## 2 Preliminaries
### Hesitant Fuzzy Linguistic Term Sets  
As an extended form of Hesitant Fuzzy Sets, [PERSON] et al. introduced the concept of HFLTS to address scenarios in which decision-makers are indecisive when selecting suitable linguistic terms for evaluating expressions. In the subsequent text, some fundamental definitions of HFLTS are provided.  
**Definition 1**[16].: _Let \(S\) be a linguistic term set, \(S=\{s_{i}:i=0,1,2,\cdots,g\}\), and HFLTS, \(H_{s}\), be an ordered finite subset of the consecutive linguistic term of \(S\). \(S\) can be formulated as follows:_  
\[S=\left\{\begin{array}{l}s_{0}: remarkably\;low,s_{1}:low,s_{2}:\;slightly\; low,\\ s_{3}:medium,s_{4}:slightly\;high,\\ s_{5}:high,s_{6}:remarkably\;high\end{array}\right\},\]  
_let \(\vartheta\) be a linguistic variable, then two hesitant fuzzy sets of different granularity might be \(\widetilde{H}_{\rm s}(\vartheta_{1})=\{s_{3},s_{4},s_{5}\}\) and \(\widetilde{H}_{\rm s}(\vartheta_{2})=\{s_{3},s_{4}\}\)._  
**Definition 2**[16].: _The upper bound and lower bound of the HFLTS are expressed as follows:_  
\[\widetilde{H}_{\rm s^{+}}=\max(s_{i})=s_{j},\;s_{i}\in H_{s}\;and\;s_{i}\leq s _{j}\forall i, \tag{1}\]  
\[\widetilde{H}_{\rm s^{-}}=\min(s_{i})=s_{j},\;s_{i}\in H_{s}\;and\;s_{i}\geq s _{j}\forall i, \tag{2}\]  
It is assumed that a team of \(l\) groups of experts \(E_{k}=(k=1,2,\cdots,l)\) evaluates the linguistic variable item \(\vartheta_{i}(i=1,2,\cdots,m)\). The hesitant fuzzy linguistic term set obtained is \(\widetilde{H}_{\rm s}^{k}(\vartheta_{i})\). \(\widetilde{H}_{\rm s_{-}}(\vartheta_{i})\) and \(\widetilde{H}_{\rm s^{+}}(\vartheta_{i})\), respectively, stand for the lower and upper bounds of the HFLTS obtained by the \(l\)th expert's evaluation. The min-upper and max-lower operators [27] aggregate expert evaluation information. The steps are as follows:  
Step 1: Calculate the optimal semantic evaluation of the HFLTS lower bound set and the worst semantic evaluation of the upper bound set, respectively, as follows.  
\[\widetilde{H}_{\rm s_{max}}(\vartheta_{i})=\max\Bigl{\{}\widetilde{H}_{\rm s_ {-}}^{1}(\vartheta_{i}),\widetilde{H}_{\rm s_{-}}^{2}(\vartheta_{i}),\cdots \widetilde{H}_{\rm s_{-}}^{k}(\vartheta_{i})\Bigr{\}}, \tag{3}\]  
\[\widetilde{H}_{\rm s_{min}^{+}}(\vartheta_{i})=\min\Bigl{\{}\widetilde{H}_{\rm s ^{+}}^{1}(\vartheta_{i}),\widetilde{H}_{\rm s^{+}}^{2}(\vartheta_{i}),\cdots \widetilde{H}_{\rm s^{+}}^{k}(\vartheta_{i})\Bigr{\}}, \tag{4}\]  
Step 2: Integrate the semantic evaluation information to determine the aggregated result.  
\[[s_{r_{i}},s_{t_{i}}]=\widetilde{H}_{\rm s}(\vartheta_{i})=\Bigl{\{} \widetilde{H}_{\rm min}(\vartheta_{i}),\widetilde{H}_{\rm max}(\vartheta_{i}) \Bigr{\}}. \tag{5}\]  
where \(\widetilde{H}_{\rm max}(\vartheta_{i})=\max\Bigl{\{}\widetilde{H}_{\rm s_{min }^{+}}(\vartheta_{i}),\widetilde{H}_{\rm s_{max}^{-}}(\vartheta_{i})\Bigr{\}}\), \(\widetilde{H}_{\rm min}(\vartheta_{i})=\min\Bigl{\{}\widetilde{H}_{\rm s_{min }^{+}}(\vartheta_{i})\,,\,\widetilde{H}_{\rm s_{max}^{-}}(\vartheta_{i}) \Bigr{\}}\), and \(s_{t_{i}}\) and \(s_{r_{i}}\) represent the upper and lower bounds of the semantic interval, respectively.