Interval valued q -rung orthopair fuzzy sets and their properties

Abstract

Yager [1] introduced the concept of q-rung orthopair fuzzy sets (q-ROFSs) in which the sum of the q^th exponent of the support for membership and the q^th exponent of the support against membership is bounded by one. Thus, the q-ROFSs are an important way to express uncertain information in broader space, and they are superior to the intuitionistic fuzzy sets (IFSs) and the Pythagorean fuzzy sets (PFSs). However, in dealing with many real life situations, it is not appropriate for experts to precisely quantify their judgements with a crisp number due to insufficiency in available information. In such situation it is advisable for decision makers to provide their judgements by the subset of the closed interval [0, 1]. The notion of interval-valued q-rung orthopair fuzzy sets (IVq-ROFSs) is presented in this paper, which allows decision makers to provide their satisfying degrees and non-satisfying degrees to a given set of alternatives by an interval value. Some of its important operations such as: negation, union and intersection are also given. Based on these operations, the aggregation of IVq-ROFSs is also studied.

Keywords

Intuitionistic fuzzy set (IFS)interval-valued aggregation operator

1 Introduction

To deal with uncertain and vague situations, Zadeh [2] introduced fuzzy sets (FSs) in 1965 as the extension of classical sets. Then, Atanassov [3] in 1986 established intuitionistic fuzzy set (IFS) and this new theory has paid more attention to practitioner Xu and Yager [4], Xu [5], Chen and Chang [6, 7], Joshi [8], Xu and Gou [9], Liu and Chen [10] and etc. Yager [1] introduced a new concept called it q-rung orthopair fuzzy sets (q-ROFSs), in which the sum of the q^th exponent of the support for membership and the q^th exponent of the support against membership is bounded to one, and further proved that the q-ROFS is more general because IFS and PFS are all its special cases. We have to also note that as the rung q raises the space of acceptable orthopairs raises and thus provides the observers more liberty in expressing their belief in order to support for membership degree. Therefore, the q-ROFSs express a wider range of fuzzy information and are more flexible and more suitable tool to handle the uncertain environment. Yager and Alajlan [11] discussed basic properties of these q-ROFSs and use these sets in knowledge representation. Recently, Liu and Wang [12] presented the q-rung orthopair (fuzzy) weighted averaging (q-ROFWA) operator and the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator, and multi-criteria decision making (MCDM) problems are solved by using these operators.

However, while dealing with much real life decision-making problems under q-rung orthopair fuzzy environment, it is not appropriate for experts to precisely quantify their judgements with a crisp number due to insufficiency in the provided information. In such situation it is advisable for decision makers to provide their judgements by the subset of the closed interval [0, 1]. This can be achieved by introducing the idea of interval-valued q-rung orthopair fuzzy sets (IVq-ROFSs), which allow decision makers to provide their satisfying degrees and non-satisfying degrees to a given set of alternatives by an interval value. It can be easily analysed that under the equality conditionof the upper and lower limits of the interval values, IVq-ROFS becomes q-ROFS. Further, it also indicates that the latter one is a particularcondition of the former one. Some of its important operations such as: negation, union and intersection are also given. Based on these operations, the aggregation of IVq-ROFSs is also studied.

The remaining part of this work is organised as shown:

Section 2 briefly shows definition of q-ROFS. The idea of IVq-ROFSs is presents in section 3 along with some of its operations. How the IVq-ROFSs are combined is explained in section 4. Finally, the obtained key features are described in section 5 under conclusion heading.

2 q-rung orthopair fuzzy set [1]

A q-ROFSR in the universal set Y is explained in the form R ={〈 y, u_R (y) , v_R (y) 〉 : y ∈ Y }, where the functions u_R : Y → [0, 1] and v_R : Y → [0, 1] define the “degree of membership” and the “degree of non-membership” for member y ∈ Y respectively, with the restriction 0 ≤ u_R (y) ^q + v_R (y) ^q ≤ 1, (q ≥ 1). The non-determinacy index for each member y of Yin the q-ROFSR, is presented by π_R (y) = (1 - u_R (y) ^q - v_R (y) ^q) ^1/-q, where π_R (y) gives outputs in the closed interval from zero to one.

For convenience, 〈u_R (y) , v_R (y)〉 is called a q-rung orthopair fuzzy number (q-ROFN) and it can be put as a = 〈u_a, v_a〉. Liu and Wang [12] presented the following score and accuracy to measure two q-ROFNs.

Let a =〈 u_a, v_a 〉 and b =〈 u_b, v_b 〉 be two q-ROFNs, then $S (a) = u_{a}^{q} - v_{a}^{q}$ , $S (b) = u_{b}^{q} - v_{b}^{q}$ shows score functions and $H (a) = u_{a}^{q} + v_{a}^{q}, H (b) = u_{b}^{q} + v_{b}^{q}$ shows accuracy values for a and b. If score of a is greater than that of b, then conclusion is: a is superior to b, and is indicated by a > b. In the case of same scores, we need to calculate accuracy values and if H (a) > H (b) then a is superior to b, and is indicated by a > b. Under the equality of both score and accuracy values, the result represents that a and b indicate the identical values, hence indicated by a = b.

3 Interval-valued q-rung orthopair fuzzy sets

Let Closed ([0,1]) denote the set of all closed subintervals of [0,1]. An interval-valued q-rung orthopair fuzzy set (IVq-ROFS) A in the universal set X is defined as: A = {〈y, u_A (y) , v_A (y) 〉 : y ∈ Y}, where the functions u_A : X → Closed [0, 1] and v_A : X → Closed [0, 1] define the “degree of membership” and the “degree of non-membership” of the element y ∈ Y respectively. Then for every y ∈ Y, 0 ≤ sup {u_A (y) ^q} + sup {v_A (y) ^q} ≤1, (q ≥ 1). Here, for each element y ∈ Y the “degree of membership” and “degree of non-membership” are intervals rather than crisp numbers. When lower and upper bounds are indicated by $u_{A}^{-} (y), u_{A}^{+} (y), v_{A}^{-} (y), v_{A}^{+} (y)$ respectively, then A can be also expressed in the form $\begin{matrix} A = {〈 y, [u_{A}^{-} (y), u_{A}^{+} (y)], [v_{A}^{-} (y), v_{A}^{+} (y)] 〉 \\ : y \in Y}, \end{matrix}$ where the expression is subjected to the restriction $0 \leq {u_{A}^{+} (y)}^{q} + {v_{A}^{+} (y)}^{q} \leq 1, (q \geq 1)$ . The non-determinacy index for each element yof Yin the IVq-ROFSA, is described by $\begin{matrix} π_{A} (y) = [π_{A}^{-} (y), π_{A}^{+} (y)] \\ = [(1 - u_{A}^{+} (y)^{q} - v_{A}^{+} (y)^{q})^{1 / - q}, \\ (1 - u_{A}^{-} (y)^{q} - v_{A}^{-} (y)^{q})^{1 / - q}] . \end{matrix}$

For convenience throughout in this paper, we call $a = ([u_{a}^{-}, u_{a}^{+}], [v_{a}^{-}, v_{a}^{+}])$ as an interval-valued q-rung orthopair fuzzy number (IVq-ROFN). It is true that IVq-ROFS can be reduced in q-ROFN when upper and lower bounds are same in membership and non-membership degrees.

3.1 Negation (Complement) operation

In order to find the negation or complement of an IVq-ROFN we first recall the definition of complement operator as proposed by Klir and Yuan [13]. A negation operator Negg is a function Negg : [0, 1] → [0, 1] which has the following properties:

Boundedness: Negg(0) = 1 and Negg(1) = 0

Inverse Monotonicity: If a, b are any two numbers from zero to one and if a is less than or equal to b then, Negg(a)≥Negg(b)

Involution: Negg(Negg(a)) is equal to the number a itself and Continuity.

For instance, the linear function Negg (a) =1?a shows a perfect example for complement operator. A family of complement operators is introduced by Yager [14, 15] as Negg (a) = (1 - a^q) ^1/-q where q ∈ (0, ∞). Here, we examine that for q = 1, we get classic linear complement operator Negg (a) =1?a which is basically intuitionistic fuzzy complement, and for q = 2, we find the Pythagorean fuzzy complement $Negg (a) = \sqrt{1 - a^{2}}$ . We also observe that for q ≥ 1, we find Negg (a) = (1 - a^q) ^1/-q and call it q-rung orthopair fuzzy complement. In the observation of Yager [1, 16], we proposed a related idea of complement with respect to s as Negg_q,s : [0, s] → [0, s] where s ∈ [0, 1] if it satisfies the followingproperties:

Boundedness: Negg_q,s (0) = s and Negg_q,s (s) =0

Inverse Monotonicity: If a, b are any two numbers from zero to one and if a is less than or equal to b then, Negg_q,s (a) ≥ Negg_q,s (b).

Involution: Negg_q,s (Negg_q,s (a)) = a and Continuity.

Now in order to extend the concept of set complement operator, let assume a q-rung orthopair fuzzy membership A = (u_A (y) , v_A (y)) where {u_A (y)} ^q + {v_A (y)} ^q ≤ 1, (q ≥ 1). A more fundamental explaining for this operation which can be from recalling strength for commitment as s (y) = ({u_A (y)} ^q + {v_A (y)} ^q) ^1/-q.

Let we define $\bar{A} = ({\bar{u}}_{A} (y), {\bar{v}}_{A} (y))$ as the complement of A = (u_A (u) , v_A (y)) such that ${\bar{u}}_{A} (y) = {({s (y)}^{q} - {v_{A} (y)}^{q})}^{1 / - q}$ and ${\bar{v}}_{A} (y) = {({s (y)}^{q} - {v_{A} (y)}^{q})}^{1 / - q}$ where {s (y)} ^q = {u_A (y)} ^q + {v_A (y)} ^q. Then, ${\bar{u}}_{A} (y)$ and ${\bar{v}}_{A} (y)$ are called the q-rung orthopair fuzzy complements of u_A (y) and v_A (y) respectively with respect to s(x). Furthermore, ${\bar{u}}_{A} (y) = {({s (y)}^{q} - {u_{A} (y)}^{q})}^{1 / - q}$ $= {({u_{A} (y)}^{q} + {v_{A} (y)}^{q} - {u_{A} (y)}^{q})}^{1 / - q}$ $= {({v_{A} (y)}^{q})}^{1 / - q}$ $\Leftrightarrow {\bar{u}}_{A} (y) = v_{A} (y) .$

Similarly, we have ${\bar{v}}_{A} (y) = u_{A} (y)$ . Thus, the complement of a q-rung orthopair fuzzy A = (u_A (y) , v_A (y)) is $\bar{A} = (v_{A} (y), u_{A} (y))$ i.e. complement is obtained by simply interchanging values of “degree of membership” and “degree of non-membership”.

In an analogues manner, $\bar{A} = ([{\bar{u}}_{A}^{-}, {\bar{u}}_{A}^{+}], [{\bar{v}}_{A}^{-}, {\bar{v}}_{A}^{+}])$ is called the complement of an interval-valued q-rung orthopair fuzzy number $A = ([t_{A}^{-}, t_{A}^{+}]$ , $[f_{A}^{-}, f_{A}^{+}])$ if ${\bar{u}}_{A}^{-} = {({s^{-}}^{q} - {u_{A}^{-}}^{q})}^{1 / - q}$ , ${\bar{u}}_{A}^{+} = {({s^{+}}^{q} - {u_{A}^{+}}^{q})}^{1 / - q}$ , ${\bar{v}}_{A}^{-} = {({s^{-}}^{q} - {v_{A}^{-}}^{q})}^{1 / - q}$ and ${\bar{v}}_{A}^{+} = {({s^{+}}^{q} - {v_{A}^{+}}^{q})}^{1 / - q}$ , where ${s^{-}}^{q} = {u_{A}^{-}}^{q} +$ ${v_{A}^{-}}^{q}$ and ${s^{+}}^{q} = {u_{A}^{+}}^{q} + {v_{A}^{+}}^{q}$ .

On simplifying, we have $\begin{matrix} {\bar{u}}_{A}^{-} = {({s^{-}}^{q} - {u_{A}^{-}}^{q})}^{1 / - q} \\ = {({u_{A}^{-}}^{q} + {v_{A}^{-}}^{q} - {u_{A}^{-}}^{q})}^{1 / - q} = {({v_{A}^{-}}^{q})}^{1 / - q} \end{matrix}$ $\Leftrightarrow {\bar{u}}_{A}^{-} = v_{A}^{-}$

Similarly, we can find ${\bar{u}}_{A}^{+} = v_{A}^{+}, {\bar{v}}_{A}^{-} = u_{A}^{-}$ and ${\bar{v}}_{A}^{+} = u_{A}^{+}$ . Thus, the interval-valued q-rung orthopair fuzzy complement is obtained by simply interchanging interval-values of membership and for non-membership degree.

Theorem 1. If A shows IVq₁-ROFS in X and further if q₂ > q₁ then A is also an IVq₂-ROFS in X.

Proof: Since, $A = {\begin{matrix} 〈 y, [u_{A}^{-} (y), u_{A}^{+} (y)], \\ [v_{A}^{-} (y), v_{A}^{+} (y)] 〉 : y \in y \end{matrix}}$ is an IVq₁-ROFS, then $u_{A}^{+} (y)^{q_{1}} + v_{A}^{+} (y)^{q_{1}} \leq 1, (q_{1} \geq 1)$ . Therefore, for q₂ > q₁, we have $u_{A}^{+} (y)^{q_{1}} + v_{A}^{+} (y)^{q_{1}} \leq u_{A}^{+} (y)^{q_{2}} + v_{A}^{+} (y)^{q_{2}} \leq 1 .$

Thus, the following conclusions are made:

Every interval-valued intuitionistic fuzzy set shows an IVq-ROFS for all q≥1.

Every interval-valued intuitionistic fuzzy set is an interval-valued Pythagorean fuzzy set.

Every interval-valued Pythagorean fuzzy subset showsIVq-ROFS for q≥2.

If A is an IVq₂-ROFS and if q₂ > q₁ then A is not necessarily an IVq₁-ROFS.

If q₂ > q₁, then liberty of choosing membership grade in IVq₂-ROFS is larger than that of space allowed in IVq₁-ROFS.

3.2 Basic set operations for IVq-ROFSs

In order to combine IVq-ROFSs, consider two IVq-ROF membership grades $A = ([u_{A}^{-}, u_{A}^{+}], [v_{A}^{-}, v_{A}^{+}])$ and $B = ([u_{B}^{-}, u_{B}^{+}], [v_{B}^{-}, v_{B}^{+}])$ , where ${u_{A}^{+}}^{q} + {v_{A}^{+}}^{q} \leq 1$ and ${u_{B}^{+}}^{q} + {v_{B}^{+}}^{q} \leq 1$ . If ${u_{A}^{+}}^{q} + {v_{A}^{+}}^{q} = {s_{A}^{+}}^{q} \leq 1$ and ${u_{B}^{+}}^{q} + {v_{B}^{+}}^{q} = {s_{B}^{+}}^{q} \leq 1$ , then we can have ${u_{A}^{-}}^{q} + {v_{A}^{-}}^{q} = {s_{A}^{-}}^{q} \leq 1$ and ${u_{B}^{-}}^{q} + {v_{B}^{-}}^{q} = {s_{B}^{-}}^{q} \leq 1$ . Now consider their intersection $C = A \cap B = ([u_{C}^{-}, u_{C}^{+}], [v_{C}^{-}, v_{C}^{+}])$ , where $u_{C}^{-} = min {u_{A}^{-}, u_{B}^{-}}$ , $u_{C}^{+} = min {u_{A}^{+}, u_{B}^{+}}$ , $v_{C}^{-} = max {v_{A}^{-}, v_{B}^{-}}$ and $v_{C}^{+} = max {v_{A}^{+}, v_{B}^{+}}$ . Now we have to show that C is also an IVq-ROF membership grades i.e. ${u_{C}^{+}}^{q} + {v_{C}^{+}}^{q} \leq 1$ , for this $\begin{matrix} (v_{C}^{+})^{q} = max {(v_{A}^{+})^{q}, (v_{B}^{+})^{q}} \\ = max {(s_{A}^{+})^{q} - (u_{A}^{+})^{q}, (s_{B}^{+})^{q} - (u_{B}^{+})^{q}} \end{matrix}$ $\leq max {\begin{matrix} (s_{A}^{+})^{q} - min {(u_{A}^{+})^{q}, (u_{B}^{+})^{q}}, \\ (s_{B}^{+})^{q} - min {(u_{A}^{+})^{q}, (u_{B}^{+})^{q}} \end{matrix}}$ $\leq \max {1 - \min {{(u_{A}^{+})}^{q}, {(u_{B}^{+})}^{q}}, 1 - \min {{(u_{A}^{+})}^{q}, {(u_{B}^{+})}^{q}}}$ $\leq 1 - min {(u_{A}^{+})^{q}, (u_{B}^{+})^{q}} = 1 - (u_{C}^{+})^{q}$ $(u_{C}^{+})^{q} + (v_{C}^{+})^{q} \leq 1 .$

Let $D = A \cup B = ([u_{D}^{-}, u_{D}^{+}], [v_{D}^{-}, v_{D}^{+}])$ , where $u_{D}^{-} = max {u_{A}^{-}, u_{B}^{-}}, u_{D}^{+} = max {u_{A}^{+}, u_{B}^{+}}, v_{D}^{-} = min {v_{A}^{-}, v_{B}^{-}}$ and $v_{D}^{+} = min {v_{A}^{+}, v_{B}^{+}}$ . In the analogous manner we can easily verify that D is also an IVq-ROF membership grades i.e. ${u_{D}^{+}}^{q} + {v_{D}^{+}}^{q} \leq 1$ .

4 Aggregation of IVq-ROFNs

In order to develop aggregation operator for IVq-ROFNs, we first look the concepts of operator (aggregation) and aggregation operator (dual) presented in Beliakov et al. [17] and Grabisch et al. [18].

Definition: Mapping Aggre [0, 1] ^m → [0, 1] shows an aggregation operator if

Aggre (0, …, 0) =0 and Aggre (1, …, 1) =1 for boundary condition

Monotonicity: If a_i ≥ b_i (∀ i), then Aggre (a₁, …, a_m) ≥ Aggre (b₁, …, b_m).

Definition: Let Aggre is said as aggregation operator for the unit interval i.e. Aggre [0, 1] ^m → [0, 1] then the dual of aggregation operator Aggre is defined as $\begin{matrix} Aggre (a_{1}, . . ., a_{m}) \\ = Negg (Aggre (Negg (a_{1}), . . ., Negg (a_{m})) \end{matrix}$

Where Negg is a negation (complement) operator.

Definition: Let A_i (i = 1, …, m) is a value form IVq-ROFSs with grades $A_{i} = ([u_{A_{i}}^{-}, u_{A_{i}}^{+}], [v_{A_{i}}^{-}, v_{A_{i}}^{+}])$ where ${u_{A_{i}}^{+}}^{q} + {v_{A_{i}}^{+}}^{q} \leq 1$ . We define E = Aggre (A₁, …, A_m) as an IVq-ROFS with membership grades $E = ([u_{E}^{-}, u_{E}^{+}], [v_{E}^{-}, v_{E}^{+}])$ such that $u_{E}^{-} = Aggre (u_{A_{1}}^{-}, u_{A_{2}}^{-}, . . ., u_{A_{m}}^{-}), u_{E}^{+} = Aggre (u_{A_{1}}^{+}, u_{A_{2}}^{+}, . . ., u_{A_{m}}^{+}), v_{E}^{-} = Aggre (v_{A_{1}}^{-}, v_{A_{2}}^{-}$ , $. . ., v_{A_{m}}^{-})$ and $v_{E}^{+} = Aggre (v_{A_{1}}^{+}, v_{A_{2}}^{+}, . . ., v_{A_{m}}^{+})$ .

As we know the Aggre operator is closed i.e. it maps the collection of IVq-ROFSs into an IVq-ROFS. This can be achieved by proving the following theorem using the monotonicity.

Theorem:IfE = Aggre (A₁, …, A_m) with grade $E = ([u_{E}^{-}, u_{E}^{+}], [v_{E}^{-}, v_{E}^{+}])$ then ${u_{E}^{+}}^{q} + {v_{E}^{+}}^{q} \leq 1$ is always hold.

Proof: For our convenience, we shall write the pair of intervals $([u_{A_{i}}^{-}, u_{A_{i}}^{+}], [v_{A_{i}}^{-}, v_{A_{i}}^{+}])$ as ([a_i, b_i] , [c_i, d_i]) where $b_{i}^{q} + d_{i}^{q} \leq 1$ then $\begin{matrix} t_{E}^{+} = Aggre (b_{1}, b_{2}, . . ., b_{m}) \\ \leq Aggre ((1 - d_{1}^{q})^{1 / - q}, (1 - d_{2}^{q})^{1 / - q}, . . ., (1 - d_{m}^{q})^{1 / - q}) \end{matrix}$ $\begin{matrix} f_{E}^{+} = Aggre (d_{1}, d_{2}, . . ., d_{m}) \\ = {(\begin{matrix} 1 - (Aggre ((1 - d_{1}^{q})^{1 / - q}, \\ (1 - d_{2}^{q})^{1 / - q}, . . ., (1 - d_{m}^{q})^{1 / - q})^{q}) \end{matrix})}^{1 / - q} \end{matrix}$ .

Therefore, $\begin{matrix} {u_{E}^{+}}^{q} + {v_{E}^{+}}^{q} \\ {\leq (\begin{matrix} Aggre ((1 - d_{1}^{q})^{1 / - q}, (1 - d_{2}^{q})^{1 / - q}, \\ . . ., (1 - d_{m}^{q})^{1 / - q}) \end{matrix})}^{q} \end{matrix}$ ${+ ({(\begin{matrix} 1 - (Aggre ((1 - d_{1}^{q})^{1 / - q}, (1 - d_{2}^{q})^{1 / - q}, \\ . . ., (1 - d_{m}^{q})^{1 / - q})^{q}) \end{matrix})}^{1 / - q})}^{q}$ $\Rightarrow {u_{E}^{+}}^{q} + {v_{E}^{+}}^{q} \leq 1 .$

Thus it has been seen that the aggregated interval value of IVq-ROFSsobtained through E = Aggre (A₁, …, A_m) is also an IVq-ROFS. Which verifies that the Aggre operator is closed.

A number of important aggregation operators were considered by Beliakov et al. [17] in their book “Aggregation Functions: A Guide for Practitioners”. Before going to extend these aggregation operators for IVq-ROFSs, Conjunctive and disjunctive operators are the two groups. Conjunctive type operator is aggregation operator when Aggre (a₁, a₂, …, a_m) ≤ min(a₁, a₂, …, a_m) and is said to be a disjunctive type operator if Aggre (a₁, a₂, …, a_m) ≥ max(a₁, a₂, …, a_m). The conjunctive operator takes a broad view to the intersection set “and” disjunctive operator is a operator which takes a broad view to the set union “or” operator. Further it is said that when Aggre is a conjunctive type operator then Aggre is a disjunctive and when Aggre is a disjunctive then Aggre is of conjunctive type. Beliakov et al. [17] have shown that if Aggre is a t-norm then Aggre is a t-conorm and vice-versa. Certain conditions are defined for aggregation operators $\begin{matrix} min (a_{1}, a_{2}, . . ., a_{m}) \\ \leq Aggre (a_{1}, a_{2}, . . ., a_{m}) \leq max (a_{1}, a_{2}, . . ., a_{m}) . \end{matrix}$

Aggre is a mean type operator, when Aggre is operator (mean).

Certain aggregation operator are considered for IVq-ROFSs derived by considering the weighted power means presented by Beliakov et al. [17], then $Aggre (a_{1}, a_{2}, . . ., a_{m}) = {(\sum_{i = 1}^{m} w_{i} a_{i}^{q})}^{1 / - q}$

and its dual is $\begin{matrix} Aggre (a_{1}, . . ., a_{m}) \\ = Negg (Aggre (Negg (a_{1}), . . ., Negg (a_{m})) \\ = {(1 - \sum_{i = 1}^{m} w_{i} (1 - a_{i}^{q}))}^{1 / - q} . \end{matrix}$

Where w_i (i = 1, 2, …, m) is the value for a_i (i = 1, …, m) such that $\sum_{i = 1}^{n} w_{i} = 1$ .

Let $A_{i} = ([u_{A_{i}}^{-}, u_{A_{i}}^{+}], [v_{A_{i}}^{-}, v_{A_{i}}^{+}]) (i = 1, . . ., m)$ is value for of m IVq-ROFSs with w_i (i = 1, 2, …, m) be the weight where ${u_{A_{i}}^{+}}^{q} + {v_{A_{i}}^{+}}^{q} \leq 1$ and $\sum_{i = 1}^{n} w_{i} = 1$ . Then, their aggregated value $E = Aggre (A_{1}, . . ., A_{m}) = ([u_{E}^{-}, u_{E}^{+}], [v_{E}^{-}, v_{E}^{+}])$ obtained by using the power means is an IVq-ROFS.Where $\begin{matrix} u_{E}^{-} = Aggre (u_{A_{1}}^{-}, u_{A_{2}}^{-}, . . ., u_{A_{m}}^{-}) \\ = {(\sum_{i = 1}^{m} w_{i} (u_{A_{i}}^{-})^{q})}^{1 / - q}, \end{matrix}$ $\begin{matrix} u_{E}^{+} = Aggre (u_{A_{1}}^{+}, u_{A_{2}}^{+}, . . ., u_{A_{m}}^{+}) \\ = {(\sum_{i = 1}^{m} w_{i} (u_{A_{i}}^{+})^{q})}^{1 / - q}, \end{matrix}$ $\begin{matrix} v_{E}^{-} = Aggre (v_{A_{1}}^{-}, v_{A_{2}}^{-}, . . ., v_{A_{m}}^{-}) \\ = {(1 - \sum_{i = 1}^{m} w_{i} (1 - (v_{A_{i}}^{-})^{q}))}^{1 / - q} \end{matrix}$ and $\begin{matrix} f_{E}^{+} = Aggre (v_{A_{1}}^{+}, v_{A_{2}}^{+}, . . ., v_{A_{m}}^{+}) \\ = {(1 - \sum_{i = 1}^{m} w_{i} (1 - (v_{A_{i}}^{-})^{q}))}^{1 / - q} . \end{matrix}$

Thus, $\begin{matrix} E = Aggre (A_{1}, . . ., A_{m}) \\ = (\begin{matrix} [{(\sum_{i = 1}^{m} w_{i} (u_{A_{i}}^{-})^{q})}^{1 / - q}, {(\sum_{i = 1}^{m} w_{i} (u_{A_{i}}^{+})^{q})}^{1 / - q}], \\ [\begin{matrix} {(1 - \sum_{i = 1}^{m} w_{i} (1 - (v_{A_{i}}^{-})^{q}))}^{1 / - q}, \\ {(1 - \sum_{i = 1}^{m} w_{i} (1 - (v_{A_{i}}^{+})^{q}))}^{1 / - q} \end{matrix}] \end{matrix}) \end{matrix}$

The concept of IVq-ROFSs gives the flexibility for taking different membership indexes in a subsets of the interval [0, 1], which is not carried in q-ROFSs. This indicates that the IVq-ROFS is generalized form of q-ROFS.

5 Conclusion

Here, certain is defined for interval-valued q-rung orthopair sets (Fuzzy) (IVqROFSs) is presented, which allow decision makers to provide their satisfying degrees and non-satisfying degrees to a given set of alternatives by an interval value. Some of its important operations such as: negation, union and intersection are also given. Based on these operations, the aggregation of IVqROFSs is also studied. Present theories will get some addition for solving with uncertainties, and getting certain fields for future.

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