Distribution Linguistic Fuzzy Group Decision Making Based on Consistency and Consensus Analysis
Abstract
:1. Introduction
 The consistency of DLFPRs is redefined, and only the probability variation is considered, so the calculation is easier to understand.
 A new iterative algorithm for consistency recognition and adjustment is proposed to improve consistency level to acceptable level.
 A new iterative algorithm for recognition and adjustment of group consensus degree is proposed to improve group consensus degree.
2. Preliminaries
2.1. Linguistic Distribution Term Sets (LDTSs)
 If ${s}_{\xi}\le {s}_{\psi}$, then $\xi \le \psi $;
 If $\xi =2\tau \psi $, then $neg({s}_{\xi})={s}_{\psi}$, especially ${s}_{\tau}=neg({s}_{\tau})$.
2.2. Distribution Linguistic Fuzzy Preference Relations (DLFPRs)
 ${p}_{ij}^{(\xi )}={p}_{jl}^{(2\tau \xi )},\forall i,j=1,2,\dots ,n,\xi =0,1,\dots ,2\tau $;
 ${h}_{ii}=\left\{({s}_{\tau},1)\right\}$.
3. ConsistencyAdjustment Algorithm for DLFPRs
3.1. Multiplicative Consistency of DLFPRs
 (1)
 $0\le {\tilde{p}}_{ij}^{(\xi )}\le 1$;
 (2)
 $\tilde{H}$ is DLFPR;
 (3)
 ${p}_{ij}^{(\xi )}{p}_{jl}^{(\xi )}{p}_{li}^{(\xi )}={p}_{il}^{(\xi )}{p}_{lj}^{(\xi )}{p}_{ji}^{(\xi )}$.
 (1)
 For $\forall i,j,l=1,2,\dots ,n$, $0\le {p}_{ij}^{(\xi )}\le 1$, we have: $0\le {p}_{il}^{(\xi )}\le 1$, $0\le {p}_{jl}^{(\xi )}\le 1$, then $\frac{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}>0$. Thus,$$0\le {\tilde{p}}_{ij}^{(\xi )}={\left(1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}\right)}^{1}/{\displaystyle \sum _{\xi =0}^{2\tau}{\left(1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}\right)}^{1}}\le 1.$$
 (2)
 For $\forall i,j=1,2,\dots ,n$, when $i=j$, ${\tilde{p}}_{ij}^{(\xi )}={p}_{ij}^{(\xi )}$, we can obtain ${\tilde{h}}_{ii}=\{({s}_{0},0),\dots ,({s}_{\tau 1},0),({s}_{\tau},1),({s}_{\tau +1},0),\dots ,({s}_{2\tau},0)\}$.Thus, $\tilde{H}={({\tilde{h}}_{ij})}_{n\times n}$ is a DLFPR.
 (3)
 For $\forall i,j,l=1,2,\dots ,n,\xi =0,1,\dots ,2\tau $, ${\widehat{p}}_{ij}^{(\xi )}+{\widehat{p}}_{ji}^{(\xi )}=1$, we can attest:$$\begin{array}{l}\frac{{\widehat{p}}_{il}^{(\xi )}}{{\widehat{p}}_{li}^{(\xi )}}\cdot \frac{{\widehat{p}}_{lj}^{(\xi )}}{{\widehat{p}}_{jl}^{(\xi )}}=\frac{1}{1{\widehat{p}}_{il}^{(\xi )}}\cdot \frac{1}{1{\widehat{p}}_{lj}^{(\xi )}}=\frac{1}{1/{\widehat{p}}_{il}^{(\xi )}1}\cdot \frac{1}{1/{\widehat{p}}_{lj}^{(\xi )}1}\\ =\frac{1}{{\displaystyle \sum _{l=1}^{n}{p}_{ll}^{(\xi )}}/{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}\cdot \frac{1}{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}/{\displaystyle \sum _{l=1}^{n}{p}_{ll}^{(\xi )}}}\\ =\frac{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}=\frac{1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}}{1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}}=\frac{{\left(1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}\right)}^{1}}{{\left(1+\frac{{\displaystyle \sum _{l=1}^{n}{p}_{il}^{(\xi )}}}{{\displaystyle \sum _{l=1}^{n}{p}_{jl}^{(\xi )}}}\right)}^{1}}=\frac{{\widehat{p}}_{ij}^{(\xi )}}{{\widehat{p}}_{ji}^{(\xi )}},\end{array}$$$$\begin{array}{l}\frac{{\tilde{p}}_{il}^{(\xi )}}{{\tilde{p}}_{li}^{(\xi )}}\cdot \frac{{\tilde{p}}_{jl}^{(\xi )}}{{\tilde{p}}_{lj}^{(\xi )}}=\frac{{\widehat{p}}_{il}^{(\xi )}}{{\displaystyle \sum {\widehat{p}}_{il}^{(\xi )}}}\cdot \frac{{\displaystyle \sum {\widehat{p}}_{li}^{(\xi )}}}{{\widehat{p}}_{li}^{(\xi )}}\cdot \frac{{\widehat{p}}_{lj}^{(\xi )}}{{\displaystyle \sum {\widehat{p}}_{lj}^{(\xi )}}}\cdot \frac{{\displaystyle \sum {\widehat{p}}_{jl}^{(\xi )}}}{{\widehat{p}}_{jl}^{(\xi )}}\\ =\frac{{\widehat{p}}_{ij}^{(\xi )}}{{\widehat{p}}_{ji}^{(\xi )}}\cdot \frac{{\displaystyle \sum {\widehat{p}}_{ji}^{(\xi )}}}{{\displaystyle \sum {\widehat{p}}_{ij}^{(\xi )}}}=\frac{{\tilde{p}}_{ij}^{(\xi )}}{{\tilde{p}}_{ji}^{(\xi )}},\end{array}$$$$\mathrm{i}.\mathrm{e}.,\forall i,j,l=1,2,\dots ,n,\xi =0,1,\dots ,2\tau ,{p}_{ij}^{(\xi )}{p}_{jl}^{(\xi )}{p}_{li}^{(\xi )}={p}_{il}^{(\xi )}{p}_{lj}^{(\xi )}{p}_{ji}^{(\xi )}$$
3.2. Consistency Index of the DLFPR
 $d({H}_{1},{H}_{2})\ge 0$,
 $d({H}_{1},{H}_{2})=d({H}_{2},{H}_{1})$
 $d({H}_{1},{H}_{1})=0$
 $d({H}_{1},{H}_{3})\le d({H}_{1},{H}_{2})+d({H}_{2},{H}_{3}).$
 $0\le CI(H)\le 1$;
 $H$ is a DLFPR of multiplicative consistency if $CI(H)=0$.
3.3. ConsistencyAdjustment Algorithm for DLFPRs
Algorithm 1. Consistencyadjustment process for DLFPRs. 
Input: The incipient DLFPR $H={({h}_{ij})}_{n\times n}$, the threshold of consistency $\overline{CI}$ and the adjusted parameter $\theta (0<\theta <1)$. 
Output: The adjusted DLFPR $\overline{H}={({\overline{h}}_{ij})}_{n\times n}$, which is of acceptable multiplicative consistency. 
Step 1. Let $t=0$ and $H(t)=H$. 
Step 2. According to Theorem 1, let $\tilde{H}={({\tilde{h}}_{ij})}_{n\times n}$ be a DLFPR with multiplicative consistency. 
Step 3. Given
$$CI({H}_{(t)})=d(\tilde{H},H)=\frac{1}{n}\sqrt{\frac{1}{2\tau +1}{\displaystyle \sum _{i=1}^{n}{\displaystyle \sum _{j=1}^{n}{\displaystyle \sum _{\xi =0}^{2\tau}{\left({\tilde{p}}_{ij(t)}^{(\xi )}{p}_{ij(t)}^{(\xi )}\right)}^{2}}}}}.$$

Step 4. Compare the level of consistency with the threshold, if $CI({H}_{(t)})\le \overline{CI}$, then jump to step 8. Otherwise, advance to Step 5. 
Step 5. Seek the element ${p}_{{i}^{\prime}{j}^{\prime}(t)}^{({\xi}^{\prime})}$ with the lowest consistency level, where ${\left({\tilde{p}}_{ij(t)}^{({\xi}^{\prime})}{p}_{ij(t)}^{({\xi}^{\prime})}\right)}^{2}=\underset{0\le \xi \le 2\tau ,i<j}{\mathrm{max}}{\left({\tilde{p}}_{ij(t)}^{(\xi )}{p}_{ij(t)}^{(\xi )}\right)}^{2}$. 
Step 6. Generate the new DLFPR ${H}_{(t+1)}={({h}_{ij(t+1)})}_{n\times n}$ with ${h}_{ij}{}_{(t+1)}=\{({s}_{\xi},{p}_{ij(t+1)}^{(\xi )})$ $\xi =0,1,\dots ,2\tau ,0\le {p}_{ij(t+1)}^{(\xi )}\le 1\}$ and
$${p}_{ij(t+1)}^{(\xi )}=\{\begin{array}{l}(1\delta )\cdot {p}_{{i}^{\prime}{j}^{\prime}(t)}^{({\xi}^{\prime})}+\delta \cdot {\tilde{p}}_{{i}^{\prime}{j}^{\prime}(t)}^{({\xi}^{\prime})},i={i}^{\prime},j={j}^{\prime},\xi ={\xi}^{\prime}\\ {p}_{ij(t)}^{(\xi )},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise\\ {p}_{ij(t+1)}^{(2\tau \xi )},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i={j}^{\prime},j={i}^{\prime},2\tau \xi ={\xi}^{\prime}\end{array}.$$

Step 7. Let $t=t+1$, then back to Step 2. 
Step 8. Let $\overline{H}={H}_{(t)}$. 
Step 9. End. 
4. Consensus Measures and Consensus Model for DLFPRs
Algorithm 2. Consensusadjustment process for DLFPRs. 
Input: The distribution linguistic information decisionmaking matrices ${D}_{k}={({d}_{ij}^{k})}_{m\times n}(k=1,2,\dots ,l)$, the threshold of group consensus index $\theta (0<\theta <1)$, the adjustment cost of DMs ${c}_{1},{c}_{2},\dots ,{c}_{n}(n=1,2,\cdots ,l).$ 
Output: The adapted distribution linguistic information decisionmaking matrices ${\tilde{D}}_{k}={({\tilde{d}}_{ij}^{k})}_{m\times n}(k=1,2,\dots ,l)$, which is of acceptable consistency and consensus degree. 
Step 1. Determine the magnitude of $GCI$ with $\theta $. If $GCI\ge \theta $, then jump to Step 5; otherwise, advance to Step 2. 
Step 2. Seek out the smallest element ${b}_{pq}$, which means there is the lowest consensus degree between ${e}_{p}$ and ${e}_{q}$, then search for the similarity matrices $S{M}^{pq}$. 
Step 3. Based on $S{M}^{pq}$, looking for the element with the smallest value $s{m}_{ij}^{pq}$, which shows ${e}_{p}$ and ${e}_{q}$ differ the most greatly on the evaluation of alternative ${A}_{i}$ in regard to attribute ${c}_{j}$. Then, ${d}_{ij}^{p}$ or ${d}_{ij}^{q}$ need be adjusted. 
Step 4. Adjust according to the adjustment cost ${c}_{p}$ and ${c}_{q}$ of DMs ${e}_{p}$ and ${e}_{q}$. 
If ${c}_{p}>{c}_{q}$, then the LDE ${d}_{ij}^{p}$ should be changed into ${d}_{ij}^{q}$. 
If ${c}_{p}<{c}_{q}$, then the LDE ${d}_{ij}^{q}$ should be changed into ${d}_{ij}^{p}$. 
Step 5. Output the new distribution linguistic information decisionmaking matrices ${\tilde{D}}_{k}={({\tilde{d}}_{ij}^{k})}_{m\times n}(k=1,2,\dots ,l)$. 
Step 6. End. 
5. Distribution Linguistic Fuzzy GDM (DLFGDM) Method
6. Numerical Examples and Comparative Discussion
6.1. Application to Select the Best Equity Incentive Mode
6.2. Comparative Discussion
6.2.1. Application of the Method in Zhang et al.
6.2.2. Application of the Method in Tang et al.
 It is essential to improve the level of consistency and consensus in GDM. However, Zhang et al. [10] did not check the consistency of the original DLFPRs, but directly carried out the consensus degree test by defining $\tilde{CI}$, ignoring the consistency adjustment within the experts, and then directly used the weighted average method to update and adjust, which results in insufficient retention of the original language information of the experts and a large range of changes. Therefore, the developed DLFGDM method includes not only the consistency identification but also the consistency improvement method, so the application of it will be more reliable.
 Compared to the method of Tang et al. [38], our scheme has a different ranking. However, Tang et al. [38] only use the goal programming model based on expected consistency for adjustment, without consensus test and consensus improvement. Due to the influence of subjective and objective factors, the evaluation information proposed by experts may differ greatly, and there may be differences between them. Therefore, the direct application of the established expert weight to the final evaluation ranking may lead to the results being not reliable and lack of rationality. Our DLFGDM method measures and adjusts the consensus degree, and uses the weighted average operator to form a comprehensive consensus matrix to comprehensively deal with the experts’ opinions, which is more reasonable, more reliable and has a wider application prospect.
7. Conclusions
 Based on the new distance formula, a new consistency index is introduced.
 The definition of multiplicative consistency of DLFPR is presented to include only the variation of distributed language evaluation probability. A new consistency adjustment algorithm is proposed, which preserves the original appraisement information as far as possible and adjusts the lowest consistency element each time.
 A new consensus degree and a consensus promotion algorithm are developed by considering the costs of experts.
 Two operators are used to integrate the distribution linguistic elements to derive alternative sorting.
 Without considering the limited knowledge and complex problems in real decision making, the evaluation information may be incomplete.
 The predetermined expert weights and attribute weights remain unchanged, which lacks certain rationality.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Jin, F.; Li, C.; Liu, J.; Zhou, L. Distribution Linguistic Fuzzy Group Decision Making Based on Consistency and Consensus Analysis. Mathematics 2021, 9, 2457. https://doi.org/10.3390/math9192457
Jin F, Li C, Liu J, Zhou L. Distribution Linguistic Fuzzy Group Decision Making Based on Consistency and Consensus Analysis. Mathematics. 2021; 9(19):2457. https://doi.org/10.3390/math9192457
Chicago/Turabian StyleJin, Feifei, Chang Li, Jinpei Liu, and Ligang Zhou. 2021. "Distribution Linguistic Fuzzy Group Decision Making Based on Consistency and Consensus Analysis" Mathematics 9, no. 19: 2457. https://doi.org/10.3390/math9192457