## 1 Introduction  
Artificial neural network (ANN), a technique for artificial intelligence and machine learning, is often applied as a tool to deal with nonlinear problems and offer predictions in civil engineering [1; 2; 3; 4], material science [5; 6], etc. The extension of its applications into the iron and steel industry is also reported [7; 8; 9; 10; 11]. Architecture establishment of a suitable ANN model is still challenging in the definition of the layer number and node number in the respective layer. Generally, a three-layer ANN model is sufficient to build the relations among variables and responses [12]. Therefore, the determination of the proper node number in the single hidden layer is the key issue. Table 1 lists the publications on the optimization of the single hidden layer node number. To seek an optimal one, many applied the trial and error method in a diverse range of the node numbers [13; 14; 15; 16; 17]; whilst others merely followed the rules of thumb proposed in the literature [18; 19; 20; 21; 22; 23; 24], and the application of rules can significantly reduce the number of trials. Six selected rules of thumb [18; 19; 20; 21; 22; 23; 24; 25] used in the respective publication are given below and indicated in Table 1 with an asterisk. The node number or number range for each study was calculated according to the six rules of thumb and compared to the optimal hidden layer node number determined in the respective study. It is evidently shown that the extended applications of six empirical equations into other fields are far less satisfying. Equations (1) and (2) show more robust applications, yet they yield a rather larger number for trials. \[N_{h}=(N_{i}+N_{o})^{1/2}+a,a\in[1,10] \tag{1}\]  
\[N_{h}=(N_{i}+N_{o})^{1/2}+a,a\in[0,10] \tag{2}\]  
\[N_{h}=(N_{i}\times N_{o})^{1/2} \tag{3}\]  
\[N_{h}=N_{train}/(N_{i}+1) \tag{4}\]  
\[N_{h}=\ 1/2\ (N_{i}+N_{o})+N_{train}^{1/2} \tag{5}\]  
\[N_{h}=2/3\ (N_{i})+N_{o} \tag{6}\]  
where, \(N_{i}\), \(N_{h}\), and \(N_{o}\) are the node numbers in the input, hidden, and output layer, \(N_{train}\) is the dataset size for training, and \(a\) is an empirical integer not larger than 10. Several reasons contribute to the diverse results of the hidden layer node number optimization with the six rules of thumb. One is the problem nature or complexity. As shown in Table 1, the optimal values of erosion of beaches and energy conservations in old buildings are out of the range defined by the empirical equations. One empirical equation may show consistent performance for the problems within similar complexity. For instance, the prediction of Equation (3) for beach erosion [13] is rather close to the optimal value that was used as the rule of thumb in the study of the mechanical behavior of mortar [22]. The second reason is the quality of the dataset for training. The optimal training dataset size was suggested to be ten times larger than the total number of weights and biases [12]. A steel ladle lining study showed that the dataset size could be 16 times larger than the variable number in the input layer, in the case of a well-distributed dataset in variation/response spaces [26]. Table 1 also shows the size of the dataset for training and the ratio of the training dataset size to the number of variables quoted in literature. Mostly, the information of dataset quality is missing and insufficient dataset size relative to the variable number could be expected. Finally, the sufficiency of important variables or factors, i.e., the variable completeness, in the problem definition shall also be taken into account. The present paper investigates the influence of variation/response space complexity and variable completeness on the required node numbers for a three-layer backpropagation (BP)-ANN model using steel ladle lining as a case study. A representative dataset was obtained by a sample screening approach applying multiple orthogonal arrays [26]. The dataset contains 160 samples constituted by ten variables and three responses.