## 3 Hesitant Probabilistic Multiplicative Sets and Relations
Then,  
\[M=\{<a,\{1/2|0.3,3|0.7\}>,<b,\{0.45|0.6,0.98|0.4\}>,<c,\{6.5|0.4,7|0.3,8|0.3\}>\}.\]6  
\(D(H_{1},H_{2})\) measures the deviation between HPMPRs in the sense that the first part \(\frac{1}{dn^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{s=1}^{d}\left(\overline{h}_{ ij,1}^{\sigma\left(s\right)}\times\overline{h}_{i,2}^{\sigma\left(s\right)}\right)\) will be 1 if \(H_{1}=H_{2}\) and the second part \(\frac{2}{nd(n+1)}\sum_{i=1}^{n}\sum_{j\geq i}^{n}\sum_{s=1}^{d}\left|\overline{ p}_{ij,1}^{\sigma\left(s\right)}-\overline{p}_{ij,2}^{\sigma\left(s\right)}\right|\) is 0. The properties of deviation are presented in the following theorem. **Theorem 4**. : _Consider two HPMPRs \(H=\left(h_{ij,1}\right)_{n\times n}\) and \(H_{2}=\left(h_{ij,2}\right)_{n\times n}\) and their NHPMPRs \(\overline{H}_{1}=\left(\overline{h}_{ij,1}\right)_{n\times n}\) and \(\overline{H}_{2}=\left(\overline{h}_{ij,2}\right)_{n\times n}\). The deviation degree defined in Equation (21) satisfies the following properties:_  
1. \(D(H_{1},H_{2})\geq 1\)_;_
2. \(D(H_{1},H_{2})=1\) _if and only if_ \(H_{1}=H_{2}\)_;_
3. \(D(H_{1},H_{2})=D(H_{2},H_{1})\)_._  
_(please see Appendix A for Proof 3)._  
Before proceeding, let us present and prove some lemmas that will be useful in providing the theoretical framework. **Lemma 1**. : _Take \(0<x<1\) and \(y>0\). Then_  
\[M=\{<a,\{1/2|0.3,3|0.7\}>,<b,\{0.45|0.6,0.98|0.4\}>,<c,\{6.5|0.4,7|0.3,8|0.3\}>\}.\]7  
_and equality holds if and only if \(y=1\) (please see Appendix A for Proof 4)._  
**Lemma 2**. : _[_52_]__: Consider \(x_{i}>0\), \(\omega_{i}>0\), and \(i=1,2,...,n\) with \(\sum_{i=1}^{n}\omega_{i}=1\). Then,_  
\[M=\{<a,\{1/2|0.3,3|0.7\}>,<b,\{0.45|0.6,0.98|0.4\}>,<c,\{6.5|0.4,7|0.3,8|0.3\}>\}.\]8  
_where equality holds iff \(x_{1}=x_{2}=...=x_{n}\)._  
While the decision-maker provides its preference in the form of an HPMPR, it should be noted that it can be used for decision-making with good results only if it has sufficient consistency. **Definition 9**. : _Suppose an HPMPR \(H=(h_{ij})_{n\times n}\), its NHPMPR \(\overline{H}=(\overline{h}_{ij})_{n\times n}\) and consistent HPMPR \(\widehat{H}=(\overline{h}_{ij})_{n\times n}\) with optimized parameter \(0\leq\zeta\leq 1\) obtained from Algorithm 1. The constancy index of \(H\) is defined to be the deviation degree between \(\overline{H}\) and \(\widehat{H}\), denoted as_  
\[M=\{<a,\{1/2|0.3,3|0.7\}>,<b,\{0.45|0.6,0.98|0.4\}>,<c,\{6.5|0.4,7|0.3,8|0.3\}>\}.\]9  
It is clear that \(CI(H)=1\) if and only if \(H\) is consistent. Ideally, the decision-maker should provide a consistent HPMPR so that it can be used for meaningful decision-making. However, some margin of error should be provided to the decision-maker, relative to the practical problems. **Definition 10**. : _Consider an HPMPR \(H=(h_{ij})_{n\times n}\). For a given tolerance value \(CI_{r}\), HPMPR \(H\) is said to be an acceptably consistent HPMPR if_  
\[s(h)=\left(\prod\limits_{i=1}^{\#h}\gamma_{i}^{p_{i}}\right)^{\frac{1}{\#h}} \tag{2}\]0  
If an HPMPR is not even acceptably consistent, then the decision-maker should revisit and modify it. The Algorithm 2 is proposed to make an HPMPR acceptably consistent. ```
Input: The HPMPR \(H=(h_{ij})_{n\times n}\), consistency tolerance value \(CI_{r}\) and the parameter \(\alpha\in(0,1)\). Output: The acceptably consistent HPMPR \(H^{(t)}\), consistency index \(CI(H^{(t)})\), and number of iterations \(t\). Step 1: Compute \(\overline{H}=(\overline{h}_{ij})_{n\times n}\) with optimized parameter \(\zeta\). Let \(t=0\), \(H^{(0)}=\left(h^{(0)}_{ij}\right)_{n\times n}=\overline{H}=(\overline{h}_{ij} )_{n\times n}\). Step 2: Calculate the consistent HPMPR \(\widetilde{H}^{(t)}=\left(\widetilde{h}^{(t)}_{ij}\right)_{n\times n}\) by applying Algorithm 1 to \(H^{(t)}\) and consistency index \(CI(H^{(t)})\), where

\[s(h)=\left(\prod\limits_{i=1}^{\#h}\gamma_{i}^{p_{i}}\right)^{\frac{1}{\#h}} \tag{2}\]1

Step 3: If \(CI(H^{(t)})\leq CI_{r}\), then go to Step 5; otherwise, go to Step 4.