3.3.1. Fuzzy GDM-AHP

AHP is a well-known MCDM method invented by Saaty as a decision-making tool; it is widely used for its simple calculation process and straightforward theory [68]. Traditional AHP has some shortcomings, such as subjective deviations, insufficient reliability, and an inability to reflect human thinking processes. To overcome these shortcomings, fuzzy set theory and GDM theory are introduced into AHP to determine the first-level criteria weights.

Fuzzy set theory was introduced by Zadeh to deal with vague, imprecise, and uncertain problems [69]. Fuzzy decision-making is a rational decision-making method that considers human subjectivity. In a fuzzy environment, linguistic variables are transformed into TFNs, which take the ambiguity and uncertainty of expert judgment into account [70]. By integrating TFNs into AHP, decision-making processes can be described more accurately.

Expert judgment is the most important part of the AHP method. To reduce the bias of individual expert evaluation and make the evaluation results more objective, GDM theory is introduced into the calculation of evaluation criteria weights [71]. By selecting experts from different fields and empowering them according to their importance, the advantages of expert judgment can be maximized, and the accuracy and reliability of results can be improved [72].

The process of calculating evaluation criteria weights through fuzzy GDM-AHP is as follows [73]:

(1) Establish a fuzzy pairwise comparison matrix:

Let *<sup>F</sup>* <sup>=</sup> [*ckv*]*n*×*<sup>n</sup>* be the matrix for *<sup>n</sup>* criteria against the goal. *ckv* is a fuzzy set representing the relative importance of criterion *k* over *v*. Then, assume *ckv* = <sup>1</sup> *cvk* .

Figure 2 shows the possible assessment values of *ckv* in the pairwise comparison matrix, represented as TFNs [74].

**Figure 2.** Degree of influence and corresponding TFNs for expert assessment.

#### (2) Synthesize judgements using GDM:

There are *t* experts forming an expert set *E* = {*E*1, *E*2, ··, *Et*}; the weights of experts are {*e*1,*e*2, ··· ,*et*}. Let *c* (*t*) *kv* = - *l* (*t*) *kv* , *<sup>m</sup>*(*t*) *kv* , *<sup>u</sup>*(*t*) *kv* be a TFN representing the relative importance of *ck* over *cv* judged by *DMt*. After GDM, let *ckv* = (*lkv*, *mkv*, *ukv*) be the aggregated relative importance of *ck* over *cv* judged by all experts. *ckv* can be calculated as follows:

*lkv* = *l* (1) *kv e*1 *l* (2) *kv <sup>e</sup>*<sup>2</sup> ··· *<sup>l</sup>* (*t*) *kv et* ; (2)

$$m\_{kv} = m\_{kv}^{(1)} {}^{c\_1} m\_{kv}^{(2)} {}^{c\_2} \cdot \cdot \cdot m\_{kv}^{(t)} {}^{c\_t}; \tag{3}$$

$$
\mu\_{kv} = \mu\_{kv}^{(1)\_{\mathcal{E}1}} \mu\_{kv}^{(2)\_{\mathcal{E}2}} \cdot \cdots \cdot \mu\_{kv}^{(t)\_{\mathcal{E}t}}.\tag{4}
$$

(3) Calculate the fuzzy weights of the criteria:

The geometric normalized average method can be used to calculate the fuzzy weights of criteria. Where the values are fuzzy, not crisp, the weight vector will be achieved through the following formula:

$$(w\_{lk'}\,w\_{mk'}\,w\_{uk}) = \frac{\left(\prod\_{\upsilon=1}^{n} (l\_{k\upsilon}\,m\_{k\upsilon}\,u\_{k\upsilon})\right)^{1/n}}{\sum\_{k=1}^{n} \left(\prod\_{\upsilon=1}^{n} (l\_{k\upsilon}\,m\_{k\upsilon}\,u\_{k\upsilon})\right)^{1/n'}}\tag{5}$$

where (*wlk*, *wmk*, *wuk*) is the fuzzy weight of the *k*-th criterion.

(4) Defuzzify the fuzzy weights:

Fuzzy sets are difficult to compare directly because they are partially ordered rather than linear or strictly ordered crisp values. So, we defuzzify the obtained weights to calculate the crisp value of each criterion weight as follows:

$$w\_{FCk} = \frac{w\_{lk} + 4w\_{mk} + w\_{uk}}{6}.\tag{6}$$

where *wFCk* is the crisp weight of the *k*-th first-level criterion.

#### 3.3.2. Entropy Method

The entropy method is used to calculate criteria weights according to the size and difference degree of the value of the sample data [75,76]. The larger the entropy, the smaller the influence of the evaluation criterion on decision-making; that is, the weight of the criterion is smaller. The process for calculating the weight of the evaluation criteria by the entropy method is as follows:

(1) Normalize the decision matrix:

Different criteria can be of different scales. A given decision matrix should first be transformed into a dimensionless space via

$$p\_{i\bar{j}} = \frac{\mathbb{x}\_{i\bar{j}}}{\sum\_{i=1}^{m} \mathbb{x}\_{i\bar{j}}} (i = 1, 2, \cdot, \cdot, m; j = 1, 2, \cdot, \cdot, n), \tag{7}$$

where *pij* is the probability of the *j*-th criteria in the *i*-th alternative.

(2) Calculate the entropy of the *j*-th criteria:

$$E\_{\bar{j}} = -\mathbb{K} \sum\_{i=1}^{m} p\_{i\bar{j}} \ln p\_{i\bar{j}\prime} \tag{8}$$

$$\mathbf{K} = \frac{1}{\ln m'} \tag{9}$$

where *Ej* is the entropy of the *j*-th criteria, and K is the coefficient.

(3) Calculation of objective weights:

$$w\_{SCj} = \frac{1 - E\_j}{\sum\_{j=1}^{n} (1 - E\_j)}.\tag{10}$$

where *wSCj* is the weight of the *j*-th second-level criterion.

#### 3.3.3. Combined Algorithm

To obtain the criteria weights for site selection, a combined weighting algorithm is proposed, which is to solve the weights of the first- and second-level criteria respectively. The first-level criteria weights are calculated by fuzzy GDM-AHP, the second-level criteria weights are calculated by the entropy method, and the combined weight is calculated by the following:

$$w\_{\mathbb{C}j}^{\*} = w\_{F\mathbb{C}k} \cdot w\_{\mathbb{S}\mathbb{C}j(k)}.\tag{11}$$

where *w*∗ *Cj* is the combined weight of the *j*-th criterion, *wFCk* is the weight of the *k*-th firstlevel criterion, and *wSCj*(*k*) is the weight of the *j*-th second-level criterion under the *k*-th first-level criterion.

#### *3.4. TOPSIS-GRA*

This study proposes a novel hybrid method integrating TOPSIS and GRA to obtain the optimal site for a wave power plant.

TOPSIS method, first developed by Hwang and Yoon [77], is commonly used for addressing the rank issue. The basic idea of TOPSIS is that the best decision is the one that is closest to the ideal situation and farthest from the non-ideal situation. Although TOPSIS is widely used in many fields, it has some shortcomings. TOPSIS introduces two reference points and ranks alternatives by comparing the distances from alternatives to these points. It can express the position similarity between alternatives, but it does not consider the shape similarity between the alternatives. The GRA method was originally developed by Deng and is suitable for making decisions in multiple-attribute situations [78]. The limitation of TOPSIS can be overcome by the grey relation coefficient of the GRA model [79,80]. The combination of TOPSIS and GRA measures the relations among alternatives based on the degree of similarity or difference in both the position and shape of the alternatives.

The process for TOPSIS-GRA is as follows:

(1) Calculate the weighted normalized decision matrix:

$$w\_{i\bar{j}} = w\_{C\bar{j}}^\* \cdot x\_{i\bar{j}}^\* (i = 1, 2, \cdot \cdot \cdot, m; j = 1, 2, \cdot \cdot \cdot, n), \tag{12}$$

$$V = \begin{pmatrix} v\_{ij} \end{pmatrix}\_{m \times n} = \begin{bmatrix} v\_{11} & v\_{12} & \dots & v\_{1n} \\ v\_{21} & v\_{22} & \dots & v\_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ v\_{m1} & v\_{m2} & \dots & v\_{mn} \end{bmatrix}' \tag{13}$$

where *vij* denotes the weighted normalized criterion value of the *j*-th criterion in the *i*-th alternative.

(2) Determine the positive ideal solutions (*A*+) and negative ideal solutions (*A*−):

$$A^{+} = \left\{ v\_1^{+}, \dots, v\_n^{+} \right\} = \left\{ \left( \max\_{i} v\_{ij}, j \in J \right) \left( \min\_{i} v\_{ij}, j \in J' \right) \right\}, \quad (i = 1, \dots, m), \tag{14}$$

$$A^- = \{v\_1^-, \dots, v\_n^-\} = \left\{ \left(\min\_i v\_{\text{ij}}, j \in J\right) \left(\max\_i v\_{\text{ij}}, j \in J'\right) \right\}, \quad (i = 1, \dots, m), \tag{15}$$

where *J* and *J* refer to the benefit criteria set and cost criteria set, respectively.

(3) Calculate the Euclidean distance *d*<sup>+</sup> *<sup>i</sup>* and *d*<sup>−</sup> *<sup>i</sup>* of each alternative from positive ideal solutions (PIS) and negative ideal solutions (NIS):

$$d\_i^+ = \sqrt{\sum\_{j=1}^n \left(v\_{ij} - v\_j^+\right)^2}, \ (i = 1, \dots, m), \tag{16}$$

$$d\_i^- = \sqrt{\sum\_{j=1}^n \left(v\_{ij} - v\_j^-\right)^2}, \ (i = 1, \dots, m), \tag{17}$$

where *d*<sup>+</sup> *<sup>i</sup>* is the distance from alternative *i* to PIS, and *d*<sup>−</sup> *<sup>i</sup>* is the distance from alternative *i* to NIS.

(4) Calculate the grey relational coefficients:

Based on the weighted normalized decision matrix, the grey relational coefficient between the *i*-th alternative and the PIS with respect to the *j*-th criterion is calculated as follows:

$$r\_{ij}^{+} = \frac{\stackrel{
\text{minimum}}{i} \left| v\_{ij} - v\_j^{+} \right| + \rho \frac{\text{max} \, \text{max}}{i} \left| v\_{ij} - v\_j^{+} \right|}{\left| v\_{ij} - v\_j^{+} \right| + \rho \frac{\text{max} \, \text{max}}{i} \left| v\_{ij} - v\_j^{+} \right|} \,\tag{18}$$

$$\mathcal{R}^{+} = \left[r\_{ij}^{+}\right]\_{m \times n'} \tag{19}$$

where ρ is the distinguishing coefficient, and *R*<sup>+</sup> is the grey relational coefficient matrix with PIS. In this study, the distinguishing coefficient is set as 0.5.

Similarly, the grey relational coefficient between the *i*-th alternative and the NIS with respect to the *j*-th criterion can be obtained as follows:

$$r\_{ij}^- = \frac{\underset{i}{\minmin}}{\underset{j}{\text{argmax}}} \left| v\_{ij} - v\_j^- \right| + \left| \begin{matrix} \maxmax \\ i & j \end{matrix} \left| v\_{ij} - v\_j^- \right| \right|}{\left| v\_{ij} - v\_j^- \right| + \left| \begin{matrix} \maxmax \\ i & j \end{matrix} \left| v\_{ij} - v\_j^- \right|} \right|}, \tag{20}$$
 
$$R^- = \left[ r\_{ij}^- \right]\_{m \times n'} \tag{21}$$

*m*×*n*

where *R*− is the grey relational coefficient matrix with NIS.

(5) Calculate the grey relational grade:

The grey relational grade is used for the overall evaluation of alternatives depending on all criteria. It is defined as the average value of relational coefficients at different criteria. For the *i*-th alternative, the grey relational grades from PIS and NIS are given as follows:

$$\log\_i^+ = \frac{1}{n} \sum\_{j=1}^n r\_{ij}^+;\tag{22}$$

$$\log\_i^- = \frac{1}{n} \sum\_{j=1}^n r\_{ij}^-. \tag{23}$$

#### (6) Calculate a new relational grade:

Normalize the Euclidean distances and grey relational grades obtained from Equations (5) and (7), as follows:

$$D\_i^+ = \frac{d\_i^+}{\max d\_i^+},\ D\_i^- = \frac{d\_i^-}{\max d\_i^-},\ G\_i^+ = \frac{\mathcal{g}\_i^+}{\max \mathcal{g}\_i^+},\ G\_i^- = \frac{\mathcal{g}\_i^-}{\max \mathcal{g}\_i^-};\tag{24}$$

$$\text{S}\_{i}^{+} = \mathfrak{a}D\_{i}^{-} + \mathfrak{B}\text{G}\_{i}^{+};\tag{25}$$

$$S\_i^- = \mathfrak{a}D\_i^+ + \mathfrak{B}G\_i^-. \tag{26}$$

Among them, the larger the values of *D*− *<sup>i</sup>* and *<sup>G</sup>*<sup>+</sup> *<sup>i</sup>* , the closer the alternative is to the positive ideal solution in position and shape. The larger the values of *D*<sup>+</sup> *<sup>i</sup>* and *G*<sup>−</sup> *<sup>i</sup>* , the closer the alternative is to the negative ideal solution in position and shape. In the above formulas, α and β are the weights of position and shape, respectively, in the calculation of the similarity degree of the alternative and ideal solutions, reflecting the decision-maker's preference for position and shape factors. In this study, α and β are both set as 0.5.

The new relational grade is as follows:

$$Z\_i = \frac{\mathcal{S}\_i^+}{\mathcal{S}\_i^+ + \mathcal{S}\_i^-}. \tag{27}$$

(7) Rank alternatives according to the values of *Zi*:

The order of alternatives is ranked according to the value of relative closeness to each of the alternatives. A greater value of *Zi* indicates a higher priority in the alternatives.
