A Decision Framework under a Linguistic Hesitant Fuzzy Set for Solving MultiCriteria Group Decision Making Problems
Abstract
:1. Introduction
 The primary challenge encountered is that there is an urgent need for a scientific decisionmaking framework under an LHFS to utilize the potential power of an LHFS.
 Following this, the idea of aggregation of LHFSbased preference information has just begun and there is a good scope for exploration. The claim by Xu and Liao [35] to produce consistent aggregated preference information is an interesting challenge to be addressed.
 Another challenge is the calculation of criteria weights using the systematic procedure for obtaining sensible weight values.
 Further, ranking of objects by using LHFSbased preference information is another interesting challenge to address for better decisionmaking under uncertain situations. Though Dong et al. [32] extended the popular VIKOR method to an LHFS, the challenge of using the method for MCGDM still needs to be addressed.
 Finally, comprehensive comparison of the proposed framework with other methods for realizing the strength and weakness of the proposal is an attractive challenge for exploration.
 (1)
 With a view of alleviating the primary challenge, a new decision framework is proposed under an LHFS context to utilize the potential power of an LHFS.
 (2)
 Following this, a new aggregation operator called simple linguistic hesitant fuzzy weighted geometry (SLHFWG) is presented with the view of producing consistent aggregated preference information by extending the operator discussed in Reference [35] under an LHFS context. This operator also uses the idea of a power geometry operator for sensible aggregation. He et al. [36] claimed that “whenever the relationship between the objects and criteria are to be aggregated, some unduly high and low information may have some bad impact on the aggregation process. In order to mitigate the effect, support measures are to be used which assign weights to information. This showcases the urge need for power operators during aggregation”. Motivated by this claim, we set our focus in this direction.
 (3)
 Further, a new method for criteria weight estimation is presented which is an extension to standard variance (SV) under an LHFS context. Previous studies on weight estimation have predominantly used entropy measures [37], optimization models [38,39,40], analytic hierarchy process (AHP) [41] method, and decision making trial and evaluation laboratory (DEMATEL) [42], etc., which often yields unreasonable and irrational weight values. Motivated by this challenge, we set our proposal towards this direction.
 (4)
 Also, the popular linguistic hesitant fuzzy visekriterijumska optimizacijai kompromisno resenje (LHFVIKOR) method is adopted for selecting a suitable hospital from a set of hospitals. This example is an MCGDM problem that clarifies the practicality and usefulness of the proposed decision framework and addresses the challenge mentioned by Dong et al. [32].
 (5)
 Finally, the strengths and weaknesses of the proposed framework is realized by comparison with other methods.
2. Preliminaries
 ${s}_{u}$and${s}_{v}$are two linguistic term sets, and the relation${s}_{u}>{s}_{v}$holds true, if$u>v$.
 Negation of${s}_{u}$is given by$neg({s}_{u})={s}_{v}$, such that$u+v=n$.
3. Proposed Decision Framework under LHFS Context
3.1. Some Operational Laws and Properties of LHFS
 Empty LHFS$L\left(h\right)=\{\varnothing \}$;
 Full LHFS$L\left(h\right)=Swithpossiblemembershipdegrees$;
 (1)
 $\lambda \left({L}_{1}\left(h\right)\bigcup {L}_{2}\left(h\right)\right)=\lambda {L}_{1}\left(h\right)\bigcup \lambda {L}_{2}\left(h\right)\lambda >0$;
 (2)
 $\left({\lambda}_{1}+{\lambda}_{2}\right){L}_{1}\left(h\right)={\lambda}_{1}{L}_{1}\left(h\right)\bigcup {\lambda}_{2}{L}_{1}\left(h\right){\lambda}_{1},{\lambda}_{2}>0$.
3.2. Proposed SLHFWG Aggregation Operator
 (1)
 The aggregation of linguistic terms using an SLHFWG operator yields a much more sensible term with no virtual set. This can be easily realized from the formulation given in Equation (14). This ensures that the aggregation of the linguistic term is consistent and rational.
 (2)
 Similarly, for the aggregation of the membership degrees, the motivation is gained from the power operator [44,45] and from the work of Xu and Liao [35]. As mentioned earlier by He et al. [34], the unduly high and low values cause bad effects in the aggregation process and the support measure (in formulation of power operator) is used to mitigate the same. Also, they claimed that the relationship between objects and criteria can be realized with the help of a support measure. Further, Xu and Liao [33] proposed a variant of the weighted geometry operator and claimed that the aggregation of preferences by this operator yields consistent values. The second phase of the aggregation applies the idea of a power operator to determine the relative importance of each DM, and these values are further used for aggregating the membership degrees (motivated by the operator proposed in [33]).
 (3)
 Finally, the proposed SLHFWG operator produces consistent nonvirtual aggregated values of LHFS preferences and also helps DMs to better understand the relationship between objects and criteria.
3.3. Proposed LHFSV Method
 (1)
 Unlike previous studies on criteria weight estimation (for example analytical hierarchy process (AHP) [41], decision making trial and evaluation laboratory (DEMATEL) [42], entropy based method [37], optimization model [36,37,38], etc.), the proposed method does not produce unrealistic and unreasonable weight values.
 (2)
 Also, the proposed method is simple and straightforward, and pays significant attention to those data points (criteria) that are highly conflicting. This property of SV further motivated our focus in this direction.
 (3)
 Rao et al. [48] pointed out that unlike other statistical methods that concentrate only on the boundary points, the SV method concentrates on every data point for determining the distribution. This property of SV helps DMs to estimate criteria weights in a rational manner.
 (4)
 Generally, relative importance is interpreted as the importance of a criterion relative to the hesitation that exists among DMs during preference elicitation. Thus, DMs’ personal characteristics and stimuli play a significant role in the interpretation of relative importance [49]. Thus, criteria with a high variation in preferences are given high importance and the SV method captures and reflects this idea in a better way. Further, Kao [50] presented a geometric proof for the same claim by using the idea of frontiers and projection. This work provides sufficient mathematical justification for realizing the strength of the SV method.
3.4. Procedure for LHFVIKOR Method
 (1)
 Based on the work of Opricovic and Tzeng [51], it can be clearly observed that both VIKOR and TOPSIS (technique for order preference by similarity to ideal solution) are compromise ranking methods. However, VIKOR performs better than TOPSIS in the following ways: (i) The VIKOR method considers a relative distance measure that is much more rational than the rank index of the TOPSIS method. (ii) The VIKOR method considers the attitude of the DM as a key parameter in its formulation, which is missing in TOPSIS.
 (2)
 Further, from the work of Opricovic and Tzeng [52], it can be observed that the ranking order from PROMETHEE (preference ranking organization method for enrichment evaluation) and ELECTRE (ELimination Et Choix Traduisant la REalité) can be easily realized from the S and R parameters of the VIKOR method respectively.
 (3)
 The VIKOR method also selects the compromise solution based on two conditions viz. acceptable stability and acceptable advantage. Also, along with the ranking order, the VIKOR method provides a rank value set (advantage rate) for backup management during uncertain situations.
 (1)
 The proposed LHFS concept extends the HFLTS concept by reflecting the hesitancy and vagueness of the DM by using possible membership degrees. This concept allows DMs to associate possible membership degrees for each linguistic term, which motivates sensible and rational decision making. Moreover, the concept circumvents the drawback of HFLTS by handling uncertainty and vagueness to a reasonable extent.
 (2)
 Following this, a new decision framework is put forward under an LHFS context that uses LHFS information for rating objects. Initially, a new aggregation operator called SLHFWG is proposed that sensibly aggregated DMs’ viewpoints without producing virtual sets.
 (3)
 Further, a new criteria weight estimation method is proposed which is an extension to the SV method under an LHFS context. The LHSV method produces reasonable criteria weights by focusing on every data point rather than only the extreme values.
 (4)
 Finally, a new ranking method is presented, which is an extension to the VIKOR method under an LHFS context. The method does the following: (a) PIS and NIS are calculated by using Equations (23) and (24), which identify a suitable LHFS value for each criterion and hence, the PIS and NIS is a vector of order $\left(1\times n\right)$, where n is the number of criteria. (b) The parameters ${S}_{i}$, ${R}_{i}$, and ${Q}_{i}$ are estimated using Equations (25)–(28), which is of order $\left(m\times 1\right)$ where m is the number of alternatives. (c) The stability of the ranking method is realized by performing a sensitivity analysis by varying the strategy parameter ($v$).
4. Numerical Example
5. Comparative Analysis: Proposed Versus others
 The proposal presented a new concept (structure) to the decisionmaking context by extending HFLTS with possible membership degrees to better reflect hesitation and vagueness. We also investigated some attractive properties of LHFS.
 A twophase scientific decisionmaking framework was further presented under an LHFS context for rational decisionmaking. The framework put forward a new aggregation operator that was motivated from the work in Reference [35] and power operators for the sensible aggregation of DMs’ preference information. Following this, a new criteria weight estimation method was presented for the reasonable estimation of criteria weights, which is an extension of SV method under an LHFS environment. Finally, the framework presented an extension to the popular VIKOR ranking method for MCGDM problems to select a suitable object from the set of objects.
 As mentioned earlier, the superiority of the proposal was realized from theoretic and numerical perspectives. Clearly, Table 9 brought out the superiority of the proposal and showed that the proposed framework was a powerful aid for critical and rational decisionmaking.
 Certain key factors discussed in Table 9 are: stability, which is ensured by sensitivity analysis on parameters (like weights, strategy etc.); consistency, which is ensured from Spearman correlation; robustness, which is realized from an adequacy test motivated by Reference [55]; and scalability, which is motivated by Reference [57].
 In order to demonstrate the practicality and usefulness of the proposal, an interesting hospital evaluation problem was presented. From Table 8, we observed that the ranking order obtained from the proposal was consistent with its close counterpart. The compromise solution selected by the proposal and its close counterpart were the same and was given by ${H}_{4}$. Though the order coincides, the proposed LHFSbased decision framework was much superior in various factors discussed in Table 9 and also, the proposal handled the weakness of HFLTS in a much better manner.
6. Concluding Remarks
Author Contributions
Acknowledgments
Conflicts of Interest
References
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DMs  Hospitals  Criteria Evaluation  

${\mathit{C}}_{\mathbf{1}}$  ${\mathit{C}}_{\mathbf{2}}$  ${\mathit{C}}_{\mathbf{3}}$  ${\mathit{C}}_{\mathbf{4}}$  
${D}_{1}$  ${H}_{1}$  $\left\{\begin{array}{c}3,\left(0.33,0.42,0.45\right)\\ 2,\left(0.4,0.3,0.33\right)\\ 4,\left(0.4,0.35,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.35,0.44,0.48\right)\\ 0,\left(0.42,0.4,0.5\right)\\ 3,\left(0.44,0.48,0.54\right)\end{array}\right\}$  $\left\{\begin{array}{c}0,\left(0.34,0.4,0.46\right)\\ 1,\left(0.25,0.45,0.36\right)\\ 3,\left(\mathrm{0.44.0.4},0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.42,0.46,0.5\right)\\ 0,\left(0.33,0.35,0.44\right)\\ 3,\left(0.4,0.35,0.44\right)\end{array}\right\}$ 
${H}_{2}$  $\left\{\begin{array}{c}2,\left(0.33,0.42,0.27\right)\\ 3,\left(0.25,0.44,0.5\right)\\ 1,\left(0.35,0.45,0.4\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.25,0.35,0.42\right)\\ 2,\left(0.33,0.44,0.5\right)\\ 0,\left(0.45,0.52,0.37\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.34,0.43,0.46\right)\\ 1.\left(0.42,0.44,0.52\right)\\ 0,\left(0.4,0.3,0.36\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.45,0.52,0.54\right)\\ 2,\left(0.44,0.4,0.36\right)\\ 0,\left(0.24,0.4,0.35\right)\end{array}\right\}$  
${H}_{3}$  $\left\{\begin{array}{c}1,\left(0.35,0.42,0.5\right)\\ 0,\left(0.42,0.48,0.54\right)\\ 2,\left(0.44,0.36,0.4\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.44,0.36,0.4\right)\\ 2,\left(0.35,0.42,0.4\right)\\ 3,\left(0.4,0.5,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.42,0.35,0.5\right)\\ 2,\left(0.44,0.4,0.36\right)\\ 4,\left(0.34,0.5,0.48\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.4,0.35,0.42\right)\\ 4,\left(0.4,0.5,0.45\right)\\ 2,\left(0.35,0.45,0.4\right)\end{array}\right\}$  
${H}_{4}$  $\left\{\begin{array}{c}3,\left(0.35,0.44,0.5\right)\\ 4,\left(0.42,0.46,0.35\right)\\ 2,\left(0.3,0.4,0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.3,0.4,0.44\right)\\ 2,\left(0.3,0.42,0.35\right)\\ 1,\left(0.4,0,33,0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.3,0.4,0.35\right)\\ 3,\left(0.35,0.42,0.44\right)\\ 1,\left(0.22,0.33,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.34,0.4,0.45\right)\\ 3,\left(0.33,0.35,0.44\right)\\ 0,\left(0.42,0.4,0.3\right)\end{array}\right\}$  
${D}_{2}$  ${H}_{1}$  $\left\{\begin{array}{c}2,\left(0.3,0.35,0.4\right)\\ 3,\left(0.4,0.44,0.5\right)\\ 0,\left(0.4,0.35,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.4,0.35,0.42\right)\\ 2,\left(0.44,0.33,0.3\right)\\ 0,\left(0.3,0.5,0.38\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.44,0.4,0.35\right)\\ 2,\left(0.35,0.3,0.4\right)\\ 4,\left(0.4,0.44,0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.35,0.44,0.4\right)\\ 3,\left(0.35,0.3,0.44\right)\\ 2,\left(0.28,0.34,0.45\right)\end{array}\right\}$ 
${H}_{2}$  $\left\{\begin{array}{c}1,\left(0.34,0.44,0.5\right)\\ 2,\left(0.5,0.42,0.38\right)\\ 3,\left(0.4,0.35,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.4,0.35,0.42\right)\\ 0,\left(0.34,0.22,0.25\right)\\ 2,\left(\mathrm{0.33.0.44.0.36}\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.33,0.35,0.42\right)\\ 4,\left(0.45,0.5,0.4\right)\\ 1,\left(0.4,0.3,0.36\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.45,0.54,0.4\right)\\ 0,\left(0.38,0.42,0.44\right)\\ 2,\left(0.35,0.3,0.45\right)\end{array}\right\}$  
${H}_{3}$  $\left\{\begin{array}{c}3,\left(0.4,0.38,0.5\right)\\ 2,\left(0.35,0.4,0.3\right)\\ 0,\left(0.3,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.3,0.35,0.38\right)\\ 3,\left(0.35,0.4,0.38\right)\\ 1,\left(0.33,0.4,0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.3,0.42,0.45\right)\\ 3,\left(0.4,0.35,0.38\right)\\ 1,\left(0.42,0.4,0.35\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.33,0.44,0.48\right)\\ 1,\left(0.4,0.5,0.45\right)\\ 0,\left(0.4,0.35,0.3\right)\end{array}\right\}$  
${H}_{4}$  $\left\{\begin{array}{c}1,\left(0.4,0.3,0.5\right)\\ 0,\left(0.35,0.44,0.4\right)\\ 2,\left(0.3,0.5,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.45,0.5,0.4\right)\\ 2,\left(0.4,0.42,0.46\right)\\ 1,\left(0.35,0.4,0.5\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.44,0.3,0.35\right)\\ 4,\left(0.44,0.35,0.3\right)\\ 1,\left(0.33,0.38,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.4,0.33,0.35\right)\\ 3,\left(0.4,0.5,0.45\right)\\ 0,\left(0.33,0.43,0.25\right)\end{array}\right\}$  
${D}_{3}$  ${H}_{1}$  $\left\{\begin{array}{c}4,\left(0.33,0.4,0.5\right)\\ 2,\left(0.4,0.3,0.2\right)\\ 1,\left(0.2,0.24,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.4,0.44,0.5\right)\\ 3,\left(0.35,0.4,0.42\right)\\ 1.\left(0.3,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.33,0.4,0.42\right)\\ 1,\left(0.34,0.4,0.3\right)\\ 0,\left(0.3,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.42,0.4,0.38\right)\\ 1,\left(0.44,0.33,0.25\right)\\ 3,\left(0.5,0.43,0.4\right)\end{array}\right\}$ 
${H}_{2}$  $\left\{\begin{array}{c}3,\left(0.3,0.4,0.35\right)\\ 2,\left(0.45,0.5,0.48\right)\\ 0,\left(0.35,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.4,0.3,0.35\right)\\ 2,\left(0.44,0.35,0.5\right)\\ 0,\left(0.35,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.44,0.3,0.4\right)\\ 2,\left(0.35,0.5,0.4\right)\\ 0,\left(0.15,0.24,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.44,0.33,0.36\right)\\ 3,\left(0.4,0.45,0.35\right)\\ 0,\left(0.3,0.25,0.33\right)\end{array}\right\}$  
${H}_{3}$  $\left\{\begin{array}{c}3,\left(0.44,0.4,0.35\right)\\ 2,\left(0.33,0.38,0.42\right)\\ 1,\left(0.35,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.32,0.35,0.45\right)\\ 2,\left(0.4,0.35,0.3\right)\\ 4,\left(0.3,0.5,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.42,0.35,0.4\right)\\ 3,\left(0.44,0.5,0.4\right)\\ 1,\left(0.3,0.25,0.35\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.4,0.3,0.28\right)\\ 4,\left(0.35,0.4,0.45\right)\\ 1,\left(0.3,0.4,0.44\right)\end{array}\right\}$  
${H}_{4}$  $\left\{\begin{array}{c}4,\left(0.4,0.5,0.45\right)\\ 2,\left(0.44,0.33,0.22\right)\\ 0,\left(0.4,0.2,0.32\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.44,0.3,0.4\right)\\ 2,\left(0.33,0.35,0.4\right)\\ 1,\left(0.35,0.42,0.33\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.33,0.3,0.4\right)\\ 4,\left(0.44,0.52,0.38\right)\\ 3,(0.35,0.45,0.54\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.4,0.33,0.35\right)\\ 3,\left(0.4,0.42,0.44\right)\\ 0,\left(0.32,0.24,0.28\right)\end{array}\right\}$ 
DMs  Hospitals  Criteria Evaluation  

${\mathit{C}}_{\mathbf{1}}$  ${\mathit{C}}_{\mathbf{2}}$  ${\mathit{C}}_{\mathbf{3}}$  ${\mathit{C}}_{\mathbf{4}}$  
${D}_{1}$  ${H}_{1}$  $\left\{0.30,0.30,0.31\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$ 
${H}_{2}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.31\right\}$  $\left\{0.30,0.30,0.30\right\}$  
${H}_{3}$  $\left\{0.30,0.30,0.31\right\}$  $\left\{0.31,0.31,0.31\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.31,0.30,0.31\right\}$  
${H}_{4}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.31\right\}$  
${D}_{2}$  ${H}_{1}$  $\left\{0.40,0.39,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$ 
${H}_{2}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.39,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  
${H}_{3}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  
${H}_{4}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  $\left\{0.40,0.40,0.40\right\}$  
${D}_{3}$  ${H}_{1}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.3,0.30,0.31\right\}$ 
${H}_{2}$  $\left\{0.30,0.31,0.31\right\}$  $\left\{0.30,0.31,0.30\right\}$  $\left\{0.30,0.31,0.30\right\}$  $\left\{0.30,0.30,0.31\right\}$  
${H}_{3}$  $\left\{0.30,0.31,0.31\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.31\right\}$  $\left\{0.30,0.30,0.30\right\}$  
${H}_{4}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$  $\left\{0.30,0.31,0.30\right\}$  $\left\{0.30,0.30,0.30\right\}$ 
Hospitals  Criteria Evaluation  

${\mathit{C}}_{\mathbf{1}}$  ${\mathit{C}}_{\mathbf{2}}$  ${\mathit{C}}_{\mathbf{3}}$  ${\mathit{C}}_{\mathbf{4}}$  
${H}_{1}$  $\left\{\begin{array}{c}3,\left(0.32,0.39,0.44\right),\\ 2,\left(0.4,0.35,0.33\right),\\ 2,\left(0.33,0.31,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.39,0.40,0.46\right),\\ 2,\left(0.40,0.37,0.39\right),\\ 1,\left(0.34,0.46,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.38,0.4,0.40\right),\\ 1,\left(0.31,0.37,0.36\right),\\ 2,\left(0.38,0.42,0.47\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.39,0.43,0.41\right),\\ 1,\left(0.37,0.32,0.37\right),\\ 3,\left(0.37,0.37,0.43\right)\end{array}\right\}$ 
${H}_{2}$  $\left\{\begin{array}{c}2,\left(0.32,0.42,0.37\right),\\ 2,\left(0.39,0.45,0.44\right),\\ 1,\left(0.37,0.39,0.37\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.35,0.33,0.4\right),\\ 2,\left(0.37,0.31,0.38\right),\\ 0,\left(0.37,0.45,0.38\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.36,0.36,0.43\right),\\ 2,\left(0.41,0.48,0.43\right),\\ 0,\left(0.3,0.28,0.34\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.45,0.46,0.42\right),\\ 2,\left(0.40,0.42,0.39\right),\\ 0,\left(0.3,0.31,0.38\right)\end{array}\right\}$ 
${H}_{3}$  $\left\{\begin{array}{c}3,\left(0.4,0.4,0.45\right),\\ 2,\left(0.36,0.42,0.4\right),\\ 1,\left(0.35,0.39,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.34,0.35,0.41\right),\\ 2,\left(0.37,0.39,0.36\right),\\ 3,\left(0.34,0.46,0.46\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.37,0.38,0.45\right)\\ 3,\left(0.42,0.41,0.38\right),\\ 1,\left(0.36,0.37,0.38\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.37,0.37,0.39\right),\\ 4,\left(0.38,0.47,0.45\right),\\ 1,\left(0.35,0.39,0.37\right)\end{array}\right\}$ 
${H}_{4}$  $\left\{\begin{array}{c}3,\left(0.38,0.39,0.48\right),\\ 2,\left(0.4,0.40,0.32\right),\\ 1,\left(0.33,0.35,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.4,0.4,0.41\right),\\ 2,\left(0.35,0.4,0.41\right),\\ 1,\left(0.36,0.38,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.36,0.33,0.37\right),\\ 2,\left(0.41,0.42,0.36\right),\\ 1,\left(0.3,0.38,0.45\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.38,0.35,0.38\right),\\ 3,\left(0.38,0.43,0.44\right),\\ 0,\left(0.35,0.35,0.27\right)\end{array}\right\}$ 
DMs  Criteria Evaluation  

${\mathit{C}}_{\mathbf{1}}$  ${\mathit{C}}_{\mathbf{2}}$  ${\mathit{C}}_{\mathbf{3}}$  ${\mathit{C}}_{\mathbf{4}}$  
${D}_{1}$  $\left\{\begin{array}{c}2,\left(0.2,0.3,0.35\right)\\ 3,\left(0.25,0.33,0.4\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.3,0.33,0.36\right)\\ 1,\left(0.3,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.35,0.42,0.44\right)\\ 3,\left(0.33,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.3,0.4,0.45\right)\\ 2,\left(0.25,0.35,0.4\right)\end{array}\right\}$ 
${D}_{2}$  $\left\{\begin{array}{c}2,\left(0.3,0.35,0.4\right)\\ 4,\left(0.25,0.35,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.33,0.35,0.4\right)\\ 3,\left(0.35,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.35,0.4,0.45\right)\\ 4,\left(0.3,0.4,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.35,0.4,0.44\right)\\ 3,\left(0.33,0.4,0.45\right)\end{array}\right\}$ 
${D}_{3}$  $\left\{\begin{array}{c}3,\left(0.4,0.44,0.5\right)\\ 4,\left(0.35,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.35,0.4,0.44\right)\\ 3,\left(0.33,0.36,0.42\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.24,0.35,0.45\right)\\ 3,\left(0.35,0.4,0.44\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.3,0.32,0.34\right)\\ 4,\left(0.4,0.45,0.5\right)\end{array}\right\}$ 
IS  Evaluation Criteria  

${\mathit{C}}_{\mathbf{1}}$  ${\mathit{C}}_{\mathbf{2}}$  ${\mathit{C}}_{\mathbf{3}}$  ${\mathit{C}}_{\mathbf{4}}$  
PIS  $\left\{\begin{array}{c}3,\left(0.32,0.39,0.44\right)\\ 2,\left(0.4,0.35,0.33\right)\\ 2,\left(0.32,0.31,0.3\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.34,0.35,0.41\right)\\ 2,\left(0.36,0.39,0.36\right)\\ 2,\left(0.34,0.46,0.46\right)\end{array}\right\}$  $\left\{\begin{array}{c}1,\left(0.37,0.4,0.4\right)\\ 1,\left(0.31,0.37,0.35\right)\\ 2,\left(0.37,0.42,0.47\right)\end{array}\right\}$  $\left\{\begin{array}{c}2,\left(0.38,0.35,0.38\right)\\ 3,\left(0.38,0.43,0.44\right)\\ 0,\left(0.35,0.35,0.27\right)\end{array}\right\}$ 
NIS  $\left\{\begin{array}{c}2,\left(0.32,0.42,0.37\right)\\ 2,\left(0.39,0.45,0.44\right)\\ 1,\left(0.37,0.4,0.37\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.35,0.33,0.4\right)\\ 2,\left(0.36,0.31,0.38\right)\\ 0,\left(0.37,0.45,0.38\right)\end{array}\right\}$  $\left\{\begin{array}{c}4,\left(0.37,0.38,0.45\right)\\ 3,\left(0.42,0.41,0.38\right)\\ 1,\left(0.35,0.37,0.38\right)\end{array}\right\}$  $\left\{\begin{array}{c}3,\left(0.37,0.37,0.39\right)\\ 4,\left(0.38,0.47,0.45\right)\\ 1,\left(0.35,0.39,0.37\right)\end{array}\right\}$ 
Hospitals  Parameter(s)  

S  R  
b  ub  B  ub  
${H}_{1}$  1.2984  1.1679  1.2297  1.0248 
${H}_{2}$  0.9162  0.9243  0.3141  0.2618 
${H}_{3}$  0.6821  0.5851  0.3  0.25 
${H}_{4}$  0.2341  0.2844  0.1058  0.1429 
$\mathit{v}\text{}\mathbf{Values}$  Hospitals  Q  Ranking order  

B  ub  B  ub  
0.1  ${H}_{1}$  1  0.8743  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.2213  0.1768  
${H}_{3}$  0.1887  0.1281  
${H}_{4}$  0  0  
0.2  ${H}_{1}$  1  0.8883  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.2679  0.2376  
${H}_{3}$  0.2145  0.1516  
${H}_{4}$  0  0  
0.3  ${H}_{1}$  1  0.9023  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.3145  0.2985  
${H}_{3}$  0.2403  0.1752  
${H}_{4}$  0  0  
0.4  ${H}_{1}$  1  0.9162  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.3612  0.3593  
${H}_{3}$  0.2661  0.1988  
${H}_{4}$  0  0  
0.5  ${H}_{1}$  1  0.9302  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.4078  0.4201  
${H}_{3}$  0.2919  0.2224  
${H}_{4}$  0  0  
0.6  ${H}_{1}$  1  0.9442  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.4544  0.481  
${H}_{3}$  0.3177  0.246  
${H}_{4}$  0  0  
0.7  ${H}_{1}$  1  0.9581  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.501  0.5418  
${H}_{3}$  0.3435  0.2696  
${H}_{4}$  0  0  
0.8  ${H}_{1}$  1  0.9721  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.5476  0.6026  
${H}_{3}$  0.3693  0.2932  
${H}_{4}$  0  0  
0.9  ${H}_{1}$  1  0.986  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$  
${H}_{2}$  0.5942  0.6635  
${H}_{3}$  0.3951  0.3168  
${H}_{4}$  0  0 
Method(s)  Hospital(s)  Ranking Order  

${\mathit{H}}_{\mathbf{1}}$  ${\mathit{H}}_{\mathbf{2}}$  ${\mathit{H}}_{\mathbf{3}}$  ${\mathit{H}}_{\mathbf{4}}$  
Proposed  4  3  2  1  ${H}_{4}\succ {H}_{3}\succ {H}_{2}\succ {H}_{1}$ 
LHFSaggregate [30]  2  1  4  3  ${H}_{2}\succ {H}_{1}\succ {H}_{4}\succ {H}_{3}$ 
HFLTSTOPSIS [16]  1  2  4  3  ${H}_{1}\succ {H}_{2}\succ {H}_{4}\succ {H}_{3}$ 
HFLTSVIKOR [21]  1  2  4  3  ${H}_{1}\succ {H}_{2}\succ {H}_{4}\succ {H}_{3}$ 
Context(s)  Method(s)  

Proposed  LHFSAggregate  HFLTSTOPSIS  HFLTSVIKOR  
Input  HFLTS + possible membership degrees  HFLTS + possible membership degrees  HFLTS only  HFLTS only 
Association information  Membership degrees  Membership degrees  No  No 
Weight calculation  Proposed LHFSV method  Only DM defined  Only DM defined  Only DM defined 
Aggregation  Proposed SLHFWG operator  Ordered weighted arithmetic/geometry  yes  N/A 
Rank value set  Broad and sensible  Narrow  Narrow  Broad in nature 
Backup  Possible  Not possible  Not possible  Possible 
Complexity  $O\left(nmt\left(\beta i\right)\right)$ where n is number of objects, m is number of criteria, t is number of terms and $\beta i$ is number of probability instances  $O\left(nmt\left(\beta i\right)\right)$  $O\left(nmt\right)$  $O\left(nmt\right)$ 
Stability  Highly stable  Moderately stable  Moderately stable  Highly stable 
Consistency  Highly consistent  Moderately consistent  Inconsistent  Inconsistent 
Adequacy test  Satisfies the test  Satisfies the test when objects are repeated  Only partial adequacy test is satisfied  Satisfies the test with respect to criteria 
Scalability  yes, up to max. 9 items [57]  yes, up to max.9 items  yes, up to max. 9 items  
Strengths 


 
Weakness 



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Krishankumar, R.; Ravichandran, K.S.; Premaladha, J.; Kar, S.; Zavadskas, E.K.; Antucheviciene, J. A Decision Framework under a Linguistic Hesitant Fuzzy Set for Solving MultiCriteria Group Decision Making Problems. Sustainability 2018, 10, 2608. https://doi.org/10.3390/su10082608
Krishankumar R, Ravichandran KS, Premaladha J, Kar S, Zavadskas EK, Antucheviciene J. A Decision Framework under a Linguistic Hesitant Fuzzy Set for Solving MultiCriteria Group Decision Making Problems. Sustainability. 2018; 10(8):2608. https://doi.org/10.3390/su10082608
Chicago/Turabian StyleKrishankumar, R., K. S. Ravichandran, J. Premaladha, Samarjit Kar, Edmundas Kazimieras Zavadskas, and Jurgita Antucheviciene. 2018. "A Decision Framework under a Linguistic Hesitant Fuzzy Set for Solving MultiCriteria Group Decision Making Problems" Sustainability 10, no. 8: 2608. https://doi.org/10.3390/su10082608