In practical linguistic multicriteria group decisionmaking (MCGDM) problems, words may indicate different meanings for various decision makers (DMs), and a high level of group consensus indicates that most of the group members are satisfied with the final solution. This study aims at developing a novel framework that considers the personalized individual semantics (PISs) and group consensus of DMs to tackle linguistic singlevalued neutrosophic MCGDM problems. First, a novel discrimination measure for linguistic singlevalued neutrosophic numbers (LSVNNs) is proposed, based on which a discriminationbased optimization model is built to assign personalized numerical scales (PNSs). Second, an extended consensusbased optimization model is constructed to identify the weights of DMs considering the group consensus. Then, the overall evaluations of all the alternatives are obtained based on the LSVNN aggregation operator to identify the ranking of alternatives. Finally, the results of the illustrative example, sensitivity and comparative analysis are presented to verify the feasibility and effectiveness of the proposed method.
A multicriteria group decisionmaking (MCGDM) problem is defined as a decision problem where several experts (judges, decision makers (DMs), etc.) provide their evaluations on a set of alternatives regarding multiple criteria and seek to achieve a common solution that is most acceptable by the group of experts as a whole (Kabak and Ervural,
Although FSs and IFSs have been extended to manage various MCGGDM problems, they failed to effectively handle a situation where the indeterminate and inconsistent information are involved. To manage such issues, Smarandache (
Over the past years, lots of studies have been witnessed focusing on MCDM and MCGDM based on SVNSs and their extensions, which can be roughly grouped into two categories (Nguyen
The abovementioned approaches are effective when copying with MC(G)DM problems with SVNNs. However, specific numerical evaluations cannot always accurately reflect DMs’ behaviour and opinions because of the limitation of their cognition. Actually, DMs usually prefer to elicit their evaluations with linguistic terms, such as “poor”, “good” and “perfect” due to the prominent advantages of linguistic terms for characterizing ambiguous and inexact assessments (Zadeh,
Although great efforts have been made to improve and extend the application of linguistic neutrosophic MCGDM methods, there still exist some challenges. The existing methods seem to overlook the semantics of individual DMs and the consensual solution. In the existing methods, numerical values are identified through calculating the index values of linguistic terms. In this way, the numerical values cannot indicate experts’ personalized individual semantics with respect to linguistic terms.
When tackling linguistic MCGDM problems, it is argued and accepted that words indicate different meanings for various DMs in computing with words (Mendel
Recently, Li
A novel discrimination measure for SVNNs is proposed, based on which a discriminationbased optimization model is constructed to assign PNSs. The proposed framework is the first attempt to manage PISs in linguistic GDM, where DMs’ assessments are presented with linguistic neutrosophic MCGDM matrices.
An extended consensusbased optimization model is constructed to identify the weights of DMs considering group consensus. The proposed approach can cautiously assign DMs’ weights to guarantee a level of agreement among members regarding the final solution, and reveal the differences among alternatives with the optimal discrimination degrees.
The rest of the paper is organized as follows. Section
Let
The inverse function of Δ,
Let
If
The inverse operator of
Let
An SVNS in
Let
Let
Let
This section presents the concepts of distance and discrimination measures of LSVNNs, based on which a programming model is constructed to derive the PNSs of each DM.
For convenience, assume that
Assume that the standardized matrices are denoted by
Let
Particularly,
It is obvious that Properties (1) and (2) hold. Thus, the proof of Property (3) is provided.
Since
Thus, we have
In GDM, a panel of experts are invited to provide their evaluations about a set of alternatives. It is required that an expert should be qualified with the ability to differentiate between cases which are similar but not identical (Herowati
Let
Although the discrimination measures defined in this study and Tian
Assume that
LSVNN evaluations of Example
As mentioned above, an expert is expected to be skilled and have the ability to discriminate the differences between cases. When experts are required to express linguistic ratings, their PISs of linguistic terms are embedded in the evaluations, which implicitly indicate the subtle differences among alternatives distinguished by experts. Therefore, an optimization model by maximizing the discrimination degree can be considered as a good solution to derive the PISs of DMs.
By resolving Model (
Assume that
Without loss of generality,
By resolving Model (
Results derived from without considering and considering PISs.
This section develops a comprehensive linguistic neutrosophic MCGDM framework that considers the PIS and group consensus.
Let
For a MCGDM problem, let
In order to merge the group consensus into the MCGDM, an optimization model is established to derive the weights of DMs by maximizing the group consensus as follows:
Set
By resolving Model (
Take the numerical example with LSVNN decision matrices
First, the PNSs of linguistic terms for each DM can be derived based on Model (
Second, the weights of each DM can be identified based on Model (
By resolving the above model with the software MATLAB or LINGO, the weight vector of DMs is
PNSs of linguistic terms for DMs in Fang and Ye (
0  0.05  0.375  0.45  0.5  0.563  0625  0.95  1  
0  0.05  0.125  0.45  0.5  0.55  0.875  0.95  1  
0  0.05  0.375  0.45  0.5  0.55  0.625  0.95  1  
0  0.05  0.311  0.45  0.5  0.553  0.689  0.95  1 
This subsection presents a framework for managing MCGDM with LSVNNs considering the PIS and consensus, which is described in Fig.
Each member provides evaluation information with LSVNNs and the individual decision matrices are normalized based on Eq. (
The PNSs
The weight vector
The overall evaluations
The ranking of alternatives can be identified based on the score values
Flowchart of the proposed approach.
This section presents an illustrative example adapted from Garg and Nancy (
A panel involving five experts are invited to express their evaluations and select the best internet service provider(s). After conducting the preliminary investigation, four internet service providers, namely, Bharti Airtel (
The proposed approach is employed to deal with the above linguistic neutrosophic MCGDM problem. Let the parameter
LSVNN evaluations given by DM
LSVNN evaluations given by DM
LSVNN evaluations given by DM
Each member provides evaluation information with LSVNNs and the individual decision matrices are normalized based on Eq. (
The PNSs
By resolving the above model with the software MATLAB or LINGO, the PNS
LSVNN evaluations given by DM
LSVNN evaluations given by DM
PNSs of linguistic terms.
0  0.05  0.201  0.45  0.5  0.55  0.875  0.95  1  
0  0.05  0.125  0.45  0.5  0.651  0.875  0.95  1  
0  0.05  0.125  0.25  0.5  0.75  0.875  0.95  1  
0  0.05  0.375  0.45  0.5  0.55  0.875  0.95  1  
0  0.05  0.375  0.45  0.5  0.55  0.751  0.95  1  
0  0.05  0.262  0.406  0.5  0.605  0.854  0.95  1 
A consensusbased optimization model can be established based on Model (
By resolving the above model with the software MATLAB or LINGO, the weight vector of DMs is
Overall evaluations of alternatives in terms of each criterion.
The overall evaluations
Overall values of each alternative.
Rankings  
0.807  0.597  1  
0.751  0.522  4  
0.765  0.600  3  
0.776  0.571  2 
The score and accuracy values
In order to investigate the influence of parameter
PNSs of linguistic terms and discrimination degrees with
PNSs of linguistic terms and discrimination degrees with
PNSs of linguistic terms and discrimination degrees with
A comparative analysis is conducted between the existing MCGDM and the proposed approaches with LSVNNs. Two common MCGDM methods are employed in this comparison, and they are the LSVNN WAA and WGA operatorbased method (M1 for short) (Fang and Ye,
The proposed approach is employed to solve the MCGDM problems in Fang and Ye (
Rankings of alternatives with different values of
Rankings  
Rankings of alternatives yielded by different methods.
Methods  Rankings  Discrimination degrees 
M1 (Fang and Ye, 

The proposed approach  
M2 (Garg and Nancy, 

The proposed approach 
Comparisons between the existing and the proposed methods.
Methods  Aggregation operators  Ways of addressing LSVNNs  PISs considered  Group consensus considered 
M1 (Fang and Ye, 
LSVNN WAA and WGA operators  Consider indices of linguistic terms  No  No 
M2 (Garg and Nancy, 
LSVNN prioritized WAA and WGA operators  Consider indices of linguistic terms  No  No 
Method in Li Y. 
LSVNN geometric Heronian mean and prioritized geometric Heronian mean operators  Consider indices of linguistic terms  No  No 
Method in Liang 
LSVNN power WAA and WGA operators  Consider indices of linguistic terms  No  No 
Method in Li 
LSVNN power WAA and WGA operators  Consider indices of linguistic terms  No  No 
The proposed approach  LSVNN WAA operator  NSbased 2tuple linguistic model  Yes  Yes 
In order to highlight the characteristics of considering PISs, the discrimination degrees of decision matrices are calculated. Since the PISs of DMs are overlooked in M1 and M2, the fixed NSs for LTS
Comparisons between M1 and the proposed approach.
Comparisons between M2 and the proposed approach.
Furthermore, comparisons are conducted between the existing LSVNN MCGDM methods and the proposed approach. The comparison results are summarized in Table
Based on the discussion in the illustrative example, sensitivity and comparative analysis, the prominent features of the developed framework are summarized as follows:
An effective solution for addressing PISs. The proposed PIS model can provide an effective solution to assign PNSs of linguistic terms for DMs, characterizing their personalized semantic preferences regarding linguistic MCGDM with LSVNNs.
A cautious method to assign the weights of DMs considering group consensus. The developed consensusdriven optimization model is utilized to identify the weights of DMs, guaranteeing a high level of agreement among members in terms of the final solution.
A robust method to determine the differences among alternatives. The proposed approach can not only consider the PISs of DMs, but also provide a robust method to reveal the differences among alternatives with the optimal discrimination degrees.
However, although the proposed approach equips outstanding characteristics in dealing with linguistic MCGDM problems with LSVNNs, DMs may have to derive the PNSs and weights of DMs by resolving some mathematic programming models. Compared to the existing methods without considering PISs, the proposed approach is intricate and timeconsuming.
Discrimination degrees derived from considering and without considering PISs.
LSVNNs are valuable for describing qualitative ratings involving uncertain, incomplete, and inconsistent information. When eliciting linguistic evaluations, words may be assigned different meanings for various people, that is, DMs have PISs with regard to linguistic terms. Considering PISs of DMs can lead to a realistic and effective methodology for addressing linguistic neutrosophic MCGDM problems. This study firstly develops a discriminationbased optimization model to assign PNSs of linguistic terms on LTS for DMs, and effectively describe their personalized semantic preferences regarding linguistic MCGDM with LSVNNs. Then, an optimization model on the basis of group consensus is constructed to identify the weights of DMs, which guarantees a high level of agreement among members in terms of the final solution. Subsequently, an LSVNN WAA aggregation operator and PNSsbased score and accuracy functions are utilized to determine the ranking of alternatives. Finally, by comparing with existing methods, the results demonstrate that the proposed approach which developed PIS can effectively derive PNSs of linguistic terms on LTS for DMs and lead to higher discrimination degrees than those without considering PISs.
In the future study, it would be an interesting topic to investigate the PISbased approach for addressing incomplete MCGDM problems with LSVNNs. Moreover, complex MCGDM involving largescale members and considering their social relationships has attracted much attention (Liao
The notation used in this study is summarized in Table
Notation in this study.
Indicators  Meanings 
Set of linguistic terms  
Numerical index of linguistic term 

LSVNS with linguistic truth degree 

Collection of LSVNNs  
LSVNN WAA operator  
Score function of LSVNN 

Accuracy function of LSVNN 

Set of alternatives  
Set of criteria  
Weight of criterion 

Set of experts  
Weight of expert 

Original decision matrix of expert 

LSVNN evaluation given by expert 

Standardized decision matrix of expert 

Distance measure between LSVNNs 

Parameter of distance measure 

Discrimination measure of expert 

Set of possible PNSs of linguistic terms for expert 

Group consensus measure 
The authors are very grateful to the editors and the anonymous referees for their valuable comments and suggestions.