*3.1. Advantages of the Hybrid Weights*

The DANP-mV model is derived from the integration of DEMATEL, DANP and modified VIKOR, three models. It has a strong comprehensive effect due to the synergy of these three methods. Recently, the effectiveness of the DANP-mV model has been proven in many studies in different fields [47–52]. The model has several advantages such as consideration of the interdependencies between the criteria, needing fewer pairwise comparisons and ease of calculation. The DANP-mV model can be applied to solve decision problems in the real world, treating the process of decision-making as a whole evaluation system and focusing on the fundamental cause of the problem. Although the DANP-mV model has some advantages, it still relies on the subjective opinions of experts. Zavadskas and Podvezko [53] pointed out that the criterion weight is critical in MCDM problems. If the criterion weights are only dependent on expert judgements, there will be potential uncertainty which affects the results [20,54]. Therefore, some have proposed the use of objective weights in the decision models [21–23,55].

Among the methods for determining objective weights, the entropy method has a solid theoretical foundation and has proven to be suitable for decision-making problems in different fields. The criterion weight is mainly determined based on the relationship between the original data and does not need decision-maker opinions [20,21,23,54].

During the process of decision-making, if only relies on subjective preferences of make decisions, the final decision-making results are easily influenced by subjective preferences and lose objectivity [19]. The empirical case of this study is to choose the manufacturing location of the factory. In the past, the company only relied on the subjective preferences of senior management to make decisions.

Thus, in this study, the entropy method is combined with the DANP method, to effectively reduce the limitation of subjective weighting in the DANP method. In practice, we can also adjust the ratio between subjective and objective weights based on decision needs. Thus, in this study, the entropy method is combined with the DANP method, to effectively reduce the limitation of subjective weighting in the DANP method. In practice, we can also adjust the ratio between subjective and objective weights based on decision needs.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 5 of 23

The DANP-mV model proposed in this study retains the characteristics and advantages of the original DANP-mV model and at the same time considers the objective weights into the system. Therefore, the proposed model will be applicable to real-world decision-making. However, this study assumes that the integrated strategy coefficient of the combined weight is 0.5, which means that subjective and objective preferences are considered equally important. The DANP-mV model proposed in this study retains the characteristics and advantages of the original DANP-mV model and at the same time considers the objective weights into the system. Therefore, the proposed model will be applicable to real-world decision-making. However, this study assumes that the integrated strategy coefficient of the combined weight is 0.5, which means that subjective and objective preferences are considered equally important.

## *3.2. Proposed DANP-mV Model 3.2. Proposed DANP-mV Model*

The proposed DANP-mV model is a hybrid research tool that contains the followed methods—DANP, entropy and modified VIKOR. DANP is used to evaluate the network relationship between the criteria and the influential weights of the criteria. The entropy method is mainly used for confirmation of the objective weights and the modified VIKOR method is applied for alternative selection. The complete operating process illustrated in Figure 1 can be divided into four phases: The proposed DANP-mV model is a hybrid research tool that contains the followed methods— DANP, entropy and modified VIKOR. DANP is used to evaluate the network relationship between the criteria and the influential weights of the criteria. The entropy method is mainly used for confirmation of the objective weights and the modified VIKOR method is applied for alternative selection. The complete operating process illustrated in Figure 1 can be divided into four phases:


**Figure 1.** Analytical processes. **Figure 1.** Analytical processes.

3.2.1. Phase 1: Construct the Network Relationship 3.2.1. Phase 1: Construct the Network Relationship

Step 1: Establish the initial direct influence relationship matrix *E* Step 1: Establish the initial direct influence relationship matrix *E*

This step encodes the data obtained from the questionnaire responses of the *K* experts to get an initial direct influence relationship matrix *E* (Equation (1)) for each expert. Data collection is conducted through interviews with experts. The questionnaire scale is evaluated using scores from 0 to 4—(0) no influence, 1 (low influence), 2 (medium influence), 3 (high influence), 4 (extremely high influence). Experts are asked to specify the degree of influence between all criteria through pairwise comparisons. This step encodes the data obtained from the questionnaire responses of the *K* experts to get an initial direct influence relationship matrix *E* (Equation (1)) for each expert. Data collection is conducted through interviews with experts. The questionnaire scale is evaluated using scores from 0 to 4—(0) no influence, 1 (low influence), 2 (medium influence), 3 (high influence), 4 (extremely high influence). Experts are asked to specify the degree of influence between all criteria through pairwise comparisons.

*ij n n*

*e for k k* <sup>×</sup> = = *<sup>E</sup>* :

The initial direct influence matrix is expressed as 1,2, , *K k*

The initial direct influence matrix is expressed as *E <sup>K</sup>* = *e k ij n*×*n f or k* = 1, 2, · · · , *k*:

$$E = \begin{bmatrix} e\_{11} & \cdots & e\_{1j} & \cdots & e\_{1n} \\ \vdots & & \vdots & & \vdots \\ e\_{i1} & \cdots & e\_{ij} & \cdots & e\_{in} \\ \vdots & & \vdots & & \vdots \\ e\_{n1} & \cdots & e\_{nj} & \cdots & e\_{nn} \end{bmatrix} . \tag{1}$$

Step 2: Calculate the average direct influence relationship matrix *A*

The direct influence relationship matrixes for the *K* experts are aggregated and divided by *K* to obtain the average direct influence relationship matrix *A* as shown in Equation (2).

$$A = \mathbf{E}^{\rm AVG} = a\_{ij} = \frac{1}{k} \sum\_{k=1}^{k} \epsilon\_{ij}^{k} = \begin{bmatrix} a\_{11} & \cdots & a\_{1j} & \cdots & a\_{1n} \\ \vdots & & \vdots & & \vdots \\ a\_{i1} & \cdots & a\_{ij} & \cdots & a\_{in} \\ \vdots & & \vdots & & \vdots \\ a\_{n1} & \cdots & a\_{nj} & \cdots & a\_{nn} \end{bmatrix} . \tag{2}$$

Step 3: Calculate the normalized directly influence relationship matrix *N<sup>d</sup>*

The average direct influence relationship matrix is normalized to obtain the normalized direct influence relationship matrix *N<sup>d</sup>* as shown in Equation (3).

$$\mathbf{N}^d = \mathbf{A}/\mathbf{x} \tag{3}$$

$$\infty = \max\left[\max\_{1 \le i \le n} \sum\_{j=1}^{n} a\_{ij\prime} \max\_{1 \le j \le n} \sum\_{i=1}^{n} a\_{ij}\right]. \tag{4}$$

Step 4: Derive total influence relationship matrix *T*

Use the normalized direct influence relationship matrix to obtain the total influence relationship matrix *T* (Equation (5)). The total influence relation matrix is an *n* by *n* matrix, *T* = *T<sup>C</sup>* = h *tij*i *n*×*n f or i*, *j* = 1, 2, · · · , *n*.

$$T = A + A^2 + \cdots + A^Z = A(I - A)^{-1} \\ for \lim\_{z \to \infty} A^Z = [0]\_{n \times n}. \tag{5}$$

Step 5: Build (criteria/dimension) total influence relationship matrix *T<sup>C</sup>* and *T<sup>D</sup>*

From the total influence relationship matrix *T*, the total influence relationship matrix of the criterion *T<sup>C</sup>* and the total influence relationship matrix of the dimensions *T<sup>D</sup>* can be obtained. The calculation of the total influence relationship matrix for criterion *T<sup>C</sup>* is expressed as in Equation (6), where *D<sup>m</sup>* is the *m*-th dimension (cluster); *cmm* is the *m*-th criterion in the *m*-th dimension; *t ij c* is the sub-matrix of the criterion influence relationship obtained by comparing the *i*-th dimension with the *j*-th dimension.

*T<sup>C</sup>* = *D*<sup>1</sup> . . . *Di* . . . *D<sup>m</sup> c*<sup>11</sup> *c*<sup>12</sup> . . . *c*1*m*<sup>1</sup> . . . *ci*1 *ci*2 . . . *cim<sup>i</sup>* . . . *cm*<sup>1</sup> *cm*<sup>2</sup> . . . *cmm<sup>m</sup> D*<sup>1</sup> *D<sup>j</sup> D<sup>m</sup> c*11···*c*1*m*<sup>1</sup> · · · *cj*1···*cjm<sup>j</sup>* · · · *cm*1···*cmmm t* 11 *C* · · · *t* 1*j C* · · · *t* 1*m C* . . . . . . . . . *t i*1 *C* · · · *t ij C* · · · *t im C* . . . . . . . . . *t m*1 *C* · · · *t mj C* · · · *t mm C mj*×*m<sup>j</sup>* |*m*<*n*, P*m <sup>j</sup>*=<sup>1</sup> *mj*=*n* . (6)

The total influence relationship matrix of the dimension *T*<sup>D</sup> is shown in Equation (7).

$$T\_D = \begin{bmatrix} t\_{11} & \cdots & t\_{1j} & \cdots & t\_{1m} \\ \vdots & & \vdots & & \vdots \\ t\_{l1} & \cdots & t\_{lj} & \cdots & t\_{lm} \\ \vdots & & \vdots & & \vdots \\ t\_{m1} & \cdots & t\_{mj} & \cdots & t\_{mm} \end{bmatrix}\_{m \times m} \tag{7}$$

Step 6: Degree of influence and the degree it is influenced between systems

The total influence relationship matrix is summed up to obtain the degree of influence and the degree it is influenced between systems. As shown in Equations (8) and (9), *r<sup>i</sup>* represents the sum of the rows on the *i*-th row of matrix *T*, which means the sum of the direct and indirect effects of criterion *i* on the other criteria; *c<sup>j</sup>* represents the sum of the columns in the *j*-th column of matrix *T* which means the sum of the direct and indirect influence on criterion *j* of the other criteria.

$$\mathbf{r} = (r\_1, \dots, r\_i, \dots, r\_n)' = (r\_i)\_{n \ge 1} = \left[ \sum\_{j=1}^n r\_{ij} \right]\_{n \ge 1} \text{ for } i, j = 1, 2, \dots, n. \tag{8}$$

$$\mathbf{c} = (\mathbf{c}\_1 \cdot \dots \cdot \mathbf{c}\_{\bar{\jmath}} \cdot \dots \cdot \mathbf{c}\_n)' = (\mathbf{c}\_{\bar{\jmath}})\_{n \times 1} = (\mathbf{c}\_{\bar{\jmath}})'\_{1 \times n} = \left[ \sum\_{i=1}^n \mathbf{c}\_{i\bar{\jmath}} \right]'\_{n \times 1} \text{ for } i, j = 1, 2, \dots, n. \tag{9}$$

Finally, the degree of influence to and from the dimensions are calculated for the total influence relation matrix (*TD*).

Step 7: Draw the influence relationship map INRM

The degree of influence to and from are marked in coordinates to obtain the influence relationship map. Here, (*r<sup>i</sup>* + *ci*) represents the sum of the influence of the criterion and the influence, which is also called the total influence degree which represents the importance of criterion *i* in the entire system. The (*r<sup>i</sup>* − *ci*) represents the difference between the degree of influence of the criteria minus the degree it is influenced or the degree of net influence. This index represents the causal relationship between the criteria. Taking (*r<sup>i</sup>* + *ci*) as the *x* axis and (*r<sup>i</sup>* − *ci*) as the *y* axis, we can draw the INRM.

3.2.2. Phase 2: Derive the Subjective and Objective Weights

Step 1: Define the unweighted super matrix *S*

The total influence relationship matrix of the criterion is normalized and transposed to generate an unweighted super matrix *S*. The normalization is expressed as in Equation (10) and the unweighted super matrix *S* is shown in Equation (11).

*T* β *C* = *D*<sup>1</sup> . . . *Di* . . . *D<sup>m</sup> c*<sup>11</sup> *c*<sup>12</sup> . . . *c*1*m*<sup>1</sup> . . . *ci*1 *ci*2 . . . *cim<sup>i</sup>* . . . *cm*<sup>1</sup> *cm*<sup>2</sup> . . . *cmm<sup>m</sup> D*<sup>1</sup> *D<sup>j</sup> D<sup>m</sup> c*11···*c*1*m*<sup>1</sup> . . . *cj*1···*cjm<sup>j</sup>* · · · *<sup>c</sup>n*1···*cmmm T* β11 *C* · · · *T* β1*j C* · · · *T* β1*m C* . . . . . . . . . *T* β*i*1 *C* · · · *T* β*ij C* · · · *T* β*im C* . . . . . . . . . *T* β*m*1 *C* · · · *T* β*mj C* · · · *T* β*mm C mj*×*m<sup>j</sup>* |*m*<*n*, P*m <sup>j</sup>*=<sup>1</sup> *mj*=*n* . (10) *S* = (*T* β *C* ) 0 = *D*<sup>1</sup> . . . *Dj* . . . *Dm c*<sup>11</sup> *c* 12 . . . *c* 1*m*<sup>1</sup> . . . *cj*1 *c j*2 . . . *c jm<sup>j</sup>* . . . *cm*<sup>1</sup> *c m*2 . . . *cmm<sup>m</sup> D*<sup>1</sup> *D<sup>i</sup> D<sup>m</sup> c*11···*c*1*m*<sup>1</sup> . . . *ci*1···*cim<sup>i</sup>* · · · *<sup>c</sup>m*1···*cmmm s* 11 · · · *s i*1 · · · *s m*1 . . . . . . . . . *s* 1*j* · · · *s ij* · · · *s mj* . . . . . . . . . *s* 1*m*, · · · *s im* · · · *<sup>s</sup> mm n*×*n*|*m*<*n*, P*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *mj*=*n* . (11)

Step 2: Construct a weighted super matrix *S w*

First, normalize the total influence relationship matrix of the dimensions. Normalization is done by dividing each element by *d<sup>i</sup>* , as shown in Equation (12).

$$\begin{aligned} \mathbf{T}\_{\mathcal{D}}^{\beta} &= \begin{bmatrix} t\_{11}^{\mathcal{D}} & \cdots & t\_{1j}^{\mathcal{D}} & \cdots & t\_{1m}^{\mathcal{D}} \\ \vdots & & \vdots & & \vdots \\ t\_{i1}^{\mathcal{D}} & \cdots & t\_{ij}^{\mathcal{D}} & \cdots & t\_{im}^{\mathcal{D}} \\ \vdots & & \vdots & & \vdots \\ \vdots & & \vdots & & \vdots \\ t\_{m1}^{\mathcal{D}} & \cdots & t\_{mj}^{\mathcal{D}} & \cdots & t\_{mm}^{\mathcal{D}} \end{bmatrix}\_{\mathsf{m}\times\infty} &= \begin{bmatrix} t\_{\mathcal{D}}^{\mathcal{1}}/\mathsf{d}\_{1} & \cdots & t\_{\mathcal{D}}^{\mathcal{1}}/\mathsf{d}\_{1} & \cdots & t\_{\mathcal{D}}^{\mathcal{1}m}/\mathsf{d}\_{1} \\ \vdots & & \vdots & & \vdots \\ t\_{\mathcal{D}}^{\mathcal{1}}//\mathsf{d}\_{i} & \cdots & t\_{\mathcal{D}}^{\mathcal{1}j}/\mathsf{d}\_{i} & \cdots & t\_{\mathcal{D}}^{\mathcal{m}}/\mathsf{d}\_{i} \\ \vdots & & \vdots & & \vdots \\ t\_{\mathcal{D}}^{\mathcal{m}}//\mathsf{d}\_{m} & \cdots & t\_{\mathcal{D}}^{\mathcal{m}}/\mathsf{d}\_{m} & \cdots & t\_{\mathcal{D}}^{\mathcal{m}}/\mathsf{d}\_{m} \end{bmatrix}\_{\mathsf{m}\times\infty} \end{aligned} \tag{12}$$

Next, calculate the weighted super matrix *s <sup>w</sup>* (Equation (13)).

*S <sup>w</sup>* = *T* β *D S* = *D*<sup>1</sup> . . . *Dj* . . . *D<sup>m</sup> c*<sup>11</sup> *c* 12 . . . *c* 1*m*<sup>1</sup> . . . *cj*1 *c j*2 . . . *c jm<sup>j</sup>* . . . *cm*<sup>1</sup> *c m*2 . . . *cmm<sup>m</sup> D*<sup>1</sup> *D<sup>i</sup> D<sup>m</sup> c*11···*c*1*m*<sup>1</sup> . . . *ci*1···*cim<sup>i</sup>* · · · *<sup>c</sup>m*1···*cmmm t* β*D* <sup>11</sup> × *s* 11 · · · *t* β*D i*1 × *s i*1 · · · *t* β*D m*1 × *s m*1 . . . . . . . . . *t* β*D* 1*j* × *s* 1*j* · · · *t* β*D ij* × *s ij* · · · *t* β*D mj* × *s mj* . . . . . . . . . *t* β*D* 1*m* × *s* <sup>1</sup>*<sup>m</sup>* · · · *<sup>t</sup>* β*D im* × *s im* · · · *<sup>t</sup>* β*D mm* × *s mm* . (13)

Step 3: Derive the influence weight of the entire system *WIWS*

Limit the derivation of the weighted super matrix *S w* to obtain the overall influence weight *WIWS* as shown in Equation (14). The matrix will eventually become stable and a set of overall priority vectors *w iw* 1 , · · · , *w iw* 2 , · · · , *w iw* 3 obtained, which is called the influential weight *WIWS* .

$$\mathbf{W}^{\text{JWS}} = \lim\_{q \to \infty} (\mathbf{s}^{\text{w}})^q. \tag{14}$$

Step 4: Establish the performance evaluation matrix *F*

Extract performance data from the database to obtain a performance evaluation matrix *F* as shown in Equation (15).

$$\mathbf{F} = \begin{bmatrix} \mathbf{C}\_1 & \cdots & \mathbf{C}\_j & \cdots & \mathbf{C}\_n \\ a\_1 & & & \\ \vdots & & \begin{bmatrix} f\_{11} & \cdots & f\_{1j} & \cdots & f\_{1n} \\ \vdots & & \vdots & & \vdots \\ f\_{1i} & \cdots & f\_{ij} & \cdots & f\_{in} \\ \vdots & & \vdots & & \vdots \\ \vdots & & \vdots & & \vdots \\ f\_{m1} & \cdots & f\_{mj} & \cdots & f\_{mn} \end{bmatrix} \\ \tag{15}$$

Step 5: Calculate the normalized performance evaluation matrix *N<sup>e</sup>*

Normalize the performance evaluation matrix to obtain the normalized performance evaluation matrix *N<sup>e</sup>* as shown in Equation (16).

$$\mathbf{N}^{\varepsilon} = n\_{ij}^{\varepsilon} = \frac{f\_{ij}}{\sum\_{i=1}^{m} f\_{ij}}.\tag{16}$$

Step 6: Derive the variation degree of the criterion *e<sup>j</sup>*

The normalized performance evaluation matrix is deduced from the variation degree to obtain the entropy value *e<sup>j</sup>* for the degree of variation for each criterion (Equation (17). The *p* is a constant. Let *p* = (*ln*(*q*))−<sup>1</sup> be used to ensure that *ej*(*j* = 1, 2, · · · , *n*) belongs from 0 to 1.

$$e\_{\hat{l}} = -p \sum\_{j=1}^{n} n\_{ij}^{\varepsilon} \ln n\_{ij}^{\varepsilon}. \tag{17}$$

Step 7: Calculate the degree of the divergence coefficient *e<sup>j</sup>*

The entropy vector is used to calculate the degree of deviation and each degree of the divergence coefficient *e<sup>j</sup>* is obtained, as shown in Equation (18). The *ej*(*j* = 1, 2, · · · , *n*) represents the inherent intensity of contrast between *j* criteria. The higher the value of *e<sup>j</sup>* in the criteria, the greater the relative importance of the role it plays in the whole system.

$$
\overline{\mathbf{e}}\_{\overline{\rangle}} = \mathbf{1} - \mathbf{e}\_{\overline{\rangle}}.\tag{18}
$$

Step 8: Derive the objective weight of the entire system *WOWS*

The divergence coefficient *e<sup>j</sup>* is deduced by simple additive normalization to obtain the objective weight *WOWS* of the entire system as shown in Equation (19).

$$\mathbf{W}^{\rm OWS} = \mathbf{e}\_{j} / \sum\_{k=1}^{n} \mathbf{e}\_{j}. \tag{19}$$

#### 3.2.3. Phase 3: Integrate the Subjective and Objective Weight *w* ∗

The influential weight and the objective weight are combined to obtain the integrated weight *w* ∗ of the entire system. As shown in Equation (20), the µ is a strategic coefficient which can be adjusted according to different cases. The preset value is 0.5, which indicates equal importance between the subjective and objective weights.

$$w^\* = \mu \mathbf{W}^{\text{WWS}} + (1 - \mu)\mathbf{W}^{\text{OWS}}.\tag{20}$$

3.2.4. Phase 4: Use the Modified VIKOR to Perform the Evaluation

The concept of VIKOR originated from the problem of multi-objective planning [56]. Opricovic [57] applied it to the research of civil engineering. Opricovic and Tzeng [58] made a comparison between VIKOR and TOPSIS and their results showed that performance evaluation using VIKOR would be more reasonable and effective. For the detailed operation processes of original VIKOR, please refer to References [58–60]. In this study, modified VIKOR will be used as the following steps:

Step 1: Define the aspiration level and the worst value

Decision-makers define the aspiration level and the worst value based on their expectations. In past performance evaluation methods using the positive and negative ideal solutions as the basis for evaluation, one may be caught in the dilemma of finding a good apple in a barrel of rotten apples. Therefore, it is better to replace those "ideal" solutions with the aspiration level and the worst value. In this study, the scales range from 0 to 100, where *f asp* = 100 indicates the aspiration level and the *f wst* = 0 is set as the worst value.

Step 2: Calculate the normalized performance evaluation matrix *N<sup>v</sup>*

Normalize the performance evaluation matrix to obtain the normalized performance evaluation matrix *N<sup>v</sup>* as shown in Equation (21). Normalize the performance of *j* criteria in *q* alternatives and calculate the distance between each performance and the aspiration level at the same time.

$$\mathbf{N}^{v} = \left( \left| \mathfrak{f}^{\mathrm{asp}} - f\_{qj} \right| \right) / \left( \left| \mathfrak{f}^{\mathrm{asp}} - \mathfrak{f}^{\mathrm{wst}} \right| \right). \tag{21}$$

Step 3: Evaluate the overall performance of each alternative

The normalized performance evaluation matrix is weighted to obtain the overall benefit evaluation matrix *G* and the average group utility vector *rqj* as shown in Equations (22) and (23). Hence, the normalized performance evaluation matrix means the difference between each criterion and the aspiration level for each alternative. The *w* ∗ *j* is the integrated weight and the overall performance evaluation will be generated through the interaction of the two matrices.

$$\mathbf{G} = \mathbf{N}^{\upsilon} w^\* \tag{22}$$

$$\mathbf{r}\_{qj} = \sum\_{j=1}^{n} \mathbf{N}^{v}.\tag{23}$$

The original VIKOR considers two types of differences, the average group utility and the maximum regret. Since the purpose of the DANP-mV model is to focus on the decision-making process it can incorporate more references. The model uses the mean group utility *rqj* only. Here,*rqj* means the comprehensive difference between the various alternatives and the aspiration level, this difference will be based on the average group utility.
