Hybrid Grey Wolf Optimization-Based Gaussian Process Regression Model for Simulating Deterioration Behavior of Highway Tunnel Components

Abstract

Highway tunnels are one of the paramount infrastructure systems that affect the welfare of communities. They are vulnerable to higher limits of deterioration, yet there are limited available funds for maintenance and rehabilitation. This state of circumstances entails the development of a deterioration model to forecast the performance condition behavior of critical tunnel elements. Accordingly, this research paper proposes an integrated deterioration prediction model for five highway tunnel elements, namely, cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade. The developed deterioration model is envisioned in two fundamental components, which are model calibration and model assessment. In the first component, an integrated model of Gaussian process regression and a grey wolf optimization algorithm (GWO-GPR) is introduced for deterioration behavior prediction of highway tunnel elements. In this regard, the grey wolf optimizer is exploited to improve the prediction accuracies of the Gaussian process through optimal estimation of its hyper parameters and to automatically interpret the significant deterioration factors. The second component involves three tiers of performance evaluation comparison, statistical significance comparisons, and consolidated ranking to assess the prediction accuracies of the developed GWO-GPR model. In this regard, the developed model is validated against six widely acknowledged machine learning models, which are back-propagation artificial neural network, Elman neural network, cascade forward neural network, generalized regression neural network, support vector machines, and regression tree. Results demonstrate that the developed GWO-GPR model significantly outperformed other deterioration prediction models in the five tunnel elements. In cast-in-place tunnel liners it accomplished a mean absolute percentage error, mean absolute error, root mean square percentage error, root relative squared error, and relative absolute error of 1.65%, 0.018, 0.21%, 0.018, and 0.147, respectively. In this context, it was inferred that the developed GWO-GPR model managed to reduce the prediction errors of the back-propagation artificial neural network, Elman neural network, and support vector machines by 84.71%, 76.91%, and 69.6%, respectively. It can be concluded that the developed deterioration model can assist transportation agencies in creating timely and cost-efficient maintenance schedules of highway tunnels.

1. Introduction

Infrastructure systems in the United States are prone to large levels of deterioration and require investment. According to the American Society of Civil Engineers [1], the condition status of transit systems is “D”, which corresponds to poor condition. In addition, it has been revealed that a rehabilitation backlog of USD 90 billion is exhibited. Highway tunnels are fundamental underground structures that significantly contribute to the prosperity of society. It is reported that there are 350 highway tunnels in the United States, approximately 40% of which are more than 50 years old, and 5% exceeded the service life of 100 years [2]. As such, the criticality of highway tunnels coupled with their old age and the increase in traffic volumes necessitate efficient monitoring of the structural condition of highway tunnels over time to ensure their timely maintenance, repair, and rehabilitation.

Deterioration modeling is a vital analytical component of infrastructure asset management because it captures and predicts the future performance condition of infrastructure components as a function of time [3]. In the recent years, artificial intelligence techniques have been successfully implemented in maintenance management [4,5]. However, these techniques are of high data-demanding character due to the significant data needed in building, learning, and validating a reliable prediction model [6,7]. In view of the aforementioned, the primary objectives of the present research paper lie in the following:

• Develop a hybrid Gaussian process regression–grey wolf optimization model for simulating the deterioration process of a set of highway tunnel components.
• Validate the developed deterioration model against a set of widely used machine learning models using performance evaluation and statistical comparisons.

2. Literature Review

Few studies were reported in the literature that tackled the deterioration modeling of highway tunnels. Yamany and Elwakil [8] proposed a probabilistic model for predicting the performance condition of cast-in-place tunnel liners. Ordered-probit modeling was utilized to determine the probabilities that cast-in-place liners were in a designated condition state. It was also used to measure the marginal effects of explanatory variables on the probabilities of condition states. It was inferred that the developed model could achieve a mean absolute percentage error of 29.91%. It was also urged that interstate functional class is the only explanatory variable that was positively correlated with the condition of cast-in-place tunnel liners. Hassan and Elwakil [9] introduced an operational-based stochastic model for forecasting the future condition rating of highway tunnels. Clustering analysis was first utilized to categorize the input dataset into groups with similar features. Regression modelling was then incorporated to express the relationship between the tunnel condition and five independent variables of service of tunnel, ground condition, roadway width, annual average daily traffic, and annual average daily truck traffic. Sensitivity analysis was carried out using Monte Carlo simulation to quantify the significance of the input variables on the performance condition of highway tunnels. It was inferred that the developed model could achieve an average validity percentage, root-mean squared error, and mean absolute error of 83%, 0.066, and 0.046, respectively. It was also inferred that roadway width, annual average daily traffic, and annual average daily truck traffic showed a negative trend with the predicted tunnel condition.

Several models targeted the performance condition assessment of highway tunnel elements. Some of these models relied on digital imaging technologies for evaluating the condition of surface defects. Hou et al. [10] presented an image-based model for the recognition of cracks in tunnel linings using a residual U-net network. The left partition of the network was an encoding path and the right partition was the decoding path. The developed model outperformed the typical U-net network, yielding a pixel accuracy, intersection over union, and dice coefficient of 98.67%, 54.65%, and 68.09%, respectively. Li et al. [11] introduced a computer vision-based model for detecting cracks in highway tunnels. A deep convolutional encoder–decoder was designed that comprised the use of Unet architecture for pixel-wise semantic segmentation of cracks. A spatial constraint was added to ensure the crack continuity in eight neighbor crack maps. The developed deep learning model managed to obtain a recall, precision, and F-measure of 0.922, 0.892, and 0.907, respectively.

Miao et al. [12] proposed a deep learning-based model for the automated recognition of cracks and spalls in highway tunnels. An improved U-net architecture that included the combination of squeeze and excitation and ResNet blocks was introduced to deliver details of low-level finer details to high-level semantic features. An under-sampling strategy was incorporated to deal with the problem of an imbalanced dataset. The developed network was able to perform better than a Gabor filter, multi-scale deep convolutional neural network, fully convolutional network, and convolutional U-net by yielding dice coefficients of 93.4% and 82.91% in cracks and spalls, respectively. Xue and Li [13] proposed a region-based fully convolutional neural network for the detection of defects in tunnel linings. In their network, two loss functions of bounding box regression and softmax were utilized to probabilistically identify the type and location of defects. It was highlighted that the proposed network managed to accomplish a detection rate and detection accuracy of 94.4% and 86.6%, respectively.

Other studies exploited ground-penetrating radar to analyze components of tunnels. Cao et al. [14] studied the use of ground-penetrating radar in detecting grouting quality. A set of pre-processing procedures was carried out, including trace stacking, point stacking, digital filtering, and deconvolution. By studying the radar images, the thickness of the lining concrete and the separation between the lining concrete and the underground rock mass was able to be interpreted. Xisheng et al. [15] evaluated the thickness and voids of tunnel lining using ground-penetrating radar. The acquisition was carried out using an 800 MHZ shielded antenna, and pre-processing of radar images was performed to diminish noise and improve the signal-to-noise ratio. It was argued that ground-penetrating radar could be successfully utilized in damage detection and quality detection of tunnels.

In another branch of studies, Ye et al. [16] investigated defects in highway tunnels based on statistical analysis. The defects were classified based on their risk, visibility, correlations, progressive nature, and regional characteristics. It was revealed that cracks and water leakage are the most important defects, constituting 81% and 69%, respectively. In addition, it was urged that a large portion of the tunnels experienced three or more defects. Zhang et al. [17] studied and classified the main causes of cracks in highway tunnel linings. It was inferred that circumferential cracks are the most dominant type of cracks, representing 34% of the total number of cracks. It was also found that longitudinal cracks exhibit the highest influence on tunnel safety, followed by inclined cracks.

In light of the previous studies, it can be observed that most of them tackled detection and classification of defects like cracks, spalls, and voids. In this regard, there is a lack of deterioration models that can simulate and monitor the future performance condition of highway tunnel elements over time. It is also noted that the reported deterioration models did not look into the performance condition of some important tunnel elements like interior walls, portals, ceiling slabs, and slabs on grade despite them having a significant impact on the integrity of the entire highway tunnel system. In this regard, the absence of deterioration models for these elements may lead to inaccurate and impractical budget assignment models of highway tunnels. In addition, it can be also remarked that some of the deterioration models predicted the future condition rating of the whole tunnel as one single unit. However, this may be misleading because different components of highway tunnels follow different deterioration trends and failure is usually encountered when one of the critical components of a highway tunnel reaches critical stages. Therefore, monitoring the deterioration of highway tunnels need to be carried out at the element level. Furthermore, it is noted that some of the developed models overlooked important explanatory variables that influence the deterioration behavior of highway tunnels, such as tunnel length, tunnel shape, number of lanes, and functional classification. These variables need to be studied to understand their influence on the deterioration behavior of highway tunnels, and ignoring them may detrimentally implicate the reliability of the created deterioration models. Another shortcoming is that regression-based deterioration models fail to capture the relationships between the influential deterioration factors and future condition rating over a long-term planning horizon, particularly when taking into consideration the complex deterioration mechanism of highway tunnels [18,19].

It is also worth mentioning that some defect assessment models rely on manual tuning approaches to calibrate the hyper parameters of the machine learning and deep learning algorithms used. Hyper parameters of deep learning algorithms may include the number of convolutional layers, filter size, stride size, number of filters, type of pooling operation, etc. With regards to the Elman neural network, its hyper parameters encompass the type of transfer function, number of hidden neurons, number of context neurons, number of hidden layers, and number of context layers. Manual tuning is a challenging and time-consuming process that can cause sub-optimal solutions, inferior prediction performance, and expensive computational effort [20,21].

3. Research Framework

The ultimate objective of this research paper is to establish an integrated model for simulating the deterioration behavior of five critical elements of highway tunnels. These elements encompass cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade. In this context, a separate model is constructed for each tunnel element. The framework of the developed tunnel deterioration model is composed of two main components, namely, model calibration and model assessment (see Figure 1). Several manuals and standards were reviewed to identify the potentially relevant factors to tunnel deterioration [22,23,24]. The data used in this research paper were retrieved from the national tunnel inventory (NTI) database created by the Federal Highway Administration [25]. The inspection work of the tunnel elements was carried out as per the requirements and specifications of the national tunnel inspection standards (NTIS) [22].

Table 1. Descriptions of prospective deterioration factors and output condition state [22].

Variable	Description
Condition state	Condition state can be good (condition state 1), fair (condition state 2), poor (condition state 3), or severe (condition state 4).
Age	Age in years since the construction of the tunnel was completed.
Number of lanes	Number of highway traffic lanes that are carried out by the tunnel.
Annual average daily traffic	Annual average daily traffic for the inventory route.
Annual average daily truck traffic	Annual average daily truck traffic for the inventory route.
Detour length	Total additional travel for the vehicle resulting from closing the tunnel.
Service in tunnel	Service can be highway (1); highway and railroad (2); highway and pedestrian (3); highway, railroad, and pedestrian (4); or other (5).
NHS designation	NHS stands for the National Highway System. It can be an inventory route on the National Highway System (1) or not on the National Highway System (0).
STRAHNET designation	STRAHNET stands for the Strategic Highway Network. STRAHNET is a structure of interstate and principal highways connected to United States military installations. STRAHNET designation can be 1 if the inventory route is a STRAHNET route or 0 if it is not.
Functional classification	Functional classification can be interstate (1), principal arterial (2), freeways and expressways (3), principal arterial (4), major collector (5), minor collector (6), or local (7).
Tunnel length	It is length of the tunnel measured along the centerline of the roadway.
Roadway width	It is the most restrictive minimum distance between curbs or rails on the tunnel roadway.
Hazardous material restriction	It can be 1 if the tunnel has hazardous material or 0 if it does not.
Number of bores	It denotes number of bores in the tunnel.
Tunnel shape	The type of tunnel shape can be oval (1), horseshoe (2), rectangular (3), or circular (4).
Ground conditions	The type of ground conditions can be soil (1), rock (2), or mixed face (3).
Complexity	A complex tunnel is distinguished by the presence of advanced functional systems or structural components. The complexity can be either 1 if the tunnel is complex or 0 if it is not.

reports descriptions of the condition state and prospective 16 factors affecting tunnel deterioration. The highway tunnel elements are evaluated based on four levels of condition state of good, fair, poor, and severe. The studied deterioration factors encompass age, number of lanes, annual average daily traffic, annual average daily truck traffic, detour length, service in tunnel, NHS designation, STRAHNET designation, functional classification, tunnel length, roadway length, hazardous material restriction, number of bores, tunnel shape, ground conditions, and complexity.

In recent years, utilizing meta-heuristics emerged as an efficient mechanism to improve the prediction accuracies of machine learning models through their iterative training, such as using support vector machines and a moth-flame optimizer to detect damages in bridges [26], using an adaptive neuro-fuzzy inference system and a particle swarm optimizer to predict monthly river streamflow [27], and using an artificial neural network and a particle swarm optimizer to estimate landslide susceptibility mapping [28]. Hence, the developed deterioration model relies on the accommodation of Gaussian process regression and the grey wolf optimization algorithm to forecast the future deterioration pattern of tunnel elements. In this regard, the grey wolf optimization algorithm is deployed for two main purposes:

• To improve the prediction performance of Gaussian process regression by optimizing its hyper parameters.
• To select the most significant subset of deterioration features.

Hence, the hyper parameters of Gaussian process regression and the deterioration features listed in Table 1 are regarded as decision variables when managing to optimize Gaussian process regression. In this research paper, the training process of Gaussian process regression is carried out by stepping on designing a single-objective optimization function that minimizes the mean absolute percentage error of predicted condition ratings in the training dataset. Gaussian process regression is selected because it has low computational time and memory requirements [29,30]. In addition, it has provided promising predictive results in several civil engineering implementations such as assessment of reservoir permeability and porosity [31], evaluation of resilient modules of stabilized materials [32], soil moisture prediction [33], and fragility estimation of bridges [34].

The grey wolf optimizer is a nature-inspired meta-heuristic that capitalizes on modeling the leadership hierarchy and hunting mechanism of grey wolves. Its basic steps encompass social hierarchy creation, encircling the prey, hunting, and attacking the prey. The simulation process of the grey wolf optimizer is discussed in detail in Section 4.2. The grey wolf optimization algorithm has gained significant attention in recent years due to its simplicity, robustness, and ease of implementation [35,36]. It requires fewer control parameters to be tuned and provides proper balance between exploration and exploitation [37,38,39]. Furthermore, the grey wolf optimization algorithm has been successfully applied to solving several real-world problems such as damage identification of skeletal structures [40], design of reinforced concrete cantilevers [41], optimization of reservoir systems [42], and simulation of soil water content [43]. The basic operations of the grey wolf optimization algorithm are repeated during each iteration (Num_iter) until reaching the maximum number of iterations (Maxnum_iter). To this end, the optimum solution found by the grey wolf optimization algorithm is appended to be further used in validating the developed GWO-GPR model for deterioration prediction of highway tunnel elements. There are two types of output in the model calibration component. The first type is the optimum hyper parameters and deterioration factors in the GWO-GPR model. The second is the predicted condition rating for the training and testing datasets based on the optimized GWO-GPR model.

The second component of the developed framework is model assessment. The developed deterioration prediction model is validated using three folds of performance evaluation, statistical significance tests, and integrative ranking. In the first fold, the developed model is validated against a set of widely used machine learning models, namely, the back-propagation artificial neural network (BPANN), Elman neural network (ENN), cascade forward neural network (CFNN), generalized regression neural network (GRNN), support vector machines (SVM), and regression tree (RT). The comparative analysis is carried out based on the performance indicators of mean absolute percentage error, mean absolute error, root mean square percentage error, root relative squared error, and relative absolute error. The second fold is designated for studying the statistical significance of the differences between the prediction accuracies provided by the deterioration models. In this fold, the Shapiro–Wilk test is employed to explore the normality of the predicted scores. In this regard, the Shapiro–Wilk test is acknowledged as the most powerful normality test of measurement [44,45]. According to the outcome of the Shapiro–Wilk test, parametric or non-parametric tests are deployed to compare the significance of the predicted values of the deterioration models. Some models may perform well according to some evaluation metrics. However, they may underperform with regards to others. As such, the third fold aims at using the Copeland algorithm to establish a consolidated assessment of the deterioration models based on their obtained scores in the five performance indicators across the five tunnel elements. All the codes and conducted computations are implemented using Matlab R2017a.

4. Model Development

This section describes the procedures of the grey wolf optimizer, Gaussian process regression, and Copeland algorithm. It also demonstrates the training mechanism of Gaussian process regression.

4.1. Gaussian Process Regression

Gaussian process regression is a kernel-based non-parametric structure that is stemmed from probability theory. Gaussian process regression is usually specified using mean and covariance functions. A covariance function is a kernel function that captures the smoothness of the response variable [46,47]. A Gaussian process regression model can be expressed as follows [48].

$${\mathrm{Y}}_{\mathrm{i}}=\mathrm{f}({\mathrm{x}}_{\mathrm{i}})+\mathsf{\epsilon }$$

(1)

where

${\mathrm{Y}}_{\mathrm{i}}$ and ${\mathrm{x}}_{\mathrm{i}}$ represent the output and input variables, respectively. $\mathsf{\epsilon }$ is the Gaussian noise with variance σ2.

In order to improve the performance of the Gaussian process regression, the following properties are varied:

• Type of kernel function.
• Kernel function parameters.
• Type of basis function.
• Gaussian noise standard deviation.

4.2. Grey Wolf Optimization

Grey wolf optimization is a bio-inspired meta-heuristic that was introduced by Mirjalili et al. [49]. It simulates the social leadership hierarchy and hunting behavior of grey wolves in nature. The social leadership hierarchy is composed of four main levels of wolves, namely, alpha (α), beta (β), delta (δ), and omega (ω). The first level of the hierarchy is alpha, where the alpha wolf is the dominant wolf and leader of the pack. The alpha wolf is mostly concerned with making decisions pertinent to sleeping, waking, and hunting. The second level of the hierarchy of grey wolves is beta, where beta wolves act as advisors to reinforce and aid the alpha wolf in the decision-making process, and provide it with feedback. Delta wolves obey the dictated orders of alpha and beta wolves, and they dominate omega wolves. They encompass sentinels, hunters, scouts, and caretakers. Omega wolves are the lowest dominant level in the hierarchy, and they submit to the other dominant alpha, beta, and delta wolves. They act as scapegoats and are the last ones permitted to eat. The main phases of the grey wolf hunting process comprise tracking the prey, encircling, and attacking.

In the social hierarchy, encircling and hunting are mathematically expressed in the following lines. In the grey wolf optimization algorithm, alpha is considered the fittest solution, followed by beta and delta. The remainder of the candidate solutions are regarded as omega. After the initialization of the positions of the grey wolves, they manage to encircle the prey, which enables to update their positions around it. The encircling behavior of grey wolves can be mathematically described using Equations (2) and (3).

$$\overrightarrow{\mathrm{D}}=[\overrightarrow{\mathrm{C}}.\overrightarrow{{\mathrm{X}}_{\mathrm{p}}}(\mathrm{t})-\overrightarrow{\mathrm{X}}(\mathrm{t})]$$

(2)

$$\overrightarrow{\mathrm{X}}(\mathrm{t}+1)=[\overrightarrow{{\mathrm{X}}_{\mathrm{p}}}(\mathrm{t})-\overrightarrow{\mathrm{A}}.\overrightarrow{\mathrm{D}}]$$

(3)

where

$\overrightarrow{\mathrm{X}}$ and X_p are the position vectors of the grey wolf and prey, respectively. $\mathrm{t}$ is the current iteration. $\overrightarrow{\mathrm{D}}$ is the distance between position of the grey wolf and the position of the prey. $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ are coefficient vectors and can be interpreted using Equations (4)–(6).

$$\overrightarrow{\mathrm{a}}=2\overrightarrow{\mathrm{a}}.\overrightarrow{{\mathrm{r}}_1}-\overrightarrow{\mathrm{A}}$$

(4)

$$\overrightarrow{\mathrm{C}}=2.\overrightarrow{{\mathrm{r}}_2}$$

(5)

$$\overrightarrow{\mathrm{a}}=2-\mathrm{t}(\frac{2}{\mathrm{T}})$$

(6)

where

$\overrightarrow{\mathrm{a}}$ is a motion vector that is linearly decreasing from two to zero over the iterations. $\overrightarrow{{\mathrm{r}}_1}$ and $\overrightarrow{{\mathrm{r}}_2}$ are two randomly generated values between 0 and 1. $\overrightarrow{\mathrm{C}}$ is a coefficient vector that emulates the influence of obstacles near the prey in nature. $\mathrm{T}$ is the total number of optimization iterations. $\mathrm{t}$ refers to the current number of iterations.

Grey wolves are able to find the location of their prey and encircle it. In this regard, the hunting process is primarily directed by alpha wolves. In addition, beta and delta wolves can occasionally be involved in the hunting process. Alpha wolves are assumed to be the best candidate solutions, whereas beta and delta wolves are assumed to be in the second and third best positions, respectively. It is also assumed that alpha, beta, and delta wolves to have more information regarding the position of the prey. Thus, the best three candidate solutions generated so far are appended. Furthermore, other search agents are forced to update their positions based on the positions of the first three best-performing search agents. The positions of the candidate grey wolves can be mathematically represented as follows.

$$\overrightarrow{{\mathrm{D}}_{\mathsf{\alpha }}}=|\overrightarrow{{\mathrm{C}}_1}.\overrightarrow{{\mathrm{X}}_{\mathsf{\alpha }}}-\overrightarrow{\mathrm{X}}|$$

(7)

$$\overrightarrow{{\mathrm{D}}_{\mathsf{\beta }}}=|\overrightarrow{{\mathrm{C}}_2}.\overrightarrow{{\mathrm{X}}_{\mathsf{\beta }}}-\overrightarrow{\mathrm{X}}|$$

(8)

$$\overrightarrow{{\mathrm{D}}_{\mathsf{\delta }}}=|\overrightarrow{{\mathrm{C}}_3}.\overrightarrow{{\mathrm{X}}_{\mathsf{\delta }}}-\overrightarrow{\mathrm{X}}|$$

(9)

$$\overrightarrow{{\mathrm{X}}_1}=\overrightarrow{{\mathrm{X}}_{\mathsf{\alpha }}}-\overrightarrow{{\mathrm{A}}_1}.(\overrightarrow{{\mathrm{D}}_{\mathsf{\alpha }}})$$

(10)

$$\overrightarrow{{\mathrm{X}}_2}=\overrightarrow{{\mathrm{X}}_{\mathsf{\beta }}}-\overrightarrow{{\mathrm{A}}_2}.(\overrightarrow{{\mathrm{D}}_{\mathsf{\beta }}})$$

(11)

$$\overrightarrow{{\mathrm{X}}_3}=\overrightarrow{{\mathrm{X}}_{\mathsf{\delta }}}-\overrightarrow{{\mathrm{A}}_3}.(\overrightarrow{{\mathrm{D}}_{\mathsf{\delta }}})$$

(12)

$$\overrightarrow{\mathrm{X}}(\mathrm{t}+1)=\frac{\overrightarrow{{\mathrm{X}}_1}+\overrightarrow{{\mathrm{X}}_2}+\overrightarrow{{\mathrm{X}}_3}}{3}$$

(13)

where

$\overrightarrow{\mathrm{X}}$ represents the position of the candidate grey wolf. $\overrightarrow{\mathrm{X}}(\mathrm{t}+1)$ is the updated position of the candidate grey wolf in the next iteration. $\overrightarrow{{\mathrm{X}}_{\mathsf{\alpha }}}$, $\overrightarrow{{\mathrm{X}}_{\mathsf{\beta }}}$, and $\overrightarrow{{\mathrm{X}}_{\mathsf{\delta }}}$ are the position vectors of the alpha, beta, and delta fray wolves, respectively. The coefficient vectors of $\overrightarrow{{\mathrm{C}}_1}$, $\overrightarrow{{\mathrm{C}}_2}$, $\overrightarrow{{\mathrm{C}}_3}$, $\overrightarrow{{\mathrm{A}}_1}$, and $\overrightarrow{{\mathrm{A}}_3}$ are obtained using Equations (4)–(6).

The coefficients of $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{C}}$ heavily influence the exploration and exploitation behavior of the grey wolf optimization algorithm. The hunting process is finalized when the prey is attacked, and it stops movement. The gradual approach to the prey is mathematically formulated by assuming that the fluctuation of $\overrightarrow{\mathrm{A}}$ is decreasing by value $\overrightarrow{\mathrm{a}}$ over the course of the iterations. In this regard, $\overrightarrow{\mathrm{A}}$ is set as a random value within the interval range [−$2\mathrm{a}$, $2\mathrm{a}$] and $\overrightarrow{\mathrm{a}}$ decreases from 2 to 0 over the iterations. In addition, setting $|\overrightarrow{\mathrm{A}}|\ge$1 causes the grey wolves to leave the current target and search for a better prey target, which promotes exploration of the search space. On the contrary, setting $|\overrightarrow{\mathrm{A}}|<$1 pushes the grey wolf to attack the target prey, which implies exploitation of the search space. The coefficient vector $\overrightarrow{\mathrm{C}}$ contains random values between 0 and 2, and it provides random values for the prey target. The optimization process is terminated when the convergence criteria are satisfied.

4.3. Training of the Gaussian Process Regression Model

The developed model utilized the grey wolf optimization algorithm for the automated tuning of the hyper parameters of Gaussian process regression, and to identify the most significant deterioration factors. In this regard, the developed model optimizes the hyper parameters of the type of kernel function, type of basis function, values of kernel parameters, and Gaussian noise standard deviation. As such, the length of optimization problem is 21, 5 of the variables are the hyper parameters of Gaussian process regression, and the remaining 16 variables are for the relevant deterioration factors. The solution structure of the designed grey wolf optimization-based model is depicted in Figure 2. The variable ${\mathrm{X}}_{\mathrm{ij}}$ refers to the values of the hyper parameters of Gaussian process regression and to the values of input deterioration factors in the tunnel element. The counter $\mathrm{j}$ is an integer number that ranges from one to five, reflecting the five highway tunnel elements. For the hyper parameters, the value of ${\mathrm{X}}_{\mathrm{ij}}$ is an integer number for the type of kernel function and type of basis function. The value of ${\mathrm{X}}_{\mathrm{ij}}$ is a decimal number for the kernel parameters and Gaussian process standard deviation. As for the deterioration factors, the variable ${\mathrm{X}}_{\mathrm{ij}}$ takes the form of binary number, whereas 1 implies that the deterioration factor is significant and needs to be considered. On the other hand, a value of 0 indicates that the input factor is not influential on the deterioration process of the highway tunnel element.

The studied kernel parameters were characteristic length scale and signal standard deviation. Three types of kernel functions were studied: exponential, Matern 3/2, and Matern 5/2. The developed model also investigates the basis function of linear, constant, and pure quadratic in addition to no basis function. The minimum and maximum boundaries of Gaussian noise standard deviation, characteristic length scale, and signal standard deviation were set between 0 and 10, respectively. The automated calibration of the hyper parameters of Gaussian process regression was carried out based on a single-objective optimization function that minimizes mean absolute percentage error of the condition rating of tunnel elements, as shown in Equation (14). Mean absolute percentage error was selected for training the Gaussian process regression because it is a broadly utilized indicator to evaluate the accuracies of forecasting models [50,51,52]. In addition, it is independent of the scale or unit of the predicted value, which endorses the robustness of the developed deterioration prediction model. The training process of Gaussian process regression was executed for each highway tunnel element due to their varying deterioration behavior. As such, different optimum structures of Gaussian process regression were obtained for each highway tunnel element.

$${\mathrm{MAPE}}_{\mathrm{T}}=\frac{100}{\mathrm{T}}\times {\displaystyle \sum }_{\mathrm{t}=1}^{\mathrm{T}}\frac{|{\mathrm{PT}}_{\mathrm{t}}-{\mathrm{AT}}_{\mathrm{t}}|}{{\mathrm{AT}}_{\mathrm{t}}}$$

(14)

where

${\mathrm{MAPE}}_{\mathrm{T}}$ is the mean absolute percentage error of the condition rating in the training dataset. $\mathrm{T}$ is the number of training cases in the dataset of the designated highway tunnel element. ${\mathrm{PT}}_{\mathrm{t}}$ and ${\mathrm{AT}}_{\mathrm{t}}$ refer to the predicted and actual condition rating of $\mathrm{t}-\mathrm{th}$ data instance in the training dataset.

4.4. Copeland

Copeland is a modified incarnation of the Borda count algorithm [53,54]. It considers both the number of dominance or number of wins and the number of defeats of each alternative. In this research paper, the Copeland algorithm was deployed to create a consolidated final ranking of the deterioration models. In this regard, each pair of deterioration models was compared against each other with respect to “p” performance evaluation metrics. The deterioration model that obtained the largest number of predominance over the other deterioration model accomplished a win and was appended as a “Winner.” On the other hand, the other deterioration model is known as the “Loser” [55,56]. A final sore of each deterioration model was yielded by subtracting the number of losses from the number of dominances. In this context, a higher final score signifies a better deterioration model. The final score of each deterioration model was generated using Equation (15).

$${\mathrm{FS}}_{\mathrm{i}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{W}}_{\mathrm{i}}-{\mathrm{L}}_{\mathrm{i}}$$

(15)

where

${\mathrm{W}}_{\mathrm{i}}$ and ${\mathrm{L}}_{\mathrm{i}}$ denote the number of wins and losses accomplished by each deterioration model, respectively. ${\mathrm{FS}}_{\mathrm{i}}$ is the final score of each deterioration model, which is utilized for its ranking. It worth noting that the deterioration models were sorted in descending order, where the highest performing deterioration model is the one accompanied by the highest final score.

5. Performance Assessment

Five performance metrics were adopted in this research paper to assess the deterioration prediction models and compare their prediction accuracies. The equations of mean absolute percentage error, mean absolute error, root mean square percentage error, root relative squared error, and relative absolute error can be mathematically expressed using Equations (16)–(20), respectively [57,58,59].

$$\mathrm{MAPE}=\frac{100}{\mathrm{k}}\times {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{K}}\frac{|{\mathrm{P}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{i}}|}{{\mathrm{A}}_{\mathrm{i}}}$$

(16)

$$\mathrm{MAE}=\frac{1}{\mathrm{K}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{K}}|({\mathrm{A}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}})|$$

(17)

$$\mathrm{RMSPE}=\frac{100}{\mathrm{k}}\times {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{K}}{\left(\frac{{\mathrm{P}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{i}}}{{\mathrm{A}}_{\mathrm{i}}}\right)}^2$$

(18)

$$\mathrm{RRSE}=\sqrt{\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{K}}{({\mathrm{A}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}})}^2 }{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{K}}{({\mathrm{A}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{i}}{}^{-})}^2 }}$$

(19)

$$\mathrm{RAE}=\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{K}}|({\mathrm{A}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}})| }{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{K}}|({\mathrm{A}}_{\mathrm{i}}-{\mathrm{A}}_{\mathrm{i}}{}^{-})| }$$

(20)

where

${\mathrm{A}}_{\mathrm{i}}$ and ${\mathrm{P}}_{\mathrm{i}}$ represent the actual and predicted condition ratings, respectively.${ \mathrm{A}}_{\mathrm{i}}{}^{-}$ denotes the mean of the actual condition ratings. $\mathrm{K}$ is the number of available observations for each tunnel component. It is worth mentioning that lower values of $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ imply a higher-performing deterioration prediction model.

6. Model Implementation

The validation process of the developed deterioration model was conducted using 2020 inspection records of cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade. The dataset of cast-in-place tunnel liners is composed of 428 inspection records such that 342 and 86 records were used for training and testing purposes, respectively. The inventory database of concrete interior walls consisted of 91 records, where 73 and 18 records were used for training and testing, respectively. In concrete portal, there was a total of 362 data points, and 290 training points and 72 testing points were used. The entire dataset of concrete ceiling slab comprised 85 data points, and training and testing subsets used 68 and 17 data points, respectively. Historical inspection records of 285 concrete slab on grade were used, and it encompassed 228 training records and 57 testing records.

The hyper parameters of the conventional-based deterioration models were tuned manually based on trial-and-error experiments. In the $\mathrm{BPANN}$, the numbers of hidden layers and hidden neurons were assumed to be two and five, respectively. In addition, the transfer function was sigmoid, and the learning rate and momentum were set to 0.001 and 0.8, respectively. With regards to $\mathrm{ENN}$, there were five hidden and context layers, and three hidden and context neurons. The spread of Gaussian function is one in Gaussian regression neural network. The hidden layer size is five in the cascade forward neural network. As for support vector machines, Gaussian kernel function is used with kernel scale and epsilon of 1 and 0.1, respectively. The minimum parent size is ten in the regression tree model.

With respect to the developed $\mathrm{GWO}-\mathrm{GPR}$ model, the numbers of iterations and search agents in the grey wolf optimization algorithm were assumed to be 100 and five, respectively. Figure 3 displays the convergence curves of the developed $\mathrm{GWO}-\mathrm{GPR}$ model in cast-in-place tunnel liners and concrete slab on grade. It was revealed that the developed $\mathrm{GWO}-\mathrm{GPR}$ model converged to 9.48 × 10⁻⁴ at iteration 3 in cast-in-place tunnel liners. In addition, the developed $\mathrm{GWO}-\mathrm{GPR}$ model converged to 8.78 × 10⁻⁴ at iteration 10 in concrete slab on grade. Figure 4 displays the convergence curves of the developed $\mathrm{GWO}-\mathrm{GPR}$ model in concrete ceiling slab and concrete interior walls. It was found that the developed $\mathrm{GWO}-\mathrm{GPR}$ model converged to a value of 6.45 × 10⁻⁴ at iteration 11 in concrete ceiling slab, and it converged to a value of 6.36 × 10⁻⁴ at iteration 4 in concrete interior walls. This illustrates the higher search capability of the grey wolf optimizer. In cast-in-place tunnel liners, the optimized Gaussian process regression was composed of an exponential kernel function with a constant basis function, and the length scale, signal standard deviation, and Gaussian noise standard deviation were found to be equal to 0.3, 0.54 and 0.3, respectively. The most influential factors were found to be age, annual average daily traffic, service in tunnel, functional classification, number of bores, tunnel shape, and ground conditions. With respect to concrete slab on grade, the optimum structure of Gaussian process regression encompasses exponential kernel function with a constant basis function, and the values of length scale, signal standard deviation, and Gaussian noise standard deviation were equal to 0.32, 0.61, and 0.37, respectively. The significant factors were found to be age, number of lanes, detour length, NHS designation, STRAHNET designation, functional classification, tunnel shape, ground conditions, and complexity.

In concrete ceiling slab, the optimum architecture of Gaussian process regression was comprised of an exponential kernel function with a constant basis function, and the values of length scale, signal standard deviation, and Gaussian noise standard deviation were 0.37, 0.3, and 0.62, respectively. The most influential deterioration factors were age, roadway width, number of bores, tunnel shape, and ground conditions. As for concrete interior walls, the optimized Gaussian process regression consisted of Matern 3/2 with a constant basis function with a length scale, signal standard deviation, and Gaussian noise standard deviation of 0.79, 0.64, and 0.32, respectively. The significant deterioration parameters involved age, number of bores, tunnel shape, and ground conditions. With regards to concrete portal, the optimum structure of Gaussian process regression was formed of an exponential kernel function with a constant basis function, and the length scale, signal standard deviation, and Gaussian noise standard deviation were equal to 0.41, 0.4, and 0.42, respectively. The influential deterioration factors were age, service in tunnel, NHS designation, number of bores, tunnel shape, and ground conditions.

Figure 5 demonstrates illustrations of the prediction performances using a back-propagation artificial neural network and the developed $\mathrm{GWO}-\mathrm{GPR}$ model. In this figure, 20 inspection records of the testing dataset are plotted. It can be noted that the back-propagation artificial neural network predicted condition ratings were much departed from the actual condition ratings. However, the developed $\mathrm{GWO}-\mathrm{GPR}$ model showed more promising results, generating predicted condition ratings closer to the actual ones.

Table 2. Sample of the predicted condition ratings for cast-in-place tunnel liners based on the developed $\mathrm{GWO}-\mathrm{GPR}$ model.

Age	Annual Average Daily Truck Traffic	Service in Tunnel	Functional Classification	Number of Bores	Tunnel Shape	Ground Conditions	Actual Condition	Predicted Condition
47	75,320	1	1	2	4	1	1	1
68	8250	1	3	1	2	2	1	1
62	4822	1	3	1	2	2	2	2
86	1500	1	6	1	2	2	3	3
86	1000	1	6	1	2	2	2	2
110	5	3	7	1	2	2	2	2
110	5	1	7	1	2	2	2	3
61	6650	1	3	1	2	3	3	3
48	7100	1	7	1	2	3	4	4
61	1400	1	6	1	2	2	2	1

reports a sample of the results obtained by the developed $\mathrm{GWO}-\mathrm{GPR}$ model in cast-in-place tunnel liners. The predicted condition rating was computed by capitalizing on the significant deterioration factors identified by the developed model. It can be observed that the developed $\mathrm{GWO}-\mathrm{GPR}$ model was able to simulate the actual condition ratings of cast-in-place, producing very promising low-error values.

Table 3. Performance comparison between deterioration prediction models of cast-in-place tunnel liners.

Deterioration Model	$\mathbf{MAPE}$	$\mathbf{MAE}$	$\mathbf{RMSPE}$	$\mathbf{RRSE}$	$\mathbf{RAE}$
$\mathrm{GWO}-\mathrm{GPR}$	1.65%	0.018	0.21%	0.018	0.147
$\mathrm{BPANN}$	17.74%	0.204	1.68%	0.864	1.702
$\mathrm{ENN}$	11.98%	0.149	0.79%	0.425	1.24
$\mathrm{CFNN}$	17.11%	0.199	1.28%	0.64	166
$\mathrm{GRNN}$	2.69%	0.028	1.02%	0.24	0.23
$\mathrm{SVM}$	9.71%	0.132	0.65%	0.5	1.09
$\mathrm{RT}$	7.25%	0.083	1.35%	0.49	0.69

reports the performance prediction comparison between the deterioration models of cast-in-place tunnel liners. It can be observed that the developed $\mathrm{GWO}-\mathrm{GPR}$ obtained the highest prediction accuracies, followed by $\mathrm{GRNN}$ and then $\mathrm{RT}$. On the other hand, $\mathrm{BPANN}$ yielded the lowest prediction performance. In this regard, $\mathrm{GWO}-\mathrm{GPR}$ accomplished a $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 1.65%, 0.018, 0.21%, 0.018, and 0.147, respectively. In addition, $\mathrm{BPANN}$ scored a $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 17.74%, 0.204, 1.68%, 0.864, and 1.702, respectively.

Table 4. Performance comparison between deterioration prediction models of concrete interior walls.

Deterioration Model	$\mathbf{MAPE}$	$\mathbf{MAE}$	$\mathbf{RMSPE}$	$\mathbf{RRSE}$	$\mathbf{RAE}$
$\mathrm{GWO}-\mathrm{GPR}$	1.24%	0.018	0.55%	0.445	0.17
$\mathrm{BPANN}$	13.95%	0.157	2.1%	1.051	1.511
$\mathrm{ENN}$	6.71%	0.073	2.07%	0.949	0.701
$\mathrm{CFNN}$	14.37%	0.148	2.62%	1.134	1.427
$\mathrm{GRNN}$	3.84%	0.044	1.98%	0.92	0.423
$\mathrm{SVM}$	2.31%	0.029	0.92%	0.553	0.276
$\mathrm{RT}$	7.14%	0.084	2.17%	1.029	0.811

displays the prediction accuracies for the deterioration models of concrete interior walls. $\mathrm{GWO}-\mathrm{GPR}$ showed the highest prediction performance in terms of all five indicators, whereas $\mathrm{BPANN}$ and $\mathrm{CFNN}$ had the lowest prediction accuracies. In this context, the obtained scores of $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ by $\mathrm{GWO}-\mathrm{GPR}$ were 1.24%, 0.018, 0.55%, 0.445, and 0.17, respectively.

Table 5. Performance comparison between deterioration prediction models of concrete portal.

Deterioration Model	$\mathbf{MAPE}$	$\mathbf{MAE}$	$\mathbf{RMSPE}$	$\mathbf{RRSE}$	$\mathbf{RAE}$
$\mathrm{GWO}-\mathrm{GPR}$	1.73%	0.017	0.21%	0.125	0.132
$\mathrm{BPANN}$	12.03%	0.146	0.94%	0.283	1.117
$\mathrm{ENN}$	10.78%	0.128	0.92%	0.773	0.974
$\mathrm{CFNN}$	12.63%	0.154	0.95%	0.251	1.177
$\mathrm{GRNN}$	5.01%	0.064	0.85%	0.365	0.49
$\mathrm{SVM}$	11.29%	0.125	0.63%	0.235	0.957
$\mathrm{RT}$	8.5%	0.113	0.95%	0.278	0.863

records a forecasting performance comparison for the deterioration models of concrete portal. It is derived that the developed $\mathrm{GWO}-\mathrm{GPR}$ model outperformed other deterioration models, whereas $\mathrm{CFNN}$ produced the highest prediction error. $\mathrm{GWO}-\mathrm{GPR}$ resulted in an $\mathrm{MAPE}$ of 1.73%, $\mathrm{MAE}$ of 0.017, $\mathrm{RMSPE}$ of 0.21%, $\mathrm{RRSE}$ of 0.125, and $\mathrm{RAE}$ of 0.132. In addition, $\mathrm{CFNN}$ generated an $\mathrm{MAPE}$ of 12.63%, $\mathrm{MAE}$ of 0.154, $\mathrm{RMSPE}$ of 0.95%, $\mathrm{RRSE}$ of 0.251, and $\mathrm{RAE}$ of 1.177. The prediction performances for the deterioration models of concrete ceiling slab are listed in

Table 6. Performance comparison between deterioration prediction models of concrete ceiling slab.

Deterioration Model	$\mathbf{MAPE}$	$\mathbf{MAE}$	$\mathbf{RMSPE}$	$\mathbf{RRSE}$	$\mathbf{RAE}$
$\mathrm{GWO}-\mathrm{GPR}$	2.81%	0.038	0.88%	0.362	0.22
$\mathrm{BPANN}$	25.81%	0.283	4.02%	1.024	1.615
$\mathrm{ENN}$	9.19%	0.108	1.82%	0.56	0.619
$\mathrm{CFNN}$	17.49%	0.191	4.62%	1.157	1.092
$\mathrm{GRNN}$	7.05%	0.082	3.99%	0.991	0.471
$\mathrm{SVM}$	11.73%	0.17	1.74%	1.01	0.971
$\mathrm{RT}$	8.38%	0.125	2.09%	0.904	0.717

. It can be noticed that $\mathrm{GWO}-\mathrm{GPR}$ model performed better than other deterioration models, and the models of $\mathrm{BPANN}$ and $\mathrm{CFNN}$ attained the lowest levels of prediction accuracies. $\mathrm{GWO}-\mathrm{GPR}$ managed to induce an $\mathrm{MAPE}$ of 2.81%, $\mathrm{MAE}$ of 0.038, $\mathrm{RMSPE}$ of 0.88%, $\mathrm{RRSE}$ of 0.362, and $\mathrm{RAE}$ of 0.22. The performance evaluation results for the deterioration prediction models of concrete slab on grade are presented in

Table 7. Performance comparison between deterioration prediction models of concrete slab on grade.

Deterioration Model	$\mathbf{MAPE}$	$\mathbf{MAE}$	$\mathbf{RMSPE}$	$\mathbf{RRSE}$	$\mathbf{RAE}$
$\mathrm{GWO}-\mathrm{GPR}$	1.21%	0.012	0.17%	0.114	0.138
$\mathrm{BPANN}$	16.22%	0.177	1.63%	1.207	2.016
$\mathrm{ENN}$	9.19%	0.11	0.82%	0.868	1.257
$\mathrm{CFNN}$	13.22%	0.151	1.23%	1.066	1.726
$\mathrm{GRNN}$	7.74%	0.101	0.71%	0.99	1.149
$\mathrm{SVM}$	10.77%	0.114	0.66%	0.61	1.304
$\mathrm{RT}$	6.84%	0.089	0.81%	0.958	1.02

. It can be observed that $\mathrm{GWO}-\mathrm{GPR}$ yielded the lowest values of $\mathrm{MAPE}$ (1.21%), $\mathrm{MAE}$ (0.012), $\mathrm{RMSPE}$ (0.17%), $\mathrm{RRSE}$ (0.114), and $\mathrm{RAE}$ (0.138). Moreover, $\mathrm{BPANN}$ produced the highest values of of $\mathrm{MAPE}$ (16.22%), $\mathrm{MAE}$ (0.177), $\mathrm{RMSPE}$ (1.63%), $\mathrm{RRSE}$ (1.207), and $\mathrm{RAE}$ (2.016).

Figure 6, Figure 7 and Figure 8 display the error histograms with 20 bins in the developed $\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{BP}-\mathrm{ANN}$, and $\mathrm{SVM}$. The error histograms are plotted for cast-in-place tunnel liners and concrete portal. The error herein represents the difference between the actual and predicted condition ratings. The yellow line indicates the zero error. In the developed $\mathrm{GWO}-\mathrm{GPR}$ model, it is observed that most of the data instances exhibited errors of 0.023 in cast-in-place tunnel liners and 0.002 in concrete portal. As for $\mathrm{BPANN}$, most of the errors ranged between 0.06 and 0.33 in cast-in-place tunnel liners and the errors ranged between 0.04 and 0.22 in concrete portal. In $\mathrm{SVM}$, most of the data instances sustained errors between 0.06 and 0.32 in cast-in-place tunnel liners. In addition, the largest portion of errors varied between 0.04 and 0.37 in concrete portal.

Figure 9 and Figure 10 depict the obtained rankings for the seven deterioration models of concrete portal and concrete ceilings across the different performance indicators. It can be interpreted that the developed $\mathrm{GWO}-\mathrm{GPR}$ model outranked other deterioration models in both tunnel elements according to all the indicators. It can be also observed that different performance indicators triggered different rankings of alternatives. In concrete portal, $\mathrm{GRNN}$ achieved the second rank in mean absolute percentage error and the sixth rank in root relative squared error. In concrete ceiling slab, $\mathrm{SVM}$ obtained the second rank in root mean square percentage error and provided the fifth rank in the remaining indicators. It can be also noticed that some perturbations were experienced between the performances of the deterioration models across the different tunnel elements. For instance, $\mathrm{ENN}$ achieved the second and seventh ranks in predicting root relative squared error of concrete portal and concrete ceiling slab, respectively. In addition, $\mathrm{RT}$ yielded the second and fourth ranks in forecasting relative absolute error of concrete portal and concrete ceiling slab, respectively.

In the statistical comparison analysis, the Shapiro–Wilk test was first applied to test the normality of the values of $\mathrm{MAPE}$ and $\mathrm{RRSE}$ at a significance level of 5%. The null hypothesis is that the random variables (performance indicators) follow normal distribution.

Table 8. $p-\mathrm{values}$ of some performance indicators of the deterioration models using the Shapiro–Wilk test for normality.

Model	Description	$\mathit{p}-\mathbf{Value}$
$\mathrm{GWO}-\mathrm{GPR}$	Mean absolute percentage error	1.83 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{BPANN}$	Mean absolute percentage error	1.41 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{ENN}$	Mean absolute percentage error	1.72 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{CFNN}$	Mean absolute percentage error	8.08 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{GRNN}$	Mean absolute percentage error	2.95 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{SVM}$	Mean absolute percentage error	7.08 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{RT}$	Mean absolute percentage error	9.67 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{GWO}-\mathrm{GPR}$	Root-mean square percentage error	1.85 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{BPANN}$	Root-mean square percentage error	6.89 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{ENN}$	Root-mean square percentage error	2.41 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{CFNN}$	Root-mean square percentage error	3.96 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{GRNN}$	Root-mean square percentage error	6.76 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{SVM}$	Root-mean square percentage error	1.47 × 10−1 (${\mathrm{H}}_0$)
$\mathrm{RT}$	Root-mean square percentage error	5.52 × 10−1 (${\mathrm{H}}_0$)

presents the results of the Shapiro–Wilk test of normality. As can be seen, all the $p-\mathrm{values}$ of all performance indicators were more than the significance level. Hence, the null hypothesis (${\mathrm{H}}_0$) was accepted and the performance scores followed normal distribution. For instance, the $p-\mathrm{values}$ of $\mathrm{GWO}-\mathrm{GPR}$ were 1.83 × 10⁻¹ in mean absolute percentage error, and 1.85 × 10⁻¹ in root relative squared error. As such, the parametric Student’s t-test was deployed to study the significance levels of the prediction accuracies of the deterioration models. This test investigated the null hypothesis (${\mathrm{H}}_0$), which is that there is no significant difference between the prediction performances of the deterioration models.

Table 9. Statistical comparison of the developed deterioration model against other models using $\mathrm{MAPE}$ based on Student’s t-test.

Pair of Deterioration Models	$\mathit{p}-\mathbf{Value}$
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{BPANN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 8.7 × 10−4)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{ENN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.97 × 10−3)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{CFNN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 9.35 × 10−5)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{GRNN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.79 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{SVM}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.01 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{RT}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.31 × 10−5)

and

Table 10. Statistical comparison of the developed deterioration model against other models using $\mathrm{RRSE}$ based on Student’s t-test.

Pair of Deterioration Models	$\mathit{p}-\mathbf{Value}$
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{BPANN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.21 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{ENN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 6.53 × 10−3)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{CFNN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.03 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{GRNN}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 1.64 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{SVM}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 2.79 × 10−2)
$(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{RT}$)	${\mathrm{H}}_1$ ($p-\mathrm{value}$ = 9.52 × 10−3)

demonstrate the results of Student’s t-test based on the indicators of $\mathrm{MAPE}$ and $\mathrm{RRSE}$. It can be seen that all $p-\mathrm{values}$ of $\mathrm{GWO}-\mathrm{GPR}$ against other deterioration models were less than 5%. Thus, the null hypothesis (${\mathrm{H}}_0$) is rejected, and the alternative hypothesis (${\mathrm{H}}_1$) is accepted. For example, the $p-\mathrm{values}$ of the pairs $(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{BPANN}$) and $(\mathrm{GWO}-\mathrm{GPR}$, $\mathrm{SVM}$) in $\mathrm{MAPE}$ and $\mathrm{RRSE}$ were 8.7 × 10⁻⁴ and 2.79 × 10⁻², respectively. Therefore, it can be concluded that the developed $\mathrm{GWO}-\mathrm{GPR}$ model accomplished significantly better prediction performance than the remainder of the machine learning-based deterioration models.

The consolidated final rankings of deterioration models based on the Copeland algorithm are recorded in

Table 11. Final rankings of the Copeland algorithm.

Deterioration Model	Loss	Win	Difference	Final Rank
$\mathrm{GWO}-\mathrm{GPR}$	0	30	30	1
$\mathrm{BPANN}$	28	2	−26	7
$\mathrm{ENN}$	17	13	−4	5
$\mathrm{CFNN}$	27	3	−24	6
$\mathrm{GRNN}$	8	22	14	2
$\mathrm{SVM}$	13	17	4	4
$\mathrm{RT}$	12	18	6	3

. The wins and losses were obtained based on the results of five performance indicators in the five tunnel elements. It can be inferred that the developed $\mathrm{GWO}-\mathrm{GPR}$ model was the best performing deterioration prediction model, with zero losses and 30 wins. $\mathrm{GRNN}$ attained the second rank with eight losses and 22 wins. $\mathrm{RT}$ had the third rank with 12 losses and 18 wins. $\mathrm{RT}$ was in the fourth rank with 13 losses and 17 wins. $\mathrm{BPANN}$ had the lowest ranking (seventh) with 28 losses and two wins.

The developed $\mathrm{GWO}-\mathrm{GPR}$ model was further validated against the deterioration model proposed by Yamany and Elwakil [8]. The comparative analysis was carried out using 2018 inspection records of cast-in-place liners. The dataset was composed of 404 inspection records, and training and testing was performed using 90% and 10% of the size of the dataset, respectively. In this comparison, the numbers of iterations and search agents in the grey wolf optimization algorithm were set as 100 and five, respectively. The plotted convergence curve of the developed $\mathrm{GWO}-\mathrm{GPR}$ model is presented in Figure 11. The developed $\mathrm{GWO}-\mathrm{GPR}$ model stabilized at 8.64 × 10⁻⁴ at iteration 80.

Table 12. Performance comparison of the developed model against previous research study.

Deterioration Model	Total Size of Dataset	Training Dataset	Testing Dataset	$\mathbf{MAPE}$
$\mathrm{Developed} \mathrm{GWO}-\mathrm{GPR}$ model	404	366	38	7.69%
Yamany and Elwakil [8]	404	366	38	29.91%

shows the results of the two deterioration models. As can be seen, using the same size dataset (404 records), the developed $\mathrm{GWO}-\mathrm{GPR}$ model was able to produce a mean absolute percentage error (7.69%) that was less than the mean absolute percentage error (29.91%) of the model proposed by Yamany and Elwakil [8] by 22.22%.

7. Conclusions

This research paper presented an integrated model ($\mathrm{GWO}-\mathrm{GPR}$) to emulate the deterioration trend of five critical highway tunnel elements: cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade. The performance evaluation analysis exemplified that the developed $\mathrm{GWO}-\mathrm{GPR}$ model significantly outperformed the six other investigated machine learning-based deterioration models in the five tackled highway tunnel elements. In cast-in-place tunnel liners, the developed $\mathrm{GWO}-\mathrm{GPR}$ model produced an $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 1.65%, 0.018, 0.21%, 0.018, and 0.147, respectively. In concrete interior walls, it exhibited an $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 1.24%, 0.018, 0.55%, 0.445, and 0.17, respectively. With regards to concrete portal, it managed to yield an $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 1.73%, 0.017, 0.21%, 0.125, and 0.132, respectively. With respect to concrete ceiling slab, the developed $\mathrm{GWO}-\mathrm{GPR}$ model generated an $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ equal to 2.81%, 0.038, 0.88%, 0.362, and 0.22, respectively. As for concrete slab on grade, the developed $\mathrm{GWO}-\mathrm{GPR}$ model achieved an $\mathrm{MAPE}$, $\mathrm{MAE}$, $\mathrm{RMSPE}$, $\mathrm{RRSE}$, and $\mathrm{RAE}$ of 1.21%, 0.012, 0.17%, 0.114, and 0.138, respectively. In the grand scheme of things, it is manifested that the developed $\mathrm{GWO}-\mathrm{GPR}$ model accomplished improvements in the prediction accuracies of $\mathrm{BPANN}$, $\mathrm{ENN}$, $\mathrm{CFNN}$, $\mathrm{GRNN}$, $\mathrm{SVM}$, and $\mathrm{RT}$ by 84.71%, 76.99%, 83.84%, 67.23%, 69.6%, and 76.59%, respectively.

Across the highway tunnel elements, it was also deduced that the performances of conventional machine learning-based deterioration models varied from one highway element to the other. For instance, $\mathrm{SVM}$ provided satisfactory results in concrete interior walls, and underperformed in cast-in-place tunnel liners, concrete portal, concrete ceiling slab, and concrete slab on grade. Moreover, $\mathrm{GRNN}$ obtained good predictive results in cast-in-place tunnel liners, concrete interior walls, and concrete portal. However, it diverged in concrete ceiling slab and concrete slab on grade. The performance comparative analysis also revealed that the developed $\mathrm{GWO}-\mathrm{GPR}$ model managed to produce consistent results over the five performance indicators. Nevertheless, the prediction capabilities of the conventional machine learning-based deterioration models differed drastically based on the performance indicator. For example, $\mathrm{GRNN}$ in concrete portal was ranked second and sixth according to mean absolute percentage error and root relative squared error, respectively. In concrete ceiling slab, $\mathrm{SVM}$ was ranked in the fifth place and second place according to mean absolute error and root mean square percentage error. This evinces the need to study a wide range of performance indicators while assessing deterioration models.

The results of Student’s t-test illustrated that the $p-\mathrm{values}$ between the developed $\mathrm{GWO}-\mathrm{GPR}$ model and the remainder of the conventional-based deterioration models were less than 5%, which suggests that the developed $\mathrm{GWO}-\mathrm{GPR}$ model was able to accomplish significantly better prediction accuracies than the conventional-based deterioration models. The Copeland algorithm demonstrated that the developed $\mathrm{GWO}-\mathrm{GPR}$ model (0 wins, 30 losses) outranked other deterioration models, followed by $\mathrm{GRNN}$ (22 wins, 8 losses) in the second rank and then $\mathrm{GRNN}$ (18 wins, 12 losses) in the third rank, whereas $\mathrm{BPANN}$, despite its wide use in asset management practices (2 wins, 28 losses), was ranked in the lowest position. The main contributions of this research paper lie in (1) designing an integrated $\mathrm{GWO}-\mathrm{GPR}$ model that managed to accurately simulate the deterioration mechanisms of cast-in-place tunnel liners, concrete interior walls, concrete portal, concrete ceiling slab, and concrete slab on grade and (2) exploiting the grey wolf optimizer to improve the prediction capabilities of Gaussian process regression through automated calibration of its hyper parameters, and to select the most significant subset of deterioration factors. Reliable element-level deterioration models are essential for highway agencies to devise optimal maintenance and rehabilitation polices. It is anticipated that the promising results of the developed deterioration model could assist highway administrations in efficient and cost-effective planning of their maintenance schedules, mostly in the light of available squeezed maintenance budgets.