Some induced aggregation operators based on interval-valued Pythagorean fuzzy numbers

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Abstract

Interval-valued Pythagorean fuzzy set is one of the successful extensions of the interval-valued intuitionistic fuzzy set for handling the uncertainties in the data. Under this environment, in this paper, induced interval-valued Pythagorean fuzzy ordered weighted averaging aggregation operator and induced interval-valued Pythagorean fuzzy hybrid averaging aggregation operator have been introduced a long with their desirable properties namely, idempotency, boundedness and monotonicity. The main advantage of using the proposed methods and operators is that these operators and methods give a complete view of the problem to the decision makers. These methods provide more general, more accurate and precise results as compared to the existing methods. Therefore, these methods play a vital role in real world problems. Finally, the proposed operators have been applied to decision making problems to show the validity, practicality and effectiveness of the new approach. At the end of application, we have considered an example for the section of a television from different televisions.

Keywords

I-IVPFOWA aggregation operator I-IVPFHA aggregation operator Decision making 

1 Introduction

Fuzzy theory is an important tool to process fuzzy information. Zadeh (1965) first proposed the fuzzy set theory, and then Atanassov (1986) proposed the intuitionistic fuzzy set by adding a non-membership function. Later on, Atanassov and Gargov (1989) extended it to the interval-valued intuitionistic fuzzy sets, which are characterized by a membership degree and a non-membership degree, whose values are intervals rather than real numbers. Over the last four decades, the intuitionistic fuzzy set and interval-valued intuitionistic fuzzy sets have received more and more attention by introducing the various kinds of aggregation operators, information measures and employed them to solve the decision-making problems under the different environments (Chen et al. 2001, 2012a, b; Chen and Li 2013; Chen and Chiou 2015; Chen and Tsai 2016; Chen and Huang 2017a, b; Horng et al. 2005; Chen and Kao 2013; Chen 1996; Chen and Chung 2006; Chen and Chang 2001; Chen and Han 2018) Like the above mention aggregation operators and methods a large number of aggregation operators have been published (Liu 2017; Liu et al. 2017a, b, 2018a, b; Liu and Li 2017; Wang and Chen 2017a, b; Pedrycz and Chen 2011, 2015a, b). Su et al. introduced the notion of induced aggregation operators and applied the to group decision making.

But the limitation of their studies is that they are valid only for those environments whose degrees sum is less than one. However, in day-to-day life, there are many situations where this condition is ruled out. For instance, if a person gives their preference in the form of membership and non-membership degrees towards a particular object is 0.8 and 0.6, and then clearly this situation is not handling with intuitionistic fuzzy set. To resolve it, Yager (2013, 2014) proposed the Pythagorean fuzzy set by relaxing this sum condition to its square sum less than one. For instance, corresponding to the above-considered example, we see that (0.8)2 + (0.6)2 = 1 and hence PFS is an extension of the existing IFS. After their pioneer work, Yager and Abbasov (2013) studied the relationship between the Pythagorean numbers and the complex numbers. Xu (2010), Tan and Chen (2010), used the Choquet integral to develop some intuitionistic fuzzy aggregation operators, which not only consider the importance of the elements or their ordered positions, but also can reflect the correlations among the elements or their ordered positions. Peng and Yang (2015a) developed some important results for Pythagorean fuzzy sets. Garg (2016a, b, 2017) used the Einstein sum and Einstein product and introduced the notion of Pythagorean fuzzy Einstein arithmetic aggregation operators and Pythagorean fuzzy Einstein geometric aggregation operators such as Pythagorean fuzzy Einstein weighted averaging operator, Pythagorean fuzzy Einstein ordered weighted averaging operator, generalized Pythagorean fuzzy Einstein weighted averaging operator, generalized Pythagorean fuzzy Einstein ordered weighted averaging operator, Pythagorean fuzzy Einstein weighted geometric operator, Pythagorean fuzzy Einstein ordered weighted geometric operator, generalized Pythagorean fuzzy Einstein weighted geometric operator, generalized Pythagorean fuzzy Einstein ordered weighted geometric operator and also applied them to group decision making.

But, in some real decision-making problems, due to insufficiency in available information, it may be difficult for decision makers to exactly quantify their opinions with a crisp number, but they can be represented by an interval number within [0, 1]. Therefore, it is so important to present the idea of interval-valued Pythagorean fuzzy sets, which permit the membership degrees and non-membership degrees to a given set to have an interval value. Peng and Yang (2015b) introduced the notion of, interval-valued Pythagorean fuzzy weighted averaging operator, interval-valued Pythagorean fuzzy weighted geometric operator and also introduced some of their fundamental and important properties. Rahman et al. (2017, 2018) introduced the notion of interval-valued Pythagorean fuzzy ordered weighted averaging (I-IVPFOWA) operators, interval-valued Pythagorean fuzzy hybrid weighted averaging operator, interval-valued Pythagorean fuzzy ordered weighted geometric operators, interval-valued Pythagorean fuzzy hybrid weighted geometric operator, interval-valued Pythagorean fuzzy Einstein aggregation operators and applied them to multiple attribute group decision making.

Thus, keeping the advantages of the above mention aggregation operators in this paper, we introduce the notion of two new induced aggregation operators based on interval-valued Pythagorean fuzzy numbers, such as induced I-IVPFOWA aggregation operator, induced interval-valued Pythagorean fuzzy hybrid averaging (I-IVPFHA) aggregation operator and apply them to group decision making. We also discuss some of their basic properties including idempotency, boundedness, commutativity and monotonicity. We also give some examples to develop these proposed operators.

The remainder paper can be constructed as. In Sect. 2, we present some basic definition and results which will be used in our later sections. In Sect. 3, we develop some algebraic operations and relations. In Sect. 4, we develop I-IVPFOWA aggregation operator and I-IVPFHA aggregation operator. In Sect. 5, we develop the advantage of the proposed operators. In Sect. 6, we have conclusion.

2 Preliminaries

Definition 1

(Peng and Yang 2015a and Garg 2016a, b) Let K be a universal set, then an interval-valued Pythagorean fuzzy sets, I in K can be defined as:
$$I=\left\{ {\left\langle {k,{\mu _I}\left( k \right),{v_I}\left( k \right)} \right\rangle |k \in K} \right\},$$
(1)
where
$${\mu _I}\left( k \right)=\left[ {\mu _{I}^{a}\left( k \right),\mu _{I}^{b}\left( k \right)} \right] \subset \left[ {0,1} \right],$$
(2)
$${v_I}\left( k \right)=\left[ {v_{I}^{a}\left( k \right),v_{I}^{b}\left( k \right)} \right] \subset \left[ {0,1} \right].$$
(3)
Also
$$\mu _{I}^{a}\left( k \right)=\mathop {\inf }\limits_{}^{} \left( {{\mu _I}\left( k \right)} \right)$$
(4)
$$\mu _{I}^{b}\left( k \right)=\mathop {\sup }\limits_{}^{} \left( {{\mu _I}\left( k \right)} \right)$$
(5)
$$v_{I}^{a}\left( k \right)=\mathop {\inf }\limits_{}^{} \left( {{v_I}\left( k \right)} \right)$$
(6)
$$v_{I}^{b}\left( k \right)=\mathop {\sup }\limits_{}^{} \left( {{v_I}\left( k \right)} \right),$$
(7)
and
$$0 \leqslant {\left( {\mu _{I}^{b}\left( k \right)} \right)^2}+{\left( {v_{I}^{b}\left( k \right)} \right)^2} \leqslant 1.$$
(8)
Also
$${\pi _I}\left( k \right)=\left[ {\pi _{I}^{a}\left( k \right),\pi _{I}^{b}\left( k \right)} \right],{\text{ for all }}k \in K.$$
(9)

Definition 2

(Peng and Yang 2015a and Garg 2016a, b) Let \(\lambdabar =\left( {\left[ {{p_\lambdabar },{q_\lambdabar }} \right],\left[ {{r_\lambdabar },{t_\lambdabar }} \right]} \right)\) be interval-valued Pythagorean fuzzy number, then score function and accuracy function can be defined as following, respectively:
$$S\left( \lambdabar \right)=\frac{1}{2}\left[ {{{\left( {{p_\lambdabar }} \right)}^2}+{{\left( {{q_\lambdabar }} \right)}^2} - {{\left( {{r_\lambdabar }} \right)}^2} - {{\left( {{t_\lambdabar }} \right)}^2}} \right],$$
(10)
and
$$H\left( \lambdabar \right)=\frac{1}{2}\left[ {{{\left( {{p_\lambdabar }} \right)}^2}+{{\left( {{q_\lambdabar }} \right)}^2}+{{\left( {{r_\lambdabar }} \right)}^2}+{{\left( {{t_\lambdabar }} \right)}^2}} \right],$$
(11)

If \({\lambdabar _1},{\lambdabar _2}\) be two interval-valued Pythagorean fuzzy number, then the following conditions hold

  1. 1.

    If \(S({\lambdabar _1}) \prec S({\lambdabar _2}),\) then \({\lambdabar _1} \prec {\lambdabar _2}\),

     
  2. 2.

    If \(S({\lambdabar _1})=S({\lambdabar _2}),\) then

     
  3. 1.

    If \(H({\lambdabar _1})=H({\lambdabar _2}),\) then \({\lambdabar _1}={\lambdabar _2},\)

     
  4. 2.

    If \(H({\lambdabar _1}) \prec H({\lambdabar _2}),\) then \({\lambdabar _1} \prec {\lambdabar _2},\)

     
  5. 3.

    If \(H({\lambdabar _1}) \succ H({\lambdabar _2}),\) then \({\lambdabar _1} \succ {\lambdabar _2},\)

     

Definition 3

(Su et al. 2011) Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuple, then an I-IVIFOWA operator can be defined as:
$$\begin{aligned} & {\text{I-IVIFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( \begin{gathered} \left[ {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {p_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {q_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right), \\ \end{aligned}$$
(12)

where \({\lambdabar _{\sigma \left( j \right)}}\) is the jth largest value of \({\lambdabar _{\sigma \left( j \right)}}\) and \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) be the weighted vector of \({\lambdabar _{\sigma \left( j \right)}}\left( {j=1,2,3,...,n} \right)\).

Example 1

Let
$$\begin{gathered} \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.7,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.4} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.5} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
and \(\varpi ={\left( {0.1,0.2,0.3,0.4} \right)^T}\)be the weighted vector of \({\lambdabar _j}\,\left( {j=1,2,3,4} \right).\)
Performing the ordering of the IVIFOWA pairs with respect to the first component, we have
$$\begin{gathered} \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.5} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.7,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.4} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle . \hfill \\ \end{gathered}$$
This ordering includes the ordered interval-valued intuitionistic fuzzy arguments
$$\begin{gathered} \left\langle {{u_{\sigma \left( 1 \right)}},{\lambdabar _{\sigma \left( 1 \right)}}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.5} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 2 \right)}},{\lambdabar _{\sigma \left( 2 \right)}}} \right\rangle =\left\langle {0.7,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 3 \right)}},{\lambdabar _{\sigma \left( 3 \right)}}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 4 \right)}},{\lambdabar _{\sigma \left( 4 \right)}}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.4} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle . \hfill \\ \end{gathered}$$
Thus,
$$\begin{aligned} & {\text{I}}-{\text{IVIFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,\left\langle {{u_4},{\lambdabar _4}} \right\rangle } \right) \\ & \quad =\left( \begin{gathered} \left[ {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {p_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {q_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{4} {{\left( {{r_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{4} {{\left( {{t_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right) \\ & \quad =\left( {\left[ {0.341,0.518} \right],\left[ {0.247,0.401} \right]} \right) \\ \end{aligned}$$

Definition 4

(Su et al. 2011) Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuple, then an I-IVIFHA operator can be defined as:
$$\begin{aligned} & {\text{I-IVIFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( \begin{gathered} \left[ {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {p_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {q_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right) \\ \end{aligned}$$
(13)
where \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\)be the weighted vector of \({\lambdabar _j}\left( {j=1,2,3,...,n} \right)\).

Example 2

Let
$$\begin{gathered} \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.5} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.3,0.5} \right],\left[ {0.2,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.6,\left( {\left[ {0.3,0.4} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.5} \right],\left[ {0.1,0.2} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
be the four IVIFVs and\(\varpi ={\left( {0.1,0.2,0.3,0.4} \right)^T}\), then we have
$$\begin{gathered} {{\dot {\lambdabar }}_1}=\left( {\left[ {0.132,0.242} \right],\left[ {0.617,0.693} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_2}=\left( {\left[ {0.381,0.425} \right],\left[ {0.275,0.480} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_3}=\left( {\left[ {0.348,0.458} \right],\left[ {0.235,0.333} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_4}=\left( {\left[ {0.558,0.670} \right],\left[ {0.025,0.076} \right]} \right). \hfill \\ \end{gathered}$$
Performing the ordering with respect to the first element, we have
$$\begin{gathered} \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.5} \right],\left[ {0.1,0.2} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.6,\left( {\left[ {0.3,0.4} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.3,0.5} \right],\left[ {0.2,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.5} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
Hence,
$$\begin{gathered} \left\langle {{u_{\sigma \left( 1 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 1 \right)}}} \right\rangle =\left( {\left[ {0.558,0.670} \right],\left[ {0.025,0.076} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 2 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 2 \right)}}} \right\rangle =\left( {\left[ {0.348,0.458} \right],\left[ {0.235,0.333} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 3 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 3 \right)}}} \right\rangle =\left( {\left[ {0.381,0.425} \right],\left[ {0.275,0.480} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 4 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 4 \right)}}} \right\rangle =\left( {\left[ {0.132,0.242} \right],\left[ {0.617,0.693} \right]} \right). \hfill \\ \end{gathered}$$
Thus,
$$\begin{gathered} I - {\text{IVIFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,\left\langle {{u_4},{\lambdabar _4}} \right\rangle } \right) \hfill \\ =\left( \begin{gathered} \left[ {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {p_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {q_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{4} {{\left( {{r_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{4} {{\left( {{t_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right) \hfill \\ =\left( {\left[ {0.307,0.399} \right],\left[ {0.289,0.429} \right]} \right) \hfill \\ \end{gathered}$$

3 Some basic algebraic operations and relations

Definition 5

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2} \right)\) be a collection of 2-tuple, and \(\delta \succ 0\), then the following operational laws always hold:
$$\delta {\lambdabar _1}=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_{{\lambdabar _1}}}} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_{{\lambdabar _1}}}} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_{{\lambdabar _1}}}} \right)}^\delta },{{\left( {{t_{{\lambdabar _1}}}} \right)}^\delta }} \right]} \right),$$
(14)
$${\left( {{\lambdabar _1}} \right)^\delta }=\left( {\left[ {{{\left( {{p_{{\lambdabar _1}}}} \right)}^\delta },{{\left( {{q_{{\lambdabar _1}}}} \right)}^\delta }} \right],\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{r_{{\lambdabar _1}}}} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {\left( {{t_{{\lambdabar _1}}}} \right)} \right)}^2}} \right)}^\delta }} } \right]} \right),$$
(15)
$${\lambdabar _1} \otimes {\lambdabar _2}=\left( {\left[ {{p_{{\lambdabar _1}}}{p_{{\lambdabar _2}}},{q_{{\lambdabar _1}}}{q_{{\lambdabar _2}}}} \right],\left[ {\begin{array}{*{20}{c}} {\sqrt {{{\left( {{r_{{\lambdabar _1}}}} \right)}^2}+{{\left( {{r_{{\lambdabar _2}}}} \right)}^2} - {{\left( {{r_{{\lambdabar _1}}}} \right)}^2}{{\left( {{r_{{\lambdabar _2}}}} \right)}^2}} ,} \\ {\sqrt {{{\left( {{t_{{\lambdabar _1}}}} \right)}^2}+{{\left( {{t_{{\lambdabar _2}}}} \right)}^2} - {{\left( {{t_{{\lambdabar _1}}}} \right)}^2}{{\left( {{t_{{\lambdabar _2}}}} \right)}^2}} } \end{array}} \right]} \right),$$
(16)
$${\lambdabar _1} \oplus {\lambdabar _2}=\left( {\left[ {\begin{array}{*{20}{c}} {\sqrt {{{\left( {{p_{{\lambdabar _1}}}} \right)}^2}+{{\left( {{p_{{\lambdabar _2}}}} \right)}^2} - {{\left( {{p_{{\lambdabar _1}}}} \right)}^2}{{\left( {{p_{{\lambdabar _2}}}} \right)}^2}} ,} \\ {\sqrt {{{\left( {{q_{{\lambdabar _1}}}} \right)}^2}+{{\left( {{q_{{\lambdabar _2}}}} \right)}^2} - {{\left( {{q_{{\lambdabar _1}}}} \right)}^2}{{\left( {{q_{{\lambdabar _2}}}} \right)}^2}} } \end{array}} \right],\left[ {{r_{{\lambdabar _1}}}{r_{{\lambdabar _2}}},{t_{{\lambdabar _1}}}{t_{{\lambdabar _2}}}} \right]} \right){\text{.}}$$
(17)

Remark 1

In the following, we introduce some special cases of \(\delta\) and \(\lambdabar\).

  1. 1.
    If \(\lambdabar =\left( {\left[ {p,q} \right],\left[ {t,r} \right]} \right)=\left( {\left[ {1,1} \right],\left[ {0,0} \right]} \right)\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( 1 \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( 1 \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( 0 \right)}^\delta },{{\left( 0 \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {1,1} \right],\left[ {0,0} \right]} \right) \\ \end{aligned}$$
     
  2. 2.
    If \(\lambdabar =\left( {\left[ {p,q} \right],\left[ {t,r} \right]} \right)=\left( {\left[ {0,0} \right],\left[ {1,1} \right]} \right)\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( 0 \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( 0 \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( 1 \right)}^\delta },{{\left( 1 \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {0,0} \right],\left[ {1,1} \right]} \right) \\ \end{aligned}$$
     
  3. 3.
    If \(\lambdabar =\left( {\left[ {p,q} \right],\left[ {t,r} \right]} \right)=\left( {\left[ {0,0} \right],\left[ {0,0} \right]} \right)\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( 0 \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( 0 \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( 0 \right)}^\delta },{{\left( 0 \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {0,0} \right],\left[ {0,0} \right]} \right) \\ \end{aligned}$$
     
  4. 4.
    If\(\delta \to 0\), \(0 \leqslant \left[ {p,q} \right],\left[ {t,r} \right] \leqslant 1\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^0}} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^0}} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^0},{{\left( {{t_\lambdabar }} \right)}^0}} \right]} \right) \\ &=\left( {\left[ {0,0} \right],\left[ {1,1} \right]} \right) \\ \end{aligned}$$
     
  5. 5.
    If \(\delta \to +\infty\), \(0 \leqslant \left[ {p,q} \right],\left[ {t,r} \right] \leqslant 1\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\infty }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\infty }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\infty },{{\left( {{t_\lambdabar }} \right)}^\infty }} \right]} \right) \\ &=\left( {\left[ {1,1} \right],\left[ {0,0} \right]} \right) \\ \end{aligned}$$
     
  6. 6.
    If\(\delta =1\), \(0 \leqslant \left[ {p,q} \right],\left[ {t,r} \right] \leqslant 1\), then
    $$\begin{aligned} \delta \lambdabar &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^\delta }} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^\delta }} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^\delta },{{\left( {{t_\lambdabar }} \right)}^\delta }} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - {{\left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)}^1}} ,\sqrt {1 - {{\left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)}^1}} } \right],\left[ {{{\left( {{r_\lambdabar }} \right)}^1},{{\left( {{t_\lambdabar }} \right)}^1}} \right]} \right) \\ &=\left( {\left[ {\sqrt {1 - \left( {1 - {{\left( {{p_\lambdabar }} \right)}^2}} \right)} ,\sqrt {1 - \left( {1 - {{\left( {{q_\lambdabar }} \right)}^2}} \right)} } \right],\left[ {\left( {{r_\lambdabar }} \right),\left( {{t_\lambdabar }} \right)} \right]} \right) \\ &=\lambdabar \\ \end{aligned}$$
     

Theorem 1

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2} \right)\) be a collection of 2-tuples, and \(\delta \succ 0,\) then the following conditions hold:

  1. 1.

    If \(\alpha ={\lambdabar _1} \oplus {\lambdabar _2},\) then \(\alpha\) is also PFV,

     
  2. 2.

    If \(\beta ={\lambdabar _1} \otimes {\lambdabar _2},\) then \(\beta\) is also PFV,

     
  3. 3.

    If \(\gamma =\delta \left( \lambdabar \right),\) then \(\gamma\) is also PFV,

     
  4. 4.

    If \(\varphi ={\left( \lambdabar \right)^\delta },\) then \(\varphi\) is also PFV.

     

Proof

The proof is trivial, so it is omitted here.

4 Some induced interval-valued averaging aggregation operators

In this section, we introduce the notion of two induced interval-valued averaging aggregation operators such as induced I-IVPFOWA aggregation operator and I-IVPFHA aggregation operator. We also discuss some desirable properties of these propose operators such as idempotency, boundedness, commutatively, monotonicity.

4.1 Induced interval-valued Pythagorean fuzzy ordered weighted averaging aggregation operator

Definition 6

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuple, then an induced I-IVPFOWA aggregation operator can be define as:
$$\begin{aligned} & {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( \begin{gathered} \left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{p_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{q_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} } \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right), \\ \end{aligned}$$
(18)
where \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) is the weighted vector of \({\lambdabar _{\sigma \left( j \right)}}\left( {j=1,2,3,...,n} \right)\), with some conditions such as \({\varpi _j} \in \left[ {0,1} \right]\) and \(\mathop \sum \nolimits_{{j=1}}^{n} {\varpi _j}=1.\) Also \({\lambdabar _{\sigma \left( j \right)}}\) is the \({\lambdabar _j}\) value of the IVPFOWA pairs \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle\) having the jth largest \({u_j}\) and \({u_j}\) in \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle\) is referred to as the order inducing variable and \({\lambdabar _j}\) as the Pythagorean fuzzy argument variable.

Example 3

Let
$$\begin{gathered} \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.7} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.7} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.6,\left( {\left[ {0.3,0.8} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.9} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
be the four IVPFVs and let \(\varpi ={\left( {0.1,0.2,0.3,0.4} \right)^T}\) be the weighted vector of \({\lambdabar _j}\left( {j=1,2,3,4} \right)\).
Performing the ordering of the IPFOWA pairs with respect to the first component, we have
$$\begin{gathered} \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.9} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.6,\left( {\left[ {0.3,0.8} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.7} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.7} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
This ordering includes the ordered interval-valued intuitionistic fuzzy arguments
$$\begin{gathered} \left\langle {{u_{\sigma \left( 1 \right)}},{\lambdabar _{\sigma \left( 1 \right)}}} \right\rangle =\left\langle {0.8,\left( {\left[ {0.4,0.9} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 2 \right)}},{\lambdabar _{\sigma \left( 2 \right)}}} \right\rangle =\left\langle {0.6,\left( {\left[ {0.3,0.8} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 3 \right)}},{\lambdabar _{\sigma \left( 3 \right)}}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.4,0.6} \right],\left[ {0.3,0.7} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_{\sigma \left( 4 \right)}},{\lambdabar _{\sigma \left( 4 \right)}}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.7} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
Thus,
$$\begin{gathered} {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,\left\langle {{u_4},{\lambdabar _4}} \right\rangle } \right) \hfill \\ =\left( \begin{gathered} \left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {p^2}_{{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {q^2}_{{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} } \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{4} {{\left( {{r_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{4} {{\left( {{t_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right) \hfill \\ =\left( {\left[ {0.344,0.703} \right],\left[ {0.228,0.601} \right]} \right). \hfill \\ \end{gathered}$$

Theorem 2

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples, then their aggregated value using the I-IVPFOWA aggregation operator is also an interval-valued Pythagorean fuzzy value, and
$$\begin{aligned} & {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( \begin{gathered} \left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{p_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{q_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} } \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{\lambdabar _{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right), \\ \end{aligned}$$
(19)
where \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) is the weighted vector of \({\lambdabar _{\sigma \left( j \right)}}\left( {j=1,2,3,...,n} \right)\), with some conditions such as \({\varpi _j} \in \left[ {0,1} \right]\) and \(\mathop \sum \nolimits_{{j=1}}^{n} {\varpi _j}=1.\) Also \({\lambdabar _{\sigma \left( j \right)}}\) is the \({\lambdabar _j}\) value of the IVPFOWA pairs \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle\)having the jth largest \({u_j}\) and \({u_j}\) in \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle\) is referred to as the order inducing variable and \({\lambdabar _j}\) as the Pythagorean fuzzy argument variable.

Proof

Straightforward.

Theorem 3

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) and \(\left\langle {{u^ * }_{j},{\lambdabar ^ * }_{j}} \right\rangle \left( {j=1,2,...,n} \right)\) be two set of 2-tuples, then
$$\begin{aligned} & {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =1 - {\text{IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},\lambdabar _{1}^{ * }} \right\rangle ,\left\langle {{u_2},\lambdabar _{2}^{ * }} \right\rangle ,\left\langle {{u_3},\lambdabar _{3}^{ * }} \right\rangle ,...,\left\langle {{u_n},\lambdabar _{n}^{ * }} \right\rangle } \right), \\ \end{aligned}$$
(20)
where \(\left\langle {{u^ * }_{j},{\lambdabar ^ * }_{j}} \right\rangle \left( {j=1,2,...,n} \right)\) is a permutation of \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,...,n} \right)\) and \(\varpi ={\left( {{\varpi _1},{\varpi _2},...,{\varpi _n}} \right)^T}\)be the weighted vector with some conditions such as \({\varpi _j} \in \left[ {0,1} \right]\) and \(\mathop \sum \nolimits_{{j=1}}^{n} {\varpi _j}=1.\)

Proof

Straightforward.

Theorem 4

If \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples, where \({\lambdabar _{{\sigma _{\left( j \right)}}}}=\lambdabar\) for all j, then
$${\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right)=\lambdabar .$$
(21)

Proof

Straightforward.

Theorem 5

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples whose weighted vector is given by \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T},\) then
$${\lambdabar _{\hbox{min} }} \leqslant {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \leqslant {\lambdabar _{\hbox{max} }}{\text{,}}$$
(22)

where \({\lambdabar _{\hbox{max} }}=\mathop {\hbox{max} \left( {{\lambdabar _j}} \right)}\limits_{j} ,{\lambdabar _{\hbox{min} }}=\mathop {\hbox{min} \left( {{\lambdabar _j}} \right)}\limits_{j}\).

Proof

Straightforward.

Theorem 6

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,...,n} \right)\) and \(\left\langle {{u_j},{\lambdabar ^ * }_{j}} \right\rangle \left( {j=1,2,...,n} \right)\) where \({\lambdabar _j} \leqslant \lambdabar _{j}^{ * }\) for all j, then
$$\begin{aligned} & {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_3},{\lambdabar _3}} \right\rangle } \right) \\ & \quad \leqslant {\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {\left\langle {{u_1},\lambdabar _{1}^{ * }} \right\rangle ,\left\langle {{u_2},\lambdabar _{2}^{ * }} \right\rangle ,\left\langle {{u_3},\lambdabar _{3}^{ * }} \right\rangle ,...,\left\langle {{u_n},\lambdabar _{n}^{ * }} \right\rangle } \right). \\ \end{aligned}$$
(23)

Proof

Straightforward.

4.2 Induced interval-valued Pythagorean fuzzy hybrid averaging aggregation operator

Definition 7

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right),\) be a collection of 2-tuples, then an I-IVPFHA aggregation operator can be define as:
$$\begin{aligned} & {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( {\begin{array}{*{20}{c}} {\left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{p_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{q_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} } \right],} \\ {\left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right]} \end{array}} \right), \\ \end{aligned}$$
(24)
where \({\dot {\lambdabar }_{\sigma \left( j \right)}},\) is the jth largest of the weighted interval-valued Pythagorean fuzzy values\({\dot {\lambdabar }_{\sigma \left( j \right)}},\) \(\left( {{{\dot {\lambdabar }}_{\sigma \left( j \right)}}=n{\varpi _j}{\lambdabar _j}} \right)\), \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) is the weighted vector of \({\lambdabar _{\sigma \left( j \right)}}\) and also \({\varpi _j} \in \left[ {0,1} \right]\), \(\mathop \sum \nolimits_{{j=1}}^{n} {\varpi _j}=1,\) and n is the balancing coefficient, which plays a role of balance. If the vector \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) approaches to \({\left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},...,\frac{1}{n}} \right)^T}\), then \(\left( {n{\varpi _1}{\lambdabar _1},n{\varpi _2}{\lambdabar _2},...,n{\varpi _n}{\lambdabar _n}} \right)\) approaches to \({\left( {{\lambdabar _1},{\lambdabar _2},{\lambdabar _3},...,{\lambdabar _n}} \right)^T}\).

Example 4

Let
$$\begin{gathered} \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.4,0.7} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.4,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.3,0.7} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.9,\left( {\left[ {0.4,0.8} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
be the four IVPFVs whose weighted vector is \(\varpi ={\left( {0.4,0.3,0.2,0.1} \right)^T},\) then we have
$$\begin{gathered} {{\dot {\lambdabar }}_1}=\left( {\left[ {0.259,0.485} \right],\left[ {0.617,0.693} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_2}=\left( {\left[ {0.269,0.547} \right],\left[ {0.275,0.480} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_3}=\left( {\left[ {0.327,0.744} \right],\left[ {0.235,0.435} \right]} \right), \hfill \\ {{\dot {\lambdabar }}_4}=\left( {\left[ {0.493,0.897} \right],\left[ {0.025,0.145} \right]} \right), \hfill \\ \end{gathered}$$
Performing the ordering with respect to the first element, we have
$$\begin{gathered} \left\langle {{u_4},{\lambdabar _4}} \right\rangle =\left\langle {0.9,\left( {\left[ {0.4,0.8} \right],\left[ {0.1,0.3} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_3},{\lambdabar _3}} \right\rangle =\left\langle {0.5,\left( {\left[ {0.3,0.7} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_2},{\lambdabar _2}} \right\rangle =\left\langle {0.4,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.4} \right]} \right)} \right\rangle , \hfill \\ \left\langle {{u_1},{\lambdabar _1}} \right\rangle =\left\langle {0.3,\left( {\left[ {0.4,0.7} \right],\left[ {0.3,0.4} \right]} \right)} \right\rangle , \hfill \\ \end{gathered}$$
Hence,
$$\begin{gathered} \left\langle {{u_{\sigma \left( 1 \right)}},{{\dot {\lambdabar }}_{\sigma (1)}}} \right\rangle =\left( {\left[ {0.4933,0.8972} \right],\left[ {0.0251,0.1456} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 2 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 2 \right)}}} \right\rangle =\left( {\left[ {0.3271,0.7444} \right],\left[ {0.2358,0.4352} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 3 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 3 \right)}}} \right\rangle =\left( {\left[ {0.2695,0.5479} \right],\left[ {0.2759,0.4804} \right]} \right), \hfill \\ \left\langle {{u_{\sigma \left( 4 \right)}},{{\dot {\lambdabar }}_{\sigma \left( 4 \right)}}} \right\rangle =\left( {\left[ {0.2595,0.4859} \right],\left[ {0.6178,0.6931} \right]} \right), \hfill \\ \end{gathered}$$
Thus,
$$\begin{gathered} {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,\left\langle {{u_4},{\lambdabar _4}} \right\rangle } \right) \hfill \\ =\left( \begin{gathered} \left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {p^2}_{{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{4} {{\left( {1 - {q^2}_{{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} } \right], \hfill \\ \left[ {\mathop \prod \limits_{{j=1}}^{4} {{\left( {{r_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{4} {{\left( {{t_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right] \hfill \\ \end{gathered} \right) \hfill \\ =\left( {\left[ {0.705,0.793} \right],\left[ {0.109,0.300} \right]} \right) \hfill \\ \end{gathered}$$

Theorem 7

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples, then their aggregated value using the I-IVPFHA aggregation operator is also an interval-valued Pythagorean fuzzy value, and
$$\begin{aligned} & {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad =\left( {\begin{array}{*{20}{c}} {\left[ {\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{p_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} ,\sqrt {1 - \mathop \prod \limits_{{j=1}}^{n} {{\left( {1 - {{\left( {{q_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^2}} \right)}^{{\varpi _j}}}} } \right],} \\ {\left[ {\mathop \prod \limits_{{j=1}}^{n} {{\left( {{r_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}},\mathop \prod \limits_{{j=1}}^{n} {{\left( {{t_{{{\dot {\lambdabar }}_{\sigma \left( j \right)}}}}} \right)}^{{\varpi _j}}}} \right]} \end{array}} \right), \\ \end{aligned}$$
(25)
where \({\dot {\lambdabar }_{\sigma \left( j \right)}},\) is the jth largest of the weighted interval-valued Pythagorean fuzzy values \({\dot {\lambdabar }_{\sigma \left( j \right)}},\) \(\left( {{{\dot {\lambdabar }}_{\sigma \left( j \right)}}=n{\varpi _j}{\lambdabar _j}} \right)\), \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) is the weighted vector of \({\lambdabar _{\sigma \left( j \right)}}\) and also \({\varpi _j} \in \left[ {0,1} \right]\), \(\mathop \sum \nolimits_{{j=1}}^{n} {\varpi _j}=1,\) and n is the balancing coefficient, which plays a role of balance. If the vector \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T}\) approaches to\({\left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},...,\frac{1}{n}} \right)^T}\), then \(\left( {n{\varpi _1}{\lambdabar _1},n{\varpi _2}{\lambdabar _2},...,n{\varpi _n}{\lambdabar _n}} \right)\)approaches to\({\left( {{\lambdabar _1},{\lambdabar _2},{\lambdabar _3},...,{\lambdabar _n}} \right)^T}\).

Proof

Straightforward.

Theorem 8

If \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples, where \({\dot {\lambdabar }_{\sigma \left( j \right)}}=\lambdabar\) for all j, then
$${\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle , \ldots ,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right)=\dot {\lambdabar }.$$
(26)

Proof

Straightforward.

Theorem 9

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,3,...,n} \right)\) be a collection of 2-tuples whose weighted vector is given by \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _n}} \right)^T},\) then
$${\dot {\lambdabar }_{\hbox{min} }} \leqslant {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle , \ldots ,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \leqslant {\dot {\lambdabar }_{\hbox{max} }},$$
(27)
where \({\dot {\lambdabar }_{\hbox{max} }}=\hbox{max} \left( {{{\dot {\lambdabar }}_j}} \right),{\dot {\lambdabar }_{\hbox{min} }}=\hbox{min} \left( {{{\dot {\lambdabar }}_j}} \right)\).

Proof

Straightforward.

Theorem 10

Let \(\left\langle {{u_j},{\lambdabar _j}} \right\rangle \left( {j=1,2,...,n} \right)\) and \(\left\langle {{u_j},{\lambdabar ^ * }_{j}} \right\rangle \left( {j=1,2,...,n} \right)\) where \({\dot {\lambdabar }_j} \leqslant \dot {\lambdabar }_{j}^{ * }\) for all j, then
$$\begin{aligned} & {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle ,\left\langle {{u_3},{\lambdabar _3}} \right\rangle ,...,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad \leqslant {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},\lambdabar _{1}^{ * }} \right\rangle ,\left\langle {{u_2},\lambdabar _{2}^{ * }} \right\rangle ,\left\langle {{u_3},\lambdabar _{3}^{ * }} \right\rangle ,...,\left\langle {{u_n},\lambdabar _{n}^{ * }} \right\rangle } \right). \\ \end{aligned}$$
(28)

Proof

Straightforward.

Theorem 11

An IVPFWA operator is a specials case of I-IVPFHA operator.

Proof

Let \(\varpi ={\left( {\tfrac{1}{n},\tfrac{1}{n},\tfrac{1}{n},...,\tfrac{1}{n}} \right)^T},\) then
$$\begin{aligned} & {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {\left\langle {{u_1},{\lambdabar _1}} \right\rangle ,\left\langle {{u_2},{\lambdabar _2}} \right\rangle , \ldots ,\left\langle {{u_n},{\lambdabar _n}} \right\rangle } \right) \\ & \quad ={\varpi _1}\left( {{{\dot {\lambdabar }}_{\sigma \left( 1 \right)}}} \right) \oplus {\varpi _2}\left( {{{\dot {\lambdabar }}_{\sigma \left( 2 \right)}}} \right) \oplus {\varpi _3}\left( {{{\dot {\lambdabar }}_{\sigma \left( 3 \right)}}} \right) \oplus \ldots \oplus {\varpi _n}\left( {{{\dot {\lambdabar }}_{\sigma \left( n \right)}}} \right) \\ & \quad =\frac{1}{n}\left( {{{\dot {\lambdabar }}_{\sigma \left( 1 \right)}}} \right) \oplus \frac{1}{n}\left( {{{\dot {\lambdabar }}_{\sigma \left( 2 \right)}}} \right) \oplus \frac{1}{n}\left( {{{\dot {\lambdabar }}_{\sigma \left( 3 \right)}}} \right) \oplus \ldots \oplus \frac{1}{n}\left( {{{\dot {\lambdabar }}_{\sigma \left( n \right)}}} \right) \\ & \quad =\frac{1}{n}\left( {n{\varpi _1}\left( {{\lambdabar _1}} \right) \oplus n{\varpi _2}\left( {{\lambdabar _2}} \right) \oplus n{\varpi _3}\left( {{\lambdabar _3}} \right) \oplus \ldots \oplus n{\varpi _n}\left( {{\lambdabar _n}} \right)} \right) \\ & \quad ={\text{IVPFW}}{{\text{A}}_\varpi }\left( {{\lambdabar _1},{\lambdabar _2},{\lambdabar _3}, \ldots ,{\lambdabar _n}} \right) \\ \end{aligned}$$

Theorem 12

An I-IVPFOWA operator is a specials case of the I-IVPFHA operator.

Proof

Let \(\varpi ={\left( {\tfrac{1}{n},\tfrac{1}{n},\tfrac{1}{n},...,\tfrac{1}{n}} \right)^T},\) and \({\dot {\lambdabar }_j}=n{\varpi _j}\left( {{\lambdabar _j}} \right)={\lambdabar _j},\) then
$$\begin{aligned} & {\text{I-IVPFH}}{{\text{A}}_{\varpi ,\varpi }}\left( {{\lambdabar _1},{\lambdabar _2},{\lambdabar _3}, \ldots ,{\lambdabar _n}} \right) \\ & \quad ={\varpi _1}\left( {{{\dot {\lambdabar }}_{\sigma \left( 1 \right)}}} \right) \oplus {\varpi _2}\left( {{{\dot {\lambdabar }}_{\sigma \left( 2 \right)}}} \right) \oplus {\varpi _3}\left( {{{\dot {\lambdabar }}_{\sigma \left( 3 \right)}}} \right) \oplus \ldots \oplus {\varpi _n}\left( {{{\dot {\lambdabar }}_{\sigma \left( n \right)}}} \right) \\ & \quad ={\varpi _1}\left( {{\lambdabar _{\sigma \left( 1 \right)}}} \right) \oplus {\varpi _2}\left( {{\lambdabar _{\sigma \left( 2 \right)}}} \right) \oplus {\varpi _3}\left( {{\lambdabar _{\sigma \left( 3 \right)}}} \right) \oplus \ldots \oplus {\varpi _n}\left( {{\lambdabar _{\sigma \left( n \right)}}} \right) \\ & \quad ={\text{I-IVPFOW}}{{\text{A}}_\varpi }\left( {{\lambdabar _1},{\lambdabar _2},{\lambdabar _3}, \ldots ,{\lambdabar _n}} \right). \\ \end{aligned}$$

5 An application of the proposed aggregation operators to multiple attribute decision-making problem

Let \(A=\left\{ {{A_1},{A_2},{A_3},...,{A_n}} \right\}\) be a finite set of \(n\) alternative, and \(C=\left\{ {{C_1},{C_2},{C_3}...,{C_m}} \right\}\) be a finite set of \(m\) attributes and \(\varpi ={\left( {{\varpi _1},{\varpi _2},{\varpi _3},...,{\varpi _m}} \right)^T}\) be the weighted vector of the criteria \({C_i}\left( {i=1,2,3...,m} \right)\) such that \({\varpi _i} \in \left[ {0,1} \right]\) and \(\mathop \sum \nolimits_{{i=1}}^{m} {\varpi _i}=1.\)

The method proposed in this paper having the following steps.

Step 1 Construct decision matrix, \(D={\left[ {{\alpha _{ij}}} \right]_{m \times n}}\) for decision.

Step 2 Utilize the I-IVPFOWA aggregation operator to derive the overall preference values.

Step 3 Calculate the score functions.

Step 4 Arrange the scores function of the all alternatives in the form of descending order and select that alternative, which has the highest score function value.

Example 5

Suppose a customer wants to buy a television from different Televisions, let \({{\text{A}}_1},{{\text{A}}_2},{{\text{A}}_3},{{\text{A}}_4}\) represent the four televisions of different companies. Let \({{\text{C}}_1},{{\text{C}}_2},{{\text{C}}_3},{{\text{C}}_4},\) be the criteria of these televisions. In this method, selecting one of the televisions, four factors are considered. C1, price of each television, C2, Model of each Television, C3, Design of each television C4, Display of each television. Suppose the weight vector of C i is \(\varpi ={\left( {0.1,0.2,0.3,0.4} \right)^T},\) and the interval-valued Pythagorean fuzzy values of the alternative A j are denoted by the following decision matrix.

For I-IVPFOWA aggregation operator

Step 1 Construct the decision matrix

Table 1 Pythagorean fuzzy decision matrix

 

\({\text{A}}{_{1}}\)

\({\text{A}}{_{2}}\)

\({\text{A}}{_{3}}\)

\({\text{A}}{_{4}}\)

\({\text{C}}{_{1}}\)

\(\left\langle {0.6,\left( {\left[ {0.3,0.5} \right],\left[ {0.4,0.8} \right]} \right)} \right\rangle\)

\(\left\langle {0.7,\left( {\left[ {0.3,0.6} \right],\left[ {0.2,0.7} \right]} \right)} \right\rangle\)

\(\left\langle {0.8,\left( {\left[ {0.3,0.5} \right],\left[ {0.5,0.8} \right]} \right)} \right\rangle\)

\(\left\langle {0.9,\left( {\left[ {0.2,0.6} \right],\left[ {0.5,0.7} \right]} \right)} \right\rangle\)

\({\text{C}}{_{{\text{2}}}}\)

\(\left\langle {0.5,\left( {\left[ {0.2,0.6} \right],\left[ {0.3,0.7} \right]} \right)} \right\rangle\)

\(\left\langle {0.5,\left( {\left[ {0.4,0.5} \right],\left[ {0.3,0.6} \right]} \right)} \right\rangle\)

\(\left\langle {0.6,\left( {\left[ {0.2,0.5} \right],\left[ {0.2,0.6} \right]} \right)} \right\rangle\)

\(\left\langle {0.7,\left( {\left[ {0.5,0.8} \right],\left[ {0.3,0.5} \right]} \right)} \right\rangle\)

\({\text{C}}{_{3}}\)

\(\left\langle {0.3,\left( {\left[ {0.3,0.7} \right],\left[ {0.2,0.5} \right]} \right)} \right\rangle\)

\(\left\langle {0.4,\left( {\left[ {0.2,0.6} \right],\left[ {0.2,0.7} \right]} \right)} \right\rangle\)

\(\left\langle {0.4,\left( {\left[ {0.3,0.7} \right],\left[ {0.4,0.7} \right]} \right)} \right\rangle\)

\(\left\langle {0.5,\left( {\left[ {0.3,0.6} \right],\left[ {0.4,0.7} \right]} \right)} \right\rangle\)

\({\text{C}}{_{4}}\)

\(\left\langle {0,2,\left( {\left[ {0.4,0.5} \right],\left[ {0.4,0.6} \right]} \right)} \right\rangle\)

\(\left\langle {0.1,\left( {\left[ {0.4,0.6} \right],\left[ {0.4,0.5} \right]} \right)} \right\rangle\)

\(\left\langle {0.2,\left( {\left[ {0.4,0.4} \right],\left[ {0.2,0.8} \right]} \right)} \right\rangle\)

\(\left\langle {0.4,\left( {\left[ {0.2,0.4} \right],\left[ {0.4,0.8} \right]} \right)} \right\rangle\)

Step 2 Utilize the I-IVPFOWA aggregation operator to derive the overall preference values
$$\begin{gathered} {\alpha _1}=\left( {\left[ {0.343,{\text{ }}0.603} \right],{\text{ }}\left[ {0.195,{\text{ }}0.602} \right]} \right), \hfill \\ {\alpha _2}=\left( {\left[ {0.466,{\text{ }}0.584} \right],{\text{ }}\left[ {0.284,{\text{ }}0.589} \right]} \right), \hfill \\ {\alpha _3}=\left( {\left[ {0.343,{\text{ }}0.570} \right],{\text{ }}\left[ {0.291,{\text{ }}0.719} \right]} \right), \hfill \\ {\alpha _4}=\left( {\left[ {0.355,{\text{ }}0.629} \right],{\text{ }}\left[ {0.385,{\text{ }}0.679} \right]} \right), \hfill \\ \end{gathered}$$
Step 3 Calculate the scores functions of\({\alpha _j}\left( {j=1,2,3,4} \right)\), using Eq. 10.
$$\begin{gathered} S\left( {{\alpha _1}} \right)=\frac{1}{2}\left[ {{{\left( {0.343} \right)}^2}+{{\left( {0.603} \right)}^2} - {{\left( {0.195} \right)}^2} - {{\left( {0.602} \right)}^2}} \right]=0.040 \hfill \\ S\left( {{\alpha _2}} \right)=\frac{1}{2}\left[ {{{\left( {0.466} \right)}^2}+{{\left( {0.584} \right)}^2} - {{\left( {0.284} \right)}^2} - {{\left( {0.589} \right)}^2}} \right]=0.063 \hfill \\ S\left( {{\alpha _3}} \right)=\frac{1}{2}\left[ {{{\left( {0.343} \right)}^2}+{{\left( {0.570} \right)}^2} - {{\left( {0.291} \right)}^2} - {{\left( {0.719} \right)}^2}} \right]= - 0.079 \hfill \\ S\left( {{\alpha _4}} \right)=\frac{1}{2}\left[ {{{\left( {0.355} \right)}^2}+{{\left( {0.629} \right)}^2} - {{\left( {0.385} \right)}^2} - {{\left( {0.679} \right)}^2}} \right]= - 0.043 \hfill \\ \end{gathered}$$

Hence, \(S\left( {{\alpha _2}} \right) \succ S\left( {{\alpha _1}} \right) \succ S\left( {{\alpha _4}} \right) \succ S\left( {{\alpha _3}} \right).\)

Step 4 Thus, \({{\text{A}}_2}\) is the best option for customer.

For I-IVPFHA aggregation operator

Step 1 Utilize \({\dot {\alpha }_{\sigma \left( j \right)}}=n{\varpi _j}{\alpha _j},\), for Table 1, we have
$$\begin{aligned} ~\dot{\alpha }_{{11}} & = \left( {\left[ {0.238,{\text{ }}0.569} \right],{\text{ }}\left[ {0.614,{\text{ }}0.876} \right]} \right),\dot{\alpha }_{{21}} = \left( {\left[ {0.189,{\text{ }}0.569} \right],{\text{ }}\left[ {0.361,{\text{ }}0.643} \right]} \right) \\ \dot{\alpha }_{{31}} & = \left( {\left[ {0.313,{\text{ }}0.727} \right],{\text{ }}\left[ {0.154,{\text{ }}0.454} \right]} \right),\dot{\alpha }_{{41}} = \left( {\left[ {0.449,{\text{ }}0.559} \right],{\text{ }}\left[ {0.270,{\text{ }}0.495} \right]} \right) \\ ~\dot{\alpha }_{{12}} & = \left( {\left[ {0.238,{\text{ }}0.482} \right],{\text{ }}\left[ {0.127,{\text{ }}0.814} \right]} \right),\dot{\alpha }_{{22}} = \left( {\left[ {0.378,{\text{ }}0.473} \right],{\text{ }}\left[ {0.361,{\text{ }}0.643} \right]} \right) \\ \dot{\alpha }_{{32}} & = \left( {\left[ {0.209,{\text{ }}0.625} \right],{\text{ }}\left[ {0.154,{\text{ }}0.670} \right]} \right),\dot{\alpha }_{{42}} = \left( {\left[ {0.449,{\text{ }}0.669} \right],{\text{ }}\left[ {0.270,{\text{ }}0.378} \right]} \right) \\ ~\dot{\alpha }_{{13}} & = \left( {\left[ {0.238,{\text{ }}0.399} \right],{\text{ }}\left[ {0.685,{\text{ }}0.876} \right]} \right),\dot{\alpha }_{{23}} = \left( {\left[ {0.189,{\text{ }}0.473} \right],{\text{ }}\left[ {0.259,{\text{ }}0.643} \right]} \right) \\ \dot{\alpha }_{{33}} & = \left( {\left[ {0.313,{\text{ }}0.727} \right],{\text{ }}\left[ {0.350,{\text{ }}0.670} \right]} \right),\dot{\alpha }_{{43}} = \left( {\left[ {0.449,{\text{ }}0.449} \right],{\text{ }}\left[ {0.092,{\text{ }}0.744} \right]} \right) \\ ~\dot{\alpha }_{{14}} & = \left( {\left[ {0.159,{\text{ }}0.482} \right],{\text{ }}\left[ {0.685,{\text{ }}0.643} \right]} \right),\dot{\alpha }_{{24}} = \left( {\left[ {0.473,{\text{ }}0.765} \right],{\text{ }}\left[ {0.154,{\text{ }}0.551} \right]} \right) \\ \dot{\alpha }_{{34}} & = \left( {\left[ {0.313,{\text{ }}0.625} \right],{\text{ }}\left[ {0.350,{\text{ }}0.670} \right]} \right),\dot{\alpha }_{{44}} = \left( {\left[ {0.224,{\text{ }}0.449} \right],{\text{ }}\left[ {0.270,{\text{ }}0.744} \right]} \right) \\ \end{aligned}$$
Step 2 Utilize the I-IVPFHA aggregation operator to derive the overall preference values
$$\begin{gathered} {\alpha _1}=\left( {\left[ {0.375,{\text{ }}0.623} \right],{\text{ }}\left[ {0.257,{\text{ }}0.530} \right]} \right), \hfill \\ {\alpha _2}=\left( {\left[ {0.382,{\text{ }}0.614} \right],{\text{ }}\left[ {0.223,{\text{ }}0.526} \right]} \right), \hfill \\ {\alpha _3}=\left( {\left[ {0.375,{\text{ }}0.588} \right],{\text{ }}\left[ {0.202,{\text{ }}0.707} \right]} \right), \hfill \\ {\alpha _4}=\left( {\left[ {0.344,{\text{ }}0.616} \right],{\text{ }}\left[ {0.282,{\text{ }}0.665} \right]} \right), \hfill \\ \end{gathered}$$
Step 3 Calculate the scores functions of \({\alpha _j}\left( {j=1,2,3,4} \right)\), using Eq. 10.
$$\begin{gathered} S\left( {{\alpha _1}} \right)=\frac{1}{2}\left[ {{{\left( {0.375} \right)}^2}+{{\left( {0.623} \right)}^2} - {{\left( {0.257} \right)}^2} - {{\left( {0.530} \right)}^2}} \right]=0.090 \hfill \\ S\left( {{\alpha _2}} \right)=\frac{1}{2}\left[ {{{\left( {0.382} \right)}^2}+{{\left( {0.614} \right)}^2} - {{\left( {0.223} \right)}^2} - {{\left( {0.526} \right)}^2}} \right]=0.098 \hfill \\ S\left( {{\alpha _3}} \right)=\frac{1}{2}\left[ {{{\left( {0.375} \right)}^2}+{{\left( {0.588} \right)}^2} - {{\left( {0.202} \right)}^2} - {{\left( {0.707} \right)}^2}} \right]= - 0.027 \hfill \\ S\left( {{\alpha _4}} \right)=\frac{1}{2}\left[ {{{\left( {0.344} \right)}^2}+{{\left( {0.616} \right)}^2} - {{\left( {0.282} \right)}^2} - {{\left( {0.665} \right)}^2}} \right]= - 0.011 \hfill \\ \end{gathered}$$

Hence, \(S\left( {{\alpha _2}} \right) \succ S\left( {{\alpha _1}} \right) \succ S\left( {{\alpha _4}} \right) \succ S\left( {{\alpha _3}} \right).\)

Step 4 Thus, \({{\text{A}}_2}\) is the best option for customer.

6 Conclusions

The objective of this paper is to develop some induced aggregation operators based on interval-valued Pythagorean fuzzy numbers and applied them to the multi-attribute group decision-making problems where attribute values are the interval-valued Pythagorean fuzzy numbers. First, we have developed two induced aggregation operators a long with their properties, namely, induced I-IVPFOWA aggregation operator and I-IVPFHA aggregation operator. These methods provide more general, more accurate and precise results as compared to the existing methods. Furthermore, we have developed a method for multi-criteria group decision making based on these operators, and the operational processes have illustrated in detail. The suggested methodology can be used for any type of selection problem involving any number of selection attributes. We ended the paper with an application of the new approach in a decision making problem.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHazara University MansehraDhodialPakistan
  2. 2.Department of MathematicsAbdul Wali Khan University MardanMardanPakistan

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