**3. Proposed Methodology**

### *3.1. IVPHFS Based MM Operator and Its Properties*

This section presents a new extension to the popular and powerful MM operator in the IVPHFS context. The MM operator [22] is a generalized operator which aggregates preferences by properly capturing the interrelationship between attributes. The operator can be used to realize other operators as mentioned above. The MM operator also considers the risk appetite values of DMs along with their relative importance (weights) in its formulation which intuitively produces a rational aggregation of preference information.

Table 1 provides a review on MM operators that are proposed for different fuzzy sets. This provides an idea on the basic concept of MM operator, its practical use in MADM problems. Moreover, the challenges mentioned above are clearly supported by this tabular analysis.


**Table 1.** Review of MM operator on different fuzzy sets.

From Table 1, it can be inferred that (i) preference styles either associate a single value as a probability or ignore probability, which is not reasonable for decision-making; (ii) the MM operator is popularly used for aggregating preference information by effectively capturing the interrelationship between attributes and (iii) finally, attributes' weight values are directly (not calculated) obtained from the DMs which causes inaccuracies in the decision-making process.

Motivated by the power of MM operator and IVPHFS concept, in this paper, we extend the MM operator to IVPHFS and the definition and properties are given below:

**Definition 5.** *The aggregation of IVPHFEs using IVPHFMM (interval-valued probabilistic hesitant fuzzy Muirhead mean) operator is a mapping from T<sup>k</sup>* → *T for k* = 1, 2, ... , *nd which is given by,*

$$= \begin{pmatrix} \text{IVPHF}MM^{(\lambda\_1, \lambda\_2, \dots, \lambda\_{nd})}(h\_1, h\_2, \dots, h\_{nd}) \\ \begin{pmatrix} \prod\_{k=1}^{nd} \left( \prod\_{j=1}^{nd} \boldsymbol{\gamma}\_j^{\lambda\_j} \right)^{w\_k} \boldsymbol{\gamma}^{\lambda\_j} \\ \prod\_{k=1}^{nd} \left( \prod\_{j=1}^{nd} \left( \boldsymbol{p}\_i^{\lambda\_j} \right)^{w\_k} \right)^{\sum\_j \lambda\_j} \end{pmatrix} \end{pmatrix} \tag{8}$$
 
$$= \left\{ \left[ \left( \prod\_{k=1}^{nd} \left( \prod\_{j=1}^{nd} \left( \boldsymbol{p}\_i^{\lambda\_j} \right)^{w\_k} \right)^{\sum\_j \lambda\_j} \right), \left( \prod\_{k=1}^{nd} \left( \prod\_{j=1}^{nd} \left( \boldsymbol{p}\_i^{\lambda\_j} \right)^{w\_k} \right)^{w\_k} \right)^{\sum\_j \lambda\_j} \right] \right\}$$

 1

*where* λ1, λ2, ... , λ*k is risk appetite values of each DM from the set* {1, 2, ... , *nd*}*, nd is the total number of DMs, wk is the weight of the kth DM.*

It must be noted that Equation (8) provides the MM operator in the IVPHFS context. The operator aggregates the membership values, followed by the probability values (in the interval fashion). That is, the lower limit of the probability value is aggregated and then the upper limit of the probability value is aggregated. The square bracket represents the interval values that we obtain upon aggregation of probability values. Further, we present a theorem below to show that aggregation of di fferent IVPHFEs by using IVPHFMM operator produces an IVPHFE.

**Remark 3.** *The MM operator is initially proposed in Reference [22] and it is given by* ⎛⎜⎜⎜⎜⎝ 1*nd*! *nd k*=1 2 *nd j*=1 *a* λ*j k* ⎞⎟⎟⎟⎟⎠ *j* λ*j where nd is the number of DMs and* λ*j is the risk appetite values for j* = 1, 2, ... , *nd. Risk appetite is defined by ISO 31000 as the amount of risk pursued, retained or taken by an organization. In this case, it is the risk pursued, retained or taken by a DM. The possible values are from the set* {1, 2, ... , *nd*}*. The higher the value of* λ *indicates a higher risk appetite value for the DM.*
