Predicting Car Rental Prices: A Comparative Analysis of Machine Learning Models

Abstract

In modern times, people predominantly use personal vehicles as a means of transportation, and, as this trend has developed, services that enable consumers to rent vehicles instead of buying their own have emerged. These services have grown into an industry, and the demand for predicting rental prices has arisen with the number of consumers. This study addresses the challenge in accurately predicting rental prices using big data with numerous features, and presents the experiments conducted and results obtained by applying various machine learning (ML) algorithms to enhance the prediction accuracy. Our experiment was conducted in two parts: single- and multi-step forecasting. In the single-step forecasting experiment, we employed random forest regression (RFR), multilayer perceptron (MLP), 1D convolutional neural network (1D-CNN), long short-term memory (LSTM), and the autoregressive integrated moving average (ARIMA) model to predict car rental prices and compared the results of each model. In the multi-step forecasting experiment, rental prices after 7, 14, 21 and 30 days were predicted using the algorithms applied in single-step forecasting. The prediction performance was improved by applying Bayesian optimization hyperband. The experimental results demonstrate that the LSTM and ARIMA models were effective in predicting car rental prices. Based on these results, useful information could be provided to both rental car companies and consumers.

1. Introduction

Automobiles are one of the most important innovations in human history, and, as of 2023, nearly 76 million cars were produced worldwide [1]. In modern society, private vehicles have become a central part of daily transportation. However, as modern lifestyles evolve, owning a car may be impractical or uneconomical. Therefore, the need for a service that allows individuals to rent a vehicle rather than purchasing their own has become evident. These changes are not merely individual choices. With the development of travel and tourism services, vehicle rental services have evolved into an important industry, which is gaining popularity among consumers. However, consumers using car rental services often spend a significant amount of time selecting and finalizing the right vehicle, which also depends on the rental price; however, fluctuating daily prices add to their confusion during this selection process. Moreover, accurate forecasting of rental prices is crucial for companies that offer rental services. Accurate price prediction enables companies to raise rates when the demand is high and offer discounts during low-demand periods to optimize profits. The vehicle utilization rate can also be regulated to prevent overuse. Moreover, precise market analysis and price forecasting enable companies to preserve or increase their market share by establishing competitive pricing. Furthermore, this understanding of customer behavior patterns facilitates the formulation of targeted marketing strategies and promotions. Consequently, given these considerations, forecasting rental prices has emerged as a significant subject.

The prediction of vehicle rental prices is affected by various factors; thus, an appropriate ML algorithm should be chosen and designed to extract the best features to predict the rental price based on multiple variables. In this study, we elaborately analyzed these challenges to enhance the accuracy of rental price predictions using ML algorithms. The experiment was conducted using two approaches: single- and multi-step forecasting models. The single-step forecasting model employed random forest (RF) regression (RFR) [2], multilayer perceptron (MLP) [3], a convolutional neural network (CNN) [4], long short-term memory (LSTM) [5], and the autoregressive intergrated moving average (ARIMA) model [6] to predict vehicle rental prices, and the performances of these models were compared. In multi-step forecasting, rental prices were predicted after 7, 14, 21, and 30 days using the algorithms that were applied in the single-step forecasting, and these predictions were compared. Furthermore, we attempted to enhance the accuracy of the rental price predictions by employing Bayesian optimization hyperband (BOHB), which is popular for its superior performance among optimization algorithms [7]. The results were then compared with those of previous experiments. Predictive models are either statistical tools or ML algorithms that predict future outcomes or trends by analyzing past data and patterns. Predictive models are utilized in various fields such as business-related decision making, scientific research, and engineering. For example, they have been applied to weather forecasting, real estate pricing, credit score estimation, and price prediction. Price prediction has been explored in various domains, and the research on stock price prediction [8] has been particularly active. Predictions in other areas such as housing [9], oil [10], and vehicle price, have also been continuously studied, and a relatively recent study has even focused on vehicle price prediction.

Early studies on vehicle price prediction [11] were conducted by applying supervised ML algorithms. For example, a linear regression analysis model was employed to predict used car prices by considering independent variables such as price, mileage, vehicle type, and brand. With the increase in the number of related studies, various supervised learning models, including k-nearest neighbors, naive Bayes, RFR, and support vector machines were employed. These advancements have significantly enhanced the accuracy of vehicle price predictions [11]. Beyond the conventional supervised learning algorithms, deep learning techniques, particularly artificial neural network models, have significantly improved the prediction performance of algorithms such as CNNs and recurrent neural networks [5]. Deep learning models have been extensively utilized in research, owing to their high predictive accuracy for intricate data structures. In particular, recurrent neural network models such as LSTM and gated recurrent units [12], which handle time-series data with their sequential architecture, have enhanced studies in these fields. Moreover, studies have used recurrent neural networks for both multi- [13] and single-step forecasting [14].

Based on the studies focusing on prediction models, researchers are considering various methods to predict car rental prices, which is a pivotal aspect of the tourism industry. It is worth noting that studies of the rental vehicle market are still relatively limited compared to those of the vehicle market or vehicle price prediction. While most vehicle price prediction studies utilize the post information about used cars, car rental price prediction research often employs real-time datasets. Most research on vehicle price prediction has focused on the estimation of the rental sale price. However, for car rentals, various factors affect the prices, making the prediction more complicated. For instance, during the period of the COVID-19 pandemic, throughout the world, the tourism industry suffered, and both car rental and car-sharing services were significantly impacted [15]. Additionally, car rental price prediction becomes difficult as seasonal variations and various factors like vacation periods and regional events could influence the demand and price of car rentals.

Consequently, owing to its complexity and numerous challenges, the field of car rental price prediction has witnessed limited developments as compared to those in vehicle price prediction. However, owing to its significance in the tourism industry, this field requires further analysis, and the prediction accuracy must be improved by leveraging the existing vehicle price prediction approaches and optimization techniques.

In this study, vehicle rental prices were predicted for either the subsequent day or on multiple future days. Owing to the complexity of variations in the daily prices of car rentals, various ML algorithms were tested. The proposed experimental results were obtained based on real data. This study aimed to assist consumers of car rental services by providing more accurate price predictions to enable their decision-making process. The primary contributions of this study are as follows:

• We demonstrated the potential application of our study to rental car companies by performing rental car price predictions, which have not been effectively studied owing to several factors, based on real industry data.
• The results confirm that car rental service consumers can be provided with more accurate price predictions to enable their decision-making process.
• The proposed method will assist rental car companies in deciding important business plans such as pricing and marketing strategies.
• The proposed approach can be applied to price prediction and related research in various industrial fields beyond the rental car industry.

The remainder of this paper is organized as follows. Section 1 briefly introduces the current situation and background of the development of the vehicle rental industry, explaining the economic importance of this industry and summarizing the primary contents of this paper. Section 2 provides a brief description of the data and the preprocessing that was performed and introduces the methods for predicting rental prices using various ML algorithms. Here, we introduce an experiment that predicts rental car prices using two methods, single- and multi-step forecasting, and describe the algorithms used in each experiment. It also explains the use of BOHB optimization when performing multi-step forecasting. Section 3 explains the performance metrics used in the experiment and presents the results, and compares the performances. Section 4 reviews the experimental results and Section 5 presents the conclusions.

2. Method

This section provides a concise overview of the experimental data and algorithms used to predict car rental prices after one or more days at different time intervals. An experiment was conducted using three approaches: single-step forecasting, multi-step forecasting, and multi-step forecasting using hyperparameter optimization.

2.1. Data Sources

The dataset used in this study was provided by KAFLIX, a leading company that operates a car rental platform based in Jeju, Republic of Korea. KAFLIX provides a platform called JEJUPASS, which manages and handles reservation services for approximately 105 rental car companies in Jeju. Using this platform, convenient and easy reservation services are provided to consumers who use rental car services. The data encapsulate the actual usage of car rental services by consumers in Jeju, which were sourced from various small- and medium-sized car rental companies between January 2021 and November 2023. Moreover, a summary of historical rental prices according to the period and detailed information on the vehicles rented by customers was obtained.

2.2. Data Preprocessing

Predicting car rental prices using only data obtained from a single city was quite challenging owing to the price fluctuations caused by various factors. Given the various factors influencing price fluctuations, irrelevant features must be eliminated for accurate car rental price prediction. We excluded data from January 2021 to February 2021, the months where an abnormal impact on the car rental industry was observed because of COVID-19. The reason for this exclusion is that KAFLIX reported that there was abnormal demand during these months, and they identified them as outliers. This issue also had a significant impact on car rental companies in other countries [15]. Moreover, to enhance the accuracy of price predictions, features related to public holidays and vacations were incorporated, reflecting their influence on actual car rental usage. Further, the data of individuals under the age of 20 and over the age of 70 were excluded, as people of these age groups typically do not use rental services. The price data was normalized using a standard scaler. After that, feature selection was conducted using the Permutation Feature Importance of scikit-learn to identify the features with the greatest influence on model prediction. To address potential randomness problems, features were selected by averaging the results after several repeated runs to ensure consistency. Subsequently, a more precise analysis of the impact on price was achievable by considering the specifics of the vehicle rented by the customer, rental date, and prior rental price. Through this preprocessing step, we created data that enabled more accurate predictions of car rental prices.

2.3. Single-Step Forecasting

In the single-step forecasting process, the window size was set to 14 days and the ML algorithms predicted the car rental price of the subsequent days. To proceed with the experiment, data were divided into training and test datasets. The training dataset was used to update the model, whereas the test dataset was used to evaluate the performance of the model and was not involved in the training process. We divided the training and test sets in a ratio of 8:2. Cross-validation was applied to RF to prevent overfitting [16], whereas dropout and early stopping were used in the deep learning process. Early stopping used 10% of the training dataset to monitor the validation loss during the training process, and, if the validation loss was improved for 5 epochs, the training process was stopped. The details of the proposed algorithms are presented subsequently.

2.3.1. Random Forest Regression

RF is an ML algorithm introduced in [2] that can be used for regression and classification tasks. This algorithm can be applied to a variety of prediction problems [17]. It is widely used owing to its computational simplicity and high performance in real-world applications [18,19,20]. Its other advantages include the ability to handle various data structures and problems such as a small sample size and high-dimensional characteristic spaces [21]. It can also recognize nonlinear patterns and is robust against outliers and noise in the data. Moreover, multiple input variables can be simultaneously processed using RF, providing more accurate predictions by integrating various elements of time-series data [22]. Thus, the RFR model can be used for time-series prediction tasks such as predictions using financial or medical data [23]. Using these features, studies have been conducted on the rental price prediction for houses, used car price prediction, and various other price prediction tasks. Li et al. demonstrated the effectiveness of the RF algorithm in predicting used car prices in China, where an overfitting issue could occur with noisy data [24]. To address this, they randomly sampled and trained decision trees, which were also applied in our experiments. Cross-validation was referenced to prevent overfitting in RF and various ML models [16]. Raju et al. used RF for accurate and data-driven rent prediction in the rapidly changing real estate market [25]. The RF-based prediction model was attempted using multiple variables, demonstrating the effectiveness of the RF algorithm [25]. Although it is not focused on the car rental prices, it was confirmed that the RF algorithm was effective in various price prediction tasks. We applied these insights to predict rental car prices using RF. Further details regarding the RF algorithm are elaborated in [2]. Additionally, the operation of the RF algorithm is explained in Figure 1.

2.3.2. Multilayer Perceptron

MLP is a significantly popular feedforward network structure, and has an artificial neural network architecture [3]. MLP typically consists of multiple layers including an input layer, one or more hidden layers, and an output layer [3]. Each layer is composed of multiple neurons that are connected to the subsequent layer through associated weights. These weights are iteratively adjusted and repeatedly learned through the gradient descent method of the backpropagation algorithm. After the training process, networks yield prediction inferences using unseen testing data. MLP can learn complex patterns and relationships in input data through nonlinear activation functions [26,27,28]. It also exhibits a superior ability to recognize and generalize inherent patterns in data during the learning process. Therefore, it could be applied to our prediction task based on past data patterns, and its operation is explained in Figure 2. Applying these characteristics, Al-Turjman et al. demonstrated that MLP can be used to predict vehicle prices [29]. They achieved good prediction performance for the vehicle price prediction by applying the ReLU activation function [29]. However, as a drawback of deep learning architecture, Sabiri et al. criticized the overfitting problem [30]. Considering these findings, we attempted to create a car rental price prediction model based on an MLP structure with ReLU activation function. Additionally, to prevent the overfitting problems that could occur in deep learning structures, we used dropout and early stopping strategies [30].

2.3.3. Convolutional Neural Networks

CNN was originally developed for computer vision tasks [4], where two-dimensional CNNs were designed for classification tasks such as object recognition in images [31]. CNNs can also improve the performance of prediction models, including natural language processing [32], speech recognition and modeling [33], and time-series forecasting tasks on financial data [4,34,35]. CNN can learn recurring patterns within time-series data and utilize them to yield future prediction [35]. CNN also effectively filters out various noises in time-series data using subsequent layers and extracts only the relevant features [35]. Hence, we considered these aspects and incorporated the CNN into our experiments, and its operation is explained in Figure 3. This model is highly useful for modeling the nonlinear dynamic changes that can occur in time-series data. In their study on stock price prediction, Mehtab et al. highlighted that the advantages provided by CNNs include automatic feature learning and the ability of the model to directly output a vector [36]. By leveraging these characteristics, they achieved high accuracy in stock price prediction [36]. However, Zou et al. pointed out that, while CNNs are useful for processing complex data and time-series data structures, they require significant time and resources for training [37]. Therefore, to utilize the strengths of CNNs for car rental price prediction, we applied dropout and early stopping methods to prevent overfitting and reduce training time [30].

2.3.4. Long Short-Term Memory

LSTM is a type of recurrent neural network, specifically designed to model long-term sequence dependencies [5]. In contrast to traditional recurrent neural networks, LSTM uses a memory-block structure rather than hidden layers. This memory block comprises of multiple memory cells that are controlled by sigmoid gates. These gates regulate the flow of information, preserving or discarding information based on whether the gate value is 1 or 0, respectively. Memory cells use a shared gate to minimize the number of parameters. Based on this architecture, Du et al. used an LSTM-based model to predict the airline ticket prices influenced by various factors and achieved high performance [38]. They argued that the reason for their successful predictions was for the LSTM structure to efficiently learn long-term dependencies within time-series data sequences [38]. This is because LSTM processes data in temporal order, storing past information in long-term memory cells and using it to predict future values. Consequently, LSTM could provide high prediction accuracy, considering both short-term and long-term dependencies, making it effective, even with complex temporal patterns [39]. Due to these features, we determined that LSTM would be suitable for the rental car dataset with temporal dependencies. Figure 4a,b explain the operation of single-step forecasting and multi-step forecasting, respectively. However, Shahi et al., who successfully applied an LSTM to stock price prediction, insisted that LSTM requires considerable training time and poses a risk of overfitting, although LSTM provides high prediction accuracy [40]. They claimed that early stopping could prevent these drawbacks [40]. Following this approach, we applied early stopping and dropout [30] and utilized these features in both single-step and multi-step forecasting experiments. For a given input sequence including the details of the vehicles rented by customers over different periods and previous rental prices ($x_1,x_2,x_3,\dots$), LSTM maps these to an output sequence of the future car rental prices ($y_1,y_2,y_3,\dots$).

2.3.5. Autoregressive Integrated Moving Average

The ARIMA model is a traditional approach for time-series forecasting based on the Box–Jenkins method [6], which primarily consists of autoregression (AR) and moving average (MA) components. AR forms the foundation of the prediction, while the MA is used to adjust for errors in past predictions. Thus, ARIMA is highly effective in identifying the patterns in time-series data and forecasting future values. Utilizing these advantages, Poongodi et al. successfully predicted Bitcoin prices [41], demonstrating ARIMA’s effectiveness in multivariate data price prediction tasks. At the same time, it is important to mention that setting the three parameters (p, d, and q) is crucial for effective prediction [41]. The ARIMA model is characterized by three parameters: (p, d, and q), where p, d, and q denote the order of the autoregressive components, degree of differencing, and order of the moving average components, respectively. Siami-Namini et al. also highlighted ARIMA’s effectiveness in time-series-based price prediction tasks [42]. However, they noted that ARIMA is suitable for single-step forecasting and not for the multi-step forecasting due to its simplicity [42]. Referring to these points, we attempted single-step forecasting using the ARIMA model, and during the experiment, we set the parameters to p = 1, d = 1, and q = 0.

2.3.6. Informer

The Informer model is a time-series prediction model based on the Transformer architecture [43]. This model was developed to efficiently improve the existing Transformer model in terms of computation, memory, and structure while maintaining high-performance prediction. This methodology can reduce the computational load while effectively and quickly performing predictions for long-sequence time-series forecasting (LSTF). In addition to LSTF, this method has been proven to be effective for short-term predictions [44]. The Informer consists of the following three main features in its architecture:

• ProbSparse Self-Attention: The self-attention mechanism of the existing Transformer model exhibits the limitation of high computational complexity and memory usage in proportion to the input sequence length. To solve these problems, ProbSparse Self-Attention was introduced in the Informer model. This increases the computational and memory efficiencies by selectively performing the calculations for only the most important queries, which are identified with a high probability:

  $$A(Q,K,V)=\mathrm{Softmax}\left(\frac{\overline{Q}{K}^T}{\sqrt{d}}\right)V$$

  (1)

  where Q is the query matrix, K the key matrix, V the value matrix, $\overline{Q}$ the sparse matrix (same size as Q,) and d the dimension of the key vectors.
• Distilling Operation: Through this operation, the model extracts only important information from the input sequence and compresses it into sequences that are shorter than the input. This significantly improves the computational efficiency, particularly when dealing with long-sequence data.
• Multiscale Time Embedding: The Informer model uses multiscale time embedding to learn the patterns at different timescales, which can deal with the prediction of data for multiple periods.

The Informer model significantly reduces the computational burden of the existing Transformer. Owing to these features, it can efficiently predict data sequences and its superior performance has been demonstrated in various real-world applications [45,46]. In this study, this model was applied in the single-step forecasting experiment. The operation is explained in Figure 5.

2.4. Multi-Step Forecasting

As previously mentioned, multi-step forecasting was conducted using an algorithm that was employed in single-step forecasting. Multi-step forecasting predicts future outputs that are more than one step ahead, and, in each step, the predicted output of the previous step is used as an additional input, while maintaining the same model parameters. Our goal was the prediction of car rental prices for various intervals, including 7, 14, 21, and 30 days. The window and batch sizes used for the daily price prediction are listed in

Table 1. Setting of the multi-step forecasting experiment. Unit1 and Unit2 in the table represent the number of neurons in each layer.

Forecasting Days	Window Size	Batch Size	Unit1	Unit2
7 days	20	8	64	64
14 days	30	8	64	64
21 days	40	8	64	64
30 days	50	8	64	64

.

2.5. Bayesian Optimization Hyperband (BOHB) for Multi-Step Forecasting

Bayesian optimization is an optimization method introduced in [47,48], which was developed to overcome the limitations of the conventional hyperparameter tuning methods such as grid or random search operations. According to the experimental results in [48], Bayesian optimization outperforms manual hyperparameter tuning. Hyperband [49] was designed to discard the less-effective combinations of hyperparameters that underperformed during the training process. It prunes the hyperparameter search space by applying the successive halving method [50] using the values obtained from various early terminations performed multiple times. BOHB [7] combines the sampling strategy of the tree-structured Parzen estimator [51] with the pruning strategy of hyperband [52]. In multiple studies, BOHB outperformed Bayesian optimization and hyperband in high-dimensional and diverse models. Hence, by leveraging the strengths of BOHB, we attempted to optimize the window size, batch size, and layer units to improve the car rental price prediction. The hyperparameters before and after BOHB optimization are listed in

Table 2. Results of the Bayesian optimization hyperband (BOHB) optimization of hyperparameters.

Parameter	7 Days		14 Days		21 Days		30 Days
	Before	After	Before	After	Before	After	Before	After
Window size	20	24	30	38	40	48	50	56
Batch size	8	16	8	16	8	16	8	32
Unit1	64	64	64	128	64	32	64	64
Unit2	64	128	64	128	64	64	64	128

.

3. Results

This section outlines the performance metrics employed in the experiment. Furthermore, the results of the vehicle rental price prediction experiment using ML algorithms are presented and thoroughly discussed. The experimental results are analyzed based on the single-step forecasting, multi-step forecasting, and BOHB-based multi-step forecasting experiments. The performances of the algorithms applied in each method were compared to highlight their differences and unique features.

3.1. Performance Metric

We adopted MAE, MSE, RMSE, and correlation metrics for the performance evaluation. These metrics are all intuitive and easy to interpret, clearly showing the difference between the predicted and actual values in prediction tasks [53]. These attributes make them popular performance metrics in many price prediction-related tasks [24,25,29,36,38,42]. However, a notable drawback of these values is their sensitivity to outliers [54]. To address this, we used the correlation value, which indicates how similar the patterns of the predicted and actual values are [55]. Furthermore, we validated the reliability of the correlation value using the p-value [56]. The calculation methods for each performance metric are as follows.

3.1.1. Mean Absolute Error

The mean absolute error (MAE) is a commonly used metric to assess the accuracy of time-series prediction models, and is also referred to as mean absolute deviation [57]. A specific value indicating a suitable MAE value does not exist for a time-series prediction model. However, a lower MAE is believed to indicate better accuracy. The formula for calculating the MAE value is as follows:

$$\mathrm{MAE}=\frac{1}{n}\sum _{i=1}^n|y_i-{\widehat{y}}_i|$$

(2)

where $y_i$ and ${\widehat{y}}_i$ denote the actual and predicted values, respectively, and n denotes the number of samples.

3.1.2. Mean Squared Error

Mean squared error (MSE) is a commonly used metric to assess the performance of time-series prediction models. A smaller MSE value corresponds to a more accurate prediction. MSE is calculated by squaring the difference between the predicted and actual time-series values. The formula to calculated MSE is as follows:

$$\mathrm{MSE}=\frac{1}{n}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$$

(3)

where $y_i$ and ${\widehat{y}}_i$ denote the actual and predicted values, respectively, and n the number of samples.

3.1.3. Root Mean Squared Error

The root MSE (RMSE) is the most commonly used metric to measure the accuracy of time-series forecasting models [58]. This is calculated by determining the square root of the mean of the squared differences between the actual values and the predictions of the model. Although the appropriate RMSE value is unclear, it is generally considered significant when it is as low as possible while maintaining accuracy. Equation (4) shows the formula for calculating the RMSE value:

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(4)

where $y_i$ and ${\widehat{y}}_i$ denote the actual and predicted values, respectively, and n represents the number of samples.

3.1.4. Correlation

A correlation is a statistical measure that indicates the strength of a linear relationship. It is primarily determined using the Pearson correlation coefficient, which ranges between $-1$ and 1 [59]. A value closer to 1 indicates a strong positive correlation, whereas a value closer to $-1$ indicates a strong negative correlation. A value of 0 indicates no linear correlation. We used these metrics to assess the correlation between the predicted values and ground truth. This is because the correlation indicates the similarity between the patterns of the predicted values with those of the actual values. Its formula is as follows:

$$r=\frac{{\sum }_{i=1}^n(x_i-\overline{x})(y_i-\overline{y})}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(5)

The individual values of datasets x and y are denoted by $x_i$ and $y_i$, respectively. $\overline{x}$ and $\overline{y}$ represent the mean values of the datasets x and y, respectively, and n denotes the number of data points. The correlation coefficient r has a range between $-1$ and 1.

3.1.5. p-Value of Correlation

The p-value is used in statistical hypothesis tests and indicates the significance of the correlation coefficients obtained from the given data. In general, null hypothesis and alternative hypotheses are established, and the p-value determines whether the null hypothesis should be rejected. The p-value is expressed as a value between 0 and 1; a smaller value indicates a stronger rationale for rejecting the null hypothesis. In general, p-values less than 0.05 are considered to be statistically significant, indicating that we reject the null hypothesis and accept the alternative hypothesis. We obtained the p-value through the correlation value described in Equation (5) to validate the performance indicators such as MAE, MSE, and RMSE. It was calculated to evaluate the validity of the correlation coefficient between the predicted value and the actual value. Therefore, the null hypothesis states that there is no correlation between the predicted value and the true value. In the calculations, the t-statistic is determined using the obtained correlation coefficient r, and then the p-value is derived from the t-distribution. The obtained formula and process are as follows:

$$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$

(6)

where r and n denote the correlation and amount of data, respectively.

$$df=n-2$$

(7)

where $df$ is the degrees of freedom and n denotes the amount of data.

$$p-\mathrm{value}=2\times (1-{\mathrm{CDF}}_t\left(\right|t\left|\right))$$

(8)

where ${\mathrm{CDF}}_t$ is the cumulative distribution function of the t-distribution with $df$ degrees of freedom.

3.2. Results of Single-Step Forecasting

Based on the previously described performance metrics, various ML algorithms including RFR, MLP, CNN, LSTM, and ARIMA were compared during the single-step forecasting experiment. As shown in

Table 3. Result of the single-step forecasting experiment. The numbers in bold denote the best prediction results.

Algorithm Used	MAE	MSE	RMSE	Correlation (p-Value)
RFR	0.445	0.406	0.637	0.776 ($p<0.05$)
MLP	0.475	0.436	0.660	0.750 ($p<0.05$)
CNN	0.496	0.470	0.686	0.726 ($p<0.05$)
LSTM	0.411	0.391	0.626	0.619 ($p<0.05$)
ARIMA	0.199	0.091	0.302	0.954 ($\mathit{p}<\mathbf{0.05}$)
Informer	0.547	0.536	0.732	0.617 ($p<0.05$)

, ARIMA yields the highest accuracy with MAE, MSE, RMSE, and correlation indices of 0.199, 0.091, 0.302, and 0.954, respectively. This confirms the high effectiveness of the ARIMA algorithm in the single-step forecasting of car rental prices. Figure 6 shows the ground truth and the price estimation using the ML algorithms in red and blue colors, respectively; we can note the similarity between the predictions using ARIMA and the labels in Figure 6e. Furthermore, LSTM exhibits the second-highest predictive performance of MAE, MSE, and RMSE values of 0.411, 0.391, and 0.626, respectively. Figure 6d shows that the predictions obtained using LSTM closely follow the trend of the label. Other algorithms, excluding the Informer algorithm, also exhibit price estimations that are similar to the ground truth, as shown in Figure 6a–c.

Conversely, the results of the Transformer-based model, Informer, were relatively worse than those of other models. According to a previous study [60], the poor performance may have been caused by the following two reasons. One reason may be that numerical time-series data with multiple trends and periods often contain insufficient semantic information. In the Transformer model, an increase in the look-back window size does not reduce, and, sometimes, even increases, the prediction errors; thus, it fails to extract the features of the temporal relationships in the series data, resulting in worse performance as compared to those of other models.

In conclusion, the experimental results demonstrate that the ML-based predictions of the car rental price is a highly accurate approach and can offer substantial benefits in real-world business scenarios.

3.3. Result of the Multi-Step Forecasting Experiment

Following the single-step forecasting, another crucial experiment of this study was the multi-step forecasting. In this experiment, the LSTM algorithm, which is known to be effective for time-series data analysis, exhibited the best performance when predicting car rental prices on multiple days, that is, after 7, 14, 21, and 30 days. The prediction accuracy and efficiency were evaluated using performance indicators such as MAE, MSE, RMSE, correlation, and p-value.

Unlike the previous experiments, our experimental results also confirmed that the LSTM exhibits the best performance.

Table 4. Results of the multi-step forecasting. As the number of forecasted days increases, mean absolute error (MAE), mean squared error (MSE), and root MSE (RMSE) values gradually decrease, whereas the correlation shows a sudden reduction. The numbers in bold denote the best prediction results.

Algorithm	Predicted Days	MAE	MSE	RMSE	Correlation (p-Value)
RFR	7-days	0.646	0.736	0.858	0.205 ($p>0.05$)
	14-days	0.633	0.709	0.842	0.208 ($p>0.05$)
	21-days	0.563	0.566	0.752	0.224 ($p>0.05$)
	30-days	0.497	0.437	0.661	0.179 ($p>0.05$)
MLP	7-days	0.550	0.594	0.770	0.421 ($p<0.05$)
	14-days	0.523	0.495	0.702	0.599 ($p<0.05$)
	21-days	0.452	0.409	0.634	0.478 ($p>0.05$)
	30-days	0.407	0.333	0.581	0.352 ($p>0.05$)
CNN	7-days	0.537	0.542	0.736	0.510 ($p<0.05$)
	14-days	0.483	0.431	0.667	0.595 ($p<0.05$)
	21-days	0.446	0.394	0.626	0.482 ($p>0.05$)
	30-days	0.395	0.354	0.574	0.356 ($p>0.05$)
LSTM	7-days	0.486	0.458	0.677	0.604 ($p<0.05$)
	14-days	0.439	0.393	0.627	0.658 ($p<0.05$)
	21-days	0.409	0.357	0.598	0.565 ($p<0.05$)
	30-days	0.378	0.294	0.542	0.518 ($p<0.05$)
ARIMA	7-days	0.721	0.622	0.789	−0.807 ($p>0.05$)
	14-days	0.689	0.713	0.844	−0.717 ($p>0.05$)
	21-days	0.794	1.077	1.038	−0.570 ($p>0.05$)
	30-days	0.802	0.930	0.964	−0.149 ($p>0.05$)
Informer	7-days	0.672	0.845	0.919	0.335 ($p>0.05$)
	14-days	0.665	0.779	0.882	0.223 ($p>0.05$)
	21-days	0.677	0.918	0.958	0.144 ($p>0.05$)
	30-days	0.673	0.930	0.964	0.089 ($p>0.05$)

shows that the MAE, MSE, and RMSE values exhibit a progressive decrease as the window of prediction days expanded. It should be noted that MAE decreases from 0.486 to 0.439, 0.409, and 0.378 as the prediction time increases. Furthermore, the MSE and RMSE decrease from 0.458 to 0.393, 0.357, and 0.294 and from 0.677 to 0.627, 0.598, and 0.542, respectively. This indicates that the models achieve stable performance, even in long-term predictions. Moreover, the red line in Figure 7a–d illustrates the predicted car rental prices across multiple days using LSTM. It should be noted that the predicted values are well aligned with the actual car rental prices. However, the correlations marginally increase from 0.604 to 0.658 and then decrease to 0.565 and 0.518, presumably due to the increased uncertainty of the data as the prediction range is widened. Nevertheless, the correlation values are significant based on the p-values (<0.05), indicating that the model predictions do not deviate significantly from the ground truth. The algorithm that exhibits the second-best performance is the 1D CNN. Table 4 shows that the MAE values decrease from 0.547, 0.483, 0.446, and 0.395 as the number of prediction days increase from 7 to 14, 21, and 30 days, respectively. Similarly, the MSE and RMSE values also decrease from 0.542, 0.431, 0.394, and 0.354, and 0.736, 0.667, 0.636, and 0.574, respectively, similar to the LSTM results. However, the correlation coefficient rapidly decreases for the 21- and 30-day predictions and becomes insignificant based on the p-value (<0.05), indicating that the predictions are inaccurate. As shown in Figure 7, the predictions indicate a tendency to follow the overall trend, except in the cases of the 21- and 30-day predictions. MLP and CNN exhibit similar trends to the labels of Figure 7, shown in the sky-blue and purple lines. Conversely, ARIMA, which shows the best performance in the previous experiment, is ineffective in the multi-step forecasting. The MAE values listed in Table 4 exhibit high values of 0.721, 0.689, 0.794, and 0.802, indicating its unsuitability for the multi-step forecasting, as the p-value is not significant. Additionally, the Transformer-based algorithm, Informer, also demonstrates poor performance. This is believed to be due to the same reasons [60] as those in the previous experiment. As with the ARIMA results, Figure 7 shows that the trend cannot be maintained, depicted by a black line. RF-based multi-step forecasting results are also poor. Random forests capture nonlinear relationships well but do not explicitly model temporal dependence [14]. Additionally, in multi-step forecasting where future predictions depend on the previous ones, RF could cause each step to propagate errors, resulting in inaccuracies [14]. For this reason, the results of the multi-step forecasting with RF were judged to be poor, which can be seen in Figure 7, where the green line does not follow the true trend well.

3.4. Result of BOHB-Optimized Multi-Step Forecasting

In the design of the multi-step forecasting model, we enhanced the model performance by incorporating the BOHB optimization technique. The BOHB addresses the limitations of the traditional hyperparameter optimization approaches, applying a more accurate parameter search approach. Through this method, an optimal combination of hyperparameters such as an appropriate activation function in the neural network layer, ideal number of hidden layers in the neural network, and optimum batch size could be defined, which improved the model performance.

After applying BOHB optimization to LSTM, which performed best in the previous experiment, the experiment was repeated. The results obtained using the optimized hyperparameters (Figure 8 and results listed under “After” in Table 2) and results obtained using the initial parameters (the red lines in Figure 7 and the results listed under “Before” in Table 2) were compared. As an example, let us consider the 30-day prediction results listed in

Table 5. Results of BOHB-optimized multi-step forecasting. The MAE, MSE, RMSE, and correlation values were reduced or improved using the optimized hyperparameters, as highlighted in bold.

Predicted Days	MAE		MSE		RMSE		Correlation (p-Value)
	Before	After	Before	After	Before	After	Before	After
7-days	0.486	0.484	0.458	0.432	0.677	0.657	0.603 ($p<0.05$)	0.613 ($\mathit{p}<\mathbf{0.05}$)
14-days	0.439	0.432	0.393	0.401	0.627	0.633	0.658 ($p<0.05$)	0.660 ($\mathit{p}<\mathbf{0.05}$)
21-days	0.409	0.404	0.357	0.360	0.598	0.600	0.565 ($p<0.05$)	0.570 ($\mathit{p}<\mathbf{0.05}$)
30-days	0.378	0.367	0.294	0.287	0.542	0.535	0.518 ($\mathit{p}<\mathbf{0.05}$)	0.522 ($\mathit{p}<\mathbf{0.05}$)

; the outcomes after the optimization process demonstrate reductions in MAE from 0.378 to 0.367, MSE from 0.294 to 0.287, and RMSE from 0.542 to 0.535. These demonstrate the effect of the automatic hyperparameter optimization approach on the prediction performance as compared to that of the manual search-based approach. Moreover, Figure 8 shows the results of multi-step predictions using BOHB, which are more aligned with the labels than the red lines in Figure 7, which were designed manually. These imply that the automatic parameter optimization approach using BOHB contributes to enhancing the performance of the predictive model. This enhancement will play a crucial role in elevating the accuracy of car rental price predictions.

4. Discussion

In this study, various ML algorithms were compared and analyzed to predict car rental prices. The ARIMA algorithm yielded the best predictive performance in the single-step forecasting experiments. The superior performance of ARIMA suggests that it can be a reliable tool for predicting the trends in car prices despite its lightness. It has a simpler structure than that of the other models, offering the advantage of being more efficient and straightforward. Moreover, the relatively small number of parameters ensures that the meanings of the parameters are clear. Therefore, interpreting the model results and directly applying them to decision making in the car rental business is simpler. Another advantage is its effectiveness on small datasets. This is particularly advantageous in scenarios where data collection is challenging or costly, unlike previous models requiring large quantities of data. Furthermore, while other tested models may exhibit instability in predictions, the fixed structure of ARIMA enables it to provide stable forecasts. This consistency is a significant advantage for application in the car rental business. Given the high price volatility in the car rental industry, the accuracies of these predictions are crucial for formulating business strategies. However, despite positive outcomes and advantages in the single-step forecasting experiments, limitations were exhibited in their practical industrial applications. This is because, in an actual car rental business, consumers tend to consider prices not just a day later, but over the long term. Therefore, multi-step forecasting is required to predict long-term prices in the actual car rental business.

In the multi-step forecasting experiments, LSTM produced the most stable performance for the predictions. While the accuracy declined slightly with the extension of the forecast period, LSTM continuously yielded the best results close to the actual prices. This suggests that LSTM could be a strong candidate to analyze complex time-series data for the multi-step future prediction. Furthermore, LSTM could be applied to develop effective pricing strategies, especially for the long-term forecasting of businesses.

The implementation of the BOHB optimization technique further enhanced the performance of multi-step forecasting. This indicates that the hyperparameter optimization plays a crucial role in enhancing the accuracy of the predictive models. Particularly, given the intricate data patterns and varying prediction ranges of the time-series data, which can shift with changes in consumer behavior or economic trends, optimized parameter settings are essential for maximizing the performance of the model. This implies that, in practical business implementation, BOHB is employed only during the initial learning phase, thereby not imposing a significant burden in practical applications post-learning.

Overall, this study demonstrated the applications of the diverse ML algorithms to car rental price forecasting and proposed strategies to optimize their performances, thereby enhancing their applicability in real-world business environments. These findings are crucial for car rental service providers to formulate future pricing strategies, and contribute to delivering more equitable pricing information to consumers.

5. Conclusions

In this study, we explored the potential and effectiveness of various ML algorithms for predicting car rental prices. The experiment involved three separate trials: single-step forecasting, multi-step forecasting, and BOHB-optimized multi-step forecasting. The experimental results showed that the ARIMA algorithm was superior in single-step forecasting, whereas the LSTM-based multi-step forecasting outperformed the other multi-step predicting models. Moreover, fine-tuning of the parameters of the ML algorithm using BOHB positively influenced the enhancement of car rental price predictions, as proven by the improvement in prediction accuracy. However, the availability of a limited number of previous studies on price prediction in the rental car industry can act as a limitation of this study, owing to the lack of reference materials for a comparative analysis. Additionally, the data provided for the experiment only reflect records from rentals in Jeju, which limits the ability to consider variations from other regions and may act as a limitation of this study. To overcome this limitation, we referred to price prediction methods used in other areas and we expect that providing a more accurate and competitive pricing information for vehicle rental service users can ensure a superior service experience. Moreover, this can provide service providers with valuable insights for pivotal business decisions, such as pricing strategies and inventory management. Furthermore, the proposed methods in this study can be extended beyond the rental car industry and applied to price prediction challenges in various fields. In the future, we plan to conduct research by applying various algorithms, including exponential smoothing and Prophet, excluding those used in this experiment, for a more accurate verification. Furthermore, the research will be based on the results obtained via predictive analysis using a large language model.