Prediction of pile settlement using a hybrid Adaptive-Network-Based Fuzzy Inference System

Abstract

Precise pile settlement prediction (SP) in rock-socketed foundations is vital for designing robust bridge foundations and other civil engineering structures. In this work, the behaviors of three powerful algorithms are employed, Dynamic Differential Annealed Optimization (DDAO), Runge Kutta Optimization (RKO), and Ant Lion Optimization (ALO) to improve the performance of the Adaptive Neuro-Fuzzy Inference System (ANFIS) model. In the ANFIS model, some critical input parameters include the rock's unconfined compressive strength, pile length, and pile diameter, which predict SP with high accuracy. The primary contribution of this research is its comparative study with optimization techniques applied to the ANFIS model. Results show that the ANFIS model optimized by DDAO algorithm has the lowest Root Mean Square Error (RMSE) and highest coefficient of determination (R²). On the other side, even though the models optimized through RKO and ALO algorithms also have high predictive capabilities, ALO has extra power in generating a set of Pareto-optimal solutions. This will facilitate the engineers in selecting the most appropriate model given specific design requirements and site-specific constraints. The study provides essential development within the geotechnical engineering study by enhancing the SP prediction accuracy. All these can greatly improve the design and reliability of bridge foundations and other major civil engineering structures for performance and long-term stability.

Keywords

Pile settlement prediction Adaptive Neuro-Fuzzy Inference System ant lion optimizer Runge Kutta optimization dynamic differential annealed optimization

1. Introduction

Civil engineers often use pile foundations that transmit the building loads to the underneath soil or rock. Piles socketed in rock will transmit the loads to the ground through either resistance from shafts, end bearing, or a mix of both. When shallow soil cover is unsecured and loose, lying on top of the bedrock, rock-socketed piles become one of the most suitable solutions. In such cases, the lateral strength of the rock can develop a high bearing capacity with minimal pile displacement.¹ Greater design and higher load-carrying capacity demand of piles have increased the need for socketed piles. The empirical and analytical procedures can be used to design the rock-socketed piles.

Further full-scale model testing of piles is desired.² Based on the results, one can draw a conclusion that all the existing methods of rock-embedded pile placement are equally practical. Even though because the pile's manner is complicated, the methods might not yield precise estimations.³ Therefore, there is a need for improving a novel soft calculating framework capable of providing accurate predictions for the settlement of pile (SP), a critical factor in pile arranging. SP is a crucial element that should be taken into account during pile design due to its potential impact on the stability and safety of structures.^4,5

Geotechnical engineers face the challenge of finding a practical and novel predicting model with satisfactory performance for pile settlement prediction despite the introduction of several design methods. Research has indicated that the precision of pile performance estimation relies heavily on the intake variables’ choice. Hence, the subsequent sentences summarize pertinent research studies that identify the suitable intake components for predicting SP.^6,7

According to,⁸ the SP is affected by various factors involving the load on the pile, the pile length, the shear modulus, the diameter of the pile, and the radial distance where the shear stress becomes insignificant. In contrast, prior research has suggested that the rocks’ UCS plays a substantial role in determining pile capacity and, subsequently, in SP.^9,10 A predictive model using data mining techniques was generated to estimate the foundations’ settlement based on the standard penetration test results. The model was trained using approximately 1000 data points obtained from various sources and was subsequently expanded to incorporate field measurements.¹¹

Hence, a new soft computing model has to be developed to accurately predict SP, being one of the important factors in the design of piles. Artificial intelligence (AI) and machine learning (ML) within the discipline of geotechnical engineering have come up much stronger over the last few years.^12,13 One of the most dominant ML variants, Predicting behavior for the ANFIS, has emerged as a high point for geotechnical systems. ANFIS is a hybrid model that combines strengths from both fuzzy logic and neural networks, hence this model can handle complicated and uncertain data.^14–17

This study aims to present an effective prediction model in the estimation of SP in rocksocketed foundations using ANFIS. In the light of such a statement, the objective of the present study is to integrate three advanced optimization algorithms to enhance the accuracy and reliability of SPs. These will include RKO, DDAO, and ALO. RKO is one of the well-established numerical methods helpful in providing a solution to differential equations and yielding precise iterative solutions. DDAO represents a novel methodology that couples differential evolution principles with simulated annealing principles, hence offering enhanced global search capability and convergence speed. ALO draws its inspiration from the natural behaviors of ant colonies and lions; it merges their optimization strategies to handle multi-objective problems efficiently. In this respect, the performance of the ANFIS model, with and without the optimization algorithms, is evaluated rigorously by various statistical metrics that include Mean Square Error (MSE), RMSE, and the correlation coefficient (R²). These metrics provide a comprehensive view of predictive accuracy and model reliability. The results illustrate that the ANFIS model, when combined with proposed optimization algorithms, significantly improved from the one without optimizations. Optimized models show lower values of MSE and RMSE and higher values of R², underlining the effectiveness of the optimization techniques in refining the predictive capability of the ANFIS model.

This paper proposes developing a new approach to estimating pile settlement using the ANFIS integrated with three different optimization algorithms: FA, PSO, and GA. Based on the results obtained, the optimized ANFIS model represents high accuracy and better performance, which will be very useful for geotechnical engineers. This technique provides more accurate results in SP and can be extended for optimum design of pile foundations, thus leading to more reliable and efficient civil engineering.

2. Datasets and technique

2.1. Data collection

The most recent attempt to reduce traffic congestion in Malaysia's bustling capital city of Kuala Lumpur is the Klang Valley Mass Rapid Transit (KVMRT). After conducting a site assessment, it was discovered that numerous bored piles would be necessary for ensuring station reliability for the KVMRT in Kuala Lumpur. According to Figure 1, the project's Malaysia location encompasses diverse rock foundations, such as phyllite, limestone, granite, and sandstone, necessitating the installation of numerous piles.

Figure 1.

KVMRT location.

This research focused on 96 granite rock piles in a specific area.¹⁸

Based on the data presented in Table 1, it can analyze the input variables and their connection to the output variable for the predictive model of SP as follows:

Table 1.

Statistical properties of inputs and output.

Indicators	Variables
	Input					Targets
	Lp/D	Ls/Lr	N_SPT	UCS	Qu	Sp
	(%)	(%)	-	(MPa)	(kPa)	(mm)
Mini.	4.33	0.29	2.92	25.32	12408.76	4.49
Max.	31.56	31.71	166.42	68.49	42700.73	20.09
Avg.	15.37	7.06	80.03	43.41	24539.99	10.99
St Dev.	6.585	6.596	59.38	12.50	8071.82	3.709

The input variables are all related to the characteristics of the pile and the rock into which the pile is socketed. The relation between the input and output is as follows:

Lp/D (Pile Length to Diameter Ratio):

- This ratio affects the structural stability and load distribution of the pile. A higher ratio generally implies a deeper pile, which can affect settlement differently based on soil and rock conditions.

Ls/Lr (Length of Socket to Rock Length Ratio):

- This ratio represents the portion of the pile embedded in rock relative to the total pile length. A higher ratio usually indicates better anchorage in the rock, potentially reducing settlement.

N_SPT (Standard Penetration Test Value):

- This measure indicates the density and strength of the soil layers above the rock. Higher SPT values usually suggest denser soil, which can support more load with less settlement.

UCS (Unconfined Compressive Strength of Rock):

- UCS represents the strength of the rock itself. Stronger rock (higher UCS) provides a more stable foundation, likely reducing settlement.

Qu (Ultimate Bearing Capacity):

- This is the maximum load that the pile foundation can bear. Higher bearing capacity generally correlates with lower settlement, as the foundation can support more weight without significant deformation.

By integrating these input variables into the ANFIS model, the system can predict the output variable (Sp), which is the settlement of the pile. The optimization algorithms (RKO, DDAO, ALO) further refine this prediction by adjusting the parameters precision of the ANFIS model, hence improving reliability.

Moreover, granite rock was found nearby and is thought to have come from the San Trias formation. The geological characteristics that prevailed in the subsurface materials were examined at the pile positions. In light of the findings, it was discovered that the subsoil profiles consisted of residual rocks. The maintained data revealed that the bedrock depth ranges from 70 centimeters to almost 1400 meters below the surface. The bore log details and information on field sampling are elaborated further in the subsequent paragraphs.

1) The study's observed rock masses showed a range of weathering conditions, from lightly to heavily worn. The strength and stability of the rocks are influenced by their mechanical and physical properties, which can be understood from the degree of weathering.

2) The International Society for Rock Mechanics (ISRM) determined UCS values ranging from 25 to 68 (MPa). Notably, the lowest and highest values, which represent the variations in rock strength within the observed formations, were determined empirically. These numbers are important for comprehending the rock masses’ structural integrity and load-bearing capacity. The primary crucial action for creating an estimation framework is gathering a dataset with influential independent variables. Identifying and outlining the crucial features that affect the framework's outcome is imperative.

3) Bore log data revealed highly weathered soil down to a depth of 16.5 meters, providing additional insight into the subsurface conditions. Hard sandy mud was the predominant soil composition in this layer; this conclusion was corroborated by Standard Penetration Test (N_SPT) values, which ranged from 4 to 167 blows per 300 mm, at minimum. These N_SPT values provide information about the soil's resistance to penetration, which is helpful when evaluating foundation design and geotechnical factors.

4) Most subsoil materials showed N_SPT values greater than 50 blows per 300 mm in the 7.5 to 27.0 m depth range. This suggests a shift to more compact and possibly stable geological formations, indicating a notable increase in soil resistance and density.

Moreover, a computer equipped with an NVIDIA GeForce GT 640 GPU, 1 terabyte of storage, a 64-bit operating system (Windows 10 Enterprise), an 11th Gen Intel(R) Core(TM) i7-1165G7 @ 2.80 GHz CPU, and 16.0 GB of RAM was used for the research. Python was used as the programming language, sci-kit-learn was used for machine learning frameworks, and Pandas, NumPy, and Matplotlib were used for data analysis and visualization. The runtime trains and assesses each model on the 96 samples. The training times for ANFIS, ANAL, ANRK, and ANDD were one second, fifty-two seconds, ninety seconds, and sixty seconds, respectively.

2.2. Adaptive-Network-Based Fuzzy Inference System (ANFIS)

The ANFIS network structure comprises two components: the consequence and the premise. Training ANFIS involves using an optimization algorithm to identify the parameters related to these parts. Within the training phase, ANFIS utilizes input-output data pairs to obtain IF-THEN fuzzy rules that connect the premise and consequence parts.^19,20

The ANFIS structure has five layers, four rules and four membership functions. In this structure of the ANFIS, there are two inputs and one output. The layer structure of ANFIS based on the given structure shown in Figure 2 will be explained as follows:

a)
layer 1

Figure 2.
The ANFIS structure.

The first layer in ANFIS is known as a fuzzification layer. It uses membership functions to get fuzzy clusters from the input values.²¹ The shape of the membership function depends on parameters that are also referred to as premise parameters. The set of parameters {a, b, c} is termed the premise parameter set, which is utilized in the fuzzification layer to obtain fuzzy clusters from the values of input. These parameters are used to resolve each membership function's degrees as given by equations (1) and (2). The resulting membership degrees from this layer are represented as $γ_{x}$ and $γ_{y}$ .
$\begin{aligned} γ_{A i} (x) & = g b e l l m f (x; a, b, c) = \frac{1}{1 + {| \frac{x - c}{a} |}^{2 b}} \end{aligned}$
(1)

$\begin{aligned} O_{i}^{1} & = γ_{A i} (x) \end{aligned}$
(2)
layer 2

It is called the rule layer, and firing strengths ( $w_{i}$ ) each rule is computed here, depending on the membership values achieved from the fuzzification layer. The computation of $w_{i}$ values involve multiplying the membership values matching to the below formula:
$O_{i}^{2} = w_{i} = γ_{A i} (x) \times γ_{B i} (y) i = 1, 2$
(3)
layer 3

The layer that realizes the normalization of firing strengths is called normalization layer. It computes the normalized firing strengths of each rule. The normalized firing strength of the ith rule is the ratio of its firing strength to the sum of all firing strengths as in equation (4):
$O_{i}^{3} = \bar{w_{i}} = \frac{w_{i}}{w_{1} + w_{2} + w_{3} + w_{4}} i ϵ 1, 2, 3, 4$
(4)
layer 4

This layer is referred to as the defuzzification layer. At each node of this layer, the weighted values of rules are computed using a first-order polynomial, which is represented in equation (5):
$O_{i}^{4} = \bar{w_{i}} f_{i} = \bar{w_{i}} (Z_{i} x + q_{i} y + r_{i})$
(5)

The set of parameters { $Z_{i}$ , $q_{i}$ , $r_{i}$ } are known as consequence parameters, and they determine the output $\bar{w_{i}}$ of the normalization layer. The number of inputs is one less than the number of consequence parameters for each rule. b)
layer 5

The summation layer is the layer that sums the outputs of each rule in the defuzzification layer to obtain the final output of ANFIS.
$O_{i}^{5} = \sum_{i} \bar{w_{i}} f_{i} = \frac{\sum_{i} w_{i} f_{i}}{\sum_{i} w_{i}}$
(6)
2.3. Antlion optimizer (ALO)

It is necessary to provide the basic version of the ALO algorithm before delving into its multi-objective iteration. The ALO algorithm originates from their engagement with ants and derives its design from the tracking methods of antlions.²² This method involves a dual population of ants and antlions, where the key duty of the former is to traverse the search area through random motion.²³ The antlions in the algorithm are responsible for preserving the finest location acquired by the ants and adjusting their location as the ants’ position progresses. An elite antlion in the exploration area also influences the ants’ movement despite their spatial range.²⁴

If an antlion discovers a more favorable position than the elite antlion, it becomes the new elite. Once the stopping criterion is met, the spot of the superior antlion is deemed the ultimate ideal solution.²⁵ To replicate the dynamics between antlions and ants, the movement of ants is initially simulated within the search space, followed by the predatory conduct of the antlions. The subsequent equation formulates the stochastic motion of ants:

X (t) = [0, c u m s u m (2 r (t_{1}) - 1), c u m s u m (2 r (t_{2}) - 1), \dots, c u m s u m (t r (t_{n}) - 1)]

(7)

The function $c u m s u m$ computes the cumulative sum of the elements, while n denotes the highest number of iterations, and t indicates the random walk step. The stochastic function $r (t)$ is determined by the subsequent formula:

r (t) = {\begin{array}{lllll} 1 if r a n d > 0.5 \\ 0 if r a n d \leq 0.5 \end{array}, r a n d \in [0, 1]

(8)

Due to the inherent constraints of any search space, the variables range of $E q . (8)$ is not able to be straightly utilized to update the ants’ position. The ants’ motion is regularized using Equation to navigate at random. (9)

x_{i}^{t} = \frac{(x_{i}^{t} - e_{i}) \times (d_{i}^{t} - c_{i}^{t})}{(l_{i} - e_{i})} + c_{i}^{t}

(9)

In Eq. (9), $e_{i}$ denotes the lowest possible value for the ith variable, while $l_{i}$ represents the highest possible value for the random walk. The $d_{i}^{t}$ is the maximum value of the $i t h$ variable at the $t t h$ iteration, while $c_{i}^{t}$ represents the minimum value of the $i t h$ variable at the $t t h$ iteration. Eq. (10) simulates antlion pits:

{\begin{array}{lllll} c_{i}^{t} = A n t l i o n_{j}^{t} + c^{t} \\ d_{i}^{t} = A n t l i o n_{j}^{t} + d^{t} \end{array}

(10)

The equation involves $A n t l i o n_{j}^{t}$ , which refers to the $j t h$ antlion at the $t t h$ iteration. It uses a roulette wheel to simulate the hunting skills of ants. If an ant falls into a pit and becomes stuck, the antlion will throw rocks at the pit's periphery. The mathematical model is given below:

{\begin{array}{lllll} c^{t} = \frac{c^{t}}{I} \\ d^{t} = \frac{c^{t}}{I} \end{array}

(11)

This equation defines I as a ratio and can be defined in the following manner:

I = 10^{w} \frac{t}{T}, w = {\begin{array}{lllll} 2 when t \leq 0.1 T \\ 2 when t > 0.1 T \\ 3 when t > 0.5 T \\ 4 when\; t > 0.75 T \end{array}, \begin{array}{lllll} 5 when t > 0.9 T \\ 6 when t > 0.95 T \end{array}

(12)

The t represents the current iteration, while the maximum iteration number is shown by T, and the w represents a constant value in Eq. (12).

When the ant becomes trapped in the pit, the final stage of hunting of this algorithm occurs. At this point, the antlion must update its position about where the ant was trapped to improve future hunts’ chances. This process is illustrated by the equation below:

\begin{aligned} A n t l i o n_{j}^{t} = A n t_{i}^{t} i f f (A n t_{j}^{t}) < f (A n t l i o n_{j}^{t}) \end{aligned}

(13)

Elitism is a common technique used in evolutionary algorithms to preserve the best solution found throughout the optimization process. The top-performing antlion is kept as the elite in the ALO algorithm at every level. Then, ants—regardless of how close they are to the elite—move in response to the elite. The equation below illustrates how the elite is simulated in the ALO algorithm:

A n t_{i}^{t} = \frac{Z_{A}^{t} + Z_{E}^{t}}{2}

(14)

The equation presented here involves two random walks, namely $Z_{E}^{t}$ and $Z_{A}^{t}$ , representing the movement around the antlions selected by the elite antlion and the roulette wheel in the $t t h$ iteration.²⁶

2.4. Runge Kutta optimization (RKO)

According to (RKM), It was employed to compute first-order differential equation solutions, comes the RUN optimization algorithm.^27,28 The Runge Kutta method (RKM) generally yields a precise numerical result based on functions alone, without the need for any gradient information. Calculating slope within the RKM framework is crucial to the functioning of the RUN optimization algorithm to simulate the exploration abilities of swarm-based optimization as it enables the algorithm. The stages that make up the mathematical formulation of the RUN algorithm are expanded upon below:

The initialization stage entails generating the first solutions for N agents based on the search space boundaries $[L B, U B]$ . This is accomplished by employing the subsequent equation:

\begin{aligned} Z_{i j} = L B_{j} + r_{1} \times (U B_{j} - L B_{j}) \\ i = 1, 2, \dots, N, j = 1, 2, \dots, P \end{aligned}

(15)

The formula considers the problem's dimension, indicated by P, $L B_{j}$ and $U B_{j}$ represents the lower and upper limits of the $j t h$ variable in the solution set $Z_{i j}$ , where i ranges from 1 to N, indicating the total number of search agents.

At each iteration, the RKO algorithm uses a search mechanism (SM) during the updating solutions stage that uses the RKM to change the location of the current solution. This mechanism is described in the following:

Z_{i} = {\begin{array}{lllll} Z_{C F} + S_{F M} + μ \times r a n d n \times Z_{m c}, if r a n d \leq 0.5 \\ Z_{m F} + S_{F M} + μ \times r a n d n \times Z_{r a}, otherwise \end{array}

(16)

In Eq. (18), $Z_{C F} = (Z_{c} + r \times S F \times g \times Z_{c})$ and $S_{F M} = S F \times S M$ . $Z_{r a} = (Z_{r 1} - Z_{r 2})$ , $Z_{M F} = (\; Z_{M} + r \times S F \times g \times Z_{m}) \;$ and $Z_{M C} = (Z_{m} - Z_{c})$ . The search direction can be changed by using the integer number r, which is between −1 and 1. Conversely, the random numbers represented by the symbols $μ$ and g range from 0 to 1 and 0 to 2, alternatively. The following determines the adaptation factor SF:

\begin{aligned} SF & = 2 \times (0.5 - rand) \times f \\ f & = a \times exp (- b \times r a n d \times (\frac{t}{t m a x})) \end{aligned}

(17)

$t m a x$ represents the total iteration numbers. The values of $Z_{m}$ and $Z_{c}$ utilized in Eq. (16) are described by:

\begin{aligned} Z_{c} & = φ \times Z_{i} + (1 - φ) \times Z_{r 1} \end{aligned}

(19)

\begin{aligned} Z_{m} & = φ \times Z_{b} + (1 - φ) \times Z_{p b} \end{aligned}

(20)

A randomly developed integer, denoted with the $φ$ , ranging from 0 and 1, is included in Eq. (20). Here, $Z_{p b}$ and $Z_{b}$ denote the best-so-far agent and the best agent at each iteration, respectively. With the use of the following formula, the SM parameter found in Eq. (18) is updated:

\begin{aligned} SM & = \frac{1}{6} (Z_{R K}) Δ Z; \\ Z_{R K} & = k_{1} + 2 \times k_{2} + 2 \times k_{3} + k_{4} \\ k_{1} & = \frac{1}{2 Δ Z} (rand \times Z_{w} - u \times Z_{b}) \\ k_{2} & = \frac{1}{2 Δ Z} (rand \times (Z_{w} + r a n d_{1} \times k_{1} \times Δ Z) - UZ \\ k_{3} & = \frac{1}{2 Δ Z} (rand \times (Z_{w} + r a n d_{1} \times (\frac{1}{2} k_{2}) \times Δ Z) - U Z_{b} \\ k_{4} & = \frac{1}{2 Δ Z} (rand \times (Z_{w} + r a n d_{1} \times k_{3} \times Δ Z) - U Z_{b 2} \\ u & = r o u n d (1 + r a n d) \times (1 - r a n d) \\ U Z & = (U \times Z_{b} + r a n d_{2} \times k_{1} \times Δ Z) \\ U Z_{b} & = (U \times Z_{b} + r a n d_{2} \times (\frac{1}{2} k_{2}) \times Δ Z) \\ U Z_{b 2} & = (U \times Z_{b} + r a n d_{2} \times k_{3} \times Δ Z) \end{aligned}

(21)

The symbols $r a n d_{1}$ and $r a n d_{2}$ represent the numbers of random. The $Δ Z$ value is computed by:

\begin{aligned} Δ Z & = 2 \times r a n d \times | S t p | \\ S t p & = r a n d \times ((Z_{b} - r a n d \times Z_{a v g}) + y) \\ y & = r a n d (Z_{n} - r a n d \times (u - l)) \times exp (- 4 \times \frac{t}{t m a x}) \end{aligned}

(22)

The following equations are used to update the values of $Z_{w}$ and $Z_{b}$ :

\begin{aligned} i f f (Z i) & < f (Z p b) \end{aligned}

\begin{aligned} Z b & = Z i \end{aligned}

\begin{aligned} Z w & = Z p b \end{aligned}

\begin{aligned} e l s e \end{aligned}

\begin{aligned} Z b & = Z p b \end{aligned}

\begin{aligned} Z w & = Z i \end{aligned}

\begin{aligned} e n d \end{aligned}

Various operators are employed to ameliorate avoiding local optima and the convergence rate during the stage of the Enhanced Solution Quality. The goal is to improve the quality of solutions, and the following procedure helps to accomplish that:

\begin{aligned} \begin{aligned} Z_{n e w 2} & = {\begin{array}{lllll} Z_{n e w 1} + r \times ω \times | (Z_{n e w 1} - Z_{a v g}) + r a n d n |, & if ω < 1 \\ (Z_{n e w 1} \times Z_{a v g}) + r \times ω \times Z_{n a} & otherwise \end{array} \\ Z_{n a} & = | (u \times Z_{n e w 1} - Z_{a v g}) + r a n d n |, c = 5 \times r a n d \\ ω & = r a n d (0, 2) . exp (- c (\frac{t}{t m a x})), Z_{a v g} = \frac{Z_{r 1} + Z_{r 2} + Z_{r 3}}{3} \end{aligned} \end{aligned}

(23)

\begin{aligned} \begin{aligned} Z_{n e w 1} & = δ \times Z_{a v g} + (1 - δ) \times Z_{b} \end{aligned} \end{aligned}

(24)

In the formula in Eq. (25), a random number between 0 and 1 and an integer number, r, which might have the values 1, 0, or −1, are utilized. According to,²⁷ there is another chance to update the value of $\; Z_{i}$ in the event that Znew2's fitness value is not higher than Zi's. This may be accomplished by applying the following equation:

\begin{aligned} Z_{n e w 3} & = (Z_{n e w 2} - r_{1} \times Z_{n e w 2}) + S F \times D_{Z} \\ D_{Z} & = (r_{2} \times Z_{R K} + (v \times Z_{b} - Z_{n e w 2})) \end{aligned}

(25)

There are three random values in this equation $r_{1}$ , $r_{2}$ and $r_{3}$ . Two times the difference between $r_{3}$ and 0.5 is the value of v, here $r_{3}$ is a random number between 0 and 1.¹⁵

2.5. Dynamic differential annealed optimization (DDAO)

The procedure of enhancing the quality of metal from low to high has inspired the development of the dynamic differential annealed optimization (DDAO) algorithm. In this section,¹⁶ Mathematically, DDAO is able to be expressed as the following:

a)
Initially, the steel's mass comprises molecule groups representing the set of potential solutions. The aim is to improve these solutions and converge them into a single combination of martensite and ferrite, representing the optimum solution.
b)
During the manufacturing process of dual-phase steel, different phases of steel can form as the temperature decreases. This process is similar to the iterative procedure used in mathematical optimization, where an optimum solution is searched for at each iteration.
c)
The internal energy of each metal phase corresponds to the objective function value in mathematical optimization.
d)
The following equation has been proposed to represent the cooling process involving various cooling rates, such as air cooling, slow cooling, and accelerated cooling.
$T^{k} = (T_{c i} - T_{c j}) + T_{r}$
(26)

The proposed equation for the cooling operation involves a new solution, $T^{k}$ , which is suggested for each iteration number (k), ranging from 1 to n. Here, n shows the total number of iterations. Additionally, the equation involves two randomly selected solutions, $T_{c i}$ and $T_{c j}$ , from the population, with indices $(i)$ and $(j)$ , respectively. The solution, $T_{r}$ , is generated randomly within the problem's search area and is not part of the existing population. e)
To mathematically model the mechanical operation that occurs during the differential cooling process of the metal, it is necessary to account for the rolling process. However, the metal is assumed to undergo forging rather than rolling for programming convenience. The parameter that represents the dynamic behavior of the forging hammer can be modeled as a fluctuating value among a random number and 1.
$f = {\begin{array}{lllll} 1 & if r e m (i t e r a t i o n, 2) = 1 \\ r a n d o m [0, 1] & if r e m (i t e r a t i o n, 2) = 0 \end{array}$
(27)

The forging process is mathematically modeled by representing the dynamic behavior of the hammer with a parameter that varies between 1 and a random number. This parameter, denoted as f, is calculated using Eq. (28), where rem represents the remainder after dividing by 2. When the iteration number is odd (1, 3, 5, etc.), f is set to 1; when the iteration number is even, f is a random number between 0 and 1. Due to the online forging process with differential cooling, the employed force via the hammer fluctuates between a constant and a random value. Therefore, Eq. (27) is modified as follows:
$T^{k} = (T_{c i} - T_{c j}) + T_{r} \times f$
(28)
f)
Adopting a similar approach to the real annealing process is suggested to improve the optimization process, which is more likely to accept the creation of new phases at higher than lower temperatures. This can be achieved using the probability formula outlined in the SA algorithm.²⁹
$\begin{aligned} P & = e^{\frac{- Δ E}{v}} \end{aligned}$
(29)

$\begin{aligned} Δ E & = \frac{c o s t (T^{k}) - c o s t (T_{L})}{c o s t (T_{L})} \end{aligned}$
(30)

The formula for determining the acceptance of the chance of a novel solution is given by the equation, where P represents the probability, ΔE is the difference among the objective value of the solution SL, which has an index of L within the population, and the objective value of the suggested solution from Eq. (29). The value of L ranges from 1 to the size of the population. During each iteration, the temperature variable is represented by v, which must have a high initial value and be consistently updated to decrease. A proposed solution will only be accepted if its acceptance probability, P, exceeds a random number ranging from 0 to 1. At the beginning of the search, v is set to a high value, causing P to approach 1 according to the formula in Eq. (29). To clarify, a broad random numbers range is able to be less than one, leading to the selection of a solution. However, at low values of v, the probability P approaches zero, making it less likely for a narrow random numbers range to be less than P, resulting in a lower likelihood of selecting a solution. This can be understood by referring to Eq. (30). g)
The described procedure is repeated beginning from step d, and the most favorable solution is retained at the end of each iteration.

2.6. Performance evaluation methods

Indicators are presented in this section to assess the hybrid models. These indicators reveal the error level and correlation of the evaluated models. The indicators involve coefficient correlation (R²), mean absolute error (MAE), root mean square error (RMSE), various account factors (VAF), and weight absolute percentage error (WAPE), which the mathematical equations have been shown as:

\begin{aligned} R M S E & = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(m_{i} - b_{i})}^{2}} \end{aligned}

(31)

\begin{aligned} M A E & = \frac{1}{n} \sum_{i = 1}^{n} | b_{i} - m_{i} | \end{aligned}

(32)

\begin{aligned} R^{2} & = {(\frac{\sum_{i = 1}^{n} (b_{i} - \bar{b}) (m_{i} - \bar{m})}{\sqrt{[\sum_{i = 1}^{n} {(b_{i} - \bar{b})}^{2}] [\sum_{i = 1}^{n} {(m_{i} - \bar{m})}^{2}]}})}^{2} \end{aligned}

(33)

\begin{aligned} V A F & = (1 - \frac{v a r (b_{i} - m_{i})}{v a r (b_{i})}) * 100 \end{aligned}

(34)

\begin{aligned} W A P E & = max [\frac{| b_{i} - m_{i} |}{b_{i}}] \end{aligned}

(35)

In Eqs. (31–35), n determine the sample number, $b_{i}$ determines the foresaw value, $\bar{b}$ and $\bar{m}$ are the average value of predicted and measured value, alternatively, and $m_{i}$ defines the measured value.

2.7. Hyperparameter

Hyperparameter optimization was carried out in this study to improve the predictive performance of the ANFIS model. Thorough research and analysis were used to identify the ideal hyperparameters.

The outcome of the ANFIS hyperparameter optimization is displayed in Table 2. Three variants (ANKR, ANDD, and ANAL) of the ANFIS models were optimized, with unique hyperparameter configurations made for each model. Most notably, the optimization process produced optimized values for parameters, including the minimum improvement threshold, maximum number of iterations, and number of clusters. These comprised specifications like the maximum depth, minimum sample leaf size, and minimum sample split size.

Table 2.
The results of hyperparameters optimization for ANFIS.

Models Hyperparameter

NumClusters MaxNumIteration MinImprovement

ANRK 19 19 15.348

ANDD 15 23 19.523

ANAL 17 11 12.548

Models	Hyperparameter
ANRK	19	19	15.348
ANDD	15	23	19.523
ANAL	17	11	12.548

2.8. Convergence

Since convergence indicates the consistency and effectiveness of algorithms in resolving problems, it is a crucial property in many mathematical and computational processes. Comprehending convergence facilitates comprehension of algorithmic behavior, performance assessment, and application suitability. The result of convergence using developed hybrid models is displayed in Figure 3.

Figure 3.

Result of convergence of hybrid models based on MSE.

3. Result and discussion

3.1. Models comparison

Numerous hybrid models have been developed to predict the SP, including ANFIS - Multi-objective ant lion optimizer (ANFAL), ANFIS - Dynamic Differential Annealed Optimization (ANFDA), and ANFIS- Runge Kutta optimization (ANFRK). During the training and testing phases, the researchers compared the measures derived from experiential tests with the predictions made by three models: ANFAL, ANFDA, and ANFRK. The training phase utilized 70% of the experimental results, whereas the testing phase used the remaining 30% of the workforce, as demonstrated in Table 3. Five statistical evaluators (RMSE, MAE, R², and WAPE) were employed to compare the efficacy of the algorithms utilized and comprehensively assess them. A model with an R² value approaching 1 indicates superior performance in both the training and testing phases. Other parameters, such as RMSE, MAE, and WAPE, demonstrate the extent of error, a lower number in the model denoting a more tolerable degree of mistake. In Table 3, these metrics were employed for comparing the methods’ effectiveness and comprehensively evaluated employed, and their outcomes are compiled. The created models’ statistical performance criterion values were quite near to each other in both the training and testing phases. However, the hybrid model ANFDA displayed the highest level of accuracy with an R² of 0.998 in the testing phase and 0.989 in the training phase. This model's RMSE, MAE, and WAPE are the lowest among the other models, indicating that ANFDA demonstrated the most significant level between the expected and actual values’ degree of agreement. However, ANFAL models perform the worst, with R² values of 0.973 and 0.983 in the train and test phases. This model also has the worst RMSE, MAE, and WAPE, all of which point to its poor performance.

Table 3.
The outcomes were achieved from the hybridized models.

Model

Metric Phase ANFRK ANFDD ANFMO

$R^{2}$ Train 0.983 0.989 0.973

Test 0.987 0.998 0.983

$R M S E$ Train 0.507 0.379 0.617

Test 0.434 0.165 0.490

$M A E$ Train 0.446 0.316 0.494

Test 0.358 0.135 0.401

$W A P E$ Train 0.039 0.027 0.043

Test 0.035 0.013 0.039

$V A F$ Train 98.25 98.93 97.28

Test 98.72 99.80 98.28

		Model
$R^{2}$	Train	0.983	0.989	0.973
Test	0.987	0.998	0.983
$R M S E$	Train	0.507	0.379	0.617
Test	0.434	0.165	0.490
$M A E$	Train	0.446	0.316	0.494
Test	0.358	0.135	0.401
$W A P E$	Train	0.039	0.027	0.043
Test	0.035	0.013	0.039
$V A F$	Train	98.25	98.93	97.28
Test	98.72	99.80	98.28

3.2. Comparative analysis of related studies

Table 4 presents a comparative analysis of models utilized in related studies, focusing on two critical metrics: the R² and RMSE. These will provide the evaluation for the precision and reliability of predictive models in geotechnical engineering. Four models are considered: ANFDD, SVR-HGSO, GEP, and RF. In each model's case, R2 and RMSE values were given, considering training and testing phases that reflect the performance in different scenarios.

Table 4.
Comparative analysis of models in related studies.

Models

Metric Phase ANFDD SVR-HGSO³⁰ GEP³¹ RF¹⁰

R² Train 0.989 0.996 0.872 0.969

Test 0.998 0.998 0.861 0.866

RMSE Train 0.379 0.289 1.293 47.333

Test 0.165 0.285 1.656 98.161

		Models
R²	Train	0.989	0.996	0.872	0.969
Test	0.998	0.998	0.861	0.866
RMSE	Train	0.379	0.289	1.293	47.333
Test	0.165	0.285	1.656	98.161

The performance of the models varied in the training phase. ANFDD yielded an impressive R2 of 0.989, thus showing that the predicted and actual values correlated well. Similarly, SVR-HGSO yielded an even higher R2 of 0.996. In their turn, GEP and RF were able to attain respective R2 values of 0.872 and 0.969. Similarly, RMSE values also reflected similar inconsistencies whereby ANFDD and SVR-HGSO yielded notably low RMSE values compared to GEP and RF models. For the testing, SVR-HGSO does not disappoint as it comes in at an R2 value of 0.998-a very powerful result-whereas ANFDD's performance was high. However, GEP and RF came up with lower R2 values. The RMSE during testing is similar, with SVR-HGSO and ANFDD performing very well, whereas GEP and RF came up with a higher RMSE value, depicting reduced predictive accuracy over unseen data.

3.3. Visual representation of model performance

In Figure 4, the R2 and RMSE are the main criteria for evaluating the hybrid model performance in the scatter plot. $X = Y$ coordinates give the centerline of the plot that serves as the reference point for comparison of model performance. The accuracy of a model is based on how close data points will be to that centerline.

Figure 4.

The scatter plot of presented frameworks in the training and testing phase.

Closer observation shows the model ANFDA gives the least dispersion; the data points are closely clustered around the centerline. The models ANFRK and ANFALO give relatively similar performances, as can also be obtained from their wider dispersions that their data points give, which insinuate less accuracy in alignment. This analysis gives rather good insight into the comparative strengths and weaknesses of these hybrid models.

The observed dispersion was influenced by the difference between the measured and predicted values, which was found to be significantly lower during the testing phase, as shown in Figure 5. The ANFDA model demonstrated minimal dispersion during the training phase, and the angle difference between the center line and linear fits was greater in training than in testing. Although discrepancies were observed among the measured and predicted values of specific samples during the training phase, resulting in significant differences, Performance gains, and desired learning have somewhat made up for this shortcoming.

Figure 5.

The scatter plot to compare the values that were measured with those that were expected.

Another analysis that needs to be conducted is monitoring the percentage error per pile, which reflects the deviation of the actual settlement rate from the model. Figures 6 and 7 aim to demonstrate each type of model's effectiveness in forecast pile settlement and compare their performance. On the basis of the error distribution chart, the accuracy of the ANFDA, ANFAL, and ANFRK models in predicting pile settlement varied. ANFDA showed the least amount of error, with the majority of the predicted values closely aligning with the observed values. ANFRK had a moderate level of inaccuracy, exhibiting a wider range of anticipated values.

Figure 6.

The box-normal plot for error of improved hybrid ones.

Figure 7.

The error percentage of the hybrid models.

In contrast, ANFAL demonstrated the highest level of inaccuracy, with some projected values that diverge significantly from the measured values. Overall, the ANFDA model displayed the most reliable performance, while ANFAL had the poorest performance, and ANFDA fell in between. The error distribution chart provided valuable information on the relative advantages and restrictions of each model's capacity for prediction, assisting researchers in choosing the best model to forecast $S p$ in real-world scenarios.

3.4. Statistical analyses

The Wilcoxon signed-rank test was applied in order to check the differences in predictive performances of the hybrid ANFIS models. This non-parametric test is appropriate for paired data on any significant statistical difference between models. Results of the Wilcoxon test are shown in Table 5. The p-values for all pairwise comparisons are comfortably above 0.05 to confirm that there is no significant statistical difference among the models. Specifically, paired comparisons between ANFRK-ANFDD (p = 0.131), ANFRK-ANFMO (p = 0.247), and ANFDD-ANFMO (p = 0.945) favor that the predictive performance of the models is comparable, with no model statistically performing better than another.

Table 5.
Statistical analyses result based on the Wilcoxon method.

Index

Models p_value stat

ANFRK-ANFDD 0.131 1915

ANFRK-ANFMO 0.247 2011

ANFDD-ANFMO 0.945 2309

	Index
ANFRK-ANFDD	0.131	1915
ANFRK-ANFMO	0.247	2011
ANFDD-ANFMO	0.945	2309

4. Conclusion

This article's key goal is to assess the effectiveness of 3 hybrid ANFIS frameworks for forecasting the stability of rock-embedded piles. The ANFIS frameworks are built by 3 optimizing methods, namely Ant lion optimizer (ALO), DDAO, and RKO. The assessment was conducted using a dataset comprising pile driving evaluator test outcomes and the characteristics of both the earth and the piles. The study yielded the following main results:

This study found that all three ANFIS models show a high potential in predicting SP, with the lowest R² of 0.983 in training and 0.973 during the testing phase. The distribution of data samples around the best-fit line, however, indicated that the ANFDA model outperformed the ANFAL and ANFRK models in estimating tiny SP values. The DDAO optimization technique showed better results over the whole SP value spectrum.

The performance of ANFAL is weaker than the other ANFIS models based on all statistical indices, but its results are still acceptable with R², RMSE, MAE, WAPE, and VAF values of 0.973, 0.617, 0.494, 0.043, and 97.28, respectively. In contrast, the ANFDA model shows the most appropriate performance as it exhibits the highest R², RMSE, MAE, WAPE, and VAF values in both the testing and training phases, except for MAE in the training phase.

While all three hybrid models have given a high degree of accuracy in their predictions, statistical analysis using the Wilcoxon signed-rank test showed no significant difference among the models.

In general, machine learning models can be relied upon as an alternative to experimental methods for evaluating pile settlement, thereby saving time and energy.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data and code availability

The code and dataset used in this study are available for future research upon request.

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Models	Hyperparameter
Models	NumClusters	MaxNumIteration	MinImprovement
ANRK	19	19	15.348
ANDD	15	23	19.523
ANAL	17	11	12.548

		Model
Metric	Phase	ANFRK	ANFDD	ANFMO
$R^{2}$	Train	0.983	0.989	0.973
$R^{2}$	Test	0.987	0.998	0.983
$R M S E$	Train	0.507	0.379	0.617
$R M S E$	Test	0.434	0.165	0.490
$M A E$	Train	0.446	0.316	0.494
$M A E$	Test	0.358	0.135	0.401
$W A P E$	Train	0.039	0.027	0.043
$W A P E$	Test	0.035	0.013	0.039
$V A F$	Train	98.25	98.93	97.28
$V A F$	Test	98.72	99.80	98.28

	Index
Models	p_value	stat
ANFRK-ANFDD	0.131	1915
ANFRK-ANFMO	0.247	2011
ANFDD-ANFMO	0.945	2309

Prediction of pile settlement using a hybrid Adaptive-Network-Based Fuzzy Inference System

Abstract

Keywords

1. Introduction

2. Datasets and technique

2.1. Data collection

Table 2. The results of hyperparameters optimization for ANFIS. Models Hyperparameter NumClusters MaxNumIteration MinImprovement ANRK 19 19 15.348 ANDD 15 23 19.523 ANAL 17 11 12.548

3.1. Models comparison

Table 4. Comparative analysis of models in related studies. Models Metric Phase ANFDD SVR-HGSO 30 GEP 31 RF 10 R2 Train 0.989 0.996 0.872 0.969 Test 0.998 0.998 0.861 0.866 RMSE Train 0.379 0.289 1.293 47.333 Test 0.165 0.285 1.656 98.161

Table 5. Statistical analyses result based on the Wilcoxon method. Index Models p_value stat ANFRK-ANFDD 0.131 1915 ANFRK-ANFMO 0.247 2011 ANFDD-ANFMO 0.945 2309

Footnotes

Funding

Declaration of conflicting interests

Data and code availability

References

Table 2.
The results of hyperparameters optimization for ANFIS.

Models Hyperparameter

NumClusters MaxNumIteration MinImprovement

ANRK 19 19 15.348

ANDD 15 23 19.523

ANAL 17 11 12.548

Table 4.
Comparative analysis of models in related studies.

Models

Metric Phase ANFDD SVR-HGSO³⁰ GEP³¹ RF¹⁰

R² Train 0.989 0.996 0.872 0.969

Test 0.998 0.998 0.861 0.866

RMSE Train 0.379 0.289 1.293 47.333

Test 0.165 0.285 1.656 98.161

Table 5.
Statistical analyses result based on the Wilcoxon method.

Index

Models p_value stat

ANFRK-ANFDD 0.131 1915

ANFRK-ANFMO 0.247 2011

ANFDD-ANFMO 0.945 2309