CONSTRUCTING A SALES FORECASTING MODEL BY INTEGRATING GRA AND ELM:A CASE STUDY FOR RETAIL INDUSTRY

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1 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2, pp (2011) 107 COSTRUCTIG A SALES FORECASTIG MODEL BY ITEGRATIG GRA AD ELM:A CASE STUDY FOR RETAIL IDUSTRY Fe-Long Chen and Tsung-Yn Ou * Department of Industral Engneerng and Engneerng Management atonal Tsng Hua Unversty Hsnchu (300), Tawan ABSTRACT Due to the strong competton and economc hardshp, sales forecastng s a challengng problem as the demand fluctuaton s nfluenced by many factors. A good forecastng model leads to mprove the customers satsfacton, reduce destructon of fresh food, ncrease sales revenue and make producton plan effcently. In ths study, the GELM forecastng model ntegrates Grey Relaton Analyss (GRA) and extreme learnng machne (ELM) to support purchasng decsons n the retal ndustry. GRA can seve out the more nfluental factors from raw data and transforms them as the nput data n a novel neural network such as ELM that can abandon the slow gradent-based learnng speed and parameters tuned teratvely. The proposed system evaluated the real sales data of fresh food n the retal ndustry. The expermental results ndcate the GELM model outperforms than other tme seres forecastng models, such as GARCH, GBP and the GMFL model n predctng accuracy and tranng speed. Otherwse, the dfferent actvaton functons of the GELM model have sgnfcant dfferences n tranng tme and performance durng our experments. Keywords: Sales Forecastng, Grey Relaton Analyss, Extreme Learnng Machne, Retal Industry, Actvaton Functons * 1. ITRODUCTIO In retal ndustry actual operatons, sales forecastng plays a more and more promnent role as part of the commercal enterprse. However, sales forecastng s usually a hghly complex problem due to the nfluence of nternal and external factors. If decson-makers could estmate ther sales quanttes properly, the demands of customers would be satsfed and the cost of spoled fresh food would be substantally reduced. Actually, the varatons n consumers demand are caused by many factors lke prce, promoton, changng consumer preference or weather changes, especally n fresh food [34]. Both shortage and surplus of fresh tems, whch can only be sold for a lmted perod, would lead to loss revenue for the retal company. An effectve and tmely forecastng model s an urgent and ndspensable tool for handlng the nventory level n the retal busness. On the other hand, poor forecastng methods would result n redundant or nsuffcent stock that wll affect the ncome and compettve advantage drectly. Therefore, t s a very * Correspondng author: [email protected] crtcal ssue to fgure out the nfluental factors then obtan accurate forecastng results about the fresh food wthn a modern retal ndustry. Snce managers n retals usually lack an accurate forecastng tool, they have to rely on ther own experence or consult the pont of sales system (POS system) to predct the future demand and place purchasng orders. Few decson makers adopt statstcal methods, such as the movng average method or exponental smoothng to deal wth the tme daly problems. LeVee [27] ndcated that accurate sales forecastng was obtanable and that t can help the decson-makers to calculate the producton and materal costs and determne the sales prce. In fact, most conventonal sales forecastng methods used ether factors or tme seres data to determne the sales predcton. The relatonshp between the past tme seres data (ndependent varables) and the sales predcton (dependent varable) s always too complcated to acqure an advantageous orderng suggestons by usng the unsuted statstcal approaches. Practcally, the POS system actually provdes some forecastng suggestons for the managers to place orders. However, most decson-makers stll prefer to place the same quantty as usual or depend on ther own

2 108 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) ntuton nstead of model-based approaches. In ths paper, we present a relatvely novel neural network methodology, Grey relaton analyss ntegrated wth extreme learnng machne (GELM) to construct a forecastng model n the fresh food sector of the retal ndustry. Sales n the retal sector exhbt strong seasonal varatons. Hstorcally, modelng and forecastng seasonal data s one of the major research efforts and many theoretcal and heurstc methods have been developed n the last several decades. The avalable tradtonal quanttatve approaches nclude heurstc methods such as tme seres decomposton and exponental smoothng as well as tme seres regresson and autoregressve and ntegrated movng average (ARIMA) models that have formal statstcal foundatons [7]. evertheless, ther forecastng ablty s lmted by ther assumpton of a lnear behavor and thus, t s not always satsfactory [37]. Recently, artfcal neural network (A) have been appled comprehensvely n sales forecastng [17,31], pattern recognton [26], aggregate retal [7], PCB ndustry [11]. Most studes ndcate that A have the better performance than conventonal methodology [23,24]. Ths flexble data-drven modelng property has made the A model an attractve tool for many forecastng tasks. However, most A and ts varetes used gradent-bases learnng algorthms, such as back-propagaton network (BP), and faced many dffcultes n stoppng crtera, learnng rate, learnng epochs, over-tunng, local mnma and long computng tme. A new learnng algorthm for sngle-hdden-layer feed-forward neural network (SLF) called the extreme learnng machne (ELM) has been proposed recently and overcome the prevous dsadvantages as we mentoned [18,19,30, 32,34]. The rest of ths study wll llustrate the GELM model for mprovng the accuracy of forecastng fresh foods n the retal ndustry. Secton 2 revews the related sales forecastng lteratures ncludng the tradtonal statstcal model and the A model. Secton 3 presents the methodology of ths study n solvng the real forecastng problems. Secton 4 descrbes the development of varous forecastng models and dscusses the comparson results. Then the concluson wll be provded n Secton LITERATURES REVIEW The avalable tradtonal tme seres forecastng approaches are dvded nto two groups.e. the unvarate tme seres model and multvarate tme seres model. One of the major lmtatons of tradtonal statstcal methods s that they are essentally lnear methods. The sales status of fresh food s often nfluenced by uncertan factors such as weather, promoton, compettve market, etc. Therefore, tradtonal methodologes requre some mprovements for provdng better forecastng suggestons. ext, we wll brefly ntroduce the tradtonal statstcal forecastng models and A model n sales forecastng applcatons. 2.1 Tradtonal Statstcal Model for Tme Seres Data Forecastng In the past several decades, many researchers had used many knds of forecastng methods to study tme seres events. Unvarate tme seres models nclude the movng average model, exponental smoothng model, and auto-regressve ntegrated movng average (ARIMA) model. Box and Jenkns [9] developed ARIMA, a basc prncple of ths model s the assumpton of lnearty among the varables. However, many tme seres events may not hold on the lnearty assumpton. Clearly, ARIMA models could not be effectvely used to capture and explan non-lnear relatonshps, especally for handlng actual sales forecastng problems. When t s appled to processes that are non-lnear, forecastng errors often ncrease greatly as the forecastng horzon becomes longer. For mprovng forecastng non-lnear tme seres events, many researchers have developed alternatve modelng approaches. These approaches nclude non-lnear regresson models, the blnear model, the threshold auto-regressve model, the auto-regressve heteroscedastc model (ARCH) [16] and generalzed auto-regresson condtonal heterskedastcty (GARCH) model [4]. Although the tradtonal methods have been proved somewhat effectve, they stll have certan shortcomngs. Zhang [36] ndcated that although these methods had dsplayed some mprovements over the lnear models n some specfc cases, they tended to be appled to specal events, and lacked generalty and were poorly mplement. 2.2 A Model n Tme Seres Data Forecastng The A model s a model-free approach that was been recently appled n forecastng due to ts competent performance n forecastng and pattern recognton. In general, t conssts of a collecton of smple non-lnear computng elements whose nputs and outputs are ted together to form a network. Many studes have attempted to apply A model to tme seres forecastng. Wegend. et al. [35] ntroduced the ''weght-elmnaton'' back-propagaton learnng procedure and appled t to sunspots and exchange-rate tme seres. Tang and Han [33] compared the A model wth the ARIMA model by usng nternatonal arlne passenger traffc, domestc car sales and foregn car sales n the USA. Chakraborty et al. [10] presented an A approach based on multvarate tme-seres analyss, whch can

3 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng GRA and ELM 109 accurately predct the flour prces n three ctes n the USA. Lachtermacher et al.[20] developed a calbrated A model. In ths model, the Box-Jenkns methods are used to determne the lag components of the nput data. Moreover, t employed a heurstcs method to choose the number of hdden unts. Ansuj et al. [5] expressed a comparson made for the tme seres model wth nterventons related to the A model for analyzng the sales behavor of a medum-sze enterprse. The results showed that the A model was more accurate. Furthermore, Bgus [7] used promoton, tme of year, end of month age, and weekly sales as nputs for the A model to forecast the weekly demand wth promsng results. Kuo and Chen [22] beleved that the tradtonal statstc approaches had hgher performance dealng wth data of seasonalty and trends, but they are napproprate for unexpected stuatons. In the ELM method, the nput weghts (lnkng the nput layer to the hdden layer) and hdden bases are randomly chosen and the output weghts (lnkng the hdden layer to the output layer) are analytcally determned by usng the Moore-Penrose (MP) generalzed nverse. As ths new learnng algorthm can be easly mplemented, t tends to dentfy the smallest tranng error, obtans the smallest norm of weghts and the good generalzaton performance, and runs extremely fast. 2.3 Demand Forecastng of the Retal Industry Chu and Zhang [13] and Alon et al.[4] developed the artfcal networks for forecastng the aggregate retal sales. Alon et al.[21] compared wth tradtonal methods ncludng Wnter exponental smoothng, Box-Jenkns ARIMA model, and multvarate regresson. The dervatve analyss shows that the nonlnear neural networks model s able to capture the dynamc nonlnear trend and seasonal patterns, as well as the nteractons between them. Chu et al.[7] found the non-lnear models are able to outperform lnear counterparts n out-of-sample forecastng, and pror seasonal adjustment of the data can sgnfcantly mprove performance of the neural network model. The overall best model s the neural network bult on deseasonalzed tme seres data. Dogans et al. [15] also presented a evolutonary sales forecastng model whch s a combnaton of two artfcal ntellgence technologes, namely the radal bass functon and genetc algorthm. The methodology s appled successfully to sales data of fresh mlk provded by a major manufacturng company of daly product. Aburto and Weber [1] presented a hybrd ntellgent system combng ARIMA model and MLP neural networks for demand forecastng. It shows mprovements n forecastng accuracy and a replenshment system for a Chlean supermarket, whch leads smultaneously to fewer sales and lower nventory levels. Au et al. [6] and Sun et al.[37] developed dfferent sales forecastng models n fashon retalng. Au et al. [6] llustrated evolutonary neuron network for sales forecastng and showed that when guded wth the BIC and the pre-search approach, the non-fully connected neuron network can converge faster and more accurate n forecastng for tme seres than the fully connected neuron network and tradtonal SARIMA model. Forecastng s often tme crucal, the mprovement of convergence speed makes wdely applcable to decson-makng problems. Sun et al [37] appled ELM neural network model to nvestgate the relatonshp between sales amount and some sgnfcant factors whch affect demand. The experment results demonstrate that the proposed methods outperform than back-propagaton neural network model. Al et al. [3] explored forecastng accuracy versus data and model complexty tradeoff n the grocery retalng sales forecastng problem, by consderng a wde spectrum n data and technque complexty. The experment results ndcated that smple tme seres technques perform very well for perods wthout promotons. However, for perods wth promotons, regresson trees wth explct features mprove accuracy substantally. More sophstcated nput s only benefcal when advanced technques are used. Chen et al. [12] developed the GMFL forecastng model by ntegratng GRA and MFL neural networks. GRA seves out the more nfluental factors from raw data then transforms them as the nput data n the MFL model. The expermental results ndcated the proposed forecastng model outperforms than MA, ARIMA and GARCH forecastng model of the retal goods. Accordng to the above lterature revew, the retal forecastng problems are usually a tme and accuracy crucal ssue. Ths paper ams to construct a more effcently sales forecastng model that could perform more accurate and faster than the unvarate and multvarate tme seres model for retal goods. As we know, sales wll be affected by many dynamc factors. GRA and the expert knowledge wll seve the more nfluental factors out as the nput varables of the ELM model. Provdng an mproved forecastng method that can help the managers to make decsons for orderng the approprate amounts wll be the focal pont of ths research. 3. METHODOLOGY The followng secton presents the purposed sales forecastng model by ntegratng GRA and ELM. The GRA computes the Grey Relaton Grades (GRG), whch are the nfluental degree of a compared seres by relatve dstance. Subsequently, the data composed of these nput and output pars are dvded nto tranng, testng and predctng data. All

4 110 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) the data sets should be normalzed nto a specfc range [-1,1]. The ELM would offer predctng results then process the unnormalzaton step, to convert the data back nto unnormalzed outcomes. 3.1 Grey Relaton Analyss (GRA) Deng [14] proposed the Grey Relaton Analyss (GRA) mathematcs. It has been successfully appled n many felds such as management, economcs, and engneerng. The Grey Relaton Grades (GRG) s the nfluence degree of a compared seres on the reference seres that can be represented by the relatve dstance. The smaller dstance would have more nfluence. The degree of nfluence descrbes the relatve varatons between two factors that ndcate the magntude and gradent n a gven system. The GRG between two seres, the compared seres and the reference seres, s called relatonal coeffcent r( x 0, x ). Before calculatng the Grey relatonal coeffcents, each data seres must be normalzed by dvdng the respectve data from the orgnal seres wth ther averages. After performng Grey data processng, the transformed reference sequence s x 0 ={x 0 (1), x 0 (2),..., x 0 (n)}. The compared sequences are denoted by x ={x (1), x (2),,x (n)}, =1 to m. The relatonal coeffcent r( x 0, x ) between the reference seres x 0 (t) and the compared seres x (t) at tme t=k can be calculated usng the followng equaton [20]: r ( x 0, x ) mnmn x0 x maxmax x0 x k k x0 x maxmax x0 x k k=1,2,,m;=1,2,,m (1) Whle s a dstngushng coeffcent (0< 1) that s used to adjust the range of the comparson envronment and control level of dfferences n the relatonal coeffcents. When =1, the comparson envronment s altered. When =0, the comparson envronment dsappears. In cases, when the data varaton s large, usually ranges from 0.1 to 0.5 for reducng the nfluence of extremely large mn mnk x0 x. where x0 x denotes the absolute dfference between the two sequences, whch represent the dstance x 0( k) after data transformaton s the mnmum (maxmum) dstance for the tme n all compared sequences whch form the comparson envronment. Whle mn mnk x0(k)-x(k) equals zero snce the transformed seres wll crsscross at a certan pont. 3.2 ormalzaton and Unnormalzaton The normalzed method for the nput and output data set s descrbed as follows: j Max{ j} ( j Mn{ j}) normalze ( Max{ j} Mn{ j}) 1,2,..., n; j 1,2,..., (3) The unnormalzed method for the predctng result s descrbed as follows: P unnormalze Pj ( Max{ j} Mn{ j}) Max{ j} Mn{ j 2 1,2,..., n; j 1,2,..., (4) 3.3 Extreme Learnng Machne (ELM) ELM s a sngle hdden-layer feed-forward neural network (SLF). It randomly chooses the nput weght matrx W and analytcally determnes the output weght matrx of SLF. Suppose that we are tranng a SLF wth K hdden neurons and an actvaton functon vectors g x) [ g ( x), g ( ),..., g k (x)] to learn dstnct samples ), where } ( 1 2 x ( x, t [ 1 2 m. T n T m x [ x 1, x2,..., xn ] R and t t, t,... t ] R If the SLF can approxmate the samples wth a zero error then we have j 1 y j t 0 (5) Where y s the actual output value of the SLF. There also exst parameters, w and b such that K 1 g ( w x b ) t j 1,2,..., (6) j j T Where w [ w 1, w 2,..., wm ] s the weght vector connectng the th hdden node and the nput T nodes, [ 1, 2,..., m ] s the weght vector connectng the th hdden node and the output node, and b s the threshold of the th hdden node. The operaton w x j n Equaton (6) denotes the nner product of w and x j. The above equatons can be wrtten compactly as: H T Where H ( w1,..., w, b1,..., b ~, x1,... x ~ j ) max maxk x0 x [max maxk x0( k) x ] (2)

5 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng GRA and ELM 111 g( w1 x1 b1 ) g( w1 x b1 ) ~ m T 1 and T ~ g( w g( w ~ ~ x x 1 T t 1 T T t b ~ b m ~ ) ) (7) In ELM, the nput weghts and hdden bases are randomly generated nstead of tuned. Thus the determnaton of the output weghts s as smple as fndng the least-square (LS) soluton to the lnear s ˆ H (8) where H s the MP generalzed nverse of the matrx H. The mnmum norm LS soluton s unque and has the smallest norm among all the LS solutons. 3.4 Steps of Constructng the GELM Forecastng Model Ths secton wll descrbe how to constructng a Grey relaton analyss and Extreme Learnng Machne (GELM) forecastng model systematcally. The basc elements of the present study are presented n Fgure1, and can be brefly descrbed as follows: Step 1: Data collecton Collect the daly sales and prce data from the target store and the other relatve seres data provded by neghborng stores or some government agences as the forecastng references. One of the data s the forecastng target ( x 0 ), and the other s the m comparson seres data ( x 0, 1,2,..., m) where ( { x 1,2,..., m}. Step 2: ormalze the ntal data All ntal data s composed of a movng wndow of fxed length along wth the seres and the nput data wll be normalzed by Equaton (3). After normalzng all the collected data, each data set wll fall nto the nterval between -1 and 1. Step 3: Calculate the grey relaton grades (GRG) The grey relatonal grades between the two seres at a certan tme pont t s represented by grey relatonal coeffcent r( x 0, x ), defne as Equaton (1). The range of GRG s closed nterval between 0 and 1. The great GRG between two data sets, the closer the relatonshp between these data sets are. Step 4: Select the more affectve factors Accordng to the rankng of the GRG, an expert who owns the doman knowledge can select the mportant factors that affect the sales amounts more sgnfcant. Step 2 and Step 3 not only can provde a ratonal analyss but also avod the preconceved opnons of experts. Step 5: Dvde the nput and output data nto tranng data, testng data and predctng data The ELM s a SLF wth three man layers, nput layers, hdden layers and output layers. Dfferent from tradtonal learnng algorthms the proposed learnng algorthm tends to reach the smallest tranng error and obtans the smallest norm of weghts. The ELM can be summarzed as follows: Algorthm ELM: Gven a tranng set n m {( x, t) x R, t R }, actvaton functon g(x), and hdden node number ~, Step 5.1 Randomly assgn nput weght w and ~ bas b, 1,2,...,. Step 5.2 Calculate the hdden layer output matrx H. Step 5.3 Calculate the output weght. H T T where T [ t 1,..., t ] H s the MP generalzed nverse of matrx H (See Appendx) Step 6: Select dfferent actvaton functons and neuron number of hdden nodes The ELM randomly chooses hdden nodes and analytcally determnes the output weghts. There are three actvaton functons (sgmodal sne hardlm) and four knds of hdden node ( ) can be select. Step 7: Input tranng and testng data and predct the further sales amounts Obtan the predcted results of tranng and testng data then unnormalze the outcomes by Equaton (4) As dscussed n the Appendx, we have the followng mportant propertes: 1. Mnmum tranng error The specal soluton H T s one of the least-square soluton of a general lnear system H T, meanng that the smallest tranng error can be reached by ths specal soluton: ˆ H T HH T T mn H T (9) Although almost all learnng algorthms wsh to reach the mnmum tranng error, however, most of them cannot reach t because of local mnmum or nfnte tranng teraton s usually not allowed n applcatons. 2. Smallest norm of weght Further, the specal soluton ˆ H T has the smallest norm among all the least-square solutons of H T :

6 112 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) ˆ H T, H T Hz T, z R { : (10) The mnmum norm least-square solutons of H T s unque, whch s ˆ H T Step 8: Measure the accuracy of the forecastng results Measure the accuracy of the forecastng results by MAD and MSE crteron. 1. MSE (Mean Square Error) applcatons wll dvde the raw data nto a tranng set and a testng set. The tranng set s used for neural network constructon whle the test set s used for measurng the model predctve ablty. The tranng process for determnng the functon s determned by usng the lnkng arc weghts of the network. The structural sze of the GELM model depends on the number of hdden nodes. The nput data should provde strong representatve after the GRA and opnons accordng to expert knowledge. ( ) 1 MSE= A F t t t 1 2 (11) 2. MAD (Mean absolute devaton) Where 1 MAD= A t F t 1 A t s the actual amount and t (12) F t s the forecastng amount, respectvely. Step 9: Repeat step 6~8 for the same data The GELM model wll offer the best predcted results then measure the accuracy of those results. We wll do some statstcal tests (pared t-test) on obtaned results of the sgmodal actvaton functon, sne actvaton functon and hardlm actvaton functon. Fgure1: Outlne of present study Fgure 2 shows the framework and non-lnear transformaton of the GELM network that ncorporates nput layers, hdden layers and output layers. Generally, GELM model n practcal Fgure 2: The framework and non-lnear transformaton of the GELM network 3.5 The Propertes of the GELM Forecastng Model of the Retal Industry The GELM forecastng model combned Grey relaton analyss (GRA) and Extreme learnng machne (ELM) methodologes. The GRA n the grey system s an mportant problem-solvng method that s used when dealng wth the smlarty measures of complex relatons. The man purpose of GRA n the proposed hybrd-forecastng model s to realze the relatonshp between two sets of tme seres data n relatonal space [25]. The Grey relatonal grade (GRG) s a globalzed measure adopted for GRA. It s used to descrbe and explan the relaton between two sets. If the data for the two sets at all ndvdual tme ponts were the same, then all the relatonal coeffcents would equal one. The great GRG between two sets, the closer the relatonshp between the sets are. The hgher GRG of the canddate data sets would be the delegates as the nput data sets of the GELM model for enhancng the predct ablty. Owng to the learnng speed of the feedforward neural network s far slower than requred and t has been a major bottleneck n practcal applcaton for past decades. Ths study appled ELM for sngle-hdden layer feed-forward neural networks that randomly chooses hdden nodes and analytcally determnes the output weghts of the networks. The major property of the ELM can abandon the slow gradent-based learnng speed and parameters tuned teratvely algorthms that are extensvely used to tran neural network then provde good generalzaton forecastng performance at extremely fast learnng speed. The lmtaton of the purposed GELM forecastng model s that lacks to consder the nfluence of the fnancal crss, free trade agreement,

7 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng GRA and ELM 113 consumers behavor and advertsements. Besdes, t s more sutable regardng the mature product's forecast but not the new announcement product on the market. 4. EPERIMET RESULTS AD DISCUSSIOS In operatonal management n the retal ndustry, t s ndspensable to forecast the further demand and place orders at varous tmes of the day. If the system can offer more accurate predcton functons that can assst managers to cater for the demand of customers and reduce scraped quanttes of fresh food. Usng the GELM model to predct sales amounts can ncrease the accuracy n the proposed system. The procedures of the experments and the results are descrbed sequentally n the followng subsectons. Ths study compared the GELM forecastng model wth the multvarate statstcal forecastng methods such as the GARCH model, the Back-Propagaton etwork (BP) as well as the GBP and the GMFL model by forecastng 120 days sales. The GBP model ntegrates the GRA and Back-Propagaton etworks and the GMFL model ntegrate the GRA and Multlayer Functonal Lnk etworks. The GARCH model s bult by E-vew and the smulaton of relatve BP model are conducted n MATLAB runnng on an ordnary notebook wth a 1.4 GHz CPU and 760MB RAM. 4.1 Data Collecton and Analyss Well-known retalers and a government organzaton n Tawan provded the ntal data that can be separated nto three dfferent groups. Frstly, the target store collected the daly sales data and prce of 960ml contaners of mlk. The total numbers of the data was 334 as shown n Fgure 3. We also collected the sales amounts of other two dfferent brands and ther prces respectvely. Ordnarly, the sales prce would not be a fxed number, as t wll be adjusted due to many reasons such as promoton, the hot/cold season or some specfc actvtes. Secondly, the sales data was also obtaned from two neghborng stores. Those neghborng stores are n the same dstrbuton area. Stores were close to each other and they servced the same customers. We also collected the sales amounts and prce data from other stores. Thrdly, the Central Weather Bureau provded the local weather records. Fgure 3: The sale quanttes of the target brand As we know, many factors wll affect consumers behavor n the actual retal ndustry. Among those factors that would be descrbed nclude how to select the most nfluental ndces by usng the analytcal methodology to be the nput data of the ELM model as below. After normalzng the raw data and calculatng the GRG of each ndex. The expert selected three factors wth hgher GRG to be the nput data of the multvarate tme seres model and ELM model. The GRG of each factor s shown n Table 1. The selected factors wll represent the more nfluental n the sales amounts of fresh food. The three selected factors are W, TAs and TB S. 4.2 Experment Results The expermental algorthms of the GARCH, GBP, GMFLM, and GELM mport the same data sets ncludng three ndces (W, TAs and TB S ) selected by GRA and the last 7 days lagged data GARCH Forecastng Model Bollerslev [8] proposed the GARCH (Generalzed ARCH) condtonal varance specfcaton that allows for a parsmonous parameterzaton of the lag structure. In analyzng the tme seres model, several sutable models could explan the nput data. We adopt two statstcs to be the crterons for choosng the best statstcal forecastng model. 1. AIC (Akake s Informaton Crteron) Akake [2] provded the followng crteron to evaluate the ftness of the proposal statstcal models. (Data set ftted by P parameters of the statstcs models.) 2 AIC(P)= n Ln( ˆ a ) 2P (13) 2. SBC (Schwartz s Bayesan Crteron) Schawrtz [28] provded the smlar crteron to evaluate the ftness of the statstcal models. 2 SBC(P)= n Ln( ˆ a ) P Ln( n) (14) The best GARCH forecastng model wll use the same tme seres data and three ndces (W, TAs

8 114 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) and TB S ) to predct the 120 days demand. After examnng AIC( ) and SBC( ) the best adapted model s descrbed below. y t = y t y t y t W t TAs TB S t t t t-5 + t t ~(0, 2t ) 2t = t-2 (15) Table 1: The GRG of collected data I. Data from target store II. Data from neghborng two stores Target Brand GRG Target Brand n eghborng A GRG Sales amount T S - Sales amount TA S Prce T P Prce TA p Compettve Brand 1 Compettve Brand 1n eghborng A Sales amount C1 S Sales amount C1A S Prce C1 P Prce C1A p Compettve Brand 2 Compettve Brand 2 n eghborng A Sales amount C2 S Sales amount C1B S Prce C2 P Prce C1B p Target Brand n eghborng B III. Weather data Sales amount TB S Weather records W Prce TB p Compettve Brand 1n eghborng B Sales amount C1B S Prce C1B p Compettve Brand 2 n eghborng B Sales amount C2B S Prce C2B p GBP Forecastng Model Generally, the BP s a typcal type of artfcal neural networks model, whch s a class of generalzed non-lnear nonparametrc model that was nspred by studes of the bran and nervous system. BP s composed of several layers of nput, hdden and output nodes. It s a challenge to develop approprate sze of BP model for combnng the avalable data n the tranng data and the testng data. The structure sze of the model depends on the number of nput nodes and the number of hdden nodes. There are no systematc reports on the decson of nput and hdden nodes. Dfferent nput and hdden nodes have a sgnfcant mpact on the learnng and predcton ablty of the network. As mentoned before, the purpose of GRA s to realze the relatonshp between two sets of tme seres data n a relatonal space. In the GBP model, the nput nodes of the neural network are usually the past, lagged observatons and more nfluental factors that wll affect the sales amounts, and the output node s the real sales data. We expect to obtan an applcable GBP forecastng model that has generalzaton and good forecastng capablty GMFL Forecastng Model The MFL ncorporates basc nput nodes, logarthmc nput nodes, and exponental nput nodes n the nput layer for mprovng the forecastng ablty and reducng the learnng cycle tme of the nervous networks [12]. It s composed of one or two hdden layers that have competent contnuous functon n a theoretcal tme-seres. In the analogous models, the hdden nodes are used to capture the non-lnear structures. Makng the decson for how many hdden nodes should be used s another dffcult ssue n the neural network forecastng model constructon process. In practce, the numbers of hdden nodes were chosen through experments or by tral-and-error wthout any theoretcal bass to gude the decson. Some theores suggest that more hdden nodes can ncrease the accuracy n approxmatng a functonal relatonshp but t stll causes the over-fttng problem. Ths problem s more lkely to happen n the GMFL model than n other statstcal models. The over-fttng problem soluton s to fnd a parsmonous model that fts the data well. Another way to tackle the over-fttng problem s to dvde the tme seres nto three sets; tranng, testng and valdaton [21]. The frst two sets are used for model buldng and the last s used for model valdaton or evaluaton. The best GMFL model s the one that gves the best results n the predctng set GELM Forecastng Model The learnng speed of ELM s faster than other tradtonal classc gradent-based learnng algorthms. Ths advantage has already been recognzed n many further studes. In order to obtan hgher predcton

9 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng GRA and ELM 115 accuracy, we desgned the experments wth dfferent actvaton functons for the number of hdden nodes. In the GELM forecastng model, we compare the accuracy wth sgmodal actvaton functon, and hardlm actvaton functon n the dfferent numbers of hdden nodes. The numbers of hdden nodes are selected from the 20, 50, 100 and 200. The GELM forecastng model wll use the same tme seres data and three ndces (W, TAs and TB S ) to predct the 120 days demand. Table 2 shows the tranng tme of GELM, GBP and GMFLM. The GELM learnng algorthm spent s CPU tme wth the Sgmodal actvaton functon and 200 hdden nodes. The tradtonal gradent-based learnng algorthm as GBP and GMFL cost too much tranng tme compared wth GELM. Hdden nodes Table 2: Tranng tme of dfferent algorthms GELM GBP GMFLM Actvaton functon Sgmodal (Sg.) Sne (Sn.) Hardlm (Har.) Table 3 shows the performance of the GELM forecastng model n dfferent actvaton functons and hdden nodes. The more hdden node has a better ablty to predct the sales amounts. The best forecastng results have MAD of and MSE of wth sgmodal actvaton functon and 200 hdden nodes. In the GELM model, the nput weghts and hdden bases are randomly chosen and the output weghts are analytcally determned by usng the Moore-Penrose generalzed nverse. In order to compare tranng tme and performance wth dfferent actvaton functons we tested 30 tmes each run and dd some statstcal tests (pared t-test) on obtaned results to examne the statstcally sgnfcant dfference. The pared t-test s a wdely used method to examne whether the average dfference of performance between two methods over varous data sets s sgnfcantly from zero. If the p-value generated by a pared t-test s lower than the sgnfcant level (0.05) that ndcate the dfference between the two methods. Table 3: Performance of dfferent actvaton functons and hdden nodes Hdden nodes Actvaton functon Crteron Sgmodal (Sg.) MAD MSE Sne (Sn.) MAD MSE Hardlm (Har.) MAD MSE Table 4 shows the tranng tme of the GELM wth dfferent actvaton functons and dfferent hdden nodes. There s no sgnfcant dfference when the numbers of hdden nodes are 20 and 50. When hdden nodes are 100, the tranng tme of the hardlm actvaton functon s sgnfcantly dfferent between sgmodal and sne actvaton functon. But the sgmodal and sne actvaton functon have no dfference. When hdden nodes are 200, these three actvaton functons have sgnfcant dfferences. The hardlm actvaton functon s better than the sgmodal actvaton functon and the sgmodal actvaton functon s better than the sne actvaton functon. Table 5 shows the MAD of GELM wthn dfferent actvaton functons and dfferent hdden nodes. The p-value n sgmodal and sne actvaton functon s always lower than 0.05, whch means there s a sgnfcant dfference between these two actvaton functons and the performance of the sgmodal actvaton functon s always better than the sne actvaton functon. The hdden nodes are 20, 50 and 100, the p-value n the sgmodal and the hardlm actvaton functon are lower than 0.05, the sgmodal actvaton functon s sgnfcantly better than hardlm actvaton functon. The hdden nodes are 20, 100 and 200, the p-value n the sne and the hardlm actvaton functon are lower than 0.05, the hardlm actvaton functon s sgnfcantly better than sne actvaton functon.

10 116 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) Hdden nodes Hdden nodes Table 4: Pared t-test of tranng tme between dfferent actvaton functon Pared Dfferences Pared Methods Mean StDev 95% Confdence Interval of the Dfference t P values Lower Upper Sg.-Sn Sg.-Har Sn.-Har Sg.-Sn Sg.-Har Sn.-Har Sg.-Sn Sg.-Har * Sn.-Har * Sg.-Sn * Sg.-Har * Sn.-Har * Table 5: Pared t-test results of predctng between dfferent actvaton functon n MAD Pared Dfferences Pared Methods Mean StDev 95% Confdence Interval of the Dfference t P values Lower Upper Sg.-Sn * Sg.-Har * Sn.-Har * Sg.-Sn * Sg.-Har * Sn.-Har Sg.-Sn * Sg.-Har * Sn.-Har * Sg.-Sn * Sg.-Har Sn.-Har * Table 6 shows the MSE of GELM wthn dfferent actvaton functons and dfferent hdden nodes. The p-value n sgmodal and sne actvaton functon are always lower than 0.05, whch means there s a sgnfcant dfference between these two actvaton functons and the performance of sgmodal actvaton functon s always better than sne actvaton functon. The hdden nodes are 50 and 100, the p-value n sgmodal and hardlm s are lower than 0.05, the sgmodal actvaton functon s sgnfcantly better than hardlm actvaton functon. The hdden nodes are 20 and 200, the p-value n sne and hardlm actvaton functon are lower than 0.05, the hardlm actvaton functon s sgnfcantly better than sne actvaton functon. From above results, the sgmodal actvaton functon has sgnfcant dfferences between the sne actvaton functon. But, there s no sgnfcant dfference between sgmodal vs. hardlm or sne vs. hardlm. 4.3 Dscusson Table 7 presents the results of dfferent forecastng models. The best GARCH model has MAD of and MSE of The best forecastng result of GBP model has MAD of and MSE of The best forecastng result has MAD of and MSE of The best GELM model has MAD of and MSE of The GELM forecastng model we proposed has the smallest predctng errors and the learnng speed s extremely faster than others.

11 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng GRA and ELM 117 Hdden nodes Table 6: Pared t-test results of predctng between dfferent actvaton functon n MSE Pared Dfferences Pared Methods Mean StDev 95% Confdence Interval of the Dfference t P values Lower Upper Sg-Sn * Sg-Har Sn-Har * Sg-Sn * Sg-Har * Sn-Har Sg-Sn * Sg-Har * Sn-Har Sg-Sn * Sg-Har Sn-Har * Table 7: The compared results of dfferent forecastng models Model Type MAD MSE Tranng Tme Statstcal tme seres model GARCH GBP Artfcal neural network model GMFL GELM COCLUSIOS Recently, many researches and ndustral managers are nterested n applyng data mnng and artfcal ntellgence algorthms to deal wth routne problems. Sales forecastng plays a more and more mportant role n operatng management of commercal enterprses especally n the retal dustry. In ths paper, we present a relatvely novel neural network methodology, Grey relaton analyss ntegrated wth extreme learnng machne (GELM) to construct a forecastng model for fresh food. The proposed GELM model ncludes several major characterstcs as followng: (1) Ths study appled GRA, whch s a problem-solvng method that used when dealng wth smlarty measures of complex relatons. The man purpose of GRA n ths model s to realze the relatonshp between two sets of tme seres data n the relatonal space and seve out the more nfluental factors as the nput data to the ELM. (2) The learnng speed of GELM s extremely fast than GBP and GMFL. The learnng phase of GELM can be completed less than a second wthn dfferent actvate functons and hdden nodes. (3) The proposed GELM has better generalzaton performance than the gradent-based algorthms such as GBP and GMFL. (4) The GELM method can avod many harmful ssues that happened n the tradtonal gradent-based algorthms, such as stoppng crtera, local mnma, mproper learnng rate and over-fttng problems. (5) The GELM tends to reach the solutons straghtforward wthout trval ssue and looks much smpler than most feed-forward neural networks algorthms. The experment results demonstrated the effectveness of the GELM was superor to other forecastng models. In summary, ths research would provde the followng contrbutons n practcal forecastng problems n the retal ndustry. (1) Influental factor selectons The Grey relaton analyss (GRA) s able to dentfy the approprate factors for forecastng future values. These nfluental factors can elucdate and ncorporate nto the nput data. (2) Forecastng effcency The effcency of GELM s better than other GBP or GMFL methods. For the demand of fresh food fluctuates usually, the faster learnng speed can provde tmely and frequent forecastng results for the manager s reference. When hdden nodes are bgger, the learnng speed of the hardlm actvaton functon s better than the sgmodal actvaton functon and the sgmodal actvaton functon s better than the sne actvaton functon. (3) Forecastng performance Ths research apples many forecastng models to be the compared benchmark. Accordng to the results, the GELM model has the smallest MAD and MSE than GARCH, GBP, and GMFL models.

12 118 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) Therefore, GELM s a vald and effectve forecastng tool that can be further appled n smlar feld for applcatons. Examnng the performance wth dfferent actvaton functons by a pared t-test, the sgmodal actvaton functon has sgnfcant dfferences wth the sne actvaton functon n MAD and MSE crterons. In ths paper, our experments have successfully demonstrated the GELM can be well employed n sales forecastng for the retal ndustry. It not only provdes smaller predctng errors but also mproves the tranng speed more than other forecastng models. Future research wll focus on the dfferent temperature levels of fresh food n the retal ndustry and mprove the stablty and learnng speed of the GELM model. REFERECES 1. Aburto, L. and Weber, R., 2007, Improved supply chan management based on hybrd demand forecasts, Appled Soft Computng, Vol. 7, o. 1, pp Akake, H., 1974, A new look at the statstcal model dentfcaton, IEEE Transactons on Automatc Control, Vol. 19, o. 6, pp Al, Ö. G., Sayn, S., Woensel, T. V. and Fransoo, J., 2009, SKU demand forecastng n the presence of promotons, Expert Systems wth Applcaton, Vol. 36, o. 10, pp Alon, I., Q, M. and Sadowsk, R. J., 2001, Forecastng aggregate retal sales: A comparson of artfcal neural networks and tradtonal methods, Journal of Retalng and Consumer Servces, Vol. 8, o. 3, pp Ansuj, A. P., Camargo, M. E., Radharamanan, R. and Petry, D. G., 1996, Sales forecastng usng tme seres and neural networks, Computers and Industral Engneerng, Vol. 31, o. 1-2, pp Au, K. F., Cho, T. M. and Yu, Y., 2008, Fashon retal forecastng by evolutonary neural networks, Internatonal Journal of Producton Economcs, Vol. 114, o. 2, pp Bgus, J. P., 1996, Data Mnng wth eural etworks: Solvng Busness Problems - From Applcaton Development to Decson Support, McGraw-Hll, ew York. 8. Bollerslev, T., 1986, Generalzed autoregressve condtonal heteroskedastcty, Journal of Econometrcs, Vol. 31, o. 3, pp Box, G. E. P. and Jenkns, G. M., 1976, Tme seres analyss forecastng and control, Management Scence, Vol. 17, o. 4, pp Chakraborty, K., Mehrotra, K. and Mohan, C. K., 1992, Forecastng the behavor of multvarate tme seres usng neural networks, eural etworks, Vol. 5, o. 6, pp Chang, P. C. and Wang, Y. W, 2006, Fuzzy Delph and back-propagaton model for sales forecastng n PCB ndustry, Expert Systems wth Applcatons, Vol. 30, o. 4, pp Chen, F. L. and Ou, T. Y., 2009, Grey relaton analyss and multlayer functon lnk network sales forecastng model for pershable food n convenence store, Expert Systems wth Applcaton, Vol. 36, o. 3, pp Chu, C. W. and Zhang, G. P., 2003, A comparatve study of lnear and nonlnear models for aggregate retal sales forecastng, Internatonal Journal of Producton Economcs, Vol. 86, o. 3, pp Deng, J. L., 1982, Control problems of Grey systems, System Control Letter, Vol. 1, o. 4, pp Dogans, P., Alexandrds, A., Patrnos, P. and Sarmves, H., 2006, Tme seres sales forecastng for short shelf-lfe food products based on artfcal neural networks and evolutonary computng, Journal of Food Engneerng, Vol. 75, o. 2, pp Engle, R. F., 1982, Autoregressve condtonal heteroskedastcty wth estmates of the varance of U.K. nflaton, Econometrca, Vol. 50, o. 4, pp Frank, C., Garg, A., Sztandera, L. and Raheja, A., 2003, Forecastng women s apparel sales usng mathematcal modelng, Internatonal Journal of Clothng Scence and Technology, Vol. 15, o. 2, pp Huang, G. B., 2003, Learnng capablty and strong capacty of two-hdden-layer feedforward networks, IEEE Transactons on eural etworks, Vol. 14, o. 2, pp Huang, G. B., Zhu, Q. Y. and Sew, C. K., 2006, Extreme learnng machne: Theory and applcatons, eurocomputng, Vol. 70, o. 1-3, pp Huang, S. T., Chu,. H. and Chen, L. W., 2008, Integraton of grey relatonal analyss wth genetc algorthm for software effort estmaton, European Journal of Operatonal Research, Vol. 188, o. 3, pp Kaastra, I. and Boyd, M., 1996, Desgnng a neural network for forecastng fnancal and economc tme seres, eurocomputng, Vol. 10, o. 3, pp Kuo, R. J. and Chen, J. A., 2004, A decson support system for order selecton n electronc commerce based on fuzzy neural network supported by real-coded genetc algorthm, Expert Systems wth Applcaton, Vol. 26, o. 2, pp Kuo, R. J., 2001, A sales forecastng system

13 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng Gra and Elm 119 based on fuzzy neural network wth ntal weghts generated by genetc algorthm, European Journal of Operatonal Research, Vol. 129, o. 3, pp Lachtermacher, G. and Fuller, J. D., 1995, Back-propagaton n tme seres forecastng, Journal of Forecastng, Vol. 14, o. 4, pp La, H. H., Ln, Y. C. and Yeh, C. H., 2005, Form desgn of product mage usng grey relatonal analyss and neural network models, Computers & Operatons Research, Vol. 32, o. 10, pp Legh, W., Purvs, R. and Ragusa, J. M., 2002, Forecastng the YSE composte ndex wth techncal analyss, pattern recognzer, neural network, and genetc algorthm: A case study n romantc decson support, Decson Support System, Vol. 32, o. 4, pp LeVee, G. S., 1993, The key to understandng the forecastng process, Journal of Busness Forecastng, Vol. 11, o. 4, pp Schawrtz, G., 1978, Estmatng the dmenson of a model, Annals of Statstcs, Vol. 6, o. 2, pp Serre, D., 2002, Matrces: Theory and Applcatons, Sprnger, ew York. 30. Sun, Z. L., Cho, T. M., Au, K. F. and Yu, Y., 2008, Sales forecastng usng extreme learnng machne wth applcatons n fashon retalng, Decson Support Systems, Vol. 46, o. 1, pp Sztandera, L. M., Frank, C. and Vemulapal, B., 2004, Predctng women s apparel sales by soft computng, Lecture otes n Artfcal ntellgence, Vol. 3070, pp Tang,. and Han, M., 2009, Partal lanczos extreme learnng machne for sngle output regresson problems, eurocomputng, Vol. 72, o , pp Tang, Z., Almeda, C., and Fshwck, P. A., 1991, Tme seres forecastng usng neural networks vs. Box-Jenkns methodology, Smulaton, Vol. 57, o. 5, pp Van der Vorst, J. G. A. J., Beulens, A. J. M., De Wt, W. and Van Beek, P., 1998, Supply chan management n food chans: Improvng performance by reducng uncertanty, Internatonal Transactons n Operaton Research, Vol. 5, o. 6, pp Wegend, A. S., Rumelhart, D. E. and Huberman, B. A., 1991, Generalzaton by weght-elmnaton wth applcaton to forecastng, Advances n eural Informaton Processng Systems, Vol. 3, pp Zhang, G. P., 2001, An nvestgaton of neural networks for lnear tme-seres forecastng, Computers and Operatons Research, Vol. 28, o. 12, pp Zhang, G. P., 2003, Tme seres forecastng usng a hybrd ARIMA and neural network model, eurocomputng, Vol. 50, pp ABOUT THE AUTHORS Fe-Long Chen s a Professor of Industral Engneerng and Engneerng Management at atonal Tsng-Hua Unversty (THU), Hsnchu, Tawan. He receved the B.S. degree n Industral Engneerng from atonal Tsng-Hua Unversty, Tawan, n 1982, and the M.S. and Ph.D degrees n Industral Engneerng from Aubrun Unversty, USA, n 1988 and respectvely. Hs currently research nterests nclude statstcal process control, total qualty management, 6-sgma, engneerng data analyss, enterprse ntegraton, enterprse resource plannng, and global logstcs management. Currently he s temporally transferred to Lteon Corp. and serves as the Dean of IE Acodemy. Tsung-Yn Ou s currently a Ph.D. canddate n the Department of Industral Engneerng and Engneerng Management at atonal Tsng Hua Unversty, Tawan. He receved hs B.S. degree at atonal Chao Tung Unversty and M.S. degree at Tungha Unversty n Ta Chung. He s also an engneer of IE Department n Chna Steel Corporaton, Tawan. Hs research nterestng ncludes Data Mnng, Operaton Management and ERP. (Receved September 2009, revsed December 2009, accepted December 2009)

14 120 Internatonal Journal of Electronc Busness Management, Vol. 9, o. 2 (2011) APPEDI Appendx 1 Moore-penrose Generalzed Inverse The resoluton of a general lnear system Ax y, where A may be sngular and may even not be square, can be made very smple by the use of Moore-Penrose generalzed nversed [29]. Defnton 1: A matrx G of order n m s the Moore-penrose generalzed nverse matrx A of order m n, f Appendx 2 Mnmum orm Least-square Solutons of General Lnear System For general lnear system Ax y, we say that xˆ s a least-square solutons f Axˆ y mn x Ax y where s a norm n Eucldean space (15) T T AGA A, GAG G, ( AG) AG, ( GA) GA (14) For the sake of convenence, the Moore-Penrose generalzed nverse matrx A wll be denote by A

15 F. L. Chen and T. Y. Ou: Constructng a Sales Forecastng Model by Integratng Gra and Elm 121 整 合 灰 關 聯 分 析 及 快 速 學 習 器 建 構 銷 售 預 測 模 式 零 售 業 之 實 證 研 究 * 陳 飛 龍 歐 宗 殷 國 立 清 華 大 學 工 業 工 程 與 工 程 管 理 學 系 新 竹 市 光 復 路 二 段 101 號 摘 要 近 來 商 業 競 爭 激 烈 且 經 濟 環 境 不 佳, 零 售 業 如 何 在 需 求 驟 變 的 環 境 下 進 行 銷 售 預 測 乃 是 為 一 大 難 題, 好 的 銷 售 預 測 可 以 提 高 顧 客 滿 意 度 減 少 鮮 食 商 品 的 報 廢 和 提 昇 營 業 額 並 有 利 於 制 定 生 產 計 劃 本 研 究 所 提 出 的 GELM 銷 售 預 測 模 式 整 合 了 灰 關 聯 分 析 與 快 速 學 習 器, 並 以 零 售 業 鮮 食 商 品 為 主 要 驗 證 對 象, 目 的 在 於 提 供 零 售 業 一 個 迅 速 且 正 確 地 預 測 模 式, 進 而 成 為 管 理 者 的 決 策 支 援 系 統 整 合 灰 關 聯 分 析 與 快 速 學 習 器 的 主 要 因 素 在 於, 灰 關 聯 分 析 可 以 在 資 訊 不 充 分 的 條 件 下 將 影 響 銷 售 量 的 重 要 因 子 篩 選 出 來, 作 為 倒 傳 遞 網 路 多 層 函 數 連 結 網 路 以 及 快 速 學 習 器 等 類 神 經 網 路 之 輸 入 資 料, 而 其 中 快 速 學 習 器 的 學 習 效 率, 經 驗 證 後 確 實 優 於 傳 統 的 類 神 經 網 路 演 算 法, 因 此 得 以 建 構 出 一 個 良 好 的 預 測 模 式 本 研 究 利 用 零 售 業 實 際 的 銷 售 數 據 進 行 驗 證,GELM 模 式 所 得 的 預 測 結 果 與 學 習 速 度 均 較 GARCH GBP 及 GMFL 等 預 測 模 型 為 佳, 此 外, 更 進 一 步 證 實, 在 GELM 預 測 模 式 中 採 用 不 同 的 活 化 函 數 (actvaton functon) 對 於 預 測 結 果 及 訓 練 速 度 是 有 顯 著 差 異 關 鍵 詞 : 銷 售 預 測 灰 關 聯 分 析 快 速 學 習 器 零 售 業 活 化 函 數 (* 聯 絡 人 :[email protected])