Consensus Building in Multi-criteria Group Decision-Making with Single-Valued Neutrosophic Sets

Abstract

In order to obtain high satisfaction from experts, the consensus reaching process (CRP) is an essential requirement for dealing with multi-criteria group decision making (MCGDM) problems. Single-valued neutrosophic number (SVNN) is an effective tool to describe the uncertainty of the expert cognition. Thus, we develop a consensus reaching model for single-valued neutrosophic MCGDM in this paper. First, each expert makes his/her judgment on each alternative with respect to multiple criteria by SVNNs, and the group solution is obtained by the generalized Shapley single-valued neutrosophic Choquet integral (GS-SVNCI) operator to consider the correlations among elements comprehensively. Second, the projection-based consensus measure is proposed to reflect the agreement between the individual and collective opinions. Then, a threshold value is used to determine the CRP whether to be executed based on the expert’s consensus level. If yes, the feedback mechanism provides the experts with personalized adjustment advices based on their psychic utility to group pressure. Finally, we illustrate the feasibility of the proposed consensus model by an example and analyze the superiority by comparing with some existing MCGDM methods and different CRP models. The developed consensus model can consider interrelationships between experts, which is more effective and reasonable to obtain the collective resolution. Further, the consensus measure based on the projection can comprehensively reflect the closeness between the individual and collective opinions. In addition, the personalized adjustment advices considering the experts’ psychic utility to group pressure improve their acceptance of these advices.

Appendix

Based on the Definition 6, $X = \left\{ {R_{1} ,R_{2} ,R_{3} } \right\}$, we can obtain the following cases of Shapely values:

• (1) $S = \left\{ {R_{1} } \right\}$, $T = \emptyset$, $T = \left\{ {R_{2} ,R_{3} } \right\}$, $T = \left\{ {R_{2} } \right\}$, $T = \left\{ {R_{3} } \right\}$,

  $\varphi_{{R_{1} }} \left( {\mu ,E} \right) =$$\frac{1}{3}\left[ {\mu \left( {R_{1} } \right) + 1 - \mu \left( {R_{2} ,R_{3} } \right)} \right] +$$\frac{1}{6}\left[ {\mu \left( {R_{1} ,R_{2} } \right) - \mu \left( {R_{2} } \right) + \mu \left( {R_{1} ,R_{3} } \right) - \mu \left( {R_{3} } \right)} \right]$;

• (2) $S = \left\{ {R_{2} } \right\}$, $T = \emptyset$, $T = \left\{ {R_{1} ,R_{3} } \right\}$, $T = \left\{ {R_{1} } \right\}$, $T = \left\{ {R_{3} } \right\}$

  $\varphi_{{R_{2} }} \left( {\mu ,E} \right) =$$\frac{1}{3}\left[ {\mu \left( {R_{2} } \right) + 1 - \mu \left( {R_{1} ,R_{3} } \right)} \right] +$$\frac{1}{6}\left[ {\mu \left( {R_{1} ,R_{2} } \right) - \mu \left( {R_{1} } \right) + \mu \left( {R_{2} ,R_{3} } \right) - \mu \left( {R_{3} } \right)} \right]$;

• (3) $S = \left\{ {R_{3} } \right\}$, $T = \emptyset$, $T = \left\{ {R_{1} ,R_{2} } \right\}$, $T = \left\{ {R_{1} } \right\}$, $T = \left\{ {R_{2} } \right\}$ $\varphi_{{R_{3} }} \left( {\mu ,E} \right) =$$\frac{1}{3}\left[ {\mu \left( {R_{3} } \right) + 1 - \mu \left( {R_{1} ,R_{2} } \right)} \right] +$$\frac{1}{6}\left[ {\mu \left( {R_{1} ,R_{3} } \right) - \mu \left( {R_{1} } \right) + \mu \left( {R_{2} ,R_{3} } \right) - \mu \left( {R_{2} } \right)} \right]$;

• (4) $S = \left\{ {R_{1} ,R_{{2}} } \right\}$, $T = \left\{ {R_{3} } \right\}$, $T = \emptyset$, $\varphi_{{\left\{ {R_{1} ,R_{{2}} } \right\}}}$$\left( {\mu ,E} \right) = \frac{1}{{2}}\left[ {1 - \mu \left( {R_{3} } \right) + \mu \left( {R_{1} ,R_{{2}} } \right)} \right]$;

• (5) $S = \left\{ {R_{1} ,R_{{3}} } \right\}$,$T = \left\{ {R_{2} } \right\}$, $T = \emptyset ,\varphi_{{\left\{ {R_{1} ,R_{3} } \right\}}}$$\left( {\mu ,E} \right) = \frac{1}{{2}}\left[ {1 - \mu \left( {R_{2} } \right) + \mu \left( {R_{1} ,R_{3} } \right)} \right]$; and

• (6) $S = \left\{ {R_{2} ,R_{{3}} } \right\}$, $T = \left\{ {R_{1} } \right\}$, $T = \emptyset$, $\varphi_{{\left\{ {R_{2} ,R_{3} } \right\}}} \left( {\mu ,E} \right) = \frac{1}{{2}}\left[ {1 - \mu \left( {R_{1} } \right) + \mu \left( {R_{2} ,R_{3} } \right)} \right]$.

Then, the grey relational coefficient $\varsigma_{ij}^{k}$ can be obtained by Eq. (14) ($\varepsilon = 0.5$), and the objective function is calculated as follows:

$$\begin{aligned}\sum\limits_{i = 1}^{5} {\sum\limits_{k = 1}^{3} {\varsigma_{ij}^{k} \varphi_{j} \left( {\mu_{j} ,R} \right)} } & = \varphi_{{R_{1} }} \sum\limits_{i = 1}^{5} {\varsigma_{i1}^{1} } + \varphi_{{R_{2} }} \sum\limits_{i = 1}^{5} {\varsigma_{i1}^{2} } \varsigma_{11}^{2} \varphi_{{R_{2} }} + \varphi_{{R_{3} }} \sum\limits_{i = 1}^{5} {\varsigma_{i1}^{3} } \\ &= 4.64\varphi_{{R_{1} }} \left( {\mu ,E} \right) + 4.533\varphi_{{R_{2} }} \left( {\mu ,E} \right) + 4.075\varphi_{{R_{{3}} }} \left( {\mu ,E} \right) \\ &= \frac{4.64}{3} \left[ {\mu \left( {R_{1} } \right) + 1 - \mu \left( {R_{2} ,R_{3} } \right)} \right] + \frac{4.64}{6}\left[ {\mu \left( {R_{1} ,R_{2} } \right) - \mu \left( {R_{2} } \right) + \mu \left( {R_{1} ,R_{3} } \right) - \mu \left( {R_{3} } \right)} \right] \\ &{}\quad\begin{aligned} &+ \frac{4.533}{3}\left[ {\mu \left( {R_{2} } \right) + 1 - \mu \left( {R_{1} ,R_{3} } \right)} \right] \\ &+ \frac{4.533}{6}\left[ {\mu \left( {R_{1} ,R_{2} } \right) - \mu \left( {R_{1} } \right) + \mu \left( {R_{2} ,R_{3} } \right) - \mu \left( {R_{3} } \right)} \right] \frac{4.075}{3}\left[ {\mu \left( {R_{3} } \right) + 1 - \mu \left( {R_{1} ,R_{2} } \right)} \right] \\ &+ \frac{4.075}{6}\left[ {\mu \left( {R_{1} ,R_{3} } \right) - \mu \left( {R_{1} } \right) + \mu \left( {R_{2} ,R_{3} } \right) - \mu \left( {R_{2} } \right)} \right] \end{aligned} \\ &= \left( {\frac{4.64}{3} - \frac{4.533}{6} - \frac{4.075}{6}} \right)\mu \left( {R_{1} } \right) + \left( {\frac{4.533}{3} - \frac{4.64}{6} - \frac{4.075}{6}} \right)\mu \left( {R_{2} } \right) + \left( {\frac{4.075}{3} - \frac{4.64}{6} - \frac{4.533}{6}} \right)\mu \left( {R_{3} } \right) \\ &{}\quad\begin{aligned} &+ \frac{4.64}{3} + \frac{4.533}{{3}} + \frac{4.075}{{3}} + \left( {\frac{4.64}{6} + \frac{4.533}{6} - \frac{4.075}{3}} \right)\mu \left( {R_{1} ,R_{2} } \right) + \left( {\frac{4.64}{6} + \frac{4.075}{6} - \frac{4.533}{3}} \right)\mu \left( {R_{1} ,R_{3} } \right) \\ &+ \left( {\frac{4.533}{6} + \frac{4.075}{6} - \frac{4.64}{3}} \right)\mu \left( {R_{2} ,R_{3} } \right) = 0.112\left( {\mu \left( {R_{1} } \right) - \mu \left( {R_{2} ,R_{3} } \right)} \right) \\ &+ 0.0585\left( {\mu \left( {R_{2} } \right) - \mu \left( {R_{1} ,R_{3} } \right)} \right) - 0.1705\left( {\mu \left( {R_{3} } \right) - \mu \left( {R_{1} ,R_{2} } \right)} \right) + {4}{\text{.416}}\end{aligned}\end{aligned}$$

According to the programming model (15), there is

$$\begin{array}{l}\max = 0.112\left[ {\mu_{{1}} \left( {R_{1} } \right) - \mu_{{1}} \left( {R_{2} ,R_{3} } \right)} \right] + 0.059\left[ {\mu_{{1}} \left( {R_{{2}} } \right) - \mu_{{1}} \left( {R_{{1}} ,R_{3} } \right)} \right] - 0.171\left[ {\mu_{{1}} \left( {R_{{3}} } \right) - \mu_{{1}} \left( {R_{{1}} ,R_{{2}} } \right)} \right] + 4.416\\ \quad s.t.\left\{ \begin{array}{l} \mu_{1} \left( \phi \right) = 0,\mu_{1} \left( {R_{1} ,R_{2} ,R_{3} } \right) = 1, \hfill \\ \mu_{1} \left( S \right) \le \mu_{1} \left( T \right),\forall S,T \subseteq R,S \subseteq T, \hfill \\ \mu_{1} \left( {R_{1} } \right) \in \left[ {0.3,0.4} \right],\mu_{1} \left( {R_{2} } \right) \in \left[ {0.35,0.45} \right],\mu_{1} \left( {R_{3} } \right) \in \left[ {0.2,0.3} \right]. \hfill \\ \end{array} \right.\end{array}$$

We can solve the above model by Lingo software and obtain fuzzy measure of experts as follows:

$$\begin{array}{l}\mu_{{1}} \left( {R_{1} } \right) = \mu_{{1}} \left( {R_{2} } \right) = \mu_{{1}} \left( {R_{1} ,R_{3} } \right) = \mu_{{1}} \left( {R_{2} ,R_{3} } \right) = 0.35,\\\mu_{{1}} \left( {R_{{3}} } \right) = 0.2,\mu_{{1}} \left( {R_{{1}} ,R_{2} } \right) = \mu_{{1}} \left( {R_{{1}} ,R_{2} ,R_{3} } \right) = 1\end{array}$$

In the same way, we also get the optimal fuzzy measures on the expert set R for the criteria $C_{j}$, $\left( {j = 2,3,{4}} \right)$.