2.1 Sparrow search algorithm

The SSA is a recently developed swarm intelligence optimisation technique proposed by Xue et al. [Reference Xue and Shen21] in 2020. The algorithm models the foraging and anti-predatory behaviour of sparrows and categorises them into three types: finders, joiners and detectors, each with their corresponding behavioural rules. The algorithm assumes that the search space has $d$ dimensions and there are $n$ sparrows in the swarm.

The position of the finder is updated as shown in Equation (1):

(1) \begin{equation}X_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}{l}}{X_{i,j}^t \cdot {\rm{exp}}\left( {\frac{{ - i}}{{\alpha \cdot T}}} \right),{R_2} \lt ST}\\[9pt]{X_{i,j}^t + Q \cdot L,{R_2} \geqslant ST}\end{array}} \right.\end{equation}

where $t$ is the current number of iterations; $T$ is the maximum number of iterations; $X_{i,t}^t$ is the current sparrow position; $\alpha $ is a random number between $\left( {0,1} \right]$ ; $Q$ is a random number that fits the normal distribution; $L$ is the unit matrix of $1 \times d$ ; ${R_2}$ is the alert value; and $ST$ is the safety value.

The position of the joiner is updated as shown in Equation (2):

(2) \begin{equation}X_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}{l}}{Q \cdot {\rm{exp}}\left( {\frac{{{X_{{\rm{worst}}}} - X_{i,j}^t}}{{{i^2}}}} \right),{\rm{\;\;\;\;}}i \gt \frac{n}{2}}\\[10pt]{X_p^{t + 1} + |X_{i,j}^t - X_p^{t + 1} \cdot {A^ + } \cdot L,{\rm{otherwise}}}\end{array}} \right.\end{equation}

where ${X_{{\rm{worst}}}}$ denotes the global worst position; ${X_p}$ denotes the best position occupied by the finder; $A$ denotes a matrix of $1 \times d$ where each element is randomly assigned a value of 1 or −1 and ${A^ + } = {A^{\rm T}}{\left( {A{A^{\rm T}}} \right)^{ - 1}}$ .

The detectors make up 10%–20% of the total number of sparrows and their position is updated as shown in Equation (3):

(3) \begin{equation}X_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}{l}}{X_{{\rm{best}}}^t + \beta \cdot \left| {X_{i,j}^t - X_{{\rm{best}}}^t} \right|,{f_i} \gt {f_g}}\\[10pt]{X_{i,j}^t + K\left( {\dfrac{{\left| {X_{i,j}^t - X_{{\rm{worst}}}^t} \right|}}{{\left( {{f_i} - {f_w}} \right) + \varepsilon }}} \right),{f_i} = {f_g}}\end{array}} \right.\end{equation}

where $X_{{\rm{best}}}^t$ denotes the current global optimum position; $\beta $ is the step control parameter for a normally distributed random number obeying a mean of 0 and a variance of 1; $K$ is a random number between −1 and 1; ${f_i}$ denotes the current sparrow’s fitness value; ${f_g}$ and ${f_w}$ denote the current global optimum fitness and worst fitness, respectively; and $\varepsilon $ is the smallest constant to avoid zero in the denominator.

2.2 Improved approach for sparrow search algorithm

The limitation of the SSA algorithm in tackling complex optimisation problems is its low convergence accuracy and the tendency to converge to a local optimum. To counteract these drawbacks, ISSA was introduced to enhance the global optimisation capabilities of the algorithm. Three key improvements were made to the SSA algorithm: (1) the initial population was optimised using the tent chaotic mapping; (2) dynamic adaptive weights were integrated into the finder to balance the algorithm’s global search and local exploitation ability; (3) the detector was enhanced with a Levy flight strategy to disrupt the current optimal solution and reinforce the local search ability.

2.2.1 Tent chaos map

Chaotic mapping is a term used to describe the random motion that arises from a deterministic equation. This motion exhibits both periodicity and inherent randomness within a phase space. By introducing chaotic mapping to an algorithm, the diversity of the initial population can be increased, leading to improved optimisation capabilities. Compared to logistic chaos, the tent chaotic map demonstrates more pronounced chaotic features [Reference Jiang, Yang and Huang22]. Therefore, the tent chaotic map was utilised in this study to initialise the population, generating an initial population with high diversity. The equation for the tent chaotic map is provided in Equation (4):

(4) \begin{equation}{y_{i + 1}} = \left\{ {\begin{array}{*{20}{l}}{\dfrac{{{y_i}}}{a},0 \le {y_i} \le a}\\[10pt]{\dfrac{{1 - {y_i}}}{{1 - a}},a \lt {y_i} \le 1}\end{array}} \right.\end{equation}

where $a \in \left( {0,1} \right)$ ; $i$ is the number of iterations.

In the tent mapping formula, $a$ is generally taken to be 0.5. The tent mapping is used to generate a chaotic sequence matrix to initialise the sparrow population, as shown in Equation (5):

(5) \begin{equation}{X_i} = {X_{lb}} + \left( {{X_{lb}} - {X_{ub}}} \right) \times {y_i}\end{equation}

where ${X_{lb}}$ and ${X_{ub}}$ are the upper and lower bounds for each individual in each dimension and ${X_i}$ is the mapped individual.

2.2.2 Dynamic adaptive weights

This study introduces dynamic adaptive weights for the finder to enhance the global search capability and convergence rate of SSA. This is done to mitigate the likelihood of an individual falling into a locally optimal solution due to insufficient search capability when it is near the optimal solution. The dynamic adaptive weights are represented mathematically as $w$ and are illustrated in Equation (6):

(6) \begin{equation}X_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}{l}}{X_{i,j}^t \cdot w \cdot {\rm{exp}}\left( {\dfrac{{ - i}}{{\alpha \cdot T}}} \right)} {,{R_2} \lt ST}\\[14pt]{X_{i,j}^t + Q \cdot L} {,{R_2} \geqslant ST}\end{array}} \right.\end{equation}

The dynamic adaptive weights $w$ equation is shown in Equation (7):

(7) \begin{equation}w = 1 - \frac{{{e^{\left( {t/{T_{{\rm{max}}}}} \right)}} - 1}}{{e - 1}}\end{equation}

where $t$ is the current number of iterations and ${T_{{\rm{max}}}}$ is the maximum number of iterations.

Implementing dynamic adaptive weights enables adaptive regulation of the finder’s position. As depicted in Fig. 1, higher initial values of $w$ correspond to a larger search range for the algorithm. Towards the end of the iteration, lower values of $w$ facilitate the algorithm’s local development.

Figure 1.

Dynamic adaptive weight change curves.

2.2.3 Levy flight strategy

Levy flight [Reference Deepa and Venkataraman23] is a unique random wandering model that characterises motion in terms of random step lengths and directions. In Levy flight, these step lengths and directions are random and possess a long-tailed distribution, occasionally including extreme values. Adding the Levy flight strategy to the detector can now perturb the optimal solution and prevent getting trapped in a locally optimal solution. The revised formulation of the detector is given by Equation (8):

(8) \begin{equation}X_{i,j}^{t + 1} = \left\{ {\begin{array}{*{20}{l}}{Levy\left( \lambda \right) \cdot X_{{\rm{best}}}^t + \beta \cdot \left| {X_{i,j}^t - Levy\left( \lambda \right) \cdot X_{{\rm{best}}}^t} \right|,{f_i} \gt {f_g}}\\[9pt]{X_{i,j}^t + K\left( {\dfrac{{\left| {X_{i,j}^t - X_{{\rm{worst}}}^t} \right|}}{{\left( {{f_i} - {f_w}} \right) + \varepsilon }}} \right),{f_i} = {f_g}}\end{array}} \right.\end{equation}

The symbol $Levy\left( \lambda \right)$ denotes the random number that follows the Levy distribution. To calculate $Levy\left( \lambda \right)$ , the Mantegna algorithm can be used, as shown in Equation (9).

(9) \begin{equation}S = \frac{\mu }{{|v{|^{1/\tau }}}}\end{equation}

where $s$ is the Levy flight stage length; $\tau = 1.5$ ; $\mu $ and $v$ obey a normal distribution, $\mu \sim N\left( {0,\sigma _\mu ^2} \right)$ , $v\sim N\left( {0,\sigma _v^2} \right)$ , $\sigma _v^2 = 1$ ; $\sigma _\mu ^2$ can be calculated from Equation (10).

(10) \begin{equation}{\sigma _\mu } = \left\{ {\frac{{{\rm{\Gamma }}\left( {1 + \tau } \right) \cdot {\rm{sin}}\left( {\pi \cdot \tau /2} \right)}}{{{\rm{\Gamma }}\left[ {\left( {1 + \tau } \right)/2} \right] \cdot \tau \cdot {2^{\left( {\tau - 1} \right)/2}}}}} \right\}\end{equation}

where ${\rm{\Gamma }}$ is the gamma function.

The detailed steps of ISSA are presented in pseudocode form in Algorithm 1.

Algorithm 1

The framework of the ISSA.

2.3 Benchmark function experiments

To evaluate the effectiveness of ISSA in solving the optimisation problems presented in this paper, six test functions were simulated and are listed in Table 1. Among them, ${F_1}$ and ${F_2}$ were unimodal functions, ${F_3}$ and ${F_4}$ were multimodal functions, and ${F_5}$ and ${F_6}$ were fixed-dimensional functions. The performance of ISSA was compared with that of standard SSA, GWO and PSO using a population size of 30 and 300 iterations for all algorithms.

Table 1.

Benchmarking functions

The simulation experiments were conducted in Python 3.9. The parameters of the four algorithms were set as presented in Table 2, and 30 independent experiments were performed for each function to eliminate any chance bias. The optimal values, average values and standard deviation (STD) of the results are presented in Table 3. To better visualise the optimisation performance of the algorithms, the convergence curves of ISSA, SSA, GWO and PSO for the six tested functions are included in Fig. 2.

The analysis presented in Table 3 indicates that the ISSA algorithm outperforms other algorithms regarding the optimal solutions for unimodal and multimodal functions. For the unimodal functions, the optimal solutions obtained by the ISSA algorithm are all zero, whereas other algorithms produce larger values. Similarly, the ISSA algorithm’s optimal solutions for the multimodal functions are also all zero, while other algorithms produce larger values. This demonstrates that the ISSA algorithm has higher accuracy and more robust global search capability. Regarding fixed dimensional functions, the ISSA algorithm performs comparably to other algorithms, but its mean and standard deviation are relatively small, suggesting better stability. The ISSA algorithm performs well in solving optimisation problems and exhibits significant advantages, especially in unimodal and multimodal functions.

Table 2.

Algorithm parameter setting

Table 3.

Benchmark function simulation results

Figure 2.

Benchmark function convergence curves.

In Fig. 2, the convergence curves for the six test functions are displayed. It is evident that ISSA outperforms the other optimisation algorithms in terms of accuracy and convergence speed, except for ${F_5}$ . The superiority of ISSA is not as apparent in ${F_5}$ because the test functions are not very intricate, and thus all the algorithms can find optimal solutions. Nonetheless, ISSA is still more efficient in this case.

3.1 Multilayer Perceptron

The Multilayer Perceptron [Reference Zhou, Moayedi and Bahiraei24] is a feed-forward neural network architecture, as depicted in Fig. 3. Comprising an input layer, multiple hidden layers and an output layer, it acquires the mapping association between the input and output signals by applying a nonlinear transformation to the input signal. The formula for its forward propagation is expressed as Equation (11).

(11) \begin{equation}{h^{\left( l \right)}} = f\left( {{W^{\left( l \right)}}{h^{\left( {l - 1} \right)}} + {b^{\left( l \right)}}} \right)\end{equation}

where ${h^{\left( l \right)}}$ denotes the output of the layer $l$ neuron, ${W^{\left( l \right)}}$ denotes the connection weight of the layer $l$ neuron, and ${b^{\left( l \right)}}$ denotes the bias of the layer $l$ neuron. $f$ denotes the nonlinear activation function.

Figure 3.

Multilayer Perceptron model.

Every neuron within the MLP is a fundamental processing unit that obtains input from a neuron in the previous layer and produces an output value. The interconnections between these neurons are characterised by connection weights, which may be deemed parameters within the learning model. During MLP training, a backpropagation algorithm is utilised to determine the gradient of each connection weight, thereby reducing the loss function of the training sample. The loss function, represented by Equation (12), is minimised. Theoretically, a neural network can conform to any function [Reference Hashem Samadi, Ghobadian and Nosrati25]. However, the neural network’s architecture significantly affects the model, and the hyperparameters to be adjusted are the number of neurons per layer, the learning rate and the batch_ size.

(12) \begin{equation} L(y, \hat{y}) = \frac{1}{n}\sum^{n}_{i=1} l(y^{(i)}, \hat{y}^{(i)})\end{equation}

where $l$ represents the loss value of each sample, ${y^{\left( i \right)}}$ is the true value and is the predicted value.

3.2 ISSA-MLP prediction model

When creating a neural network model, selecting appropriate hyperparameters is essential as they directly impact the model’s predictive performance. Conventional methods of hyperparameter search, such as grid search or random search, are typically employed to test different combinations of hyperparameters. However, these methods have several limitations, including a vast search space and high computational expense. To enhance the prediction accuracy of MLP, the ISSA algorithm is utilised to explore the hyperparameters of the MLP model, which consist of the number of neurons in the first and second layers, the probability of randomly dropping neurons (dropout), and the data block size for batch training (batch_size). The flow diagram for the constructed ISSA-MLP model is presented in Fig. 4.

Figure 4.

Flow chart of the ISSA-MLP prediction model.

4.1 Ground delay programs

The GDP initiation process is presented in Fig. 5. Before GDP commencement, traffic flow management (TFM) issues an initial GDP plan to the affected airport based on GDP operating parameters such as GDP start time, end time and geographical scope. During a GDP, the ground waiting time corresponds to the duration between the estimated time of departure (ETD) and the controlled time of departure (CTD).

Table 4.

Weather and traffic variables description

Figure 5.

GDP occurrence process.

There are three primary reasons for delays caused by GDP, the first being airspace congestion. When airspace is congested, flights may need to wait a certain period before entering the take-off and landing sequence. Secondly, weather conditions may result in delayed or canceled take-offs and landings, leading to further delays. Finally, ATC restrictions may lead to postponed take-off and flight landing times.

4.2 Data set

This study used flight operation data and meteorological aerodrome report (METAR) messages obtained from the International Nanjing Lukou Airport (ICAO four-character code: ZSNJ) between January and June 2021. Each flight data included details on the scheduled departure time, actual departure time and planned landing time. Based on the literature [Reference Liu, Liu and Hansen13], the number of scheduled arrival flights and the number of scheduled departure flights were selected as traffic characteristics. Furthermore, considering that the actual take-off and landing flights are also essential factors for GDP implementation, the actual number of arriving and departing flights was also included in the traffic characteristics.

The GDP records hourly are sourced from the logbooks of the Jiangsu ATC branch. These records furnish information on the GDP commencement and cessation times, based on which the instances of departure flight delays during GDP implementation are tabulated. The ultimate data features constructed are explicitly outlined in Table 4. This paper employs a partitioning scheme to allocate 20% of the initial dataset as a distinct test dataset. The remaining data is then divided into training and validation datasets utilising a five-fold cross-validation method.

We acquired 117,467 flight data and tallied the hourly departure flight delays during the GDP implementation period. The delay time distribution is illustrated in Fig. 6, indicating that most delays are within 25 minutes.

Figure 6.

Delay distribution.

4.3 Data standardisation

Table 4 presents the data features used in this study. However, due to the differences in unit magnitudes among the features and the neural network’s sensitivity to data values, direct analysis of the raw data is not possible. Therefore, we normalised the data using Equation (13) to transform the data into the range of [0,1] during the model training process.

(13) \begin{equation}{x^{\rm{*}}} = \frac{{x - {x_{{\rm{min}}}}}}{{{x_{{\rm{max}}}} - {x_{{\rm{min}}}}}}\end{equation}

4.4 Performance measures

In assessing the model’s prediction performance, mean absolute error (MAE) and root mean squared error (RMSE) are used to assess the overall prediction accuracy of the model. The equations for these two metrics are shown in Equations (14) and (15).

(14) \begin{equation}MAE = \frac{1}{n}\sum_{i=1}^{n} |\hat{y}_{i} - y_{i}| \end{equation}

(15) \begin{equation}RMSE = \sqrt{\frac{1}{n}\sum_{i=1}^{n} (\hat{y}_{i} - y_{i})^2} \end{equation}

where $\hat{y}_i$ is the predicted value, ${y_i}$ is the true value and $n$ is the length of the data.

5.1 Comparison of prediction results of different models

To analyse MLP’s performance in predicting departure flight delays, we compared it with other models such as support vector regression (SVR) [Reference Hazarika, Gupta and Natarajan26], extreme learning machine (ELM) [Reference Zhou, Moayedi and Bahiraei24] and extreme gradient boosting tree (XGBoost) [Reference Chen and Guestrin27]. We used MAE and RMSE as the performance evaluation metrics for all models. To ensure fairness, we used grid search [Reference Fayed and Atiya28] to select hyperparameters for all models, and the grid search results are shown in Table 5.

Table 5.

Grid search range

According to the optimal hyperparameters specified in Table 5, the model’s training outcomes are depicted in Table 6. Upon examination of Table 6, it becomes evident that the MLP model exhibits superior predictive capabilities compared to the alternative regression models, as evidenced by its minimal MAE and RMSE values on both the training and test datasets. Following the MLP model, XGBoost and ELM models demonstrate relatively good performance. In contrast, the SVR model exhibits inferior results on both the training and test datasets, with higher MAE and RMSE values than the other models. This discrepancy indicates the SVR model’s inadequacy in effectively fitting the provided data.

Table 6.

Prediction performances of different models

5.2 Comparison of hyperparametric merit search

The Python 3.9.0 programming language was utilised to write the simulation environment in this study, operating under the win10 64-bit system, and implemented the Keras 2.11.0 deep learning framework. The computer had an Intel core 710,700 processor, 2.92 Hz central frequency, and 16GB of RAM. ISSA was utilised to optimise the hyperparameters of the MLP model [Reference Zhang, Liu and Yan29] by defining a search range for each hyperparameter to cover the hyperparameter search space thoroughly. The fitness of the MLP model was assessed using the MAE metric, the Adam algorithm was used as an optimiser, the learning rate was set to 0.001 and the number of model iterations was 300. Table 7 depicts the optimisation range of model hyperparameters and the four optimisation algorithms acquired for optimal hyperparameters.

The PSO, GWO and SSA optimisation outcomes were subjected to a comparative analysis, as depicted in Fig. 7. The PSO exhibited a sluggish convergence rate and produced the least optimal fitness value compared to the other three optimisation algorithms. On the other hand, the SSA algorithm displayed the initiation of convergence after the 13th generation. In contrast, the GWO algorithm exhibited a faster convergence rate than the SSA and achieved a better fitness value. Notably, the ISSA exhibited the most optimal fitness value despite having a comparatively slower convergence point, thus underscoring the superior prediction accuracy of the ISSA-MLP model.

The results presented in Fig. 8 indicate that the ISSA-optimised MLP model outperforms the models optimised using PSO, GWO and standard SSA algorithms in terms of MAE and RMSE. Specifically, the MAE and RMSE of the ISSA-MLP model are 16.8 and 24.2, respectively, the smallest among the four models. The ISSA algorithm is more effective in optimising the MLP model than the standard SSA algorithm, as it improves the accuracy and speed of the population search. Overall, the ISSA-MLP model can provide a better decision basis for the relevant stakeholders in predicting the departure flight delays generated by GDP.

Table 7.

Range of adjustment parameters and results of the optimisation algorithm

Figure 7.

Fitness change curves.

Figure 8.

Prediction errors for four optimisation methods.

To assess the significance of the predictive performance disparity between ISSA-MLP and MLP, the Wilcoxon rank sum test was employed by comparing the rank sums of the two data sets. The evaluation focused on RMSE and MAE values obtained through five-fold cross-validation of the models. The significance level for this experiment was set at $\alpha $ = 0.05. If P < 0.05, it indicates a statistically significant difference between the experimental outcomes of the two algorithms; conversely, if P > 0.05, the disparity is deemed insignificant.

Our analysis calculated the P-values for RMSE and MAE as 0.043 and 0.039, respectively. Since both P-values are less than 0.05, it demonstrates a statistically significant difference between ISSA-MLP and MLP in terms of predictive performance.

5.3 Feature importance analysis

The intricate nature of MLP models hinders the ability to comprehend the relationship between input features and output outcomes, resulting in a lack of explanatory insights for prediction results. These models are commonly referred to as “black box” models. To address this limitation, we employ SHAP (Shapley additive explanations) to elucidate the prediction outcomes concerning the variables.

SHAP, proposed by Lundberg et al. in 2017 [Reference Lundberg and Lee30], leverages the concept of Shapley values derived from game theory. It serves as a method to explain the prediction results of machine learning models by evaluating the individual contributions of each feature towards the model’s prediction outcomes.

The significance of a feature can be determined by calculating the sum of the absolute SHAP values for each feature and then averaging them. Figure 9 illustrates the importance of the features based on this methodology. The results indicate that time-related features, such as the day of the month and the hour of the day, hold the highest importance in delay prediction. This observation suggests a strong correlation between flight delays and temporal factors. Subsequently, the weather feature of cloud base height emerges as the next most influential, as it impacts visibility and can lead to prolonged flight delays under reduced to severe visibility conditions. All the remaining variables contribute to the predicted outcome to a similar degree.

Figure 9.

SHAP feature importance.

5.4 Analysis of ISSA-MLP prediction results

The optimal hyperparameters for the MLP model were obtained from Section 5.2 and subsequently integrated into the model for training. Figure 10 depicts the loss variation on the training and test sets during the MLP’s training process. We carried out 300 epochs of experimentation and observed that the loss on the test set attained a state of convergence at the 100th epoch, followed by stable fluctuations in the subsequent stages, ultimately stabilising between 0.09 and 0.1. The loss gradually achieved convergence in the training set, reaching a stable state at the 200th epoch. These results indicate that the iterative training process of the MLP is rational and does not exhibit any significant overfitting.

Figure 10.

Changes in training loss for ISSA-MLP.

As per Table 5, the ISSA-MLP model exhibits the highest prediction accuracy, with MAE and RMSE values of 16.8 and 24.2, respectively. To visually demonstrate the predictive performance of the ISSA-MLP model, Fig. 11 portrays the predictive performance of select test sets. From the figure, it can be discerned that the prediction errors for most points fall within an acceptable range, albeit for some points with significant changes, the prediction effect falls short of the ideal. The probable reason for this situation is twofold. Firstly, the number of extreme test points is relatively small, and thus the model cannot fully capture the coupling relationship and change characteristics of variables. Secondly, the reasons for flight delays are multifarious, and the features discussed in this paper need to be more comprehensive, which negatively affects the model’s learning and prediction performance. Therefore, enhancing the model’s generalisation ability will be the focal point of the following research stage.

Figure 11.

Comparison between predicted and true values of ISSA-MLP model.