## 3 Hesitant Probabilistic Multiplicative Sets and Relations  
With the introduction of probability in hesitant fuzzy sets [15] and linguistic term sets [18], it is important to improve other sets in this context. HME expresses the opinion of the decision-maker in [PERSON]'s scale. The aim of this section is first to introduce the HPMS and then define and build on its basis a basic framework for the HPMPR. **Definition 4**. : _For a fixed set \(X\), the HPMS is defined as_  
\[M=\{<x,h(x)>:x\in X\}\]  
_where \(h(x)=\{\gamma_{i}|p_{i}:i=1,\,2,\,3,...,\#h\}\) denotes the hesitant probabilistic multiplicative element (HPME) such that \(\gamma_{i}\in[1/9,9]\) and \(p_{i}\in[0,1]\) with \(\sum\limits_{i=1}^{\#h}p_{i}=1\), namely \(p_{j}\) is the probability of \(\gamma_{j}\) among all others \(\gamma_{i}\)._  
For clarification, the following example is given. **Example 2**. : _Let \(X=\{a,b,c\}\). Consider_  
\[M=\{<a,\{1/2|0.3,3|0.7\}>,<b,\{0.45|0.6,0.98|0.4\}>,<c,\{6.5|0.4,7|0.3,8|0.3\}>\}.\]  
_Then, \(M\) is HPMS and \(\{1/2|0.3,3|0.7\}\), \(\{0.45|0.6,0.98|0.4\}\), \(\{6.5|0.4,7|0.3,8|0.3\}\) are HPMEs._  
For comparison between two HPMEs, the score and deviation for the HPME is defined below, based on the definition of the score function of the HME and the variance of he random variable, respectively. **Definition 5**. : _Take HPME \(h=\{\gamma_{i}|p_{i}:i=1,2,...,\#h\}\). Then,_  
\[s(h)=\left(\prod\limits_{i=1}^{\#h}\gamma_{i}^{p_{i}}\right)^{\frac{1}{\#h}} \tag{2}\]  
_is defined as the score function. Additionally,_  
\[d(h)=\sum\limits_{i=1}^{\#h}\left(\gamma_{i}^{p_{i}}-s(h)\right)^{2}. \tag{3}\]  
Consider two HPMEs, \(h_{1}\) and \(h_{2}\). The order of the HPMEs is decided as follows:  
* If \(s(h_{1})>s(h_{2})\), then \(h_{1}>h_{2}\) (\(h_{1}\) is superior to \(h_{2}\)). * For the case \(s(h_{1})=s(h_{2})\), compare the deviations. If \(d(h_{1})>d(h_{2})\), then \(h_{2}>h_{1}\); if \(d(h_{2})>d(h_{1})\), then \(h_{1}>h_{2}\); if \(d(h_{1})=d(h_{2})\), then \(h_{1}=h_{2}\). The main aim of this study is to introduce the HPMPR and to make a decision-making model with it. For this purpose, the following operations are made. Let \(h_{1}\) and \(h_{2}\) be HPMEs of the same length, i.e., \(\#h_{1}=\#h_{2}\). Then,  
\[h_{1}\otimes h_{2}=\cup_{\gamma^{1}_{\sigma(s)}|p^{1}_{\sigma(s)}\in h_{1}, \gamma^{2}_{\sigma(s)}|p^{2}_{\sigma(s)}\in h_{2}}\{\gamma^{1}_{\sigma(s)} \times\gamma^{2}_{\sigma(s)}|p^{1}_{\sigma(s)}+p^{2}_{\sigma(s)}\} \tag{4}\]  
\[h^{\omega}=\cup_{\gamma_{\sigma(s)}|p_{\sigma(s)}\in h}\{\gamma^{\omega}_{ \sigma(s)}|\omega p_{\sigma(s)}\}:\omega>0 \tag{5}\]  
where \(\gamma^{1}_{\sigma(s)}|p^{1}_{\sigma(s)}\) and \(\gamma^{2}_{\sigma(s)}|p^{2}_{\sigma(s)}\) are \(s\)th elements of \(h_{1}\) and \(h_{2}\), respectively. These operations satisfy the following properties. **Theorem 1**. : _Take three HPMEs, \(h_{1}\), \(h_{2}\) and \(h_{3}\), with \(\#h_{1}=\#h_{2}=\#h_{3}\) and \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}>0\). Then,_  
1. \(h_{1}\otimes h_{2}\)_._
2. \((h_{1}\otimes h_{2})\otimes h_{3}=h_{1}\otimes(h_{2}\otimes h_{3})\)_._
3. \((h_{1}\otimes h_{2})^{\omega}=h^{\omega_{1}}_{1}\otimes h^{\omega_{2}}_{2}\)_._
4. \(h^{(\omega_{1}+\omega_{2})}=h^{\omega_{1}}\otimes h^{\omega_{2}}\)_._  
_(please see Appendix A for Proof 1)._  
To allow decision-makers to provide the preferences in a hesitant probabilistic environment, we define HPMPR. **Definition 6**. : _Let \(X=\{x_{1},x_{2},x_{3},...,x_{n}\}\) be the set of alternatives.