# A New Flexible Method for Solving Multi-Expert Multi-Criterion Decision-Making Problems
## 3 Proposed Integration of Induced LOWGA Operator and HFLTS  
Multi-expert MCDM problems are complex, and include uncertainty in group DM. In real decision-making, the criterion weights and criterion rating values for alternatives usually simultaneously contain certain, uncertain, or incomplete information. This is because the experts' opinions include personal preferences according to their different backgrounds and experience.  
Traditional linguistic computing methods may cause information distortion and loss in the information aggregation process. To address this problem, this paper presents the integration of an induced LOWGA operator and HFLTS to augment the assessment ability in group DM problems. In contrast to other methods, the proposed method can effectively deal with the criterion weights and criterion rating values for alternative or uncertain and incomplete information. For uncertain information provided by experts, the HFLTS approach is used to increase the flexibility of linguistic expression in assessments. For incomplete information provided by experts, existing complete information is used with supplementing information to fill gaps in incomplete information.  
The process of the proposed novel group DM method is divided into the following seven steps (shown in Figure 1).  
Step 1. Identify the goal, criteria, and alternatives of group DM problems to hierarchical structure.  
According to the problems to be evaluated, the decision-making committee must disassemble problems into hierarchical structures that include alternatives, criteria, and goals.  
Step 2. Determine criterion weights and criterion rating values for all alternatives.  
Step 3. Fill in missing weights and rating values of criteria for all alternatives  
Step 4. Defuzzify hesitant values of assessment criteria data  
Step 5. Use fuzzy linguistic quantifier to calculate weighting vector  
Step 6. Compute the overall \(zi\) value of alternative \(xi\), respectively  
Step 7. Rank order of the alternatives and provide suggestions  
Figure 1: The flow chart of proposed novel group decision-making (DM) method.  
Step 3. Fill in missing weights and rating values of criteria for all alternatives.  
If the information provided by experts is incomplete, incomplete information can be supplemented by the mean value of all existing complete information. If all information is complete, move to Step 4.  
Step 4. Defuzzify hesitant values of assessment criterion data.  
In this step, the parametric mean (PM) is used to defuzzify the hesitant values of assessment criterion data [45], computed as shown in Equation (6).  
\[\text{PM}=\left(\frac{a+b}{2}\right) \tag{6}\]  
where \(a\) and \(b\) represent left boundaries and right boundaries, respectively.  
Step 5. Use fuzzy linguistic quantifier to calculate weighting vector.  
The aggregation weighting vector \(w=(w_{1},w_{2},\ldots,w_{n})^{T}\) of the proposed method using fuzzy quantifiers, which for a non-decreasing relative quantifier \(Q\), is defined as [48; 49]  
\[w_{j}=Q\!\left(\frac{j}{n}\right)-Q\!\left(\frac{j-1}{n}\right),\ j=1,2,\ldots,n \tag{7}\]  
\[(r)=\left\{\begin{array}{cc}0&\quad if\ r<a\\ \frac{r-a}{b-a}&\quad if\ a\leq r\leq b\quad a,b,r\in[0,1]\\ 1&\quad if\ r<b\end{array}\right. \tag{8}\]  
Three frequently used fuzzy quantifiers \"at least\", \"most\", and \"as many as possible\" are utilized to obtain the weighting vector \(w=(w_{1},w_{2},\ldots,w_{n})^{T}\), of which the parameters \((a,b)\) are \((0,0.5)\), \((0.3,0.8)\) and \((0.5,1)\), respectively.  
Step 6. Compute the overall \(z_{i}\) value of alternative \(x_{i}\), respectively [43].  
\[z_{i}=ILOWGA_{\overline{w},w}\!\left(s_{\alpha_{1}},s_{\alpha_{2}},\ldots,s_{ \alpha_{in}}\right) \tag{9}\]  
Using the results of Steps 3 to 5, use Equation (9) to calculate overall value \(z_{i}\) of alternative \(A_{i}\).  
Step 7. Rank order of the alternatives and provide suggestions.  
According to the calculation results of Step 6, rank the order of the alternatives and provide feasible suggestions for decision-makers.