Study on Strength Prediction and Material Scheme Optimization for Modified Red Mud Based on Artificial Neural Networks

Abstract

Focusing on the complex nonlinear problems of strength prediction and the material scheme design of modified red mud for use as a road material in engineering applications, a strength prediction neural network is established and utilized to optimize the material scheme, including the compound-solidifying agent ratio, water content, and curing age, based on experimental data accumulated during years of engineering practice and an artificial neural network. In this study, a backpropagation (BP) neural network is adopted, and 114 sets of experimental data are used to train the parameters of the unconfined compressive strength prediction model. Then, using the BP strength prediction model, the material scheme optimization process is carried out, with the strength and material costs as the objectives. The results show that the BP neural network model has a high prediction accuracy, the relative prediction error is basically within 10%, the root-mean-squared error is less than 0.04, and the correlation coefficient is more than 0.99. According to the strength requirements of modified red mud in different road projects and the constraints of each property, an optimal material scheme with a lower cost and higher 7 d target strength is obtained using a mix of polymer agent–fly-ash–cement–speed-cement in a ratio of 0.02%:1.96%:4.78%:0%, with a 33.93% water content of raw red mud, so that the target strength and material cost are 2.987 MPa and 17.099 CNY/T. This study creates an optimal material scheme, incorporating the compound-solidifying agent ratio, curing age, and water content of the modified red mud road material according to the strength requirements of different projects, thereby promoting the popularization of the utilization of red mud with better engineering practicability and economy.

1. Introduction

Red mud is the industrial solid waste discharged from alumina extraction in the aluminum industry. It is alkaline and can pollute surface water and groundwater. With the development of the aluminum industry and the reduction in bauxite grade, the amount of red mud is increasing, and so the comprehensive utilization of red mud is becoming an increasingly important research topic [1,2,3]. The modified red mud is a kind of resource utilization with wide application prospects. Roadbed filling, stabilizing roadbed gravel, improving roadbed soil, prefabricating road construction components, and filling hollow areas are among its applications [4,5,6]. These engineering uses have specific technical requirements for the strength of modified red mud road material. However, the strength of modified red mud road material is affected by many things, such as the amount and ratio of compound-solidifying agents, the amount of water in the material, the curing age [7,8], and more. Some of these factors have an interaction effect on the strength. Addressing these issues will incur significant costs in terms of manpower. Therefore, it is vital to reasonably optimize the material scheme of modified red mud road material.

In previous studies, researchers used linear and nonlinear regression models to predict the unconfined compressive strength of modified red mud road materials based on a substantial number of experiments [9]. Liu [10] utilized reliable and mature multivariate linear equations to fit the long-term compressive strength test data for steel slag fine-aggregate red-mud-based concrete, including five influence factors affecting red mud admixtures, curing age, curing temperature, curing humidity, and 28 d unconfined compressive strength. Fu et al. [11] used a Gauss–Markov linear model to create a binary prediction model that took into account the curing age and the amount of coal system metakaolin. The model was 5% accurate for predicting the compressive strength of red mud that had been changed by the coal metakaolin. The studies, within the confines of the research scope, establish a regression prediction model for the strength of a specific modified red mud material, utilizing a moderate amount of experimental data. However, the strength requirements for using modified red mud as a road material vary due to the different sources of raw red mud, engineering projects, and road types. When it comes to predicting the strength of modified red mud road materials based on different application conditions, the existing prediction models struggle to meet all requirements.

The excellent adaptive learning ability, high parallelism, and fault tolerance of artificial neural networks (ANN) have made them a popular choice for predicting the strength of engineering materials in recent years. ANN, inspired by biological neural systems, are computational models designed to mimic the human brain’s mechanisms for information processing and learning. These networks consist of numerous interconnected artificial neurons, forming network structures through weighted connections. The learning process of an ANN typically employs the backpropagation algorithm, which minimizes the error between outputs and expected values by adjusting the connection weights. Compared to traditional engineering design methods, ANN excel in tasks involving complex nonlinear relationships, pattern recognition, classification, and prediction. Due to their superior adaptability and efficient information-processing capabilities, ANN are considered essential tools for solving complex problems, especially in the context of big data analysis and machine learning. ANN have shown a lot of promise for use, especially in predicting the buckling capacity and compressive strength of solidified soil, checking the performance of structures, and finding the best materials for a job.

For example, Abarkan et al. [12] used an ANN to find the best design of circular hollow-section stainless steel short columns and then compared their findings to Eurocode 3. This showed that their model accurately predicted how different design parameters would affect the columns’ ability to hold weight. Rabi et al. [13,14,15] applied ANN to predict the web-post-buckling capacity of high-strength steel beams with elliptical openings, the sectional load-bearing capacity of cold-formed circular hollow sections (CHS), and the buckling capacity of hot-formed CHS beam-columns. Their results showed that ANN models provided accurate predictions in terms of large-scale data analysis and performed excellently when addressing nonlinear relationships between structural response and material properties. Moreover, they also validated the effectiveness of ANN models with experimental data and existing design standards, emphasizing their potential in complex structural analysis. A study by Jweihan et al. [16] created an empirical model for the rutting-susceptibility index (RSI) of asphalt mixtures. They used a hybrid machine-learning approach that combined the strengths of artificial neural networks (ANN) and gene expression programming (GEP). Their study demonstrated the potential of ANN to improve asphalt pavement design and performance evaluation under various environmental conditions. Kewalramani et al. [17] used ANN for the non-destructive testing of concrete compressive strength via ultrasonic pulse velocity, showing that ANN were more effective than other methods in terms of predicting concrete compressive strength. Yoon et al. [18] made a prediction model that was more accurate than most regression models by looking at the complicated and nonlinear relationships between the parts of lightweight aggregate concrete and its compressive strength and elastic modulus. Ridho et al. [19] constructed three highly accurate ANN models using the Levenberg–Marquardt algorithm, with recycled aggregate content as input and compressive strength as output. These studies indicate that ANN exhibits significant advantages and application potential in improving mixture-design efficiency and accuracy and predicting the mechanical properties of materials.

The relationship between the strength of solidified soil and the ratio is a complex, non-linear one. The BP neural network, an artificial neural network, is capable of powerful self-learning and nonlinear mapping, which makes it an effective tool for predicting the relationship between the strength of solidified soil and the ratio. Consequently, the BP neural network is frequently employed for the purpose of predicting the properties of solidified soil. Li et al. [20] constructed a compressive strength prediction model for coral concrete using BP neural network and optimized the model parameters and found that the optimized model produced reliable results. Han et al. [21] conducted an analysis on the impact of various fly ash dosages on the strength of concrete, leading to the establishment of a BP-neural-network-based compressive strength prediction model for fly ash concrete. Ma [22] created a BP neural network to predict the compressive strength of fiber-reinforced-polymer-confined concrete and compared it with the traditional regression model. The neural network model is more accurate and stable in its predictions. Li et al. [23] used a BP neural network to establish a prediction model for the compressive strength and flexural strength of hybrid fiber-reinforced concrete. These studies have achieved favorable results, which can replace part of the test screening work, reduce the test workload, and save human and financial resources. However, because these prediction models rely on the mapping relationship between parameters established by concrete or cemented gravel test data, they cannot accurately predict the strength of red mud road materials due to insufficient red mud test data support. Currently, there are few studies on the strength prediction of modified red mud road materials using neural network models. Zhang [24] employed a feed-forward neural network to develop a predictive model for the strength of red-mud-based grouting materials. The model was trained using two inputs: the amount of raw red mud and the ratio of solidifying agents. The objective of the study was to develop a prediction model for the strength of red-mud-based grouting materials utilized in tunnel construction and civil-engineering-disaster prevention and control projects. However, the model was not applicable to the strength prediction of modified red mud road materials due to the differing performance requirements for fluidity and strength.

When it comes to optimizing the road material scheme, most scholars choose the material scheme that exhibits the highest strength and best performance based on experimental tests. An et al. [25] conducted an experimental study on the road performance, durability, and environmental performance of red-mud–steel slag–cement pavement base material. They ultimately determined two sets of optimal ratios for this material. Wu et al. [26] employed an orthogonal test method and a linear regression model in MATLAB to derive a strength-prediction equation for modified red mud and identify the optimal ratio. A review of the literature reveals a lack of studies investigating the use of artificial neural networks (ANN) for the optimization of solidifying-agent ratios. Li et al. [27] established a BP model for optimizing SMA material schemes, taking into account the stability, fluidity, volumetric parameters of the Marshall test, and the properties of both asphalt and minerals. Compared to the traditional method, the artificial neural networks (ANN) can significantly reduce time and test costs while also offering a new approach to optimize mix ratios.

To solve the problems of finding the best red mud road materials for different engineering projects, this study uses the BP neural network as a deep-learning algorithm to predict the strength of modified red mud and suggests a way to use the trained BP model to find the best material scheme. First, we establish an unconfined compressive strength prediction model for modified red mud road materials, using the ratio of compound-solidifying agent (polymer agent, fly ash, cement, speed cement), water content, and curing age as input. We then train this model using accumulated experimental data from multiple regions over several years. The model’s prediction performance is validated using 10 sets of experimental data, with evaluation metrics including correlation coefficient (R), maximum and minimum relative error, standard deviation of errors (SD), mean absolute error (MAE), mean-squared error (MSE), and root-mean-squared error (RMSE). Next, we use the BP strength-prediction model to estimate a mapping set of strength with various material schemes, taking into account the strength requirements of various road-engineering projects and the constraints on the material components. We then calculate the material cost of each material scheme based on the engineering market. Finally, we utilize a double-objective decision method to optimize material schemes, aiming to achieve both strength and cost objectives. This study offers a method for predicting the unconfined compressive strength of modified red mud road materials within a specific range, as well as a technique for optimizing the material scheme. Its use will reduce the amount of experimental work and offer valuable insights for the strength prediction and material scheme optimization of modified red mud.

2. Experimental Dataset on the Strength of Modified Red Mud Road Material

2.1. Experimental Data Analysis and Selection

The experimental data on the unconfined compressive strength of modified red mud road materials used in this study were from the experiments carried out by our team in a couple of engineering projects located at Longkou in Shandong province, Taiyuan in Shanxi province, and Beihai in Guangxi province of China. We experimentally investigated the effects of the amount and material scheme of the compound-solidifying agent, water content, and curing age on the unconfined compressive strength of the modified red mud road materials in these tests.

Table 1. Characteristics of the experimental dataset.

Property	Minimum Value	Maximum Value	Average Value
Polymer agent (%)	0.05	0.54	0.35
Fly ash (%)	0	2.40	0.80
Cement (%)	0.56	11.46	6.34
Speed cement (%)	0	1.80	0.27
Water content (%)	24.10	33.40	27.25
Curing age (d)	3	11.00	5.19
Unconfined compressive strength (MPa)	0.15	4.99	2.94

displays the value ranges of each property, from which we randomly selected 114 experimental groups for analysis in this study (

Table 2. Representative values of the experimental dataset.

Material Scheme	Polymer Agent (%)	Fly Ash (%)	Cement (%)	Speed Cement (%)	Curing Age (d)	Water Content (%)	Strength (MPa)
S1	0.05	0.1	0.855	0	3	25	0.36
S2	0.36	0.8	6.84	0	7	29.6	3.82
S3	0.36	0	7.64	0	4	29.6	3.31
S4	0.135	0.3	2.565	0	6	25	1.27
S5	0.54	0	9.66	1.8	3	25	3.03
…	…	…	…	…	…	…	…
S51	0.225	0.5	4.275	0	3	25	1.73
S52	0.225	0.5	2.775	1.5	3	25	0.97
S53	0.36	1.6	6.04	0	11	29.6	3.01
S54	0.09	0.2	1.11	0.6	6	25	0.61
S55	0.36	1.2	6.44	0	7	24.1	4.06
…	…	…	…	…	…	…	…
S110	0.36	0	6.44	1.2	7	26.7	3.56
S111	0.54	2.4	9.06	0	11	29.6	4.69
S112	0.36	0	6.44	1.2	7	33.4	3.1
S113	0.045	0.1	0.855	0	3	25	0.46
S114	0.36	1.2	6.44	0	3	26.7	3.39

), and Figure 1 displays the frequency distribution histograms of each influence factor.

2.2. Influence of the Compound-Solidifying Agent

To give an illustration of the effect of each component in the compound-solidifying agent on the unconfined compressive strength of the modified red mud, four groups of compound-solidifying agent ratio A~D are carefully selected from the 114 data points of experiment tests, as listed in

Table 3. Four groups of compound-solidifying agent ratio A~D.

Ratio	Polymer Agent (%)	Fly Ash (%)	Cement (%)	Speed Cement (%)
A	4.5	10	85.5	0
B	4.5	10	55.5	30
C	4.5	15	80.5	0
D	4.5	0	80.5	15

. Specifically, we prepared A and B to compare the effects of cement and speed cement in the compound-solidifying agent on the unconfined compressive strength of modified red mud road material. The ratio A was polymer agent/fly ash/cement in a ratio of 4.5:10:85.5, while the ratio B was polymer agent/fly ash/cement/speed cement in a ratio of 4.5:10:55.5:30. Ratio B differs from A in that it replaces 30% of the cement with speed cement. We fixed the water content at 25% and set the curing age at 3 and 7 days. Figure 2 presents and compares the unconfined compressive strength results for the four groups, plotting the results of the pure cement for comparison.

Figure 2a demonstrates that the unconfined compressive strength of the modified red mud road material increases proportionally with the amount of the compound-solidifying agent. Additionally, ratio A exhibits a superior solidifying effect compared to ratio B, and the gap between their effectiveness widens as the amount of compound-solidifying agent increases, indicating that speed cement (in ratio B) performs worse than regular cement (in ratio A).

We selected two solidifying agent ratios, C and D, to compare their effects on the unconfined compressive strength of modified red mud road material with pure cement in Figure 2b. Ratio C comprises a polymer agent, fly ash, and cement in a ratio of 4.5:15:80.5, whereas ratio D maintains the same ratio but substitutes 15% of the fly ash with speed cement. We set the water content of the two groups at 26.7% and the amount of the compound-solidifying agent at 8%. Figure 2b displays the comparison results at the curing ages of 3 and 7 days, respectively. It shows that at the early curing age, ratio D exhibits a stronger strength than ratio C, primarily due to the faster solidifying rate of speed cement compared to fly ash. Meanwhile, ratio C shows a little more strength than the pure cement, meaning that a small amount of fly ash can make red mud road material stronger. At the curing ages of 7 days, the difference in the unconfined compressive strength between ratios C and D and C and pure cement is reduced to about 7% and 9%, respectively, suggesting that their strengths sufficiently meet general engineering requirements. Considering both cost and environmental benefits, ratio C, which incorporates a polymer ratio and fly ash as a partial replacement for cement, is an optimal choice.

2.3. Data Normalization

It is notable that each influence property has different units and demonstrates a considerable degree of variation in its range of values. Normalizing the initial input data is necessary to eliminate the impact of these magnitude differences to ensure that all values are scaled in a consistent interval range, between 0 and 1. In this study, the following functions are applied for data preprocessing:

$${\widehat{X}}_{\mathrm{i}}={\displaystyle \frac{\left(X_{\mathrm{i}}-X_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}{\left(X_{\mathrm{m}\mathrm{a}\mathrm{x}}-X_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)}}$$

(1)

where X_i is the original value of the sample sequence, X_min is the minimum value of the sample sequence, X_max is the maximum value of the sample sequence, and ${\widehat{X}}_{\mathrm{i}}$ is the normalized value.

3. BP Prediction Model for Modified Red Mud Road Material Strength

3.1. Architecture of the BP Strength Prediction Model

The BP neural network, a type of multilayer feedforward neural network, consists of an input layer, hidden layers, and an output layer. The input layer serves as the first layer, while the output layer is the final layer, with the hidden layers functioning as intermediate layers. A key feature of the BP neural network is the forward transmission of signals in conjunction with the backward propagation of errors. The network continuously adjusts its weights and thresholds based on the predicted error values when the output layer produces results that deviate from the expected outputs. This iterative process allows the BP neural network to progressively approximate the desired output values [28]. Backpropagation neural networks (BP neural networks) are very good at predicting stabilized soil strength with the compound-solidifying agent. This is primarily due to their capacity for nonlinear modeling, flexibility, and accuracy. The network effectively captures the complex nonlinear relationships inherent in the soil stabilization process, making it suitable for various soil types and solidifying agents. The trained BP model could be used in the optimization of the compound-solidifying agent ratio and thereby reduce the number of physical experiments required and the time-consuming process associated with experiments in engineering implementation.

The input variables consist of six influence properties for modified red mud road materials: polymer agent, fly ash, cement, speed cement, water content, and curing age. The output variable is the unconfined compressive strength of the modified red mud road material. Therefore, a three-layer neural network structure is established by using six neurons in the input layer and one neuron in the output layer in Figure 3. We first determine the number of neurons in the hidden layer by using an empirical formula to define a range, and then we choose the optimal value based on the training and fitting results.

The empirical formula for the number of neurons in the hidden layer is as follows:

$$l=\sqrt{m+n}+a$$

(2)

where n is the number of neurons in the input layer, l is the number of neurons in the hidden layer, m is the number of neurons in the output layer, and a is a constant between 0 and 10.

According to the empirical formula, the number of neurons in the hidden layer is determined to be a range of 3 to 12, the training set data are used to repeatedly train the neural network, and the number of neurons in hidden layer with the smallest mean-squared error is selected as the optimal number of neurons in hidden layer.

Table 4. The mean-squared error of different neurons in the hidden layer.

Number of Neurons in the Hidden Layer	Mean Squared Error (MSE) of the Training Set
3	0.0327
4	0.0225
5	0.0206
6	0.0754
7	0.0175
8	0.0107
9	0.0117
10	0.0083
11	0.0188
12	0.0296

displays the results of the mean-squared error for different numbers of neurons in the hidden layer.

Table 4 demonstrates that, as the number of neurons in the hidden layer increases, the mean-squared error (MSE) exhibits a trend where it first decreases and then increases. When the number of neurons reaches 10, the MSE in the training set reaches a minimum value of 0.0083. The increase in neurons may also result in more complex formulas, potentially leading to overtraining of the model. Therefore, we select 10 as the optimal number of neurons in the hidden layer, given its high accuracy with MSE. The final ANN architecture of the unconfined compressive strength prediction model for the modified red mud road material is 6-10-1, as shown in Figure 3.

The commonly used BP neural network training algorithms are trainlm, traingd, traindm, trainda, and traindx, among which the one with the optimal iteration accuracy and number of iterations is the trainlm algorithm [29]. Therefore, this study adopts the trainlm algorithm as the B-neural-network-training algorithm.

A hyperbolic tangent sigmoid transfer function is utilized to predict the output-parameter-based normalized values [30], as shown in Equation (3). This transfer function was chosen, as it provided accurate predictions when it was used in other research studies [31,32]. It is one of the most widely used transfer functions when constructing neural networks.

$$y_{\mathrm{j}}=f\left({net}_{\mathrm{j}}\right)=\left({\displaystyle \frac{2}{1+e^{\left(-2{net}_{\mathrm{j}}\right)}}}\right)-1$$

(3)

where ${net}_{\mathrm{j}}={\sum }_{i=1}^n{w}_{\mathrm{i}\mathrm{j}}{x}_{\mathrm{i}}+bia{s}_{\mathrm{j}}$; x_i and y_j are the input and output values, respectively; ω_ij and bias_j are the weights and biases, respectively.

The specific principle of choosing the purelin linear function as the transfer function of the neurons in the output layer is shown in Equation (4):

$$y=f\left(\sum _i^n{w}_{\mathrm{i}}{x}_{\mathrm{i}}+b\right)$$

(4)

where w_i represents the weight parameter corresponding to the ith input x_i of this neuron, b represents the deviation of this neuron, and f(x) is the activation function of the neuron.

3.2. Accuracy Metrics of Neural Network Output

This study uses a number of accuracy metrics to assess the accuracy of the output predicted by the neural network model. These include the correlation coefficient (R), maximum and minimum relative errors, standard deviation of the errors (SD), mean absolute error (MAE), mean-squared error (MSE), and root-mean-squared error (RMSE). These metrics provide a comprehensive assessment of the ANN model’s accuracy and reliability in predicting outputs.

The correlation coefficient (R) measures the degree of correlation between the predicted and experimental values in the sample set. As the value of R approaches 1, the accuracy of the neural network prediction increases. The correlation coefficient is calculated using Equation (5):

$$R=1-{\displaystyle \frac{\sum _{=1}^n\left({\overline{y}}_{\mathrm{i}}-y_{\mathrm{i}}\right)}{{\sum }_{i=1}^n\left({\overline{y}}_{\mathrm{i}}-\overline{y}\right)}}$$

(5)

where y_i is the experimental value of unconfined compressive strength, ${\overline{y}}_{\mathrm{i}}$ is the predicted value of unconfined compressive strength, $\overline{y}$ is the average of the experimental values, and n is the amount of sample set.

We use the mean-squared error (MSE) as an evaluation index to quantify the deviation of the predicted values from the true value in the sample set. A smaller MSE indicates higher prediction accuracy for the neural network model [33]. The MSE is calculated using the following formula:

$$MSE={\displaystyle \frac{\sum _{i=1}^n{\left(y_{\mathrm{i}}-{\overline{y}}_{\mathrm{i}}\right)}^2}{n}}$$

(6)

Relative error is used as an evaluation index of the deviation rate between the predicted value and the experimental value for a specific sample datum, which is calculated with Equation (7). A lower value indicates greater credibility for the neural network model’s predictive capabilities.

$$f_{\mathrm{u}}={\displaystyle \frac{f-f_0}{f_0}}={\displaystyle \frac{{\overline{y}}_{\mathrm{i}}-y_{\mathrm{i}}}{y_{\mathrm{i}}}}\times 100\%$$

(7)

The standard deviation (SD) of the errors is an important statistical measure for assessing the differences between predicted values and true values. A smaller value indicates that the model’s predictions are relatively consistent across multiple samples, thereby demonstrating better reliability. The formula for the standard deviation is as follows:

$$SD=\sqrt{{\displaystyle \frac{1}{n-1}}{\sum }_{n=1}^n{(x_{\mathrm{i}}-{\overline{x}}_{\mathrm{i}})}^2}$$

(8)

where $x_{\mathrm{i}}$ represents the individual error values, and ${\overline{x}}_{\mathrm{i}}$ denotes the mean of the error values.

The mean absolute error (MAE) is the average of the absolute differences between the true values and the predicted values. It reflects the absolute magnitude of the errors, providing a straightforward measure of the prediction accuracy. The formula for MAE is as follows:

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_{\mathrm{i}}-{\overline{y}}_{\mathrm{i}}\right|$$

(9)

The root-mean-squared error (RMSE) is the square root of the mean-squared error. It shows the average deviation between the predicted value and the actual value. The formula for calculating RMSE is as follows:

$$RMSE=\sqrt{MSE}=\sqrt{{\displaystyle \frac{\sum _{i=1}^n{\left(y_{\mathrm{i}}-{\overline{y}}_{\mathrm{i}}\right)}^2}{n}}}$$

(10)

3.3. Training of the BP Strength Prediction Model

The 114 experimental data are randomly divided into 2 categories: the training sample set with 104 data groups and the evaluating sample set with 10 data groups. The training sample set are divided into 3 categories. The training dataset (70%, 72 groups) is used to train the network and adjust according to the errors; the validation dataset (15%, 16 groups) is used to measure network generalization and stop the training when it stops improving; and the test dataset (15%, 16 groups) refers to data that have not been used in training or validation to independently assess the accuracy of the trained prediction model. We set the number of training samples at 1000, the target error at 10⁻⁷, and the learning rate at 0.01. Figure 4 illustrates the BP-neural-network-model-training process.

The predicted and true values of the unconfined compressive strength are shown in Figure 5. These values are compared in the training set, the validation set, the test set, and the total training sample set. Figure 5 reveals a strong correlation between the predicted and true values in the test set and the entire training samples, reaching R = 0.9186 and R = 0.9753, respectively. This strong correlation suggests that the trained BP neural network model performs well.

The mean-squared error (MSE) varies with the number of iterations during the training, validation, and testing stages of the BP neural network model as shown in Figure 6. Figure 6 demonstrates that the MSE decreases at each stage as the number of iterations increases, reaching its minimum value at 9–10 iterations. At this juncture, the MSE reaches a point of equilibrium and exhibits no further decline with the addition of further iterations. Notably, the MSE in the validation stage converges to 0.0076, which satisfies the accuracy requirement. This indicates that the neural network model has been well trained and is suitable for practical engineering applications in predicting the unconfined compressive strength of modified red mud road materials.

3.4. Accuracy Evaluation of BP Strength Prediction Model

In order to evaluate the prediction accuracy of the trained BP strength prediction model, 10 data groups from the evaluating sample set were employed. This process compared the experimental value and the predicted results by the trained BP strength prediction model using the same input in the evaluating sample set. Figure 7 depicts the comparison between the predicted and actual values from the BP neural network model, with the numbers marked on the figure representing the relative errors between the actual and predicted values.

Table 5. Prediction accuracy evaluation index.

Maximum RE	Minimum RE	SD	RMSE	MAE	MSE	R
12.20%	0.21%	0.148	0.149	0.119	0.0222	0.995

gives the accuracy-evaluation results, with a maximum relative error of 12.20% and a minimum relative error of 0.21%. The average relative error was found to be 5.58%. The standard deviation of the errors (SD) was 0.148, the mean absolute error (MAE) was 0.119, the mean-squared error (MSE) was 0.022, and the root-mean-squared error (RMSE) was 0.149. These accuracy results meet the engineering prediction accuracy requirements [34]. In addition, we conducted a regression analysis to assess the correlation between the predicted and actual values, as shown in Figure 8. The correlation coefficient (R) is 0.9950, indicating high prediction accuracy for the BP neural network strength prediction model.

The total sample set, comprising the training sample set and the evaluating predicted sample set, is employed as the test sample for the purpose of evaluating the prediction error of the model. The trained BP neural network model is then used to predict the unconfined compressive strength of the test sample for comparisons. The errors of the predicted values and the values are analyzed, and the relative error and the distribution of the error range are presented in

Table 6. Relative error distributions by the trained BP prediction model with total sample set.

Relative Error X Distribution Range	BP Sample Size	Relative Error as a Percentage/%
X < 5%	59	51.75
5% ≤ X < 10%	26	22.81
10% ≤ X < 15%	18	15.79
15% ≤ X < 25%	7	6.14
X > 25%	4	3.51
(grand) total	114	100

.

As demonstrated in Table 6, 51.75% of the BP neural network prediction results exhibit a relative error of less than 5%, while 74.56% demonstrate a relative error of less than 10%. On the whole, it is notable that this BP prediction model for the unconfined compressive strength of modified red mud road material tends to predict with an error of basically less than 10%, indicating that the use of a BP neural network for this purpose is a feasible approach.

4. Material Scheme Optimization

The main process of the optimization is divided into four steps. First, the mapping set of strength and the material scheme are calculated using the trained BP model in the constraint conditions of the six variables h₁~h₆. Second, the mapping set of material cost and the material scheme is calculated with Equation (11) for all the material schemes in the strength-mapping set. Hence, we obtain the double-mapping set of strength and material cost from each material scheme. Third, based on the strength needs of red mud road materials for various road projects, the curing age h₅ and its design-unconfined compressive strength are found. Next, the mapping sets of strength and material cost are chosen using the constraints for the curing age h₅ and the strength in

Table 7. Unit price of each variable for modified red muds road materials.

Variable	Unit Price (CNY/T)
Polymer agent	850
Fly ash	20
Cement	290
Speed cement	900
1% reduction in red mud water content	2.5

. Fourth, a double-objective decision method is utilized to make an optimization of material schemes with the target objectives of the strength and cost.

4.1. Objective Material Cost Function

In this optimization, the material cost and the unconfined compressive strength of the modified red mud are considered the objectives. The material scheme is defined as each combination of the six variables, including polymer agent, fly ash, cement, speed cement, curing age, and the water content. Set the material cost as C, and set the unconfined compressive strength as S. The unconfined compressive strength is calculated using the trained BP strength model in Section 3. The material cost function of modified red mud road material is composed of the six variables in the BP strength model, and the cost calculation formula for 1 ton of modified red mud road material is as follows:

C = 850h₁ + 20h₂ + 290h₃ + 900h₄ + 0h₅ + 250(35% − h₆)

(11)

where h₁ is the percentage of polymer agent, h₂ is the percentage of fly ash, h₃ is the percentage of cement, h₄ is the percentage of speed cement, h₅ is the curing age, and h₆ is the water content of raw red mud. The material cost is considered for the first four variables h₁~h₄; the curing cost of h₅ is not included. According to the cost accounting in engineering practice in the Jinan-Qingdao Expressway in China [35], the construction cost for reducing water content of raw red mud is utilized as 2.5 CNY/T for every 1% reduction in water content from the natural water content of 35% in average. Table 7 lists all factors and its unit price the considered in the cost function.

4.2. Constraint Conditions for the Mapping Set

The constraint conditions are divided into four types, one is for the content of each component in the compound-solidifying agent, h₁~h₄, as listed in

Table 8. Constraint conditions for the optimization.

Item of Constraint	Constraint Condition
Component of the compound-solidifying agent	0.0% ≤ hi ≤ 10%, i = 1, 2, 3, 4
Red mud water content	17% < h6 < 34%
Relative error of curing age	$0\%<{\displaystyle \frac{h_{5-\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}-h_{5-\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}}{h_{5-\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}}}$< 5%
Relative error of strength	$0\%<{\displaystyle \frac{y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}-y_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}}{y_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}}}$< 5%

. The second is the water content of the raw red mud h₁ listed in Table 8, which can be reduced from 34% to 17% in the construction process by special construction technique. The third is for the relative error of the curing age between the prediction value $h_{5-\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ in the BP model and the design value $h_{5-\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}$ in engineering, which is in the range of 3~14 d in the BP model but usually fixed as 7 d or 14 d in an engineering project. The fourth is for the relative error of the unconfined compressive strength of the modified red mud between the prediction value $y_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ in the BP model and the design value $y_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}}$ in engineering, which has the minimum design standard value in the industry specification.

4.3. Optimization Process

Assuming that the 7-day unconfined compressive strength requirement y of road materials in a project is 2.9~3.0 MPa, about 8028 material schemes are selected from the mapping set predicted by using the trained BP strength model, as shown in the first five columns of

Table 9. The mapping set of strength and material cost in a project.

Material Scheme	Polymer Agent (%)	Fly Ash (%)	Cement (%)	Water Content (%)	Strength (MPa)	Cost (CNY/T)	Normalized Strength S	Normalized Cost C	$\mathit{Z}$
S1	0.02	1.96	4.78	33.93	2.989	17.099	0.893	0.002	442.253
S2	0.02	2.18	4.78	33.93	2.907	17.143	0.067	0.002	292.410
S3	0.08	1.75	4.78	33.93	2.970	17.567	0.699	0.010	68.821
S4	0.08	1.96	4.78	33.93	2.994	17.609	0.938	0.011	64.058
S5	0.08	2.18	4.78	33.93	2.962	17.653	0.621	0.012	59.578
S6	0.08	2.4	4.78	33.93	2.906	17.697	0.056	0.013	55.588
S7	0.02	2.4	4.98	33.93	2.981	17.767	0.808	0.014	50.654
S8	0.18	2.4	4.57	33.93	2.918	17.938	0.181	0.017	41.147
S9	0.13	1.96	4.78	33.93	2.999	18.034	0.987	0.019	37.525
S10	0.13	1.75	4.78	33.93	2.911	17.992	0.106	0.018	38.858
……	……	……	……	……	……	……	……	……	……
S8028	0.08	0.44	5.19	20	2.902	69.519	0.020	0.967	0.730

. The material cost of each material scheme is calculated with Equation (11) as listed in the sixth column of Table 9. Then, for the convenience of a double-objective decision analysis, the strength and cost are normalized using Equation (1), as listed in the seventh and eighth columns of Table 9.

There are two objectives in this optimization: the strength is a positive objective which is needed to be maximized, and the material cost is a negative objective which is needed to be minimized. A weighted-utility optimization approach is adopted in this study using the following formula:

$$\max Z=w_{1}S+{\displaystyle \frac{w_{2}0.7}{C}}$$

(12)

where Z is the aggregate utility as the sum of the two objective utility functions, S is the normalized unconfined compressive strength by the BP strength model, C is normalized material cost by Equation (11), and w₁ and w₂ are the weights for the strength and cost, which are set as w₁ = 0.3 and w₂ = 0.7 with the reference to the engineering experience.

With the substitution of normalized material cost C and the normalized unconfined compressive strength I into the Formula (12), the mapping set Z is calculated and sorted from maximum to minimum in Table 7. The maximum of Z is chosen from the mapping set Z listed in the first row of Table 7. Therefore, the optimal material scheme is the one corresponding to the maximum Z, which is 0.02% polymer agent, 1.96% fly ash, 4.78% cement, 0% speed cement for the compound-solidifying agent, and 33.93%water content of raw red mud, generating a modified red mud road material with the 7 d unconfined compressive strength of 2.987 MPa and material cost of 14.424 CNY/T. Looking carefully at the data in Table 9, it can be seen that the aggregate utility Z is dominated by the water content, followed by the polymer agent. To demonstrate this tendency, the water content of the material scheme versus material cost and 7-day unconfined compressive strength in the mapping set are plotted in Figure 9. It can be observed that with the reduction in water content, the material cost in every strength level increases, indicating that the reduction in water content is the most expensive procedure and high water content of 30%~33.93% should be chosen to minimize the material cost. However, it should be pointed out that only the material cost is considered in this study, and the other limitations of construction conditions are neglected, for example, the construction cost of red mud with different water content and construction schedule limit. These costs and limits should be counted in a practical engineering project application.

5. Conclusions

In this study, the BP neural network was used to establish the unconfined compressive strength of modified red mud road materials and trained by the experimental data, with the ratio of the compound-solidifying agent (polymer agent, fly ash, cement, speed cement), water content, and curing age as the input, defined as material scheme. Then, the mapping set of strength and cost of each material scheme is calculated using the trained BP model in the constraint conditions of the six variables. Finally, a double-objective decision method is utilized to make an optimization of material schemes with the target objectives of strength and cost. With the established BP strength model, the designer can adjust and optimize the material scheme, including the material scheme of the compound-solidifying agent and water content of the raw red mud according to the target strength and cost, which can reduce the huge experimental work and save the manpower and material resources. The main conclusions are as follows:

(1)

The analysis of the influence on the unconfined compressive strength of modified red mud road material shows that the effect of cement is more pronounced than that of quick-hard cement, and its advantage is increasingly evident with an increase in amount. The addition of 15% fly ash improves the solidification effect, acting as a substitute for cement while offering cost and environmental protection.

(2)

The verification results of the trained BP model by the prediction sample set demonstrate that the prediction model exhibits high prediction accuracy with a relative error of less than 10%, a root mean-squared error of less than 0.04, and a correlation coefficient approaching 1.

(3)

With the trained BP model, the mapping set of strength and cost of each material scheme is calculated in the constraint conditions of the six variables. A double-objective decision method with a weighted utility optimization approach is utilized to make an optimization of material schemes with the target objectives of the strength and cost.

(4)

An optimal material scheme is achieved as polymer composite:fly ash:cement:speed cement = 0.02%:1.96%:4.78%:0%, with a 33.93% water content of raw red mud, resulting in a strength of 2.987 MPa and cost of 17.099 CNY/T, with the aim of achieving a 7-day unconfined compressive strength requirement of 2.9~3.0 MPa for the subbase of extremely heavy and extra heavy traffic roads for second-class highways.