIMAL WEIGHTS UTILIZING GA 6. Definiion of he problem We search of () which minimizes (4) by uilizing GA. By (), we only have o deermine and. ((4)) is a funcion of and, herefore we express hem as. Now, we pursue he following: (, ),, Minimize:, ) ( i subjec o:,, (7) We do no necessarily have o uilize GA for his problem which has small member of variables. Considering he possibiliy ha variables increase when we use logisics curve ec in he near fuure, we wan o ascerain he effeciveness of GA. 6. The srucure of he gene Gene is expressed by he binary sysem using {,} bi. Domain of variable is [,] from (). We suppose ha variables ake down o he second decimal place. As he lengh of domain of variable is =, seven bis are required o express variables. The binary bi srings <bi6, ~,bi> is decoded o he [,] domain real number by he following procedure. [4] Procedure :Conver he binary number o he binarycoded decimal. bi, bi, bi, bi, bi, bi, bi 6 6 i bii X 5 4 i (8) Procedure : Conver he binarycoded decimal o he real number. The real number = (Lef hand saring poin of he domain) + ( X ' ((Righ hand ending poin of he domain)/ 7 )) (9) The decimal number, he binary number and he corresponding real number in he case of 7 bis are expressed in Table 6. Table 6: Corresponding able of he decimal number, he binary number and he real number The decimal number The binary number The Corresponding real number Posiion of he bi variable is expressed by 7 bis, herefore variables needs 4 bis. The gene srucure is exhibied in Table 6. Table 6: The gene srucure Posiion of he bi The flow of Algorihm The flow of algorihm is exhibied in Figure 6.

5 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Figure 6: The flow of algorihm A. Iniial Populaion Generae M iniial populaion. Here, M. Generae each individual so as o saisfy (). B. Calculaion of Finess Firs of all, calculae forecasing value. There are 6 monhly daa for each case. We use 4 daa(s o 4h) and remove rend by he mehod saed in secion. Then we calculae monhly raio by he mehod saed in secion 4. Afer removing monhly rend, he mehod saed in secion is applied and Exponenial Smoohing Consan wih minimum variance of forecasing error is esimaed. Then sep forecas is execued. Thus, daa is shifed o nd o 5h 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 5h o 6h daa. Final forecasing daa is obained by muliplying monhly raio and rend. Variance of forecasing error is calculaed by (4). Calculaion of finess is exhibied in Figure 6. Figure 6: The flow of calculaion of finess Scaling [5] is execued such ha finess becomes large when he variance of forecasing error becomes small. Finess is defined as follows. f (, ) U (, ) () Where U is he maximum of (, ) during he pas generaion. Here, W is se o be 5. W C. Selecion Selecion is execued by he combinaion of he general eliis selecion and he ournamen selecion. Eliism is execued unil he number of new elies reaches he predeermined number. Afer ha, ournamen selecion is execued and seleced. D. Crossover Crossover is execued by he uniform crossover. Crossover rae is se as follows.

Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp://www.esjournals.org P c.7 () E.

Applicaion o he manufacurer s daa of saniary maerials Manufacurer s daa of saniary maerials from Sepember 9 o Augus are analyzed.

6 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// P c.7 () E. Muaion Muaion rae is se as follows. P m.5 () Muaion is execued o each bi a he probabiliy P m, herefore all muaed bis in he populaion M becomes P m M NUMERICAL EXAMPLE 7. Applicaion o he manufacurer s daa of saniary maerials Manufacurer s daa of saniary maerials from Sepember 9 o Augus are analyzed. Furhermore, GA resuls are compared wih he calculaion resuls of all considerable cases in order o confirm he effeciveness of GA approach. Firs of all, graphical chars of hese ime series daa are exhibied in Figure Execuion Resuls Figure 7: C GA execuion condiion is exhibied in Table 7. Table7: GA Execuion Condiion GA Execuion Condiion Populaion Maximum Generaion 5 Crossover rae.7 Figure 7: A Muaion raio.5 Scaling window size 5 The number of elies o reain Tournamen size We made imes repeiion and he maximum, average, minimum of he variance of forecasing error and he average of convergence generaion are exhibied in Table 7 and 7. Figure 7: B 4

7 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Table7: GA execuion resuls (Monhly raio is no used) The variance of forecasing error Minimum Maximum Average Average of convergence generaion Produc A,654,588,77,,576,4 4,999,5,5. Produc B 8,75,96,49,6,4,66 9,6,677,8 9.5 Produc C 59,97,9,84 6,87,64,48 59,4,96, Table7: GA execuion resuls (Monhly raio is used) The variance of forecasing error Average of convergence generaion Minimum Maximum Average Produc A 4,58,4,586 5,775,7,46 5,9,54,94.6 Produc B 4,97,5,69 7,46,95,897 4,58,64,596. Produc C 7,557,6,4 44,78,79,6 8,6,464,56 8. In all cases, he variance of forecasing error for he case monhly raio is used is smaller han he case monhly raio is no used. The minimum variance of forecasing error of GA coincides wih hose of he calculaion of all considerable cases and i shows he heoreical soluion. Alhough i is a raher simple problem for GA, we can confirm he effeciveness of GA approach. Furher sudy for complex problems should be examined hereafer. Figure77: A (Monhly raio is used) Figure76: A (Monhly raio is no used) 5

8 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Figure78: B (Monhly raio is no used) Figure7: C (Monhly raio is no used) Figure79: B (Monhly raio is used) Figure7: C (Monhly raio is used) Nex, opimal weighs and heir genes are exhibied in Table 7 4,75. Table74: Opimal weighs and heir genes (Monhly raio is no used) Daa posiion of he bi Produc A. Produc B.64.6 Produc C.8. 6

9 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Table75: Opimal weighs and heir genes (Monhly raio is used) Daa posiion of he bi Produc A.96.4 Produc B Produc C.9.9 In he case monhly raio is used, he s + nd order funcion model is bes in Produc A and Produc C, while Produc B has seleced s + rd order funcion as he bes one. Parameer esimaion resuls for he rend of equaion () using leas square mehod are exhibied in Table 76 for he case of s o 4h daa. Table76: Parameer esimaion resuls for he rend of equaion () Daa Produc A a Produc B Produc C b a b c a b c d Trend curves are exhibied in Figure Figure76: Trend of A 7

10 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Figure77: Trend of B Figure78: Trend of C 8

11 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Calculaion resuls of Monhly raio for s o 4h daa are exhibied in Table 77. Table77: Parameer Esimaion resul of Monhly raio Dae Produc A Produc B Produc C Esimaion resul of he smoohing consan of minimum variance for he s o 4h daa are exhibied in Table 78, 79. Table 78: Smoohing consan of Minimum Variance of equaion (7) (Monhly raio is no used) Dae ρ α Produc A Produc B Produc C Table 79:Smoohing consan of Minimum Variance of equaion (7) (Monhly raio is used) Dae, ρ α Produc A Produc B Produc C Forecasing resuls are exhibied in Figure Figure 7: Forecasing Resul of A 9

12 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// Figure 7: Forecasing Resul of B Figure 7: Forecasing Resul of C 4

13 Volume 4 No. 4, April 4 ISSN 4985 Inernaional Journal of Informaion and Communicaion Technology Research 4 ICT Journal. All righs reserved hp:// 7. Remarks In all cases, he variance of forecasing error for he case monhly raio is used is smaller han he case monhly raio is no used. In he case monhly raio is used, he s + nd order funcion model is bes in Produc A and Produc C, while Produc B has seleced s + rd order funcion as he bes one. The minimum variance of forecasing error of GA coincides wih hose of he calculaion of all considerable cases and i shows he heoreical soluion. Alhough i is a raher simple problem for GA, we can confirm he effeciveness of GA approach. Furher sudy for complex problems should be examined hereafer. 8. 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 us which saisfied minimum variance of forecasing error. Generally, smoohing consan was seleced arbirary. Bu in his paper, we uilized above saed heoreical soluion. Firsly, we made esimaion of ARMA model parameer and hen esimaed smoohing consans. Thus heoreical soluion was derived in a simple way and i migh be uilized in various fields. Furhermore, combining he rend removal mehod wih his mehod, we aimed 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 removal. Geneic Algorihm was uilized o search he opimal weigh for he weighing parameers of linear and nonlinear funcion. 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 non monhly rend removing daa. Then forecasing was execued on hese daa. The new mehod shows ha i is useful for he ime series ha has various rend characerisics. I is our fuure works o ascerain he effeciveness of his mehod by examining many oher cases. REFERENCES []. Box Jenkins. (994) Time Series Analysis Third Ediion,Prenice Hall. []. R.G. Brown. (96) Smoohing, Forecasing and Predicion of Discree Time Series, Prenice Hall. []. Hidekasu Tokumaru e al. (98) Analysis and Measuremen Theory and Applicaion of Random daa Handling, Baifukan Publishing. [4]. Kengo Kobayashi. (99) Sales Forecasing for Budgeing, ChuokeizaiSha Publishing. [5]. Peer R.Winers. (984) Forecasing Sales by Exponenially Weighed Moving Averages, Managemen Science,Vol6,No., pp. 44. [6]. Kasuro Maeda. (984) Smoohing Consan of Exponenial Smoohing Mehod, Seikei Universiy Repor Faculy of Engineering, No.8, pp [7]. M.Wes and P.J.Harrison. (989) Baysian Forecasing and Dynamic Models,SpringerVerlag,New York. [8]. Seinar Ekern. (98) Adapive Exponenial Smoohing Revisied,Journal of he Operaional Research Sociey, Vol. pp [9]. F.R.Johnson. (99) Exponenially Weighed Moving Average (EWMA) wih Irregular Updaing Periods, Journal of he Operaional Research Sociey, Vol.44,No.7 pp.776. []. Spyros Makridakis and Robea L.Winkler. (98) Averages of Forecass;Some Empirical Resuls, Managemen Science,Vol.9, No.9, pp []. Naohiro Ishii e al. (99) Bilaeral Exponenial Smoohing of Time Series, In.J.Sysem Sci., Vol., No.8, pp []. Kazuhiro Takeyasu. (996) Sysem of Producion, Sales and Disribuion, ChuokeizaiSha Publishing. []. Kazuhiro Takeyasu and Kazuko Nagao.(8) Esimaion of Smoohing Consan of Minimum Variance and is Applicaion o Indusrial Daa, Indusrial Engineering and Managemen Sysems, vol.7, no., pp [4]. Masaosi Sakawa. Masahiro Tanaka. (995)Geneic Algorihm Asakura Pulishing Co., Ld. [5]. Hioshi Iba.()Geneic Algorihm Igaku Publishing. [6]. Daisuke Takeyasu and Kazuhiro Takeyasu.() Esimaion of Smoohing Consan wih Opimal Parameers of Weigh in he Medical Case of A Tube and A Caheer, (Inernaional Journal of research In Medical and Healh Science (IJRMHS), Vol., No.4, pp.) 4

UTILIZATION OF GENETIC ALGORITHM TO IMPROVE FORECASTING ACCURACY AN APPLICATION TO THE DATA OF A TUBE AND A CATHETER

Sep. 0.Vol., No. ISSN07-08 0 IJRMHS & K.A.J. All righs reserved UTILIZATION OF GENETIC ALGORITHM TO IMPROVE FORECASTING ACCURACY AN APPLICATION TO THE DATA OF A TUBE AND A CATHETER Daisuke Takeyasu, Kazuhiro