1 Introduction

Multi-criteria decision-making (MCDM) method [1] is a decision-making approach to select the best alternative or rank the alternatives considering multi-criteria, which has a wide range of applications in many fields [2, 3]. Due to the complexity and uncertainty of objective things and the ambiguity of human thinking, the study of MCDM problems in uncertain environments has attracted great attention. To solve problems in uncertain environments and provide better solutions, the concept of fuzzy sets [4] and its related fuzzy decision-making methods are proposed. In recent years, the construction of intuitionistic fuzzy sets (IFSs) [5], Pythagorean fuzzy sets (PFSs) [6] and other fuzzy decision-making theories and methods mainly benefits from the continuous development and improvement of early fuzzy set theory. Based on the representation of uncertain information by membership degree \(\mu\) of fuzzy sets, the ability of IFSs to deal with uncertain decision-making problems is further improved by introducing non-membership degree \(v\), accompanied by constraint \(\mu + v \le 1\), and provide a paradigm for the development of PFSs. As an extension of the IFSs, PFSs with wider membership and non-membership values (\(\mu^{{2}} { + }v^{2} \le 1\)) are proposed. By taking advantage of the more flexible uncertainty representation brought by the expansion of the value range, N-soft model, VIKOR method and rough set model are successively extended to PFSs environment, and a series of Pythagorean fuzzy decision-making approaches are developed [7,8,9,10]. Moreover, as an extension of PFSs, the concept of q-rung orthopair fuzzy sets (q-ROFSs) is defined, which can be degenerated into PFSs and IFSs. The WASPAS, QUALIFLEX and other traditional methods are extended to q-ROFSs environment to form some new decision-making methods, which enriches the PFSs theory and method system [11, 12].

Frequent interactions between attributes in real decision-making problems [13,14,15], such as the correlation between the interior decoration and comfort of automobile products, promoted the interactive decision-making of criteria to become one of the important branches of multi-criteria decision-making [13,14,15]. Distortion of decision-making results can be caused by the information superposition effect between related criteria. From the perspective of aggregation operators, there are two types of operators mainly focusing on the study of criterion interaction: the Choquet integral (CI) operator [16, 17] and the Bonferroni mean (BM) operator [18, 19]. CI operator is characterized by portraying the interaction between criteria through weights. BM operator is characterized by the interaction operation between information to portray the interaction between criteria. However, these existing operators rely heavily on the subjective settings of the decision-maker to determine the interaction or independence between criteria. For example, the degree of interaction between criterion \(c_{i}\) and \(c_{j}\), and whether criterion \(c_{i}\) and criterion \(c_{j}\) are independent, are subjectively decided by experts or decision-making makers.

Based on the research foundation of BM and CI operators, Dutta [20] proposed an extended weighted 2-tuple linguistic BM operator to overcome the shortcomings of the above operators. This operator is characterized by two different operation structures. Although this operator overcomes the difficulty of traditional BM operators to describe the interaction and independence of criteria at the same time, it focuses less on the scientific quantification of the correlation of criteria. Therefore, based on the good foundation laid by these operators, the Bonferroni mean with weighted interaction (WIBM) operator [21] emerge as the times require, which can handle both independent and non-independent criterion information and quantifiably characterize the degree of interaction. Although the WIBM operator has embedded the interaction coefficients of criteria into the internal structure of the BM operator, it is mainly applied in the MCDM problem with a small amount of data or expert evaluation information, resulting in the generation of its interaction coefficients mainly determined by the subjective settings of experts. It is worth noting that existing data-driven MCDM methods, such as MCDM problems for products based on user reviews [22,23,24], with many evaluation information sources, which provide new ideas for the generation methods of interaction coefficients. In this paper, a data-driven interaction operator method to generate interaction coefficients is used in the context of user rating. An interactive coefficient generation method that combines big data with weighted interactive BM operator by taking advantage of data and is driven by expert knowledge and data synergy is proposed.

In recent years, with the continuous development of PFSs, the BM operators are expanded to develop some Pythagorean fuzzy BM operators [25, 26] and related multi-criterion interactive decision-making methods. However, after case analysis, it is found that the above operators are difficult to process the independent and non-independent criterion information at the same time, and this deficiency is made up by real data. Inspired by Andrea [21], the Bonferroni mean with weighted interaction operator is extended into the Pythagorean fuzzy environment, and the interaction mechanism of the interaction coefficients of the original weighted interaction BM operator is discussed, and the Pythagorean fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is developed to fusion multidimensional online ratings. With help of the transformation method of ratings and PFSs, the interaction coefficient generation method of PFWIBM operator is proposed by combining expert knowledge and user ratings, which can overcome the disadvantages of completely relying on subjective methods and make full use of the advantages of data resources.

The current study is arranged as the following. In Sect. 2, we briefly review some basic concepts of IFSs, PFSs and BM operators. The PFWIBM operator is presented, and some properties of PFWIBM operator are explored in Sect. 3. Section 4 provides an online multi-dimensional rating aggregation decision-making approach for solving product ranking problems. In Sect. 5, a passenger car ranking case is furnished to show the effectiveness and scientific of the proposed methods and compare with some of the existing operators. Section 6 concludes the study and elaborates on future studies.

2 Preliminaries

The current section presents some basic concepts, including the IFSs, PFSs and the BM operators.

2.1 Some Basic Concepts of IFSs and PFSs

Definition 1

[5] Let \(X\) be a universe of discourse. An intuitionistic fuzzy set (IFS) \(I\) in \(X\) is given by

$$ I = \, \left\{ {\left\langle {x, \, \mu_{I} \left( x \right), \, v_{I} \left( x \right)} \right\rangle \left| {x \in } \right.X} \right\}, $$
(1)

where \(\mu_{I} :X \to \left[ {0,1} \right]\) is the membership function,\(v_{I} :X \to \left[ {0,1} \right]\) is the non-membership function, and \(0 \le \mu_{I} \left( x \right) + v_{I} \left( x \right) \le 1\). For simplicity, we call the two tuples \(\left( {\mu_{I} \left( x \right), \, v_{I} \left( x \right)} \right)\) as intuitionistic fuzzy number (IFN) and simply express it as \(P = \left( {\mu_{P} ,v_{P} } \right)\).

Definition 2

[6] Let \(X\) be a universe of discourse. A Pythagorean fuzzy set (PFS) \(P\) in \(X\) is given by

$$ P = \, \left\{ {\left\langle {x, \, \mu_{P} \left( x \right), \, v_{P} \left( x \right)} \right\rangle \left| {x \in } \right.X} \right\}, $$
(2)

where \(\mu_{P} :X \to \left[ {0,1} \right]\) is the membership function,\(v_{P} :X \to \left[ {0,1} \right]\) is the non-membership function, and \(0 \le \left( {\mu_{P} \left( x \right)} \right)^{2} + \left( {v_{P} \left( x \right)} \right)^{2} \le 1\). And \(\pi_{P} \left( x \right) = \sqrt {1 - \left( {\mu_{P} \left( x \right)} \right)^{2} - \left( {v_{P} \left( x \right)} \right)^{2} }\) is the hesitant degree. For simplicity, we call the two tuples \(\left( {\mu_{P} \left( x \right), \, v_{P} \left( x \right)} \right)\) as Pythagorean fuzzy number (PFN) and simply express it as \(P = \left( {\mu_{P} ,v_{P} } \right)\).

The main difference between PFN and IFN is their corresponding constraint conditions, are shown in Fig. 1.

Fig. 1
figure 1

Comparison of spaces of the PFNs and IFNs

Full size image

Definition 3

[25] Let \(\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)\) be an PFN, the score function of \(\alpha\) be defined as

$$ s\left( \alpha \right) = \mu_{\alpha }^{2} - v_{\alpha }^{2} ,\quad {\text{where}}\quad s\left( \alpha \right) \in \left[ { - 1,1} \right]. $$
(3)

Based on the above definition, Zhang and Xu [27] defined the following concept of the distance measure for PFNs.

Definition 4

[27] Let \(\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)\) and \(\beta = \left( {\mu_{\beta } ,v_{\beta } } \right)\) be two PFNs, the distance between \(\alpha\) and \(\beta\) is defined as

$$ d\left( {\alpha ,\beta } \right) = \frac{1}{2}\left( {\left| {\left( {\mu_{\alpha } } \right)^{2} - \left( {\mu_{\beta } } \right)^{2} } \right| + \left| {\left( {v_{\alpha } } \right)^{2} - \left( {v_{\beta } } \right)^{2} } \right| + \left| {\left( {\pi_{\alpha } } \right)^{2} - \left( {\pi_{\beta } } \right)^{2} } \right|} \right). $$
(4)

Definition 5

[25] Let \(p = \left( {\mu ,v} \right),p_{1} = \left( {\mu_{1} ,v_{1} } \right)\) and \(p_{2} = \left( {\mu_{2} ,v_{2} } \right)\) be three PFNs, then we have:

$$ (1)\;p_{1} \oplus p_{2} = \left( {\sqrt {\mu_{1}^{2} + \mu_{2}^{2} - \mu_{1}^{2} \mu_{2}^{2} } ,v_{1} v_{2} } \right)$$
$$(2)\;p_{1} \otimes p_{2} = \left( {\mu_{1} \mu_{2} ,\sqrt {v_{1}^{2} + v_{2}^{2} - v_{1}^{2} v_{2}^{2} } } \right) $$
$$ (3)\;\lambda p = \left( {\sqrt {1 - \left( {1 - \mu^{2} } \right)^{\lambda } } ,v^{\lambda } } \right);\lambda > 0$$
$$ (4)\;p^{\lambda } = \left( {\mu^{\lambda } ,\sqrt {1 - \left( {1 - v^{2} } \right)^{\lambda } } } \right);\lambda > 0 $$
$$ (5)\;p_{1} \oplus p_{2} = p_{2} \oplus p_{1}$$
$$ (6)\;p_{1} \otimes p_{2} = p_{2} \otimes p_{1} $$
$$ (7)\;\lambda \left( {p_{1} \oplus p_{2} } \right) = \lambda p_{1} \oplus \lambda p_{2} ;\lambda > 0 $$
$$(8)\;\lambda_{1} p \oplus \lambda_{2} p = \left( {\lambda_{1} + \lambda_{2} } \right)p;\lambda_{{1}} ,\lambda_{2} > 0 $$
$$ (9)\;\left( {p_{1} \otimes p_{2} } \right)^{\lambda } = p_{1}^{\lambda } \otimes p_{2}^{\lambda } ;\lambda > 0$$
$$(10)\;p^{{\lambda_{1} }} \otimes p^{{\lambda_{2} }} = p^{{\left( {\lambda_{1} + \lambda_{2} } \right)}} ;\lambda_{{1}} ,\lambda_{2} > 0 $$

2.2 Bonferroni Mean Operators

The desirable characteristic of the Bonferroni mean (BM) [20] is that it can capture the interrelationship between input arguments.

Definition 6

[20] Let \(p,q \ge 0\), and \(a_{i} \ge 0\,\,\left( {i = 1,2, \ldots ,n} \right)\) be a collection of crisp data. The BM is defined as:

$$ {\text{BM}}^{p,q} \left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) = \left( {\frac{1}{{n\left( {n - 1} \right)}}\sum\limits_{i,j = 1;i \ne j}^{n} {a_{i}^{p} a_{j}^{q} } } \right)^{{\frac{1}{p + q}}} . $$
(5)

Based on the above BM operator, the normalized Bonferroni mean with weighted interaction (NWIBM) is proposed [21].

Definition 7

[21] Let \(p,q \ge 0\), and \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of crisp data. The NWIBM operator is given by:

$$ {\text{NWIBM}}^{p,q} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} x_{i}^{p} \left( {\sum\limits_{j = 1}^{n} {w_{i,j} x_{j}^{q} } \bigg/\sum\limits_{j = 1}^{n} {w_{i,j} } } \right)} } \right)^{{\frac{1}{p + q}}} , $$
(6)

where \(w_{i,j} \in [0,1]\,\,\left( {i \ne j;\,\, i,j = 1,2, \ldots ,n} \right)\) be a collection of weights such that \(w_{i,j} \ge 0\) for all \(i,j = 1, \ldots ,n\) and \(w_{i,i} = 0\left( {i = 1,2, \ldots n} \right)\). Furthermore, \(w_{i} \in [0,1]\left( {i = 1,2, \ldots ,n} \right)\) be a collection of weights such that \(w_{i} \ge 0\) and \(\sum\nolimits_{i = 1}^{n} {w_{i} = 1}\).

Note that the weights \(w_{i,j} \in [0,1]\left( {i \ne j;i,j = 1,2,...,n} \right)\) from the previous definition can be seen as a quantitative evaluation of the positive interaction between different criteria i and j.

Specially, when \(w_{i,j} = w_{i} \cdot w_{j}\), the NWIBM operator reduces to the NWBM operator [28]:

$$ {\text{NWBM}}^{p,q} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} x_{i}^{p} \left( {\sum\limits_{j = 1,i \ne j}^{n} {\frac{{w_{j} x_{j}^{q} }}{{1 - w_{i} }}} } \right)} } \right)^{{\frac{1}{p + q}}} . $$
(7)

When \(w_{i} = {1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}\left( {i = 1,2, \ldots ,n} \right)\), the NWIBM operator reduces to the WBM operator [19]:

$$ {\text{WBM}}^{p,q} \left( X \right) = \left( {\frac{1}{n}\sum\limits_{{i{ = }1}}^{n} {x_{i}^{p} } \frac{1}{n - 1}\left( {\sum\limits_{j = 1;j \ne i}^{n} {x_{j}^{q} } } \right)} \right)^{{\frac{1}{p + q}}} . $$
(8)

Theorem 1:

[21] The Bonferroni mean with weighted interaction is an averaging n-ary aggregation function. Therefore, the NWIBM operator satisfies boundedness, idempotency and monotonicity.

Let \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of crisp data, then

  1. (1)

    (Boundedness) if \(x_{u} = \max_{i} \left( {x_{i} } \right),x_{l} = \min_{i} \left( {x_{i} } \right)\) for all \(i = 1,2, \ldots ,n\), then

    $$ x_{l} \le {\text{NWIBM}}^{p,q} \left( X \right) \le x_{u} $$
  2. (2)

    (Idempotency) if \(x_{i} = x_{0} \left( {i = 1,2, \ldots ,n} \right)\), then \({\text{NWIBM}}^{p,q} \left( X \right) = x_{0}\)

  3. (3)

    (Monotonicity) Let \(Y = \{ y_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collections of crisp data, and if \(x_{i} \le y_{i}\) for all \(i = 1,2, \ldots ,n\), then

    $$ {\text{NWIBM}}^{p,q} \left( X \right) \le {\text{NWIBM}}^{p,q} \left( Y \right). $$

3 PFWIBM Operator

In this section, the internal structure and the shortcomings of the NWIBM operator are exposed, and an improved operator is proposed to develop the PFWIBM operator.

3.1 Improved Normalized Bonferroni Mean with Weighted Interaction Operator

The NWIBM operator (p = q = 1) is analyzed as follows (Table 1):

$$ {\text{NWIBM}}^{1,1} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} \cdot \underbrace {{x_{i} \underbrace {{\left( {\sum\limits_{j = 1}^{n} {\underbrace {{\underbrace {{\frac{{w_{i,j} }}{{\sum\nolimits_{j = 1}^{n} {w_{i,j} } }}}}_{{{\text{interaction weight }}\Delta_{ij} \in \left[ {0,1} \right]{\text{ of }}x_{i} {\text{ and }}x_{j} ,}}x_{j} }}_{{{\text{interactive information value }}\Delta_{{x_{i} x_{j} }} \in \left[ {0,1} \right]{\text{ of }}x_{i} {\text{ and }}x_{j} }}} } \right)}}_{{x_{i} {\text{ interaction value }}\Delta_{{x_{i} }} \in \left( {0,1} \right)}}}}_{{x_{i} {\text{ information value }}\Theta_{{x_{i} }} }}} } \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} . $$
(9)
Table 1 Indicator description for the NWIBM operator with \(p = q = 1\)
Full size table

Property 1

  1. (1)

    When \(x_{i}\) is independent i.e., all the interaction coefficients satisfy \(w_{i,j} = 0\), the overall information of \(x_{i}\) satisfies \(\Theta_{{x_{i} }} = \Theta_{{x_{i} }}^{indep} = x_{i}\);

  2. (2)

    when \(x_{i}\) is non-independent i.e., there is at least one interaction coefficient satisfies \(w_{i,j} > 0\left( {j \ne i} \right)\), the overall information of \(x_{i}\) satisfies \(\Theta_{{x_{i} }} = \Theta_{{x_{i} }}^{{\text{non - indep}}} < x_{i}\).

According to Property 1, the \(x_{i}\) independent information is greater than the information when interacting i.e., \(\Theta_{{x_{i} }}^{{{\text{indep}}}} > \Theta_{{x_{i} }}^{{\text{non - indep}}}\).

Property 2

\(\Delta_{{x_{i} x_{j} }} = \Delta_{ij} x_{j}\) monotonically increases with respect to interaction coefficient \(w_{i,j}\).

Property 3

\(\frac{{\Delta_{{x_{i} x_{j} }} }}{{\Delta_{{x_{i} }} }}\) and \(\frac{{\Delta_{{x_{i} x_{j} }} }}{{\Theta_{{x_{i} }} }}\) are monotonically increasing with respect to \(\Delta_{ij}\).

Obviously, there is a contradiction between Property 1 and Property 3. Therefore, the rationality of the interaction mechanism for the NWIBM operator is discussed through the following examples.

Example 1

Let \(x_{{1}}\) and \(x_{{2}}\) be the criterion values of the two interactive criteria \(C_{{1}}\) and \(C_{2}\), respectively. If \(w_{1,2}^{\left( 1 \right)} > w_{1,2}^{\left( 2 \right)}\), by Property 2, we obtain \(\Delta_{{x_{1} ,x_{2} }}^{\left( 1 \right)} > \Delta_{{x_{1} ,x_{2} }}^{\left( 2 \right)}\), then \(\Delta_{{x_{1} }}^{\left( 1 \right)} > \Delta_{{x_{1} }}^{\left( 2 \right)}\) and \(\Theta_{{x_{1} }}^{\left( 1 \right)} > \Theta_{{x_{1} }}^{\left( 2 \right)}\). It can be concluded that \(w_{i,j}\) is proportional to \(\Theta_{{x_{i} }}\).

The above analysis shows that the existing operators have problems in setting the interaction weight \(w_{i,j}\). Therefore, the following adjustments are made for the interaction weight:

$$ \overline{w}_{i,j} = 1 - w_{i,j} ,w_{i,j} \in \left[ {0,1} \right],i \ne j $$
(10)

The rationality of \(\overline{w}_{i,j}\) is proved by Example 2.

Example 2

(Continuation of Example 1) Let \(w_{1,2}^{\left( 1 \right)} > w_{1,2}^{\left( 2 \right)}\), by Property 2, we have \(\Delta_{{x_{1} ,x_{2} }}^{\left( 1 \right)} > \Delta_{{x_{1} ,x_{2} }}^{\left( 2 \right)}\), then \(\Delta_{{x_{1} }}^{\left( 1 \right)} > \Delta_{{x_{1} }}^{\left( 2 \right)} ,\Theta_{{x_{1} }}^{\left( 2 \right)} > \Theta_{{x_{1} }}^{\left( 1 \right)}\). It can be concluded that \(w_{i,j}\) is inversely proportional to \(\Theta_{{x_{i} }}\).

The result of Example 2 is consistent with the conclusion of Property 1. Accordingly, the improved NWIBM operator is defined as follows.

Definition 8

Let \(p,q \ge 0\), and \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of crisp data, the improved NWIBM operator is given by

$$ {\text{INWIBM}}^{p,q} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} x_{i}^{p} \left( {\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)x_{j}^{q} /\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} } } \right)} } \right)^{{\frac{1}{p + q}}} , $$
(11)

where \(w_{i,j} \in [0,1]\left( {i \ne j;i,j = 1,2,...,n} \right)\) be a collection of weights such that \(w_{i,j} \ge 0\) for all \(i,j = 1, \ldots ,n\) and \(w_{i,i} = 0\left( {i = 1,2,...n} \right)\). Furthermore,\(w_{i} \in [0,1]\left( {i = 1,2,...,n} \right)\) be a collection of weights such that \(w_{i} \ge 0\) and \(\sum\nolimits_{i = 1}^{n} {w_{i} = 1}\).

Obviously, the INWIBM operator still satisfies the properties of Theorem 1.

3.2 PFWIBM Operator

According to the INWIBM operator proposed in Sect. 3.1, the dual of the INWIBM operator is defined. By combining INWIBM and its dual operator, the Pythagorean fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is proposed.

  1. 1.

    Dual INWIBM operator

Definition 9

Let \(p,q \ge 0\), and \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of crisp data, the dual INWIBM (DINWIBM) operator with respect to Pythagorean negation \(N_{d} \left( x \right) = \sqrt {\left( {1 - x^{2} } \right)}\) is defined as follows.

$$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( X \right)} \right)} \right). $$
(12)

The following result is obtained based on Definitions 8 and 9.

$$ \begin{gathered} {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) \hfill \\ { = }\sqrt {1 - \left( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \left( {\sqrt {\left( {1 - x_{i}^{2} } \right)} } \right)^{p} \left( {\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)\left( {\sqrt {\left( {1 - x_{j}^{2} } \right)} } \right)^{q} /\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} } } \right)} } \right)^{{\frac{1}{p + q}}} } \right)^{2} } \hfill \\ \end{gathered} $$
(13)

Some desirable properties of the DINWIBM operator are investigated in detail.

Theorem 2

Let \(p,q \ge 0\) and \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of numbers, the following are valid:

  1. (1)

    (Idempotency) If \(x_{i} = x_{0} \left( {i = 1,2, \ldots ,n} \right)\), then

    $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) = x_{0} $$
  2. (2)

    (Monotonicity) If \(Y = \{ y_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) and \(x_{i} \le y_{i} \left( {i = 1,2, \ldots ,n} \right)\), then

    $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( Y \right) $$
  3. (3)

    (Boundedness) If \(x_{u} = \max_{i} \left( {x_{i} } \right),x_{l} = \min_{i} \left( {x_{i} } \right)\), then

    $$ x_{l} \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( X \right) \le x_{u} . $$

Proof

  1. (1)

    (1) Since \(x_{i} = x_{0} \left( {i = 1,2, \ldots ,n} \right)\), then \(N_{d} \left( {x_{i} } \right) = N_{d} \left( {x_{0} } \right)\). Therefore,

    $$ \begin{aligned} {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) & = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right)} \right) \\ & = N_{d} \left( {N_{d} \left( {x_{0} } \right)} \right) = x_{0} \\ \end{aligned} $$
  2. (2)

    Since \(x_{i} \le y_{i} \left( {i = 1,2, \ldots ,n} \right)\) and \(N_{d}\) is a monotone decreasing function, then

    $$ \begin{aligned} & {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right)} \right) \\ & \quad \le {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( Y \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {y_{1} } \right),N_{d} \left( {y_{1} } \right), \ldots ,N_{d} \left( {y_{n} } \right)} \right)} \right) \\ & \quad \Leftrightarrow \\ & \quad {\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right) \\ & \quad \ge {\text{INWIBM}}^{p,q} \left( {N_{d} \left( {y_{1} } \right),N_{d} \left( {y_{1} } \right), \ldots ,N_{d} \left( {y_{n} } \right)} \right) \\ & \quad \Leftrightarrow \\ & \quad N_{d} \left( {x_{i} } \right) \ge N_{d} \left( {y_{i} } \right)\left( {i = 1,2, \ldots ,n} \right) \\ & \quad \Leftrightarrow \\ & \quad x_{i} \ge y_{i} \left( {i = 1,2, \ldots ,n} \right) \\ \end{aligned} $$
  3. (3)

    According to Proofs (1) and (2), we have:

    $$ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{l} ,x_{l} , \ldots ,x_{l} } \right) = x_{l} ,{\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{u} ,x_{u} , \ldots ,x_{u} } \right) = x_{u} $$
    $$ \begin{gathered} {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{l} ,x_{l} , \ldots ,x_{l} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{u} ,x_{u} , \ldots ,x_{u} } \right). \hfill \\ \end{gathered} $$

Thus, we can get

$$ x_{l} \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( X \right) \le x_{u} . $$

The proof is completed.

  1. 2.

    PFWIBM operator.

According to the INWIBM operator and its dual operator, the Pythagorean fuzzy INWIBM (PFWIBM) operator is proposed.

Definition 10

Let \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \ldots ,n} \right)\) be a collection of PFNs, and \(p,q \ge 0\), the PFWIBM operator is defined as follows:

$$ \begin{aligned} & {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \\ & \quad { = }\left( {{\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right),{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right), \\ \end{aligned} $$
(14)

where \(w_{i,j} \ge 0\left( {i \ne j;i,j = 1,2, \ldots ,n} \right)\).

According to Definitions 8 and 9, the concrete form of the operator can be obtained:

$$ \begin{aligned} & {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \\ & \quad = \Bigg( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \mu_{i}^{p} \left( {\sum\limits_{j = 1}^{n} {\Delta_{ij} \mu_{j}^{q} } } \right)} } \right)^{{\frac{1}{p + q}}} ,\sqrt {1 - \left( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \left( {\sqrt {\left( {1 - v_{i}^{2} } \right)} } \right)^{p} \left( {\sum\limits_{j = 1}^{n} {\Delta_{ij} \left( {\sqrt {\left( {1 - v_{j}^{2} } \right)} } \right)^{q} } } \right)} } \right)^{{\frac{1}{p + q}}} } \right)^{2} } } \Bigg), \\ \end{aligned} $$
(15)

where \(\Delta_{ij} = \frac{{1 - w_{i,j} }}{{\sum\nolimits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} }}\left( {i \ne j;\,\, i,\,\, j = 1,2, \ldots ,n} \right).\)

Theorem 3

Let \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \ldots ,n} \right)\) be a collection of PFNs,

  1. (1)

    (Idempotency) If \(\alpha_{i} = \alpha_{0} = \left( {\mu_{0} ,v_{0} } \right)\left( {i = 1,2, \ldots ,n} \right)\), then

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right){ = }\alpha_{{0}} . $$
  2. (2)

    (Monotonicity) Let \(\alpha_{i} = \left( {\mu_{{\alpha_{i} }} ,v_{{\alpha_{i} }} } \right),\beta_{i} = \left( {\mu_{{\beta_{i} }} ,v_{{\beta_{i} }} } \right),i = 1,2, \ldots ,n\) be two collections of PFNs, and \(\mu_{{\alpha_{i} }} \ge \mu_{{\beta_{i} }} ,v_{{\alpha_{i} }} \le v_{{\beta_{i} }} \left( {i = 1,2, \ldots ,n} \right)\), then

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \ge {\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right). $$
  3. (3)

    (Boundness) Let \(\alpha^{ + } = \left( {\max_{j} \left\{ {\mu_{j} } \right\},\min_{j} \left\{ {v_{j} } \right\}} \right)\), \(\alpha^{ - } = \left( {\min_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\}} \right)\), then

    $$ \alpha^{ - } \le {\text{PFWIBM}}^{p,q} \left( {\alpha_{i} } \right) \le \alpha^{ + } . $$

Proof

  1. (1)

    According to Theorems 1 and 2, we have

    $$ {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right) = {\text{INWIBM}}^{p,q} \left( {\mu_{0} ,\mu_{0} , \ldots \mu_{0} } \right) = \mu_{0} $$
    $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right) = {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{0} ,v_{0} , \ldots ,v_{0} } \right) = v_{0} . $$

    Therefore,

    $$ \begin{aligned} & {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \\ & \quad { = }\left( {{\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right),{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right) \\ & \quad = \left( {\mu_{0} ,v_{0} } \right) = \alpha_{0} . \\ \end{aligned} $$

    The proof is completed.

  2. (2)

    Since \(\mu_{{\alpha_{i} }} \ge \mu_{{\beta_{i} }}\) and \(v_{{\alpha_{i} }} \le v_{{\beta_{i} }}\), and based on Theorems 1 and 2, we have

    $$ \begin{gathered} {\text{INWIBM}}^{p,q} \left( {\mu_{{\alpha_{1} }} , \ldots ,\mu_{{\alpha_{n} }} } \right) \ge {\text{INWIBM}}^{p,q} \left( {\mu_{{\beta_{1} }} , \ldots ,\mu_{{\beta_{n} }} } \right), \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\alpha_{1} }} , \ldots ,v_{{\alpha_{n} }} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\beta_{1} }} , \ldots ,v_{{\beta_{n} }} } \right). \hfill \\ \end{gathered} $$

    Divided into the following two cases to discuss:

    Case 1: At least one \(\alpha_{k} \left( {k \in \left\{ {1,2, \ldots ,n} \right\}} \right)\) exists such that \(\mu_{{\alpha_{k} }} > \mu_{{\beta_{k} }}\) or \(v_{{\alpha_{k} }} < v_{{\beta_{k} }}\), we have:

    $$ \begin{gathered} {\text{INWIBM}}^{p,q} \left( {\mu_{{\alpha_{1} }} , \ldots ,\mu_{{\alpha_{n} }} } \right) > {\text{INWIBM}}^{p,q} \left( {\mu_{{\beta_{1} }} , \ldots ,\mu_{{\beta_{n} }} } \right), \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\alpha_{1} }} , \ldots ,v_{{\alpha_{n} }} } \right) < {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\beta_{1} }} , \ldots ,v_{{\beta_{n} }} } \right). \hfill \\ \end{gathered} $$

    Then,

    $$ S\left( {{\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right)} \right) > S\left( {{\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right)} \right). $$

    Therefore,

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) > {\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right). $$

    Case 2: When \(\alpha_{i} = \beta_{i} = \left( {\mu_{{\beta_{i} }} ,v_{{\beta_{i} }} } \right),i = 1,2, \ldots ,n\), we can get:

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) = {\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right). $$

    Based on Case 1 and Case 2, we have:

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \ge {\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right). $$

    This completes the proof of the Theorem.

  3. (3)

    According to Theorem 3, we have

    $$ {\text{PFWIBM}}^{p,q} \left( {\alpha^{ - } ,\alpha^{ - } , \ldots ,\alpha^{ - } } \right) = \alpha^{ - } ,{\text{PFWIBM}}^{p,q} \left( {\alpha^{ + } ,\alpha^{ + } , \ldots ,\alpha^{ + } } \right) = \alpha^{ + } . $$

    Because \(\alpha^{ + } = \left( {\max_{j} \left\{ {\mu_{j} } \right\},\min_{j} \left\{ {v_{j} } \right\}} \right),\alpha^{ - } = \left( {\min_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\}} \right)\), then

    $$ \min_{j} \left\{ {\mu_{j} } \right\} \le \mu_{i} \le \max_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\} \ge v_{i} \ge \min_{j} \left\{ {v_{j} } \right\}. $$

    According to Theorem 3, we have

    $$ \begin{gathered} {\text{PFWIBM}}^{p,q} \left( {\alpha^{ - } ,\alpha^{ - } , \ldots ,\alpha^{ - } } \right) \le {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right), \hfill \\ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right) \le {\text{PFWIBM}}^{p,q} \left( {\alpha^{ + } ,\alpha^{ + } , \ldots ,\alpha^{ + } } \right). \hfill \\ \end{gathered} $$

    Hence, we get \(\alpha^{ - } \le {\text{PFWIBM}}^{p,q} \left( {\mu_{i} ,v_{i} } \right) \le \alpha^{ + } .\).

    The proof is completed.

Theorem 4

Let \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right),i = 1,2, \ldots ,n\) be the set of PFNs and \(p,q \ge 0\), then the aggregated value by using the PFWIBM operator is also a PFN.

Proof

To prove the PFWIBM operator is also a PFN i.e.,

$$ \begin{aligned} & \left( {{\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right)} \right)^{2} + \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right)^{2} \le 1 \\ & \quad \Leftrightarrow {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right) \le N_{d} \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right). \\ \end{aligned} $$

The above formula is proved below:

According to Definition 8, we have

$$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {v_{1} } \right),N_{d} \left( {v_{2} } \right), \ldots ,N_{d} \left( {v_{n} } \right)} \right)} \right). $$

Then

$$ \begin{aligned} & N_{d} \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right) \\ & \quad = N_{d} \left( {N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {v_{1} } \right),N_{d} \left( {v_{2} } \right), \ldots ,N_{d} \left( {v_{n} } \right)} \right)} \right)} \right) \\ & \quad {\text{ = INWIBM}}^{p,q} \left( {N_{d} \left( {v_{1} } \right),N_{d} \left( {v_{2} } \right), \ldots ,N_{d} \left( {v_{n} } \right)} \right). \\ \end{aligned} $$

Because the PFN \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \cdot \cdot \cdot ,n} \right)\) satisfies \(\mu_{i} \le N_{d} \left( {v_{i} } \right)\), according to the monotonicity of \({\text{INWIBM}}^{p,q}\), we have

$$ \begin{aligned} & N_{d} \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right) \\ & \quad {\text{ = INWIBM}}^{p,q} \left( {N_{d} \left( {v_{1} } \right),N_{d} \left( {v_{2} } \right), \ldots ,N_{d} \left( {v_{n} } \right)} \right) \ge {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right). \\ \end{aligned} $$

Thus,

$$ {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right) \le N_{d} \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right). $$

This completes the proof.

4 Multidimensional Online Rating Aggregation Decision-Making Approach

In this section, a multidimensional online rating aggregation decision-making approach is developed by using the proposed operator, which including criteria interaction coefficient learning and a multi-criteria information fusion model.

4.1 Multidimensional Online Rating Aggregation Problem

Online platforms, such as Autohome and PCauto, can provide consumers with channels to comment and make such information public, which undoubtedly helps consumers make more rational purchase decisions. Consumers can use the public information platforms to make initial screening before purchasing products and obtain reference information to provide decision reserves for offline store selection. However, the explosive growth of word-of-mouth information on the platform makes it more difficult for consumers to identify the authenticity of the information. In this paper, we address the problem of multi-criteria of products and uncertainty of user ratings to effectively aggregate online ratings to facilitate information screening by consumer groups.

For the multidimensional online rating aggregation problem, we let \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{m} } \right\}\) be a set of m products; \(E = \left\{ {e_{1} ,e_{2} , \ldots ,e_{t} } \right\}\) be a set of t experts; \(U = \left\{ {u_{1} ,u_{2} , \ldots ,u_{m} } \right\}\) be a set of the number of users who provide ratings for m products, where \(u_{i}\) indicates the number of users of product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\); \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\}\) be a set of n criteria; \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\) be a matrix of user rating information, where \(cu_{g}^{i,j}\) indicates that the ratings of the \(g{\text{th}}\left( {g = 1,2 \ldots ,u_{i} } \right)\) user for the product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\) under criteria \(C_{j} \left( {j = 1,2, \ldots ,n} \right)\).

4.2 Multi-criteria Information Fusion Model

4.2.1 Criteria Interaction Coefficient Learning

The interaction coefficients of the existing BM operators are often determined directly by experts. In this study, the operation mechanism of interaction coefficients with the support of a large number of actual ratings is optimized, and a two-stage interaction coefficient calculation method that combines expert knowledge with actual ratings is proposed.

Phase 1: The generation method for interactive criteria set based on expert knowledge: Although experts are often clustered based on the similarity of their preferences to each other [29]. However, filtering by expert knowledge for interaction attributes first is still a valid approach. The DEMATEL method is an effective method for factor analysis and identification through expert judgment [30]. However, in an uncertain environment, it is quite difficult to use the criteria of the DEMATEL method or any other decision-making method for accurate assessment. PFSs can effectively handle the uncertain environment in decision-making. In this section, the DEMATEL method is combined with PFSs and linguistic term sets transformation method [31] to construct a method for determining whether the interaction coefficients are independent.

The following steps are involved:

Step 1. Obtain the comprehensive impact matrix.

Step 1.1 Obtain the criteria interaction linguistic information matrix \(\Omega^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n} \left( {k = 1,2, \ldots ,t} \right)\) provided by the experts, where \(\alpha_{{j_{0} j}}^{k}\) indicates expert \(E_{k}\) provides information about the association between criteria \(C_{{j_{0} }}\) and \(C_{j} \left( {j_{0} \ne j;\,\, j_{0} ,\,\, j = 1,2, \ldots ,n} \right)\), and \(\alpha_{{j_{0} j}}^{k} \in \left\{ {L,ML,M,MH,H} \right\}\), and the conversion method between linguistic information and Pythagorean fuzzy information is provided in Table 2.

Table 2 Quantitative method of linguistic information [32]
Full size table

Step 1.2. Convert the linguistic values in the criteria interaction linguistic information matrix \(\Omega^{k}\) to Pythagorean fuzzy matrix \(\widetilde{\Omega }^{k} = \left( {\widetilde{\alpha }_{{j_{0} j}}^{k} } \right)_{n \times n}\), where \(\widetilde{\alpha }_{{j_{0} j}}^{k} = \left( {\mu_{{j_{0} j}}^{k} ,v_{{j_{0} j}}^{k} } \right)\) is PFN.

Step 1.3 The decision-maker indicate the importance of experts through the linguistic term set in Table 2, which is represented by \(b_{k} \in \left\{ {VU,U,G,T,VT} \right\}\left( {k = 1,2, \ldots ,t} \right)\), and get the expert influence rating \(E_{{b_{k} }}\). The expert weights are computed as follows:

$$ w_{k}^{e} = \frac{{E_{{b_{k} }} }}{{\sum\nolimits_{k = 1}^{t} {E_{{b_{k} }} } }},\left( {k = 1,2, \ldots ,t} \right),w_{k}^{e} \in \left[ {0,1} \right],\sum\limits_{k = 1}^{t} {w_{k}^{e} = 1} $$
(16)

Step 1.4 Set the criteria PFN matrix \(\widetilde{\Omega }^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n}\) into a comprehensive correlation matrix \(\widetilde{\Omega }\).

$$ \widetilde{\Omega } = \left( {\alpha_{{j_{0} j}} } \right)_{n \times n} ,\quad {\text{where}}\quad \alpha_{{j_{0} j}} = \left( {\widetilde{\mu }_{{j_{0} j}} ,\widetilde{v}_{{j_{o} j}} } \right) = \left( {\sqrt {\sum\limits_{k = 1}^{t} {w_{k}^{e} \mu_{{j_{0} j}}^{2} } } ,\sqrt {\sum\limits_{k = 1}^{t} {w_{k}^{e} v_{{j_{0} j}}^{2} } } } \right). $$
(17)

Step 1.5 The comprehensive correlation matrix \(\widetilde{\Omega } = \left( {\alpha_{{j_{0} j}} } \right)_{n \times n}\) is defuzzied to obtain the defuzzied relation matrix \(X\).

$$ X = \left( {x_{{j_{0} j}} } \right)_{n \times n} ,\quad {\text{where}}\quad x_{{j_{0} j}} = \frac{1}{2}\left( {1 + \widetilde{\mu }_{{j_{0} j}}^{2} - \widetilde{v}_{{j_{0} j}}^{2} } \right). $$
(18)

Step 1.6 The matrix \(X\) is normalized to obtain the normalized matrix \(N\).

$$ N = \lambda X,\quad {\text{where}}\quad \lambda = \frac{1}{{\max_{{1 \le j_{0} \le n}} \sum\nolimits_{j = 1}^{n} {x_{{j_{0} j}} } }}\left( {j_{0} ,j \in \left\{ {1,2, \ldots ,n} \right\}} \right). $$
(19)

Step 1.7 Obtain the comprehensive impact matrix \(M\).

$$ M = N\left( {E - N} \right)^{ - 1} = \left( {m_{{j_{0} j}} } \right)_{n \times n} \quad {\text{where}}\quad E\;{\text{is}}\;{\text{the}}\;n{\text{th}}\;{\text{order}}\;{\text{unit}}\;{\text{matrix}} $$
(20)

Step 2. The sum of rows and the sum of columns are separately denoted as D and R within the comprehensive impact matrix \(M\). Calculate the center factors and the causal factors.

$$ R_{{j_{0} }} = \sum\limits_{j = 1}^{n} {m_{{j_{0} j}} } $$
(21)
$$ D_{{j_{0} }} = \sum\limits_{{j_{0} = 1}}^{n} {m_{{j_{0} j}} } . $$
(22)

Then, we obtain the center factors \(R_{{j_{0} }} + D_{{j_{0} }}\) and the causal factors \(R_{{j_{0} }} - D_{{j_{0} }}\).

Step 3. Obtain the set of criteria interaction \(I_{j}\) for criterion \(j\).

$$ \begin{aligned} I_{j} & = \left\{ {c_{{j_{0} }} \left| {m_{{j_{0} j}} \ge \theta } \right.} \right\} \\ \theta & = {\text{Mean}}(M) + SD(M) = \frac{1}{{n^{2} }}\sum\limits_{{j_{0} = 1}}^{n} {\sum\limits_{j = 1}^{n} {m_{{j_{0} j}} } } + \sqrt {\frac{1}{{n^{2} }}\sum\limits_{{j_{0} = 1}}^{n} {\sum\limits_{j = 1}^{n} {\left( {m_{{j_{0} j}} - {\text{Mean}}(M)} \right)^{2} } } .} \\ \end{aligned} $$
(23)

Step 4. Obtain the criteria interaction matrix \(r\_expert\) provided by the expert.

$$ r\_expert = \left( {re_{{j_{0} j}} } \right)_{n \times n} ,re_{{j_{0} j}} = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{if }}j_{0} = j{\text{ or }}m_{{j_{0} j}} < \theta } \\ {m_{{j_{0} j}} ,} & {{\text{if }}m_{{j_{0} j}} \ge \theta } \\ \end{array} } \right.. $$
(24)

Step 5. Calculate criterion weights \(w_{{j_{0} }}^{c}\)[33].

$$ w_{{j_{0} }}^{c} = \frac{{\sqrt {\left( {R_{{j_{0} }} + D_{{j_{0} }} } \right)^{2} + \left( {R_{{j_{0} }} - D_{{j_{0} }} } \right)^{2} } }}{{\sum\nolimits_{{j_{0} = 1}}^{n} {\sqrt {\left( {R_{{j_{0} }} + D_{{j_{0} }} } \right)^{2} + \left( {R_{{j_{0} }} - D_{{j_{0} }} } \right)^{2} } } }},\quad {\text{where}}\quad w_{{j_{0} }}^{c} \in \left[ {0,1} \right],\sum\limits_{{j_{0} = 1}}^{n} {w_{{j_{0} }}^{c} = 1} . $$
(25)

Phase 2: Generation of criteria interaction coefficients based on ratings: Grey relational analysis (GRA) is widely used because of its convenience and reliability in calculating correlations. However, for two obviously independent attributes, the use of GRA still makes the independent attributes have correlation, which will make the decision face irrationality. The interaction set \(I_{j}\) obtained in Phase 1 has narrowed the scope of GRA by eliminating the criteria that are independent of each other in the criterion set to avoid the above situation. Combined with the above analysis, a GRA method based on PFSs [34] is proposed.

Let \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\) be the product user rating information matrix, and the set of criteria interaction for criterion \(C_{j}\) is known to be \(I_{j}\). The interaction coefficient generation method for criterion \(C_{j}\) is given below.

Step 1. Convert linguistic ratings information matrix \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\) into a Pythagorean fuzzy matrix \(\widetilde{CU} = \left( {\widetilde{cu}_{g}^{i,j} } \right)_{{u_{i} \times n}}\), where \(\widetilde{cu}_{g}^{i,j} = \left( {\mu_{i,gj}^{{}} ,v_{i,gj}^{{}} } \right)\) is obtained by Table 3.

Table 3 Transformation between linguistic linguistic ratings information and PFSs [32]
Full size table

Step 2. The grey relational coefficient between the criterion \(c_{{j_{0} }}\) and its associated set of criteria \(c_{j}^{i} \in I_{{j_{0} }}\) can be calculated as follows

$$ \gamma \left( {c_{{j_{0} }}^{i} ,c_{j}^{i} } \right) = \sum\limits_{g = 1}^{{u_{i} }} {\gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) \bigg/u_{i} } ,\left( {i = 1,2, \ldots ,m} \right), $$
(26)

where \(j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} \right\}\), \(\gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\) indicate the grey relational coefficient between \(\widetilde{cu}_{{i,gj_{{0}} }}^{{}} \in \widetilde{cu}_{{i,gj_{{0}} }}^{{}}\) and \(\widetilde{cu}_{i,gj}^{{}} \in \widetilde{cu}_{i,gj}^{{}}\), and satisfy

$$ \gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) = \frac{{\mathop {\min }\limits_{i} \mathop {\min }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} \right\}}} d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) + \rho \mathop {\max }\limits_{i} \mathop {\max }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} \right\}}} d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)}}{{d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) + \rho \mathop {\max }\limits_{i} \mathop {\max }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} \right\}}} d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)}}, $$
(27)

where the identification coefficient \(\rho\) is given as 0.5, and \(d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\) is the distance measure between PFNs \(\widetilde{cu}_{{i,gj_{{0}} }}^{{}}\) and \(\widetilde{cu}_{i,gj}^{{}}\) based on Eqs. (4) [27].

Step 3. By aggregating the grey relational coefficients of all criteria, the comprehensive relational coefficient can be computed.

$$ \overline{\gamma }\left( {c_{{j_{0} }} ,c_{j} } \right) = \sum\limits_{i = 1}^{n} {\gamma \left( {c_{{j_{0} }}^{i} ,c_{j}^{i} } \right)/n} . $$
(28)

Step 4. Based on \(\overline{\gamma }\left( {c_{{j_{0} }} ,c_{j} } \right)\), the user criterion interaction coefficient matrix \(r\_user\) is obtained.

$$ r\_user = \left( {ru_{{j_{0} j}} } \right)_{n \times n} ,{\text{where}}\quad ru_{{j_{0} j}} = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{if }}j_{0} = j \, or \, j \notin I_{{j_{0} }} } \\ {\overline{\gamma }\left( {c_{{j_{0} }} ,c_{j} } \right),} & {{\text{if }}j \in I_{{j_{0} }} } \\ \end{array} } \right. $$
(29)

Step 5. The comprehensive interaction coefficients \(w_{{j_{0} j}} \left( {j_{0} \ne j;j_{0,} j = 1,2, \ldots ,n} \right)\) required by the proposed operator to aggregate the associated criterion information is obtained by the interaction coefficient generation method.

$$ w_{{j_{0} j}} = \varepsilon ru_{{j_{0} j}} + \left( {1 - \varepsilon } \right)re_{{j_{0} j}} ,\varepsilon \in \left[ {0,1} \right]. $$
(30)

4.2.2 Multi-criteria Information Fusion Model

Based on the \(w_{{j_{0} j}}\) obtained in Sect. 4.2.1, the user ratings can be aggregated by the following steps.

Step 1. Aggregate user comprehensive ratings.

Step 1.1 Obtain criterion weights \(w_{i}\) and interaction coefficients \(w_{i,j}\).

$$ w_{i} = w_{{j_{0} }}^{c} ,w_{i,j} = w_{{j_{0} j}} . $$
(31)

Step 1.2 The comprehensive Pythagorean fuzzy matrix \(D = \left( {d_{g}^{i} } \right)_{{u_{i} \times m}}\) is obtained using the PFWIBM operator to aggregate \(\widetilde{CU} = \left( {\widetilde{cu}_{g}^{i,j} } \right)_{{u_{i} \times n}}\):

$$ d_{g}^{i} = \left( {\mu_{i,g} ,v_{i,g} } \right) = {\text{PFWIBM}}\left( {\widetilde{cu}_{g}^{i,1} ,\widetilde{cu}_{g}^{i,2} , \ldots ,\widetilde{cu}_{g}^{i,n} } \right). $$
(32)

Step 1.3 The user comprehensive rating matrix \(\widetilde{D}\) is obtained by Eq. (3).

$$ \widetilde{D} = \left( {\widetilde{d}_{g}^{i} } \right)_{{u_{i} \times m}} ,{\text{where}}\quad \widetilde{d}_{g}^{i} = \mu_{i,g}^{2} - v_{i,g}^{2} . $$
(33)

Step 2. Based on \(\widetilde{D}\), the comprehensive rating \(sc_{i}\) of product \(a_{i}\) is obtained by aggregation.

$$ sc_{i} = \sum\limits_{g = 1}^{{u_{i} }} {\widetilde{d}_{g}^{i} /u_{i} } . $$
(34)

Step 3. Rank product \(a_{i}\) according to comprehensive rating \(sc_{i}\).

$$ a_{\sigma \left( i \right)} \succ a_{{\sigma \left( {i + 1} \right)}} ,{\text{where}}\quad sc_{\sigma \left( i \right)} > sc_{{\sigma \left( {i + 1} \right)}} . $$
(35)

4.3 Online Multi-dimensional Rating Aggregation Decision-Making Approach

To solve the problem in Sect. 4.1, an online multi-dimensional rating aggregation decision-making approach is developed.

Step 1. Determine the set of products to be evaluated \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{m} } \right\}\) and the set of product evaluation criteria \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\}\).

Step 2. Collect word-of-mouth ratings from product websites to get user ratings matrix \(CU\).

Step 3. Obtain the expert criterion interaction coefficient matrix \(r\_expert\) and criterion weights \(w_{{j_{0} }}^{c}\) based on the criteria interaction linguistic information matrix \(\Omega^{k}\) provided by the experts.

Step 4. Calculate the user criterion interaction coefficient matrix \(r\_user\) based on the user rating information \(CU\). Then, calculate the comprehensive interaction coefficients \(w_{{j_{0} j}}\) based on \(r\_expert\) and \(r\_user\).

Step 5. Based on the criterion weights \(w_{{j_{0} }}^{c}\) and the interaction coefficients \(w_{{j_{0} j}}\), the Pythagorean fuzzy matrix \(\widetilde{CU}\) and the user comprehensive rating matrix \(\widetilde{D}\) are obtained by using the PFWIBM operator. Then, the comprehensive rating \(sc_{i}\) of product \(a_{i}\) is obtained by aggregation.

Step 6. Based on the comprehensive ratings \(sc_{i}\) to get the ranking of alternatives \(a_{\sigma \left( i \right)} \succ a_{{\sigma \left( {i + 1} \right)}}\).

5 Application Examples

To illustrate the use of the proposed method, a case study about ranking automobiles and an comparative analysis are given in this section.

5.1 Case Study

In this section, the proposed approach is used to ranking passenger cars from autohome.com. It should be noted that, in order to highlight the role and effect of the proposed aggregation operator in the decision-making process, during case analysis, it is assumed that the ratings provided by all users are reliable, so there is no division of user groups, and these related work will also be targeted in future research. The specific steps are given below:

Step 1. The evaluation criteria \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{8} } \right\}\) of the passenger car are described in Table 4, and alternatives is \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{5} } \right\}\), as shown in Table 5.

Table 4 Evaluation criteria from autohome.com
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Table 5 Number of ratings
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Step 2. The original number of data items is shown in Table 4. Due to the large amount of data, the information matrix \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times 8}}\)\(\left( {i = 1,2, \ldots ,5;\,\,j = 1,2, \ldots ,8} \right)\) is not provided in detail.

Step 3. According to the experts rating of the criteria in Table 3, the expert criteria rating matrix \(\Omega^{k} \left( {k = 1,2,3,4} \right)\) is obtained, and the criteria Pythagorean fuzzy matrix \(\widetilde{\Omega }^{k}\) is obtained through Table 1. The influence rating of experts is shown in Table 6, and the expert weights \(w_{k}^{e} = \left( {0.29,0.21,0.14,0.36} \right)\) are calculated using Eq. (16). Then, using Eqs. (17–20), the comprehensive impact matrix \(M\) is obtained, as shown in Table 7. Afterwards, using Eq. (23), the set of criteria interaction \(I_{j}\) is obtained, as shown in Table 8, where the \(\theta = 0.6815\). Finally, the expert criteria interaction matrix \(r\_expert\) is obtained through Eq. (24) and shown in Table 9, and then the criteria weights \(w_{{j_{0} }}^{c} = \left( {0.12,0.12,0.13,0.12,0.13,0.12,0.11,0.15} \right)\) are obtained through Eqs. (21–22) and (25).

Table 6 The influence ratings of experts
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Table 7 The comprehensive impact matrix \(M\)
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Table 8 The set of criteria interaction \(I_{j}\) with respect to \(C_{j}\)
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Table 9 The expert criteria interaction matrix \(r\_expert\)
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Step 4. According to Table 2, the matrix \(CU\) is transformed into the Pythagorean fuzzy matrix \(\widetilde{CU}\), then the user correlation matrix \(r\_user\) is calculated through Table 7 and Eqs. (26–29), which are shown in Table 10. Based on Table 9 and Eq. (30), the comprehensive interaction coefficients \(w_{{j_{0} j}}\) can be calculated, which are shown in Table 11 and the \(\varepsilon = 0.5\).

Table 10 The user criteria interaction matrix \(r\_user\)
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Table 11 The comprehensive criteria interaction matrix
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Step 5. According to the criteria weights \(w_{{j_{0} }}^{c}\) and the comprehensive interaction coefficients \(w_{{j_{0} j}}\), the Pythagorean fuzzy matrix \(\widetilde{CU}\) is aggregated by Eqs. (31–33) to obtain the user comprehensive rating matrix \(\widetilde{D}\), where \(p = q = 1\). Due to the large amount of data, the user comprehensive rating matrix is no longer provided in detail. Using Eqs. (31–34), the comprehensive rating \(sc_{i}\) of alternative vehicle model \(a_{i}\) is calculated, which is shown in Table 12.

Table 12 The comprehensive rating \(sc_{i}\) of alternative vehicle model \(a_{i}\)
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Step 6. According to the \(sc_{i}\), the ranking results of cars can be determined as follow:

$$ a_{4} \succ a_{1} \succ a_{3} \succ a_{2} \succ a_{5} $$

5.2 Comparison Analysis

The proposed operators and related parameters in this paper will be compared and analyzed in this section.

5.2.1 Comparison of the Pythagorean Fuzzy BM Operators

In order to investigate the scientific nature of the proposed operator in this paper, some Pythagorean fuzzy BM operators in the literature [25, 26, 35, 36] is compared with the PFWIBM operator through simple examples.

Example 3

Let \(C = \left\{ {C_{i} \left| {i = 1,2, \ldots ,5} \right.} \right\}\) be the set of product criteria, and the related weights is \(w_{i} = 0.2\left( {i = 1,2, \ldots ,5} \right)\), and suppose all the comprehensive interaction coefficients satisfy \(w_{ij} = 0.5\left( {i \ne j} \right)\), and \(X = \left\{ {x_{i} \left| {i = 1,2, \ldots ,5} \right.} \right\} = \left\{ {\left( {1,0} \right),\left( {0.8,0.3} \right),\left( {0.6,0.5} \right),\left( {0.3,0.9} \right),\left( {0,1} \right)} \right\}\) be the Pythagorean fuzzy information with respect to criteria set \(C\). Different BM operators with \(p = q = 1\) are used to obtain the aggregation results, as shown in Table 13.

Table 13 Aggregation results of different Pythagorean fuzzy BM operators
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The results in Table 2 show that when the extreme PFNs (1, 0) or (0, 1) appears in the Pythagorean fuzzy array, some operators [25, 26, 35, 36] will have aggregation failure. In contrast, the operators proposed in this paper can better handle such situations.

5.2.2 Comparison of Interaction Coefficients of BM Operator

To explore the advancement of the operator, the criteria interaction coefficients of BM operators in the literature [19, 21, 25, 37] is analyzed. The advancement of the BM operator is judged from the presence or absence of wij correlation coefficients and the method of wij interaction coefficient generation which is listed in Table 14.

Table 14 Comparison of interaction coefficients of different BM operators
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According to Table 14, it can be concluded that, compared with some existing BM operators, the operator proposed in this paper generates attribute interaction coefficients by combining expert knowledge with user ratings, which is helpful to improve the rationality of multi-attribute interaction decision-making process.

5.2.3 Comparison of the Parameters \(p\) and \(q\) on the Decision-Making

According to the above introduction of the PFWIBM operator, the parameters \(p\) and \(q\) play an important role in the decision-making process. Next, the effect of parameters \(p\) and \(q\) on the ranking results are observed, and the ratings of the alternatives using different values of \(p\) and \(q\) in Step 5 of Sect. 4.3 are calculated. The total ratings and the corresponding ranking results for each alternative are presented in Table 15.

Table 15 The parameters \(p\) and \(q\) on the decision-making results
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According to Table 15, it can be concluded that the total ratings of the cars increases with increasing values of \(p\) and \(q\). The ranking results by using the PFWIBM operator are different when the parameters \(p = {1},q = {40}\) and \(p = {1},q = {50}\). The reason is that the ratings of the two alternatives are similar (\(\left| {sc_{2} - sc_{3} } \right| < 0.1\)). With the increase of parameter values, the growth range of the corresponding comprehensive values of the two alternatives is changed. However, the best car is always \(a_{4}\). In general, we can take some simple values from the point of view of computational difficulty, such as \(p = q = 1\).

6 Conclusions

The mutual coefficient structure embedded in NWIBM operator lays a good foundation for the operator proposed in this paper, and provided a better solution for quantifying the degree of attribute interaction. In this paper, the PFWIBM operator is constructed under the Pythagorean fuzzy environment by adjusting the internal structure of the operator and giving play to the advantages of data driving, and it is used to develop an online multi-dimensional attribute scoring aggregation decision-making model. Through comparative analysis, the method proposed in this paper has two advantages: (1) the operator can effectively deal with the extreme value failure phenomenon in the input information group; (2) the interactive coefficient generation method embedded in the operator is constructed by combining expert knowledge with user ratings. In addition, because Pythagorean fuzzy sets contain intuitionistic fuzzy sets, that is, any IFSs are PFSs. Therefore, the study of the PFWIBM operator in this paper provides a reference paradigm for the study of intuitionistic fuzzy BM operators, and it can also provide a basis for the subsequent expansion of BM operators in the q-rung orthopair fuzzy environment. However, the operation mechanism of the PFWIBM is the focus of the decision-making model designed in this paper, and the division and processing of online ratings are not involved. This will be the next work. Besides, we can focus on expanding the INWIBM to proportional interval type-2 hesitant fuzzy environment [38] to better handle linguistic ratings.