ASSESSING PREDICTION INTERVALS FOR DEMAND RATES OF SLOW-MOVING PARTS FOR A NATIONAL RETAILER

Transcription

1 ASSESSING PREDICTION INTERVALS FOR DEMAND RATES OF SLOW-MOVING PARTS FOR A NATIONAL RETAILER Ma Lindsey, Nelson Rusche College of Business, Sephen F. Ausin Sae Universiy, Nacogdoches, TX 75965, (936) , lindseymd@sfasu.edu Rober Pavur, ITDS Dep, P.O. Box 30549, Universiy of Norh Texas, Denon, TX 7603 Phone (940) , pavur@un.edu ABSTRACT Many reail companies carry service pars for producs ha are ou of producion or have accessories. A quesion arises as o wheher he fuure demand rae for each individual componen is high enough o jusify he cos of carrying he service par. Many of hese componen pars have inermien demand. A proposed mehod for creaing predicion inervals for esimaing he fuure demand of producs wih no pas sales or one sale over a fixed ime frame is assessed using daa from a naional reail elecronics company wih a large invenory of replacemen and accessory pars. The performance of he predicion inervals on he invenory of componens for 30 sores is mixed wih he one-sided predicion inerval for esimaing he fuure demand of componens showing no demand over various periods being he mos reliable. INTRODUCTION Invenory opimizaion is a managerial objecive o mee expeced cusomer demand and preven shorages of reailers. However, holding large levels of invenory presens real coss o companies who are under pressure o reduce invenories o reduce coss (Masers, 1993). Slowmoving iems ofen provide he bulk of invenory bu no he bulk of sales. Techniques o minimize invenory coss, wihou impacing cusomer saisfacion, have been widely sudied, wih he excepion of producs wih low demand raes (Hollier, Mak, & Lai, 00). This sudy is moivaed by a renewed ineres in forecasing demand for reail producs characerized by infrequen ransacions (Willemain, Smar, Shockor, & DeSauels, 1994; Johnson & Boylan, 1996; Syneos & Boylan, 001). This paper uses daa from a naional elecronics reailer o examine how a mehodology of forecasing demand raes (Lindsey & Pavur, 009) performs for compuing a demand rae for a family of slow-moving producs wih lile demand hisory. A daabase of sales from 676 slow-moving componen producs was obained for 30 represenaive sores across he Unied Saes from he naional reailer. This daabase was used o assess he performance of predicion inervals for producs wih eiher no pas sales or wih no more han one pas sale. The analysis was inended o abulae he number of reliable predicion inervals and display descripive saisics and aemps o explain why he predicion inervals under cerain condiions may no perform reliably. This sudy illusraes how he one-sided and wo-sided confidence inervals may differ in heir reliabiliy. One-sided predicion inervals (OSPIs) could provide a hreshold amoun of invenory o sock for a given family of producs. The OSPI provides an upper limi such ha he fuure demand rae should be no more han han upper limi wih a cerain confidence level. The Two-sided predicion inervals (TSPIs) provide a lower and upper limi for he expeced demand rae for he family of producs

2 FORECASTING DEMAND RATES FOR SLOW-MOVING INVENTORY Miraglioa and Saudacher (004) sugges ha o mainain a given service level organizaions can compensae for poor forecass by increasing asses or working capial, boh of which are cosly choices. Silver (1965) proposes ha invenory levels be firs based on he desired level of service and hen updaed based on hisorical usage. Croson (197) seminal work proposed a widely acceped mehodology by forecasing demand sizes and ime beween demands separaely inended for demand daa in which many periods have zero demand. Johnson and Boylan (1996) idenify exponenially weighed moving average (EWMA) adjused by he mean absolue deviaion (MAD) of he forecas errors as he viable mehods o esimae demand in a variey of low demand invenory siuaions. Ghobbar and Friend, (00) classify demand paerns ino erraic, lumpy, smooh and inermien demand and sugges appropriae echniques for each caegory. Snyder (00) inroduced wo variaions of Croson s mehod applicable o eiher slow moving or fas moving ime series. Boosrapping has been inroduced as a relaively new approach for forecasing inermien demand ha produced beer resuls han Croson s mehod or exponenial smoohing on several large indusrial daa ses (Willemain, Smar & Schwarz, 004). Boylan, Syneos, and Karakosas (007) provide he definiions for erms used in he sudy of slowmoving invenory. However, no dominan mehodology for forecasing reail invenory wih low raes of demand has emerged. The mehodology considered in his paper focuses on esimaing fuure demand raes when a produc s pas demand is zero or no more han one sale and has no been examined wih real world daa. The predicion inervals used in his sudy will esimae he fuure demand for all componen producs ha have a hisory of zero sales or no more han one sale. Trying o obain a reliable predicion inerval for each iem individually would be a much more challenging problem. Many of he componen pars experiencing almos no demand may in fac have similar fuure demand. Bu esablishing ha heir demand raes are similar is no an objecive of his sudy. The mehodology for he predicion inervals does no require he raes for he individual producs o be similar. An imporan assumpion abou he demand is ha i follows a Poisson process which will be discussed in he nex secion. Assumpion of Poisson Disribuion for Demand Raes While he normal disribuion is ofen assumed for demand of mos iems in invenory sysems, he Poisson disribuion is ofen used o model he demand for slow-moving iems (Vereecke & Versraeen, 1994; Bagchi e al., 1983). Heyvaer and Hur (1956), Hadley and Whiin (1963), Gelders and Van Looy (1978), Schulz (1987), and Silver e al. (1998) recommend he Poisson disribuion for modeling he demand paerns for slow-moving invenory. This disribuion is generally appropriae provided ha he demand variance falls wihin 10 percen of he mean (Silver e al., 1998). However, Vereecke and Versraeen (1994) remark ha use of he Poisson disribuion o model business daa ofen violaes his condiion. Silver (1970) describes applicaions of he Poisson disribuion wihin miliary and indusrial organizaions and finds i o be mos useful when boh he demand and he number of orders are large. One poenial problem - 7 -

3 in applying mehodology requiring he assumpion of Poisson disribuion is ha his assumpion is no ofen checked. Consrucion of Predicion Inervals See Lindsey and Pavur (009) for he explanaion for consrucing he predicion inerval M 1 () / ± 1.96 (M 1 () + M ( ) ) / for he fuure demand rae of producs wihou sales. The exension for he fuure sales rae of producs having no more han one sale over a specified M1 () + M () ime frame is which is an unbiased esimaor of he underlying demand rae. Furhermore, he unbiased esimaor for he expeced squared difference of his esimaor and he M 1() + M () + 6M 3 () fuure demand rae is. The end poins of he predicion inerval for he fuure demand rae of producs having no more han one sale by ime period are M1 () + M () M 1() + M () + 6M 3 () ± Z α/.the firs esimaor described will be referred o as he Zero Sales predicion inervals. Tha is, he Zero Sales predicion inervals deermine fuure demand rae for producs exhibiing no sales over a specified ime frame. The second predicion inerval will be referred o as he Zero and One Sales predicion inerval. Curren mehods provide robus forecass for producs wih healhy sales. The objecive of his research is o assess he use of proposed predicion inervals on acual demand daa o deermine heir usefulness in making decisions on producs ha have ye o experience sales or only experienced sales of single unis in a given ime period. Thus, he model only examined periods when he reail sales for each produc was eiher 0 or 1 in he period o deermine sales raes for hese producs. M 1 (), M (), and M 3 () are couns of he number of producs wih one, wo, or hree sales during a period (Lindsey & Pavur, 009). DESCRIPTION OF STORE ANALYSIS In his analysis of acual sore daa from a naional reailer, a random sample of 30 represenaive sores (10 small, 10 medium, and 10 large) was seleced and he observed produc sales were used o consruc predicion inervals for he pool of producs wih zero sales and he pool of producs wih no more han one sale. The number of predicion inervals compued for his analysis was 360. Tha is, 30 sores imes wo ypes of produc sales (producs wih zero sales and producs wih no more han one sale) imes wo ypes of predicion inervals (wo-sided and one-sided) imes hree confidence levels (90%, 95%, and 99%). In all a family of 676 producs, available in all 30 sores, wih similar demand raes were used in he analysis. An observaion period of 103 weeks (abou years of daa) was spli beween a daa se of observed produc sales and a daa se of fuure produc sales. The predicion inervals for he fuure demand rae were esimaed using he daa se of observed produc sales from eiher 1, 30, or 50 weeks. The reliabiliy of he predicion inervals for fuure demand rae was assessed using he daa se of fuure produc sales from he remaining periods in he wo-year daa se. The

4 lenghs of he remaining number of weeks were 91, 73, or 53. From hese weeks, he fuure demand rae of he pool of producs having zero sales or no more han one sale was esimaed. As expeced, predicion inervals are no necessarily reliable for all siuaions when real daa are used. The OSPIs conained esimaed fuure demand raes in a high percenage of he cases wih a pool of producs exhibiing zero sales, bu were reliable only abou 60% of he ime for a pool of producs having zero or one sale. For all 99% predicion inervals, he percenage of fuure demand raes inside he predicion inervals is larger han a he oher wo confidence levels. This occurs because he 99% predicion inervals are wider. The TSPIs for producs wih zero sales, performed poorly. Less han 10 percen of he sores had prediced demand raes inside of he predicion inervals. However, he Zero and One Sales TSPIs performed beer wih percenages of reliable inervals beween 53% and 87%. Ordinarily, one would expec ha 50 weeks of hisory would provide a more reliable predicion inerval han a 1 week hisory. However, a limiaion o his sudy is ha only 103 weeks of daa are available for analysis. A longer hisorical period implies ha he ime inerval for esimaing he fuure demand rae will be shorer. Thus, he longer hisorical period predicion inervals may be more reliable han indicaed because of he shorer ime frame of evaluaing hem. A quesion migh arise as o how he TSPIs for producs wih zero sales can perform so poorly when he OSPIs for producs wih zero sales performs very well. The reason is because he fuure sales of several producs during he remaining periods in he wo year ime frame were zero or close o zero. The OSPIs has a lower limi of zero, so hese producs do no affec is performance. Anoher possibiliy is ha he daa se dramaically violaed he assumpion ha he producs demand raes followed a Poisson disribuion. TEST OF POISSON DISTRIBUTION To es his possible violaion, an analysis was performed o check if he assumpion ha he produc demands follow a Poisson disribuion. A chi-squared saisic was compued for each of he 30 sores using he GENMOD procedure in SAS. The oupu provides a chi-square saisic, which, if sufficienly large, suggess ha he daa do no follow he Poisson disribuion. The daa se for each sore was sufficienly large, however, o promoe an addiional es. The deviance divided by he degrees of freedom was also compued. For his es, l(y,µ) is he loglikelihood funcion expressed as a funcion of he esimaed mean values µ and he vecor y of he response values, hen he Deviance is defined by D*(y,m) = (l(y,y) l(y,µ)). For his es, for he disribuion o be a Poisson disribuion, he Deviance/df will be close o one or a leas a relaively small number. A hisogram of he raio is shown afer he able wih he values. Three sores have excepionally high Deviances/df. This indicaes ha he daa clearly do no follow a Poisson disribuion. An analysis of he produc demand occurrences for hem reveal ha each had, on occasion, a small number of producs wih an unusually large demand. For his reason, he saisic compued for hese sores was no included in he hisogram. The hisogram shows ha assuming a Poisson disribuion for some of he sores, wih a low Deviance/df value, is probably appropriae; however for several of he sores he assumpion migh no be suiable. The chi-square is always significan in esing for he goodness-of-fi o a Poisson disribuion. Ofen imes, when here are many observaions he daa will clearly depar from a hypohesized

5 disribuion somewhere for he observed values. These low p-values are common in esing for goodness-of-fi when here are a large number of observaions. Thus, he Deviance/df saisic is a beer gauge as o he deparure from he hypohesized disribuion. CONCLUSION This work suggess ha grouping similar iems and compuing an expeced sales rae is accepable for some applicaions. When producs wih no sales are used, one-sided confidence inervals can be consruced o provide hreshold raes for socking decisions. While he model offers poenial, more work is required before he reailer can safely rely upon i for assisance. This sudy illusraes ha he proposed predicion inervals may be useful as an addiional saisic in esimaing he fuure demand rae of slow-moving producs even in he mids of violaion of he Poisson disribuion. Only he wo-sided predicion inerval for producs wih zero sales performed unaccepably poorly due o producs ha have almos zero for a rue fuure demand rae. The percenage of correcly prediced predicion inervals was ofen low, bu his is no surprising given ha several sores violaed he Poisson disribuion assumpion. The one-sided predicion inervals are recommended are managers who wish o use he upper limi in making a deerminaion as o wheher heir invenory componens should be coninued. Addiional research is needed o deermine how useful he proposed predicion inervals are, especially from producs wih a large hisory and a large ime frame o esimae he rue fuure demand rae. Tess for he underlying assumpions of he predicion inervals are recommended for managers who wish o use hese predicion inervals. REFERENCES :Available upon reques