learning, Repair and maintenance costs.

Agricultural tractors are essential assets in modern farming systems, and their availability directly affects the timeliness and efficiency of agricultural operations. Because tractors are capital-intensive machines, their operating and maintenance costs constitute a significant share of total machinery ownership and use costs. As tractors age and their cumulative operating hours increase, repair needs, lubrication requirements, fuel consumption patterns, and downtime risks may also increase. Therefore, predicting future maintenance-related costs is an important requirement for machinery budgeting, maintenance planning, and replacement-related decision-making.

Repair and maintenance cost estimation has traditionally been studied in agricultural machinery management through empirical and regression-based cost models. Many classical studies present accumulated repair and maintenance costs as a percentage of the machine's purchase or list price and relate these normalized costs to cumulative use, working hours, or age. Such models are useful because they enable comparison of machines with different acquisition prices, operating conditions, and service lives. Previous studies on tractors and other agricultural machines have commonly evaluated linear, logarithmic, exponential, power, and polynomial regression models for predicting accumulated repair and maintenance costs.

In tractor-specific studies, cumulative operating hours have frequently been identified as one of the most important predictors of repair and maintenance cost. Studies on MF285, John Deere, Massey Ferguson, Universal, and other tractor models have shown that nonlinear models, particularly power and polynomial functions, often provide a strong fit to accumulated cost data. Power models are commonly preferred because of their simplicity and consistency with classical machinery cost equations. However, polynomial models can also provide accurate predictions, especially when maintenance costs grow more rapidly at higher cumulative use levels.

More recently, machine learning methods have been introduced into the prediction of maintenance costs for agricultural machinery. Artificial neural networks, including radial basis function networks, have demonstrated high predictive accuracy for components of tractor repair and maintenance costs. These methods can capture complex nonlinear relationships among cost variables, operating hours, and machine characteristics. However, neural network models are often less transparent than regression-based models and may require larger datasets, additional preprocessing, and more computational effort. For practical machinery management, especially in contexts where simple and interpretable decision-support tools are needed, regression-based machine learning approaches remain useful.

Among regression-based methods, polynomial regression provides a practical balance between predictive capability and interpretability. Polynomial regression can model nonlinear cost trajectories while preserving a clear mathematical form that can be implemented in spreadsheets or simple software tools. It is particularly useful for farm managers and maintenance planners who need understandable cost curves rather than black-box predictions. In addition, polynomial regression can be applied to standardized cost indices, enabling separate analysis of different cost components.

In this study, a polynomial regression-based machine learning approach is applied to predict indices of agricultural tractor maintenance costs. The study focuses on three cumulative cost indices: the cumulative repair cost index, the cumulative oil cost index, and the cumulative fuel cost index. These indices are used to standardize cost values and reduce the influence of inflation, differences in tractor purchase prices, and variations in operating context. Cumulative operating hours are used as the main explanatory variable because they represent the actual physical use of the tractor more directly than calendar age alone.

The main objective of this paper is to evaluate whether polynomial regression can provide an accurate and interpretable model for predicting tractor maintenance cost indices. Specifically, the study compares second-and third-degree polynomial and exponential models across different cost indices and evaluates their predictive performance using error and goodness-of-fit measures. The study's contribution is modest but practical: it applies a simple polynomial regression-based approach to multiple tractor cost indices. It discusses its usefulness for maintenance budgeting and for decision-making related to replacements.

The remainder of this paper is organized as follows. Section 2 reviews previous studies on agricultural machinery repair and maintenance cost modeling, tractor maintenance cost prediction, and machine learning applications in agricultural machinery maintenance. Section 3 introduces the polynomial regression model used in the study. Section 4 describes the data preparation procedure and the construction of cumulative cost indices. Section 5 presents the model evaluation criteria. Section 6 reports the case study results and compares the predictive performance of the selected models. Section 7 discusses the findings and their practical implications. Finally, Section 8 concludes the paper.

In supervised machine learning, regression models are used when the target variable is numerical. The objective is to learn a mathematical relationship between one or more independent variables and a continuous dependent variable. In this study, the independent variable is cumulative operating hours, and the dependent variables are cumulative maintenance cost indices.

A simple linear regression model may not be sufficient when the relationship between tractor operating hours and cumulative cost is nonlinear. Maintenance costs often increase slowly in the early stages of a machine's life and then accelerate as components wear out and failures become more frequent. Polynomial regression is a practical method for modeling such nonlinear behavior while keeping the model mathematically simple and interpretable.

In polynomial regression, the independent variable is transformed into higher-order terms. For example, a third-degree polynomial regression model can be written as Eq. (1):

y = β0 + β1. x + β2. x + β3. x + ε (1)

where y represents the maintenance cost index, x represents cumulative operating hours, β0 is the intercept, β1, β2, and β3 are model coefficients, and ε is the error term.

In this study, polynomial regression is used to estimate the relationship between cumulative operating hours and each maintenance cost index. Second-degree and third-degree polynomial models are compared with an exponential model. The objective is not to introduce a new algorithm, but to evaluate whether polynomial regression can provide an accurate and interpretable model for forecasting tractor maintenance cost indices.

Raw maintenance cost data cannot usually be used directly in predictive modeling. Several preparation steps are required before model estimation. First, cost values must be checked for consistency and adjusted when necessary. Second, cost values measured at different times may be affected by inflation and price changes. Third, tractors may differ in purchase price, model, age, operating conditions, and usage intensity. These factors can make the direct comparison of raw costs misleading.

To address these issues, this study uses cumulative cost indices. The use of indices allows repair, oil, and fuel costs to be standardized relative to the tractor's initial purchase price. This argument makes the cost variables more comparable across tractors and over time. The standardized cost indices considered in this study are:

I. Cumulative repair cost index (CCIrepair).

II. Cumulative oil cost index (CCIoil).

III. Cumulative fuel cost index (CCIfuel).

Cumulative operating hours are used as the main explanatory variable. Operating hours provide a practical representation of tractor use because they reflect the machine's physical workload more directly than calendar age alone. In this study, cumulative operating hours were estimated based on the frequency of monthly engine oil changes. Although this approach provides a useful proxy for tractor use, it should be considered an approximation and is noted as one of the study's limitations.

Before model fitting, the dataset was split into training and test sets. The training subset was used to estimate model parameters, and the testing subset was used to evaluate predictive performance. In the original implementation, 75% of the data were used for model training and 25% for testing. This procedure provides a basic method for evaluating how well the model predicts observations not used directly in parameter estimation.

To evaluate the predictive performance of the proposed models, the dataset was split into training and test sets. The training subset was used to estimate model parameters, while the testing subset was used to assess the models' ability to predict unseen observations. This procedure provides a practical basis for comparing different regression specifications and identifying the model with the most reliable predictive performance. In this study, model performance was evaluated using three common regression metrics: Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and the coefficient of determination (R²).R ). MAPE expresses the average prediction error as a percentage of the actual value and is useful for comparing errors across cost indices with different scales. RMSE measures the average magnitude of prediction errors and gives greater weight to larger deviations. The coefficient of determination (R ) measures the proportion of variance in the observed data that the regression model explains.

Let yi denote the actual observed value of the cost index for observation i, y ̂i denote the predicted value obtained from the model, y ̅ denote the mean of the observed values, and (n) denotes the number of observations. The evaluation metrics are defined as Eq. (2)-(4):

100 yi − y ̂i

MAPE = ∑ | |. (2)

n yi

√ ( )2 (3)

RMSE = ∑ yi − y ̂i .

∑i=1(yi − y ̂i)

R = 1 − . (4)

∑ (yi − y ̅i) Lower values of MAPE and RMSE indicate better predictive accuracy, while higher values of R indicate better goodness-of-fit. However, R alone is not sufficient for model selection, especially when nonlinear or higher-degree polynomial models are considered. A model may achieve a high R by fitting the training data very closely, but this does not necessarily guarantee strong generalization performance. Therefore, the reported prediction error was also considered when comparing candidate models.

The case study focuses on predicting indices of agricultural tractor maintenance costs. The cost components considered in this study include repair, oil, fuel, and total maintenance costs. Because tractors may differ in purchase price, operating conditions, and cost levels, the raw cost values were standardized before model estimation. This standardization was carried out using cumulative cost indices, which express cumulative costs relative to the tractor's initial purchase price. For each cost component c, the cumulative cost index at time t can be represented as in Eq. (5):

where CCIc,t is the cumulative cost index for cost component c at time t, Cc,k is the cost of component c in ∑ Cc,k

CCI = , (5)

period k, and P0 is the initial purchase price of the tractor. In this study, three main indices were considered: the cumulative repair cost index (CCIrepair), cumulative oil cost index (CCIoil), and cumulative fuel cost index (CCIfuel). Cumulative operating hours were used as the primary independent variable because they reflect the tractor's actual use more directly than calendar age alone. The regression models were estimated separately for each cost index. Second-degree and third-degree polynomial and exponential specifications were compared to identify the most suitable functional form for predicting each maintenance cost index.

The findings of this study are consistent with previous tractor repair and maintenance cost studies, which have shown that nonlinear regression models are generally more suitable than simple linear models for representing accumulated maintenance costs. Previous research on MF285, John Deere, Universal, and Massey Ferguson tractors has reported that accumulated repair and maintenance costs increase nonlinearly with cumulative operating hours and that power or polynomial models often provide a strong statistical fit. The results of the present study support this general conclusion by showing that polynomial regression can effectively model cumulative repair, oil, and fuel cost indices.

A key finding of this study is that the third-degree polynomial model outperformed the second-degree polynomial and exponential models for all three cost indices. The improvement was particularly clear for the repair cost index, where the third-degree model substantially reduced the reported percentage error. The nature of repair costs may explain this. Unlike fuel and oil costs, which are more directly tied to regular operation and service routines, repair costs may increase more sharply as the tractor ages and components wear. Therefore, a higher-degree polynomial may better represent the accelerating behavior of repair costs at higher cumulative use levels.

The oil cost index showed lower errors across polynomial models, suggesting that oil-related expenses may follow a more predictable trajectory. This argument is reasonable because oil consumption and oil filter replacement are often tied to service schedules and periodic maintenance routines. Fuel cost behavior was also better represented by the third-degree polynomial model. However, it may be affected by additional factors, such as load, field conditions, tractor efficiency, operator behavior, and fuel price variation.

From a practical perspective, the proposed polynomial regression approach has two advantages. First, it is simple and interpretable. The model coefficients can be reported, examined, and implemented in spreadsheet-based decision tools. Second, the use of cumulative cost indices allows different cost categories to be compared on a normalized basis. This argument is useful for maintenance planners and machinery managers because it allows them to identify which cost components grow more rapidly over the tractor's operating life. At the same time, the results should not be interpreted as a complete replacement of the optimization model. The predicted cost indices can support maintenance budgeting and inform decision-making related to replacements. Still, a full replacement analysis would also require additional variables, such as resale value, downtime costs, productivity losses, interest and inflation rates, and the cost of alternative machines. Therefore, the present model should be viewed as a cost forecasting tool that can support, rather than fully determine, replacement decisions.

This study has several limitations that should be considered when interpreting the results. First, the analysis is based on a specific dataset of tractor maintenance costs, and the estimated coefficients may not be directly generalizable to other tractor models, regions, farming systems, or maintenance regimes. Local spare part prices, labor costs, fuel prices, operator skill, maintenance quality, and field conditions influence repair and operating costs. Therefore, the proposed models should be recalibrated before being applied to other contexts.

Second, cumulative operating hours were used as the main explanatory variable. Although operating hours are a meaningful indicator of tractor use, maintenance costs may also be influenced by other variables, including tractor age, engine power, workload intensity, soil conditions, operator behavior, and the quality of preventive maintenance. Including such variables may improve prediction accuracy in future studies.

Third, this study compares a limited set of functional forms, including second- and third-degree polynomials and exponential models. Although the third-degree polynomial model produced the lowest reported error in the present case, the comparison does not cover a broader set of machine learning methods, such as support vector regression, random forest, gradient boosting, or artificial neural networks. Future research can compare these methods using open or larger datasets while considering both predictive accuracy and interpretability. Fourth, the very high R values reported for the polynomial models should be interpreted carefully. High goodness-of-fit does not automatically imply strong generalization performance. Future studies should use larger datasets, cross-validation, and independent test samples to evaluate predictive reliability better and reduce the risk of overfitting.

Future research can extend this study in three directions. First, additional cost components, such as downtime, labor, spare parts, and resale value, can be incorporated into the modeling framework. Second, the predicted cumulative cost indices can be integrated with engineering economic models to estimate economic life and replacement timing more rigorously. Third, interpretable machine learning methods can be compared with polynomial regression to determine whether higher predictive accuracy can be achieved without sacrificing practical usability.

This study applied a polynomial regression-based machine learning approach to predict indices of agricultural tractor maintenance costs. The analysis focused on three cumulative cost indices: the cumulative repair cost index, the cumulative oil cost index, and the cumulative fuel cost index. Cumulative operating hours were used as the main explanatory variable because they reflect the tractor's physical use and provide a practical basis for modeling cost growth over time.

The results showed that the third-degree polynomial regression model achieved the lowest reported percentage error for all three cost indices. The improvement was most notable for the repair cost index, suggesting that repair costs may follow a more strongly nonlinear pattern than oil and fuel costs. The results also showed that exponential models were less accurate than polynomial models for the studied indices. The findings suggest that polynomial regression can provide a simple, interpretable, and practically useful method for forecasting trajectories of tractor maintenance costs. By modeling repair, oil, and fuel costs separately as cumulative indices, the proposed approach provides more detailed information than a single aggregate maintenance cost model. These predictions can support maintenance budgeting, cost monitoring, and preliminary decision-making for replacement. However, the results should be interpreted within the limitations of the study. The model was developed using a specific dataset, and its coefficients may not be directly transferable to other tractor fleets or operating environments. In addition, high R values in polynomial regression may indicate a strong fit to the available data, but should be validated using additional datasets and cross-validation procedures. Future studies can extend this work by incorporating additional explanatory variables, comparing polynomial regression with other machine learning methods, and integrating predicted cost indices into full engineering economic replacement models.

The authors appreciate the valuable insights and contributions of the experts who participated in this study.

This research received no external funding.

The data used in this study are available from the corresponding author upon reasonable request.