**4. Parameters Optimization**

In this section, the optimal model is constructed to solve the second problem. The optimization of model parameters is conducted in Section 4.1. The detailed implementation process of the whole model is introduced in Section 4.2.

### *4.1. Optimization of Model Parameters*

Due to the initial values of the evidence weight, indicator reference grades, expected utility, and transformation matrixes in the assessment model are given by experts. Thus, to obtained accurate assessment results, these parameters need to be optimized based on the observation data. The optimization process is shown as Figure 3.

**Figure 3.** Optimization process of model parameters.

It should be noted that the assessment of true value of overall health is set based on experts' overall judgment in prior. According to the observation data, combined with the method of expert scoring, expert panels are set to determine the health status of the research object. The optimization objective function of the health status model is established as follows. 

$$\text{min. RMSE}(\Psi) = \sqrt{\left(y - \hat{y}\right)^2} \tag{24}$$

where, *y* denotes the real health condition of the complex system, *y*ˆ denotes the assessment model output, Ψ = " *<sup>H</sup>*1,*i*,..., *H Ni*,*i*, *ω*1,..., *ωl*, *A*1,..., *Al*, *<sup>u</sup>*(*<sup>D</sup>*1),..., *u*(*<sup>D</sup> N*) # is the parameter in the optimal model, and RMSE denotes the root mean square error, which is used to measure the difference between the model output and the actual output.

To ensure the accuracy of the assessment results without changing the physical meaning of the optimization parameters, the optimization range of parameters is designed according to expert knowledge, as follows.

$$b\_{k,i} \prec H\_{k,i} \prec c\_{k,i} \tag{25}$$

$$d\_{j,k} \le a\_{j,k} \le f\_{j,k\prime} \sum\_{j} a\_{j,k} = 1 \tag{26}$$

$$
\mathcal{C}\_{\hat{l}} \prec^\* \mathcal{L}\mathcal{O}\_{\hat{l}} \prec^\* \mathcal{G}\_{\hat{l}} \tag{27}
$$

$$p\_j < u(D\_j) < q\_j \quad j = 1, 2, \dots, N,\\ k = 1, 2, \dots, N\_{\text{i}},\\ i = 1, 2, \dots, L \tag{28}$$

where, *Hn*,*<sup>i</sup>* denotes the indicator reference of the *i*th *Ni* indicator; *cn*,*<sup>i</sup>* and *bn*,*<sup>i</sup>* denote the indicator reference upper and lower bounds; *aj*,*<sup>k</sup>* denotes the elements in row *j* and column *k* of the transformation matrix *Ai*; *dj*,*<sup>k</sup>* and *fj*,*<sup>k</sup>* denote respectively the lower and upper bounds; *ωi* denotes evidence weight; *fi* and *di* denote the weight upper and lower bounds; *u*(*Dn*) denotes the utility of the *n*th assessment grade; *en* and *gn* denote the assessment grade upper and lower bounds.

### *4.2. Process of Health Assessment Based on the ER Rule*

The specific steps of health assessment using the ER assessment model proposed are as follows, shown in Figure 4.

**Figure 4.** The implementation process of the assessment model.

Step 1: The health assessment indicator system of a complex system is established based on expert knowledge and observation data.

Step 2: Transformation matrixes are determined, then input information can be transformed into the form of initial evidence.

Step 3: The evidence weight and static reliability are given according to industry standards and expert knowledge, and the dynamic reliability is calculated based on the observation data. Then, the reliability is determined by the weighting of static reliability and dynamic reliability.

Step 4: The ER rule is employed to aggregate the initial evidence, evidence weight, and reliability, to obtain the health assessment results. The expected utility *u*(*Dn*) of the assessment result grades is introduced to obtain the expected utility of the assessment result.

Step 5: The optimization of the assessment model is constituted to improve the accuracy of assessment results.
