A Hybrid Method to Improve Forecasting Accuracy Utilizing Genetic Algorithm An Application to the Data of Operating equipment and supplies

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1 A Hyrd Mehod o Improve Forecasng Accuracy Ulzng Genec Algorhm An Applcaon o he Daa of Operang equpmen and supples Asam Shara Tax Corporaon Arkne, Shzuoka Cy, Japan, e-mal: a-shara@arkne.nfo Dasuke Takeyasu The Open Unversy of Japan, e-mal: ake@homal.co.jp Kazuhro Takeyasu Tokoha Unversy, e-mal: akeyasu@fj.okoha-u.ac.jp Asrac - 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, a hyrd mehod s nroduced and plural mehods are compared. Focusng ha he equaon of exponenal smoohng mehod(esm) s equvalen o (,) order ARMA model equaon, new mehod of esmaon of smoohng consan n exponenal smoohng mehod s proposed efore y us whch sasfes mnmum varance of forecasng error. Trend removng y he comnaon of lnear and nd order non-lnear funcon and rd order non-lnear funcon s execued o he daa of Operang equpmen and supples for hree cases (An njecon devce and a puncure devce, A serlzed hypodermc needle and A serlzed syrnge). Genec Algorhm s ulzed o search he opmal wegh for he weghng parameers of lnear and non-lnear funcon. For he comparson, monhly rend s removed afer ha. Theorecal soluon of smoohng consan of ESM s calculaed for oh of he monhly rend removng daa and he non monhly rend removng daa. Then forecasng s execued on hese daa. The new mehod would e useful for he me seres ha has varous rend characerscs. Key Words: mnmum varance, forecasng, operang equpmen and supples. INTRODUCTION Many mehods for me seres analyss have een presened such as Auoregressve model (AR Model), Auoregressve Movng Average Model (ARMA Model) and Exponenal Smoohng Mehod (ESM). In hs paper, a revsed forecasng mehod s proposed. In makng forecas such as producon daa, rend removng mehod s devsed. Trend removng y he comnaon of lnear and nd order non-lnear funcon and rd order nonlnear funcon s execued o he daa of Operang equpmen and supples for hree cases (An njecon devce and a puncure devce, A serlzed hypodermc needle and A serlzed syrnge). These Operang equpmen and supples are used for medcal use. Genec Algorhm s ulzed o search he opmal wegh for he weghng parameers of lnear and non-lnear funcon. For he comparson, monhly rend s removed afer ha. Theorecal soluon of smoohng consan of ESM s calculaed for oh of he monhly rend removng daa and he non monhly rend removng daa. Then forecasng s execued on hese daa. Ths s a revsed forecasng mehod. The new mehod would e useful for he me seres ha has varous rend characerscs. 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. The Monhly Rao s referred n secon. Forecasng Accuracy s defned n secon 5. Opmal weghs are searched n secon. Forecasng s carred ou n secon, and esmaon accuracy s examned.

2 . DESCRIPTION OF ESM USING ARMA MODEL In ESM, forecasng a me + s saed n he followng equaon. Here, ˆ xˆ xˆ x xˆ () x : forecasng a x : realzed value a xˆ x () : smoohng consan () s re-saed as l 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 e j j Here, x : Sample process of Saonary Ergodc Gaussan Process x,,, N, e :Gaussan Whe Nose wh mean MA process n (5) s supposed o sasfy converly condon. Ulzng he relaon ha E e e, e, we ge he followng equaon from (). (5) xˆ x e () Operang hs scheme on +, we fnally ge xˆ xˆ xˆ e x xˆ 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 (,,) order ARIMA model varance e () ecause s order AR parameer s. Comparng wh () and (5), we oan From (), (), Therefore, we ge a a 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 (5) e p () ~ x x a x () q j ~ x e e () j j We express he auocorrelaon funcon of x~ as ~ r k and from (), (), we ge he followng non-lnear equaons whch are well known. ~ r k ~ r qk e j q e j j k j j ( k q) ( k q ) () 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 () (5) () (), we ge If we se q a ~ r e ~ r e ()

3 ~ r ~ k k () r he followng equaon s derved. We can ge as follows. () (5) In order o have real roos, mus sasfy () From nverly condon, mus sasfy From (), usng he nex relaon, () always holds. As s whn he range of Fnally we ge () whch sasfes aove condon. Thus we can oan a heorecal soluon y a smple way. 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 h and s order auocorrelaon funcon.. TREND REMOVAL METHOD As rend removal mehod, we descre he comnaon of lnear and non-lnear funcon. [] Lnear funcon We se as a lnear funcon. [] Non-lnear funcon We se y a x () y a () x x c y a () x x cx d as a nd and a rd order non-lnear funcon. a,, ) ( c and a,, c, ) are also parameers for a nd and a rd ( d order non-lnear funcons whch are esmaed y usng leas square mehod. [] The comnaon of lnear and non-lnear funcon. We se y a x a x x c a x x c x d (),,, () as he comnaon lnear and nd order non-lnear and rd order non-lnear funcon. Trend s removed y dvdng he orgnal daa y (). The opmal weghng parameer,,are deermned y ulzng GA. GA mehod s, precsely descred n secon.. MONTHLY RATIO For example, f here s he monhly daa of L years as saed ellow:,, L j,, x j Where, 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 ~ x j j,, s calculaed as follows.

4 L xj ~ L x j L () xj L j Monhly rend s removed y dvdng he daa y (). Numercal examples oh of monhly rend removal case an d non-removal case are dscussed n. 5. FORECASTING ACCURACY Forecasng accuracy s measured y calculang he varance of he forecasng error. Varance of forecasng error s calculaed y: N () N Where, forecasng error s expressed as: xˆ x (5) N N (). SEARCHING OPTIMAL WEIGHTS UTILIZING GA lengh of doman of varale s -=, seven s are requred o express varales. The nary srngs <, ~,> s decoded o he [,] doman real numer y he followng procedure. [] Procedure :Conver he nary numer o he nary-coded decmal.,,,,,, 5 X () Procedure : Conver he nary-coded decmal o he real numer. The real numer = (Lef hand sarng pon of he doman) + X ' ((Rgh hand endng pon of he doman)/ ( )) () varale s expressed y s, herefore varales needs s.. The flow of Algorhm The flow of algorhm s exhed n Fgure -.. Defnon of he prolem We search,, of () whch mnmzes () y ulzng GA. By (), we only have o deermne and. (()) s a funcon of and, herefore we express hem as, ). Now, we pursue he followng: ( Mnmze:, ) ( sujec o:,, () We do no necessarly have o ulze GA for hs prolem whch has small memer of varales. Consderng he possly ha varales ncrease when we use logscs curve ec n he near fuure, we wan o asceran he effecveness of GA.. The srucure of he gene Gene s expressed y he nary sysem usng {,}. Doman of varale s [,] from (). We suppose ha varales ake down o he second decmal place. As he Fgure -: The flow of algorhm A. Inal Populaon Generae M nal populaon. Here, M. Generae each ndvdual so as o sasfy ().

5 B. Calculaon of Fness Frs of all, calculae forecasng value. There are monhly daa for each case. We use daa(s o h) and remove rend y he mehod saed n secon. Then we calculae monhly rao y he mehod saed n secon. Afer removng monhly rend, he mehod saed n secon 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 h daa s execued consecuvely, whch fnally reaches forecas of h daa. To examne he accuracy of forecasng, varance of forecasng error s calculaed for he daa of 5h o h daa. Fnal forecasng daa s oaned y mulplyng monhly rao and rend. Varance of forecasng error s calculaed y (). Calculaon of fness s exhed n Fgure -. Selecon s execued y he comnaon of he general els selecon and he ournamen selecon. Elsm s execued unl he numer of new eles reaches he predeermned numer. Afer ha, ournamen selecon s execued and seleced. D. Crossover Crossover s execued y he unform crossover. Crossover rae s se as follows. E. Muaon Muaon rae s se as follows. P. () c P.5 () m Muaon s execued o each a he proaly P m, herefore all muaed s n he populaon M ecomes P m M.. NUMERICAL EXAMPLE. Applcaon o he orgnal producon daa o f Wheelchars The daa of Operang equpmen and supples for hree cases (An njecon devce and a puncure devce, A serlzed hypodermc needle and A serlzed syrnge) from January o Decemer are analyzed. These daa are oaned from he Annual Repor of Sascal Invesgaon on Sascal-Survey-on-Trends-n- Pharmaceucal-Producon y Mnsry of Healh, Laour and Welfare n Japan. Furhermore, GA resuls are compared wh he calculaon resuls of all consderale cases n order o confrm he effecveness of GA approach.. Execuon Resuls Fgure -:The flow of calculaon of fness Scalng [] s execued such ha fness ecomes large when he varance of forecasng error ecomes small. Fness s defned as follows. Where U f, ) U (, ) () ( s he maxmum of, ) durng he pas ( W generaon. Here, W s se o e 5. C. Selecon GA execuon condon s exhed n Tale -. Tale-: GA Execuon Condon GA Execuon Condon Populaon Maxmum Generaon 5 Crossover rae. Muaon rao.5 Scalng wndow sze 5 The numer of eles o rean Tournamen sze

6 We made mes repeon and he maxmum, average, mnmum of he varance of forecasng error and he average of convergence generaon are exhed n Tale - and -. Tale-: GA execuon resuls(monhly rao s no used) Food No The varance of forecasng error Average of An njecon devce and a puncure devce A serlzed hypodermc needle A serlzed syrnge Maxmum Average Mnmum 55,55, 5,,,,,5,,5 5,, 5,,,,,5,,,,,,,5,,555,, convergence generaon Fgure-:Convergence Process n he case of An njecon devce and a puncure devce (Monhly rao s no used) Fgure-:Convergence Process n he case of A serlzed hypodermc needle (Monhly rao s used) Tale-: GA execuon resuls(monhly rao s used) Food No The varance of forecasng error Average of An njecon devce and a puncure devce A serlzed hypodermc needle A serlzed syrnge Maxmum Average Mnmum,5,,,,, 5,,,5,5,,,, 55,5,,, 5,,,5,,5, 5,,, convergence generaon The varance of forecasng error for he case monhly rao s no used s smaller han he case monhly rao s used n A serlzed hypodermc needle. Oher cases had good resuls n he case monhly rao was used. The mnmum varance of forecasng error of GA concdes wh hose of he calculaon of all consderale cases and shows he heorecal soluon. Alhough s a raher smple prolem for GA, we can confrm he effecveness of GA approach. Furher sudy for complex prolems should e examned hereafer.... Fgure-:Convergence Process n he case of A serlzed syrnge (Monhly rao s no used) Fgure-:Convergence Process n he case of An njecon devce and a puncure devce (Monhly rao s used)

7 Fgure-5:Convergence Process n he case of A serlzed hypodermc needle (Monhly rao s no used) Fgure-:Trend of An njecon devce and a puncure devce Fgure-:Convergence Process n he case of A serlzed syrnge (Monhly rao s used) The lnear funcon model s es n A serlzed syrnge. An njecon devce and a puncure devce seleced s + nd order funcon as he es one. A serlzed hypodermc needle seleced s + nd + rd order funcon as he es one. These resuls were same for oh of Monhly rao s no used and Monhly rao s no used. Parameer esmaon resuls for he rend of equaon () usng leas square mehod are exhed n Tale - for he case of s o h daa. Tale-: Parameer esmaon resuls for he rend of equaon () Daa a a c a c d An njec - - on dev. 5 5 ce and...5. a puncu.... re devc e A serl zed hyp odermc needle A serl zed syr nge Trend curves are exhed n Fgure Fgure-:Trend of A serlzed hypodermc needle Fgure-:Trend of A serlzed syrnge Calculaon resuls of Monhly rao for s o h daa are exhed n Tale -5. Tale-5: Parameer Esmaon resul of Monhly rao Dae. 5 An nj eco n devc e and a punc 5 5 ure d evce.. 5

8 A ser lzed hypod ermc needle A ser lzed s yrnge Forecasng resuls are exhed n Fgure was used was es for A serlzed syrnge case. s + nd funcon model n he case Monhly rao was used was es for An njecon devce and a puncure devce case. s + nd + rd funcon model n he case Monhly rao was no used was es for A serlzed hypodermc needle case. The mnmum varance of forecasng error of GA concdes wh hose of he calculaon of all consderale cases and shows he heorecal soluon. Alhough s a raher smple prolem for GA, we can confrm he effecveness of GA approach. Furher sudy for complex prolems should e examned hereafer..conclusion Fgure -: Forecasng Resul of An njecon devce and a puncure devce Fgure -: Forecasng Resul of A serlzed syrnge Fgure -: Forecasng Resul of A serlzed hypoder mc needle. Remarks The lnear funcon model n he case Monhly rao Comnng he rend removal mehod wh hs mehod, we amed o mprove forecasng accuracy. An approach o hs mehod was execued n he followng mehod. Trend removal y a lnear funcon was appled o he daa of Operang equpmen and supples for hree cases (An njecon devce and a puncure devce, A serlzed hypodermc needle and A serlzed syrnge). The comnaon of lnear and non-lnear funcon was also nroduced n rend removal. Genec Algorhm was ulzed o search he opmal wegh for he weghng parameers of lnear and non-lnear funcon. For he comparson, monhly rend was removed afer ha. Theorecal soluon of smoohng consan of ESM was calculaed for oh of he monhly rend removng daa and he non monhly rend removng daa. Then forecasng was execued on hese daa. The new mehod shows ha s useful for he me seres ha has varous rend characerscs. The effecveness of hs mehod should e examned n varous cases. REFERENCES [] Kazuhro Takeyasu and Kazuko Nagao.() Esmaon of Smoohng Consan of Mnmum Varance and s Applcaon o Indusral Daa, Indusral Engneerng and Managemen Sysems, vol., no., pp. -5. [] Masaos Sakawa. Masahro Tanaka. (5)Genec Algorhm Asakura Pulshng Co., Ld. [] Hosh Ia.()Genec Algorhm Igaku Pulshng. [] H.Takeyasu, Y.Hguch and K.Takeyasu. () A Hyrd Mehod o Improve Forecasng Accuracy Ulzng Genec Algorhm An Applcaon o he Daa of Processed Cooked Rce-, Indusral Engneerng and Managemen Sysems Vol., No., pp.-5