Figure 1 shows the MCDA problem-solving flowchart of this study.

## 3.2 Methods

## 3.2.1 Using entropy weight method to determine each indicator’s weight

The entropy weight method, a method to determine the weights of indicators by evaluating the values of indicators under objective conditions[52], was recommended in our study to calculate each indicator’s weight according to the following 4 steps.

Step 1: Establishing the judgment matrix

According to Table 1, we establish the following judgment matrix A :

A=(\({a}_{ij}{)}_{m,n}\)(i = 1,2,...,m; j = 1,2,...n)

A=(\({a}_{ij}{)}_{m,n}=\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& \begin{array}{cc}\cdots & {a}_{1n}\end{array}\\ {a}_{21}& {a}_{22}& \begin{array}{cc}\cdots & {a}_{2n}\end{array}\\ \begin{array}{c}⋮\\ \begin{array}{c}{a}_{i1}\\ ⋮\\ {a}_{m1}\end{array}\end{array}& \begin{array}{c}⋮\\ \cdots \\ \begin{array}{c}⋮\\ {a}_{m2}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}⋮\\ \begin{array}{c}{a}_{ij}\\ ⋮\\ {a}_{m3}\end{array}\end{array}& \begin{array}{c}⋮\\ \begin{array}{c}\cdots \\ ⋮\\ {a}_{mn}\end{array}\end{array}\end{array}\end{array}\right]\)

In which m = 21, n = 6,\(\text{a}\text{n}\text{d} {a}_{ij}\) represent the *j*-th indicator’s value in the *i*-th year.

Step 2: Normalize the judgment matrix.

The criteria are generally classified into 2 types: benefit and cost. The benefit criteria mean that the higher value it is the better result would be, while the cost criteria is valid the opposite. In our study, I1, I2, and I3 are cost criteria, while others are benefit criteria. We use equations (1) and (2) to normalize the benefit criteria and cost criteria value, respectively.

$${S}_{ij}=\frac{{a}_{ij}-\text{m}\text{i}\text{n}\left\{{a}_{ij}\right\}}{\text{max}\left\{{a}_{ij}\right\}-\text{m}\text{i}\text{n}\left\{{a}_{ij}\right\}}$$

1

$${S}_{ij}=\frac{\text{m}\text{a}\text{x}\left\{{a}_{ij}\right\}-{a}_{ij}}{\text{max}\left\{{a}_{ij}\right\}-\text{m}\text{i}\text{n}\left\{{a}_{ij}\right\}}$$

2

Step 3: Calculating the indicator’s entropy

In an evaluation problem that has m evaluated object with n indicators, the entropy for the *j-*th indicator is calculated as the Eq. (3):

\({E}_{j}=-k\sum _{i=1}^{m}{f}_{ij}\text{ln}{f}_{ij}\) , i = 1,2,...,m; and j = 1,2,...,n; (3)

Where ,\({f}_{ij}=\frac{{S}_{ij}}{\sum _{i=1}^{m}Sij}\), k=\(\frac{1}{\text{ln}m}\).

Among them, \({f}_{ij}\) is the characteristic proportion of the *i*-th object.

Step 4: Calculating the entropy weight

The *j*th indicator’s entropy weight \(\left({w}_{j}\right)\) was then calculated based on the Eq. (3)

$${w}_{j}=\frac{1-{E}_{j}}{n-\sum _{j=1}^{n}{E}_{j}}$$

4

## 3.2.2 Entropy-weighted TOPSIS evaluation method

The entropy-weighted TOPSIS evaluation model has been widely used in MCDA applications due to its objectiveness, rationality, and effectivity. It is an effective MCDA method to evaluate the performance of alternatives through similarity with the ideal solution [53]. Its basic concept is that the chosen alternative should have the shortest distance from the ideal solution and the farthest from the negative-ideal solution [54]. The detailed processes of applying the entropy-weighted TOPSIS method are given below:

Step 1: Build the co-trending decision matrix

TOPSIS method requires all the criteria should have the same type, which is benefit type or cost type, in other words, the decision matrix must be the co-trending matrix. Thus, we first convert all the cost indicators (I1, I2, and I3) in Table 1 into the benefit indicators by replacing each cost indicator’s value with 100 minus it, respectively.

Table 1

The original data on child health care in China from 2000 to 2020.

Year | I1 | I2 | I3 | I4 | I5 | I6 |

**2000** | 2.40 | 13.99 | 3.09 | 85.8 | 73.8 | 73.4 |

**2001** | 2.35 | 13.28 | 3.01 | 86.7 | 74.7 | 74.5 |

**2002** | 2.39 | 12.47 | 2.83 | 86.1 | 73.9 | 74.0 |

**2003** | 2.26 | 12.24 | 2.70 | 84.7 | 72.8 | 72.7 |

**2004** | 2.20 | 11.08 | 2.56 | 85.0 | 73.7 | 74.1 |

**2005** | 2.21 | 10.27 | 2.34 | 85.0 | 73.9 | 74.8 |

**2006** | 2.22 | 9.68 | 2.10 | 84.7 | 73.9 | 75.0 |

**2007** | 2.26 | 8.71 | 2.02 | 85.6 | 74.4 | 75.9 |

**2008** | 2.35 | 8.74 | 1.92 | 85.4 | 75.0 | 77.4 |

**2009** | 2.40 | 7.70 | 1.71 | 87.1 | 77.2 | 80.0 |

**2010** | 2.34 | 7.02 | 1.55 | 89.6 | 81.5 | 83.4 |

**2011** | 2.33 | 6.32 | 1.51 | 90.6 | 84.6 | 85.8 |

**2012** | 2.38 | 5.89 | 1.44 | 91.8 | 87.0 | 88.9 |

**2013** | 2.44 | 5.53 | 1.37 | 93.2 | 89.0 | 90.7 |

**2014** | 2.61 | 5.37 | 1.48 | 93.6 | 89.8 | 91.3 |

**2015** | 2.64 | 4.99 | 1.49 | 94.3 | 90.7 | 92.1 |

**2016** | 2.73 | 5.05 | 1.44 | 94.6 | 91.1 | 92.4 |

**2017** | 2.88 | 4.58 | 1.40 | 93.9 | 91.1 | 92.6 |

**2018** | 3.13 | 4.26 | 1.43 | 93.7 | 91.2 | 92.7 |

**2019** | 3.24 | 4.02 | 1.37 | 94.1 | 91.9 | 93.6 |

**2020** | 3.25 | 4.14 | 1.19 | 95.5 | 92.9 | 94.3 |

Step2: Normalize the co-trending matrix.

The co-trending matrix was then normalized by Eq. (5), which eliminated the influence of the different measurement units. Then, a normalized matrix R was established.

$${r}_{ij}={a}_{ij}/\sqrt{\sum _{i=1}^{n}{a}_{ij}^{2}}$$

5

Where \({r}_{ij}\) represent the normalized value of *j*-*th* indicator’s value in the *i*-*th* year.

Step3: Build the normalized matrix of weight.

We built the normalized matrix of weight X by Eq. (6).

$${x}_{ij}={w}_{j}{·r}_{ij}$$

6

Where i = 1, 2,...,21, and j = 1,2,...6.

Namely, each index \({r}_{ij}\) multiply its the corresponding weight \({w}_{j}\) which is calculated by the entropy weight method mentioned above. Then, a normalized matrix of weight X was obtained as below:

X=\(\left[\begin{array}{ccc}{{w}_{1}r}_{11}& {{w}_{2}r}_{12}& \begin{array}{cc}\cdots & {{w}_{6}r}_{16}\end{array}\\ {{w}_{1}r}_{21}& {{w}_{2}r}_{22}& \begin{array}{cc}\cdots & {{w}_{6}r}_{26}\end{array}\\ \begin{array}{c}⋮\\ {{w}_{1}r}_{211}\end{array}& \begin{array}{c}⋮\\ {{w}_{2}r}_{212}\end{array}& \begin{array}{cc}\begin{array}{c}⋮\\ \cdots \end{array}& \begin{array}{c}⋮\\ {{w}_{6}r}_{216}\end{array}\end{array}\end{array}\right]\)

Step4: Identify the ideal solution A+ and negative-ideal solution A-.

The positive-ideal solutional X+ and negative-ideal solution X- were determined by matrix X as follows:

X+ =(max{\({x}_{i1}\)}, max{\({x}_{i2}\)},... max{\({x}_{i6}\)}) (7)

X- =(min{\({x}_{i1}\)}, min{\({x}_{i2}\)},... min{\({x}_{i6}\)}) (8)

Where max{\({x}_{ij}\)} and min{\({x}_{ij}\)} means the max and min value in the *j-th* column, respectively.

Step4: Calculate Euclidean distance

We calculated Euclidean distance from X+ and X- for each alternative \({x}_{i}\), respectively as follows:

$${D}_{i}^{+}=\sqrt{\sum _{j=1}^{n}{(x}_{j}^{+}-{x}_{ij}}{)}^{2}$$

9

$${D}_{i}^{-}=\sqrt{\sum _{j=1}^{n}({{x}_{j}^{-}-x}_{ij}}{)}^{2}$$

10

Where \({D}_{i}^{+}\) are Euclidean distances between *i-th* objective and positive-ideal solution, and \({D}_{i}^{-}\) are Euclidean distances between *i-th* objective and negative-ideal solution.

Step5: Calculate the relative closeness coefficient

The relative closeness coefficient of *i-th* objective is calculated by using Eq. (11) :

$${C}_{i}=\frac{{D}_{i}^{-}}{{D}_{i}^{+}+{D}_{i}^{-}}$$

11

Where 0\(\le {C}_{i}\le 1\), and the larger the \({C}_{i}\) value, the better the performance of CHC in that year. Then, we ranked all the objectives according to their \({C}_{i}\) values.

## 3.2.3 Entropy-weighted RSR evaluation method

Entropy-weighted RSR is another comprehensive evaluation method that uses a rank transformation to calculate dimensionless statistical indexes from the matrix. And the distribution of WRSR can be explored by the parameter statistical method. Generally, the WRSR indicators range from 0 (worst) to 1 (best), which was used to assess the state of the subjective [55]. The detailed processes are given below:

Step 1: Rank the indicators.

We first rank all the indicators in Table 1 based on the rules that indicators of benefit type are ranked in ascending order while indicators of cost type are ranked in descending order.

Step 2: Calculate the WRSR.

We then calculate the WRSR of each evaluation object (i.e work quality of the CHC in a year) by Eq. (12).

*WRSR* *i* =\(\frac{1}{n}\sum _{j=1}^{m}{w}_{j}{S}_{ij}\) (12)

Where \({S}_{ij}\) is the rank of CHC indicators in China from 2000 to 2021, i = 1, 2, ...,21; *m = 6*, which is the index number of CHC, and \({w}_{j}\) is the weight of *j-th* indicator.

Step 3: Sort the objectives

The last step is sorting the objective according to the WRSR values. The greater the value of *WRSR**i*, the better the performance of CHC.

## 3.2.4 Fuzzy comprehensive evaluation method

The FCE method is an application of the fuzzy set theory to make a synthetic assessment in a fuzzy decision environment with multiple criteria [56]. The FCE method used in our study is given below:

Step 1: Calculate the coefficient *C**i* and *WRSR**i*

The coefficient *C**i* and *WRSR**i* of each alternative can be obtained by using the entropy-weighted TOPSIS and weighted RSR method, respectively.

Step 2: Calculate the rank of each alternative based on the Fuzzy Set theory.

The coefficient *C**i* and *WRSR**i* were substituted to the following formula:

*W* *1* *C* *i* *+ W**2**WRSR**i* (13)

Where *W**1*:*W**2* is the weight ratio for *C**i* and *WRSR**i*, respectively. According to the previous study applying fuzzy set theory to a comprehensive evaluation work[57], the weight ratio *W**1*:*W**2* is set to 0.1:0.9, 0.5:0.5, and 0.9:0.1, respectively.

Step 3: Rank the alternative comprehensively

Since the weight ratio has 3 sets of values (i.e., 0.1:0.9, 0.5:0.5, and 0.9:0.1), we ranked all the alternatives 3 times based on the result calculated by the formula (13), respectively. Correspondingly, each alternative has 3 orders and we selected the order that appeared most frequently as the comprehensive order of the alternative. The greater the value, the better the performance of CHC.

## 3.3 Sensitivity analysis through criteria weight change

Sensitivity analysis is an effective method to observe variations in the final result that was caused by the changes in the model’s parameters. In our study, sensitivity analysis was conducted by changing each criterion’s weight according to the changing rate \({\delta }_{k}\). The designed scheme was also applied in the previous study[18]. Specially, supposing *W**i* changes to \({W}_{i}^{*}\), i = 1,2,...,6 and \({W}_{i}^{*} is\)calculated by the Eqs. (14) and (15).

$${W}_{k}^{*}={\delta }_{k}{W}_{k}$$

14

$${ \delta }_{k}=\frac{{\gamma }_{k}-{\gamma }_{k}{W}_{k}^{{\prime }}}{1-{\gamma }_{k}{W}_{k}^{{\prime }}}$$

15

Where *k* = 1,2,...n ( n = 6), \({\gamma }_{k}\)=0.01, 0.03, 0.06, 0.1, 0.2, 0.5, 0.8,1.0, 1.3, 1.8, 2.1, 2.5, 3, 3.5,4 and 4.5.\({\delta }_{k}\) is the changing rate of \({W}_{k}\). The variable \({\gamma }_{k}\) is defined as the unitary variation rate of the variation of \({W}_{k}\).\(\)

Since the sum of the 6 indicator’s weights should be equal to 1 when *W**k* changes \({W}_{k}^{*}\), other weights will also change, which was calculated as Eq. (16).

\({W}_{1}^{{\prime }}\) *=* \(\frac{{w}_{1}}{{w}_{1}+{w}_{2}+\dots +{W}_{k}^{*}+\dots {w}_{i}}\)

\({W}_{2}^{{\prime }}\) *=* \(\frac{{w}_{2}}{{w}_{1}+{w}_{2}+\dots +{W}_{k}^{*}+\dots {w}_{i}}\)

\({W}_{i}^{{\prime }}\) *=* \(\frac{{W}_{k}^{*}}{{w}_{1}+{w}_{2}+\dots +{W}_{k}^{*}+\dots {w}_{i}}\) (16)

\({W}_{n}^{{\prime }}\) *=* \(\frac{{w}_{n}}{{w}_{1}+{w}_{2}+\dots +{W}_{k}^{*}+\dots {w}_{i}}\)

Where \({W}_{k}^{{\prime }}\) is the *k-th* indicator’s weight after changing.

Taking *W**1* as an example, because the unit change rate \({\gamma }_{1}\) was designed to 16 different values, a total of 16 sets of changed weights \({W}_{k}^{*} and {W}_{k}^{{\prime }}\) can be derived from the above formulas\(.\) Correspondingly, *C**i* also has 16 changed values which will be analyzed further. These changes were all based on the variation of *W**1.* With the same algorithms, the recalculated *C**i* based on the variation of other weights (i.e., *W**2*, *W**3*, *W**4*, *W**5*, *W**6*) can be obtained. All the calculations in our study were implemented by Matlab 2019b and Microsoft Excel 2010.