**Property 3: Monotonicity**

Consider IVPHFEs *h* = *h i* and *h* = *hi* for *i* = 1, 2, ... , *k* such that *h i* ≥ *hi*. Then, *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)*<sup>h</sup>*1, *<sup>h</sup>*2, ... , *h k*.

$$\textbf{Proof. Let } \boldsymbol{\gamma}\_{i}^{\prime} = \left(\prod\_{k=1}^{nd} \left(\prod\_{j=1}^{nd} \left(\boldsymbol{\gamma}\_{i}^{\prime}\right)^{\lambda\_{j}}\right)^{\boxtimes \boldsymbol{\gamma}\_{j}} \text{ and } \left[\boldsymbol{p}\_{i}^{\prime\prime}, \boldsymbol{p}\_{i}^{\prime\nu}\right] = \left[\left(\prod\_{k=1}^{nd} \left(\prod\_{j=1}^{nd} \left(\boldsymbol{p}\_{i}^{\prime\lambda\_{j}}\right)^{\lambda\_{j}}\right)^{\boxtimes \boldsymbol{\gamma}\_{j}}\right)^{\frac{1}{\sum\_{j}^{d}\lambda\_{j}}} \left(\prod\_{k=1}^{nd} \left(\boldsymbol{p}\_{i}^{\prime\nu}\right)^{\lambda\_{j}}\right)^{\boxtimes \boldsymbol{\gamma}\_{j}}\right]^{\frac{1}{\sum\_{j}^{d}\lambda\_{j}}}.$$

Similarly, *hi* is defined as above. Now score and deviation measure is adopted from Reference [19] for IVPHFEs. Since *h i* ≥ *hi* ∀*i* = 1, 2, ... , *k*, *s*(*h*) ≥ *s*(*h*) which concludes that *h* ≥ *h* and if *s*(*h*) = *<sup>s</sup>*(*h*), then calculate deviation and if σ(*h*) ≥ σ(*h*) *h* ≥ *h*. Thus, *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)*<sup>h</sup>*1, *<sup>h</sup>*2, ... , *h k*. -
