### 1 Introduction

^{TM}) software is used to simulate different cases of the 3D wire rod rolling process and investigate the relationship between the input parameters and the output. Second, all the input and output parameters are changed to dimensionless values in which they are not related to a specific rolling stand, so the proposed model can be transferred to other setups and some level of transferability can be achieved which increases the generality of the model. Third, multiple AI methods including MLP and ANFIS are employed to predict the output based on the dimensionless data obtained from FE analysis which results in an accurate and robust prediction. Forth, some level of reasoning is provided for the user by using the ANFIS, thus, the effect of different input parameters can be understood. In addition, data fusion is employed to combine the result of trained MLP and ANFIS models for the prediction of those cases that are out of the range of simulations, thus, the reliability and explainability of this model will be increased.

### 2 Methodology

### 2.1 FE analysis and Data Generation

^{TM}to simulate the steel wire rod rolling process [28,29]. Based on the symmetry of the rollers and billet, only a quarter of them are modeled to decrease the simulation time. In addition, material density is increased artificially by using a fixed mass scaling factor to increase the computational efficiency of the simulations. A mass scaling factor of 16 is considered after increasing the mass scaling factor from 1 to 1,000 and comparing the results of FE analysis in terms of the deformation, stress, kinetic and internal energies [30].

*ɛ*is the equivalent plastic strain,

*ε̇*is the strain rate,

*ε̇*

*is the reference strain rate,*

_{ref}*T*

*is the reference temperature,*

_{ref}*T*

*is the melt temperature,*

_{m}*A, B, C, n*, and

*m*are the five material constants [32]. The JC parameters of the employed steel are shown in Table 1. Since the effect of radiation and convection is negligible in the rolling process, it is assumed that conduction is the main factor for heat loss in the current work [33]. The other parameters which are used for the simulation are shown in Table 2.

*Round-Oval-Round*calibers) is changed at five levels (17.97, 18.44, 19.91, 19.391, and 19.86 rev./min), thus, the backward tension is changed at five levels and its effect can be understood; the same method is used to change the speed of the third roller and investigate the effect of forward tension while the speed of the first roller is kept constant. The same approach is used for the other setup of three-stand rolling (

*Oval-Round-Oval*calibers). The reference friction coeff icient is 0.4 for all cases. To consider more cases for the training, all the simulations for one-stand and three-stand rolling are repeated with two other friction coeff icients (

*μ*= 0.25 and 0.3).

### 2.2 Preprocessing

*W*

*is the maximum width after rolling,*

_{max}*W*

*is the initial width (before rolling),*

_{i}*α*is a coefficient that is recommended to be 0.97 for the round-to-oval pass and 0.83 for the oval-to-round pass,

*R*

*is the effective radius of the roller,*

_{eff}*H̄*

*is the mean height of the material before rolling,*

_{i}*H̄*

_{0}is the mean height of caliber,

*H*

*is the initial height,*

_{i}*X*is the original data,

*X*

_{mi}_{n}is the minimum value of

*X*,

*X*

*is the maximum value of*

_{max}*X*, and

*X*

*is the normalized data. Thus, the inputs and outputs are scaled between −1 and 1.*

_{i}### 2.3 Multi-Layer Perceptron (MLP)

*y*is the output of the neural network,

*x*is the input,

*w*

*are the weights, and*

_{i}*b*is the bias [37,38].

^{TM}is used to design a feedforward neural network. The first layer of this MLP model is the input layer with five nodes. Three hidden layers are used with different number of neurons in each of the layer. The output layer has only one neuron since this model has only one output which is spread ratio in one case and Shinokura coefficient in the other case. The prediction of the spread for oval and round caliber are done by separate MLPs and for each case two MLP model is used with the same inputs but different outputs. Thus, MLP-1 and MLP-2 are designed to predict the spread in oval caliber, MLP-3 and MLP-4 are used to predict the spread in round caliber. The input layer for all MLPs has 5 input variables which are rolling velocity (with a unit) and dimensionless values of friction, geometry, backward and forward tension. The output for two MLPs is the spread ratio (

*S*

*) and for the other two MLPs, the output is the α coefficient of Shinokura’s equation which are both dimensionless values. The inputs and outputs of all MLPs are shown in Table 7.*

_{r}*TANSIG*” except the output layer which is “

*PURELIN*”. “

*TRAINLM*” function which is based on the Levenberg-Marquardt optimization is used as the training method to adjust weights and biases. Since each input parameter is changed 5 or 6 levels and the total number of data set is limited, the random data division may result in a biased test data set. Thus, manual data division is used to ensure that the validation and test sets demonstrate good representatives of the whole data set.

### 2.4 Adaptive Neuro-Fuzzy Inference Systems

*X*

_{1}and

*X*

_{2}are the inputs,

*A*

*and*

_{i}*B*

*are fuzzy sets,*

_{i}*y*

*is the variable of consequence,*

_{i}*p*

_{i}*, q*

*, and*

_{i}*r*

*are the consequence parameters of the T-S model [41]. In the first layer, the membership functions are generated so the inputs are fuzzified. Each node in this layer is an adaptive node with a node function as follows:*

_{i}*μ*

_{Ai}(

*X*

_{1}) is the membership degree of

*X*

*in the fuzzy set of*

_{1}*A*and

*O*

_{i}^{1}is the output of the node in the

*i*

^{th}position in the first layer. The Gaussian membership function used in this study with the following equation:

*x*is a variable,

*σ*and

*c*are premise parameters. In the second layer, the strength of each rule (

*w*

*) is calculated by multiplying all inputs of a node and then is normalized considering the firing strength of all rules (*

_{i}*w̄*

*) in the third layer.*

_{i}### 2.5 Data Fusion

*α*coefficient can be converted to the spread ratio by reverse calculation. Therefore, the result of all MLP and ANFIS models will be the spread ratio which can be used for data fusion.

### 3 Results and Discussions

### 3.1 Verification of FE Analysis

^{2}, respectively. The position of the input cross-section is before the first stand of rolling and the position of the output cross-section is after the sixth stand of rolling which is shown in Fig. 7(a). The same input area is used for the simulation and the area of the output cross-section obtained from the FE analysis by ABAQUS

^{TM}is 229.08 mm

^{2}which shows a 1.5% deviation from the experimental values. It is also worth mentioning that there may be some errors in the measurements and the values from the factory may be a little different from the real values. However, the simulated results are perfectly matched with the real data which shows the employed ABAQUS

^{TM}model is correct and can be used for the rest of the simulations and case studies.

### 3.2 FE Analysis

*α*factors of 0.83 and 0.97 are shown in Fig. 9 in which the roll gap, the input material size, and the roller diameter are changed. Shinokura’s model can approximately predict the spread when the geometrical parameters are changed. Thus, it approximately has a similar trend to the result of FE analysis, however, there is a vertical shift when different

*α*factors are used. If the correct α factor is selected, it can predict the spread ratio with good accuracy. The problem is the lack of a standard approach for finding the correct

*α*factor.

*α*coefficient is a challenge in these cases. Based on the effect of input parameters on the spread ratio, the geometrical parameters, speed, friction, and interstand tensions are used for the training, the temperature and material strength are not considered due to their small effects on the spread ratio. For the output parameter, the spread ratio and the

*α*coefficient of Shinokura’s model are considered which can be changed to the spread and width of the material after rolling.

### 3.3 MLP Methods

*α*coefficient by MLP models for oval caliber (MLP-1 and MLP-2) are compared in Fig. 11. Furthermore, the performance of the MLP-1 and MLP-2 are shown in Fig. 12.

*α*coefficients are normally higher than the values of the spread ratio. Finally, it can be inferred that the trained MLP models can predict the spread in wire rod rolling with good accuracy.

### 3.4 ANFIS Methods

*α*coefficient, the RMSE of ANFIS-2 and ANFIS-4 are higher than the other two ANFIS models which is related to the prediction of the spread ratio. The results demonstrate that ANFIS can be used to predict spread in steel wire rod rolling, however, the result of MLP models shows higher accuracy in comparison to ANFIS models.

### 4 Explainability and Transferability of the Model

### 4.1 Explainability of the Model

*H̄*

*, backward and forward tension, and the output is also the maximum width after the rolling process. Based on Fig. 3,*

_{o}*H̄*

*is considered as the input instead of the roll gap since the effect of both the roll gap and roll groove radius is included in this parameter. Thus, it can represent the effect of both parameters on the output.*

_{o}*H̄*

*), backward and forward tension the output width decreases. The results of the ANFIS model are also in line with the FE analysis results which show the ANFIS model can understand the correct effect of different input parameters on the output and provide these relationships for the user.*

_{o}