Abstract. 260 Business Intelligence Journal July IDENTIFICATION OF DEMAND THROUGH STATISTICAL DISTRIBUTION MODELING FOR IMPROVED DEMAND FORECASTING

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1 260 Busness Intellgence Journal July IDENTIFICATION OF DEMAND THROUGH STATISTICAL DISTRIBUTION MODELING FOR IMPROVED DEMAND FORECASTING Murphy Choy Mchelle L.F. Cheong School of Informaton Systems, Sngapore Management Unversty, 80, Stamford Road, Sngapore Emal: Abstract Demand functons for goods are generally cyclcal n nature wth characterstcs such as trend or stochastcty. Most exstng demand forecastng technques n lterature are desgned to manage and forecast ths type of demand functons. However, f the demand functon s lumpy n nature, then the general demand forecastng technques may fal gven the unusual characterstcs of the functon. Proper dentfcaton of the underlyng demand functon and usng the most approprate forecastng technque becomes crtcal. In ths paper, we wll attempt to explore the key characterstcs of the dfferent types of demand functon and relate them to known statstcal dstrbutons. By fttng statstcal dstrbutons to actual past demand data, we are then able to dentfy the correct demand functons, so that the the most approprate forecastng technque can be appled to obtan mproved forecastng results. We appled the methodology to a real case study to show the reducton n forecastng errors obtaned. Demand forecastng s an mportant aspect of busness operaton. It s applcable to many dfferent functonal areas such as sales, marketng and nventory management. Proper demand forecastng also allows for more effcent and responsve busness plannng. Because of the benefts t can brng, many ndustres have pad great attenton to demand varablty management and forecastng. Toursm and manufacturng are the two major ndustres who adopt a wde range of demand forecastng and varablty management solutons. There are many factors that affect demand varablty. These fluctuatons can be attrbuted to external factors such as changes n trends (rapd change n consumer preference) or events affectng that geographcal regon (such as major earthquakes or natural dsasters, major sports games). Occasonally, fluctuatons may also be due to marketng efforts whch has successfully pqued the consumer's nterest n the products. The supply structure n the economy can also affect the nature of the demand for a good. There are huge amounts of lterature dedcated to demand forecastng as well as demand varablty management. Most demand forecastng technques dscussed n the exstng lterature assumes that the demand functon s cyclcal n nature wth trend. The tme varyng nature of some demand functons also ncreased the dffculty n establshng the demand functon type and the rght model to be used. Lumpy demand functon also creates a varety of forecastng problems whch are dffcult to model usng common forecastng technques. In ths paper, we wll descrbe three types of demand functons and ther mathematcal representatons. We wll then smulate demand data usng the mathematcal representatons and model the smulated data to dentfy the statstcal dstrbutons. As such, we would have establshed the relatonshp between demand type and statstcal dstrbuton of demand data. Wth actual demand data, we map t to the statstcal dstrbuton to dentfy the demand functon. Our proposed methodology s represented n Fgure 4 below. Ths research s motvated by the need to reduce forecastng errors due to the wrongful applcaton of forecastng models wthout proper dentfcaton of the demand functon. At the same tme, ths paper also seeks to demonstrate the reducton n forecastng error for dfferent demand functons usng the approprate forecastng technque. Busness Intellgence Journal - July, 2012 Vol.5 No.2

2 2012 Murphy Choy, Mchelle L.F. Cheong 261 Fgure 1: Proposed Methodology Demand Functon Equaton of Demand Functon Smulated Data from Equaton Sutable Forecastng Method Hstogram of Smulated Data Actual Demand Data Kolmogorov- Smrnov Test Statstcal Dstrbuton In Secton 2.0, we defne three dfferent types of demand functons and ther respectve mathematcal representatons. In Secton 3.0, we revew exstng lteratures on the dentfcatons of the varous types of demand functons and the varous crtera used to examne them. In Secton 4.0, we perform data smulaton to relate statstcal dstrbuton to demand functon. We appled our methodology n Secton 5.0 on a real case wth demand data whch are mapped to dfferent statstcal dstrbuton usng Kolmogorov-Smrnov Goodness of Ft test and then determne the demand functons. Wth the dentfed demand functons, dfferent forecastng methods are appled and ther respectve forecastng errors are tabulated. Types of Demand Functons TYPE 1 The frst type of demand functon s the generc cyclcal model wth trend. Ths type of demand functon can be generalzed nto the followng form as Equaton (1). Let, Y = demand of product at tme T = upward or downward trend component of demand at tme C j = cyclcal component of type j at tme, where j = 1 to J (1) Y = T + C + C + + C + e 1 2 TYPE 2 The second type of demand functon s commonly known as stochastc demand. Stochastc demand can be consdered to be random values where the startng value s derved from prevous value or values. It often J occurs as tme seres whch s serally correlated. Thus, ths type of demand functon can be generalzed nto the followng mathematcal form as Equaton (2). Let Y = demand level at tme Then, Y = F(Y ) + e -1 TYPE 3 The last type of demand functon s the lumpy demand functon. Ths type of demand resembles stochastc processes but has ts own unque characterstcs. In the lterature on lumpy demand forecastng (Bartezzagh et al, 1999; Wemmerlöv and Why-bark, 1984), there were several dfferent types of lumpy demand functons dscussed. Three man characterstcs were summarzed from the lterature. Varable: Fluctuatons are present and related to some common factors (Wemmerlöv, 1986; Ho, 1995; Syntetos and Boylan, 2005). Sporadc: Demand can be non-exstent for many perods n hstory (Ward, 1978; Wllams, 1982; Fldes and Beard, 1992; Vereecke and Verstraeten, 1994; Syntetos and Boylan, 2005) Nervous: Each successve observatons s dfferent whch mples low cross tme correlaton (Wemmerlöv and Whybark,1984; Ho, 1995; Bartezzagh and Vergant, 1995). A lumpy demand dstrbuton s defned as a demand whch s extremely rregular wth hgh level of volatlty coupled wth extensve perods of zero demand (Guterrez, 2004). Whle there are several other versons of ths (2) Choy M., Cheong M. L. F. - Identfcaton of Demand Through Statstcal Dstrbuton Modelng for Improved Demand Forecastng.

3 262 Busness Intellgence Journal July defnton, all defntons essentally retaned the three key characterstcs mentoned above. However, there s a fundamental problem wth defnton of the lumpy dstrbuton. The sporadc characterstc suggests that demand wll be zero for many perods n hstory. However, one could stll have a lumpy demand f there exsts a fxed base level of demand greater than zero. Below s a demonstraton of the equvalence of the two. Let, Y = the level of a normal lumpy demand for tme Z = the level of a modfed lumpy demand for tme F() = the dstrbuton whch created the lumpy demand at tme F () = dstrbuton F() shfted by a constant value A A = fxed base level demand > 0 Y = F(I)at I = Addng a fxed base level demand A to Y to get Z, Z = Y + A at I = Thus, Z = F (I)at I = Gven the equvalence n terms of form for (3) and (5), there are no mathematcal reasons to exclude any potental lumpy demand functon wth a fxed base level demand > 0, from the famly. So far, we have not found any lterature whch has a specfc mathematcal form to explan the phenomena. Unlke stochastc demand functon, lumpy demand functon can be dentfed as a compound dstrbuton between a fxed base level demand and a postve demand functon whch s usually defned as Geometrc dstrbuton or Exponental dstrbuton, represented by F() n Equaton (3). After mathematcally defnng the three demand functons that are commonly encountered, let us examne exstng lteratures on the dentfcatons of the varous types of demand functons and the varous crtera used to examne them. (3) (4) (5) Lterature Revew Current lteratures n ths area typcally focused on the forecastng soluton gven a partcular type of demand functon. There are many papers whch talk about varous technques n managng the level of uncertanty (Bartezzagh et. Al., 1995; Bartezzagh et. Al., 1999; Syntetos et. Al., 2005). Such papers focused on the development of sngle algorthm or framework and attempted to measure the performance of such framework aganst exstng ones (Fledner, 1999). The second group of papers focused on solvng the problems of ntermttent demand or lumpy demand wth a varety of tools (Ward, 1978; Wemmerlöv et. Al., 1984; Wemmerlöv, 1986) and suggested framework (Vereecke et. Al., 1984, 1994). Whle some of the papers (Syntetos, 2001) are focused on the problems of the technques employed, they are stll descrbng the ssues gven a fxed context. The papers do not generally dscuss any of the ponts about the proper dentfcaton of the demand functons whle some have mentoned the mportance of dentfcaton and descrpton of the demand functon (Rafael, 2002). The last group of papers focus on the system that oversees the operaton and how mprovement to these systems does help (Fldes, 1992). Whle there are papers descrbng how the processes are affected by the lumpy demand (Ho, 1995), they agan do not attempt to classfy the type of demand wth respect to the ndvdual characterstcs. In our paper, we wll attempt to explore the key characterstcs of the dfferent types of demand functon and relate them to known statstcal dstrbutons. By fttng statstcal dstrbutons to actual past demand data, we are then able to dentfy the correct demand functons, so that the the most approprate forecastng technque can be appled to obtan mproved forecastng results. Statstcal Dstrbuton of Smulated Demand Data From the mathematcal formulatons n Secton 2.0, we can smulate some data that best represent each group of demand functon. It s mportant to note that the smulatons made use of random number generators to demonstrate the dstrbuton of the demand data for a gven demand functon, so the absolute values are mmateral. For TYPE 1 generc cyclcal model wth trend, we can represent the model n Fgure 2. Busness Intellgence Journal - July, 2012 Vol.5 No.2

4 2012 Murphy Choy, Mchelle L.F. Cheong 263 Fgure 2: Generc Cyclcal Model wthout Trend Chart 1: Hstogram of Smulated Data from a Cyclc Data Form From Equaton (1) and Fgure 2, we can observe that the cyclcal form contans elements from trend and seasonal nfluence. Gven the trends are snusodal or regular n nature, whch essentally places a constrant on the possble values for the demand data. Let, C j be the type j cyclcal component of the demand at tme A = Mn (Σ j C j ) for all B = Max (Σ j C j ) for all Thus, A Yj Bfor j = 1to J Snce from Equaton (1), Y = T + C + C + + C + e 1 2 J From Chart 1, we can see that the hstogram from a smulated strong cyclcal dataset demonstrated the overall balance of the cycle ndcatng that demand data seems to be relatvely well dstrbuted and appears to be unform dstrbuton. In essence, we can determne that a partcular demand functon s cyclcal f the hstogram of the dataset fts a unform dstrbuton. The explanaton for ths s the followng. For a cyclcal demand functon, for each tme, + S, + 2S, and so on, where S s the tme perod for 1 complete cycle, the value of the demand Y should be the same value, wthout consderng the trend component. If ths characterstc s appled to all Y for all, then each Y wll occur wth equal probablty. For TYPE 2 stochastc demand, we can represent the model n Fgure 3. Fgure 3: Stochastc Demand Model Therefore, T + A Y T + B For a gven tme, ) P(Y P(Y ) = 1/(T + B T A) = 1/(B A) whch sa unform dstrbuton Choy M., Cheong M. L. F. - Identfcaton of Demand Through Statstcal Dstrbuton Modelng for Improved Demand Forecastng.

5 264 Busness Intellgence Journal July From Equaton (2) and Fgure 3, we can observe that the stochastc demand contans ponts whch have hgher frequency around the mean and lower frequency further away from the mean. To represent a tme seres, we assgn a modfer α n Equaton (2) such that, Y = F(Y ) + e = α Y + e -1-1 (2a) At the same tme, due to the nature of such process beng dependent on the prevous observaton, the observatons can have hgh seral correlatons. For TYPE 3 lumpy demand, we can represent the model n Fgure 4. Fgure 4: Lumpy Demand Model e We can expand Equaton (2a) as follow, Y = α Y α -1 + e -1-1 = = α 0, and Y e = α( α Y (Y ) + e 1-2 ) + e = α( α( α Y -3 )) + For 0 α 1, when, Thus, the dstrbuton of Y effectvely follows that of the error term whch s essentally a normal dstrbuton. Chart 2: Hstogram of Smulated Data from a Stochastc Data Form As defned n Secton 2.0, lumpy demand s a compounded demand of a statstcal dstrbuton and a fxed base level. As the base level s fxed, the dstrbuton of the demand data wll follow the dstrbuton that s generatng the demand level above the fxed level. In Chart 3 below, we can see the effect of an exponental dstrbuton on the demand data. Thus, we can determne that a partcular demand functon s lumpy f the hstogram of the dataset fts a lumpy stochastc process compounded wth a base level demand 0. Chart 3: Hstogram of Smulated Data from an Exponental Data Form Compounded wth a Non-Zero Base Level Demand of 1000 From Chart 2, we can see that the hstogram of a smulated stochastc dataset s almost normally dstrbuted. Ths s dstnctly dfferent from the hstogram of a cyclcal demand functon whch s unformly dstrbuton. Thus, we can determne that a partcular demand functon s stochastc f the hstogram of the dataset fts a normal dstrbuton. One key reason for ths dstrbuton s that the mathematcal form of the demand functon would dctate that the behavor of the model s more akn to the exponental weghted movng average whch fluctuates around a certan average value. Busness Intellgence Journal - July, 2012 Vol.5 No.2

6 2012 Murphy Choy, Mchelle L.F. Cheong 265 Applcaton of Methodology on Real Case We apply our methodology on a real case wth demand data from a retaler specalzng n luxury watches. The retaler s currently facng problems wth forecastng the demand for luxury watches n several countres. The hgh level of volatlty n growng economes has thrown most of the forecasts off by a huge error. Currently, the practce s to use a smple movng average n conjuncton wth manual adjustment. However, ths approach only works for demands whch are stochastc n nature. We ftted the past demand data from each country to statstcal dstrbutons usng the Kolmogorov-Smrnov goodness of ft test. Table 1 below shows the best ftted dstrbutons for each country and ther correspondng dentfed demand type. Table 1: Luxury Watches Demand Classfcaton for 9 Countres Country Dstrbuton Demand Type A Exponental Lumpy B Normal Stochastc C Unform Cyclcal D Exponental Lumpy E Normal Stochastc F Normal Stochastc G Normal Stochastc H Exponental Lumpy I Exponental Lumpy From Table 1, we observed that four out of nne demand functons are lumpy demand whch supported the reason for large demand forecastng errors due to the use of napproprate forecastng technque. We appled four dfferent tme seres forecastng methods to a selected demand of each demand type and calculated the average Mean Squared Error n Table 2. Table 2: Mean Squared Error of Dfferent Forecastng Methods Appled to Dfferent Demand Types Demand Type Forecastng Method Cyclcal Stochastc Lumpy Exponental Smoothng Holts Wnter (Multplcatve) Holts Wnter (Addtve) Stepwse Auto-Regressve From Table 2, we can observe that Lumpy demand s best predcted by Holts Wnter Addtve model as opposed to Stepwse Auto-Regressve model wth the lowest average MSE. In fact, the drop n MSE corresponded to approxmately 43% mprovement n the accuracy of the forecast. Concluson We have attempted to characterze three types of demand functons and usng smulated data to establsh the relatonshp between demand functon and statstcal dstrbutons. We appled the relatonshp to real demand data so as to correctly dentfy the demand type and to determne the most approprate forecastng method wth the smallest Mean Squared Error. We have llustrated that a good forecast does not depend solely on the forecastng technque, but also on the correct dentfcaton of demand functon. At the same tme, the paper provded a smple approach to classfyng demand functons whch does not requre complex calculatons or evaluaton crtera. References Bartezzagh E., R. Vergant, 1995, Managng demand uncertanty through order overplannng, Internatonal Journal of Producton Economcs, 40, Bartezzagh E., Vergant V., Zotter G., 1999, A smulaton framework for forecastng uncertan lumpy demand, Int. J. Producton Economcs, 59: Fldes R., C. Beard, 1992, Forecastng system for producton and nventory control, Internatonal Journal of Operaton and Producton Management, 12 (5), 4-27 Fledner, G., An nvestgaton of aggregate varable tme seres forecast strateges wth specfc subaggregate tme seres statstcal correlaton. Computers & Operatons Research 26, Rafael S. Guterrez, Adrano O. Sols, Nthn R. Bendore, 2004, Lumpy Demand Characterzaton and Forecastng Performance: An Exploratory Case Study, WDIS 2004 Proceedngs Ho C.J., 1995, Examnng the mpact of demand lumpness on the lot-szng performance n MRP systems, Internatonal Journal of Producton Research, 33 (9), Choy M., Cheong M. L. F. - Identfcaton of Demand Through Statstcal Dstrbuton Modelng for Improved Demand Forecastng.

7 266 Busness Intellgence Journal July Syntetos A.A., Boylan J.E., 2001, On the bas of ntermttent demand estmates, Internatonal Journal of Producton Economcs, 71: Syntetos A.A., Boylan J.E., 2005, The accuracy of ntermttent demand estmates, Internatonal Journal of Forecastng, 21: Vereecke A., P. Verstraeten, 1994, An nventory model for an nventory consstng of lumpy tems, slow movers and fast movers, Internatonal Journal of Producton Economcs 35, Ward J.B., 1978, Determnng reorder ponts when demand s lumpy, Management Scence, 24 (6), Wemmerlöv U., D.C. Whybark, 1984, Lot-szng under uncertanty n a rollng schedule envronment, Internatonal Journal of Producton Research, 22 (3), Wemmerlöv U., 1986, A tme phased order-pont system n envronments wth and wthout demand uncertanty: A comparatve analyss of no-monetary performance varables, Internatonal Journal of Producton Research, 24 (2), Wllams T.M., 1982, Reorder levels for lumpy demand, Journal of the Operatonal Research Soce-ty, 33, Busness Intellgence Journal - July, 2012 Vol.5 No.2