Performance Comparison between ARIMAX , ANN and ARIMAX-ANN Hybridization in Sales Forecasting for Furniture Industry

Manufacturing fi rms aim to increase their profi ts and reduce costs in a competitive and rapidly changing market. One of the most important ways to reach these goals is to forecast sales correctly. Furniture manufacturing, which is considered a prosperous and growing industry in Turkey, has an increasing trend related to the growth in construction and associated industries, increase in urban migration and increase in per capita income. Accuracy of sales forecasting in furniture industry is affected by external factors, such as consumer confi dence index, producer price index, month of the year and number of vacation days as well as the time factor itself. This study aims to develop an Autoregressive Integrated Moving Average with external variables (ARIMAX) to forecast the total monthly sales of furniture products of a well-known manufacturer in Turkey. As a follow up study, a performance comparison between ARIMAX, artifi cial neural networks (ANNs) and ARIMAX-ANN hybridization is performed. In conclusion, results of performance measures demonstrate that hybrid model developed for each amount of product sales give better accuracy values than single methods. Overall, it is proved that using the ARIMAX and hybridization of this method with ANN are applicable for forecasting monthly sales of furniture products.


INTRODUCTION 1. UVOD
The demand for furniture products in Turkey has increased rapidly with the recent development in construction industry, increase in urban population and personal income level (Hazır et al., 2016; Turkey Furniture Products Council Industry Report, 2013).However, it is diffi cult to forecast this increase considering the multiplicity and internal interactions of the affecting factors.Statistical methods, such as regression or ARIMAX and data mining methods as ANNs have been frequently applied in sales forecasting by numerous researchers since they have the ability to solve complicated interactions affected by internal and external environments (Kuo et al., 2002;Luxhøj et al., 1996).Bearing this issue in mind, this study attempts to develop a time series model with external variables to forecast monthly sales of furniture products using ARIMAX, ANN and ARIMAX-ANN model.
In the related literature, researchers proposed new forecasting methods, evaluated the performance of existing ones or modifi ed the existing ones depending on applications in sales forecasting (Arunraj and Ahrens, 2015).The methods include regression, timeseries related methods and advanced machine learning related methods.Although sales forecasting in the context of all sectors has been studied by many researchers, in the area of furniture sales forecasting, the publications are very few when compared to other fi elds.Hazır et al. (2016) studied sales forecasting of Turkish furniture industry by ANN and multiple linear regression (MLR), and provided a road map for vision 2023.Data between 2004 and 2013 was used for analysis.As a result of the study, 24 billion dollars and 21 billion dollars demands were predicted by MLR analysis and ANN methodology, respectively.They stated that these results may be evaluated as the possible retail values for Turkish furniture industry within the target of vision 2023.Mahbub et al. (2013) proposed an ANN model to forecast the optimum demand considering time variables of the year, festival period, promotional programmes, holidays, number of advertisements, cost of advertisements, number of workers and availability.A feed-forward backpropagation ANN with 13 hidden neurons in one hidden layer as the optimum network was preferred.The model was confi rmed with a furniture product data of a renowned furniture company.The model was compared with Brown's double smoothing.The results of the study show that ANN model performs much better than the linear Brown's double smoothing model.Oblak et al. (2012) applied two quantitative methods named Holt-Winters method of exponent smoothing of higher orders and linear regression of the 1st order for forecasting of parquet sales.Parquet sales data by month in the years 2000 to 2009 was used in the models and the best result was obtained with the use of Holt-Winters multiplicative model of exponent smoothing of higher orders.In conclusion, it was suggested that the proposed model can be applied for forecasting optimum demand level of furniture products in any furniture company.As stated by Arunraj and Ahrens (2015), there is no common forecasting model that can be applied to different kinds of problems.Accuracy of the forecasting models can be improved by hybridized models rather than by a single model.Therefore, in this study, an ARIMAX model was developed to forecast the total monthly sales of furniture products of a well-known Turkish manufacturer and a performance comparison was performed between the models of ARIMAX, ANN and ARIMAX-ANN hybridization in order to improve the accuracy of the proposed framework.
Apart from forecasting sales in the furniture industry, the forecasting methods are used in different fi elds of application.Fabianová et al. (2016) presented a sale forecasting using the software tool for risk analysis.It was carried out based on time series analysis, forecasting and statistics analysis.Forecasting was performed using two different approaches: when considering simple seasonality of sales and when considering sale seasonality along with the impact of known events on demand.Subsequent forecasting is focused on the identifi cation and analysis of risk factors using Monte-Carlo Simulation (MCS).Anggraeni et al. (2015) and Lee and Hamzah (2010) studied sales forecasting for clothing industry.While the fi rst study made a performance comparison between ARIMA and ARIMAX in Moslem kids' clothes sales forecasting, the second one dealt with forecasting sales data with Ramadhan effect by ARIMAX modelling.Results of studies by Anggraeni et al. (2015) show that ARIMAX model is better than ARIMA model in accuracy level of training, testing and next time forecasting processes.In conclusion of the study by Lee and Hamzah (2010), it is observed that ARIMAX yields better forecast at outsample data compared to the decomposition method and Seasonal Autoregressive Integrated Moving Average (SARIMA), and neural networks.Arunraj and Ahrens (2015) proposed hybridization of SARIMA-Quantile Regression (QR) model to food sales forecasting.The results show that the SARIMA-MLR and -QR models yield better forecasts at out-sample data when compared to seasonal naïve forecasting, traditional SARIMA, and multi-layered perceptron neural network models.Unlike the SARIMA-MLR model, models (Doganis et al., 2006;Kuo, 2001).Doganis et al. (2006) applied ANN-GA combination for short shelf-life food products sales forecasting.Kuo (2001) proposed a model based on fuzzy neural network with initial weights generated by GA.Similarly, Kuo et al. (2002) integrated ANNs and fuzzy neural networks the SARIMA-QR model provides better prediction.Murlidharan and Menezes (2013) proposed a different concept in sales forecasting "Frequent pattern miningbased sales forecasting".In some studies, ANN was used in combination with Genetic Algorithm (GA) in order to select the appropriate input variables to the  1.

Data set 2.1. Set podataka
This study uses monthly sales of furniture products (dining room, bedroom, teen room, sitting group and armchair) measured in numbers from one of the biggest furniture factories as a case study from January 2009 to December 2015.The selected factory is located in Black sea region of Turkey.Figure 1 illustrates the monthly sales data of furniture products for a seven-year time period in a time series plot.The mean sales per month is about 140, 114, 60, 115 and 531, respectively, for the products of dining room, bedroom, teen room, sitting group and armchair.This time series is highly periodic, but it is easy to observe the monthly patterns.Especially in summer, the sales are observed at the peak point.This type of month effects can be recognized by considering the months from January to November as dummy variables (0 and 1).The refer-ence month is December, so coeffi cients of other months may be interpreted relative to this month.
In Figure 2, the box plot displays the median monthly sales from January to December.From this fi gure, it is evident that the highest sales usually occur in August.May, June, July, September and October are the next highest sales months.The box-plot also shows information about extreme values and outliers, which occur mainly due to holidays and other variables.However, the median of monthly sales for May & June and September & October are similar; sales dispersion is higher in June than in July.
In addition to the month of the year affect, three variables are used as input parameters to forecast sales value of a furniture company.The fi rst one is consumer confi dence index (CCI).CCI is an aggregate of four sub-indices.Two of them are based on expectations regarding household fi nances, while the other two are based on expectations about economy-wide developments (Jansen and Nahuis, 2003).The second input variable is producer price index (PPI).PPI is an aggregate of over 1500 components.Each component is a monthly index of the national average price for some producer goods.The price pertains to the fi rst transaction after production of the good.This is a transaction between fi rms rather than between businesses and consumers (Peltzman, 2000).Other input parameters are number of vacation days in the related month of the year in Turkey.Distribution of number of vacation days for the data set is given Figure 3.
The output parameters are monthly sales of dining room, bedroom, teen room, sitting group and armchair.Inputs, outputs and applied methods are shown in Figure 4.
Data about input and output variables were collected for 84 months from January 2009 to December 2015 from the furniture company.Descriptive statistics for each output variable are presented in Table 2.The values of CCI and PPI are ratios that have a mean of 72.85 and 209.2, respectively.Mean number of vacation days in the studied data set is 9.45 per month.The reason for considering this input variable in our proposed comparative framework model stems from the fact that decision makers in furniture industry are of the opinion that sales increase in vacation days.

Metode istraživanja
In this study, three forecasting methods including single (ARIMAX, ANN) and hybrid (ARIMAX-ANN) methods were investigated in order to fi nd the best method that accurately fi ts the data.Detailed descriptions of each method are presented in the following subsections.

ARIMAX 2.2.1. ARIMAX
Recently, ARIMA has been studied by many researchers who used time series.However, when using ARIMA model, only one variable can be used, so it is not adequate to express real problems.Complex problems always need more the one variable in order to explain problems effectively.Therefore, it is necessary to build a multivariate ARIMAX model (Fan et al. 2009;Jalalkamali et al. 2015).
The ARIMAX model (Bierens, 1987) is a generalization of the ARIMA model, which is capable of incorporating an external input variable (X).The ARI-MAX model assumes the form (1) where L is usual lag operator.γ s L s = y (t-s) , ∆y t =y t -y t-1 , μ ∈, α s ∈ R, β s ∈ R k and γ s ∈ R are the unknown param-eters and e t ʹs are the errors (Arya et al., 2015).ARI-MAX model consists of four parts.These parts are Auto Regressive (AR), Integrated (I), Moving Average (MA), and Exogenous Variable (X) (Sutthichaimethee and Ariyasajjakorn, 2017).

ANN
ANNs are machine learning algorithms that aim to solve the computational processes in specifi c areas by using a large number of interconnected processing elements (Gul and Guneri, 2015; 2016a; 2016b; Yucesan et al. 2017; Onat and Gul, 2018).They are applied to the problems on prediction, clustering, classifi cation, and detection of abnormalities (Pusat et al. 2016).The computational elements used in different ANN models are known as artifi cial neurons (Guneri and Gumus, 2008;2009).The model of an artifi cial neuron is given in Figure 5.
Where x 1 ,x 2 , . . .,x p are the input signals; w k1 ,w k2 , . . .,w kp are the weights of neuron k, and, u k is the linear combiner output, while θ k denotes the threshold.Furthermore, Φ () is the activation function; and y k is the output of the neuron.The fi rst layer known as the ''input'' layer and the last one, which is called the ''output'' layer, are used to get information from inside and outside the network, respectively.The middle layers considered as ''hidden'' layers are vital for the network to convert certain input patterns into appropriate output patterns (Akkoyunlu et al., 2015; Somoza and Somoza, 1993).The fl ow of information is passed through the network by linear connections and linear or nonlinear transformations.The error between the actual and predicted values is calculated.Then, a minimization procedure is used to adjust the weights between two connection layers, i.e. for back propagation model starting backwards from the output layer to input layer.There are many minimization procedures based on different optimization algorithms, such as Quasi-Newton, and Levenberg-Marquardt, gradient descent and conjugate gradient methods.In ANN models, there is a practical problem in network architecture (number of hidden layers and units in each layer) and network properties (error and activation functions).The design of hidden layer is dependent on the selected learning algorithm (Kröse et al., 1993).The more layers and neurons, the more complex dependencies the network can model.One of the other important properties of an ANN model is the activation function for the hidden layer.Linear, logistic and hyperbolic tangent are the most common functions followed in the literature.In ANNs, some controllable factors are available in order to aid the learning of selected algorithm such as Learning Rate and Momentum.They are control parameters used by several learning algorithms, which affect the changing of weights.The higher learning rates cause higher weight changes during each iteration.The greater the momentum, the more the current weight change is affected by the weight change that took place during the previous iteration.

ARIMAX-ANN hybrid method 2.2.3. Hibridna metoda ARIMAX-ANN
In time series forecasting, hybrid models are developed apart from single models in order to reduce risk of failure and also obtain more accurate results (Khashei and Bijari, 2010).ARIMAX models are insuffi cient to solve complex nonlinear problems.On the other hand, using ANNs to solve problems yields dicey results.For that reason, if a distinctive framework of problems is not know, hybrid methodology that contains advance sides of ARIMAX and ANN models can be a good solution.Therefore, we combined ARIMAX and ANN model in sales forecasting for furniture industry.In the fi rst step, ARIMAX model is used to calculate residual of the model.In the second step, ANN model is applied with independent variables including residuals of ARIMAX model as input variables.
Since ARIMAX model cannot capture the nonlinear structure of the data, the residuals of the linear model will contain information about nonlinearity.The results from the neural network can be used as predictions of the error terms for the ARIMA model (Zhang 2003;Xu et al., 2016).
Step 1: ARIMAX model is applied to analyze the linear part of the problem.First, model identifi cation is performed by using graphs, statistics, autocorrelation function (ACF), partial autocorrelation function (PACF) and transformations.The dependent variable is determined as stationary.Least squares are used to determine the valid model and variables.Finally, forecast verifi cation and reasonableness is performed in order to track performance and determine the validity of forecast, then fi t (y t , y t-1 ... y t-n ) and calculate residuals (e t , e t-1 ... e t-n ) (Areekul et al., 2010).
Step 2: ANN model is applied considering the variables of month of the year, number of vacation days, PPI, CCI and residuals which are calculated in ARI-MAX model as input variables of the model.To measure performance of ANN model, mean squared error (MSE), network error (average train error and average test error), absolute relative error (ARE), mean absolute percentage error (MAPE) and R-squared (R 2 ) can be used.R 2 and MAPE are defi ned as in Eqs. ( 2)-( 3).(2) where SS total refers to the total sum of squares (proportional to the variance of the data), and SS residuals indicate the sum of squares of residuals, also called the residual sum of squares.
(3) Where f i is vector of n forecasting, and y i is the vector of actual values.

Proposed comparative framework 2.2.4. Predloženi usporedbeni okvir
The proposed comparative framework used in this study consists of four stages as given in Figure 6.It includes a time series (ARIMAX), an ANN and a hybrid (ARIMAX-ANN) model.The "best" model for each stage was identifi ed, and then these models were compared at the last stage of the framework.Measures of solution adequacy are evaluated under MAPE values.Moreover, plots regarding actual vs. predicted values for each furniture product are demonstrated.In this study, a comparative framework to forecast total monthly sales of furniture products was developed.To achieve this aim, a case study was carried out in a furniture product manufacturing company located in Black Sea region of Turkey.In the fi rst stage, an ARIMAX model is proposed.We collected the monthly sales data detailed above.It includes 84 data points for each output variable corresponding to sevenyear data.For all of the ARIMAX modeling processes, EViews 10.0 was used.The ARIMAX model used for forecasting was split into three steps: (1) Transformations of the dependent variables were made and the level of differencing was determined.Transformations of monthly sales of dining room, bedroom, teen room and armchair were made by differencing.The transformation of the sales of sitting group was arranged based on both logarithmic and differencing operations.
(2) The external independent variables were determined as mentioned in previous sub-sections (Subsection 2.2.1).The ARIMAX model developed to forecast teen room sales in our case study has the following external variables (Eq.4).(4) where y t refers to teen rooms sales, M j,t month of the year, CCI t , CCI, PPI t , PPI and V t number of vacation days.ρ p and θ q are the model parameters for the autoregressive and moving average terms, respectively, and ε t are the residual term representing random disturbances that cannot be predicted.
(3) The order of the autoregressive moving average (ARMA) terms was selected.EViews uses model selection to determine the appropriate ARMA order.The orders of AR and MA were limited to four.Model selection is a way of determining which type of model fi ts best a set of data and is often used to choose the best model for forecasting that data (EViews 10 tutorial, 2017).We used Akaike Information Criterion (AIC) in ARIMAX modeling process.As an example, the best ARMA terms for teen room sales were determined according to the AIC criteria.The results are given in Figure 7.The best ARMA term for this case is obtained as (3,2).The same steps were followed for each furniture product to make transformations and select ARMA terms based on AIC values.Best ARMA terms and related ARIMAX models with some model performance measures such as R 2 , adjusted R 2 and AIC are given in Table 3.
The plots of actual versus forecasted values for ARIMAX models are provided by means of the EViews as shown in Figure 8.The results show that using the ARIMAX model is an applicable choice for forecasting monthly sales of the products in the observed furniture factory.The MAPE value of the armchair sales is calculated as 5.37 %, which means that a reasonable result is obtained (Çelik et al. 2016).On the other hand, MAPE values are also calculated for the sales of dining room, bedroom, teen room and sitting group as 10.36 %, 8.70 %, 10.67 % and 8.04 %, respectively.
The ARIMAX-ANN hybrid model integrated the ARIMAX model with the neural network model and tested with the raw data.We built the hybrid ARIMAX-ANN model with the following input layers: (1) the dependent variables used in ARIMAX modeling, (2) the residuals of the ARIMAX model.This process is followed by data partition and normalization by minmax normalization method.A training and testing model based on ANN is then employed after determining the design architecture and network properties.A logistic input and output activation function are used.Three layers which include input, hidden and output   The performance indicator of R 2 is obtained as 98.97 %, which shows a good accurateness of the method.A commercial software, Alyuda NeuroIntelligence software, was used in developing the hybrid models.After the data entry to the software, it random-

Rasprava rezultata
The best model for this study was the ARIMAX-ANN hybrid model in each furniture product sales forecasting (see Table 4).Çelik et al. (2016) reported that, in practical applications, MAPE < 10 % means high accuracy.Accordingly, it must be noted that MAPE values of ARIMAX-ANN hybridization and single ANN models for monthly furniture product sales fi gures are within the range reported by Çelik et al. (2016).Forecasting accuracy is dependent on the choice of forecasting model.ANN and ARIMAX models have different advantages and disadvantages.ARIMAX model is very fl exible and it can also represent AR, MA, differencing and inclusion of external factors.ARIMAX model is very good when dealing with the linear part but it is weak when working with nonlinear data.ANN model is good at dealing with nonlinear parts.When using hybrid models, the residuals of the ARIMAX model as input variables can solve this problem.Also, the results show that a combination of ARIMAX and ANN models gives better forecasting accuracy.

ZAKLJUČAK
In this study, a comparative forecasting framework was developed and applied to solve the problem of sales forecasting for furniture products.The applied models include ARIMAX, ANN and ARIMAX-ANN hybridization.The models were used to forecast monthly sales fi gures of a corporate furniture manufacturing company located in Black Sea region of Turkey.In conclusion of this comparative study, the results of performance measures demonstrate that ARIMAX-ANN hybrid model, developed for each amount of product sales, gives better accuracy values than single models.Overall, it is proved that using the ARIMAX and hybridization of this method with ANN are applicable for forecasting monthly sales of furniture products.
The aim of sales forecasting is to determine the demand level of products in a certain time horizon and it is an important part of production planning.In aggregate production planning, sales forecasting is the starting point.This step is one of the most challenging problems for stakeholders in furniture industry.Decision makers should evaluate the costs and benefi ts of each model before choosing an appropriate forecasting method.In our opinion, the ARIMAX-ANN hybrid model was suitable for the observed company since this method gives effective solutions.This comparative framework can be adapted to any company competing in forestry industry or other industries.In addition to its methodological contributions, this study has some benefi ts for the forestry industry.First, it further encourages stakeholders to forecast sales fi gures in other forestry products such as timber, parquet, oriented standard board, plywood, etc.Secondly, advanced manufacturing methods have been increasingly employed in forestry sector due to today's technological development.Due to the possible demand for wood products, comprehensive and effective forecasting approaches are required for monitoring the trend.Therefore, a comparative forecasting outline has been developed, and it has been used to forecast the sales fi gures for furniture industry in Turkey.
By the current research, the model proposed by Yucesan et al. (2017) was improved by developing a more effective hybrid model.In future, the authors are planning to extend the hybrid model considering external factors to provide a more accurate result, although it has not been considered in this particular case study.

Figure 4
Figure 4 Presentation of inputs and output variables and applied methods Slika 4. Prikaz ulaznih i izlaznih varijabli i primijenjenih metoda

Figure 7
Figure 7 AIC values of ARMA terms proposed for teen room sales Slika 7. AIC vrijednosti ARMA uvjeta predloženih za prodaju soba za mlade

Table 1
Luxhøj et al. (1996)casting application studied in literature Tablica 1. Pregled primjene metoda za predviđanje prodaje iz literature According to the results of the study, the proposed model performs more accurately than the conventional statistical method and single ANN model.Alon et al. (2001) made a comparison of ANNs and traditional methods in forecasting aggregate retail sales.Yip et al. (1997) studied an application of ANNs in sales forecasting.Unlike Yip et al. (1997),Luxhøj et al. (1996)focused on a hybrid econometric-neural network model for sales forecasting.Using of this hybrid method yielded a modest 2.3 % reduction in the mean absolute percentage error (MAPE) when compared with the current qualitative approach used by the company.The above-mentioned literature review is summarized in Table autoregresivni pomični prosjek s vanjskim varijablama DRVNA INDUSTRIJA 69 (4) 357-370 (2018)with fuzzy weight elimination for the fi rst time in literature.

Table 3
Performance measurement results of ARIMAX models Tablica 3. Rezultati performansi ARIMAX modela Yucesan et al. (2017)atios for training, validation and testing.In some studies, this ratio may be 60:20:20(Fu et al., 2017).Experiments were run with various algorithms such as Quick Propagation, Quasi-Newton, Levenberg-Marquardt, Online Back Propagation and Conjugate Gradient Descent.The best network was obtained by Quick Propagation algorithm with some performance measures indicating as follows: while R 2 value is obtained as 0.9897 for all data set, the average test error is obtained as 3.621643.The MAPE value for overall and test data of this model was obtained as 3.22 % and 5.71 %, respectively.The same steps were followed for each furniture product to develop hybrid models.The obtained MAPE values for overall and test data of the remaining four models (dining room, bedroom, sitting group and armchair) were obtained as, 1.99 % & 2.75 %, 4.42 % & 4.47 %, 3.66 % & 4.8 % and 1.05 % & 2.71 %, respectively.The actual versus fi tted values for these fi ve models are given in Figure10.Results of ANN models, which were compared to the ARIMAX and ARIMAX-ANN hybrid models through this study, were derived fromYucesan et al. (2017).That study was carried out by the authors of this study.They applied ANN modeling based on Bayesian rules training.They obtained MAPE values of each furniture product sales forecasting as 3.588%, 5.951%, 5.954%, 5.223% and 3.057%, respectively, for dining room, bedroom, teen room, sitting group and armchair.

Table 4
MAPE values for fi ve models with respect to three methodologies Tablica 4. MAPE vrijednosti za pet modela s obzirom na tri metodologije