*6.2. Sensitivity Analysis*

To explore the impact of the parameter *λ* on the ranking results, different possible values of *λ* are used in the algorithms of two aforementioned numerical examples, such as 0.001, 0.5, 1, 2, 3, 5, 10, 20, and 50. Then, combined with the proposed methods, the different rankings of alternatives are presented in Tables 3–6. From Tables 3 and 4, we can find that the best potential ERP system in Example 1 is always *A*3 using both the GPHFWA operator and GPHFWG operator; however, some differences exist between the ranking results concerning different values of *λ*. Tables 5 and 6 show that when we utilize the GPHFPWA operator to complete the information fusion, the best foreign professor is *A*2 for 0.001 ≤ *λ* ≤ 10, but the best alternative is *A*4 for 20 ≤ *λ* ≤ 50. In addition, when the GPHFPWG operator is used in Algorithm 2, the best foreign professor is *A*2 for 0.001 ≤ *λ* ≤ 3, but the best alternative is *A*1 for 5 ≤ *λ* ≤ 50. On the other hand, the score values of all the alternatives vary with different values of *λ*; the reason is that the aggregation processes of the proposed operators have changed. For instance, when *λ* = 2, the GPHFWA operator can be reduced to the picture hesitant fuzzy weighted quadratic averaging (PHFWQA) operator as

$$PHFWQA(\overline{n}\_{i1}, \overline{n}\_{i2}, \dots, \overline{n}\_{in}) = \left(w\_1\overline{n}\_{i1}^2 \oplus w\_2\overline{n}\_{i2}^2 \oplus \dots \oplus w\_n\overline{n}\_{in}^2\right)^{1/2}.$$

when *λ* = 3, the GPHFWA operator can be reduced to the picture hesitant fuzzy weighted cubic averaging (PHFWCA) operator as

$$PHFWACK(\overline{\mathfrak{n}}\_{i1}, \overline{\mathfrak{n}}\_{i2}, \dots, \overline{\mathfrak{n}}\_{i\mathfrak{n}}) = \left(w\_1\overline{\mathfrak{n}}\_{i1}^3 \oplus w\_2\overline{\mathfrak{n}}\_{i2}^3 \oplus \dots \oplus w\_n\overline{\mathfrak{n}}\_{i\mathfrak{n}}^3\right)^{1/3}.$$

Besides, the following results can be obtained from Tables 3–6:


reflect the attitude of decision makers. In addition, the best alternative varies when the value of *λ* is relatively high, while the best alternative is always the same in Example 1. It means that the rankings obtained by the GPHFPWA and GPHFPWG operators are more affected by the parameter *λ* than those obtained by the GPHFWA and GPHFWG operators.

The aforementioned sensitivity analysis results show that the value of *λ* plays a very important role in MCDM problems, especially when the value of *λ* is relatively high. The value of *λ* can be determined based on the personal preference of decision makers to obtain different ranking results; thus, the proposed methods are highly flexible to deal with different situations in practice.


**Table 3.** Sensitivity analysis results obtained by the GPHFWA operator.

**Table 4.** Sensitivity analysis results obtained by the GPHFWG operator.


**Table 5.** Sensitivity analysis results obtained by the GPHFPWA operator.



**Table 6.** Sensitivity analysis results obtained by the GPHFPWG operator.

## *6.3. Comparative Analysis*

To prove the feasibility of the proposed MCDM methods, the rankings of Example 1 in this paper are compared with the rankings obtained by the existing MCDM methods as presented in Table 7; including the PFWA and PFWG operators [25], and the picture fuzzy cross-entropy method [16]. Similarly, a comparison of Example 2 between the GPHFPWA and GPHFPWG operators and the HFPWA and HFPWG operators [34] is presented in Table 8.

**Table 7.** Comparison result of Example 1.


**Table 8.** Comparison result of Example 2.


Table 7 shows that the best alternative of Example 2 obtained by the MCDM methods based on the GPHFWA and GPHFWG operators is always *A*3, which is consistent with the existing methods; the results can demonstrate the feasibility of the proposed method. Compared with the PFS that is used in the study of [25] and [16], PHFS proposed in this paper can convey the human opinions more effectively, including yes, abstain, no, and refusal. For instance, the evaluation information of the alternative *A*1 concerning the criteria *C*1 that are given by decision maker is expressed as a PFN (0.53,0.33,0.09) [16,25]. In practice, decision maker may feel doubtful to determine an exact value of each membership level. Obviously, PFS cannot deal with this situation; however, we can use PHFS to represent the evaluation information as a PHFE {{0.43,0.53}, {0.33}, {0.06,0.09}} as shown in Table 1. Consequently, the proposed method can solve the MCDM problems when decision makers feel difficulty to determine the accurate value of each membership level. On the other hand, when the numbers of the criteria are relatively large, the aggregation process of the proposed operators will be more complicated than the existing methods and the data size will be relatively large; it is the limitation of the proposed method. Table 8 shows that the best alternative of Example 2 obtained by the HFPWA operator is *A*5, but the result of other MCDM methods is *A*2. The main reason of the difference is that the MCDM methods combined with the HFPWA and HFPWG operators ignore

some complex evaluation information of decision makers in practice. UTHFS allows the decision makers to give several values of positive membership level, for instance, the evaluation information of the alternative *A*1 concerning the criteria *C*1 that are given by decision maker is expressed as a UTHFE (0.4,0.5,0.7) [34]. Nevertheless, in some particular situations, it is not convincing to express the evaluation information that only considers the positive membership level of decision makers; many scholars have focused on this problem and made some improvements to UTHFS [10,19]. Thus, we can overcome the limitation of UTHFS combined with the proposed method. It is worth noting that the GPHFPWA and GPHFPWG operators also have the same disadvantage as the GPHFWA and GPHFWG operators.

According to the aforementioned comparison results, we can summarize the advantages and disadvantages of the different MCDM methods (see Table 9), as well as their respective fields of application (see Table 10). In addition, the benefits of the aggregation process by using the proposed operators are presented as in the following.

#### (1) The Expansion of the Evaluation Information

The GPHFWA, GPHFWG, GPHFPWA, and GPHFPWG operators can solve the MCDM problems under PHF environment. PHFS proposed in this paper can express the different human opinions in real life and allow the decision makers to give several possible values of the different membership levels; thus, it can simultaneously depict the uncertainty and hesitancy of decision makers' evaluation information, which cannot be achieved by PFS and UTHFS. Therefore, when decision makers are not fully aware of the evaluation target and feel doubtful about each membership level, it is reasonable to deal with these MCDM problems combined with the proposed methods. Furthermore, as a generalized form of FS, IFS, PFS, and UTHFS, we can transform the proposed methods into the existing MCDM methods if necessary.

(2) The Flexibility of Information Aggregation with Different Values of *λ*

Recall the sensitivity analysis in Section 6.2, the proposed operators can be reduced to other specific PHF aggregation operators by varying the value of *λ*; thus, the proposed methods are highly flexible to deal with different situations. Furthermore, the parameter *λ* can also be regarded as a measure of the optimism and pessimism level of decision makers in the information fusion of the GPHFWA and GPHFWG operators; and the value of *λ* can be determined by decision makers according to their preferences in practice.

#### (3) The Simplicity of Dealing with Different Types of Criteria

During the MCDM process, the weight values of criteria play an important role and will affect the final ranking results. The criteria can be divided into two categories: one is in the same priority, the other is in different priorities. On the one hand, when the criteria have the same priority level, we can utilize the proposed method based on the GPHFWA and GPHFWG operators combined with the weight vector of criteria to solve the MCDM problem. On the other hand, when the criteria have different priority levels, the GPHFPWA and GPHFPWG operators can be introduced to determine the ranking of alternatives. In practice, we can use different aggregation operators in this paper to deal with different situations.

#### *6.4. Application of Web Service Selection*

To investigate the applications of the proposed methods in a more realistic scenario, we use the proposed methods to solve the Quality of Service (QoS) based web service selection problem [37]. According to the study of [37], the evaluation information of QoS is measured by a crisp number scale of 1–9, and the related criteria are availability (*C*1), throughput (*C*2), successability (*C*3), reliability (*C*4), compliance (*C*5), best practices (*C*6), documentation (*C*7), latency (*C*8), and response time (*C*9). Due to the criteria latency and response time are the cost type criteria, the closer the evaluation values concerning these two criteria are to 1, the better the alternative.


**Table 9.** Comparison of each MCDM methods.

**Table 10.** MCDM application fields of each MCDM method.


Suppose there are 20 web services to be evaluated concerning the aforementioned nine criteria, i.e., *WSi*(*i* = 1, 2, . . . , <sup>20</sup>); the evaluation information of each web service is presented in Table 11. As each evaluation value in [37] is expressed by an exact crisp number, the PHFS can be reduced to the PFS to represent the evaluation information of each web service. Based on the relationship between the linguistic variables and IFNs [38], we develop the transformation relationship between the linguistic variables and PFNs as presented in Table 12. Then, the evaluation information in Table 11 can be transformed into a PF evaluation matrix *A* = -*aij*.(*<sup>i</sup>* = 1, 2, . . . , 20; *j* = 1, 2, . . . , <sup>9</sup>), and the ranking of the 20 web services can be obtained by the Algorithm 1 in this paper. Subsequently, the ranking result will be compared with the rankings determined by AHP, TOPSIS, COPRAS, VIKOR, and SAW methods in [37]. It is worth noting that, in order to compare different MCDM methods, more effectively we suppose each criteria is considered equally important, i.e., *wj* = 1/9(*j* = 1, 2, . . . , <sup>9</sup>). Then, the ranking of the 20 web services can be determined by the following steps.

**Step 1**: According to the Definition 3, normalize the PF evaluation matrix *A* = -*aij*. to the standardized PF evaluation matrix *A* = -*aij*. as

$$
\overline{a}\_{ij} = \begin{cases}
 a\_{ij\prime} & \text{for the benefit criteria;} \\
 \binom{a\_{ij}}{\left(a\_{ij}\right)^c} & \text{for the cost criteria.}
\end{cases}
\tag{57}
$$

**Step 2**: Utilize the GPFWA (*λ* = 1) operator

$$GFFWA\_{\lambda=1}(\overline{a}\_{i1}, \overline{a}\_{i2}, \dots, \overline{a}\_{i9}) = a\_i = \left(1 - \prod\_{j=1}^{9} (1 - \overline{\mu}\_{ij})^{w\_j}, \prod\_{j=1}^{9} (\overline{\eta}\_{ij})^{w\_j}, \prod\_{j=1}^{9} (\overline{v}\_{ij})^{w\_j}\right)$$

to aggregated the PF evaluation matrix *A* = -*aij*., and the collective PFNs of each web service are obtained.

**Step 3**: Compute the score values of each web service using the equation

$$s(a\_i) = (1 + \mu\_i - \eta\_i - \upsilon\_i) / 2. \tag{58}$$

**Table 11.** Evaluation information of each web service.



**Table 12.** Transformation between linguistic variables and PFNs.

Then, the ranking of the 20 web services can be determined; the lager the score value, the better the web service. The related data of the ranking are presented in Table 13.


**Table 13.** Ranking results obtained by the proposed method.

To verify the accuracy of the ranking obtained by the proposed method, we use AHP, TOPSIS, COPRAS, VIKOR, and SAW methods to solve the web service selection problem combined with the evaluation information in Table 11. Subsequently, the ranking results of different MCDM methods are presented in Table 14. The Spearman's rank correlation coefficient is a powerful tool for measuring the similarity between two MCDM methods [39]. Then, we can calculate the Spearman's rank correlation coefficients between the proposed method and the other five MCDM methods as shown in Table 15. Table 15 shows that the Spearman's rank correlation coefficients between the proposed method and AHP and TOPSIS are 0.9722 and 0.9549, respectively, which demonstrate that the proposed method is highly correlated with these two methods. AHP and TOPSIS methods have been approved to be the most suitable two methods to solve web service selection problems [39]; thus, the comparison results above illustrate the feasibility of the proposed method.


**Table 14.** Ranking of web services of different MCDM methods.

**Table 15.** Spearman's rank correlation coefficients between the proposed method and the other MCDM methods.


From the information aggregation of the proposed method, we can find that the calculating procedure of the proposed method is more complicated than AHP and TOPSIS methods. In addition, TOPSIS method does not require the transformation of the evaluation information concerning cost and benefit type criteria. However, when decision makers are not sure if it is 3 or 4 about the evaluation information of the web service *WS*1 concerning the criteria *C*1, AHP and TOPSIS methods cannot deal with this situation in practice; we can use PHFS to express the evaluation information above, i.e., {{0.25,0.35},{0.05},{0.55,0.65}}. On the other hand, when the criteria are in different priorities, the GPHFPWA and GPHFPWG operators can be used to aggregation the evaluation information. Thus, the AHP, TOPSIS, and proposed methods have their own advantages and disadvantages; in real life, decision makers can determine to utilize which MCDM methods to solve problems according to the actual situations.
