SymOpt-CNSVR: A Novel Prediction Model Based on Symmetric Optimization for Delivery Duration Forecasting

Abstract

Accurate prediction of food delivery time is crucial for enhancing operational efficiency and customer satisfaction in real-world logistics and intelligent dispatch systems. To address this challenge, this study proposes a novel symmetric optimization prediction framework, termed SymOpt-CNSVR. The framework is designed to leverage the strengths of both deep learning and statistical learning models in a complementary architecture. It employs a Convolutional Neural Network (CNN) to extract and assess the importance of multi-feature data. An Enhanced Superb Fairy-Wren Optimization Algorithm (ESFOA) is utilized to optimize the diverse hyperparameters of the CNN, forming an optimal adaptive feature extraction structure. The significant features identified by the CNN are then fed into a Support Vector Regression (SVR) model, whose hyperparameters are optimized using Bayesian optimization, for final prediction. This combination reduces the overall parameter search time and incorporates probabilistic reasoning. Extensive experimental evaluations demonstrate the superior performance of the proposed SymOpt-CNSVR model. It achieves outstanding results with an R² of 0.9269, MAE of 3.0582, RMSE of 4.1947, and MSLE of 0.1114, outperforming a range of benchmark and state-of-the-art models. Specifically, the MAE was reduced from 4.713 (KNN) and 5.2676 (BiLSTM) to 3.0582, and the RMSE decreased from 6.9073 (KNN) and 6.9194 (BiLSTM) to 4.1947. The results confirm the framework’s powerful capability and robustness in handling high-dimensional delivery time prediction tasks.

1. Introduction

Following the conclusion of the COVID-19 pandemic, the food and beverage industry has markedly accelerated its adoption of the Online-to-Offline (O2O) model, characterized by “online ordering + offline delivery [1].” This shift has catalyzed substantial global growth for major food delivery platforms such as Meituan and Ele.me. The scope of delivery services has expanded beyond meals to include pharmaceuticals, flowers, and other commodities, contributing to continuous industry expansion.

In China, for instance, nearly 600 million people utilized online food delivery services by the end of 2024, accounting for 53.4% of the total internet user population [2]. Driven by the rapid proliferation of the O2O model, the market has demonstrated robust growth momentum, engaging over one-fifth of China’s population in using these services. The adoption rate surged remarkably from 16.5% at the end of 2015 to 52.7% by the end of 2021—an increase of 36.2 percentage points over six years—underscoring the sector’s vast developmental potential [3]. This growth has been facilitated by leading delivery platforms including Meituan, Ele.me, and Baidu, alongside continuous advancements in e-commerce and mobile technology [3,4,5]. Among these, Meituan has established a dominant market position, capturing 69% of the market share in 2020, a 5-percentage-point increase from 2019, reflecting its significant competitive advantage [6].

However, market advancement necessitates further refinement of delivery services, particularly regarding the accurate communication of delivery timeframes to enhance customer satisfaction [7,8]. Existing research indicates a correlation between accurate predicted delivery times and user satisfaction [9], as customers often need to align food orders with their personal schedules. It is noteworthy that the delivery time provided by platforms is essentially an Estimated Time of Arrival (ETA), representing the predicted duration from the point of origin to the destination—a subject that has been extensively studied [10,11,12]. Nonetheless, the final delivery duration is influenced by a multitude of features, often resulting in significant variability and unexpected fluctuations [13].

To address the challenges posed by high-dimensional feature sets in predicting food delivery times, enhance service satisfaction, and leverage technology to balance the commercial dynamics between sellers and buyers, this paper examines computational methods capable of precisely identifying predictive features and introducing new momentum into high-dimensional ETA forecasting.

In recent years, prediction technology has become a hot topic across various fields, attracting significant attention from researchers in algorithms and machine learning. For instance, traditional statistics has contributed methods such as Autoregressive (AR) models [14], which assume linear relationships, Moving Average (MA) methods [15], Autoregressive Moving Average (ARMA) models [16], Autoregressive Integrated Moving Average (ARIMA) models [17], along with their seasonal variants like SARIMA [18] and exponential smoothing techniques such as Holt-Winters [19]. These approaches have provided valuable insights into regression and forecasting from a statistical perspective. Machine learning has broken through the linear constraints inherent in traditional models, with notable examples including Support Vector Machines (SVMs) [20], Random Forests (RFs) [21], Gradient Boosting Decision Trees (GBDTs) [22], and Multi-Layer Perceptrons (MLPs) [23]. Building upon these, deep learning techniques enable end-to-end feature learning through sophisticated feature engineering and modeling. Representative deep learning architectures include Recurrent Neural Networks (RNNs) [24], Long Short-Term Memory networks (LSTMs) [25], Gated Recurrent Units (GRUs) [26], and the attention-based Transformer model [27]. These have significantly enhanced the ability to model long-term dependencies and high-dimensional dynamic systems.

Applying intelligent computational models to predict delivery times is crucial for enhancing user satisfaction and reducing operational costs for these platforms [28]. Addressing the core need for cloud resource scheduling in food delivery platforms, Lang et al. [29] innovatively applied an improved Artificial Neural Network (ANN) model to predict order volumes. This approach effectively increased prediction accuracy and successfully integrated the results into dynamic resource scheduling strategies, demonstrating the practical value of predictive models in optimizing decision-making and substantially improving platform operational efficiency [29]. However, their model primarily focused on relatively macro-level order volume forecasting and did not sufficiently account for micro-level dynamic factors affecting delivery timeliness, such as real-time traffic conditions and rider behaviors. Subsequently, Song et al. [30] focused on the “last-mile” delivery segment, which directly impacts user experience. They were among the first to systematically identify the critical issue of predicting final-leg delivery service times and explored its feasibility using big data technology, laying the groundwork for subsequent refined research in this area [30]. Nonetheless, the depth of their study and the disclosure of model details were relatively limited, and the robustness and generalization ability of their proposed framework in complex and dynamic urban environments require further validation. Against this backdrop, de Araujo et al. [31] expanded the scope to encompass the prediction of End-to-End total delivery time—from origin to destination—for packages. They innovatively applied deep learning models to process complex logistical sequence data, achieving notable improvements in prediction accuracy. Their research was explicitly geared towards serving the development of smart cities, with a clear application orientation [31]. It is noteworthy, however, that while their deep learning model is powerful, its performance is highly dependent on data quality and annotations, potentially limiting its effectiveness in scenarios with sparse or noisy data. Therefore, collecting a robust set of features that effectively reflect the factors of influencing prediction is paramount. To address the aforementioned challenges and pursue higher accuracy, the recent work by Zhu et al. [32] represents a cutting-edge direction. They proposed an innovative hybrid model integrating fuzzy systems with a Convolutional Factorization Machine (CFM). This model not only leverages fuzzy systems to effectively handle the uncertainty and semantic information prevalent in logistics processes but also captures complex high-order interactions among features efficiently through the CFM. It demonstrates superior performance and enhanced robustness in delivery time prediction tasks, marking a significant breakthrough in intelligent logistics forecasting technology [32]. Nevertheless, the architecture of such hybrid models is typically complex, accompanied by relatively high computational costs and training difficulties, necessitating the use of optimization methods for adaptive parameter tuning.

For the adaptive tuning of model hyperparameters, researchers widely employ heuristic algorithms due to their high efficiency and ease of use. Liu et al. [33] proposed the Earthworm Optimization Algorithm (EOA) to optimize Support Vector Regression (SVR), significantly enhancing the prediction accuracy of reservoir landslide displacement. Their contribution lies in using EOA to adaptively adjust SVR hyperparameters, effectively overcoming the limitations of experience-dependent traditional methods [33]. Meanwhile, Che et al. [34], addressing the issue of predicting high-speed mechanical test data, utilized a multi-strategy improved Whale Optimization Algorithm (WOA) to optimize Long Short-Term Memory (LSTM) hyperparameters, substantially boosting the model’s robustness and generalization ability. They innovatively integrated chaotic initialization and dynamic weight strategies to avoid local optima [34]. Furthermore, Zhou et al. [35] designed an improved Sparrow Search Algorithm (ISSA) to optimize LSTM hyperparameters, achieving high-precision prediction of building heating and cooling loads. The advantage of their approach is the introduction of an adaptive perturbation mechanism to enhance global search capability [35]. In the energy sector, Qiu et al. [36] combined the Particle Swarm Optimization (PSO) algorithm with Gated Recurrent Units (GRUs) and innovatively incorporated Kolmogorov-Arnold Networks (KANs) to optimize oil well production prediction. PSO efficiently identified the key hyperparameters of the GRU-KAN model, significantly reducing prediction error [36]. In a similar vein, Cui et al. [37] proposed a hybrid Whale Optimization Algorithm (WOA) to optimize a regional heat load prediction model based on CNN-LSTM-Attention. Their contribution involves the synergistic optimization of hyperparameters across multiple modules via WOA and the use of an attention mechanism to strengthen feature extraction [37]. Correspondingly, Safavi et al. [38] combined the Coati Optimization Algorithm with CNN-XGBoost for early prediction of battery lifespan. This algorithm performs a targeted search for key hyperparameters, improving prediction timeliness while reducing reliance on data, collectively demonstrating the powerful performance of models after hyperparameter optimization [38].

In summary, this research proposed a novel symmetric optimization prediction model framework (SymOpt-CNSVR) for food delivery time prediction. The specific contributions are as follows:

• Inspired by the collective intelligent behavior of Superb Fairy-wrens, three key enhancements have been made to the original Superb Fairy-wren Optimization Algorithm (SFOA) algorithm to solve complex optimization problems more efficiently. Firstly, a mechanism for group collaboration and information sharing was introduced, simulating how birds collectively avoid predators and search for food. This effectively helps the algorithm escape local optima, enhancing global exploration capabilities and stability. Secondly, a historical memory mechanism was designed, allowing individuals to remember and revisit historically optimal positions, thereby improving the algorithm’s convergence accuracy and search efficiency. Finally, a self-reinforcement learning strategy was integrated, requiring individuals to self-improve based on historical experience in each iteration, ensuring continuous enhancement of solution quality. These enhancements significantly improve the optimization performance of the algorithm, as demonstrated by the reduction in MAE and RMSE on the test set from 3.8646 and 5.0253 (using the original SFOA) to 3.0582 and 4.1947 (using ESFOA). This algorithm provides an effective new tool for automated and precise hyperparameter tuning of deep learning models.
• The core innovation of this paper lies in constructing a “symmetrical” hybrid modeling framework that skillfully combines the strengths of deep learning and statistical learning. The framework adopts a symmetrical design logic: it utilizes the powerful nonlinear feature extraction capability of Convolutional Neural Networks (CNNs) to process high-dimensional sequential data and employs the improved ESFOA for automated hyperparameter optimization. Simultaneously, to obtain more statistically meaningful prediction results, a Support Vector Regression (SVR) model is used to capture deterministic patterns in the data, with Bayesian optimization applied for probabilistic search of its key parameters. This symmetrical structure of “CNN-ESFOA” and “SVR-Bayesian optimization” achieves a balance and unification between the representational power of deep learning and the interpretability of statistical models, offering a novel solution for complex data prediction problems.
• This paper applies the proposed SymOpt-CNSVR framework to the prediction of complex high-dimensional sequential data in food delivery scenarios. Through rigorous experimental comparisons, the proposed method significantly outperforms traditional baseline models and other optimized prediction methods in terms of prediction accuracy. The results demonstrate that the framework can effectively capture nonlinear spatiotemporal dependencies and statistical patterns in food delivery data, providing a higher-precision prediction model to address the critical time prediction challenges in logistics delivery, with substantial practical application value.

The remainder of this paper is structured as follows: Section 2 describes the designed algorithm and the establishment of the SymOpt-CNSVR model based on symmetrical logic; Section 3 introduces the experimental environment and the testing and evaluation of the algorithm; Section 4 summarizes the innovations and effectiveness of this study and offers insights for future improvements.

2. Methodologies

The SymOpt-CNSVR framework developed in this research aims to enhance the prediction accuracy of complex high-dimensional sequential data in food delivery by integrating the strengths of the deep learning model CNN and the statistical learning model SVR. This is achieved by leveraging ESFOA and Bayesian optimization to balance and symmetrically optimize their key hyperparameters. The core concept of the framework lies in its symmetry: for complex high-dimensional feature data, the deep learning CNN architecture is employed for pre-training to extract nonlinear dependency features, while ESFOA is used to automate the hyperparameter tuning of CNN. Subsequently, to generate statistically informed predictions, the SVR model is applied, and Bayesian probabilistic optimization is utilized to search for the optimal parameter combinations. This symmetrical design achieves a balance between deep learning techniques and probabilistic statistics, not only optimizing model performance but also reflecting careful consideration of adapting different optimization strategies to different model types. Figure 1 illustrates the overall flowchart of the framework.

2.1. Convolutional Neural Network

The prediction of food delivery time involves multiple types of features, which contain complex spatiotemporal or pattern information, requiring a powerful nonlinear feature extractor. CNN is an ideally suited choice due to its local connectivity, weight sharing, and hierarchical feature extraction capabilities, typically, a CNN consists of multiple convolutional layers, pooling layers, and fully connected layers [39]. In this model, the CNN is used to learn the feature representation of the input data, denoted as X.

2.1.1. Convolution Operation

The function of the convolution layer is to process the input data through the convolution filter to extract local features. The convolution operation can be expressed as:

$$y=f(W\ast X+b),$$

(1)

where $\ast$ represents the convolution operation, $b$ is the bias term, $f$ is the activation function.

2.1.2. Pooling Operation

The pooling layer is used to downsample the convolutional features to reduce computational complexity and overfitting risk. Assuming the pooling operation is the maximum pooling operation:

$$z=\underset{i,j}{\max }f(y),$$

(2)

where the output $z$ of the pooling layer is the downsampled feature.

2.1.3. Fully Connected Layer

After convolution and pooling, the feature map is flattened into a one-dimensional vector and then input into the fully connected layer for processing. The calculation of the fully connected layer can be expressed as:

$$h=W_f\cdot z+b_f,$$

(3)

where $W_f$ is the weight of the fully connected layer, $b_f$ is the bias term, $h$ is the high-dimensional feature extracted by CNN.

2.2. Support Vector Regression

The problem of predicting food delivery time typically involves complex nonlinear relationships and high-dimensional features, which traditional regression models struggle to effectively capture. Leveraging the high-dimensional data features processed by CNN, Support Vector Regression (SVR) employs an epsilon-insensitive loss function and kernel functions to construct accurate regression models in high-dimensional spaces. Its strong generalization capability [40] makes it well-suited for addressing this prediction task.

2.2.1. Error Constraints and Regularization Objective

SVR uses a loss function that is insensitive to ε and penalizes predictions that differ from the expected output by more than ε, which can be expressed as follows:

$$|f(x_i)-y_i|\le \epsilon ,$$

(4)

where is the error between the function’s predicted value for a given input and the actual output. SVR attempts to find the narrowest tube centered on the surface that minimizes the prediction error while also minimizing the amount ${\displaystyle \sum _{i=1}^N}\xi$ that exceeds in the training examples, which can be expressed as follows:

$$\underset{w,b,\xi }{\min }{\displaystyle \frac{1}{2}}||w|{|}^2+C\sum _{i=1}^N{\xi }_i,$$

(5)

where $‖w‖$ is the magnitude of the normal vector of the approximated surface and C is a regularization term that gives greater weight to minimizing flatness (i.e., error).

2.2.2. Feature Maps and Kernel Functions

For nonlinear functions, the data can be mapped to a higher-dimensional space, namely the kernel space, to achieve higher accuracy, which can be expressed as follows:

$$f(x)=w^T\cdot \varphi (x)+b,$$

(6)

where $w$ represents the weight vector, $\varphi (·)$ is the transformation from feature space to kernel space, $b$ is the bias term. The kernel function defines the similarity between two samples $x$ and $x^{\prime }$ in the input space and can be expressed as follows:

$$K(x,x^{\prime })=\exp (-\gamma ||x-x^{\prime }|{|}^2).$$

(7)

2.2.3. Prediction Stage

Use the learned function $f(x)$ to make predictions:

$$\widehat{y}=f(h).$$

(8)

2.3. Enhanced Superb Fairy-Wren Optimization Algorithm (ESFOA)

Due to the excessive number of hyperparameters in CNN, manual adjustment is overly complex. Therefore, the Superb Fairy-wren Optimization Algorithm (SFOA) is employed for automated hyperparameter tuning. SFOA [41] is a novel meta-heuristic algorithm proposed in 2025, inspired by the life habits of the Superb Fairy-wren. It simulates three natural behaviors of the species: juvenile growth, reproduction and feeding, and predator avoidance. The specific model of SFOA is included in the Appendix A. Although the performance of SFOA has been proven, it is still limited when facing high-dimensional complex problems. In this section, we propose ESFOA to improve SFOA through Group-balanced Cooperative Search Mechanism, Memory Location Backtracking Strategy and Foraging Intensive Training.

2.3.1. Group-Balanced Cooperative Search Mechanism

During population evolution and reproduction, Superb Fairy-wrens exhibit highly organized collective behaviors, such as cooperative foraging and predator avoidance, which significantly enhance group survival efficiency through information sharing and collaboration. Inspired by this phenomenon, this paper innovatively introduces a group-balanced cooperative search mechanism into the ESFOA. This mechanism establishes a dynamic equilibrium between individual exploration and group collaboration: individuals contribute local information to the group while simultaneously receiving global guidance from it, thereby achieving an efficient balance between exploration and exploitation. This structure not only simulates the distributed decision-making intelligence of biological groups but also effectively suppresses premature convergence by constructing symmetric information interaction between individuals and the group, significantly enhancing the algorithm’s global search capability and convergence stability. The specific process is as follows:

For each Superb Fairy-wren, its position update depends not only on its own position and the global best position but also on the average position of the group. The position update formula is as follows:

$$X_{{new}_{ij}}=X_{{new}_{ij}}+0.5(X_{mean}-X_{ij}),$$

(9)

where $X_{mean}$ is the average position of all members in the group, $X_{mean}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{X}_i}$.

2.3.2. Memory Location Backtracking Strategy

During foraging, Superb Fairy-wrens demonstrate the ability to remember optimal feeding locations and revisit these areas to enhance foraging efficiency. Inspired by this natural behavior, the ESFOA constructs a symmetric memory pool based on individual historical best positions and group-shared information. Each individual not only updates its own position during iterations but also maintains a dynamic memory set that records its historical best position information. By introducing a memory location backtracking strategy, individuals can refer to both their own historical best positions and group collaborative information while exploring new regions. Specifically, for each Superb Fairy-wren, the memory location update formula is as follows:

$${X_{history}}_{ij}={X_{history}}_{ij}+0.1(X_{{new}_{ij}}-{X_{history}}_{ij}),$$

(10)

where ${X_{history}}_{ij}$ is best position in history for the i-th Superb Fairy-wren.

2.3.3. Foraging Intensive Training

After engaging in collective working memory, Superb Fairy-wrens undergo self-improvement to enhance their foraging capabilities. Based on the reinforcement training measurement proposed by Wang et al. [42], the ESFOA incorporates a self-improvement module after each iteration, ensuring that subsequent solutions outperform previous ones. Specifically, each Superb Fairy-wren, after collaboration and memory processes, undergoes self-learning to balance the global optimum. The update formula is as follows:

$$\left\{\begin{array}{l}{X}_{ne{w}_{ij}}=X_{ij}+0.2X_{ij}lk(X_{r1}-X_{r2}),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em} rand<0.5\\ X_{ne{w}_{ij}}=X_b+{\displaystyle \frac{1}{2}}(1+{\displaystyle \frac{1}{1000}}(1-{\displaystyle \frac{FEs}{MaxFEs}}{)}^2\sin (\mathit{rand}\pi ))(X_b-X_{ij})\hspace{1em}else\end{array}\right.$$

(11)

where $X_{r1}$ and $X_{r2}$ represent two randomly selected positions.

The complete algorithm flow chart is shown in Figure 2.

2.4. Bayesian Optimization

Since both SVR and Bayesian optimization originate from statistical probability theory, Bayesian optimization can collaboratively balance the parameters of SVR. By constructing a probabilistic model of the objective function, it automatically searches for the optimal predictive parameters of SVR with relatively few experimental evaluations. By striking a balance between exploration and exploitation, Bayesian optimization effectively enhances model performance, particularly in scenarios with high computational costs and complex parameter spaces [43].

2.4.1. Prior Distribution of Gaussian Process

Bayesian optimization usually uses a probabilistic surrogate model, usually a Gaussian process, as a surrogate $f(x)$ Gaussian process model for the unknown function, which can be defined as:

$$f(x)\sim \mathcal{G}\mathcal{P}(\mu (x),k(x,x^{\prime })),$$

(12)

where $\mu (x)=E(f(x))$ represents the average function, $k(x,x^{\prime })=E\left[(f(x)-\mu (x))(f(x^{\prime })-\mu (x^{\prime }))\right]$ represents the covariance function of the unknown process.

2.4.2. Posterior Distribution of Gaussian Process

The posterior distribution of the Gaussian process is defined as follows, which represents the conditional probability distribution of the target function $f_{\ast }$ at the predicted point $X_{\ast }$ given the training data X and the corresponding observation value y:

$$p(f_{\ast }|X_{\ast },X,y)=\mathcal{N}({\mu }_{\ast },{\Sigma }_{\ast }),$$

(13)

where $\mathcal{N}({\mu }_{\ast },{\Sigma }_{\ast })$ represents $f_{\ast }$ follows a Gaussian distribution, ${\mu }_{\ast }$ is mean value, ${\Sigma }_{\ast }$ is the covariance, The formula is as follows:

$${\mu }_{\ast }=K(X_{\ast },X)K{(X,X)}^{-1}y,$$

(14)

$${\Sigma }_{\ast }=K(X_{\ast },X_{\ast })-K(X_{\ast },X)K{(X,X)}^{-1}K(X,X_{\ast })$$

(15)

where $K(X_{\ast },X)$ is the covariance matrix of the prediction point and the training point, $K{(X,X)}^{-1}$ is the inverse of the covariance matrix of the training points, $K(X_{\ast },X_{\ast })$ is the covariance between the forecast points.

2.4.3. Expected Improvement

While it is desirable to choose x so that this improvement is as large as possible, $f(x)$ is unknown before evaluation. Therefore, we take the expected value of the improvement and choose x to maximize it, defining the expected improvement function as:

$$\mathrm{EI}(x_{\ast })=\mathbb{E}[\max (f(x)-f(x_{\mathrm{best}}),0)],$$

(16)

where $f(x_{\mathrm{best}})$ is the best target value observed so far. The improved probability, that is, the probability that the function value at candidate point $x_{\ast }$ is higher than the current best value, can be expressed as:

$$\mathrm{PI}(x_{\ast })=\mathsf{\Phi }\left({\displaystyle \frac{f(x_{\mathrm{best}})-\mu (x_{\ast })}{\sigma (x_{\ast })}}\right),$$

(17)

where $\mu (x_{\ast })$ is the mean of the Gaussian process posterior prediction, $\sigma (x_{\ast })$ is the standard deviation of the predictions. The expected improvement algorithm is then evaluated at the point with the largest expected improvement:

$$x_{n+1}=\arg \underset{x_{\ast }\in \mathcal{X}}{\max }\mathrm{A}\mathrm{c}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(x_{\ast }),$$

(18)

where $x_{n+1}$ represents input point to be selected for evaluation for the n + 1th time, $\arg \max$ represents selecting the point that maximizes the acquisition function value, $\chi$ is the search space, $\mathrm{A}\mathrm{c}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(x_{\ast })$ is the acquisition function, which measures the value of evaluating at a certain point $x_{\ast }$.

3. Experimental Analysis

3.1. Data Statistics and Processing

In this section, the present study compiles a dataset related to food delivery services. To gain a comprehensive understanding of the data’s structure and distribution, and to identify potential data issues such as missing values, outliers, and other irregularities, the study conducts a detailed statistical analysis of each variable, as presented in

Table 1. Field Quantity Statistics.

Field	Description	Count
ID	Order ID	10,000
Delivery_person_ID	Delivery driver ID	10,000
Delivery_person_Age	Delivery driver age	10,000
Delivery_person_Ratings	Customer rating of the delivery driver	10,000
Restaurant_latitude	Latitude coordinate of the restaurant’s location	10,000
Restaurant_longitude	Longitude coordinate of the restaurant’s location	10,000
Delivery_location_latitude	Latitude coordinate of the order’s delivery location	10,000
Delivery_location_longitude	Longitude coordinate of the order’s delivery location	10,000
Type_of_order	Category of the food ordered, used for preparation time analysis	10,000
Type_of_vehicle	Vehicle used for delivery	10,000
temperature	Ambient temperature during delivery	9995
humidity	Humidity level during delivery	9995
precipitation	Amount of rain or snowfall	9995
weather_description	Textual description of the weather	9995
Traffic_Level	Level of traffic congestion during delivery	9085
Distance (km)	Calculated distance (kilometers) between the restaurant and the delivery location	9080
TARGET	Target variable, representing the model’s predicted delivery time (minutes)	9459

.

As revealed in Table 1, the dataset exhibits a certain proportion of missing values across its variables. To ensure the accuracy and reliability of subsequent experimental analyses, the study presents a detailed enumeration of the missing values, as shown in

Table 2. The Count of Missing Values.

Field	Count	Percentage (%)
temperature	5	0.05
humidity	5	0.05
precipitation	5	0.05
weather_description	5	0.05
Traffic_Level	915	9.15
Distance (km)	920	9.2
TARGET	541	5.41

, in order to better understand the distribution of the missing data.

Since the missing data are textual and the proportion of missing values in this part is small, they are not suitable for interpolation filling. Therefore, excluding these samples has no significant impact on the overall distribution of the data. Furthermore, to ensure consistency within the dataset, the study eliminates these missing values to avoid any potential impact on the predictive outcomes.

For textual data, as most machine learning algorithms are unable to process raw text directly, this study employs numerical encoding to transform the text into a format that can be recognized by the model, as shown in

Table 3. Numerical Encoding.

Field	Category	Code
Type_of_order	Buffet	0
	Drinks	1
	Meal	2
Type_of_vehicle	bicycle	0
	electric_scooter	1
	motorcycle	2
weather_description	broken clouds	0
	clear sky	1
	few clouds	2
Traffic_Level	Hight	0
	Low	1
	Moderate	2

.

3.2. Experimental Environment

To guarantee the fairness and reproducibility of the experiments, the experiments were conducted on a workstation equipped with an Intel Core i7-11800H processor (Intel Corporation, Santa Clara, CA, USA), 16 GB RAM, and an NVIDIA GeForce RTX-3060 GPU with 14 GB memory (NVIDIA Corporation, Santa Clara, CA, USA).

To validate the reproducibility of the results obtained from the SymOpt-CNSVR model, its initial parameter settings are presented in

Table 4. Initial Parameter Settings.

Algorithm Name	Parameter	Initial Setting
SymOpt-CNSVR	Initial Learning Rate	Optimized by Algorithm
	Batch Size	Optimized by Algorithm
	L2 Regularization Coefficient	Optimized by Algorithm
	Maximum Number of Training Runs	50
	Learning Rate Reduction Factor	0.5
	Learning Rate Reduction Period	10
	Optimizer	Adam
	C	Optimized by Algorithm
	Gamma	Optimized by Algorithm
	Epsilon	Optimized by Algorithm
	Number of Iterations	10

.

To ensure the generalizability of the model, the dataset in this study was split into training and test sets in a 7:3 ratio.

The study employs four evaluation metrics—coefficient of determination (R²), mean absolute error (MAE), root mean square error (RMSE), and mean squared log error (MSLE)—to assess the model’s stability and accuracy. The corresponding formulas are provided in

Table 5. Evaluation Metrics.

Formulas	Explanation
$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{(y_i-{\stackrel{⌢}{y}}_i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(y_i-{\overline{y}}_i)}^2}}}$	R2 reflects the degree of correlation between the predicted values and the actual values. The closer it is to 1, the better the model’s fit.
$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n|y_i-{\widehat{y}}_i|}$	MAE (Mean Absolute Error) reflects the average deviation between the predicted values and the actual values. A smaller value indicates higher prediction accuracy of the model.
$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}$	RMSE (Root Mean Square Error) reflects the square root of the mean of the squared differences between the predicted values and the actual values. A smaller value indicates higher predictive accuracy of the model.
$MSLE(y,\widehat{y})={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(\log (1+y_i)-\log (1+{\widehat{y}}_i))}^2}$	MSLE (Mean Squared Logarithmic Error) reflects the error on a logarithmic scale. It helps mitigate the impact of large discrepancies by preventing the overemphasis of outliers when the data varies significantly. A smaller value indicates a smaller prediction error of the model.

.

3.3. Algorithm Performance Testing

In this section, ESFOA is compared with seven high-citation algorithms, including SFOA, Bermuda Triangle Optimizer (BTO) [44], Chinese Pangolin Optimizer (CPO) [45], Kepler Optimization Algorithm (KOA) [46], Improved Black-Winged Kite Algorithm (IBKA) [47], Enhanced Sea Horse Optimization (ESHO) [48], and Chaotic Mountain Gazelle Optimizer Improved by Multiple Oppositional-Based Learning Variants (HCQDOPP-MGO) [49], to demonstrate the superiority and rationality of the proposed algorithm. The function test set from IEEE CEC2022 [50] is used, the search interval is denoted as ${\left[-100,100\right]}^D$, as shown in

Table 6. CEC2022 Test Table.

	No.	Functions	${\mathit{F}}_{\mathit{i}}^{\ast }$
Unimodal function	1	Shifted and full Rotated Zakharov Function	300
Basic functions	2	Shifted and full Rotated Rosenbrock’s Function	400
	3	Shifted and full Rotated Expanded Schaffer’s $f_6$ Function	600
	4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	800
	5	Shifted and full Rotated Levy Function	900
Hybrid functions	6	Hybrid Function 1 (N = 3)	1800
	7	Hybrid Function 2 (N = 6)	2000
	8	Hybrid Function 3 (N = 5)	2200
Composition functions	9	Composition Function 1 (N = 5)	2300
	10	Composition Function 2 (N = 4)	2400
	11	Composition Function 3 (N = 5)	2600
	12	Composition Function 4 (N = 6)	2700

.

The initial population size for all swarm intelligence algorithms is set to 50, with a maximum evaluation count of 1000. To ensure fairness, 50 independent experimental runs are conducted for each algorithm. This section evaluates the performance of ESFOA using the CEC2022 (Dim = 2/10/20) benchmark test suite. In order to focus on the average performance of each algorithm under different test conditions, this subsection presents only the comparative results of the mean values obtained after running each algorithm on datasets of varying dimensions.

First, the performance of ESFOA is evaluated using the CEC2022 (Dim = 2) benchmark test suite.

Table 7. The Mean Results of Different Algorithms on the CEC2022 Test Suite (Dim = 2).

Function	ESFOA	SFOA	BTO	CPO	KOA	IBKA	ESHO	HCQDOPP-MGO
F1	319.1079	480.29	982.3148	300	367.9906	882.475	357.0258	300.0003
F2	400.0248	400.3366	400.4085	400.1223	400.0559	400.0768	400.0008	400.0057
F3	603.0041	608.7194	612.9723	600.0001	609.6699	611.3681	607.4187	600.0165
F4	801.1684	804.2182	803.4525	801.8789	802.6302	803.0522	800.5823	802.2892
F5	901.47	908.5686	902.9493	900	903.1111	900.1763	900.0348	900
F6	1800	1800	1800	1800	1800	1800	1800	1800
F7	2000	2000	2000	2000	2000	2000	2000	2000
F8	2201.718	2201.718	2201.718	2201.718	2201.718	2201.718	2201.718	2201.718
F9	2360.138	2409.278	2345.093	2420.355	2381.291	2456.745	2372.107	2390.117
F10	2427.785	2475.542	2438.678	2496.63	2460.314	2475.054	2460.386	2461.942
F11	2764.097	2848.572	2697.46	2688.423	2838.425	2715.529	2769.924	2929.522
F12	2729.684	2792.136	2796.954	2752.839	2742.455	2774.33	2782.209	2700.649

shows that ESFOA ranks first in terms of mean values on the F6, F7, and F8 hybrid functions, while it achieves a mean rank of 1 on the F10 composite function with a value of 2427.785, clearly outperforming other algorithms. This demonstrates the precise prediction capability of ESFOA in complex function environments.

Figure 3 presents the average runtime of ESFOA and other algorithms across various test functions. ESFOA is highlighted in orange, while the other algorithms are highlighted in yellow. ESFOA has an average runtime of 0.051739 s, ranking third. Notably, it shows a significant improvement compared to BTO (0.099727 s) and CPO (0.670289 s). This result indicates that ESFOA strikes a good balance between optimization performance and computational efficiency, demonstrating superior performance.

Furthermore, the performance of ESFOA is evaluated by increasing the test set dimension to 10. The mean comparison results are shown in

Table 8. The Mean Results of Different Algorithms on the CEC2022 Test Suite (Dim = 10).

Function		ESFOA	SFOA	BTO	CPO	KOA	IBKA	ESHO	HCQDOPP-MGO
F1	Mean	18,705.19019	45,428.6006	438,187.9264	803.6876464	24,248.35373	36,500.82755	12,892.41087	11,125.13063
	Std	5519.371	27650.95	1,323,020	1250.312	4863.965	7690.702	3388.615	5505.477
F2	Mean	565.4423053	1058.278909	1489.346815	406.3338772	1845.201237	1085.966493	2253.612288	412.4329987
	Std	75.21938	446.4467	1235.793	6.176012	441.4489	613.1291	939.229	20.17613
F3	Mean	645.5148773	675.6795648	658.1250773	645.2651082	670.4973776	670.5304433	665.5588955	645.5102142
	Std	10.87515	10.00441	7.757899	14.56027	14.67941	20.94238	8.189325	7.127615
F4	Mean	876.0688427	898.1642508	855.2211898	832.6346224	893.703367	900.2301954	868.4806559	856.0958679
	Std	14.2738	16.02426	4.027823	4.542487	8.21083	16.89204	10.64679	18.14969
F5	Mean	2042.104539	3038.605697	1894.203472	1504.986018	3187.691772	3236.765852	1951.222574	2015.023372
	Std	552.9362	875.7171	594.0758	140.8484	634.9104	879.9162	229.9274	567.3421
F6	Mean	57,541,020.29	233,300,677.2	430,274,328.5	4310.444628	304,057,053.7	390,781,248.9	171,578,328.4	9025.895622
	Std	39,430,455	2.86 × 108	7.35 × 108	2478.346	1.62 × 108	3.58 × 108	3.38 × 108	2736.092
F7	Mean	2115.457184	2198.051015	2155.810711	2154.30344	2146.983092	2140.407572	2157.942338	2116.015909
	Std	17.02818	48.4208	60.08971	100.7485	39.94675	30.92607	37.10217	29.91562
F8	Mean	2256.583234	2373.017682	2440.63039	2325.973827	2291.598353	2293.748276	2410.841123	2405.920927
	Std	12.79266	123.3071	229.4781	100.0627	39.41807	50.14801	114.2727	108.4196
F9	Mean	2691.158056	2842.371468	2759.755608	2554.876952	2821.471723	2830.980053	2877.885026	2620.206339
	Std	69.16598	226.6363	75.85285	52.04786	62.9376	84.92192	77.69293	53.19611
F10	Mean	2529.851816	2911.598007	2659.955911	2558.815028	2635.27378	3207.856968	2909.784838	2703.37542
	Std	54.29187	641.064	127.388	75.20515	123.8184	648.6067	292.3047	349.8366
F11	Mean	10,679.23819	30,832.89653	3700.089128	3167.708934	46,346.56892	49,636.17642	4635.058613	29,869.94737
	Std	4560.7	19,493.38	488.5566	639.5069	9873.814	19,654.55	190.5715	12,652.65
F12	Mean	2873.857064	2887.489257	3070.279093	2900.760138	3006.131615	2905.509022	3055.675105	2944.915936
	Std	3.241254	14.72067	178.6357	34.18775	43.9351	56.62088	101.9034	44.56084

.

Table 8 shows that ESFOA exhibits the best performance on the F7, F8, F10, and F12 functions, ranking first in both hybrid and composite functions. This indicates that ESFOA demonstrates high stability across these functions.

Figure 4 compares the convergence curves of ESFOA with other algorithms. The blue line represents ESFOA, which clearly demonstrates a strong ability to escape local optima in the later stages. This is a result of its self-enhancing training capability, indicating that ESFOA performs steady iterations in the early stages while exhibiting a powerful ability to escape local optima in the later stages.

Figure 5 illustrates the iterative stability of ESFOA compared to other algorithms. The blue box plot represents ESFOA, which shows a relatively small box size and a very low average fitness value, suggesting that ESFOA not only converges quickly but also exhibits a degree of stability in its search capability.

Next, the test set dimension is further increased to 20 dimensions.

As shown in

Table 9. The Mean Results of Different Algorithms on the CEC2022 Test Suite (Dim = 20).

Function	ESFOA	SFOA	BTO	CPO	KOA	IBKA	ESHO	HCQDOPP-MGO
F1	69,670.28	139,460.6	154,281.6	27,438.36	80,275.06	108,658	189,707.1	58,074.98
F2	1661.214	5898.015	3685.408	452.5394	4692.959	4506.653	3816.613	496.4041
F3	673.6049	706.5627	694.5926	659.5842	703.6169	709.9636	704.8201	672.8813
F4	1014.927	1094.113	997.5004	890.3535	1069.822	1067.724	1001.501	947.8409
F5	7104.588	11,854.94	3969.97	2920.291	9658.752	8318.398	4554.36	5718.082
F6	1.07 × 109	4.16 × 109	5.25 × 109	3038.715	4.07 × 109	3.17 × 109	2.59 × 109	414,638.6
F7	2289.718	2398.757	2300.873	2310.965	2367.315	2306.951	2309.179	2283.927
F8	2547.758	3570.951	10,697.55	2387.968	2845.255	3963.477	3473.768	2454.058
F9	2930.07	3356.926	3424.916	2510.545	3215.89	3429.896	3437.936	2594.54
F10	6534.279	7592.105	4096.087	5627.573	7123.12	7247.384	7207.4	4366.003
F11	68,912.45	154,891.4	28,511.33	2923.3	152,549.2	175,611	10,099.5	103,975.1
F12	3070.822	3317.224	3640.615	3329.902	3855.41	3247.969	3969.272	3350.177

, the ESFOA is a comprehensive and highly robust optimizer. Although it did not achieve first place on some functions, its overall performance is exceptional: it ranks first on F12, second on several challenging functions such as F2, F6, and F11, and maintains stable competitiveness across other functions. This indicates that ESFOA is not a specialized algorithm but rather a powerful tool that can be widely applied to various complex optimization problems. Especially when handling high-dimensional, multimodal, and composite functions, its balanced exploration and exploitation capabilities ensure outstanding final performance.

As shown in Figure 6, ESFOA ranks very high in the Friedman mean value for each dimension. It ranks first in the two-dimensional case and second in average, slightly lower than CPO. However, due to the very high time complexity of CPO, ESFOA has an excellent simple and efficient optimization capability, proving that the algorithm is very practical.

3.4. Model Prediction

In order to further evaluate the effectiveness of the ESFOA improvement, we brought this method into the SymOpt-CNSVR model we developed for optimization. Through the evaluation results of the training set and the test set, we can obtain the effectiveness of the model in predicting the delivery time of takeout.

Table 10. Hyperparameter Optimization Results and Ranges.

	ESOFA	SFOA	Optimization Range
Initial Learning Rate	1.25 × 10−2	5.87 × 10−3	[1 × 10−3, 1 × 10−1]
Batch Size	56	101	[8, 128]
L2 Regularization Coefficient	2.56 × 10−4	8.13 × 10−3	[1 × 10−5, 1 × 10−2]
C	22.5	22.5	[1 × 10−3, 100]
Gamma	0.0056	0.0056	[1 × 10−4, 1]
Epsilon	1.10 × 10−3	1.10 × 10−3	[1 × 10−4, 1]

shows the results and ranges of ESFOA and SFOA optimization, and the final training and testing evaluation results are shown in

Table 11. Comparison Results of Evaluation Metrics.

Optimization Method	Model		R2	MAE	RMSE	MSLE
SFOA	SymOpt-CNSVR	Training Set	0.913	3.8241	5.0067	0.1439
		Test Set	0.895	3.8646	5.0253	0.1473
ESOFA	SymOpt-CNSVR	Training Set	0.942	2.7849	4.0898	0.1025
		Test Set	0.9269	3.0582	4.1947	0.1114

.

Overall, Table 11 shows that both SFOA and ESOFA demonstrate strong predictive capabilities on the SymOpt-CNSVR model, with R² values on both the training and test sets approaching or exceeding 0.9, indicating good model fit and strong generalization. Further comparison reveals that the enhanced ESOFA optimization method significantly outperforms the basic SFOA across all evaluation metrics: its R² on the training and test sets increases by approximately 0.03 and 0.03, respectively, while its MAE, RMSE, and MSLE decrease significantly. In particular, the MAE on the training set decreases by approximately 27%, and the MSLE on the test set decreases by approximately 24%. This demonstrates that ESOFA offers significant advantages in improving prediction accuracy, stability, and error control, while avoiding significant overfitting, demonstrating enhanced optimization efficiency and model robustness.

After 10 repeated runs, the Wilcoxon test was used to verify the significance of the difference between the results of the two methods. As can be seen from

Table 12. Wilcoxon Signed-Rank Test of 2 Optimization Methods.

	ESOFA	SFOA
R2	1	3.22 × 10−3
MAE	1	8.91 × 10−4
RMSE	1	4.17 × 10−3
MSLE	1	2.05 × 10−3

, all the p-values of the four indicators of SFOA with ESFOA as the standard are less than 0.05, which is significant, indicating that there are obvious differences in the optimization results of the two methods. The results in Table 11 are reliable.

Figure 7 shows that even when the initial solution is poorly designed, ESFOA can still achieve a convergence value that exceeds the target in a short number of iterations through its powerful cruising ability. This shows that ESFOA has very strong performance in optimizing hyperparameters and is suitable for use in the problem of selecting multiple hyperparameter combinations for models in high-dimensional data prediction.

3.5. Ablation Study

Furthermore, this study will conduct ablation tests on the performance of each module of the model developed in this paper, quantitatively analyze the impact of each module on the overall performance, enhance the interpretability of the model established in this paper, and understand the module selection and importance in its establishment process.

The results presented in

Table 13. Statistical Summary of Evaluation Indicators for Different Modules.

	R2	MAE	RMSE	MSLE
CNN	0.772	5.6815	7.4057	0.2428
SVR	0.5874	7.04	9.9631	0.3074
CNN-SVR	0.8347	4.87	6.3067	0.1862
OptCNN-SVR	0.8307	4.85	6.3826	0.1828
CNN-OptSVR	0.841	4.8065	6.1843	0.1858
SymOpt-CNSVR	0.9269	3.0582	4.1947	0.1114

and Figure 8 demonstrate that SymOpt-CNSVR overwhelmingly achieves the best performance among all compared models. Its R² value, 0.9269, significantly surpasses the other models (0.77–0.84), while achieving significantly lower MAE, RMSE, and MSLE, demonstrating its superior predictive accuracy and reliability. Specifically, the SymOpt-CNSVR model achieves significant improvements in key metrics: its R² improves by over 10% compared to the next-best CNN-OptSVR model, while reducing MAE and RMSE by approximately 36% and 32%, respectively. While CNN and SVR each exhibit limited performance in isolation (R² = 0.772 and 0.5874, respectively), their integration within a unified framework yields a remarkable performance improvement. This synergy can be attributed to the complementary strengths of each module: the CNN excels at capturing complex spatial and multi-feature interactions within the data, while the SVR provides strong generalization capabilities for regression tasks based on high-level features. The significant performance gap between the individual models and the combined system underscores the importance of structural cooperation in leveraging both feature learning capacity and regression stability. Furthermore, the fact that the hybrid model (CNN-SVR and its optimized variants) does not merely average the performance of its submodules—but significantly surpasses them—suggests effective information fusion rather than error accumulation. The low MSLE value (0.1114) achieved by SymOpt-CNSVR (through symmetric optimization mechanism) further confirms that the model avoids large deviant predictions, reinforcing its practical reliability in real deployment scenarios.

3.6. Comparative Experiment

In addition to ablation studies, comparative experiments are also a crucial step in evaluating model performance. Next, we will evaluate the performance of baseline models and advanced optimization models, analyzing the predictive advantages of SymOpt-CNSVR and ensuring its superiority in the complex, high-dimensional task of predicting food delivery time.

This paper compares the predictive performance of traditional linear regression (LR), K-nearest neighbor (KNN), and advanced deep learning sequence models (GRU and BiLSTM) for this task. All results are based on the test set.

As shown in

Table 14. Statistical Summary of Evaluation Metrics for Different Baseline Models.

	R2	MAE	RMSE	MSLE
LR	0.7706	5.6603	7.4296	0.207
KNN	0.8017	4.713	6.9073	0.1969
GRU	0.7909	5.3982	7.093	0.1919
BiLSTM	0.801	5.2676	6.9194	0.1833
SymOpt-CNSVR	0.9269	3.0582	4.1947	0.1114

and Figure 9, SymOpt-CNSVR stands out among all compared prediction models, significantly outperforming traditional linear regression (LR), K-nearest neighbor (KNN), and advanced deep learning sequence models (GRU and BiLSTM), demonstrating a revolutionary improvement in predictive capabilities. Specifically, the SymOpt-CNSVR model’s superiority is evident in its significant lead across all core evaluation metrics: its coefficient of determination (R²) reaches a staggering 0.9269, an improvement of over 15 percentage points over the next-best performing KNN and BiLSTM (approximately 0.801), representing a substantial improvement in its ability to account for data variation. Its superiority in error control is even more striking, with MAE (3.0582) and RMSE (4.1947) significantly lower than those of other models. Compared to the best-performing models (MAE 4.713 for KNN and RMSE 6.9194 for BiLSTM), the error reductions reach 35% and 39%, respectively. Crucially, its MSLE indicator is only 0.1114, which is nearly 40% lower than the best value of other models (0.1833 of BiLSTM). This fully demonstrates that the model can extremely control the relative error even when predicting a large range of target values, and has unparalleled robustness and accuracy. Overall, the results strongly indicate that the symmetric optimization strategy adopted by SymOpt-CNSVR successfully integrates the advantages of different models. Its performance improvement is not a marginal improvement, but fundamentally surpasses the performance boundaries of traditional machine learning and even mainstream deep learning methods, setting a new accuracy benchmark. It also provides a more accurate model in the task of predicting the delivery time of takeout food, helping businesses and consumers to better determine the delivery process of takeout food.

Next, we will compare it with existing advanced optimization models, including BO-CNN-LSTM [51], PSO-BiGRU-BiLSTM [52], ACO-SVR-GRU [53], and IRIME-BiTCN-BiGRU-MSA [54]. These are all powerful optimization prediction models proposed in the past two years, which can effectively verify the efficiency and practicality of SymOpt-CNSVR in this paper in prediction tasks.

The comparison results of advanced optimization models in

Table 15. Statistical Summary of Evaluation Metrics for Different Optimization Models.

	R2	MAE	RMSE	MSLE
BO-CNN-LSTM	0.9195	3.1783	4.4008	0.1211
PSO-BiGRU-BiLSTM	0.7953	5.3725	7.0177	0.189
ACO-SVR-GRU	0.9249	3.0733	4.2494	0.1134
IRIME-BiTCN-BiGRU-MSA	0.8659	4.4129	5.6791	0.1711
SymOpt-CNSVR	0.9269	3.0582	4.1947	0.1114

and Figure 10 show that all the competing models utilize a hybrid architecture combining complex intelligent optimization algorithms with deep learning components. However, the SymOpt-CNSVR model stands out, achieving the best performance across most key performance metrics, firmly establishing its position as the optimal model.

SymOpt-CNSVR’s exceptional performance is reflected in its comprehensive and balanced lead. Its R² value (0.9269) is the highest among all models. While only slightly behind the similarly performing ACO-SVR-GRU model (0.9249), this demonstrates its exceptional predictive accuracy and ability to account for data variation. More importantly, in key metrics measuring prediction error, SymOpt-CNSVR demonstrates undeniable superiority: its MAE (3.0582) and RMS (3.0640) are both superior. The E (4.1947) was the lowest among all models, indicating that its predicted values had the smallest absolute deviation from the true values. Most convincingly, its MSLE (0.1114) also ranked first and was significantly lower than that of other models. This demonstrates that SymOpt-CNSVR possesses the strongest robustness and control capabilities against potentially large prediction errors, effectively avoiding extreme errors.

In summary, these comparison results strongly demonstrate that the symmetric collaborative optimization strategy employed by SymOpt-CNSVR not only rivals the performance of state-of-the-art hybrid optimization models, but also surpasses them overall. This indicates that its optimization framework is highly efficient, fully exploiting and integrating the strengths of different models, ultimately achieving the optimal combination of prediction accuracy, stability, and reliability, making it ideally suited for complex, high-dimensional food delivery time prediction tasks.

4. Conclusions

This paper proposes a new prediction framework, SymOpt-CNSVR, based on symmetric optimization, to address the problem of delivery time prediction in food delivery scenarios. This framework deeply integrates the advantages of convolutional neural networks (CNNs) in high-dimensional feature extraction with the robust performance of support vector regression (SVR) in statistical learning, achieving complementary advantages between the two models through structural symmetry. The CNN is responsible for multi-feature importance assessment and nonlinear relationship extraction, and its hyperparameters are adaptively optimized using our proposed Enhanced Superb Fairy-wren Optimization Algorithm (ESFOA). SVR further achieves accurate prediction based on this, and its parameter tuning is efficiently implemented using Bayesian optimization, significantly reducing the complexity and time cost of parameter search. At the algorithmic level, ESFOA significantly improves global search efficiency and convergence stability by introducing a group collaboration mechanism, historical memory backtracking, and a self-reinforcement learning strategy. Experimental results on the standard test set CEC2022 demonstrate that ESFOA outperforms current mainstream optimization algorithms such as BTO, CPO, KOA, IBKA, ESHO, and HCQDOPP_MGO in terms of convergence accuracy and robustness. Finally, SymOpt-CNSVR achieves exceptional results on real-world food delivery data, achieving R² = 0.9269, MAE = 3.0582, RMSE = 4.1947, and MSLE = 0.1114, comprehensively surpassing multiple baseline and cutting-edge optimization models. Notably, the model achieves a prediction MAE of only about 3 min, approaching the sub-1 min error target that is critical for high-precision logistics systems. This capability allows the platform to provide customers with highly reliable delivery time estimates, thereby significantly enhancing user trust and operational coordination efficiency. It not only validates the effectiveness and advancement of the proposed framework in handling high-dimensional spatiotemporal prediction tasks but also demonstrates its substantial practical impact in real intelligent dispatch environments.

The proposed method provides a path forward for further improving prediction performance and its broad application scope. Future work will focus on exploring more expressive Transformer architectures to capture more complex spatiotemporal dependencies and promote the universal application of ESFOA to a wider range of optimization tasks. At the same time, an online learning mechanism is developed to enhance the dynamic adaptability of the model. The ultimate goal is to integrate the framework into the actual intelligent scheduling system to achieve a closed-loop transformation from academic innovation to industrial value.