Forecasting of Intermittent Demand Data in the Case of Medical Apparatus

Transcription

1 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Forecasing of Inermien Demand Daa in he Case of Medical Apparaus Kazuhiro Takeyasu, Daisuke Takeyasu Absrac Inermien daa are ofen seen in indusries. Bu i is raher difficul o make forecasing in general. In recen years, he needs for inermien demand forecasing are increasing because of he consrains of sric Supply Chain Managemen. How o improve he forecasing accuracy is an imporan issue. There are many researches made on his. Bu here are rooms for improvemen. In his paper, a new mehod for cumulaive forecasing mehod is proposed. The daa is cumulaed and o his cumulaed ime series, he following mehod is applied o improve he forecasing accuracy. Focusing ha he equaion of exponenial smoohing mehod(esm) is equivalen o (,) order ARMA model equaion, he 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 he forecasing accuracy. An approach o his mehod is execued in he following mehod. Trend removing by he combinaion of linear and nd order non-linear funcion and 3 rd order non-linear funcion is execued o he producion daa of Medical Apparaus (Medical hermography and Bioelecric phenomenon inspecion equipmen). The weighs for hese funcions are se 0.5 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 non monhly rend removing daa. Then forecasing is execued on hese daa. The forecasing resul is compared wih hose of he non-cumulaive forecasing mehod. The new mehod shows ha i is useful for he forecasing of inermien demand daa. The effeciveness of his mehod should be examined in various cases. Key Words inermien demand forecasing, minimum variance, exponenial smoohing mehod, forecasing, rend. I. INTRODUCTION Demand forecasing is he basis in supply chain managemen. In indusries, how o improve forecasing accuracy such as sales, shipping is an imporan issue. There are cases ha inermien demand forecasing is required. Bu he mere applicaion of he pas mehod does no bear good esimaion of parameers and exquisie forecasing. There are many researchers made on his. Based upon he Croson s model (Croson 97[], Box e al.008 []), Shensone and Hyndma (005)[3] analyzed he inermien demand forecasing. Froung e al. (0)[4] applied Neural Nework o inermien demand forecasing. Ghobbar and Friend (996)[5] have made applicaion o aircraf mainenance and invenory conrol. Tanaka e al. (0)[6] has buil sales forecasing model for book publishing, where hey have devised cumulaive forecasing mehod. In his paper, we furher develop his cumulaive forecasing mehod in order o improve he forecasing accuracy for inermien demand. A new mehod for cumulaive forecasing mehod is proposed. The daa is cumulaed and o his cumulaed ime series, he following mehod is applied o improve he forecasing 545

2 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 accuracy. 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 was proposed before by us which saisfied minimum variance of forecasing error[7]. 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 he forecasing accuracy. An approach o his mehod is execued in he following mehod. Trend removing by he combinaion of linear and nd order non-linear funcion and 3 rd order non-linear funcion is execued o he daa of Medical Apparaus (Medical hermography and Bioelecric phenomenon inspecion equipmen). The weighs for hese funcions are se 0.5 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 non-monhly rend removing daa. Then forecasing is execued on hese daa. The forecasing resul is compared wih hose of he non-cumulaive forecasing mehod. The new mehod shows ha i is useful for he forecasing of inermien demand daa. The effeciveness of his mehod should be examined in various cases. 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 non-linear funcion is inroduced for rend removing in secion 3. The Monhly Raio is referred in secion 4. Forecasing is execued in secion 5, and esimaion accuracy is examined. II. DESCRIPTION OF ESM USING ARMA MODEL [7] In ESM, forecasing a ime + is saed in he following equaion. xˆ x xˆ x xˆ xˆ () Here, ˆ x : Forecasing a x : Realized value a : Smoohing consan 0 () is re-saed as l0 l x l By he way, we consider he following (,) order ARMA model. xˆ () x x e e (3) Generally, p, q order ARMA model is saed as Here, : x p q ai x i e i j b e j j x Sample process of Saionary Ergodic Gaussian Process e :Gaussian Whie Noise wih 0 mean x,,, N, variance e MA process in (4) is supposed o saisfy converibiliy condiion. Uilizing he relaion ha we ge he following equaion from (3). E e e, e, 0 (4) 546

3 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 xˆ x e (5) 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 [][3]. Comparing wih (3) and (4), we obain From (), (6), a b (6) Therefore, we ge a b (7) 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 non-linear equaions which are described below. Le (4) be We express he auocorrelaion funcion of equaions which are well known [3]. ~ r k p i ~ x x a x (8) qk e j0 q j i i ~ x e b e (9) b b x~ as r k j k j j j ~ and from (8), (9), we ge he following non-linear ( k q) ~ r 0 0 q e j0 b j ( k q ) (0) For hese equaions, a 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 (3) (4) (7) (0), we ge q If we se a b ~ r0 b e ~ r b e () 547

4 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) he following equaion is derived. We can ge b as follows. In order o have real roos, mus saisfy From inveribiliy condiion, b mus saisfy Volume 3, ~ Issue, March 04 r ~ k k () r 0 b (3) b 4 b (4) (5) b From (3), using he nex relaion, (5) always holds. As b is wihin he range of Finally we ge b 0 b 0 b b 0 b 4 4 (6) which saisfy above condiion. Thus we can obain a heoreical soluion by a simple way. Here mus saisfy 0 (7) 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. III. TREND REMOVAL METHOD[7] As rend removal mehod, we describe he combinaion of linear and non-linear funcion. 548

5 [] Linear funcion We se ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 as a linear funcion. [] Non-linear funcion We se y a x (8) b y a (9) x b x c as a nd and a 3 rd order non-linear funcion. y a (0) 3 3x b3 x c3x d3 [3] The combinaion of linear and non-linear funcion We se a x b a x b x y () c 3 a x b β a x b x c x y () β d3 y γ a x b γ a x b x c 3 γ a x b x c x d as he combinaion of linear and nd order non-linear and 3 rd order non-linear funcion. Here,, β β, γ3 ( γ γ ). Comparaive discussion concerning (), () and (3) are described in secion 5. IV. MONTHLY RATIO[7] 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 x ij is a shipping daa of i-h year, j-h monh. Then, monhly raio x~ j j,, is calculaed as follows. i j 3 (3) L xij ~ L i x j L (4) xij L Monhly rend is removed by dividing he daa by (4). Numerical examples boh of monhly rend removal case and non-removal case are discussed in 5. A. Analysis Procedure V. FORECASTING THE PRODUCTION DATA Sum oal daa of producion daa of Medical Apparaus (Medical hermography and bioelecric phenomen on inspecion equipmen) from January 00 o December 0 are analyzed. These daa are obained fro m he Annual Repor of Saisical Invesigaion on Saisical-Survey-on-Trends-in-Pharmaceuical-Produc ion by Minisry of Healh, Labour and Welfare in Japan. The original daa and accumulaed daa are 549

6 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 exhibied in Table (Medical hermography) and Table (Bioelecric phenomenon inspecion equipmen ). Table. Original Daa and Accumulaed Daa in Medical hermography Original Daa Accumulaed Daa January / February / March / April / May / June / July / Augus / Sepember / Ocober / November / December / January /0 0 4 February /0 0 4 March /0 0 4 April / May / June / July / Augus /0 46 Sepember / Ocober / November / December / January /0 57 February /0 59 March / April / May / June /

7 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 July / Augus / Sepember / Ocober / November / December / Table. Original Daa and Accumulaed Daa in Bioelecric phenomenon inspecion equipmen Original Daa Accumulaed Daa January / February / March / April / May / June / July / Augus / Sepember / Ocober / November / December / January / February / March / April / May / June /0 64 July /0 85 Augus /0 306 Sepember / Ocober / November /0 393 December / January / February /

8 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 March / April / May / June / July /0 53 Augus / Sepember / Ocober / November / December / Analysis procedure is as follows. There are 36 monhly daa for each case. We use 4 daa ( o 4) and remove rend by he mehod saed in 3. Then we calculae monhly raio by he mehod saed in 4. Afer removing monhly rend, he mehod saed in 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 36h daa. To examine he accuracy of forecasing, variance of forecasing error is calculaed for he daa of 5h o 36h daa. Final forecasing daa is obained by muliplying monhly raio and rend. Forecasing error is expressed as: Variance of forecasing error is calculaed by: xˆ x (5) i i N N i i (6) i N i (7) N i B. Trend Removing Trend is removed by dividing original daa by,(),(),(3). The paerns of rend removal are exhibied in Table 3. Table 3: The paerns of rend removal Paern, are se 0.5 in he equaion () Paern, are se 0.5 in he equaion () Paern3 is shifed by 0.0 incremen in () Paern4 is shifed by 0.0 incremen in () Paern5 γ and γ are shifed by 0.0 incremen in (3) In paern and, he weigh of,,, are se 0.5 in he equaion (),(). In paern3, 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 paern5, he weigh of and are shifed by 0.0 incremen in (3) which saisfy he range 0. 00, The bes soluion is seleced which minimizes he variance of forecasing error. 55

9 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 C. Removing rend of monhly raio Afer removing rend, monhly raio is calculaed by he mehod saed in 4. 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 (6). There are cases ha we canno obain a heoreical soluion because hey do no saisfy he condiion of (5). 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 inervals. The inermien demand daa ofen include 0 daa. If here are so many 0 daa, here is a case we canno calculae he heoreical soluion of smoohing consan In ha case, we add very iny daa which is no 0 bu close o 0 ha does no affec anyhing in calculaing parameers (i.e. negligible small). E. Forecasing and Variance of Forecasing Error Uilizing smoohing consan esimaed in he previous secion, forecasing is execued for he daa of 5h o 36h daa. Final forecasing daa is obained by muliplying monhly raio and rend. Variance of forecasing error is calculaed by (7). As we have made accumulaed daa case and iny daa close o 0 added case, we have he following cases alogeher.. Non Monhly Trend Removal () Accumulaed Daa () Non Accumulaed Daa (-) Forecasing from he Accumulaed daa (Accumulaed forecasing daa a ime n-accumulaed daa (a ime n-) ) A. Paern, B. Paern, C. Paern 3, D. Paern 4, E. Paern 5 (-) Forecasing from he iny daa close o 0 added case A. Paern, B. Paern, C. Paern 3, D. Paern 4, E. Paern 5. Monhly Trend Removal () Accumulaed Daa () Non Accumulaed Daa (-) Forecasing from he Accumulaed daa (Accumulaed forecasing daa a ime n-accumulaed daa (a ime n-) ) A. Paern, B. Paern, C. Paern 3, D. Paern 4, E. Paern 5 (-) Forecasing from he iny daa close o 0 added case A. Paern, B. Paern, C. Paern 3, D. Paern 4, E. Paern 5 We can make forecasing by reversely making he daa from he forecasing accumulaed daa, i.e., ha is shown a (-). Now, we show hem a Figure hrough 6. Figure, and 3 show he Non-monhly Trend Removal Case in Medical hermography. I includes all cases classified above. Figure shows he Accumulaed Daa Case in Non-Monhly Trend Removal. Figure shows he Forecasing from he Accumulaed Daa Case in Non-Monhly Trend Removal. Figure 3 shows he Forecasing from he iny daa close o 0 added case in Non-Monhly Trend Removal. Table 4,5 and 6 show he corresponding variance of forecasing error for each Figure, and

10 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Fig Forecasing from he Accumulaed Daa Case in Non-Monhly Trend Removal (-()) Table 4 Variance of Forecasing Error (-()) Paern Paern Paern3 Paern4 Paern Fig Forecasing from he Accumulaed Daa Case in Non-monhly Trend Removal (-(-)) Table 5 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern

11 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Fig 3 Forecasing from he Tiny Daa close o 0 Added case in Non-Monhly Trend Removal (-(-)) Table 6 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Nex, we see he Monhly Trend Removal case. Figure 4,5 and 6 show he Monhly Trend Removal Case in Medical hermography. I includes all cases classified above. Figure 4 shows he Accumulaed Daa Case in Monhly Trend Removal. Figure 5 shows he Forecasing from he Accumulaed Daa Case in Monhly Trend Removal. Figure 6 shows he Forecasing from he iny daa close o 0 added case in Monhly Trend Removal. Table 7,8 and 9 show he corresponding variance of forecasing error for each Figure 4,5 and 6. Fig 4 Accumulaed Daa case in Monhly Trend Removal (-()) 555

12 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Table 7 Variance of Forecasing Error (-()) Paern Paern Paern3 Paern4 Paern Fig 5 Forecasing from he accumulaed Daa case in Monhly Trend Removal (-(-)) Table 8 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Fig 6 Forecasing from he Tiny Daa close o 0 Added case in Monhly Trend Removal (-(-)) 556

13 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Table 9 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Table 0 shows he summary for Medical hermography by he Variance of forecasing error. Table 0 Summary for Medical hermography Monhly Trend Removal Non Monhly Trend Removal Na me Medical hermography Accumu laed Daa Forecasing Value -Accumulaed Value Tiny daa close o 0 added case Accumu laed Daa Forecasing Value -Accumulaed Value Tiny daa close o 0 added case Minimum variance of Forecasing Error Now, we proceed o he case of bioelecric phenomenon inspecion equipmen. Figure 7, 8 and 9 show he Non-monhly Trend Removal Case in Bioelecric phenomenon inspecion equipmen. I includes all cases classified above. Figure 7 shows he Accumulaed Daa Case in Non-Monhly Trend Removal. Figure 8 shows he Forecasing from he Accumulaed Daa Case in Non-Monhly Trend Removal. Figure 9 shows he Forecasing from he iny daa close o 0 added case in Non-Monhly Trend Removal. Table, and 3 show he corresponding variance of forecasing error for each Figure 7,8 and 9. Fig 7 Accumulaed Daa case in Non-Monhly Trend Removal (-()) Table Variance of Forecasing Error (-()) Paern Paern Paern3 Paern4 Paern

14 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Fig 8 Forecasing from he accumulaed Daa case in Non-Monhly Trend Removal (-(-)) Table Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Fig 9 Forecasing from he Tiny Daa close o 0 Added case in Non-Monhly Trend Removal (-(-)) Table 3 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern

15 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Nex, we see he Monhly Trend Removal case. Figure 0, and show he Monhly Trend Removal Case in Bioelecric phenomenon inspecion equipmen. I includes all cases classified above. Figure 0 shows he Accumulaed Daa Case in Monhly Trend Removal. Figure shows he Forecasing from he Accumulaed Daa Case in Monhly Trend Removal. Figure shows he Forecasing from he iny daa close o 0 added case in Monhly Trend Removal. Table 4,5 and 6 show he corresponding variance of forecasing error for each Figure 0, and. Fig 0 Accumulaed Daa case in Monhly Trend Removal (-()) Table 4 Variance of Forecasing Error (-()) Paern Paern Paern3 Paern4 Paern Fig Forecasing from he accumulaed Daa case in Monhly Trend Removal (-(-)) 559

16 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Table 5 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Fig Forecasing from he Tiny Daa close o 0 Added case in Monhly Trend Removal (-(-)) Table 6 Variance of Forecasing Error (-(-)) Paern Paern Paern3 Paern4 Paern Table 7 shows he summary for bioelecric phenomenon inspecion equipmen by he Variance of forecasing error. Table 7 Summary for Bioelecric phenomenon inspecion equipmen Monhly Trend Removal Non Monhly Trend Removal N a m e Bioelecric phenomenon inspecion equipmen Accum ulaed Daa Forecasing Value -Accumulaed Value Tiny daa close o 0 added case Accum ulaed Daa Forecasing Value -Accumulaed Value Tiny daa close o 0 added case Minimum variance of Forecasing Error

17 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 F. Remarks In boh cases, Non-Monhly Trend Removal case was beer han Monhly Trend Removal case. This is because here was no ypical monhly rend in boh cases and he resul had refleced hem. In he Non-Monhly Trend Removal case for Medical hermography, forecasing from he iny daa close o 0 added case (-(-)) was beer han hose of Accumulaed daa case (-(-)). On he oher hand, in he Non-Monhly Trend Removal case for Bioelecric phenomenon inspecion equipmen, forecasing from he accumulaed daa case (-(-)) was beer han hose of he iny daa close o 0 added case (-(-)). By he way, forecasing of accumulaed daa (-(), -()) shows raher good resul. I can be used as one of he ool o decide when and how much volume o procure he maerials ec.. I can be uilized as a new mehod o procure in supply chain managemen. VI. CONCLUSION The needs for inermien demand forecasing are increasing. In his paper, a new mehod for cumulaive forecasing mehod was proposed. The daa was cumulaed and o his cumulaed ime series, he following mehod was applied o improve he forecasing accuracy. Focusing 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 arbirarily. 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. Furhermore, combining he rend removing mehod wih his mehod, we aimed o improve he forecasing accuracy. An approach o his mehod was execued in he following mehod. Trend removing by he combinaion of linear and nd order non-linear funcion and 3 rd order non-linear funcion was execued o he producion daa of Medical Apparaus (Medical hermography and bioelecric phenomenon inspecion equipmen). The weighs for hese funcions were se 0.5 for wo paerns a firs and hen varied by 0.0 incremen for hree paerns and opimal weighs were searched. 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 forecasing resul was compared wih hose of he non-cumulaive forecasing mehod. The new mehod shows ha i is useful for he forecasing of inermien demand daa. Among hem, forecasing of accumulaed daa (-(), -()) shows raher good resul. I can be used as one of he ool o decide when and how much volume o procure he maerials ec.. I can be uilized as a new mehod o procure in supply chain managemen. The effeciveness of his mehod should be examined in various cases. REFERENCES [] Croson, J.D. (97), Forecasing and sock conrol for inermien demands, Opimal Research Quarerly 3(3), [] Box, G.E.P., Jenkins, G.M.& Reinsel, G.C. (008), Time Series analysis: forecasing and conrol, Wiley, 4 h edn. [3] Lydia Shensone and Rob J. Hyndma,(005), Sochasic models underlying Croson s mehod for inermien demand forecasing, Journal of Forecasing, 4: [4] Nguyen Khoa Vie Froung, Shin Sangmun, Vo Thanh Nha, Kwon Ichon, (January -4, 0), Inermien Demand forecasing by using Neural Nework wih simulaed daa, Proceedings of he 0 Inernaional Engineering and Operaions Managemen Kuala Lumpur, Malaysia, pp [5] Ghobbar, A.A., and Friend, C.H., (996). Aircraf mainenance and invenory conrol using he recorda poin sysem, Inernaional Journal of Producion Research, Vol.34, No.0, pp [6] Kenji Tanaka, Yukihiro Miyamura and Jing Zhang, (November 0), The Cluser Grouping Approach of Sales 56

18 ISSN: ISO 900:008 Cerified Inernaional Journal of Engineering Science and Innovaive Technology (IJESIT) Volume 3, Issue, March 04 Forecasing Model for Book Publishing, Inernaional Journal of Japan Associaion for Managemen Sysems, Vol.4, No.,pp.3-35 [7] Kazuhiro Takeyasu and Keiko Nagaa.(00) Esimaion of Smoohing Consan of Minimum Variance wih Opimal Parameers of Weigh, Inernaional Journal of Compuaional Science Vol.4,No.5, pp AUTHOR BIOGRAPHY Kazuhiro Takeyasu is a Professor of College of Business Adminisraion, Tokoha Universiy, and was a Professor of Osaka Prefecure Universiy, Japan. He received a Docoral Degree from he Graduae School of Engineering a Tokyo Meropolian Insiue of Technology, Japan in 004. His eaching and research ineress are ime series analysis, sysem idenificaion and markeing. Daisuke Takeyasu is now a Cerified Social Worker. He graduaed Tokai Universiy. He received a MBA Degree from he Open Universiy of Japan in 04. His main research ineress are ime series analysis and markeing. 56