**Property 1: Idempotent**

If IVPHFEs *h*1 = *h*2 = ... = *hk* = *h*, then *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) = *h*.

**Proof.** From Equation (8),

$$\begin{split} & \text{IVPHFMMM}^{(\lambda\_{1},\lambda\_{2},\ldots,\lambda\_{k})}(h\_{1},h\_{2},\ldots,h\_{k}) = \left( \left. \begin{pmatrix} \prod\_{k=1}^{nd} \left( \prod\_{j=1}^{nd} {p\_{i}^{\lambda\_{j}}}^{\lambda\_{j}} \right)^{\mathbf{w}\_{k}} \right\}^{\sum\_{j}^{\lambda\_{j}} j} \right. \\ & \left. \left( \prod\_{k=1}^{nd} \left( {p\_{i}^{\lambda\_{j}}}^{\text{nd}} {p\_{i}^{\lambda\_{j}}} \right)^{\mathbf{w}\_{k}} \right)^{\sum\_{j}^{\lambda\_{j}} j}, \left( \prod\_{k=1}^{nd} \left( {p\_{i}^{\lambda\_{j}}}^{\text{nd}} {p\_{i}^{\lambda\_{j}}} \right)^{\mathbf{w}\_{k}} \right)^{\mathbf{w}\_{k}} \right)^{\mathbf{w}\_{k}} \right)^{\mathbf{w}\_{k}} \\ & \left. \left( \prod\_{k=1}^{nd} \left( {p\_{i}^{\lambda\_{i}+\lambda\_{k}+\ldots+\lambda\_{kd}}}^{\lambda\_{i}+\ldots+\lambda\_{kd}} \right)^{\mathbf{w}\_{k}} \right)^{\mathbf{w}\_{k}} \right)^{\mathbf{w}\_{k}} \end{split}$$

By doing the same to DMs' weight values, we obtain,

$$\begin{aligned} \mathcal{I} &= \left( \left[ \left( p\_i^{\lambda\_1 + \lambda\_2 + \ldots + \lambda\_{nd}} \right)^{\frac{1}{\sum\_j \lambda\_j}}\_{\sum\_j \lambda\_j} \right. \right. \\ &= \left( \left[ \left( p\_i^{\lambda\_1 + \lambda\_2 + \ldots + \lambda\_{nd}} \right)^{\frac{1}{\sum\_j \lambda\_j}}\_{\sum\_j \lambda\_j} \left( \left( p\_i^{\mu} \right)^{\lambda\_1 + \lambda\_2 + \ldots + \lambda\_{nd}} \right)^{\frac{1}{\sum\_j \lambda\_j}} \right] \right) \text{ as } \sum\_k w\_k = 1. \quad \square\\ &= \left( \gamma\_{i\prime} \left[ p\_{i\prime}^{l}, p\_i^{\mu} \right] \right) = h \end{aligned}$$
