*3.3. The Application of FAHP for Weight Calculation*

*3.3. The Application of FAHP for Weight Calculation* After defining the five sustainable project criteria, as shown in the previous subsection, the first step in determining the priority weights of these criteria is collecting the opinions of experts in sustainability and sustainable development regarding the relative importance of these criteria in sustainable project selection. In this research, a number of literature publications related to sustainable project selection and sustainable development as well as some prominent project management literature covering the chosen criteria were selected and evaluated, as part of the literature review for this research, to serve as the voice of experts in determining preferences among the five different criteria shown in Table 1. These studies were closely reviewed in an effort to determine the relative importance of these criteria and preference patterns, as presented by the After defining the five sustainable project criteria, as shown in the previous subsection, the first step in determining the priority weights of these criteria is collecting the opinions of experts in sustainability and sustainable development regarding the relative importance of these criteria in sustainable project selection. In this research, a number of literature publications related to sustainable project selection and sustainable development as well as some prominent project management literature covering the chosen criteria were selected and evaluated, as part of the literature review for this research, to serve as the voice of experts in determining preferences among the five different criteria shown in Table 1. These studies were closely reviewed in an effort to determine the relative importance of these criteria and preference patterns, as presented by the authors of these publications. The list of the chosen literature publications is shown in Table 2.

authors of these publications. The list of the chosen literature publications is shown in Table 2.

**Expert Source(s)** E1 Malik et al. [20] E2 Alyamani et al. [10] E3 Sabaghi et al. [17]

E5 Chen et al. [33] E6 Wang et al. [28] E7 Işik and Aladağ [34] E8 Hatefi and Tamošaitienė [16]

**Table 2.** Selected expert literature used for the evaluation of criteria.


**Table 2.** Selected expert literature used for the evaluation of criteria.

The second step in determining the priority weights of the five sustainable project criteria is utilizing the expert opinions from the literature in Table 2 based on the linguistic variables and triangular fuzzy numbers (TFNs), shown in Table 3, as presented by Ballı and Koruko˘glu [25]. In this step, expert opinions are gathered from the literature and translated into the linguistic variables. After creating the pairwise comparison matrix representing the opinions of each of the ten experts shown in Table 1 using the linguistic variables, these ten matrices are then combined to form the combined pairwise comparison matrix shown in Table 4.

**Table 3.** Linguistic variables and triangular fuzzy number scale.


Source: adapted from Ballı and Koruko˘glu [25].

**Table 4.** Pairwise comparison matrix using linguistic variables.


**Table 4.** *Cont.*


These linguistic variables in the combined matrix are then further translated into the corresponding triangular fuzzy numbers (TFNs) and reciprocal TFNs based on the scale shown in Table 3, resulting in the combined TFN pairwise comparison matrix, shown in Table 5.


**Table 5.** Pairwise comparison matrix using TFNs.


**Table 5.** *Cont.*

Once the TFN pairwise comparison matrix is created, as shown above, it can be used to calculate the weight of importance for the five criteria. This calculation is performed in three main steps. The first step is to combine the fuzzy pairwise comparison from all ten experts for each of the five criteria. This can be done by calculating the geometric mean of the experts' opinions. To calculate the fuzzy geometric mean, the geometric mean method introduced by Buckley [37] is used leading to the fuzzy geometric mean pairwise comparison matrix shown in Table 6.


**Table 6.** Fuzzy geometric mean pairwise comparison matrix.

The second step in calculating the criteria weights of importance is determining the fuzzy relative importance weight or the fuzzy synthetic extent of each of the five criteria. To do that, the extent analysis method introduced by Chang [38] is applied in this research, as shown in Equations (2–5). Let *G* = *g*1, *g*2, *g*3, . . . , *g<sup>n</sup>* be a goal set. Each criterion is taken and the extent analysis for each goal *g<sup>i</sup>* is performed, respectively [25,39]. Accordingly, the *m* extent value for each criterion is obtained as follows: *M*<sup>1</sup> *gi* , *M*<sup>2</sup> *gi* , *M*<sup>3</sup> *gi* , . . . , *M<sup>m</sup> gi* , where *g<sup>i</sup>* (*i* = 1, 2, 3, . . . , *n*) is the goal set and *M j gi* (*j* = 1, 2, 3, . . . , *m*) are all TFNs. The value of the fuzzy synthetic extent (*S<sup>i</sup>* ) with respect to the *i*th criterion is defined as shown in Equation (2).

$$S\_{\bar{i}} = \sum\_{j=1}^{m} M\_{\mathcal{G}i}^{j} \otimes \left[ \sum\_{i=1}^{n} \sum\_{j=1}^{m} M\_{\mathcal{G}i}^{j} \right]^{-1} \tag{2}$$

In order to calculate <sup>P</sup>*<sup>m</sup> j*=1 *M j gi* , a fuzzy addition operation of the *m* extent is used for a certain matrix, as shown in Equation (3). This can be done following the addition of the fuzzy number process shown in Sun [27].

$$\sum\_{j=1}^{m} M\_{\mathcal{G}i}^{j} = \left(\sum\_{j=1}^{m} l\_{j\prime} \sum\_{j=1}^{m} m\_{j\prime} \sum\_{j=1}^{m} u\_{j}\right) \tag{3}$$

where the variables *l*, *m*, and *u* indicate the lowest possible value, the modal or most likely value, and the upper or highest possible value, respectively, as explained earlier in this research. The next logical operation is to calculate <sup>P</sup>*<sup>n</sup> i*=1 P*m j*=1 *M j gi* by performing another fuzzy addition operation of *M j gi* (*j* = 1, 2, 3, . . . , *m*), as shown in Equation (4).

$$\sum\_{i=1}^{n}\sum\_{j=1}^{m}M\_{\mathcal{G}i}^{j} = \left(\sum\_{i=1}^{n}l\_{i\prime}\sum\_{i=1}^{n}m\_{i\prime}\sum\_{i=1}^{n}u\_{i}\right) \tag{4}$$

Finally, P*n i*=1 P*m j*=1 *M j gi* −1 is determined by calculating the inverse of the vector above as shown in Equation (5).

$$\left[\sum\_{i=1}^{n}\sum\_{j=1}^{m}M\_{\mathcal{G}i}^{j}\right]^{-1} = \left(\frac{1}{\sum\_{i=1}^{n}u\_{i}}, \frac{1}{\sum\_{i=1}^{n}m\_{i}}, \frac{1}{\sum\_{i=1}^{n}l\_{i}}\right) \tag{5}$$

Equations (2)–(5) are now applied to the TFNs obtained in this research. To determine the fuzzy synthetic extent to the criteria chosen in this research, the P*<sup>m</sup> <sup>j</sup>*=<sup>1</sup> *M j gi* value is first calculated for each row of the matrix shown in Table 6. For example, for C1:

C1 = (1 + 1.676 + 1.446 + 4.143 + 3.187, 1 + 2.647 + 2.125 + 6.221 + 4.904, 1 + 3.657 + 3.071 + 8.262 + 7.020)

C1 = (11.452, 16.897, 23.010)

Accordingly, the P*<sup>n</sup> i*=1 P*m <sup>j</sup>*=<sup>1</sup> *M j gi* value is calculated for each of the five criteria in Table 6 by applying Equation (4) as follows:

 $\sum\_{i=1}^{n} \sum\_{j=1}^{m} M\_{\S i}^{j} = (11.452, 16.897, 23.010) \oplus (4.813, 6.883, 9.404) \oplus (4.760, 6.457, 8.915) \oplus (2.425, 3.205, 4.302) \oplus (2.425, 3.074, 4.345)$   $\oplus (2.425, 3.205, 4.345) \oplus (2.425, 3.6516, 4.9.976)$ 

Based on that, the reciprocal value hP*<sup>n</sup> i*=1 P*m <sup>j</sup>*=<sup>1</sup> *M j gi* i−<sup>1</sup> is calculated by applying Equation (5) as follows:

$$\left[\sum\_{i=1}^{n}\sum\_{j=1}^{m}M\_{\mathbb{S}\_{i}}^{j}\right]^{-1} = \left(\frac{1}{49.976}, \frac{1}{36.516}, \frac{1}{25.862}\right) = (0.020, 0.027, 0.039)$$

Finally, the value of the fuzzy synthetic extent (*S<sup>i</sup>* ) with respect to the *i*th criterion is calculated for each criterion, as shown in Equation (2). For example, the value of the fuzzy synthetic extent for the first criterion *S*<sup>1</sup> is calculated as follows:

$$S\_1 = (11.452, 16.897, 23.010) \text{ @ } (0.020, 0.027, 0.039) \\ = (0.229, 0.436, 0.893)$$

The fuzzy synthetic extent or the fuzzy relative importance weights resulting from applying the same process to the remaining criteria is presented in Table 7.


**Table 7.** Fuzzy synthetic extent of sustainable project selection criteria.

The third and final step in calculating the criteria weights of importance is the defuzzification of the fuzzy criteria weights shown in Table 7. To defuzzify these weights, the defuzzification method shown in Equation (6), as presented in Sun [27] and Lespier et al. [7], is used to obtain the best non-fuzzy priority (BNP) or crisp weights of the criteria.

$$\text{BNP}\_{S\_{\hat{i}}} = \frac{\left[\left(u\_{\mathbf{s}\_{\hat{i}}} - l\_{\mathbf{s}\_{\hat{i}}}\right) + \left(m\_{\mathbf{s}\_{\hat{i}}} - l\_{\mathbf{s}\_{\hat{i}}}\right)\right]}{\mathbf{3}} + l\_{\mathbf{s}\_{\hat{i}}} \qquad \text{where } \mathbf{i} = 1, 2, \dots, 5 \tag{6}$$

As an example, applying Equation (6) to calculate the BNP for criterion 1 is done as follows:

$$BNP\_{S\_1} = \frac{\left[\left(0.893 - 0.229\right) + \left(0.463 - 0.229\right)\right]}{3} + 0.229 = 0.528$$

Accordingly, the crisp weights for the remaining criteria are calculated. Using these BNP values, the criteria can be ranked based on importance, where the criterion with the highest BNP is set as the most important, while the criterion with the lowest BNP is set as the least important, as shown in Table 8.


**Table 8.** Best non-fuzzy priority (BNP) or crisp criteria weights.
