1 Introduction

As an important part of modern decision science, multiple attribute decision making has been widely used. Many operators focus on correctly aggregating information decision making problems have been developed on this issue (Zhao et al. 2010; Ye 2010; He et al. 2013; 2016; Beliakov et al. 2010; Merigó et al. 2011).

As a common form of decision information, Zadeh (1965) developed the fuzzy sets. To describe vague information more flexibly and practicably, Atanassov (1986) developed intuitionistic fuzzy sets (IFSs). Atanassov (1994) presented some basic operations on IFSs. Xu (2007) presented some aggregation operators under intutionistic fuzzy environments and obtained their detailed formulas with mathematical induction. Much more attention has been paid to Granular Computing and decision making problems (Xu and Yager 2006; Rodríguez et al. 2012, 2013; Chen 2014; Rodríguez et al. 2014; Pedrycz and Chen 2015; Livi and Sadeghian 2016; Chen and Chang 2015; He et al. 2015; Lorkowski and Kreinovich 2015; Chen et al. 2016a, b; Apolloni et al. 2016; Antonelli et al. 2016; Ciucci 2016; Kovalerchuk and Kreinovich 2017; Loia et al. 2016; Lingras et al. 2016; Liu and Cocea 2017; Liu et al. 2016; Maciel et al. 2016; Min and Xu 2016; Peters and Weber 2016; Skowron et al. 2016; Wilke and Portmann 2016; Xu and Wang 2016; Yao 2016; Sanchez et al. 2017; Song and Wang 2016; Wang et al. 2017; Zhou 2017). Xu and Xia (2011) dealt with financial decision making with induced generalized aggregation operators. Mendel (2016) synthesized the interval type-2 fuzzy set model for a word by comparing Hao-Mendel Approach, Interval Approach and Enhanced Interval Approach. Xu and Gou (2017) made an overview of interval-valued intuitionistic fuzzy information aggregations and applications. Wei and Zhao (2012) dealt with decision making by the induced correlated aggregating operators. Considering that distorted conclusions would be obtained if the decision makers don’t take account the relationships among the evaluated values (Yager 2008), Wei (2012) developed some prioritized aggregation operators. Xu and Xia (2011) proposed some induced generalized intuitionistic fuzzy operators. Yager and Filev (1999) proposed the induced ordered weighted averaging operators. Chen et al. (2016a, b) proposed a multi-criteria decision making method based on the TOPSIS method under intuitionistic fuzzy environments. Apolloni et al. (2016) proposed a neurofuzzy algorithm for learning from complex granules. Dubois and Prade (2016)  investigated the notion of extensional fuzzy set and highlighted its common features with the notion of formal concept in regard to a similarity relation. Das et al. (2017) developed a robust decision making method using intuitionistic fuzzy values. Chen and Tsai (2016) developed multiple attribute decision making method based on novel interval-valued intuitionistic fuzzy geometric averaging operators. Kreinovich (2016) solved systems of equations under uncertainty and explained how different practical problems lead to different mathematical and computational formulations.

Recently, some new operational laws on intuitionistic fuzzy values had been proposed in He et al. (2014a, b), which can be used in some special cases. As a good complement to the existing works, the new operational laws consider the interactions between membership function and non-membership function of different intuitionistic fuzzy values. However, when concerned with the decision making situations that the given intuitionistic fuzzy values and their ordered positions by the induced values should be considered with the interaction theory, very little work had been done. As a result, based on the works in He et al. (2014a, b), Xu (2007) and Xu and Yager (2006), we present some induced hybrid interaction aggregation operators on intuitionistic fuzzy values, such as the IFIHIA operator and the GIFIHIA operator. We investigate the properties of these new aggregation operators and apply them to the selection of cold chain logistics enterprises under intuitionistic fuzzy environment. The main advantage of these operators is concluded as follows, (1) the interactions of different intuitionistic fuzzy values are taken into consideration, (2) the involved intuitionistic fuzzy values are reordered according to the induced values and then are aggregated into a collective one, (3) the attitudes of decision makers are considered by taking different values of parameter according to decision makers’ preferences.

The rest of the paper is organized as follows. Section 2 reviews some basic concepts. Section 3 presents someinduced hybrid interaction averaging operators  under intuitionistic fuzzy environments and the corresponding properties are investigated. Section 4 investigates the selection of cold chain logistics enterprises based on the proposed operators under intuitionistic fuzzy environment. In Sect. 5, numerical examples show the feasibility and validity of the presented approach. Finally, Sect. 6 concludes the paper.

2 Preliminaries

Definition 1

(Atanassov 1986). Suppose that X is a fixed non-empty set.

\(A = \left\{ {\left\langle {x,{u_A}(x),{v_A}(x),{\pi _A}(x)} \right\rangle \left| {x \in X} \right.} \right\}\) indicates intuitionistic fuzzy sets (IFSs) in X, where \({u_A}(x),{v_A}(x) \in [0,1\), representing the membership degree and the non-membership degree respectively. \({\pi _A}(x) = 1 - {u_A}(x) - {v_A}(x)\), reflecting the hesitant degree of \(x \in X\).

Xu (2007) and Xu and Yager (2006) called \(A = \left\langle {u,v} \right\rangle\) intuitionistic fuzzy number (IFN) for computational convenience and all IFNs are denoted as IFNs(X) in this paper.

Atanassov (1994) and De et al. (2000) introduced some basic operations on IFNs, which have been widely used in multiple attribute decision making.

Let A = <u A ,v A > be an IFN, Chen and Tan (1994) described the suitable degree of an alternative meets the decision maker’s demand with score function S(A) = u A − v A . Hong and Choi (2000) described the accurate degree of IFN A with accuracy function H(A) = u A + v A .

Based on the score function and accuracy function, Xu (2007) and Xu and Yager (2006) defined the comparison law for IFNs as follows.

Definition 2

Let \(A = \left\langle {{u_A},{v_A}} \right\rangle \in {\text{IFNs}}(X)\) and \(B = \left\langle {{u_B},{v_B}} \right\rangle \in {\text{IFNs}}(X)\). Then \(A < B\) if and only if.

  1. i.

    \(S(A) < S(B)\) or

  2. ii.

    \(S(A) = S(B)\) and \(H(A) < H(B)\).

Recently, the new addition operation, scalar multiplication operation, multiplication operation and power operation are developed in He et al. (2014a, b) as follows.

Definition 3

Let \(A = \left\langle {{u_A},{v_A}} \right\rangle \in {\rm{IFNs}}(X)\) and \(B = \left\langle {{u_B},{v_B}} \right\rangle \in {\rm{IFNs}}(X)\).

$$A\hat \oplus B = \left\langle {1 - \left( {1 - {u_A}} \right) \cdot \left( {1 - {u_B}} \right),} \right.\left. {\left( {1 - {u_A}} \right) \cdot \left( {1 - {u_B}} \right) - \left( {1 - \left( {{u_A} + {v_A}} \right)} \right) \cdot \left( {1 - \left( {{u_B} + {v_B}} \right)} \right)} \right\rangle$$
(1)
$$\lambda A = \left\langle {1 - {{\left( {1 - {u_A}} \right)}^\lambda },{{\left( {1 - {u_A}} \right)}^\lambda } - {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right\rangle ,\lambda> 0$$
(2)
$$A\hat \otimes B = \left\langle {\left( {1 - {v_A}} \right)\left( {1 - {v_B}} \right) - \left( {1 - \left( {{u_A} + {v_A}} \right)} \right)\left( {1 - \left( {{u_B} + {v_B}} \right)} \right),} \right.\left. {1 - \left( {1 - {v_A}} \right)\left( {1 - {v_B}} \right)} \right\rangle$$
(3)
$${A^\lambda } = \left\langle {{{\left( {1 - {v_A}} \right)}^\lambda } - {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda },1 - {{\left( {1 - {v_A}} \right)}^\lambda }} \right\rangle ,\lambda> 0$$
(4)

3 Intuitionistic fuzzy induced hybrid interaction averaging (IFIHIA) operator

Definition 4

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} \in {\rm{IFNs}}(X){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\). \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) is the induced value. The IFIHIA operator is defined as

$${\text{IFIHI}}{{\text{A}}_{\omega ,w}}({A_1}, \ldots ,{A_n}) = \mathop {\hat \oplus }\limits_{i = 1}^n {w_i}{\tilde A_{index(i)}},$$
(5)

where \(\omega = {({\omega _1}, \ldots {\omega _n})^T}\) is the weight vector of \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\), satisfying \({\omega _i} \in [0,1\) and \(\sum\nolimits_{i = 1}^n {{\omega _i}} = 1\), \({\tilde A_i}\) denotes \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1, \ldots ,n),\) \({\tilde A_i}\) is reordered according to \({p_i}\) as \({\tilde A_{index(i)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\), \(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\) is the weight vector of \({\tilde A_{index(i)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\), satisfying \({w_i} \in [0,1\) and \(\sum\nolimits_{i = 1}^n {{w_i}} = 1\).

Theorem 1

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} \in {\text{IFNs}}(X){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\). \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) is the induced value. Then

$${\text{IFIHI}}{{\text{A}}_{\omega ,w}}({A_1}, \ldots ,{A_n}) = \left\langle {1 - \prod\limits_{i = 1}^n {{{(1 - {u_{{{\tilde A}_{index(i)}}}})}^{{w_i}}},\prod\limits_{i = 1}^n {{{(1 - {u_{{{\tilde A}_{index(i)}}}})}^{{w_i}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - ({u_{{{\tilde A}_{index(i)}}}} + {v_{_{{{\tilde A}_{index(i)}}}}})} \right)}^{{w_i}}}} } } \right\rangle$$
(6)

And, \({\text{IFIHI}}{{\text{A}}_{\omega ,w}}({A_1}, \ldots ,{A_n}){\kern 1pt} {\kern 1pt} {\kern 1pt} \in {\text{IFNs}}(X)\).

Theorem 2

(Idempotency) Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs, if all A i are equal, supposed as A, then \(IFIHI{A_{\omega ,w}}({A_1}, \ldots ,{A_n}) = A\).

Theorem 3

(Commutativity). Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs, if \({A'_i}(i = 1, \ldots n)\) is any permutation of \({A_i}(i = 1, \ldots n)\), then

$$IFIHI{A_{\omega ,w}}({A_1}, \ldots ,{A_n}) = IFIHI{A_{\omega ,w}}({A'_1}, \ldots ,{A'_n}).$$

Example 1

Let \({A_1} = \left\langle {0.2,0.7} \right\rangle\) \({A_2} = \left\langle {0.5,0.3} \right\rangle\) \({A_3} = \left\langle {0.4,0.5} \right\rangle\) \({A_4} = \left\langle {0.3,0.4} \right\rangle\) \({A_5} = \left\langle {0.6,0} \right\rangle\) be five IFNs. \(\omega = {\left\{ {0.25,0.20,0.15,0.18,0.22} \right\}^T}\) is the weight vector of \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,5).\)

By the operational law in Atanassov (1986), De et al. (2000) and Xu (2007), it has.

\({\tilde A_1} = \left\langle {1 - {{(1 - 0.2)}^{5 \times 0.25}},{{0.7}^{5 \times 0.25}}} \right\rangle = \left\langle {0.243,0.640} \right\rangle ,\,{\tilde A_2} = \left\langle {1 - {{(1 - 0.5)}^{5 \times 0.20}},{{0.3}^{5 \times 0.20}}} \right\rangle = \left\langle {0.500,0.300} \right\rangle ,\,{\tilde A_3} = \left\langle {1 - {{(1 - 0.4)}^{5 \times 0.15}},{{0.5}^{5 \times 0.15}}} \right\rangle = \left\langle {0.318,0.595} \right\rangle ,\,{\tilde A_4} = \left\langle {1 - {{(1 - 0.3)}^{5 \times 0.18}},{{0.4}^{5 \times 0.18}}} \right\rangle = \left\langle {0.275,0.438} \right\rangle ,\,{\tilde A_5} = \left\langle {1 - {{(1 - 0.6)}^{5 \times 0.22}},{0^{5 \times 0.22}}} \right\rangle = \left\langle {0.635,0.00} \right\rangle .\)

By Definition 2, we have

$$S({\tilde A_1}) = - 0.3969,\,S({\tilde A_2}) = 0.2000,\,S({\tilde A_3}) = - 0.2763,\,S({\tilde A_4}) = - 0.1639,\,S({\tilde A_5}) = 0.6350.$$

For a fair comparison, we suppose the score values of IFNs are the induced values, Obviously,

$$S({\tilde A_5})> S({\tilde A_2})> S({\tilde A_4})> S({\tilde A_3})> S({\tilde A_1}).$$

So

$${\tilde A_{index(1)}} = \left\langle {0.6350,0.0000} \right\rangle ,\,{\tilde A_{index(2)}} = \left\langle {0.5000,0.3000} \right\rangle ,\,{\tilde A_{index(3)}} = \left\langle {0.2745,0.4384} \right\rangle ,\,{\tilde A_{index(4)}} = \left\langle {0.3183,0.5946} \right\rangle ,\,{\tilde A_{index(5)}} = \left\langle {0.2434,0.6403} \right\rangle .$$

Suppose that \(w = {(0.112,0.236,0.304,0.236,0.112)^T}\). Then, by the intuitionistic fuzzy hybrid interaction averaging (IFHA) operator in Xu (2007), it follows that.

\(\begin{aligned} A = IFH{A_{w,\omega }}({A_1},{A_2},{A_{3,}}{A_4},{A_5}) = & \left\langle {1 - \prod\limits_{j = 1}^5 {{{\left( {1 - {u_{{{\tilde A}_{\sigma (j)}}}}} \right)}^{{w_j}}},\prod\limits_{j = 1}^5 {{{\left( {{v_{_{{{\tilde A}_{\sigma (j)}}}}}} \right)}^{{w_i}}}} } } \right\rangle = \left\langle {1 - {{\left( {1 - 0.6350} \right)}^{0.112}} \cdot {{\left( {1 - 0.5} \right)}^{0.236}} \cdot {{\left( {1 - 0.2745} \right)}^{0.304}} \cdot {{\left( {1 - 0.3183} \right)}^{0.236}} \cdot {{\left( {1 - 0.2434} \right)}^{0.112}},} \right.\,\left. {{{0.1703}^{0.112}} \cdot {{0.3}^{0.236}} \cdot {{0.4384}^{0.304}} \cdot {{0.5946}^{0.236}} \cdot {{0.0000}^{0.112}}} \right\rangle \\ = \left\langle {0.3910,0.000} \right\rangle . \\ \end{aligned}\)

While according to Definition 3, we have

$$\begin{aligned} {{\tilde A}_1} = \left\langle {1 - {{(1 - 0.2)}^{5 \times 0.25}},{{(1 - 0.2)}^{5 \times 0.25}} - {{\left( {1 - 0.9} \right)}^{5 \times 0.25}}} \right\rangle = \left\langle {0.2434,0.7004} \right\rangle ,\,{{\tilde A}_2} = \left\langle {1 - {{(1 - 0.5)}^{5 \times 0.20}},{{(1 - 0.5)}^{5 \times 0.20}} - {{\left( {1 - 0.8} \right)}^{5 \times 0.20}}} \right\rangle = \left\langle {0.500,0.300} \right\rangle ,\,{{\tilde A}_3} = \left\langle {1 - {{(1 - 0.4)}^{5 \times 0.15}},{{(1 - 0.4)}^{5 \times 0.15}} - {{\left( {1 - 0.9} \right)}^{5 \times 0.15}}} \right\rangle = \left\langle {0.3183,0.5039} \right\rangle ,\, \hfill \\ {{\tilde A}_4} = \left\langle {1 - {{(1 - 0.3)}^{5 \times 0.18}},{{(1 - 0.3)}^{5 \times 0.18}} - {{\left( {1 - 0.7} \right)}^{5 \times 0.18}}} \right\rangle = \left\langle {0.2745,0.3870} \right\rangle ,\,{{\tilde A}_5} = \left\langle {1 - {{(1 - 0.6)}^{5 \times 0.22}},{{(1 - 0.6)}^{5 \times 0.22}} - {{(1 - 0.6)}^{5 \times 0.22}}} \right\rangle = \left\langle {0.6350,0.0000} \right\rangle . \hfill \\ \end{aligned}$$

According to Definition 2, we obtain

$$S({\tilde A_1}) = - 0.4570,\,S({\tilde A_2}) = 0.2000,\,S({\tilde A_3}) = - 0.1856,\,S({\tilde A_4}) = - 0.1125,\,S({\tilde A_5}) = 0.6350.$$

obviously,

$$S({\tilde A_5})> S({\tilde A_2})> S({\tilde A_4})> S({\tilde A_3})> S({\tilde A_1}).$$

Thus,

$${\tilde A_{index(1)}} = \left\langle {0.6350,0.0000} \right\rangle ,\,{\tilde A_{index(2)}} = \left\langle {0.5000,0.3000} \right\rangle ,\,{\tilde A_{index(3)}} = \left\langle {0.2745,0.3870} \right\rangle ,\,{\tilde A_{index(4)}} = \left\langle {0.3183,0.5039} \right\rangle ,\,{\tilde A_{index(5)}} = \left\langle {0.2434,0.7004} \right\rangle .$$

Suppose that \(w = {(0.112,0.236,0.304,0.236,0.112)^T}\), which is determined by the normal distribution based method (Xu 2007). Then, by Theorem 1, it follows that

$$\begin{aligned}{l} A = IFIHI{A_{w,\omega }}({A_1},{A_2},{A_{3,}}{A_4},{A_5}) = \left\langle {1 - \prod\limits_{j = 1}^5 {{{(1 - {u_{{{\tilde A}_{index(j)}}}})}^{{w_j}}},\prod\limits_{j = 1}^5 {{{\left[ {1 - {u_{{{\tilde A}_{index(j)}}}}} \right]}^{{w_i}}}} - \prod\limits_{j = 1}^5 {{{\left[ {1 - ({u_{{{\tilde A}_{index(j)}}}} + {v_{_{{{\tilde A}_{index(j)}}}}})} \right]}^{{w_i}}}} } } \right\rangle \\ \quad = \left\langle {1 - {{\left( {1 - 0.6350} \right)}^{0.112}} \cdot {{\left( {1 - 0.5} \right)}^{0.236}} \cdot {{\left( {1 - 0.2750} \right)}^{0.304}} \cdot {{\left( {1 - 0.3183} \right)}^{0.236}} \cdot {{\left( {1 - 0.2434} \right)}^{0.112}},} \right.\,\left. {0.6090 - {{0.1703}^{0.112}} \cdot {{0.2}^{0.236}} \cdot {{0.338}^{0.304}} \cdot {{0.1778}^{0.236}} \cdot {{0.0562}^{0.112}}} \right\rangle \\ \quad = \left\langle {0.3910,0.3950} \right\rangle . \end{aligned}$$

It is obvious that \({v_{{\text{IFIH}}{{\text{A}}_{\omega ,w}}({A_1}, \ldots ,{A_n})}} = 0.3950 \ne 0\). Thus, \({v_{{A_5}}}\) doesn’t play a decisive role, which is the advantage of the proposed operator in this paper over that in Xu (2007).

The IFIHIA operator can be interpreted from three aspects as follows.

  1. 1.

    It considers not only the effects of membership function of different IFNs and effects of non-membership degree of different IFNs, but also the interactions of different IFNs.

  2. 2.

    The weighted IFNs \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1, \ldots ,n)\) are reordered according to the induced value \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\). The weighted IFNs \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1, \ldots ,n)\) are obtained by multiplies \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1, \ldots ,n)\) by the corresponding weights \(\omega = {({\omega _1}, \ldots {\omega _n})^T}\) and a balancing coefficient n.

  3. 3.

    In the process of all the weighted IFNs \({w_i}{\tilde A_{index(i)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) are aggregated into a collective one, both the given IFNs and their induced values \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) are considered.

4 Generalized intuitionistic fuzzy induced hybrid interaction averaging (GIFIHIA) operator

Definition 5

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs, \(\lambda> 0\), \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be the induced value. The generalized intuitionistic fuzzy induced hybrid interaction averaging (GIFIHIA) operator of dimension n is a mapping \({\text{GIFIHI}}{{\text{A}}_\lambda }:{\text{IF}}{{\text{N}}^n} \to {\text{IFN}}\), which has an associated vector \(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\), satisfying \({w_i} \in [0,1\) and \(\sum\nolimits_{i = 1}^n {{w_i}} = 1\) such that

$$GIFIHI{A_\lambda }({A_1}, \ldots ,{A_n}) = {\left( {\mathop {\hat \oplus }\limits_{i = 1}^n {w_i}{{\tilde A}_{index(i)}}^\lambda } \right)^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}},$$
(7)

where \(\omega = {({\omega _1}, \ldots {\omega _n})^T}\) is the weight vector of \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\), satisfying \({\omega _i} \in [0,1\) and \(\sum\nolimits_{i = 1}^n {{\omega _i}} = 1\), \({\tilde A_i}\) denotes \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\), n is the balancing coefficient, which plays a role of balance. \({\tilde A_i}\) is reordered according to \({p_i}\) as \({\tilde A_{index(i)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\).

Lemma 1

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} \left( {i = 1,2, \ldots ,n} \right)\) be a collection of IFNs. Then

$$\mathop {\hat \oplus }\limits_{i = 1}^n {w_i}{A_i}^\lambda = \left\langle {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} ,} \right.\,\left. {\prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^{\lambda {w_i}}}} } \right\rangle$$

where \(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\) is the weighting vector of \({A_i}\), satisfying \({w_i} \in [0,1\) and \(\succ\).

Proof

By using mathematical induction on n, we prove Lemma 1 as follows.

when \(n = 1\), \({w_1} = 1\), we obtain

\(\begin{aligned} {w_1}{A_1}^\lambda = {A_1}^\lambda = \left\langle {{{\left( {1 - {v_{{A_1}}}} \right)}^\lambda } - {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^\lambda },1 - {{\left( {1 - {v_{{A_1}}}} \right)}^\lambda }} \right\rangle \hfill \\ \quad = \left\langle {1 - {{\left( {1 - {{\left( {1 - {v_{{A_1}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^\lambda }} \right)}^1},{{\left( {1 - {{\left( {1 - {v_{{A_1}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^\lambda }} \right)}^1} - {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^{1 \cdot \lambda }}} \right\rangle \hfill \\ \quad = \left\langle {1 - {{\left( {1 - {{\left( {1 - {v_{{A_1}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^\lambda }} \right)}^{{w_1}}},{{\left( {1 - {{\left( {1 - {v_{{A_1}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^\lambda }} \right)}^{{w_1}}} - {{\left( {1 - \left( {{u_{{A_1}}} + {v_{{A_1}}}} \right)} \right)}^{\lambda \cdot {w_1}}}} \right\rangle . \hfill \\ \end{aligned}\)

Thus, Lemma 1 is established for \(n = 1\).

If Lemma 1 holds for n = k. Then, n = k + 1, by inductive assumption and Eq. (1), we get

$$\begin{aligned}{l} \mathop {\hat \oplus }\limits_{i = 1}^{k + 1} {w_i}{A_i}^\lambda = \left( {\mathop {\hat \oplus }\limits_{i = 1}^k {w_i}{A_i}^\lambda } \right)\hat \oplus {w_{k + 1}}{A_{k + 1}}^\lambda \\ \quad = \left\langle {1 - \prod\limits_{i = 1}^k {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} ,{\kern 1pt} {\kern 1pt} \prod\limits_{i = 1}^k {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^k {{{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^{\lambda {w_i}}}} } \right\rangle \hat \oplus \left\langle {1 - {{\left( {1 - {{\left( {1 - {v_{{A_{k + 1}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_{k + 1}}}} + {v_{{A_{k + 1}}}}} \right)} \right)}^\lambda }} \right)}^{{w_{k + 1}}}},{{\left( {1 - {{\left( {1 - {v_{{A_{k + 1}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_{k + 1}}}} + {v_{{A_{k + 1}}}}} \right)} \right)}^\lambda }} \right)}^{{w_{k + 1}}}} - {{\left( {1 - \left( {{u_{{A_{k + 1}}}} + {v_{{A_{k + 1}}}}} \right)} \right)}^{\lambda \cdot {w_{k + 1}}}}} \right\rangle \\ \quad = \left\langle {1 - \prod\limits_{i = 1}^{k + 1} {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \prod\limits_{i = 1}^{k + 1} {{{\left( {1 - {{\left( {1 - {v_{{A_i}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^{k + 1} {{{\left( {1 - \left( {{u_{{A_i}}} + {v_{{A_i}}}} \right)} \right)}^{\lambda {w_i}}}} } \right\rangle \end{aligned}$$

i.e. Lemma 1 is established for n = k + 1.

Thus, Lemma 1 holds for all n with the mathematical induction on n.

Theorem 4

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs. \(\lambda> 0\). Then \({\text{GIFIHI}}{{\text{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) \in {\text{IFNs}}(X).\) Furthermore,

$$\begin{aligned} GIFIHI{A_\lambda }({A_1},{A_2}, \ldots {A_n}) \hfill \\ = \left\langle {{{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} ,} \right.\,\left. {1 - {{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}}} \right\rangle . \hfill \\ \end{aligned}$$
(8)

Proof

By Lemma 1 we get

$$\mathop {\hat \oplus }\limits_{i = 1}^n {w_i}{\tilde A_{index(i)}}^\lambda = \left\langle {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} ,} \right.\,\left. {\prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right\rangle .$$

Then according to Eqs. (4) and (7), we have

$$\begin{aligned} {\rm{GIFIHI}}{{\rm{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) = & {\left( {\mathop {\hat \oplus }\limits_{i = 1}^n {w_i}{{({{\tilde A}_{index(i)}})}^\lambda }} \right)^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}}\\ = \left\langle {{{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} } } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} } \right.,\,\left. {1 - {{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}} \right\rangle \end{aligned}$$

Therefore, Eq. (8) is established.

Next, we prove the result that \({\text{GIFIHI}}{{\text{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) \in {\text{IFNs}}(X)\).

Let \({\text{GIFIHI}}{{\text{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) = E = \left\langle {{u_E},{v_E}} \right\rangle\), By Eq. (8), we have

$${v_E} = 1 - {\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\lambda $}}}},$$
(9)
$${u_E} = {\left( \begin{aligned} 1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} \hfill \\ + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} \hfill \\ \end{aligned} \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\lambda $}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}}$$
(10)

According to Definition 1, we have \(0 \leqslant 1 - {\left( {1 - {v_{{{\tilde A}_{\sigma (i)}}}}} \right)^\lambda } \leqslant 1\),then

$${v_E} \geqslant 1 - {\left( {1 - \prod\limits_{i = 1}^n {{{\left( {0 + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\lambda $}}}} = 0.$$
(11)

And

$$1 - {\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)^\lambda } + {\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)^\lambda } = 1 - \left( {{{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } - {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right) \in \left[ {0,1} \right].$$
(12)

Thus,

$$\begin{aligned} {v_E} \le 1 - {\left( {1 - \prod\limits_{i = 1}^n {{{\left( 1 \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} = & 1 - {\left( {1 - 1 + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{\scriptstyle{{\tilde A}_{index(i)}}\hfill\atop \scriptstyle\hfill}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}\\ = 1 - {\left( {\prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} = 1 - \left( {\prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} } \right) \le 1 \end{aligned}$$
(13)

By Eqs. (11) and (13), we have

$$0 \leqslant {v_E} \leqslant 1.$$
(14)

By Eq. (12) and \(0 \leqslant {w_i} \leqslant 1\), we obtain

$${\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)^{{w_i}}} \in \left[ {0,1} \right],$$

Thus,

$$1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} \in \left[ {0,1} \right].$$
(15)

By Eqs. (10) and (15), we have

$${u_E} \geqslant {\left( {0 + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\lambda $}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} = 0.$$
(16)

And

$$\left( {\prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} } \right) \geqslant \prod\limits_{i = 1}^n {{{\left( {0 + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{\lambda \cdot {w_i}}}} = 0,$$
(17)

By Eqs. (10) and (17), we have

$${u_E} \leqslant {\left( {1 - 0} \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$\lambda $}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} = 1 - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} \leqslant 1.$$
(18)

Then by Eqs. (16) and (18), we obtain

$$0 \le {u_E} \le 1.$$
(19)

According to Eqs. (9) and (10), we have

$${u_E} + {v_E} = 1 - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} \in \left[ {0,1} \right].$$
(20)

Therefore, according to Eqs. (14), (19) and (20) and Definition 1, we have

$${\text{GIFIHI}}{{\text{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) \in {\text{IFNs}}(X).$$

Theorem 5

Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs, \(\lambda> 0\). \(\lambda \to 0.\) Then the GIFIHIA operator approaches the following limit.

$$\mathop {Lim}\limits_{\lambda \to 0} GIFIHI{A_\lambda }({A_1}, \cdots {A_n}) = \left\langle {{e^{\sum\limits_{i = 1}^n {\left( {{w_i}\left( {ln\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right) - ln\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)} \right)} \right)} + ln\left( {\prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} } \right)}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} ,} \right.\,\left. {1 - {e^{\sum\limits_{i = 1}^n {\left( {ln\left( {1 - {v_{{{\tilde A}_{index(i)}}}}} \right) - ln\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)} \right)} + ln\prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(i)}}}} + {v_{{{\tilde A}_{index(i)}}}}} \right)} \right)}^{{w_i}}}} }}} \right\rangle .$$
(21)

And \(\mathop {Lim}\limits_{\lambda \to 0} GIFIHI{A_\lambda }({A_1},{A_2}, \ldots {A_n}) \in IFNs(X).\)

Proof

Similar to He et al. (2014b) and omitted here.

Theorem 6

(Idempotency) Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs, \(\lambda> 0\).\(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\) is the weighting vector of \({A_i}\), satisfying \({w_i} \in [0,1\) and \(\succ\). If all \({A_i}\) are equal, denoted as A, and \(\omega = \left( {\frac{1}{n}, \cdots \frac{1}{n}} \right)\), then \({\text{GIFIHI}}{{\text{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) = A\).

Proof

Let \({A_i} = A = \left\langle {{u_A},{v_A}} \right\rangle {\kern 1pt} {\kern 1pt} \left( {i = 1,2, \ldots ,n} \right)\), if \(\omega = \left( {\frac{1}{n}, \cdots \frac{1}{n}} \right)\), it has \(n\omega {A_i} = A\). By Eq. (8), taking note of \(\sum\nolimits_{i = 1}^n {{w_i}} = 1\), we obtain

$$\begin{aligned} {\rm{GIFIHI}}{{\rm{A}}_\lambda }({A_1},{A_2}, \ldots {A_n}) = & \left\langle {{{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{{w_i}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{\lambda \cdot {w_i}}}} } } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{{w_i}}}} } \right.,\,\left. {1 - {{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{{w_i}}}} + \prod\limits_{i = 1}^n {{{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{\lambda {w_i}}}} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}} \right\rangle \\ = \left\langle {{{\left( {1 - {{\left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{\sum\limits_{i = 1}^n {{w_i}} }} + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{\sum\limits_{i = 1}^n {\lambda \cdot {w_i}} }}} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} - {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{\sum\limits_{i = 1}^n {{w_i}} }}} \right.\left. {1 - {{\left( {1 - {{\left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{\sum\limits_{i = 1}^n {{w_i}} }} + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^{\sum\limits_{i = 1}^n {\lambda {w_i}} }}} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}} \right\rangle \\ = \left\langle {{{\left( {1 - \left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right) + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} - \left( {1 - \left( {{u_A} + {v_A}} \right)} \right)} \right.,\,\left. {1 - {{\left( {1 - \left( {1 - {{\left( {1 - {v_A}} \right)}^\lambda } + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right) + {{\left( {1 - \left( {{u_A} + {v_A}} \right)} \right)}^\lambda }} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}} \right\rangle \\ = \left\langle {{{\left( {{{\left( {1 - {v_A}} \right)}^\lambda }} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}} - \left( {1 - \left( {{u_A} + {v_A}} \right)} \right),1 - {{\left( {{{\left( {1 - {v_A}} \right)}^\lambda }} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 \lambda }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\lambda $}}}}} \right\rangle = \left\langle {{u_A},{v_A}} \right\rangle . \end{aligned}$$

Theorem 7

(Commutativity) Let \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) be a collection of IFNs. \(\lambda> 0\). \(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\) is the weighting vector of \({A_i}\) and \({w_i} = \frac{1}{n}{\kern 1pt} {\kern 1pt} (i = 1, \ldots ,n)\). If \(\left( {{{A'}_1},{{A'}_2}, \ldots {{A'}_n}} \right)\) is any permutation of \(\left( {{A_1},{A_2}, \ldots {A_n}} \right)\), then \(GIFIHI{A_\lambda }({A_1},{A_2}, \ldots {A_n}) = GIFIHI{A_\lambda }({A'_1},{A'_2}, \ldots {A'_n}).\).

Proof

By Eq. (8) and the condition that \(\left( {{{A'}_1},{{A'}_2}, \ldots {{A'}_n}} \right)\) is any permutation of \(\left( {{A_1},{A_2}, \ldots {A_n}} \right)\), we can get the result directly.

5 Selection of cold chain logistics enterprises under intuitionistic fuzzy environment

Intuitionistic fuzzy multiple attribute decision making (IFMADM) problems are the process of choosing the best alternative from all of the possible alternatives which are evaluated by several attributes. A cold chain is a temperature-controlled supply chain, which is used to help extend and ensure the shelf life of products. Suppose that \(X = \{ {x_1},{x_2}, \ldots ,{x_n}\}\) is a set of cold chain logistics enterprises, \(G = \{ {G_1},{G_2}, \ldots ,{G_m}\}\) is a set of attributes with the associated weighting vector \(w = {({w_1},{w_2}, \ldots ,{w_m})^T}\), satisfying \({w_i} \in [0,1\) and \(\sum\nolimits_{i = 1}^m {{w_i}} = 1\).

Suppose that the evaluated values of the cold chain logistics enterprises \({x_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) under the attribute G j are represented by IFNs \({A_{ij}} = \left\langle {{u_{{A_{ij}}}},{v_{{A_{ij}}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n\,j = 1,2, \ldots ,m),\) where \({u_{{A_{ij}}}}\) reflects the degree that the alternative \({x_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) satisfies the attribute \(G = \{ {G_1},{G_2}, \ldots ,{G_m}\} \,{v_{{A_{ij}}}}\) indicates the opposite meaning. The construction of IFNs can refer to Pedrycz and Chen (2015), Lorkowski and Kreinovich (2015) and Naim and Hagras (2015).

Then the selection of cold chain logistics enterprises under intuitionistic fuzzy environment based on the GIFIHIA operator can be listed as follows.

Step 1 Determine the weights \(w = {({w_1},{w_2}, \ldots ,{w_n})^T}\) and \(\omega = {({\omega _1}, \ldots {\omega _n})^T}\), the meaning of w and \(\omega\) please refer to Definitions 4 and 5.

Step 2 Based on the GIFIHIA operator in Deinition 5, we aggregate the decision information by experts into collective ones, and the GIFIHIA operator is listed as Eq. (22).

$$\begin{aligned} {A_i} = & {\rm{GIFIHI}}{{\rm{A}}_\lambda }({A_{i1}},{A_{i2}}, \ldots {A_{in}})\\ = \left\langle {{{\left( {1 - \prod\limits_{j = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(ij)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(ij)}}}} + {v_{{{\tilde A}_{index(ij)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_j}}} + \prod\limits_{j = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(ij)}}}} + {v_{{{\tilde A}_{index(ij)}}}}} \right)} \right)}^{\lambda \cdot {w_j}}}} } } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}} - \prod\limits_{j = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(ij)}}}} + {v_{{{\tilde A}_{index(ij)}}}}} \right)} \right)}^{{w_j}}}} ,} \right.\\ \quad \,\left. {1 - {{\left( {1 - \prod\limits_{j = 1}^n {{{\left( {1 - {{\left( {1 - {v_{{{\tilde A}_{index(ij)}}}}} \right)}^\lambda } + {{\left( {1 - \left( {{u_{{{\tilde A}_{index(ij)}}}} + {v_{{{\tilde A}_{index(ij)}}}}} \right)} \right)}^\lambda }} \right)}^{{w_j}}}} + \prod\limits_{j = 1}^n {{{\left( {1 - \left( {{u_{{{\tilde A}_{index(ij)}}}} + {v_{{{\tilde A}_{index(ij)}}}}} \right)} \right)}^{\lambda {w_j}}}} } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}}} \right\rangle . \end{aligned}$$
(22)

Step 3 Rank the final IFNs \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {i = 1,2, \ldots ,n} \right)\) by the score the accuracy functions in Definition 2.

Step 4 Rank the possible cold chain logistics enterprises \({x_i}{\kern 1pt} {\kern 1pt} \left( {i = 1,2, \ldots ,n} \right)\) and choose the best ones.

Step 5 Adjust the values of the parameter \(\lambda\) according to decision makers’ preference, and analyze the rakings of the cold chain logistics enterprises with different values of the parameter \(\lambda\).

Step 6 Draw the figure of Step 5 to illustrate the selections of the cold chain logistics enterprises further.

6 Numerical example

Suppose that a foodstuff general corporation plan to choose a cold chain logistics enterprise to store and transport its goods. After thinking the market environment, three possible cold chain logistics enterprises (x1, x2, x3) are to be considered. For the sake of choosing the best cold chain logistics enterprise to cooperate, the foodstuff general corporation has brought a panel. After careful thinking of the alternatives and decision environment, five attributes \(G = \left\{ {{G_1},{G_2},{G_3},{G_4},{G_5}} \right\}\) are concluded to evaluate the ability of the candidate cold chain logistics enterprises.

  • \({G_1}\): Storage ability.

  • \({G_2}\): The levels of processing.

  • \({G_3}\): Transportation capability.

  • \({G_4}\): Logistics support capability

  • \({G_5}\): Coordinating optimization ability of business operations.

The evaluated decision information of the three possible alternatives under the above five attributes are presented by IFNs (Table 1).

Table 1 Intuitionistic fuzzy matrix \({\left( {{A_{ij}}} \right)_{3 \times 5}}\)
Full size table

Preliminary According to the method of constructing IFNs with granularity in Sect. 5, for example, experts mark her confidence by left membership value 2 and right membership value 5 on a scale from 1 to 10 to indicate the degrees x 1 satisfy the property G 1, thus \({A_{11}}\) in the numerical example is \(\left\langle {\underline {{u_{{A_{11}}}}} ,1 - \overline {{u_{{A_{11}}}}} } \right\rangle = \left\langle {2/10,1 - 5/10} \right\rangle = < 0.2,0.5>\). Similarly, we have \({A_{ij}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2,3;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1, \ldots ,5),\) and all evaluated IFNs are listed in Table 1.

Step 1 The weights w and \(\omega\) are given by the experts as w = (0.112, 0.236, 0.304, 0.236, 0.112) and \(\omega = (0.2, \ldots 0.2)\).

Step 2 Suppose the induced values of \({\tilde A_{ij}}(i = 1,2,3;j = 1, \ldots ,5){\kern 1pt} {\kern 1pt} {\kern 1pt}\)given by the exsperts are \({p_{i1}} = 0.9,{\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{i2}} = 0.8,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{i3}} = 0.6,{\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{i4}} = 0.5,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {p_{i5}} = 0.2\quad (i = 1,2,3),\) then by Eq. (22), and taking \(\lambda = 0.5\), we obtain

$${A_1} = GIFIHI{A_{0.5}}({A_{11}}, \ldots {A_{15}}) = \left\langle {0.4387,0.3262} \right\rangle .$$

Similarly, \({A_2} = \left\langle {0.4515,0.3625} \right\rangle ,\,{A_3} = \left\langle {0.3772,0.4413} \right\rangle\).

Step 3 By the score function proposed by Chen and Tan (1994), we have

$$S\left( {{A_1}} \right) = 0.1125,\,S\left( {{A_2}} \right) = 0.0891,\,S\left( {{A_3}} \right) = - 0.0640.$$

Step 4 By Definition 2, we have \(S\left( {{A_1}} \right)> S\left( {{A_2}} \right)> S\left( {{A_3}} \right)\) and \({x_1} \succ {x_2} \succ {x_3}\).

Step 5 Taking different values of the parameter \(\lambda\) and \(\lambda = {\text{ }}0.00{\text{1}},{\text{ }}0.00{\text{2}},{\text{ }}0.00{\text{3}} \ldots ,{\text{ 2}}0,\) here we just list the scores and the rankings of involved alternatives with \(\lambda = 0.1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} 2,{\kern 1pt} {\kern 1pt} {\kern 1pt} 3,{\kern 1pt} {\kern 1pt} {\kern 1pt} 5,{\kern 1pt} {\kern 1pt} {\kern 1pt} 10,{\kern 1pt} {\kern 1pt} 15,{\kern 1pt} {\kern 1pt} 20\) as in Table 2.

Table 2 Score function obtained by the GIFIHIA operator and the rank of the alternatives
Full size table

Step 6. The scores of invovled alternatives with different values of parameter \(\lambda\) in Step 5 are shown in Fig. 1.

Fig. 1
figure 1

Scores of the alternatives with different values of parameter \(\lambda\)

Full size image

From Fig. 1, we find that \(S\left( {{A_1}} \right)\) increases as \(\lambda\) increases on (0,0.7883], and decreases as \(\lambda\) increases on (0.7883, 20]. \(S\left( {{A_2}} \right)\) and \(S\left( {{A_3}} \right)\) increase as \(\lambda\) increases on (0, 20]. Moreover, the ranking of the alternatives with different values of the parameter \(\lambda\) are concluded as follows.

  1. 1.

    When \(\lambda \in (0,1.0095)\), the ranking of the three possible cold chain logistics enterprises is \({x_1} \succ {x_2} \succ {x_3}\). Therefore, \({x_1}\) is the best alternative.

  2. 2.

    When \(\lambda \in (1.0095,20)\), the ranking of the three possible cold chain logistics enterprises is \({x_2} \succ {x_1} \succ {x_3}\). Thus, \({x_2}\) is the best alternative.

If we use the aggregation operators in Zhao et al. (2010) to aggregate a set of IFNs, when there exists only one non-membership degree of IFN equals to zero, the non-membership degree of aggregation result of n IFNs is zero even if the non-membership degrees of n − 1 IFNs are not zero, which is the weakness of the aggregation operators in Zhao et al. (2010). However, if we use the operators developed in this paper, the aggregation result can be explained reasonably. For example, Let \({A_1} = \left\langle {0.3,0.5} \right\rangle ,\,{A_2} = \left\langle {0.4,0.4} \right\rangle ,\,{A_3} = \left\langle {0.3,0.6} \right\rangle ,\,{A_4} = \left\langle {0.4,0.5} \right\rangle ,\,{A_5} = \left\langle {0.5,0} \right\rangle\) be five IFNs, \(w = {\left\{ {0.25,0.20,0.15,0.18,0.22} \right\}^T}\) be the corresponding weight vector. By the generalized intuitionistic fuzzy weighted averaging (GIFWA) operator in Zhao et al. (2010), i.e.,

$$GIFW{A_\lambda }({A_1},{A_2}, \ldots {A_n}) = \left\langle {{{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - {u_{{A_i}}}^\lambda } \right)}^{{w_i}}}} } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}},1 - {{\left( {1 - \prod\limits_{i = 1}^n {{{\left( {1 - \left( {1 - {v_{{A_i}}}^\lambda } \right)} \right)}^{{w_i}}}} } \right)}^{{1 \mathord{\left/ {\vphantom {1 \lambda }} \right. \kern-\nulldelimiterspace} \lambda }}}} \right\rangle$$
(23)

Taking \(\lambda = 0.5\), we get \(GIFW{A_{0.5}}({A_1},{A_2}, \ldots {A_5}) = \left\langle {0.3845,0} \right\rangle .\)

If we use the generalized intuitionistic fuzzy induced hybrid interaction averaging (GIFIHIA) operator in this paper, taking \(\lambda = 0.5\) and \(\omega = {(0.2, \ldots 0.2)^T}\) for the convenience of comparison, we obtain

$$GIFIHI{A_{0.5}}({A_1},{A_2}, \ldots {A_5}) = \left\langle {0.3758,0.4295} \right\rangle .$$

Obviously \({v_{{\text{GIFW}}{{\text{A}}_{0.5}}({A_1},{A_2}, \ldots {A_5})}} = 0\), while \({v_{{\text{GIFIHI}}{{\text{A}}_{0.5}}({A_1},{A_2}, \ldots {A_5})}} = 0.4295 \ne 0\), which shows that \({v_{{A_5}}} = 0\) plays a decisive role by Eq. (23),While doesn’t play a decisive role by Eq. (22). Therefore, the new generalized weighted operator developed by this paper is more practical from the averaging point of view.

The characteristics of GIFIHIA operator can be interpreted in the following five aspects.

  1. 1.

    It considers three relations of different IFNs: the interactions of membership function of different IFNs, the interactions of non-membership function of different IFNs and the interactions between membership function and non-membership function of different IFNs.

  2. 2.

    It weights the IFNs \({A_i} = \left\langle {{u_{{A_i}}},{v_{{A_i}}}} \right\rangle {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) by the associated weights \(\omega = {({\omega _1}, \ldots {\omega _n})^T}\)and multiplies these numbers by a balancing coefficient n, and then gets the weighted IFNs \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\).

  3. 3.

    It reorders the weighted IFNs \(n{\omega _i}{A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) according to the induced value \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \cdots ,n)\) as \(\left( {{{\tilde A}_{index(1)}},{{\tilde A}_{index(2)}}, \ldots ,{{\tilde A}_{index(n)}}} \right)\).

  4. 4.

    Both the weighted IFNs \({w_i}{\tilde A_{index(i)}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) and their induced value \({p_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) are considered, and all the IFNs \({A_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (i = 1,2, \ldots ,n)\) are aggregated into a collective one.

  5. 5.

    The attitude of decision makers are considered by taking different values of \(\lambda\) according to decision makers’ preferences.

7 Conclusions

In this paper, we present the IFIHIA operator and the GIFIHIA operator, taking the interactions of different IFNs into consideration, reordering the involved IFNs according to the induced values and then aggregate them into a collective one, considering the attitudes of decision makers by taking different values of parameter according to decision makers’ preferences. We investigate the properties of these operators and apply them to the selection of cold chain logistics enterprises under intuitionistic fuzzy environment. Examples are illustrated to show the validity and feasibility of the new approach. We also give some comparisons between this paper and other papers.

In the succeeding work, we will develop a class of fuzzy numbers intuitionistic fuzzy hybrid interaction averaging operators based on the existing works and apply them to the selection of cold chain logistics enterprises, decision support, recommender systems and multiple attribute group decision making.