**2. Problem Formulation**

The status of a complex system is mainly reflected by some indicators, called health status indicators. The observation information of these indicators can be obtained by placing the corresponding sensors or simulating them in the computers. Here, the health assessment of the complex system model is constructed as shown in Figure 1.

**Figure 1.** The structure of the health assessment model.

It can be seen from Figure 1 that the model mainly includes three parts: the first part establishes the mapping function to transform the input information into the initial evidence. The second part constitutes a complete assessment framework based on the calculation of parameters. Finally, the assessment model parameters need to be optimized in the third part.

The specific parameters of Figure 1 are as follows:

(1) *xi* denotes the *i*th health status indicator of the complex system, where *i* = 1, 2, . . . , *L*;

(2) *L* denotes the number of assessment indicators;

(3) *Ai* denotes the mapping function between the *i*th input indicator reference grades and assessment grades;

(4) *wi* denotes the weight of the *i*th indicator;

(5) *ri* denotes the reliability of the *i*th indicator;

(6) *ei* denotes the initial evidence of the *i*th indicator.

According to the model established in Figure 1, the following two problems need to be solved in the health assessment of complex system:

(1) When assessing the health status of a complex system, the input indicators reference grades do not correspond to the assessment results grades. Therefore, Formula (1) is mainly to establish the following mapping relationship.

$$A\left(D\_1, D\_2, \dots, D\_N\right) = A\_i\left(H\_{1,i}, H\_{2,}, \dots, H\_{N\_l}\right) \tag{1}$$

where {*Dn*|*n* = 1, 2, . . . , *N* } denotes *N* assessment result grades, {*Hn*,*<sup>i</sup>*|*<sup>n</sup>* = 1, 2, . . . , *Ni* } denotes *i*th input indicators *Ni* reference grades.

(2) The assessment model part parameters, such as indicator reference value and weight, are given by experts, which may decrease the accuracy of the assessment. Therefore, it is necessary to build an optimization model to improve the accuracy of assessment results as follows.

$$\Psi = \Psi(\mathbf{w}, H, A, D) \tag{2}$$

where w, *H*, *A*, *D* denote the indicator weight, indicator reference, transformation matrix, and assessment result grades respectively.

### **3. Health Assessment Method Based on the ER Rule with a Transformation Matrix**

In this section, an assessment model with a transformation matrix based on the ER is adopted. The transformation of input information is conducted in Section 3.1. The calculation of model parameters is introduced in Section 3.2. The aggregation of indicators is given in Section 3.3.

### *3.1. Transformation Method of Input Indicators*

First, it is necessary to establish an indicator system of health assessment, when assessing a complex system. There are *N* assessment result grades, *L* indicators and the numbers of *i*th input indicator reference grades are denoted by *Ni*, as shown in the Figure 2.

**Figure 2.** The transformation between the input and output.

Suppose *H<sup>i</sup>* = "*<sup>H</sup>*1,*i*, *<sup>H</sup>*2,*i*,..., *HNi*,*<sup>i</sup>*# and *D* = {*<sup>D</sup>*1, *D*2,..., *DN*} are sets of mutually exclusive and exhaustive propositions. Thus *H<sup>i</sup>* and *D* are regarded as frames of discernment, called the discernment frame 1 and the discernment frame 2, respectively. In order to realize the transformation from discernment frame 1 to discernment frame 2, there are process of transformation as follows:

First, the mapping relationship between the *k*th reference grade of the *i*th indicator *Hk*,*<sup>i</sup>* and assessment result grades {*<sup>D</sup>*1,..., *DN*} can be described as a "if-then" rule:

$$R\_{k,j} \;:\text{ if } \mathbf{x}\_i = H\_{k,i}, \text{ then} \left\{ \left( D\_1, a\_{1,k} \right), \dots, \left( D\_{n\_r} a\_{n,k} \right), \dots, \left( D\_N, a\_{N,k} \right) \right\}, \left( \sum\_{n=1}^N a\_{n,i} = 1, 0 \le a\_{n,i} \le 1 \;/\tag{3} \right) \tag{4}$$

where *an*,*<sup>k</sup>* denotes the belief degree to which *Dn* is regard as the consequent if, input *xi* is *Hk*,*i*. *Rk*,*<sup>i</sup>* denotes *k*th rule of the *i*th indicator. Then, the mapping relationship between the discernment frame 1 and the discernment frame 2 can be established by *Ni* rules. It can be described as a matrix:

$$A\_{i} = \begin{bmatrix} H\_{1,i} & H\_{2,i} & \cdots & H\_{k,i} & \cdots & H\_{N\_{i},i} \\ D\_{2} & \begin{bmatrix} a\_{1,1} & a\_{2,2} & \cdots & a\_{1,k} & \cdots & a\_{1,N\_{i}} \\ a\_{2,1} & a\_{2,2} & \cdots & a\_{2,k} & \cdots & a\_{2,N\_{i}} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ a\_{n,1} & a\_{n,2} & \cdots & a\_{n,k} & \cdots & a\_{n,N\_{i}} \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ a\_{N,1} & a\_{N,2} & \cdots & a\_{N,k} & \cdots & a\_{N,N\_{i}} \end{bmatrix} \tag{4}$$

where *Ai* denotes *N* × *Ni* transformation matrix, whose *Ni* columns are the *Ni* rules.

**Remark 1.** *The transformation matrix is established based on Formula (3), where the belief degree is allocated to any individual assessment grades and there is no ignorance left. It can be proved that transformation matrix retains the integrity and consistency of information transformation. The details of proof can be seen in the paper [20]. In other words, a belief distribution with no ignorance will not be transformed to a belief distribution with ignorance, and vice versa*.

Second, according to rule-based information transformation technique, the input information can be transformed as a belief distribution under discernment frame 1 as follows.

$$S'(\mathbf{x}\_i^\*) = \left\{ (H\_{k,i}, \gamma\_{k,i}), \ k = 1, 2, \dots, N\_i; \ (H\_{\Theta}, \gamma\_{\Theta, i}) \right\} \tag{5}$$

with 0 ≤ *γ<sup>n</sup>*,*<sup>i</sup>* ≤ 1 (*n* = 1, ... , *Ni*, *i* = 1, ... , *<sup>L</sup>*), where *x*∗*i* denotes input information of the *i*th indicator. *Hk*,*<sup>i</sup>* denotes the *k*th reference grade of the discernment frame 1, *γk*,*<sup>i</sup>* denotes the belief degree allocated to any individual reference grade of discernment frame 1, which can be calculated as follows.

$$\begin{cases} \gamma\_{k,i} = \frac{H\_{k+1,i} - \mathbf{x}\_i^\*}{H\_{k+1,i} - H\_{k,i}} \ \ \ H\_{k,i} \le \mathbf{x}^\* \le H\_{k+1,i} \\\\ \gamma\_{k+1,i} = 1 - \gamma\_{k,i} \ \ \ \ H\_{k,i} \le \mathbf{x}^\* \le H\_{k+1,i} \\\\ \quad \gamma\_{m,i} = 0, \ m = 1, \ldots, N\_i, m \ne k, k+1 \end{cases} \tag{6}$$

where *Hk*,*<sup>i</sup>* and *Hk*+1,*<sup>i</sup>* denote the reference values of two adjacent input indicators reference grades. If there are other information transformation techniques or qualitative indicators, the degree of global ignorance denoted by *γ*Θ,*<sup>i</sup>* may exist.

Finally, based on transformation matrix *Ai*, the input information of *i*th indicator can be transformed as a belief distribution under discernment frame 2, as follows:

$$\hat{S}^{\hat{\imath}}(\mathbf{x}\_{i}^{\*}) = \{ (D\_{n,i}\boldsymbol{\beta}\_{n,i}), \ \boldsymbol{n} = 1, 2, \ldots, N; \ (D\_{\Theta}\boldsymbol{\beta}\_{\Theta,i}) \}\tag{7}$$

with 0 ≤ *β<sup>n</sup>*,*<sup>i</sup>* ≤ 1 (*n* = 1, ... , *N*, *i* = 1, ... , *<sup>L</sup>*), *β*<sup>Θ</sup>,*<sup>i</sup>* = 1 − *N* ∑ *<sup>n</sup>*=1 *β<sup>n</sup>*,*i*, where *β<sup>n</sup>*,*<sup>i</sup>* and *β*<sup>Θ</sup>,*<sup>i</sup>* denote belief degree allocated to *n*th individual assessment result grades and global ignorance, respectively, which can be calculated as follows:

$$b\_i = A\_i \times r\_i \tag{8}$$

$$\beta\_{\Theta,i} = 1 - \sum\_{n=1}^{N} \beta\_{n,i} = \gamma\_{\Theta,i} \tag{9}$$

where, *bi* = [*β*1,*i*, *β*2,*i*,..., *β<sup>N</sup>*,*<sup>i</sup>*] is the belief degree under the discernment framework 2,*ri* = *<sup>γ</sup>*1,*i*, *γ*2,*i*,..., *<sup>γ</sup>Ni*,*<sup>i</sup>* is the belief degree under the discernment framework 1, *Ai* denotes the transformation matrix corresponding to the *i*th indicator.

**Remark 2.** *Compared with Yang's work [20], there are two contributions of this work. (1) In Yang's work, the elements of transformation matrix are only determined by the decision-makers' knowledge and experience, which may decrease the assessment accuracy. In the proposed model, the expert knowledge is used to give the initial values of the transformation matrix, and the accurate values are obtained by optimizing based on the observation data. (2) Actually, the transformation matrix makes the adjustment between different discernment frameworks realized. More importantly, this paper inherits the basic work of Yang and extends it to the field of refined health assessment*.

### *3.2. Calculation of Model Parameters*

The indicator weight is the subjective concept that reflects the relative importance among the indicators [11,21]. Thus, the indicator weight is determined by the experts' preference to the assessment results grades. Differently, the indicator reliability is the objective concept, affected by inherent disturbance or noise when measured, resulting in the unreliability of observation data [22]. Therefore, the method that the synthesis of static and dynamic reliability is adopted, can effectively combine the expert knowledge and observation data [23].

Suppose *ris* and *rid* denote the statics reliability and dynamic reliability respectively. Then the indicator reliability *ri* is determined as

$$r\_i = \delta r\_i^s + (1 - \delta)r\_i^d, 0 \le \delta \le 1\tag{10}$$

where, *δ* denotes the weighting factor given by experts. *ris* can be determined by expert experience and industry standards. *rid* can be calculated via the method of distance, as follows.

First, the average of the *i*th indicator observation data is:

$$\overline{\mathbf{x}}\_{i} = \frac{1}{k\_{i}} \sum\_{t=1}^{k\_{i}} \mathbf{x}\_{i}(k)\_{t} k = 1,2,\cdots,k\_{i} \tag{11}$$

The distance between the *i*th indicator observation data and average can be expressed as:

$$d\_i(\mathbf{x}\_i(k), \overline{\mathbf{x}}\_i) = |\mathbf{x}\_i(k) - \overline{\mathbf{x}}\_i| \tag{12}$$

Then, the average distance can be calculated as:

$$\overline{D\_i} = \frac{1}{k\_i} \sum\_{k=1}^{k\_i} |\mathbf{x}\_i(k) - \overline{\mathbf{x}}\_i| \tag{13}$$

The dynamic reliability is represented as:

$$r\_i^d = \frac{\overline{D\_i}}{\max d\_i(x\_i(k), \overline{x\_i})} \tag{14}$$

**Remark 3.** *On the one hand, the weights reflect the relative importance of indicators in the evidence aggregation process. Further, the value of the weight is strongly dependent on the decision maker. Thus, the weights can be adjusted according to actual needs. On the other hand, since the expert knowledge is limited, the initial values of the weight given by the expert may not be accurate. Thus, the weight needs to be optimized based on observation data. However, the reliability is an objective attribute of evidence, so it does not need to be optimized*.

### *3.3. Aggregation of Initial Evidence*

Once the mapping relationship from input indicator grades to assessment grades is established based on transformation matrixes, the indicator observation data can be converted into initial evidence in the form of belief degree. The indicator weight is defined by the expert, and the indicator reliability is calculated by the above method in Section 3.2. Then, multiple indicators can be aggregated by using ER rule to obtain the health assessment results as follows:

$$\beta\_{n,c(L)} = \frac{\mu \left[ \prod\_{i=1}^{L} \left( \widetilde{\omega}\_{i} \beta\_{n,i} + 1 - \widetilde{\omega}\_{i} \sum\_{n=1}^{N} \beta\_{n,i} \right) - \prod\_{i=1}^{L} \left( 1 - \widetilde{\omega}\_{i} \sum\_{n=1}^{N} \beta\_{n,i} \right) \right]}{1 - \mu \prod\_{i=1}^{L} (1 - \widetilde{\omega}\_{i})} \tag{15}$$

$$\beta\_{\Theta, \mathcal{L}(L)} = \frac{\mu \left[ \prod\_{i=1}^{L} \left( 1 - \widetilde{\omega}\_i \sum\_{n=1}^{N} \beta\_{n,i} \right) - \prod\_{i=1}^{L} (1 - \widetilde{\omega}\_i) \right]}{1 - \mu \prod\_{i=1}^{L} (1 - \widetilde{\omega}\_i)} \tag{16}$$

$$\mu = \left[ \sum\_{n=1}^{N} \prod\_{i=1}^{L} \left( \widetilde{\omega}\_{i} \boldsymbol{\beta}\_{n,i} + 1 - \widetilde{\omega}\_{i} \sum\_{n=1}^{N} \boldsymbol{\beta}\_{n,i} \right) - (N-1) \times \prod\_{i=1}^{L} \left( 1 - \widetilde{\omega}\_{i} \sum\_{n=1}^{N} \boldsymbol{\beta}\_{n,i} \right) \right]^{-1} \tag{17}$$
 
$$\simeq \quad \dots \quad \boldsymbol{\zeta} \langle 1 \quad \dots \quad \boldsymbol{\zeta} \rangle \tag{18}$$

$$
\widetilde{\omega}\_{\mathbf{i}} = \omega\_{\mathbf{i}} / (1 - \omega\_{\mathbf{i}} - r\_{\mathbf{i}}) \tag{18}
$$

where, *L* denotes the number of evidence; *N* denotes the number of assessment grades; *<sup>ω</sup>*\$*i* denotes the mixed weight considering the reliability and weight of evidence; *β<sup>n</sup>*,*<sup>i</sup>* denotes the initial belief degree allocated to assessment grades. *β<sup>n</sup>*,*<sup>e</sup>*(*L*) denotes the belief degree of the assessment result *Dn*. The residual support is allocated to the assessment framework, denoted by *β*<sup>Θ</sup>,*<sup>e</sup>*(*L*).

The aggregated belief distribution can be expressed as follows.

$$O = \left\{ (D\_{n\prime} \beta\_{n, \mathcal{c}(L)})\_{\prime} \mid (D\_{\Theta\prime} \beta\_{\Theta, \mathcal{c}(L)})\_{\prime} \text{ } n = 1, 2, \dots, N \right\} \tag{19}$$

In practical application, to obtain numerical output, the belief distribution of aggregated results can be transformed into the expected utility. Assuming that the expected utility values *u*(*Dn*) of all assessment grades are determined. If the aggregated belief distribution is complete (*β*<sup>Θ</sup>,*<sup>e</sup>*(*L*) = 0), then the expected utility of aggregated assessment result can be expressed as:

$$y = \sum\_{n=1}^{N} \beta\_{n, \mathcal{E}(L)} u(D\_n) \tag{20}$$

If the aggregated belief distribution is incomplete (*β*<sup>Θ</sup>,*<sup>e</sup>*(*L*) = 0), the global ignorance can be allocated to any assessment grades. The maximum, minimum, and average of the expected utility of aggregated assessment result can be expressed as follows:

$$y\_{\text{max}} = \sum\_{n=1}^{N-1} \beta\_{n, \mathcal{c}(L)} u(D\_n) + (\beta\_{\Theta, \mathcal{c}(L)} + \beta\_{N, \mathcal{c}(L)}) u(D\_N) \tag{21}$$

where *y*max denotes maximum of the expected utility of aggregated assessment result, when *β*<sup>Θ</sup>,*<sup>e</sup>*(*L*) is allocated to the most preferred assessment grades *Dn*.

$$y\_{\min} = (\beta\_{\Theta, c(L)} + \beta\_{1, c(L)})u(D\_1) + \sum\_{n=2}^{N} \beta\_{n, c(L)} u(D\_n) \tag{22}$$

where, *y*min denotes minimum of the expected utility of aggregated assessment result, when *β*<sup>Θ</sup>,*<sup>e</sup>*(*L*) is allocated to the least preferred assessment grades *D*1.

$$y\_{\text{average}} = \frac{y\_{\text{max}} + y\_{\text{min}}}{2} \tag{23}$$

where *yaverage* denotes the average of the expected utility of aggregated assessment result. {*u*(*Dn*), |*n* = 1, 2, . . . , *N* } cannot be given accurately, which needs to be adjusted by the optimization algorithm.
