Response Surface Methodology (RSM) is an experimental analysis method that combines a rational experimental design with a multiple regression equation [38,39]. This paper uses a multiple quadratic regression equation to fit the functional relationship between each effect factor and the response value. The principle is to search experimentally for the response surface function y = f(x) between the effect factor x and the measured value y and make y*→y. Commonly used quadratic response surface function equations are as follows:

(1) y∗=a+∑j=1mbjxj+∑k<jbkjxkxj+∑j=1mbjjxj2

In Eq. (1), y* is the response value of the measured value; a is a constant term; b_j, b_kj and b_jj are the regression coefficients of the primary, secondary and cross terms, respectively; m is the number of factors.

Extreme learning machine (ELM) has better generalization performance than traditional feed-forward neural networks [40]. While traditional neural networks use gradient descent to optimize the network parameters, ELM can randomly assign the hidden layer’s threshold and the input layer’s weights [41]. As a result, ELM can be trained faster than traditional neural networks. The ELM structure contains 3 parts: input layer, hidden layer and output layer. Each neuron between 2 neighboring layers is connected to each neuron in another layer but each neuron within the same layer is not connected as is shown in Fig. 1.

S={(xi,yi)|xi∈RD,yi∈RM,i=1,2,...,n} is a dataset of ELM, and H(xi) is the output of the hidden layer. The relevant principles are as follows [42]:

(2) H(xi)=[h1(xi),h2(xi),...,hL(xi)]

(3) h(xi)=g(wixi+bi)

(4) g(x)=11+e−x

where the wi=[wi1,wi2,...,wiN] is the weight between the i-th neuron in input layer to the N-th neuron in hidden layer; the bi=[bi1,bi2,...,biN] is the bias vector; the sigmoid function in Eq. (4) is used as the activation function in this paper. The output of ELM is derived as follows:

(5) FL(xi)=∑i=1Lβi⋅g(wixi+bi)

where the βi is the connection weight between the neurons in the hidden layer to the neurons in the output layer.

The genetic algorithm (GA) is a method to search for the optimal solution by simulating the natural evolutionary process [43,44]. GA converts the problem-solving process into a process similar to the crossover and mutation of chromosomal genes in biological evolution. It can quickly obtain good optimization results when solving more complex combinatorial optimization problems. The optimization process of GA is shown in Fig. 2.

The experimental process is shown in Fig. 5.

The representative specimens of group 3 and group 7 are selected from the experimental groups in Table 4 for microstructure analysis, as shown in Fig. 13. From Fig. 13a, it can be seen that the specimens of group 3 formed a dense structure with high strength. The hydration reaction produced a lot of C-S-H and AFt. The Aft crystals crossed each other to form a network structure and the C-S-H gels encapsulated the network structure and filled the pores to form a compact cemented structure. Fig. 13b shows that the specimens of group 7 formed a weaker pore structure. Some AFt crystals are sparsely distributed in the pores of TWSC. The C-S-H gels are produced in smaller amounts and cannot adequately encapsulate the Aft crystals to fill the pores, which results in more pores in TWSC. Comparing the experimental conditions and results of group 3 and group 7, it can be seen that the use of slag powder instead of partial cement as cementitious material and increasing the fineness modulus can improve the internal structure of TWSC and thus enhance the UCS of TWSC.

To evaluate the prediction performance of different models, root mean square error (RMSE), correlation coefficient (R) and variance account for (VAF) are used to quantify the prediction accuracy of models. The evaluation metrics are as follows [48,49]. The closer the RMSE is to 0, the closer the R is to 1 and the closer the VAR is to 100, the better the model prediction is.

(10) RMSE=1n∑i=1n[yi−f(xi)]2

(11) R=∑j=1n[(yj−yj¯)(f(xj)−f(xj)¯]∑j=1n(yj−yj¯)2⋅∑j=1n(f(xj)−f(xj)¯)2

(12) VAF=[1−VAR(yi−f(xi))VAR(yi)]×100

ELM is constructed to predict the UCS of TWSC and provide a basis for intelligent optimization of mix proportion in this study. 51 sets of data in Table 4 are used as the data set. To better evaluate the prediction performance of the model, the original data are randomly divided into a training set and a validation set in the ratio of 8:2. 41 sets of data from the training set are used to train ELM and 10 sets of data from the validation set are used to evaluate the prediction performance of ELM. Using tailings fineness modulus (TFM, X₁), replacement rate of slag powder to cement (SRSPC, X₂), binder-water ratio (BWR, X₃) and curing time (CT, X₄) as input variables and UCS (Y) as output variable. The ELM model for characterizing the nonlinear relationship between X₁, X₂, X₃, X₄ and Y is constructed. The technical route of construction for ELM is shown in Fig. 14.

The number of input layers and output layers of ELM is set to 4 and 1 according to the number of input variables and outputs, respectively. Based on the previous research results, the setting of the hidden layer nodes is determined based on the following experience equation [14,33]:

(13) Hidden layer nodes=m+n+l

In Eq. (13), m is the number of input layers, n is the number of output layers and l is the adjustable constant on the interval [1,10]. Using the mean absolute error (MAE) of the validation set to measure the ELM performance at different hidden layer nodes. In the process of determining the best hidden layer nodes, the MAE under each condition of hidden layer nodes is calculated 10 times and averaged to ensure the generalization of the results. The MAE of the validation set for ELM at different hidden layer nodes is shown in Fig. 15. It can be seen that when the hidden layer nodes of ELM are 11, the MAE of the validation set obtains the minimum value of 0.7044. Therefore, the hidden layer nodes of the ELM are set to 11.

Fig. 16 shows the prediction results of the training set and validation set of the ELM model. The R of the training set and validation set approaches 1, the VAF approaches 100, and the RMSE approaches 0, indicating that the prediction accuracy of the ELM model is high. In addition, the R, VAF and RMSE of the training set are slightly better than those of the validation set, indicating that the prediction effect of the training set is better than that of the validation set.

In this paper, ELM is compared with back proportion neural network (BPNN), radial basis function neural network (RBFNN), and generalized regression neural network (GRNN), which are widely used in engineering practice. Before the comparison of prediction performance, all models are trained and the best structural parameters of each model are obtained.

Fig. 17 shows the marginal histograms of the prediction results of different models. Overall, the prediction results of a training set of ELM and RBFNN are consistent with the actual data, while the prediction results of BPNN and GRNN are relatively poor. The prediction results of the validation set of ELM and GRNN are consistent with the actual data, while the prediction results of BPNN and RBFNN are relatively poor. The above conclusions can be verified on the R value of prediction results of the training set and validation set of different models.

Fig. 18 shows the prediction results of different models on the UCS of TWSC. The R of the ELM model is 0.99, which is 10.88%, 4.77% and 6.66% higher than BPNN, RBFNN and GRNN, respectively. The RMSE of ELM model is 0.34, which is 61.84%, 51.93% and 70.03% lower than BPNN, RBFNN and GRNN, respectively. The VAF of the ELM model is 98.40, which is 6.53%, 10.94% and 7.89% higher than BPNN, RBFNN and GRNN, respectively. The prediction accuracy ranking results (Table 5) of different algorithms based on the Zorlu method show that ELM has the best prediction effect on the UCS of TWSC.

GA is introduced to solve the optimal mix proportion of TWSC at different constraint conditions based on construction of ELM in this study. Firstly, a nonlinear mapping relationship between each affecting factor and UCS is established using ELM (as described in Section 4.2). Secondly, the constructed ELM is considered a nonlinear function f_ELM for predicting the UCS and adding constraint conditions to f_ELM for engineering applications (f_ELM is called the objective function). Finally, the above objective function is used as the fitness function in the GA and the GA is used to search for the minimum value of the fitness function. The minimum value is the optimal solution and the optimum mix proportion corresponding to this optimal solution can be obtained. The specific construction flow of ELM-GA is shown in Fig. 19.

In this paper, the individual coding method of GA uses real numbers and the values of each parameter are as follows. The population size is 100, the max iterations are 100, the crossover probability is 0.4, and the mutation probability is 0.2.

The optimal mix proportions for TWSC with UCS of 15 MPa and 20 MPa (Objective functions are f₁₅ and f₂₀) are solved using ELM-GA. It is also necessary to round off the mix proportion parameters optimized by ELM-GA to convenience for practice engineering. The ELM-GA model can optimize the mix proportions of TWSC in less than 1.5 min. The final optimization results for mix proportions are shown in Table 6. It can be seen that the errors of predicted UCS and measured UCS for the optimized mix proportions with the target strengths of 15 and 20 MPa are 1.87% and 1.46%, respectively. This shows that the ELM-GA model can quickly and accurately realize the mix proportion optimization of TWSC.