Introduction

Production Forecasting in the Automotive Industry: Performance Analysis of Time Series Models Using the Box-Jenkins Method

Chaharsoughi

Seyed Kamal

Manzari Vahed

Niloofar

Ashnavar

Hassan

Department of Industrial Engineering, Faculty of Industrial and Systems Engineering, Tarbiat Modares University, Tehran, Iran. Head of Production Control Department, SAIPA Automotive Group, Tehran, Iran.

In the realm of industrial engineering and management, production forecasting is a pivotal topic that profoundly influences productivity and the optimization of production processes. By developing a precise prediction model, companies can significantly enhance production planning, control production, minimize stoppages, optimize inventories, and boost machinery productivity. This study delves into the production processes of the SAIPA Automotive Group and proposes an accurate prediction model. Using statistical time series forecast models, including the Box-Jenkins method and the Rolling forecast approach, the study reveals that these models, particularly the Auto Regressive Integrated Moving Average eXogenous (ARIMAX) model, excel at daily production prediction. Conversely, the Auto Regressive Integrated Moving Average (ARIMA) and Autoregressive (AR) models demonstrate superior efficiency in trend-based production prediction. Additionally, the Seasonal Auto Regressive Integrated Moving Average (SARIMA) and Seasonal Auto Regressive Integrated Moving Average eXogenous (SARIMAX) models performed worse. The application of time series tools and the rolling forecast approach has also led to a notable reduction in model errors.

Production forecastingTime series modelsBox-JenkinsRolling forecasting

1Introduction

In today's highly competitive global markets, timely and cost-effective production is increasingly important. It requires the collaborative efforts of many stakeholders to develop effective strategies for supply chain management, production planning and scheduling, and the control and allocation of equipment and labor [1]. Production planning is a managerial process that involves designing and planning an organization's production activities to optimize the use of available resources and ensure the efficient production of high-quality products and services [2].

In industrial engineering and management, production forecasting is a critical topic in production planning, significantly impacting productivity and the optimization of production processes. By developing an accurate forecasting model, companies can enhance production planning and control, reduce production time, minimize stoppages, optimize inventories, and increase machinery productivity [3], [4].

All forecasting tasks must address issues such as what to forecast, the forecasting method, the level of aggregation, the data used, and the forecast accuracy. Forecasting can be at various levels of aggregation, ranging from fully aggregated to semi-aggregated or fully disaggregated forecasts for short- and long-term periods. The data used in forecasting can be based on historical production, sales, arrivals, or reservations. Additionally, forecasts should be adjusted for specific events, such as holidays and special occasions [5]. In the SAIPA Automotive Group, six companies, Saipa, Pars Khodro, Saipa Citroën, Bonro, Zamyad, and Saipa Diesel, engage in the daily production of various car models. As with any manufacturing unit, the SAIPA Automotive Group has established a Master Production Schedule (MPS) from the outset of operations, taking into account available capacity for machinery and equipment, labor hours, raw materials, market demand, and other factors. This MPS is then used to formulate monthly and weekly plans, with the weekly production plan detailing daily production targets.

However, in practice, production often deviates from the planned schedule, underscoring the necessity for accurate production forecasting to aid decision-makers and managers. This research aims to examine production in the SAIPA Automotive Group and to present an accurate production forecasting model. To this end, statistical time-series forecasting models, including the Box-Jenkins method and the Rolling forecasting approach, have been used to forecast production for the six car manufacturing companies within the SAIPA Automotive Group.

This paper is organized as follows: Section 2 reviews the related literature. Section 3 describes the research methodology, including the methods used for data collection and preprocessing, implementation of forecasting models, model evaluation, and the results obtained. Section 4 presents the discussion and conclusions of the research.

2Literature Review

Time series forecasting models use historical data to predict future values. These models include univariate linear or nonlinear time series models and multivariate linear and nonlinear time series models. Table 1 briefly reviews the literature that has employed time-series forecasting models. Based on the studied articles, among the Box-Jenkins models, the Auto Regressive Integrated Moving Average (ARIMA) model has been most frequently used for forecasting, which is certainly related to the dataset's characteristics. Additionally, in recent years, hybrid models that combine several time series models have become more prevalent.

Table 1

Examples of research on solving forecasting problems using time series models.

This table is shown in the PDF version of the article.

The time series models used in this research are briefly introduced below.

2.1Autoregressive

The Autoregressive (AR) model is a linear model that describes the relationship between a time series's previous values and its current value. In this model, the current value of the time series is modeled as a linear combination of its previous (p) values and a random variable with mean zero (usually noise or error). The formula for the AR(p) model is as follows:

Xt = ∑p i=1 φiXt−i + εt + C (1)

Here: Xt represents the current value of the time series. c is a constant. φi are the model parameters that indicate the dependency of previous values on the current value. is a random variable with a mean of zero, representing noise.

2.2Moving Average

The Moving Average (MA) is one of the primary models in time series analysis used for modeling and forecasting time series data. This model averages the values of sequential noise or errors in the time series. In an MA (q) model, the current value of the time series is modeled as a linear combination of the previous (q) values of noise or errors and a random variable with a mean of zero. The formula for the MA (q) model is as follows:

Xt = μ + εt + θ1εt−1 + ⋯ + θqεt−q = μ + ∑q i=1 θiεt−i + εt (2)

Here: Xt represents the current value of the time series. μ is the mean of the time series. εt is a random variable with a mean of zero, representing the noise or error of the model. θi are parameters indicate the dependency of previous noise or error values on the current value.

2.3Auto Regressive Integrated Moving Average

The Auto Regressive Integrated Moving Average (ARIMA) model is a linear model composed of two main components: the AR and MA models. In this model, the relationship between the previous values of the time series and the MA of the sequential noise or errors is modeled. The ARIMA model is an extension of the ARMA model, incorporating three main elements: AR, MA, and I (integration). This model is used for modeling and forecasting non-stationary time series and provides more accurate predictions by fitting parameters to real data. The formula for this model is equivalent to the following expression:

Xt = εt + ∑p i=1 φiXt−i + ∑q i=1 θiεt−i (3)

2.4Seasonal Auto Regressive Integrated Moving Average

The Seasonal Auto Regressive Integrated Moving Average (SARIMA) model is an extension of the ARIMA model used for modeling time series with seasonal patterns. This model uses AR, MA, I (integration), and seasonal AR and MA components to model and forecast time series with seasonal patterns.

2.5Auto Regressive Integrated Moving Average eXogenous

The Auto Regressive Integrated Moving Average eXogenous (ARIMAX) model is an extension of the ARIMA model that incorporates external or exogenous variables for time series modeling. This model allows for the modeling of the effect of external variables on the time series.

2.6Seasonal Auto Regressive Integrated Moving Average eXogenous

The Seasonal Auto Regressive Integrated Moving Average eXogenous (SARIMAX) model is an extension of the SARIMA model that incorporates exogenous variables to model seasonal patterns in time series. This model uses AR, MA, and I components, seasonal AR and MA components, and external variables to model time series with seasonal patterns and external-variable influences.

2.7Model Performance Evaluation Criteria

To evaluate model performance and select the best model, evaluation criteria from the time series literature, including Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Log-Likelihood (LLF), have been used. AIC and BIC are criteria for evaluating the quality of a statistical model. These criteria compare the model's simplicity with the accuracy of its predictions. Lower AIC and BIC values indicate a better, simpler model. LLF is the logarithm of the likelihood function, which indicates how well the model fits the data. Higher LLF values indicate a better fit of the model to the data. In addition to the AIC, BIC, and LLF performance evaluation criteria used in statistical models, this study uses Mean Absolute Error (MAE) which is the average absolute error between predicted and actual values; Mean Squared Error (MSE) which is the average squared error between predicted and actual values; Root Mean Squared Error (RMSE) which is the square root of the average squared error between predicted and actual values; and R-squared which shows the ratio of the MSE to the mean squared actual values, to evaluate the prediction models.

3Research Methodology

The Box-Jenkins method is a well-known approach to time series analysis and forecasting, used to model and predict time series data. In this study, the Box-Jenkins method is employed to implement time series models. This method first fits an ARIMA model to the time series under investigation and then provides better predictions by estimating the model parameters from the actual data. The research method is shown in Fig. 1. The main steps of the Box-Jenkins method are as follows:

Data understanding

In this step, the time series under investigation is analyzed to identify various structures such as stationarity, trend changes, seasonal patterns, and random values. In this study, after data cleaning, additive models and exponential smoothing are used to plot decomposed time series charts to identify trends and seasonal patterns. Additionally, to examine the stationarity of the time series and determine the appropriate time lag, the Dickey-Fuller statistical test, autocorrelation, and correlation coefficient plots are used.

Model implementation and parameter estimation

After understanding the dataset, ARIMA model parameters p, d, and q are calculated using methods such as autocorrelation functions, statistical tests, or the auto-ARIMA model. In this study, the initial parameters of the time series models are determined by implementing and running the auto-ARIMA model. Auto-ARIMA is a model that uses grey search and local optimization methods to find the best combination of hyperparameters for univariate time series models. This method searches for the best time series model that fits the data within the specified range of hyperparameter values, and reports the best model. Given the use of local optimization and top-down search concepts, the auto-ARIMA model may also identify simpler models with fewer coefficients than the reported model for the time series data. Therefore, the upper bound of the ARIMA model coefficients is determined using this model, and then models with smaller coefficients are implemented and executed.

Model validation and evaluation

Time series forecasting models are created using the estimated parameters. In this study, all time series models, including AR, MA, ARMA, ARIMA, SARIMA, ARIMAX, and SARIMAX, are implemented and evaluated over the hyperparameter range. To validate and evaluate the model, residual correlation tests and normality checks of the error distribution are performed using correlation and Q-Q plots of the model errors.

Selecting the best model

At this stage, after fine-tuning the hyperparameters, the best models are identified based on evaluation criteria, including AIC, BIC, LLF, MSE, RMSE, and MAE.

Forecasting and conclusion

Using the final time-series model, future time-series predictions are provided, and the model's validity is assessed. In this study, to examine the best model's results and improve its accuracy, a rolling (moving) forecast approach is used.

3.1Data Understanding

The data used in this study includes the daily production statistics of companies in the SAIPA automotive group, encompassing daily production characteristics, annual production plans, and weekly production plans for Saipa, Pars Khodro, Saipa Citroën, Bonro, Zamyad, and Saipa Diesel from the beginning of 2023 to the end of December 2024. Fig. 2 shows the production trend over time for the companies in the SAIPA automotive group.

Figure 1

Research method structure.

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By examining the covariance and correlation values shown in Fig. 3, contrary to expectations, there is no strong positive correlation between the production plan and the production of SAIPA group companies, especially at Zamyad and Saipa Diesel. It indicates that these companies have fallen short of their production targets. Additionally, negative correlations between the production or production plans of one company and another suggest a high likelihood of unfairness in the supply chain in some cases.

Figure 2

Production trend over time for SAIPA automotive group companies.

This figure is shown in the PDF version of the article.

Figure 3

Covariance values between production plan and production for SAIPA group companies.

This figure is shown in the PDF version of the article.

3.1.1Data cleaning

Data cleaning is the process of identifying and handling errors, duplicates, and irrelevant data from a raw dataset to prepare it for analysis. Generally, data cleaning involves three steps: 1) removing duplicate data, 2) identifying and handling missing data, and 3) identifying and handling outliers. The operations performed to clean the dataset used in this study are briefly explained below.

I. Identifying duplicate data: the dataset contains no duplicates.

II. Identifying and handling missing data: the missing data identified in the dataset pertains to holidays during the specified time period when the factories were not in production. Given the time series nature of the dataset, two scenarios are considered: 1) Missing values are replaced with zero due to factory closures. This scenario preserves the information and knowledge within the data by recording the actual event in the dataset. Still, it may complicate data patterns, potentially affecting the accuracy of predictive models, and 2) Missing values are replaced with the production value of the nearest date after the holiday, based on the assumption that no production occurs on holidays. This argument reduces data complexity and may increase the accuracy of predictive models.

In this study, a combination of these two scenarios is used to handle missing values. During official national holidays, missing values are set to 0. However, on days when one or more factories were closed for any reason, these values are replaced with the next period's value in the time series.

III. Identifying and handling outliers: given that data collection was conducted with great care and reviewed multiple times, no erroneous records are present in this dataset, and all records accurately reflect real events. Therefore, outliers identified by statistical tests contain useful information about production trends in the companies, and no action has been taken regarding these data in this study.

3.1.2Identifying seasonal patterns in time series

In the literature, the behavioral patterns or change models of a time series are divided into four components: trend, cycle, season, and irregular variations or residuals. By plotting the time series against time, these components can be distinguished, leading to a better understanding of the time series data. Identifying seasonal patterns in time series data can be done using several different methods. In this study, additive models, which consider the time series as the sum of three components: trend, seasonal patterns, and random values, and the Local Regression (LOESS) method, which decomposes the time series into seasonal and trend components using local smoothing, are used to discover and identify seasonal patterns. Fig. 4 shows the trends and patterns discovered in the daily production dataset of the Saipa company using additive models and exponential smoothing.

According to the charts plotted by these models, the dataset does not exhibit a clear upward or downward trend and shows a nearly sinusoidal behavior over time. Therefore, regarding seasonality, as indicated, the data has been identified to have seasonal patterns. This pattern is particularly associated with additive charts for weekend holidays. Irregular variations are changes caused by random and unpredictable factors. In research, these values are typically removed from the time series and treated as outliers. Still, in this study's dataset, for the reasons previously mentioned, no action has been taken regarding these values.

Figure 4

Display of production trend, seasonal patterns, and residuals using the exponential smoothing model (right) and additive model (left) for the Saipa company.

This figure is shown in the PDF version of the article.

3.1.3Stationarity

Stationarity of a time series is a necessary condition for achieving reliable forecasts. There are several statistical tests for examining the stationarity of time series data; the most popular is the Dickey-Fuller test. The Dickey-Fuller test formula generally examines the presence of a unit root in a time series. This test can be performed in various forms, but one of the most common forms, the basic Dickey-Fuller test model, is as follows:

Yt = α + βt + ρYt−1 + ϵt (4)

Yt is the value of the time series at time t, α is the intercept, β is the trend coefficient, ρ is the unit root parameter, Yt−1 is the previous value of the time series, and εt is the random error.

Test hypotheses: null hypothesis (ρ = 1) H0 the time series has a unit root and is non-stationary. Alternative hypothesis∶ (ρ < 1) H1 the time series is stationary and does not have a unit root. The Dickey-Fuller test statistic is calculated as follows:

FDA = (1 ρ− ^)ρ ES ^ , (5)

where: ρ^ is the estimated unit root parameter (using regression), and (ρ^)ES is the standard error of the estimate ρ^. In the augmented version, to control for autocorrelation, several lags are added to the model:

Yt = α + ∑p i=1 ϕiYi−1 + ρYt−1 + ϵt (6)

where p is the number of lags and ϕi are the coefficients corresponding to the lags?

By calculating the test statistic and comparing it with the critical values, the null hypothesis can be rejected or accepted. If the test statistic is less than the critical value (more negative), the null hypothesis is rejected, and we conclude that the time series is stationary. Table 2 shows the results of the Dickey-Fuller test. The results of the Dickey-Fuller test indicate that all production data columns (except Saipa Diesel, which is marginally significant at the 1% level) are stationary. It means that time-series models like Box-Jenkins can be used for forecasting.

After examining the stationarity of the time series, the lag can be determined. The lag in a time series is the time interval between observations over which meaningful relationships for predicting future values exist. There are various methods for identifying lag in a time series. In this study, the Dickey-Fuller test and plotting autocorrelation and partial autocorrelation charts have been used. Fig. 5 shows the autocorrelation and partial autocorrelation values for the Saipa company.

Table 2

Dickey-Fuller test results for examining dataset stationarity.

This table is shown in the PDF version of the article.

Figure 5

Autocorrelation charts (with 10 significant time lags) and partial autocorrelation charts (with 15 significant time lags) for the Saipa Automotive Company.

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3.2Model Implementation and Parameter Estimation

For the implementation, training, parameter estimation, and performance evaluation of the predictive models, the data is split into two sets: training (75%) and test (25%). Then, by implementing and running the auto-ARIMA model, the initial parameter values for the time series models are determined. Using this model, the upper bound of the ARIMA model coefficients is determined, so that only the performance of models with smaller coefficients is examined.

The approach to model validation and evaluation, along with a detailed explanation of the process for selecting the best predictive model, is provided. Tables 3-5 sequentially present the training and implementation results of the model, using the Box-Jenkins method, for predicting the daily production of six automotive group companies: single-target models without exogenous variables and single-target models with exogenous variables (weekly plan).

Table 3

Selected single-target time series models without exogenous variables for predicting the SAIPA automotive group production.

This table is shown in the PDF version of the article.

Table 4

Selected single-target time series models with exogenous variables (ARIMAX) for predicting the SAIPA automotive group production.

This table is shown in the PDF version of the article.

Table 5

Selected single-target time series models with exogenous variables (SARIMAX) for predicting the SAIPA automotive group production.

This table is shown in the PDF version of the article.

3.3Model Validation and Evaluation

To validate the models and examine the normality of the residual distribution, a Q-Q plot (Fig. 6) has been used. A Q-Q plot shows the quantiles of a distribution. To check the normality of the data using a Q-Q plot, if the points lie approximately on a straight line, it indicates that the data are normally distributed. Deviation from the straight line may indicate that the data distribution is not normal. By examining the plots, the error distributions in all production data columns (except Saipa Diesel, which is weaker at the 1% level) appear normal. Regarding Saipa Diesel's production, due to frequent holidays and almost no production variance, the predictive model is equal to the company's constant average production, which has also affected the error distribution. According to industry experts, this model has been accepted for predicting Saipa Diesel.

Figure 6

Q-Q Plots of residuals for predictive models in the SAIPA automotive group companies.

3.4Selecting the Best Performing Model

To fully explain the process of selecting the best predictive model, the process for selecting the best-performing model for the Saipa company is provided. Using the Box-Jenkins method and the auto-ARIMA model, time series models have been trained on the dataset. The auto-ARIMA model reports the best performing model based on BIC and AIC values. The best-performing model for predicting SAIPA's daily production, based on the auto-ARIMA results, is the ARIMA (4,1,5) model, with evaluation criteria values of AIC=5841.227, BIC=5881.795, MSE=84228.0155, RMSE=290.220, and MAE=257.60.

Then, the proposed model and models with smaller coefficients are implemented on the dataset, and the best performing model is selected based on AIC, BIC, MAE, MSE, and RMSE criteria, as well as its simplicity. As shown in Fig. 7 and Fig. 8, models with smaller coefficients have been executed on the dataset, and the evaluation criteria values for these models have been obtained. Compared to the proposed auto-ARIMA model, the ARIMA (4,1,5) model still performs best at predicting the data. Additionally, the predictive model with the exogenous variable (weekly plan) for each company was executed. The best-performing model for predicting Saipa's daily production is the ARIMAX (4,1,5) model, with evaluation criteria values of AIC=4709.518, BIC=4754.164, MSE=50939.139, RMSE=225.667, and MAE=147.098. Following this, the SARIMAX (4,1,5) model, with evaluation criteria values of AIC=4667.954, BIC=4748.375, MSE=53531.336, RMSE=231.368, and MAE=152.981, performs worse. This process has also been used to select the best-performing model for other companies.

Figure 7

Evaluation criteria values for AIC, BIC, and LLF.

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Figure 8

Evaluation criteria values for MAE, MSE, RMSE, and R-squared.

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3.5Result and Rolling Forecast

After implementation, validation, evaluation, and comparison of the results, the best time series forecasting models were selected as shown in Fig. 6. In this section, using a rolling or moving window approach, the daily production values for the SAIPA automotive group were forecast. To implement the rolling forecasting approach, the model parameters were updated every 30 days over the last three months of the dataset. Fig. 9 shows the rolling forecasting method used in the research.

Table 6

Evaluation criteria values of the best forecasting model for the SAIPA automotive group companies.

This table is shown in the PDF version of the article.

Figure 9

Rolling forecasting approach and updating parameters of the best model in October, November, and December.

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In Table 7, the mean absolute errors for the parameter update stages of the selected forecasting models are presented. It is expected that by updating the parameters based on new data, the forecasting models' error will decrease. However, this did not happen in the second update. This argument stems from an unexpected production decrease in October, highlighting the importance of patterns and data analysis for accurate forecasting. In Fig. 10, the forecasted production values for December for the update stages, the actual production statistics in this period, and the better performance of the third update are shown.

Table 7

MAE values based on parameter update stages of selected forecasting models for SAIPA.

This table is shown in the PDF version of the article.

Figure 10

Comparison of forecasted production values for update stages and actual production statistics in December.

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4Conclusion

This study examined and forecasted production for the SAIPA automotive group using statistical time-series models. The results indicate that the ARIMAX model is the best for forecasting daily production. This model has provided more accurate forecasts by incorporating an exogenous variable (weekly production schedule). However, to achieve accurate forecasting models, examining and controlling production based solely on production trends, the ARIMA and AR models have performed better. The SARIMA and SARIMAX models, which are typically used for seasonal data, performed poorly in this study. This result may be due to the lack of strong seasonal patterns in daily production data. However, this result may change when examining cumulative forecasts of weekly or monthly values.

Using the rolling forecasting approach only in the first and second stages has reduced errors and improved forecast accuracy. It indicates that not only is the selection of an appropriate model and parameter optimization crucial for achieving more accurate forecasts, but also the examination and analysis of data patterns and unexpected trends are essential for achieving an optimal forecasting model.

Suggestions for future research include using more advanced models, such as artificial intelligence and machine learning, and employing more complex models, such as neural networks and deep learning, for production forecasting. Other practical applications of this research include developing software tools to assist managers in decision-making based on production forecasts. These tools can help improve production planning and inventory management. Given the results and suggestions, this research is expected to improve production processes in the automotive industry and other similar industries and to pave the way for further research in this field.

Acknowledgments

The authors would like to express their sincere appreciation to all individuals whose valuable comments and support contributed to the successful completion of this study.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data Availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

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