**2. Preliminaries**

Let us discuss some basics of HFS, PHFS and IVPHFS concepts.

**Definition 1** [2]**.** *Consider a fixed set T and hesitant fuzzy set (HFS) H on T which is a function h that produces a subset in the interval* [0, 1]*. Mathematically, it is given by,*

$$H = (t, h\_H(t) \| t \in T) \tag{1}$$

*where hH*(*t*) *is the set of values in range 0 to 1.*

**Remark 1.** *For convenience, hH*(*t*) = *h*(*t*) *is called the hesitant fuzzy element (HFE) and the collection of HFEs is H.*

**Definition 2** [10]**.** *Consider a fixed set T. The PHFS Hp on T is given by,*

$$H\_P = \left( t\_\prime h\_{\bar{H}\_p}(\gamma\_{i\prime} p\_i) \Big| t \in T \right) \tag{2}$$

*where hHp* (γ*i*, *pi*) *is the probabilistic hesitant fuzzy element with* γ*i being the membership value of t with its associated occurrence probability pi and i* = 1, 2, ... , #*hHp .*

**Remark 2.** *For convenience, hHp* (γ*i*, *pi*) = *<sup>h</sup>*(γ*i*, *pi*) *is called the probabilistic hesitant fuzzy element (PHFE) with* γ*i and pi in the range 0 to 1.*

**Definition 3** [19]**.** *Consider a fixed set T. The IVPHFS HIP on T is given by,*

$$H\_{IP} = \left( t, h\_{H\_p} \left( \gamma\_{i\prime} \left| p\_{i\prime}^l p\_i^u \right| \right) \right) \tag{3}$$

*where hHp* <sup>γ</sup>*i*,#*pli*, *pui* \$*is the interval-valued probabilistic hesitant fuzzy element with* γ*i being the membership value of t with its associated occurrence probability value in the interval fashion as* #*pli*, *pui* \$*, for all i* = 1, 2, ... , #*hHIp .*

Here, γ*i*, *pli* and *pui* are in the range 0 to 1 and *pli* ≤ *pui* . For simplicity, *hHp* <sup>γ</sup>*i*,#*pli*, *pui* \$ = *<sup>h</sup>*<sup>γ</sup>*i*,#*pli*, *pui*\$ = *h* is called interval-valued probabilistic hesitant fuzzy element (IVPHFE).

Consider an example where a DM provides his preference for ice-creams. Initially, he uses PHFS information (refer Definition 2) to rate different ice-creams viz., vanilla, strawberry and chocolate as *HP* = (*vennila*,0.6,(0.4),*strawberry*,0.8,(0.5), *chocolate*,0.4,(0.7)). Later, he uses IVPHFS information (refer Definition 3) to provide probability values in a more flexible manner. *HIP* = (*vennila*,0.6, [0.3, 0.45],*strawberry*,0.8, [0.4, 0.6], , *chocolate*,0.4, [0.6, 0.7]). By applying the latter style for preference information, we can increase the flexibility by providing a range of values as probability values.

**Definition 4** [19]**.** *Let h*1*, h*2 *and h*3 *be three IVPHFEs; then some operations are given by,*

$$h\_1 \oplus h\_2 = \left(\gamma\_1 + \gamma\_2 - \gamma\_1 \gamma\_2, \left[p\_1^l p\_{2'}^l p\_1^u p\_2^u\right]\right) \tag{4}$$

$$h\_1 \square h\_2 = \left(\gamma\_1 \gamma\_2, \left[p\_1^l p\_{2'}^l p\_1^u p\_2^u\right]\right) \tag{5}$$

$$
\lambda h\_1 = \left(1 - (1 - \gamma\_1)^{\lambda}, \left[p\_1^l, p\_1^u\right]\right) \tag{6}
$$

$$h\_1^{\lambda} = \left(\gamma\_{1'}^{\lambda} \left[p\_{1'}^{l} p\_1^{\mu}\right]\right) \tag{7}$$

Here, Equations (4)–(7) represent addition, multiplication, scalar multiplication and power operations. Actually, these equations are algebraic Archimedean T-norm and T-conorm and the additive generators used here are *g*(*x*) = −*ln*(*x*) for T-norm *TA*(*<sup>x</sup>*, *y*) = *xy* and *h*(*x*) = −*ln*(<sup>1</sup> − *x*) for T-conorm *SA*(*<sup>x</sup>*, *y*) = *x* + *y* − *xy*.
