, pp.143-152 http://dx.doi.org/10.14257/ijhit.2015.8.8.13 A Study of Time Series Model for Forecastig of Boot i Shoe Idustry Amrit Pal sigh *, Maoj Kumar Gaur, Diesh KumarKasdekar ad Sharad Agrawal Departmet of Mechaical Egieerig, Madhya Istitute of Techology & Sciece, Gwalior, INDIA * Correspodig Author e-mail:amrit00257@gmail.com Abstract Predictig sales i a shoe idustry is a typical job due to upredictable demad of products. May models are suggested for forecastig the product i the literature over the past few decades. Most shoe maufacturig orgaizatios are i a cotiuous effort for icreasig their profits ad reducig their costs. Exact sales forecastig is certaily a iexpesive way to meet the orgaizatio goals. This paper studies ad compares differet forecastig techiques as movig average, sigle expoetial smoothig, double expoetial smoothig ad witer s method. For this, domestic sales data from a shoe idustry is collected ad the data were aalyzed by statistics techique usig Miitab 17 software. The result shows that the demad of shoes fluctuate over the period of time. The factor that iflueces the choice of forecastig model is the least value of Mea square deviatio (MSD). Keywords: time-series, sales forecastig, shoe product, MSD 1. Itroductio Idustry forecasts are maily useful to big retailers who might have a superior market share. For the retailig idustry, R. T. Peterso [1] showed that larger retailers are more likely to use time-series methods ad prepare idustry forecasts, while smaller retailers give importace to judgmetal methods ad compay forecasts. Predictig sales accurately i the shoe idustry is geerally cosidered challegig due to volatile demad of products. Regardless of the kow difficulties, predictig sales i the idustry is cosidered to be crucial due to the log lead times caused by the relatively curret tred of sourcig productio ad operatios to other coutries. The iterestig cocept of predictig sales, more commoly referred to as demad or sales forecastig, i the swiftly chagig idustry has bee greatly researched i the past decades. This has resulted i a large umber of published forecastig ideas, methods ad measures aimig to produce as exactly predicted demad as possible while admittig that a error-free forecast is ever possible. The varieties of ideas available, together with the geerally large forecastig errors, have made the process of choosig a valid forecastig method ad applyig it i a suitable maer for the compaies. Tej iteratioal group produce ad export the best quality footwear all over the world by maufacturig broad rage of comfort, casual ad busiess shoes. All the Factories are established i vast spaces ad prepared with state-of-the-art machies icludes a variety of fully Mechaized coveyor systems with o lie quality check at every stage throughout the productio. The shoes come with the fiest quality of material used i soles like Polyurethae(PU), Thermoplastic rubber(tpr), Thermo Polyurethae (TPU) ad Polyviyl chloride (PVC). (Ijectio Footwear 3000 Pairs per day, Had-Stiched ad Sa Crispio Footwear 7000 Pairs per day, Cemeted Footwear 5000 Pairs per day). This paper studies ad compares differet forecastig techiques, movig average, sigle ISSN: 1738-9968 IJHIT Copyright c 2015 SERSC
expoetial smoothig, double expoetial smoothig, ad witer s method. All the forecastig techique applied to the same real data that have varied itermittet demad. 2. Literature Review H. K. Alfares & M. Nazeeruddi [2] categorized the electric load forecastig techique. A wide rage methodologies ad model for forecastig are give i literature. These techique are classified ito ie categories: (1) multiple regressio, (2) expoetial smoothig, (3) iterative reweighted least squares, (4) adaptive load forecastig, (5) stochastic time series, (6) ARMAX model based o geetic algorithms, (7) fuzzy logic, (8) eural etworks ad (9) expert systems. The methodology for each category is briefly described, the advatage ad disadvatages discussed. W. Peter & S. Aders [3] developed flexible ad robust supply chai forecastig systems uder chagig market loads. They suggested ew tools ad models to estimate forecastig error measuremets. The paper maily studies slow movig or itermittet demad, whe for items the forecastig time periods over ad over agai have zero demad. For the difficult to forecast itermittet demad the Crosto forecastig techique is mostly regarded as a better alterative tha sigle expoetial smoothig. These two methods, Crosto ad sigle expoetial smoothig, mutually with two modificatios of the Crosto method, are discussed ad evaluated with real itermittet data. The held performace of a forecastig techique is depedet of the chose measuremet of forecast errors. The mai purpose is to examie ad estimate differet forecastig error measuremets. C. Cacatto, P. Belfiore & J. G. V. Vieira [4] itroduced the forecastig practices that have bee used by food idustries i Brazil ad detected how these compaies use forecastig methods ad what the mai factors that ifluece their selectio. The data were aalyzed by multivariate statistics techiques usig the SPSS software. They stated that the compaies did ot use sophisticated forecastig methods, the historical aalysis model was the mostly used. The factors that ifluece the choice of the models are the type of product ad the time spet durig the forecastig. M. Sawlai & M. Vijayalakshmi [5] studied the effect of decomposig time series ito multiple compoets like tred, seasoal ad irregular ad performed the clusterig o those compoets ad geerate the forecast values of each compoet separately, they worked o sales data. Statistical method ARIMA, Holt witer ad expoetial smoothig are used to forecast those compoets. Selectio of best, good ad bad forecasters are performed o the basis of cout ad rak of expert id s geerated, fially absolute percetage error (APE) is used for comparig forecast. D. P. Patil, A. P. Shrotri & A. R. Dadekar [6] aalyzed the advatage ad disadvatage of basic, i-betwee ad advaced method for visitor use forecastig where seasoality ad limited data are characteristics of the assessmet problem. Forecastig method iclude the aϊve model 1 ad 2, movig average method, double movig method, expoetial smoothig method, double expoetial method, Holt s method, Witer method ad seasoal autoregressive itegrated movig average (SARIMA) are used Milwaukee couty zoo visitor forecastig A. K. Sharma, A. Gupta & U. Sharma [7]. The series raged from Jauary 1981 to 1999 or a total of 228 moths. The last 12, 24 or 60 moth of those data were excluded from the origial aalysis ad they were used to evaluate the various methods. SARIMA ad SMA with classical decompositio procedure were foud to be roughly equivalet i performace, as judged by modified mea absolute percetage error method ad modified root mea squared error value of loger estimatio period with shorter period ahead forecasts. This study also foud that the SMA with classical decompositio method was more accurate tha other techique whe shorter estimatio period with loger period ahead forecast are icluded. C. D. Ezeliora [8] aalyzed the forecastig of plastic yield i Fioplastika maufacturig idustry by usig movig average aalysis. They gathered the data of daily plastic pipe productio 144 Copyright c 2015 SERSC
SALE Iteratioal Joural of Hybrid Iformatio Techology over the moth for three years. They fitted movig average model to predict the product total, as the coefficiet of determiatio shows a strog relatioship. This model proposed optimum mothly productio output for differet products. K. Ryu & A. Sachez [9] evaluated the forecastig method for istitutioal food service facility. They are idetifyig the most appropriate forecastig method of forecastig meal cout for a istitutioal food service facility. The forecastig method aalyzed icluded: aϊve model 1,2 ad 3; movig average method, double movig method, expoetial smoothig method, double expoetial method, Holt s method, Witer method, liear regressio ad multiple regressio method. The accuracy of forecastig methods was measured usig mea absolute deviatio, mea squared error, mea percetage error, mea absolute percetage error method, root mea squared error ad Theil s Ustatistic. The result of this study showed that multiple regressios was the most accurate forecastig method, but aϊve method 2 was selected as the most appropriate forecastig method because of its simplicity ad high level of accuracy. E. Leve' & A. Segerstedt [10] suggested a modificatio of the Crosto method where a demad rate is directly calculated whe a demad has happeed. A. A. Sytetos & J. E. Boyla [11] recommeded a adjustmet of the Crosto method due to a systematic error. J. J. Strasheim [13] itroduced variety of alterative forecastig techiques ad they were evaluated for purpose of stock repleishmet is a importat fuctio of part i the typical reorderig motor vehicle spare parts with aim of selectig oe optimal techique to be implemeted i automatic reorderig module of real time computerized ivetory maagemet system. A large umber of forecastig techique wereevaluated, amely simple movig average(averages, Movig Averages, Double Movig Averages),Expoetial Smoothig(Sigle Expoetial Smoothig, Adaptive Expoetial Smoothig, DoubleExpoetial Smoothig (Brow's oe 'parameter liear method ad Holt's two parameter method),triple Expoetial Smoothig (Brow's oe parameter quadratic method ad Witer's threeparameter tred ad seasoality method), liear Regressio, Multiple Regressio. The accuracy offorecastig methods was measured usig statistical measures (mea error, mea absolute deviatio, mea squared error), relative measures (percetage error, mea percetage error, mea absolutepercetage error method) ad others measures (Theil s U- statistic, Durba- Watso value ad forecastig idex). 3. Data Collectio The data i this study has bee collected from Tez iteratioal, Agra o mothly basis over a period of three years (April 2012 to March 2015). The domestic sales data of a product amed army boot (Article umber 741) has bee take ad forecastig models are successively applied ad the aalyzed usig statistical techique. Figure 1 shows last three years data startig from April 2012 to March 2015, the origial data has bee coverted ito graphical form usig Miitab 17. Time Series Plot of SALE 12000 10000 8000 6000 4000 2000 0 Apr Oct Apr Oct Moth Apr Oct Figure 1. Time Series Plot of Sale Copyright c 2015 SERSC 145
4. Methodology Ucertaity i demad of product ca be reduced through forecastig methods. There are various forecastig model used i the aalysis, icluded simple movig average method, sigle expoetial method, double expoetial method(holt s), witer s method. The most appropriate forecastig method was determied o the basis of accuracy. I this work, several commo accuracy methods were used: Mea Absolute Deviatio (MAD), Mea Absolute Percetage Error (MAPE), ad Mea Square Deviatio (MSD). I every method differet umbers of trials have bee used, out of which the best oe (least MSD) is take ad plotted i graph. 4.1. Forecastig Methods 4.1.1. Movig Average Method The movig average method ivolves calculatig the average of observatios ad the employig that average as the predictor for the ext period. The movig average method is highly depedet o, the umber of terms selected for costructig the average C. Floros [14] ad S. Chopra & P. Meidl [15] the equatio is as follows: F t 1 (Y Y Y... Y ) t t 1 t 2 t 1 F t+1 = the forecast value for the ext period, Y t = the actual value at period t, = the umber of term i the movig average. (1) 4.1.2. Sigle Expoetial Smoothig Method The expoetial smoothig method is a techique that uses weighted movig average of past data as the basis for a forecast. This method keeps a ruig average of demad ad adjusts it for each period i proportio to the differece betwee the latest actual demad figure ad the latest value of the average. The equatio for the simple expoetial smoothig method is: F F t t 1 ( At 1 F ) t 1 (2) F t+1 = the ew smoothig value or the forecast value for the ext period, α= the smoothig costat (0 < α <1), Y t = the ew observatio or actual value of the series i period t, F t = the old smoothed value or forecast for period t. 4.1.3. Double Expoetial Smoothig Method (Holt s method) The double expoetial smoothig model is recommeded to forecast time series data that have a liear tred P. Y Lim & C. V. Nayar [16]. A approach used to hadle a liear tred is called the Holt s two parameter method. Theoretically, this methodology is similar to Brow s expoetial smoothig, except that the techique smoothes the tred ad the slope i the time series by usig differet smoothig costats for each. Low values of α ad β should be used whe there are frequet radom fluctuatios i the data, ad high values whe there is a patter such as a liear tred i the data. Three equatios are employed: A t = αy t + (1 α) (A t 1 + T t 1 ) 146 Copyright c 2015 SERSC
T t = β(a t A t 1 ) + (1 β )T t 1 Y t+ x = A t + xt t (3) A t = smoothed value α = smoothig costat (0 <α < 1) β = smoothig costat for tred estimate (0 <β < 1) T t = tred estimate x = periods to be forecast ito future Y t+x = forecast for x periods ito the future 4.1.4. Witer s Method Witer s method provides a useful way to explai the seasoality whe time series data have a seasoal patter. The formula of this method icludes four equatios: At = αy t / I t -L + (1-α)(A t-1 + T t-1 ) Tt = β(a t - A t-1 ) + (1-β) T t-1 It = γy t / A t + (1-γ) I t-l Ad to forecast p period ito the future, we have: F t+p = (A t + pt t )I t-l+p (4) A t = smoothed value α = smoothig costat (0<α <1) Y t = actual value or ew observatio i period t β = smoothig costat for tred estimate (0<β <1) T t = tred estimate γ = smoothig costat for seasoality (0<γ <1) I t = seasoal estimate measured as a idex L = legth of seasoality p = periods to be forecast ito future F t+p = the forecast for p periods ito the future. 4.2. Measurig Forecastig Error There is o cosesus amog researcher as to which measure is best for determiig the most appropriate forecastig method. Accuracy is the criterio that determies the best forecastig method; thus, accuracy is the most importat cocer i evaluatig the quality of a forecast. The goal of the forecasts is to miimize error. Forecast error is the differece betwee a actual value ad its forecast value J. S. Armstrog & F. Collopy [17]. Some of the commo idicators used evaluate accuracy are mea forecast error, mea absolute deviatio, mea squared error, ad root mea squared error. Regardless of the measure beig used, the lowest value geerated idicates the most accurate forecastig model. 4.2.1. Mea Absolute Deviatio A commo method for measurig overall forecast error is the mea absolute deviatio (MAD). The value is computed by dividig the sum of the absolute values of the idividual forecast error by the sample size (the umber of forecast periods). The equatio is: M A D t 1 (Y F ) t t Copyright c 2015 SERSC 147
Y t = the actual value i time period t, F t = the forecast value i time period t, = the umber of periods. 4.2.2. Mea Square Deviatio The mea square error (MSD) is a geerally accepted techique for evaluatig expoetial smoothig ad other methods. The equatio is: M S D 1 t 1 (Y F ) t Y t = the actual value i time period t, F t = the forecast value i time period t, = the umber of periods. 4.2.3. Mea Absolute Percetage Error t 2 Forecasters also deped o mea absolute percetage error (MAPE) as a measure of accuracy of a forecast. This measure is similar to MAD except that it is expressed i percetage terms. The advatage of the measure is that it takes ito accout the relative size of the error term to actual uits of observatio. Symbolically, the equatio is: M A P E ( e / Y ).1 0 0 t t t 1 Y t = the actual value i time period t, e t = the predictio error i time period t, = the umber of periods. 5. Result ad Discussio The purpose of this work was to idetify a appropriate forecastig method for boot product i Tez Iteratioal, Agra Shoe Idustry. Forecastig of boot productio is a very importat because sales vary i special festival ad weather chage. I this Research work, four forecastig methods amely-movig average, sigle expoetial, double expoetial ad witers method are used. The most appropriate forecastig method was determied o the basis of accuracy. Mea absolute deviatio (MAD), Mea absolute percetage error (MAPE) ad Mea square deviatio (MSD) were adopted to estimate the accuracy of forecastig methods. The lesser the forecastig error, the more accurate forecastig method. Forecastig of each method i tabular ad plotted form is give below. I simple movig average method, five trials were take ad differet values of are put i the equatio (1). Least value of mea square deviatio (MSD) is obtaied at =4 as give i Table 1. The optimal value ca be determied by iteractive model to obtai the smallest error. I the above model, five experimets were coducted i which the optimal value is obtaied at 4MA. Table 1. Differet Values of MSD for Movig Average Method Movig average 2MA 3MA 4MA 5MA 6MA MSD 4071609 3586977 3483576 3814448 4092824 148 Copyright c 2015 SERSC
Figure 2. Movig Average Plot for Sale I sigle expoetial smoothig method, four trials were take ad differet values of α are put i the equatio (2). Least value of mea square deviatio (MSD) is obtaied at α=0.1 as give i Table 2. Table 2. Shows Differet Values of MSD for Sigle Expoetial Smoothig Method Sigle expoetial smoothig α=0.1 α=0.2 α=0.3 α=0.4 MSD 5544774 5601308 5768594 5943857 Figure 3. Sigle Expoetial Smoothig Plot for Sale I double expoetial smoothig method, ie trials were take ad differet values of αad βare put i the equatio (3). Least value of mea square deviatio (MSD) is obtaied at α =0.1,β=0.1as give i Table 3. Table 3. Differet Values of MSD for Double Expoetial Double α=0.1 α=0.1 α=0.1 α=0.2 α=0.2 α=0.2 α=0.3 α=0.3 α=0.3 expoetial β=0.1 β=0.2 β=0.3 β=0.1 β=0.2 β=0.3 β=0.1 β=0.2 β=0.3 MSD 5320925 5546724 5750542 5730281 6050028 6385211 6055795 6488527 6969637 Copyright c 2015 SERSC 149
Figure 4. Double Expoetial Smoothig Plot for Sale I Witer method, iitial values of the smoothed series L t, tred T t, ad the seasoality idices It are put i equatio (4). There are two types of models amely Multiplicative ad Additive. The multiplicative model is used whe the magitude of the seasoal patter icreases as the series goes up ad decreases as the series goes dow ad the additive model is used whe the magitude of the seasoal patter does ot chage as the series goes up or dow. I this work, additive model has bee used because the magitude of the seasoal patter i our data does ot chage as the series goes up or dow.here, 27 differet trials have bee coductedby cosiderig three cases. I the first case α =0.1 & β ad γ ragig from 0.1 to 0.3 makes ie rus. Similarly α =0.2 ad 0.3 are take for differet value of β ad γ ragig from 0.1 to 0.3 makes 9 rus each i secod ad third case respectively, seasoal legth is take as 12. Least MSD value is calculated for each case as give i Table 4.The accuracy of witer s method strogly depeds o theoptimal values of alpha (α), beta (β), ad gamma (γ). The optimal α, β ad γ were determied bymiimizig a measure of forecast error of MSD. Table 4. Least Value of MSD for Witers Method Witer s method α=0.1 α=0.2 α=0.3 β & γ=0.1 to 0.3 β & γ=0.1 to 0.3 β & γ=0.1 to 0.3 Number of 9 9 9 experimets Best combiatio α=0.1,β=0.1,γ=0.1 α=0.2,β=0.1,γ=0.1 α=0.3,β=0.1,γ=0.1 MSD 1697198 1730476 1795574 Figure 5. Witer s Method Plot for Sale I the preset work, differet forecastig models for sale of a boot i shoe idustry have bee discussed. Method estimatio techiques of these models are show i Table 5. 150 Copyright c 2015 SERSC
Method Movig Average Method Sigle Expoetial Smoothig Method Double Expoetial Smoothig Method Witer s Method 6. Coclusio Table 5. Check List for Selectig Appropriate Method Estimatio Accuracy Ease of learig techique Excel, Spreadsheet Reasoable to good Easy Excel, Spreadsheet or statistical tool (e.g. Miitab) Excel, Spreadsheet or statistical tool (e.g. Miitab) Excel, Spreadsheet or statistical tool (e.g. Miitab) Good Reasoable Reasoable to good Moderate to difficult Moderate to difficult Moderate to difficult I this work, various types of forecastig methods are used for the determiatio of future sale of the product. The forecastig method will be selected o the basis of forecast error. Lesser the forecast error, the more accurate forecastig method.the specific purpose of this study was to idetify the best quatitative forecastig method, based o level of accuracy ad the ease of use i practice, to forecast demad of the Boot for Shoe Idustry. This study idetified that may real-life forecastig situatios were more complicated ad difficult due to such variables as weather ad special festival. Therefore, Tej Iteratioal should applied appropriate quatitative methods to obtai better forecastig accuracy ad to establish their strategic productio pla of Boot product. Such forecasts ca provide valuable iformatio for productio, facility moitorig, seasoal employmet, short-term ad log-term budgetig. Refereces [1] R. T. Peterso, Forecastig practices i the retail idustry, Joural of Busiess Forecastig, vol. 12, (1993), pp. 11-14. [2] H. K. Alfares ad M. Nazeeruddi, Electric load forecastig: literature survey ad classificatio of methods, Iteratioal Joural of Systems Sciece, vol. 33, o. 1, (2002), pp. 23-34. [3] W. Peter ad S. Aders, Evaluatio of forecastig error measuremets ad techiques for itermittet demad, Iteratioal joural of productio ecoomics, vol. 128, (2010), pp. 625-636. [4] C. Cacatto, P. Belfiore ad J. G. V. Vieira, Forecastig practices i Brazilia food idustry, Joural of logistics maagemet, vol. 1, o. 4, (2012), pp. 24-36. [5] M. Sawlai ad M. Vijayalakshmi, Forecastig sales through time series clusterig, Iteratioal joural of data miig & kowledge maagemet process, vol. 3, o. 1, (2013), pp. 39-56. [6] D. P. Patil, A. P. Shrotri ad A. R. Dadekar, Maagemet of ucertaity i supply chai, Iteratioal Joural of Emergig Techology ad Advaced Egieerig, vol. 2, o. 5, (2013), pp. 303-308. [7] A. K. Sharma, A. Gupta ad U. Sharma, Electricity forecastig of Jammu & Kashmir: A methodology compariso, Iteratioal Joural of Electrical Egieerig & Techology, vol. 4, o. 2, (2013), pp. 416-426. [8] C. D. Ezeliora, Movig average aalysis of plastic productio yield i a maufacturig idustry, Iteratioal joural of multidiscipliary scieces ad egieerig, vol. 4, o. 2, (2013), pp. 65-73. [9] K. Ryu ad A. Sachez, The evaluatio of forecastig methods at a istitutioal foodservice diig facility, The Joural of Hospitality Fiacial Maagemet, vol. 11, o. 1, (2003), pp. 27-45. [10] E. Leve' ad A. Segerstedt, Ivetory cotrol with a modified Crosto procedure ad Erlag distributio, Iteratioal Joural of Productio Ecoomics, vol. 90, (2004), pp. 361-367. [11] A. A. Sytetos ad J. E. Boyla, The accuracy of itermittet demad estimates, Iteratioal Joural of Forecastig, vol. 21, (2005), pp. 303-314. [12] A. A. Sytetos ad J. E. Boyla, O the bias of itermittet demad estimates, Iteratioal Joural of Productio Ecoomics, vol. 71, (2001), pp. 457-466. Copyright c 2015 SERSC 151
[13] J. J. Strasheim, Demad forecastig for motor vehicle spare parts, A Joural of Idustrial Egieerig, vol. 6, o. 2, (1992), pp. 18-19. [14] C. Floros, Forecastig the UK uemploymet rate: model comparisos, Iteratioal Joural of Applied Ecoometrics ad Quatitative Studies, vol. 2, o. 4, (2005), pp. 57-72. [15] S. Chopra ad P. Meidl, Supply Chai Maagemet Strategy, Plaig ad Operatio, Pearso, 4th Ed, Idia, (2010). [16] P. Y. Lim ad C. V. Nayar, Solar irradiace ad load demad forecastig based o sigle expoetial smoothig method, Iteratioal Joural of Egieerig ad Techology, vol. 4, o. 4, (2012), pp. 451-455. [17] J. S. Armstrog ad F. Collopy, Error measures for geeralizig about forecastig methods: Empirical comparisos, Iteratioal Joural of Forecastig, vol. 8, o. 1, (1992), pp. 69-80. 152 Copyright c 2015 SERSC