## 4 Consensus Measure in Group Decision Making
Then, the group consensus index (GCI) of HPMPR \(H_{k}\) is defined to be the deviation measured between \(\overline{H}_{k}\) and \(H_{g}\), i.e.,  
\[GCI(H_{k})=D(\overline{H}_{k},H_{g}). \tag{32}\]The agreement between an individual decision-maker and a group decision is measured by the distance between individual HPMPR \(H_{k}\) and group HPMPR \(H_{g}\). Therefore, \(GCI(H_{k})=1\) means that the \(k\)th decision-maker has full agreement with group decision; otherwise, the smaller the value of \(GCI(H_{k})\) is, the better the consensus will be. In many real life scenarios, it is important to have consensus among all decision-makers, although we have to live with a difference of opinion sometimes, and it is often hard to reach complete consensus, so a threshold value can be decided based on the practical nature of the problem to allow a difference of opinion to some extent. **Definition 13**. : _Let \(H_{k}=(h_{ijk})_{n\times n}\)\((k=1,2,3,...,m)\) be \(m\) HPMPRs provided by decision-makers, and assume their NHPMPRs \(\overline{H}_{k}\) are obtained with optimized \(\zeta_{k}\)\((k=1,2,...,m)\). Furthermore, compute group HPMPR \(H_{g}\) and \(GCI(H_{k})\)\((k=1,2,...,m)\). Consider \(GCI_{r}\) as a tolerance value of consensus measure. Thus, HPMPR \(H_{k}\) can be said to be an acceptable consensus with group HPMPR \(H_{g}\) if_  
\[GCI(H_{k})\leq GCI_{r}. \tag{33}\]  
However, it is possible that a decision-maker has an unacceptable difference of opinion with respect to the group decision; in this regard, an algorithm is proposed to modify the dissenter's HPMPR to reach an acceptable consensus. The convergence of the Algorithm 3 follows from the following result. ```
Input: The tolerance value \(GCI_{r}\) of the consensus measure, HPMPRs \(H_{k}=(h_{ijk})_{n\times n}\)\((k=1,2,3,...,m)\), and parameter \(\beta\in(0,1)\). Output: The acceptable consensus HPMPRs \(H_{k}^{(t)}\), group consensus index \(GCI(H_{k}^{(t)})\)\((k=1,2,...,m)\), group HPMPR \(H_{g}^{(t)}\), and number of iterations value \(t\). Step 1: Compute \(\overline{H}_{k}=(\overline{h}_{ijk})_{n\times n}\) with optimized parameters \(\zeta_{k}\). Let \(t=0\), \(H_{k}^{(0)}=(h_{ijk}^{(0)})_{n\times n}=\overline{H}_{k}=(\overline{h}_{ijk})_ {n\times n}\). Step 2: Compute group HPMPR \(H_{g}^{(t)}=(h_{ijk}^{(t)})_{n\times n}\) by fusing all individual HPMPRSs \(H_{k}^{(t)}\)\((k=1,2,...,m)\) according to Equation (30), where \[h_{ij,g}^{(t)}=\bigotimes_{k=1}^{m}\left(h_{ij,k}^{(t)}\right)^{\omega_{k}}.\] (34) Step 3: Calculate \(GCI(H_{k}^{(t)})=D(H_{k}^{(t)},H_{g}^{(t)})\)\((k=1,2,...,m)\). If \(GCI(H_{k}^{(t)})\leq GCI_{r}\) for all \(k=1,2,...,m\) then go to Step 5; otherwise, go to Step 4. Step 6: Let \(H_{k}^{(t+1)}=(h_{ijk}^{(t+1)})_{n\times n}\), where \[h_{k}^{(t+1)}=\left(h_{k}^{(t)}\right)^{\beta}\otimes\left(h_{ij,g}^{(t)}\right) ^{1-\beta}.\] (35) Additionally, let \(t=t+1\), and go to Step 2. Step 5: Output the adjusted HPMPRs \(H_{k}^{(t)}=(h_{ijk}^{(t)})_{n\times n}\), the group consistency index \(GCI(H_{k}^{(t)})\)\((k=1,2,...,m)\), the group HPMPR \(H_{g}^{(t)}\), and number of iterations value \(t\). Step 6: End. ```  
**Algorithm 3** Algorithmic description to modify the dissenter's HPMPR to reach an acceptable consensus. **Theorem 8**. : _Consider \(H_{k}=(h_{ijk})_{n\times n}\)\((k=1,2,3,...,m)\) HPMPRs given by decision-makers. Let \(\{H_{k}^{(t)}\}\) be the sequence obtained from Algorithm 3. Then,_  
\[GCI\left(H_{k}^{(t+1)}\right)<GCI\left(H_{k}^{(t)}\right)\text{ for all }k=1,2,...,m.\](please see Appendix A for Proof 8)._  
Now, 1000 random sets of \(m\) HPMPRs are generated and Algorithm 3 is applied to the developed consensus. The average value of iterations of Algorithm 3 is presented in Table 3 for different values of parameters. The analysis of Table 3 suggests that the increase in the value of parameter \(\beta\) has adverse effects on the number of iterations of Algorithm 3. Thus, the value of the parameter \(\beta\) must be small. More iterations are needed to develop consensus when \(GCI_{r}\) is nearer to 1.