# Root Cause Tracing Using Equipment Process Accuracy Evaluation for Looper in Hot Rolling
## 2 EPAE Model of Looper
### EPAE Model Construction of Looper  
The division and distribution of functional areas of the hot rolling production lines have multi-level characteristics. Therefore, we considered designing a multi-level analytic hierarchy process based on the level division, to recursively deduce the global index weights. Then, the entropy weight method is used to adjust the weights, aiming to obtain subjective and objective comprehensive weights. Finally, the fuzzy comprehensive evaluation method and the membership gravity center defuzzification method are used to achieve an accurate evaluation of the equipment process accuracy. Still taking the looper area as an example, its level can be divided into: finishing rolling area, finishing rolling unit, looper, equipment process accuracy, and evaluation index (starting angle, rising time, etc.).  
Firstly, a hierarchical structure model of an analytic hierarchy process (AHP) is constructed [21], stipulating that \(t\) is the specific functional index under the looper component. The hierarchical structure can be divided upward into component-level, equipment-level, regional-level, and factory-level. The EPAE results are represented by the symbols \(e\), \(m\), \(g\), \(P\), and then a judgment matrix is constructed according to the relative importance of the indicators at each level. The subjective weight of each level of indicators is obtained by AHP:  
\[\left\{\begin{aligned} e=& A_{1}(t)\otimes E_{1}(t) \\ m=& A_{2}(e)\otimes E_{2}(e)\\ g=& A_{3}(m)\otimes E_{3}(m)\\ P=& A_{4}(g)\otimes E_{4}(g)\end{aligned}\right. \tag{14}\]  
where \(A_{n}\) is the index weight of the current level, \(E_{n}\) is the evaluation index of the current level, and \(\otimes\) is the hierarchical analysis operation process of weights and evaluation indicators.  
Secondly, the entropy weight method [22] is used to assist in the indicator weight assignment of the AHP model. Information entropy is an important indicator that reflects the degree of order and chaos of the system. According to information entropy theory, the entropy value \(H(x)\) can be expressed as  
\[H(x)=-\sum_{i=1}^{k}[p(x_{i})\ln p(x_{i})] \tag{15}\]  
where \(k\) is the number of source messages, and \(p(x_{i})\) is the probability of occurrence of event \(x_{i}\). The entropy weight method essentially uses the entropy value to judge the degree of dispersion of its indicators.  
Afterwards, the results obtained by multi-layer AHP need to be combined with the entropy weight method. AHP mainly determines the evaluation scheme based on expert experience, which makes the judgment of the relative importance of each indicator highly subjective, so entropy becomes necessary. The weight method assists in achieving objective assignment of indicator weights. The weight values of each indicator, derived through the multi-layer AHP and entropy weight method, are brought into the following formula to calculate the comprehensive weight of the evaluation indicator \(\beta_{i}\).  
\[\beta_{i}=\mu\omega_{i}+(1-\mu)\epsilon_{i} \tag{16}\]  
where \(\mu\) represents the preference factor, determined by expert experience according to the importance of subjective and objective factors; \(\omega_{i}\) is the subjective weight of various indicators obtained by multi-layer AHP; and \(\epsilon_{i}\) is the various items obtained by the entropy weight method. The indicators are objectively weighted.  
Finally, the fuzzy relationship matrix \(R\) is established based on the equipment process evaluation index and the evaluation set. Subsequently, utilizing a combined evaluation method integrating AHP and entropy weight methods, the weight vector of the evaluation factors \(W=(\beta_{1},\beta_{2},\ldots,\beta_{n})\) is determined, resulting in the ultimate evaluation outcome.  
\[result=fuzzy(W,R) \tag{17}\]  
The membership centroid method is applied to defuzzify the above results [23], and the multiple evaluation indicators of the looper are weighted and summed to obtain the final evaluation score.