**Property 4: Bounded**

For any IVPHFE *hi* ∀*i* = 1, 2, ... , *k*, *h*− ≤ *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ *h*+. Here, *h*− = *min*(*hi*) and *h*<sup>+</sup> = *max*(*hi*). Initially, γ*ipli*<sup>+</sup>γ*ipui* 2 is calculated and the IVPHFE that correspond to minimum and maximum value is considered as *h*− and *h*<sup>+</sup> respectively.

**Proof.** Based on the monotonic and idempotent property of IVPHFMM operator, we can easily conclude that *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>*<sup>−</sup>, *h*<sup>−</sup>, ... , *h*−). Similarly, *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*h*+, *h*+, ... , *<sup>h</sup>*+). Combining these two inequalities, we ge<sup>t</sup> *h*− ≤ *IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ *h*+. -

**Theorem 1.** *The aggregation of IVPHFEs by IVPHFMM operator produces an IVPHFE.*

**Proof.** To prove the above theorem, we must show that the aggregated value obeys Definition 3. Now, from Property 4, it is clear that the aggregated value is bounded within the different

IVPHFEs taken for aggregation. By extending the idea further, we ge<sup>t</sup> 0 ≤ ⎛⎜⎜⎜⎜⎝#2*DMk*=1 ⎛⎜⎜⎜⎜⎝#2*DMj*=1 γλ*j i* ⎞⎟⎟⎟⎟⎠*wk* ⎞⎟⎟⎟⎟⎠ 1 *j* λ*j* ≤

$$11 \text{ and } 0 \le \left\lceil \left( \prod\_{k=1}^{\mathsf{HDM}} \left( \prod\_{j=1}^{\mathsf{HDM}} \left( p\_i^l \right)^{\lambda\_j} \right)^{\mathsf{w}\_k} \right)^{\frac{1}{\sum\_{j}^{\mathsf{L}\_{\mathsf{L}\_j}}}} \left( \prod\_{k=1}^{\mathsf{HDM}} \left( \prod\_{j=1}^{\mathsf{HDM}} \left( p\_i^u \right)^{\lambda\_j} \right)^{\mathsf{w}\_k} \right)^{\frac{1}{\sum\_{j}^{\mathsf{L}\_j}}} \right\rceil \le 1. \quad \text{By combining these two}$$

$$\text{inequalities, we get } 0 \le \begin{pmatrix} \text{ $\!\!\!\+1$ }^{\text{\textquotedblleft}\text{ $\!\text\+1$ }} \left( \text{ $\!\text{\textquotedblleft}\text{$ \!\text\+1 $}}^{\text{\textquotedblleft}\text{$ \!\text\+1 $}} \text{$ \!\text\+1 $}^{\text{\textquotedblleft}\text{$ \!\text\+1 $}} \right)^{\text{\textquotedblleft}\text{$ \!\text\+1 $}} \right)^{\text{\textquotedblleft}\text{$ \!\text\+1 $}},\\ \left[ \begin{pmatrix} \text{$ \!\text{\textquotedblleft}\text{ $\!\text\+1$ }}^{\text{\textquotedblleft}\text{ $\!\text\+1$ }} \text{ $\!\text{\textquotedblleft}\text{$ \!\text\+1 $}}^{\text{\textquotedblleft}\text{$ \!\text\+1 $}} \end{pmatrix}^{\text{\textquotedblleft}\text{$ \!\text\-1 $}} \right)^{\text{\textquotedblleft}\text{$ \!\text\-1 $}},\\ \left[ \begin{pmatrix} \text{$ \!\text{\textquotedblleft}\text{ $\!\text\-1$ }}^{\text{\textquotedblleft}\text{ $\!\text\-1$ }} \text{ $\!\text{\textquotedblleft}\text{$ \!\text\-1 $}}^{\text{\textquotedblleft}\text{$ \!\text\-1 $}} \text{$ \!\text{\textquotedblright}}^{\text{\textquotedblleft}\text{ $\!\text\-1}} \text{$ \!\text{\textquotedblleft}\text{ $\!\text\-1$ }}^{\text{\textquotedblleft}\text{ $\!\text\-1}} \text{$ \!\text{\textquotedblleft}\text{ $\!\text\-1}}^{\text{\textquotedblleft}\text{$ \!\text\-1}} \text{ $\!\text{\textquotedblleft}\text{$ \!\text\-1}}^{\text{\textquotedblleft}\text{ $\!\text\-1}} \text{$ \!\text{\textquoted$$

*IVPHFMM*(<sup>λ</sup>1,λ2,...,λ*k*)(*<sup>h</sup>*1, *h*2, ... , *hk*) ≤ 1. Hence, aggregation of IVPHFEs yields an IVPHFE. -

### *3.2. Weight Calculation for Attributes Using the Proposed Programming Model*

This section put forwards a new mathematical programming model in the IVPHFS context for calculating the weights of attributes. There are mainly two types of weight calculation methods. In the first type, weight values are calculated with completely unknown information and some popular examples are analytical hierarchy process (AHP) [33], entropy measure [34] and so forth. In the second type, weight values are calculated with partially known information and this type of weight calculation gives DMs a chance to express their personal preference on each attribute which is considered during the weight calculation process. Whenever partial information is known for each attribute, the e ffective idea is to use the information for rational calculation of weights.

Motivated by the ability of the second type of weight calculation, in this paper, a new programming model is put forward in the IVPHFS context. The key advantages of the proposed model are (i) it uses the partial information from the DMs in a rational manner in its formulation; (ii) provides flexibility to the DMs to share their personal preferences on each attribute in the form of constraints; (iii) the nature of the attribute (benefit or cost) is also taken into consideration during formulation and (iv) the ideal solution for each attribute is considered for rational calculation of weight values which resemble closely to the human cognition process.

The systematic procedure for attribute weight calculation is presented below:


$$h\_j^{PLS} = \max\_{j \in \text{hem}} \chi\_{j \in \text{hem}} \left( \sum\_{i=1}^{\#h} \gamma\_i \left( \frac{p\_i^l + p\_i^{\mu}}{2} \right) \right) (\text{or}) \min\_{j \in \text{col}} \left( \sum\_{i=1}^{\#h} \gamma\_i \left( \frac{p\_i^l + p\_i^{\mu}}{2} \right) \right) \tag{9}$$

$$h\_j^{\text{NIS}} = \min\_{j \in \text{benefit}} \left( \sum\_{i=1}^{\#h} \gamma\_i \left( \frac{p\_i^l + p\_i^u}{2} \right) \right) (\text{or}) \max\_{j \in \text{cost}} \left( \sum\_{i=1}^{\#h} \gamma\_i \left( \frac{p\_i^l + p\_i^u}{2} \right) \right) \tag{10}$$

where *hPIS j* and *hNIS j* are PIS and NIS values of the *jth* attribute respectively. The *hPIS j* and *hNIS j* are calculated for each attribute and they contain IVPHFS information of the corresponding value obtained from Equations 9 and 10.

**Step 3:** Apply Model 1 to obtain the weights of attributes. Model 1:

$$\dim Z = \sum\_{j=1}^{n} w\_{j} \sum\_{i=1}^{m} d\left(h\_{ij\prime} h\_{j}^{PIS}\right) - d\left(h\_{ij\prime} h\_{j}^{NIS}\right)$$

Subject to:

$$0 \le w\_j \le 1; \sum\_j w\_j = 1.$$

The distance measure is calculated using Equation (11).

$$d(a,b) = \sqrt{\sum\_{i=1}^{\text{fitstance}} \left( \left( \gamma\_{ia} \left( \frac{p\_{ia}^l + p\_{ia}^u}{2} \right) \right) - \left( \gamma\_{ib} \left( \frac{p\_{ib}^l + p\_{ib}^u}{2} \right) \right) \right)^2} \tag{11}$$

where *a* and *b* are any two IVPHFEs.

### *3.3. Proposed MAGDM Method for Prioritization of Objects*

This section develops a ranking procedure for prioritizing objects based on the operational laws and newly proposed IVPHFMM operator. The procedure is presented below:


$$h\_l^{\text{single}} = \sum\_{i=1}^{\#\text{instance}} \left( \frac{\gamma\_i p\_i^l + \gamma\_i p\_i^u}{2} \right) \tag{12}$$

where *l* is the index for the object. Arrange *hsingle l* in the descending order of values to obtain ranking order.
