Analyzing Healthcare and Wellness Products’ Quality Embedded in Online Customer Reviews: Assessment with a Hybrid Fuzzy LMAW and Fermatean Fuzzy WASPAS Method

2.1.1. Satisfaction with PQ

2.1.2. Satisfaction with Online Shopping Service (OSS)

3.2.1. Delivery Quality

3.2.2. Aftersales Service Quality

4.2.1. Preliminaries for Fermatean Fuzzy Sets

4.2.2. Weighting the Criteria with the Triangular F-LMAW Method

• Step 1: Establish a group of decision makers and design the decision-making matrices: In this stage, each decision maker (DM) evaluates n criteria with the help of linguistic terms given in Table 2.
  Using the triangular fuzzy number, the first decision matrix is formed.
• Step2: Create the first (aggregated) decision-making matrix ($\tilde{X}$). The Bonferroni aggregator is used to combine initial (expert) matrices into a single aggregate matrix, as shown in Equation (11):

  $${\tilde{A}}_{ij}=\left(\frac{1}{k\left(k-1\right)}{\displaystyle \sum }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^k{\tilde{A}}_i{}^{\left(e\right)p}{\tilde{A}}_j{}^{\left(e\right)q}\right)=\left\{{\left(\frac{1}{k\left(k-1\right)}{\displaystyle \sum }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^k{\tilde{A}}_i{}^{\left(l_{{ }_e}\right)p}{\tilde{A}}_j{}^{\left(l_e\right)q}\right)}^{\frac{1}{p+q}},{\left(\frac{1}{k\left(k-1\right)}{\displaystyle \sum }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^k{\tilde{A}}_i{}^{\left(m_{{ }_e}\right)p}{\tilde{A}}_j{}^{\left(m_{{ }_e}\right)q}\right)}^{\frac{1}{p+q}},{\left(\frac{1}{k\left(k-1\right)}{\displaystyle \sum }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^k{\tilde{A}}_i{}^{\left(r_{{ }_e}\right)p}{\tilde{A}}_j{}^{\left(r_{{ }_e}\right)q}\right)}^{\frac{1}{p+q}}\right\}$$

  (11)

  where ${\tilde{A}}_{ij}$ states the averaged values derived by using the Bonferroni aggregator, p,q > 0 represent the Bonferroni aggregator’s formation settings, and e denotes the e-th expert of k expert, where l and r represent the fuzzy number’s distributions with left and right, respectively, while m represents the value at which the membership function of a fuzzy number equals 1.
• Step 3: Normalizing the initial matrix: The components of the first decision-making matrix are normalized using the formula below Equation (12), and the normalized matrix ${\tilde{A}}_{ij}{}^N$ is obtained:

  $${\tilde{A}}_{ij}{}^N=\{\begin{array}{c}1+\frac{{\tilde{A}}_{ij}}{{\tilde{A}}_j{}^{+}}=\left(1+\frac{A_{ij}{}^l}{A_j{}^{+}},1+\frac{A_{ij}{}^m}{A_j{}^{+}},1+\frac{A_{ij}{}^r}{A_j{}^{+}}\right) if j \in B, \\ 1+\frac{{\tilde{A}}_j{}^{-}}{{\tilde{A}}_{ij}}=\left(1+\frac{A_j{}^{-}}{A_{ij}{}^r},1+\frac{A_j{}^{-}}{A_{ij}{}^m},1+\frac{A_j{}^{-}}{A_{ij}{}^l}\right) if j \in C.\end{array}$$

  (12)

  where $A_j{}^{+}=\max \left({\tilde{A}}_j{}^r\right)$ and $A_j{}^{-}=\min \left({\tilde{A}}_j{}^l\right)$.
• Step 4: Calculate the weightings of the criteria: The decision makers are intended to be involved in determining the weighting values of the criterion $E=\left\{E_1,E_2, \dots ,E_k\right\}$. The n criteria are prioritized. The greater value from the linguistic variables scale is allocated to the criteria with the highest relevance, and inversely. As a result, the priority vectors ${\tilde{P}}^e=\left\{{\check{\gamma }}_{C1}{}^e,{\check{\gamma }}_{C2}{}^e, \dots ,{\check{\gamma }}_{Cn}{}^e\right\}$ are obtained. ${\check{\gamma }}_{Cn}{}^e$ corresponds to the value from the fuzzy linguistic scale ascribed to the criteria n by the expert e.
• Step 5: Compute fuzzy anti-ideal point (${\gamma }_{AIP}$) and obtain fuzzy relation vector: Using Equation (13), the absolute fuzzy anti-ideal point (${\gamma }_{AIP}$), a fuzzy value that is less than the least from the collection of all priority vectors, and fuzzy relation vector ${\tilde{R}}^e=$(${\tilde{\eta }}^e{}_{C1},{\tilde{\eta }}^e{}_{C2},\dots ,{\tilde{\eta }}^e{}_{Cn}$) are determined.

  $${\tilde{A}}_{ij}{}^N={\tilde{\eta }}^e{}_{Cn}=\left(\frac{{\check{\gamma }}_{Cn}{}^e}{{\tilde{\gamma }}_{AIP}}\right)=\left(\frac{{\gamma }_{Cn}^{\left(l\right)e}}{{\gamma }_{AIP}^{\left(r\right)}},\frac{{\gamma }_{Cn}^{\left(m\right)e}}{{\gamma }_{AIP}^{\left(m\right)}},\frac{{\gamma }_{Cn}^{\left(r\right)e}}{{\gamma }_{AIP}^{\left(l\right)}}\right)$$

  (13)
• Step 6: Aggregate fuzzy weighted and achieving final score vector: Using Equation (14), the weighting values calculated for each decision maker and the weighting vector $w^e{}_j=$(${\tilde{w}}^e{}_1,{\tilde{w}}^e{}_2,\dots ,{\tilde{w}}^e{}_n){ }^T$ are obtained.

  $${\tilde{w}}_j^e=\left(\frac{\ln \left({\tilde{\eta }}_{Cn}^e\right)}{\ln \left({{\displaystyle \prod }}_{j=1}^n{\tilde{\eta }}_{Cn}^e\right)}\right)=\left(\frac{\ln \left({\tilde{\eta }}_{Cn}^{\left(l\right)e}\right)}{\ln \left({{\displaystyle \prod }}_{j=1}^n{\tilde{\eta }}_{Cn}^{\left(r\right)e}\right)},\frac{\ln \left({\tilde{\eta }}_{Cn}^{\left(m\right)e}\right)}{\ln \left({{\displaystyle \prod }}_{j=1}^n{\tilde{\eta }}_{Cn}^{\left(m\right)e}\right)},\frac{\ln \left({\tilde{\eta }}_{Cn}^{\left(r\right)e}\right)}{\ln \left({{\displaystyle \prod }}_{j=1}^n{\tilde{\eta }}_{Cn}^{\left(l\right)e}\right)}\right)$$

  (14)

4.2.3. Ranking of Alternatives with the FF-WASPAS

• Step 7: Create a list of options. The decision makers assess the products based on the linguistic terms for each criterion. The linguistic terms were translated into the Fermatean fuzzy crisp numbers in Table 3.
• Step 8: Define linguistic concepts and the Fermatean fuzzy sets (FFSs) that correspond to them. In this phase, decision makers should specify language phrases, such as “extremely low” and “very high”, as well as their respective FFSs.
• Step 9: Obtain each decision maker’s assessment based on each criterion. Each DM should analyze options in relation to each stated criterion in this stage. The assessment technique uses the linguistic concepts created in the preceding stage based on the Fermatean fuzzy sets. The kth decision maker’s appraisal of the ith choice on the jth criterion is represented ${\widetilde{DM}}_{ijk}=\left({\alpha }_{{\widetilde{DM}}_{ijk}}, {\beta }_{{\widetilde{DM}}_{ijk}}\right)$.
• Step 10: Aggregate decision makers’ judgments with the aggregation operator defined in Equation (9) in the preceding part. The judgments produced by each DM in Step 6 were combined using the following calculation and equivalent weights ${\tilde{w}}_k=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.$. As a result, the aggregated assessments or components of the decision matrix ${}_{ℱ}{}^{ }\check{X}{}_{ij}=\left({\alpha }_{{}_{ℱ}{}^{ }\check{X}{}_{ij}},{\beta }_{{}_{ℱ}{}^{ }\check{X}{}_{ij}}\right)$ are given as follows (17):

  $${}_{ℱ}{}^{ }\check{X}{}_{ij}=FFWA\left({\widetilde{DM}}_{ij1},{\widetilde{DM}}_{ij2},\dots ,{\widetilde{DM}}_{ijp}\right)=\left(\frac{1}{p}{\displaystyle \sum }_{k=1}^p{\alpha }_{{\widetilde{DM}}_{ijk}},{\displaystyle \sum }_{k=1}^p{\beta }_{{\widetilde{DM}}_{ijk}}\right)$$

  (17)
• Step 11: Normalize the decision matrix in step eight. The decision matrix is normalized using the normalization approach in the conventional WASPAS. After employing the Fermatean fuzzy sets, it must be dealt with components with values ranging from 0 to 1. As a result, it is unnecessary to utilize a normalization method to change the value scale. In the case of cost criteria, the notion of the complement of FFSs Equation (8) is applied to change the values associated with cost criteria. Let B and C represent the sets of benefit and cost criteria. The following are the elements of the normalized decision matrix:

  $${}_{ℱ}{}^{ }\tilde{N}{{}^{ }}_{ij}=\{\begin{array}{c}{}_{ℱ}{}^{ }\check{X}{}_{ij} ; if j\in B\\ {\left({}_{ℱ}{}^{ }\check{X}{}_{ij}\right)}^{\prime } ; if j\in C\end{array}$$

  (18)
• Step 12: Determine the WSM and WPM values. WSM and WPM values are calculated using the addition, multiplication, and other operators of FFSs established in the preceding section Equations (2)–(5).

  $${\tilde{Q}}_i^S{}_{ }={\displaystyle \sum }_{j=1}^m{w}_j\otimes \left({}_{ℱ}{}^{ }\tilde{N}{{}^{ }}_{ij}\right)$$

  (19)

  $${\tilde{Q}}_i^p{}_{ }={\displaystyle \prod }_{j=1}^m{\left({}_{ℱ}{}^{ }\tilde{N}{{}^{ }}_{ij}\right)}^{w_j}$$

  (20)
• Step 13: Determine the WASPAS value. Integrating the WSM and WPM values yields the WASPAS metric using a combining parameter ($\delta$). This step’s calculation is based on the formula below in Equation (21).

  $${\tilde{Q}}_i^{ }=\delta {\tilde{Q}}_i^s{}_{ }\oplus \left(1-\delta \right){\tilde{Q}}_i^p{{}_{ }}_{ }$$

  (21)
• Step 14: Sort the options according to their positive ${\tilde{Q}}_i^{ }$ values. Definition 6, stated in the preceding section, compares and ranks the options.

5.2.1. Criteria Weights

5.2.2. FF-WASPAS Results