Forecasting women’s apparel sales using mathematical modeling Celia Frank and Ashish Garg Philadelphia University, Philadelphia, PA, USA Amar Raheja California State Polytechnic University, Pomona, CA, USA Les Sztandera Philadelphia University, Philadelphia, PA, USA Keywords Apparel, Forecasting, Computing, Time series, Modelling Abstract Traditionally, statistical time series methods like moving average (MA), auto- regression (AR), or combinations of them are used for forecasting sales. Since these models predict future sales only on the basis of previous sales, they fail in an environment where the sales are more influenced by exogenous variables such as size, price, color, climatic data, effect of media, price changes or campaigns. Although, a linear regression model can take these variables into account its approximation function is restricted to be linear. Soft computing methods such as fuzzy logic, artificial neural networks (ANNs), and genetic algorithms offer an alternative taking into account both endogenous and exogenous variables and allowing arbitrary non-linear approximation functions derived (learned) directly from the data. In this paper, two approaches have been investigated for forecasting women’s apparel sales, statistical time series modeling, and modeling using ANNs. Four years’ sales data (1997-2000) were used as backcast data in the model and a forecast was made for 2 months of the year 2000. The performance of the models was tested by comparing one of the goodness-of-fit statistics, R 2 , and also by comparing actual sales with the forecasted sales of different types of garments. On an average, an R 2 of 0.75 and 0.90 was found for single seasonal exponential smoothing and Winters’ three parameter model, respectively. The model based on ANN gave a higher R 2 averaging 0.92. Although, R 2 for ANN model was higher than that of statistical models, correlations between actual and forecasted were lower than those found with Winters’ three parameter model. Introduction What is forecasting? Forecasting is ubiquitous; nearly everyone, in almost every walk of life, forecasts to some extent. A forecast is a probabilistic estimate of a future value. The underlying assumption in most forecasting methods is that the past patterns or behavior will continue in the future. It is commonly said that a good forecast requires a good “backcast”; patterns of the past are modeled and those patterns are projected into the future. The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at http://www.emeraldinsight.com/researchregister http://www.emeraldinsight.com/0955-6222.htm This research has been supported by the United States Department of Commerce/National Textile Center Grant IP0-P10. Additional research support has been provided by Mothers Work, Inc. Forecasting women’s apparel sales 107 International Journal of Clothing Science and Technology Vol. 15 No. 2, 2003 pp. 107-125 q MCB UP Limited 0955-6222 DOI 10.1108/09556220310470097
Transcript
Forecasting women’s apparelsales using mathematical
modelingCelia Frank and Ashish Garg
Philadelphia University, Philadelphia, PA, USA
Amar RahejaCalifornia State Polytechnic University, Pomona, CA, USA
Les SztanderaPhiladelphia University, Philadelphia, PA, USA
Keywords Apparel, Forecasting, Computing, Time series, Modelling
Abstract Traditionally, statistical time series methods like moving average (MA), auto-regression (AR), or combinations of them are used for forecasting sales. Since these models predictfuture sales only on the basis of previous sales, they fail in an environment where the sales are moreinfluenced by exogenous variables such as size, price, color, climatic data, effect of media, pricechanges or campaigns. Although, a linear regression model can take these variables into accountits approximation function is restricted to be linear. Soft computing methods such as fuzzy logic,artificial neural networks (ANNs), and genetic algorithms offer an alternative taking into accountboth endogenous and exogenous variables and allowing arbitrary non-linear approximationfunctions derived (learned) directly from the data. In this paper, two approaches have beeninvestigated for forecasting women’s apparel sales, statistical time series modeling, and modelingusing ANNs. Four years’ sales data (1997-2000) were used as backcast data in the model and aforecast was made for 2 months of the year 2000. The performance of the models was tested bycomparing one of the goodness-of-fit statistics, R 2, and also by comparing actual sales with theforecasted sales of different types of garments. On an average, an R 2 of 0.75 and 0.90 was foundfor single seasonal exponential smoothing and Winters’ three parameter model, respectively. Themodel based on ANN gave a higher R 2 averaging 0.92. Although, R 2 for ANN model was higherthan that of statistical models, correlations between actual and forecasted were lower than thosefound with Winters’ three parameter model.
IntroductionWhat is forecasting?Forecasting is ubiquitous; nearly everyone, in almost every walk of life,forecasts to some extent. A forecast is a probabilistic estimate of a future value.The underlying assumption in most forecasting methods is that the pastpatterns or behavior will continue in the future. It is commonly said that a goodforecast requires a good “backcast”; patterns of the past are modeled and thosepatterns are projected into the future.
The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at
This research has been supported by the United States Department of Commerce/NationalTextile Center Grant IP0-P10. Additional research support has been provided by MothersWork, Inc.
Forecastingwomen’s apparel
sales
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International Journal of ClothingScience and Technology
Vol. 15 No. 2, 2003pp. 107-125
q MCB UP Limited0955-6222
DOI 10.1108/09556220310470097
For many organizations, their investment in forecasting has an immediate,as well as, long-term impact on profitability, customer service, productivity,etc. A good forecasting system is essential in avoiding problems such asinventory shortages and excesses, missed due dates, plant shutdowns, lostsales, lost customers, expensive expediting, and missed strategic opportunities.
Forecasting processThe process of forecasting can be relatively simple or complex depending onthe situation. A typical forecasting procedure is shown in Figure 1 and involvesthe following.
. Identify the problem. There is a need to establish clearly the targetedvariable that is to be forecasted, e.g. the demand of a particular product asa function of time.
. Assemble the data. Past data must be carefully collected to find anypatterns (as well as randomness) associated with the targeted variable.
. Formulation of the model. Based on an analysis of the data, a hypotheticalmodel is formulated that includes those factors that influence the targeted
Figure 1.Forecasting procedure
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variable. For example, the demand for a particular garment can beseasonal and trending, a function of selling price, influenced byadvertising, etc.
. Execution of the model. Based on the hypothesis, one or more modelsare then applied to the real data. The data set is generally dividedinto two subsets: one subset is used to formulate the model and theother subset is used to test how well the model performs in predictingunseen data.
. Analysis of the result. By performing statistical significance tests, andappropriate error measurements, the model may be accepted, modified orrejected.
. Ongoing improvements. A model must be constantly monitored for itsperformance and improved whenever unacceptable deviations emerge.
Classification and overview of forecasting methodsIn general, forecasting methods are divided into three categories:
(1) univariate;
(2) multivariate; and
(3) qualitative.
Univariate methods ( DeLurigo, 1998). Univariate modeling techniques,generally use time as an input variable with no other outside explanatoryvariables; this forecasting method is often called time series modeling. Forexample, a simple seasonal model might be, Yt ¼ Yt212 þ ðYt2l 2 Yt213Þ;where Yt;Yt2l;Yt212;Yt213 are sales in weeks t, t21, t 2 12; and t 2 13;respectively. A few commonly employed methods in time series models are asfollows.
. Moving averages. Time series are smoothed using moving averages thatreduce the period-to-period variation; local movements above and below along-run mean are tracked.
. Exponential smoothing (EXPOS)/Holts-Winters. Time series aresmoothed in such a way that the most recent observations receivegreater weight. Advanced methods incorporate decomposition to explaintrend and seasonality.
. Fourier series. This method models trend, seasonality, and cyclicalmovements using trigonometric sine and cosine functions. This method isused in automated forecasting systems, however, it is not without itsdetractors.
. ARIMA (Box-Jenkins). This method models a series using trend, seasonal,and smoothing coefficients that are based on moving averages, auto-regression, and difference equations. In this approach, a user is not
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constrained to a single model and, hence, to a single specific functionalform, but can select from a wide range of models (Newbold, 1983).
Models using time series methods can help managers make prudent businessdecisions, but they exclude causal relationships, and they should not befollowed automatically without consideration of other business factors(Wheelwrigth and Makridakis, 1997).
Causal/Multivariate methods. Explanatory models are used to establish“cause and effect” relationships in a system, for example, sales as a function ofprice, advertising, competition, and so forth. Although, multivariate methodsare often used to model “cause and effect”, many multivariate models arefocused primarily on the accuracy of the forecast. A few commonly used causalmodels are listed below.
. Multiple regression approach. This approach uses the method of leastsquares, to model a relationship between one dependent and manyindependent variables. From a causal standpoint, multiple regressionmodels may not be as valid as those of econometric. Nonetheless theymay forecast as accurately.
. Econometric method. In this method, using generalized least squarestechniques, relationships between one or more endogenous andexogenous variables are estimated. Small-scale, simple models aremultiple regression models; however, the theoretical foundation ofeconometric models is more rigorous. Mutual causality using severalsimultaneous equations can be modeled with econometric methods.
. Multivariate ARIMA (Box-Jenkins-MARIMA) method. This methodcombines the strengths of the econometric and ARIMA time seriesmethods. It is quite effective in applications when the effects of theindependent variables lead one or more dependent variables.
Qualitative methods. Qualitative methods of forecasting include Delphi,market research, panel consensus, historical analogy, etc. These suggestivequalitative methods are most frequently used to make long-run predictionswhen there is little objective data concerning relevant past patterns orrelationships.
Qualitative methods are useful when there is minimal data to supportquantitative methods. In business, they are used to predict the demand for newproducts, new technologies, new market shares, the cost or development timefor new products or technologies, or the best competitive strategy.
A few commonly used qualitative models are given below.. Delphi. It is an iterative process in which experts respond to
questionnaires, and the results are subsequently tabulated andmodified to reach conclusions.
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. Panel consensus. This method is based on the assumption that theconsensus of several experts will yield a better forecast than that from oneexpert. The opinions of complementary experts yield improvedpredictions.
. Historical analogy. This method models a time series using a similar eventfrom the past and is useful with new products and emerging technologieswhere there is no past data.
. Soft computing. Methods based on soft computing mimic some of theparallel processing capabilities of the human brain to model both simpleand complex situations. These models can identify non-linear andinteractive relationships, which were anticipated by the analyst. Commonmethods include artificial neural networks (ANNs), fuzzy logic, andgenetic algorithms.
Apparel sales forecastingIn the present world, all industries need to be adaptable to a changing businessenvironment in the context of a competitive global market. To comply withhigher versatility and disposability of products for consumers, firms haveadopted new forms of production behavior with names such as “just-in-time”and “quick response” (Vorman et al., 1998). To react to worldwide competition,managers are often required to make wise decisions rapidly. In fact, they mustoften anticipate events that may affect their industry. More than 80 percent ofthe US textile and apparel businesses have indicated an interest in time-basedforecasting systems and have incorporated one or more of these technologiesinto their operations (Kincade et al., 1993).
To make decisions related to the conception and the driving of any logisticstructures, industrial managers must rely on efficient and accurate forecastingsystems. Better forecasting of production, predicting in due time a sufficientquantity to produce, is one of the most important factors for the success of alean production.
Present researchForecasting garment sales is a challenging task because many endogenous aswell as exogenous variables, e.g. size, price, color, climatic data, effect of media,etc. are involved. Our approach is to forecast apparel sales in the absence ofmost of the above-mentioned factors and then use principles of fuzzy logic toincorporate the various parameters that affect sales. The forecasting modelinginvestigates the use of two statistical time series models, seasonal singleexponential smoothing (SSES) and Winters’ three-parameter model. It theninvestigates soft computing models using ANNs.
A foundation has been made for multivariate fuzzy logic based model bybuilding an expandable database and a rule base. After a substantial amount ofdata is collected, this model can be used to make predictions for sales specific toa store, color, or size of garment.
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DatabaseA US-based apparel company has been providing sales data for various types,or classes, of apparel for the current and previous years.
Data formatFrom January 1997 until February 2001, sales data were collected for each dayfor every class for every store. From March 2001 onwards, more detailedinformation about each garment is available including its size and color.
The data for the previous years (1997-2000) has much fewer independentvariables as compared to the data for the current year, 2001. This is evidentfrom Tables I and II.
Row 1 of Table I is read as: 1 unit of Class X sold for $19.90 in Store_1 on28 June 1997. Row 2 of Table I is read as: 1 unit of Class Y returned in Store_2on 28 September 1998. Similarly, Row 1 of Table II is read as: 1 unit of Class Xof size M, color 1 sold for $19.90 in Store_1 on 28 July 2001.
Building of databaseRaw data available in the formats shown in Tables I and II was processed toform a database, which was subsequently used as input for time series analysisand a fuzzy logic based multivariate forecast model.
Using the data from January 1997 till March 2000, time series analysis wasperformed. A data file for each class that contains information about its totalsales (in dollars) each day was prepared. For example, input file for class Xlooks as shown in Table III.
Total sales in terms of number of units would have been a better option butit was not possible since the number of units sold was missing from many rowsin the raw data.
From March 2001 onwards, aggregation was performed in several steps.Initially, from daily sales files, a database containing information about total
Units Price Class Store Date
1 19.9 X Store_1 28/6/199721 19.9 Y Store_2 28/9/1998
Table I.Sales data formatfor 1/1997-2/2001
Base Color Size Units Price Class Store Date Label QOH
Base_1 1 M 1 19.9 X Store_1 28/7/2001 Label_1 0Base_2 49 XL 1 19.9 Y Store_2 8/01/2001 Label_2 0
sales was formed for a particular garment-class for each color, each size, andeach store. For example, input file for class X is shown in Table IV.
The total number of rows in this format was too large to be used as an inputfile for the multivariate model. In order to reduce the dimensionality of thedatabase, two compressions were performed. First, daily sales data wereconverted into monthly sales data. Secondly, color information was compressedby aggregating sales of similar colors. For example, color code 23 representslight green, 25 dark green, 26 medium green. Instead of having different rowsfor different tones of green, we compressed this information into a single rowby assigning color code 2 for all greens.
Finally, input data file for multivariate fuzzy logic based model looks asshown in Table V.
Pre-analysisData were analyzed for trends and seasonality. This analysis helps in choosingan appropriate statistical model, although this kind of preparation is notnecessary for soft computing based models.
For pre-analysis, three classes were chosen, one each from the Spring, Falland Non-seasonal garment categories. Analysis was spread among all thecategories to remove any bias due to the type of class.
Date Size Color Store Units
28/7/2001 M 23 Store_1 628/7/2001 M 26 Store_1 228/7/2001 XL 45 Store_2 7
Weekly trendSales data often showed a weekly trend with sales volume increasing duringweekends as compared to weekdays. This trend was evident from the dailysales graphs for all the classes. Figure 2(a)-(c) shows this trend graphically forthree classes, one each from Spring, Fall and Non-seasonal category.
In all the figures, it can be seen that sales volume peaks on 4th, 11th, 18th,and 25th of January, which are all Saturdays, and generally second highestsales are on Sundays.
This observation was further supported by qualitative means by calculatingauto correlations functions (ACFs). ACF is an important tool for discerningtime series patterns. ACF were calculated for 350 observations for all the threeclasses.
ACF for a given lag k is given by equation (1):
ACFðkÞ ¼
Xn
t¼1þk
ðYt 2 �Y ÞðYt2k 2 �Y Þ
Xn
t¼1
ðYt 2 �Y Þ2ð1Þ
Figure 2.(a) Daily sales class ASfor January 1997; (b) dailysales class AF forJanuary 1997; (c) dailysales class CN forJanuary 1997
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It can be seen from Table VI that in every case there is a very high value ofACF at lag 7. This suggests that in daily sales data, sales pattern repeats afterevery 7 days.
Upon further analysis of the data, it was observed that fraction contributionstowards total sales in any week of the year and for any class remainsignificantly constant. Hence, information about fraction contribution can beused to forecast daily sales after forecasting weekly sales. Table VII andFigure 3 shows average fraction contribution towards total sales of a week.
Annual trendGarment sales are generally seasonal with demand increasing for a particulartype in one season and for a different type in another season. To investigateseasonality, the same methodology was used as was used to establish weeklytrend. Both graphically as well as using ACF, it was shown that sales of allthree classes under consideration show strong and distinct seasonal trend.Interestingly, class CN has been categorized as non-seasonal; still it showedincrease in seasonality although not as distinctively as shown in other twoclasses.
While investigating the annual seasonal trend, data were aggregatedinto weekly increments. Figure 4(a)-(c) graphically shows the seasonality ofthe three classes. Figure 5(a)-(c) graphically shows ACFs. All the graphs ofFigure 5 are sinusoidal in shape. This reflects the relationships in the low order(i.e. 1 to 12 lags) and high order (i.e. 48 to 52 lags). The sinusoidal pattern inACFs of Figure 5 is typical of many seasonal time series.
ACFs for class ASLag 1 2 3 4 5 6 7ACF 0.653 0.502 0.444 0.433 0.501 0.637 0.874
ACFs for Class AFLag 1 2 3 4 5 6 7ACF 0.767 0.676 0.685 0.659 0.650 0.757 0.87
ACFs for class CNLag 1 2 3 4 5 6 7ACF 0.670 0.514 0.486 0.470 0.490 0.655 0.871 Table VI.
Day Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Fraction (percent) 13 10 11 11 13 18 24
Table VII.Fractions of weekly
sales distributedamong 7 days
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Methodology and resultsEvident from the format of the database for years 1997-2001 only salesinformation with respect to time for various garments is available. Hence, onlyunivariate time series and soft computing models were investigated using thisdata.
Figure 4.(a) Weekly sales data forclass AS; (b) Weeklysales data for class AF;(c) Weekly sales data forclass CN
Figure 3.Fractions of weeklysales distributed among7 days
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From March 2001 onwards, much more elaborate sales data were available.Using this data set, a multivariate forecasting model was implemented whichcould prove useful for inventory maintenance.
Six classes, two each from the Spring (AS and BS), the Fall (AF and BF), andthe Non-seasonal (CN and DN) categories, were chosen for each model. Theywere built using 4 years sales data, and the next 2 months data were forecasted.The forecasted data were then compared with actual sales to estimate theforecasting quality of the model.
Figure 5.(a) ACF (k) for class AS;(b) ACF (k) for class AF;(c) ACF (k) for class CN
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Univariate time series modelSSES. EXPOS is one of the most widely used forecasting methods. Asdiscussed previously regarding seasonality in sales data, a single exponentialsmoothing (SES) model with a factor of seasonality was investigated.
A SSES model requires four pieces of data: the most recent forecast, the mostrecent actual data, a smoothing constant, and length of the seasonal cycle. Thesmoothing constant (a) determines the weight given to the most recent pastobservations and therefore controls the rate of smoothing or averaging. Thevalue of a is commonly constrained to be in the range of zero to one.
The equation for SSES is:
Ft ¼ a At2s þ ð1 2 aÞFt2s ð2Þ
where Ft is the exponentially smoothed forecast for period t, s the length of theseasonal cycle, At2 s the actual in the period t 2 s, Ft2 s the exponentiallysmoothed forecast of the period t 2 s, and a (alpha) the smoothing constant.
Weekly sales data were used for the forecast model given by the aboveequation. Hence, s was chosen to be 52 (number of weeks in a year). There aremany ways of determining alpha. Method chosen in the present work wasbased on minimum squared error (MSE). Different alpha values were tried formodeling sales of each class and the alpha that achieved the lowest SE waschosen.
After choosing the best alpha value, a forecast model was built for each classusing 4 years weekly data. Using the model, a weekly sales forecast wasconducted for January and February of 2001. In order to forecast daily sales, thefraction contribution of each day (given in Figure 3) was multiplied by the totalforecasted sales of each week.
Figure 6(a) shows the actual versus fitted values of 3 years sales data andFigure 6(b) shows the actual versus forecasted daily sales for class AS. Thedata for the remaining five classes are available on the project Web site.Table VIII gives the alpha value, R 2 of the model, and correlation coefficientbetween the actual and forecasted daily sales from 3 January 2001 to 27February 2001, for all classes.
It can been seen that even with single parameter SSES, R 2 is on an averagemore than 0.75 implying that the model is able to explain 75 percent of thevariation in the data. Correlation coefficients between the actual and forecastedsales from 3 January 2001 to 27 February 2001 are also quite high except forclass DN.
Winters’ three parameter EXPOS. Winters’ powerful method models trend,seasonality, and randomness using an efficient EXPOS process. Theunderlying structure of additive trend and multiplicative seasonality ofWinters’ model assumes that:
Ytþm ¼ ðSt þ btÞI t2Lþm ð3Þ
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where St is the smoothed non-seasonal level of the series at the end of t, bt thesmoothed trend for the period t, m the horizon length of the forecasts of Ytþm;and I t2Lþm the smoothed seasonal index for the period t þ m:
That is, Ytþm the actual value of a series, equals a smoothed level value St
plus an estimate of trend bt times a seasonal index I t2Lþm: These threecomponents of demand are each exponentially smoothed values available at theend of period t. The equations used to estimate these smoothed values are:
Figure 6.(a) Actual vs fitted salesvalue for class AS using
Note: Correlation coefficient between the actual and forecasted sales from 3 January 2001 to27 February 2001.
Table VIII.Values of alpha, R 2,
and correlationcoefficients for
SSES model
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St ¼ aðYt=I t2LÞ þ ð1 2 aÞðSt21 þ bt21Þ ð4Þ
bt ¼ bðSt 2 St21Þ þ ð1 2 bÞbt21 ð5Þ
I t ¼ g ðYt=StÞ þ ð1 2 gÞI t2Lþm ð6Þ
Ytþm ¼ ðSt þ btmÞI t2lþm ð7Þ
where Yt is the value of actual demand at the end of period t, a the smoothingconstant used for St, St the smoothed value at the end of t after adjusting forseasonality, b the smoothing constant used to calculate the trend (bt), bt thesmoothed value of trend for the period t, It2L the smoothed seasonal index Lperiods ago, L the length of the seasonal cycle (e.g. 12 months or 52 weeks),g (gamma) the smoothing constant, for calculating the seasonal index in periodt, It the smoothed seasonal index at end of period t, and m the horizon length ofthe forecasts of Ytþm:
Equation (4) calculates the overall level of the series. St in equation (5) is thetrend-adjusted, deseasonalized level at the end of period t. St is used in equation(7) to generate forecasts, Ytþm: Equation (5) estimates the trend by smoothingthe difference between the smoothed values St and St21: This estimates theperiod-to-period change (trend) in the level of Yt. Equation (6) illustrates thecalculation of the smoothed seasonal index, It. This seasonal factor is calculatedfor the next cycle of forecasting and used to forecast values for one or moreseasonal cycles ahead.
For choosing a (alpha), b (beta), and g (gamma) MSE was used as a criterion.Different combinations of alpha, beta, and gamma were tried for modeling salesof each class and the combination that achieved the lowest RSE was chosen.
After choosing the best alpha, beta, and gamma values, the forecast modelwas built for each class using 4 years of weekly sales data. Using the model, aweekly sales forecast was done for January and February of 2001. In order toforecast daily sales afterwards, the fractional contribution of each day (given inFigure 3) was multiplied by the total forecasted sales of each week. Figure 7(a)shows the actual versus fitted values of 3 year sales data and Figure 7(b) showsthe actual versus forecasted daily sales for class BS. This data for the fiveremaining classes are available on the project Web site.
Table IX gives the alpha, beta, gamma, R 2 of the model, and the correlationcoefficient between the actual and forecasted daily sales from 3 January 2001 to27 February 2001.
R 2 values for all the classes except BF are much higher than those obtainedfrom SSES. Higher R 2 values and the ability of Winters’ model to better definethis is due to the additional parameter beta utilized for trend smoothing.
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Although curve fitting is very good, correlation coefficients between the actualand forecasted sales are not as high. On observing the graphs of the actualversus forecasted values for all the classes, it can be observed that trend(growth or decay) has always been over estimated and, hence, forecasted values
Figure 7.(a) Actual vs fitted salesvalue for class BS using
Winters’ model; (b) actualvs forecasted sales value
Note: Correlation coefficient between the actual and forecasted sales from 3 January 2001 to27 February 2001.
Table IX.Values for alpha,beta, gamma, R 2,
and correlationcoefficients forWinters’ model
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are too high or too low. As in any multiplicative model, the division by verysmall numbers or multiplication by extremely large values is a problem withequation (3) which could have resulted in overestimation of the trend.
Soft computing methodsSoft computing methods are a rapidly growing area of computer science. Thesemethods are being used to solve many problems such as optimization, patternclassification, forecasting, learning, etc. ANNs, genetic algorithms, fuzzy logicbased reasoning, etc. are some of the popular soft computing methods beingused to solve many real world problems. In our current research, we have usedANN to learn the patterns of sales of garments in the past to forecast sales inthe future.
ANN modelNeural networks mimic some of the parallel processing capabilities of thehuman brain as models of simple and complex forecasting applications. Thesemodels are capable of identifying non-linear and interactive relationships andhence can provide good forecasts. In our research, one of the most versatileANNs, the feed forward, back propagation architecture was implemented. Thearchitecture of the feed forward neural network is shown in Figure 8.The hidden layers are the regions in which several input combinations from theinput layer are fed and the resulting output is finally fed to the output layer.{x1; . . .; xM } is the training vector and {z1; . . .; zM } is the output vector.
The error E of the network is computed as the difference between the actualand the desired output of the training vectors and is given in equation (8):
E ¼1
PNL
XNL
n¼l
XP
p¼1
ðtð pÞn 2 sð pÞ
n Þ2 ð8Þ
where tð pÞn is the desired output for the training data vector p and sðpÞn is the
calculated output for the same vector. The updated equation for the weights ofindividual nodes in different layers is defined using the first derivative of theerror E as given in equation (9):
Figure 8.Multilayer perceptronmodel used forforecasting sales
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wðnÞi ðlÞ ¼ wðnÞ
i ðlÞ þ DwðnÞi ðlÞ; where DwðnÞ
i ðlÞ ¼ 2h›E
›wðnÞi ðlÞ
ð9Þ
ANN consisted of three layers: input layer, hidden layer and output layer with10, 30 and 1 neuron, respectively. Two hundred and seventeen weeks salesdata were divided into three parts. The first part consisted of 198 weeks,which was used to train the network. The second part with 10 weeks datawere used to test the network for its performance. The third part with 9 weeksdata were used to compare forecasting ability of the network by comparingthe forecasted data with the actual sales data. In order to forecast daily salesafterwards, the fraction contribution of each day was multiplied by the totalforecasted sales of each week. Figure 9(a) shows the actual versus fittedvalues of 3 years sales data and Figure 9(b) shows the actual versus
Figure 9.(a) Actual vs fitted salesvalue for class CN usingANN model; (b) actual vsforecasted sales value for
class CN usingANN model
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forecasted daily sales for class CN. This data for the five remaining classesare available on the project Web site.
Table X gives the R 2 of the model, and correlation coefficient between theactual and forecasted daily sales from 3 January 2001 to 27 February 2001.
R 2 values for all the classes are much higher than those obtained from SSES,and Winters’ model. High R 2 values and the strength of the ANN model are dueto the ability of ANNs to learn non-linear patterns.
Although curve fitting is very good, correlation coefficients between theactual and forecasted sales are not that good. This might be due to overlearning of the network. A potential problem when working with noisy data isthe so-called over-fitting. Since ANN models can approximate essentially anyfunction, they can also overfit all kinds of noise perfectly. Typically, sales datahave a high noise level. The problem is intensified by a number of outliers(exceptionally high or low values). Unfortunately, all three conditions thatincrease the risk of over-fitting are fulfilled in our domain and have impactedcorrelations.
ConclusionTime series analysis seemed to be quite effective in forecasting sales. In allthe three models, R 2 and the correlation coefficients were significantly high.The three parameter Winters’ model outperformed SSES in both explainingvariance in the sales data (in terms of R 2 ) and forecasting sales (in termsof correlation coefficient).
ANN model performed best in terms of R 2 among three models. Butcorrelations between the actual and forecasted sales were not satisfactory.A potential problem when working with noisy data, a large number of inputs,and small training sets is the so-called over-fitting. Since big ANN models canapproximate essentially any function, they can also over fit all kinds of noiseperfectly. Unfortunately, all three conditions that increase the risk of over-fitting are fulfilled in our domain. Typically, sales data have a high noise level.The problem is intensified by a number of outliers (exceptionally high or lowvalues).
A multivariate fuzzy logic based model could model the sales very well, as itwould take into account many more influence factors in addition to time. This
Note: Correlation coefficient between the actual and forecasted sales from 3 January 2001 to27 February 2001.
Table X.Values of R 2,and correlationcoefficients for ANNmodel
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naturally leads to the first extension of this work. Extensions of the concept ofdiscovery learning are of current interest and are being investigated.
References
DeLurigo, S.A. (1998), Forecasting Principles and Applications, 1st ed., McGraw Hill, NY, USA.
Kincade, D.H., Cassill, N. and Williamson, N. (1993), J. Text. Inst., Vol. 84 No. 2, p. 2.
Newbold, P. (1983), Journal of Forecasting, Vol. 2, p. 28.
Vorman, P., Happiette, M. and Rabenasolo, B. (1998), J. Text. Inst., Vol. 1 No. 2, p. 78.
Wheelwrigth, S.C. and Makridakis, S. (1997), Forecasting Methods for Management, 2nd ed.,John Wiley, New York.
