GENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING

Transcription

1 forcasing, dmand, gnic algorihm Grzgorz Chodak*, Wiold Kwaśnicki* GENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING Th mhod of forcasing sasonal dmand applying gnic algorihm is prsnd. Spcific form of usd dmand funcion is shown in h firs scion of h aricl. Nx h mhod of idnificaion of h funcion paramrs using gnic algorihms is discussd. In h final scion an xampl of applying proposd mhod o forcas ral dmand procss is shown. 1. INTRODUCTION Dmand forcasing is on of h mos imporan phas of a firm s dcision making procss. Managmn of marial flow, which is basd on jus-in-im sragy, would no b possibl o achiv wihou prcis simaion of h dmand. W can disinguish wo gnral dmand forcasing mhods, namly quaniaiv and qualiaiv mhods [2, 3, 6]. Th advancd compur sofwar ha nabls daa compuing and graphic prsnaion of daa can suppor boh of hm. Arificial inllignc (AI) chniqus (such as: gnic algorihms, fuzzy logic and nural nworks) will b xnsivly applid in h nx sag of forcasing mhods dvlopmn. Applying AI chniqus should improv corrcnss of sasonal dmand forcasing. A prsnaion of sasonal dmand forcasing mhod and is applicabiliy is h main objciv of his aricl. Idnificaion of dmand funcion s paramrs is basd on gnic algorihms (GA) approach. 2. DEMAND FUNCTION Shap of a dmand funcion should b infrrd from inrnal and xrnal nvironmns of ach firm. I is impossibl o propos gnral formula of a dmand funcion applicabl o all conomic condiions of diffrn firms. In som firms dmand could dpnd mainly on on facor bu in ohr firms ns of imporan facors influncing dmand can b idnifid. Thrfor h main problm is o poin ou ha facors corrcly. *Indusrial Enginring and Managmn Insiu, Wroclaw Univrsiy of Tchnology

2 Frqunly, as h firs approximaion, i is assumd ha h dmand funcion is a funcion of im (.g. linar, logarihmic, xponnial c) rprsning a rnd of dmand. I is wll known ha h pric is on of h mos imporan facors influncing dmand of almos all producs and srvics. Wha w propos in his aricl is a combinaion of classical rnd approach (i.., dmand as a funcion of im) and h conomic xbook dmand funcion (i.., dmand as a funcion of pric). A h firs sag of dvlopmn of his approach w propos rahr simpl form h dmand funcion, namly: D = M ( ) p D dmand (of producs or srvics); p pric; im; pric lasiciy. M() is funcion of im and i can rprsn a rnd bu whn h dmand is sasonal h funcion M() is priodical. To includ priodiciy in our dmand funcion w us h following formula: (1) C + B + A sin( ω + ϕ ) D = (2) p D - dmand; A ampliud of flucuaion; im; ω - frquncy (rprsning priodiciy); ϕ - horizonal offs C vrical offs B rnd facor pric lasiciy. Th abov funcion has six paramrs (A, ω, ϕ, C, B, ) which valus ough o b idnifid using ral daa. Applying analyical mhod for valuaion (idnificaion) valus of hs paramrs is rahr impossibl. Thrfor w propos o apply wll-known and fficin approach of GA o idnify ha paramrs. 2. IDENTIFICATION VALUES OF DEMAND FUNCTION PARAMETERS Mahmaical form of our dmand funcion can b considrd as a complx on. Thrfor w nd o look for a mor ffciv mhod of idnificaion. Using chniqus of arificial inllignc sms o b vry promising o solv his problm and wihin AI

3 chniqus gnic algorihms, which ar sarching procdurs basd on naural voluion, sms o h bs on. Th GA is dfind by is gnic opraors acing on bi srings (xampls of h opraors ar: crossovr, invrsion, and muaion) and is mhod of crdi allocaion (finss valuaion and slcion). Sarching for opimal soluion is possibl hanks o h gnic opraors, which allow for small and larg sps on h rou sarch forward h opimum in a paramr spac. I is don by xchanging scions of ach individual s dscripiv bi sring (h chromosom), invring slcions of a givn bi sring, or muaing a bi sring by alring on or mor bis. Th crdi-allocaion schm is basd on a rul ha individual has diffrn probabiliis of nring ino h nx gnraion accordingly o is rlaiv finss. Advanag of applying GA for idnificaion of dmand funcion s paramrs is ha hr is no consrains on a shap of dmand funcion and also on a numbr of paramrs. Th main disadvanag of GA is a lack of guaran ha global opimum is found wihin limid priod of im. Thr is a possibiliy o adjus GA o spcific rquirmns of ral problms. W can say ha GA chniqu is vry lasic on. Hr w will no discuss dails of GA, i can b found in many publicaions,.g., [1, 4, 5, 7]. 3. APPLYING GA TO IDENTIFICATION OF DEMAND FUNCTION S PARAMETERS Firs sp in applying GA o our problm of paramrs idnificaion is o dfin a finss funcion (F). I ough o b a masur of disanc of modl gnrad daa and collcd ral daa. In our xprimns w us h following finss funcion: C + B + A sin( ω + ϕ ) F = ( D - ( )) (3) p im; D ral sll in im; p pric; A, B, C, ω, ϕ, h sam as in 2 quaion hs valus of paramrs should b idnifid. Th srucur of individuals in our GA is following: vry individual consiss of 1 chromosom dividd ino six sgmns. Each sgmn rprsns on idnifid paramr. Chromosom is composd of 6 bis, dividd ino 1 bis sgmns, h sgmns cod valus of ampliud, vrical and horizonal offs, frqunnss, pric lasiciy facor and rnd facor. W us binary cod and vry gn is codd as on bi. An xampl of individual is prsnd in Figur 1. A B C ω ϕ Figur 1 GA individual Th siz of chromosom is compromis bwn xacnss of paramrs rprsnaion and quicknss of compuing. Th grar numbr of bis for ach paramr rprsnaion

4 allows for mor xac valuaion of is valu bu lngh of calculaion (simulaion) o rach h mor xac valu grows normously. W us wo gnic opraors, namly: - muaion invr on bi in a chromosom, - crossovr - xchang of bis sring bwn wo chromosoms. Th nx crucial phas of GA applicaion is propr dfiniion of h crdi-allocaion schm. I ough o b basd on rlaiv finss ovr h nir populaion of soluions o h problm so ha a givn individual has a probabiliy of nring ino h nx gnraion according o is rlaiv finss. In our xprimns w us modifid roul mhod. Each individual (chromosom) has assignd pic of a roul whl, which siz is proporional o is finss. Th whol whl rprsns a sum of finss of all individuals of h populaion [5]. Addiionally h individual wih bs finss is chosn o h nx gnraion (h rul don los h bs ). To mak GA sarch mor ffciv i is imporan o sima h scop of h paramrs domain. W can us our hurisic knowldg abou h xplord dmand. Namly, o valua h rang of ampliud (A) on can sima maximum and minimum amoun of sll. Maximum vrical offs (C) ough o b locad bwn maximum of dmand funcion and zro. Th rang of horizonal offs (ϕ) should b placd in h rang (-2π). Th scop of variabiliy of pric lasiciy () is rlad o individual good and can b s up by a sll managr (plausibl valus ar o 2). Rang of frquncy (ω) can also b s by managr who can prdic h sasonal dmand. Accpd valus of h rnd paramr ar rlad o dynamics of incras (dcras) of sll and for saic nvironmn i can s up bwn 1 and 1. If on of paramrs valus go from opimisaion xprimn is ging closr o h limi of rang of domain, h rag should b shifd in such way ha valu of paramr will b in h cnr of nw rang. If h rangs of paramrs ar s up accuraly, h valus of idnifid paramrs will also simad quickr and mor prcisly. I is imporan whn w mak calculaions (simulaion) on no vry powrful compur (lik PC class machin). Dfiniion of h sopping condiion is h imporan for propr GA applicaion. Thr ar wo possibiliis of sopping h algorihm: whn h sld numbr of gnraions is xcdd or whn finss is lss hn sld rror. Boh criria ar usd in our algorihm. 4. SIMULATION EXPERIMENTS W mak our simulaion xprimns using ral daa from middl-siz bakry Inrpik. Th analysd good is Wrocław roll. Afr many s xprimns w s up h following paramrs of GA: gnraion_siz = 6 crossing_ovr_probabiliy =,2 muaion_probabiliy =,5 vry_faur_siz = 1 gnraions_amoun = 5

5 Tabl 1 shows monhly slls of Wrocław roll (from March 98 o April 99). Pric is a wighd avrag calculad from daily sll prics in ach monh. Collcd daa suggs ha bakry mark is rahr sabl on bu also sasonal dmand is clarly visibl. Tabl 1 Sll of Wrocław roll and i s wholsal pric Monh Sll [h] Pric [zł] For daa from Tabl 1 w run an xprimn o idnify paramrs of dmand funcion givn by quaion (2). Th rsul of idnificaion xprimn (1 bs individuals) ar prsnd in Tabl 2. Ampliud A Frquncy ω Tabl 2 Rsuls of simulaion (idnificaion) xprimn Elasiciy Horizonal offs ϕ Vrical offs C Trnd facor B Finss F Th bs individual (i.. h boldd on in Tabl 2) is usd o calcula h ovrall rror (dfind by quaion (4)) and h parial rrors ar prsnd in Tabl 3. E = Dr Dp (4) )))

6 Dr ral dmand; Dp dmand cound using idnifid valus of paramrs (q. (2)); im. Th sum of rror quals o 5863 Wrocław rolls and h avrag rror quals o 586 i mans ha h rror is rlaivly small, namly 1.28%. Tabl 3 Compar of ral sll and sll calculad wih idnifid paramrs Monh Idnificaion Sll Error Sum: 5863 Graphic rprsnaion (s Figur 2) also suggss ha h fiing of h modl and is prdicion is adqua and accpabl bu mor sysmaic sudy of vrificaion of proposd mhod nd o b don. Th proposd approach ough o b sd on diffrn yps of goods and for wid rang of mark srucurs. 55, 5, 45, 4, Idnyfid daa Ral daa Figur 2 Ral and prdicd sll of Wrocław rolls A prdicion for nx wo monhs has bn mad using idnifid valus of paramrs. Th rsuls of his prdicion ar prsnd in Tabl 4. Comparing h prdicion wih ral sll of Wrocław rolls givs h sum of rror in nx 2 monhs qual o 2548, i.., 2.8% of ral sll. Tabl 4 Dmand forcas on nx mohs wih no chang of pric

7 Monh Ral sll Dmand forcas Error Pric [zł] SUMMARY Our rsuls suggs ha in som cass idnificaion of dmand funcion paramrs using gnic algorihms can b mor ffciv han in sandard sasonal dmand forcasing (.g. Winr modl [6]). In h prsnd aricl w hav xplord a siuaion whn dmand funcion is sasonal and has linar rnd bu our proposiion is mor gnral hr is no limiaion o adjus h shap of h dmand funcion o spcific rquirmns,.g. whn doubl sasonalnss is obsrvd, or h rnd is non-linar. Rsuls of our xprimns basd on diffrn ss of ral daa suggss good applicabiliy of our proposiion bu i has o b said ha his mhod dos no work wll for all daa. Thr can b wo gnral rasons of unsaisfid rsuls of forcasing using our approach: no corrc shap of dmand funcion or no appropria mhod of paramrs idnificaion. Th rsuls of prsnd mhod wr discussd wih Inrpik managmn officrs. Th rsuls of h discussions ar vry promising, in opinion of h officrs h proposd approach can b vry usful in supporing dcisions making on amoun of Wrocław rolls producion and can mak h producion mor ffciv. REFERENCES [1] GOLDBERG D., Algorymy gnyczn i ich zasosowania, Wydawnicwo Naukowo-Tchniczn, Warszawa 1998 [2] E. NOWAK (d.), Prognozowani gospodarcz mody, modl, zasosowania, przykłady, Warszawa 1998 [3] DITTMANN P., Mody prognozowania sprzdaży w przdsiębiorswi, Wydawnicwo Akadmii Ekonomicznj im. Oskara Langgo w Wrocławiu, Wrocław 1998 [4] HOLLAND H. J. Algorymy gnyczn, Świa Nauki, 9/92 [5] RUTKOWSKA D., PILIŃSKI M., RUTKOWSKI L., Sici nuronow, algorymy gnyczn i sysmy rozmy, Wydawnicwo Naukow PWN, Warszawa 1997 [6] SARIUSZ-WOLSKI Z., Ilościow mody zarządzania logisyczngo w przdsiębiorswi, Toruńska Szkoła Zarządzania, Warszawa 1997 [7] DRESS W.B., On modling and Conroling Inllign Sysms, Insrumnaion and Conrols Division, Oak Ridg Naional Laboraory