1 Introduction

Atanassov [1, 2] introduced the concept of intuitionistic fuzzy set(IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set [3] whose basic component is only a membership function. Xu [4] developed the intuitionistic fuzzy weighted averaging (IFWA) operator, intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the intuitionistic fuzzy hybrid aggregation (IFHA) operator. Xu [5] developed some geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator and the intuitionistic fuzzy hybrid geometric (IFHG) operator and gave an application of the IFHG operator to multiple attribute group decision making. The intuitionistic fuzzy set has received more and more attention since its appearance, such as dynamic multiple attribute decision making in intuitionistic fuzzy setting [6, 7], intuitionistic fuzzy information aggregating operators [8,9,10,11,12,13], intuitionistic fuzzy entropy [14,15,16], intuitionistic fuzzy generalized Dice similarity measures [17], intuitionistic fuzzy TOPSIS [18,19,20], intuitionistic fuzzy gray relational analysis [21,22,23]. The bipolar fuzzy set (BFS) [24, 25] has emerged lately as an alternative tool to depict uncertainty in MADM problems. A pair of numbers, namely the positive membership degree and the negative membership degree, is employed to define an object in a BFS. But unlike the IFS, the range of membership degree of the bipolar fuzzy set is [−1, 1]. BFSs have been applied in many research areas including but not limited to bipolar logical reasoning and set theory [26, 27], traditional Chinese medicine theory [28, 29], computational psychiatry [30], decision analysis and organizational modeling [31, 32], quantum computing [33, 34], biosystem regulation [35], quantum cellular combinatorics [33], physics and philosophy [36] and graph theory [37,38,39,40,41]. Recently, Gul [42] defined some bipolar fuzzy aggregations operators, such as bipolar fuzzy averaging weighted aggregation operators and bipolar fuzzy geometric aggregations operators.

The information aggregation operators are an interesting and important research topic, which are receiving more and more attention. Hamacher t-conorm and t-norm, which are the generalization of algebraic and Einstein t-conorm and t-norm [43], are more general and more flexible. There is important significance to research aggregation operators based on Hamacher operations and their application to MADM problems. Liu [44] investigated the interval-valued intuitionistic fuzzy aggregation operators with the help of Hamacher operations and developed some interval-valued intuitionistic fuzzy Hamacher aggregation operators. Xiao [45] developed the induced interval-valued intuitionistic fuzzy Hamacher ordered weighted geometric (IIVIFHOWG) operator. Li [46] introduced some operations on the interval-valued intuitionistic fuzzy sets, such as the Hamacher sum and the Hamacher product, and further developed the interval-valued intuitionistic fuzzy Hamacher correlated averaging (IVIFHCA) operator. Zhou et al. [47] and Tan et al. [48] developed some hesitant fuzzy Hamacher aggregation operators for multiple attribute decision making. We note that almost all the bipolar fuzzy aggregation operators used in the literature employed the algebraic product or sum of PFSs. Constructed on the basis of general t-norm and t-conorm, Hamacher product and Hamacher sum [49] could be applied, respectively, to surrogate the algebraic product and algebraic sum. In this paper, we consider how to extend Hamacher operators to aggregate the bipolar fuzzy information. The remainder of this paper is organized as follows. In the next section, we briefly review the basic concepts of the BFSs and the fundamental operational laws of BFNs. In Sect. 3, we have proposed bipolar fuzzy Hamacher weighted average (BFHWA) operator, bipolar fuzzy Hamacher ordered weighted average (BFHOWA) operator, bipolar fuzzy Hamacher hybrid average (BFHHA) operator, bipolar fuzzy Hamacher weighted geometric (BFHWG) operator, bipolar fuzzy Hamacher ordered weighted geometric (BFHOWG) operator, bipolar fuzzy Hamacher hybrid geometric (BFHHG) operator. We also investigate the characteristics and special cases of these operators. In Sect. 4, we have utilized these operators to develop some approaches to solve the bipolar fuzzy multiple attribute decision-making problems. An illustrative example for enterprise resource planning (ERP) system selection is analyzed in Sect. 5. Some remarks are given in Sect. 6 to conclude the paper.

2 Preliminaries

2.1 The Bipolar Fuzzy Set

In this section, we present a short overview of BFSs [24, 25]. Afterward, novel score and accuracy functions for bipolar fuzzy numbers are proposed. Furthermore, a new comparison method for BFNs is developed.

Definition 1

[24, 25] Let \( X \) be a fix set. A BFS is an object having the form

$$ B = \left\{ {\left\langle {x,\left( {\mu_{B}^{ + } \left( x \right),\nu_{B}^{ - } \left( x \right)} \right)} \right\rangle \left| {x \in X} \right.} \right\} $$
(1)

where the positive membership degree function \( \mu_{B}^{ + } \left( x \right):\;X \to \left[ {0,1} \right] \) denotes the satisfaction degree of an element \( x \) to the property corresponding to a BFS \( B \) and the negative membership degree function \( \nu_{B}^{ - } \left( x \right):\;X \to \left[ { - 1,0} \right] \) denotes satisfaction degree of an element \( x \) to some implicit counter-property corresponding to a BFS \( B \), respectively, and, for every \( x \in X \).

Let \( \tilde{b} = \left( {\mu_{{}}^{ + } ,\nu_{{}}^{ - } } \right) \) be a bipolar fuzzy number (BFN). We now define a score function and an accuracy function for \( \tilde{b} \).

Definition 2

The score function \( S \) of \( \tilde{b} = \left( {\mu_{{}}^{ + } ,\nu_{{}}^{ - } } \right) \) is evaluated as

$$ S\left( {\tilde{b}} \right) = \frac{1}{2}\left( {1 + \mu_{{}}^{ + } { + }\nu_{{}}^{ - } } \right),\quad S\left( {\tilde{b}} \right) \in \left[ {0,1} \right]. $$
(2)

Definition 3

The accuracy function \( H \) of \( \tilde{b} \) is formulated as

$$ H\left( {\tilde{b}} \right) = \frac{{\mu_{{}}^{ + } { - }\nu_{{}}^{ - } }}{2},\quad H\left( {\tilde{b}} \right) \in \left[ {0,1} \right] $$
(3)

It is evident that \( S\left( {\tilde{b}} \right) \in \left[ {0,1} \right] \) and \( H\left( {\tilde{b}} \right) \in \left[ {0,1} \right] \). Note that \( H\left( {\tilde{b}} \right) \) assesses the degree of accuracy of \( \tilde{b} \). A larger value of \( H\left( {\tilde{b}} \right) \) implies a higher degree of accuracy of the BFN \( \tilde{b} \).

Applying the score function S and the accuracy function \( H \), we next define an ordered relation between two BFNs \( \tilde{b}_{1} = \left( {\mu_{1}^{ + } ,\nu_{1}^{ - } } \right) \) and \( \tilde{b}_{2} = \left( {\mu_{2}^{ + } ,\nu_{2}^{ - } } \right) \).

Definition 4

If \( S\left( {\tilde{b}_{1} } \right) < S\left( {\tilde{b}_{2} } \right) \), or \( S\left( {\tilde{b}_{1} } \right) = S\left( {\tilde{b}_{2} } \right) \) but \( H\left( {\tilde{b}_{1} } \right) < H\left( {\tilde{b}_{2} } \right) \), then \( \tilde{b}_{1} \) is smaller than \( \tilde{b}_{2} \), denoted by \( \tilde{b}_{1} < \tilde{b}_{2} \); If \( S\left( {\tilde{b}_{1} } \right) = S\left( {\tilde{b}_{2} } \right) \) and \( H\left( {\tilde{b}_{1} } \right) = H\left( {\tilde{b}_{2} } \right) \), then \( \tilde{b}_{1} = \tilde{b}_{2} \).

Some basic operations on BFNs are expressed as follows [42]:

  1. (1)

    \( \tilde{b}_{1} \oplus \tilde{b}_{2} = \left( {\mu_{1}^{ + } + \mu_{2}^{ + } - \mu_{1}^{ + } \mu_{2}^{ + } , - \left| {\nu_{1}^{ - } } \right|\left| {\nu_{2}^{ - } } \right|} \right); \)

  2. (2)

    \( \tilde{b}_{1} \otimes \tilde{b}_{2} = \left( {\mu_{1}^{ + } \mu_{2}^{ + } ,\nu_{1}^{ - } + \nu_{2}^{ - } - \nu_{1}^{ - } \nu_{2}^{ - } } \right); \)

  3. (3)

    \( \lambda \tilde{b} = \left( {1 - \left( {1 - \mu_{{}}^{ + } } \right)^{\lambda } , - \left| {\nu_{{}}^{ - } } \right|^{\lambda } } \right),\,\lambda > 0; \)

  4. (4)

    \( \left( {\tilde{b}} \right)^{\lambda } = \left( {\left( {\mu_{{}}^{ + } } \right)^{\lambda } , - 1 + \left| {1 + \nu_{{}}^{ - } } \right|^{\lambda } } \right),\,\lambda > 0; \)

  5. (5)

    \( \tilde{b}^{c} = \left( {1 - \mu_{{}}^{ + } ,\left| {\nu_{{}}^{ - } } \right| - 1} \right); \)

  6. (6)

    \( \tilde{b}_{1} \subseteq \tilde{b}_{2} \), if and only if \( \mu_{1}^{ + } \le \mu_{2}^{ + } \) and \( \nu_{1}^{ - } \ge \nu_{2}^{ - } \);

  7. (7)

    \( \tilde{b}_{1} \cup \tilde{b}_{2} = \left( {\hbox{max} \left\{ {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right\},\hbox{min} \left\{ {\nu_{1}^{ - } ,\nu_{2}^{ - } } \right\}} \right); \)

  8. (8)

    \( \tilde{b}_{1} \cap \tilde{b}_{2} = \left( {\hbox{min} \left\{ {\mu_{1}^{ + } ,\mu_{2}^{ + } } \right\},\hbox{max} \left\{ {\nu_{1}^{ - } ,\nu_{2}^{ - } } \right\}} \right). \)

Based on the basic operations on BFNs, we can introduce Theorem 1 easily.

Theorem 1

[42] Let \( \tilde{b}_{1} = \left( {\mu_{1}^{ + } ,\nu_{1}^{ - } } \right) \) and \( \tilde{b}_{2} = \left( {\mu_{2}^{ + } ,\nu_{2}^{ - } } \right) \) be two BFNs, \( \lambda ,\lambda_{1} ,\lambda_{2} > 0 \) , then

  1. (1)

    \( \tilde{b}_{1} \oplus \tilde{b}_{2} = \tilde{b}_{2} \oplus \tilde{b}_{1} ; \)

  2. (2)

    \( \tilde{b}_{1} \otimes \tilde{b}_{2} = \tilde{b}_{2} \otimes \tilde{b}_{1} ; \)

  3. (3)

    \( \lambda \left( {\tilde{b}_{1} \oplus \tilde{b}_{2} } \right) = \lambda \tilde{b}_{1} \oplus \lambda \tilde{b}_{2} ; \)

  4. (4)

    \( \left( {\tilde{b}_{1} \otimes \tilde{b}_{2} } \right)^{\lambda } = \left( {\tilde{b}_{1} } \right)^{\lambda } \otimes \left( {\tilde{b}_{2} } \right)^{\lambda } ; \)

  5. (5)

    \( \lambda_{1} \tilde{b}_{1} \oplus \lambda_{2} \tilde{b}_{1} = \left( {\lambda_{1} + \lambda_{2} } \right)\tilde{b}_{1} ; \)

  6. (6)

    \( \left( {\tilde{b}_{1} } \right)^{{\lambda_{1} }} \otimes \left( {\tilde{b}_{1} } \right)^{{\lambda_{2} }} = \left( {\tilde{b}_{1} } \right)^{{\left( {\lambda_{1} + \lambda_{2} } \right)}} ; \)

  7. (7)

    \( \left( {\left( {\tilde{b}_{1} } \right)^{{\lambda_{1} }} } \right)^{{\lambda_{2} }} = \left( {\tilde{b}_{1} } \right)^{{\lambda_{1} \lambda_{2} }} . \)

2.2 Hamacher operations

Hamacher [49] proposed a more generalized t-norm and t-conorm. Hamacher operations, i.e., Hamacher product and Hamacher sum, are respective instances of t-norms and t-conorms.

Hamacher product \( \otimes \) is a t-norm and Hamacher sum \( \oplus \) is a t-conorm, where

$$ T\left( {a,b} \right) = a \otimes b = \frac{ab}{{\gamma + \left( {1 - \gamma } \right)\left( {a + b - ab} \right)}},\gamma > 0. $$
(4)
$$ T*\left( {a,b} \right) = a \oplus b = \frac{a + b - ab - (1 - \gamma )ab}{1 - (1 - \gamma )ab},\gamma > 0. $$
(5)

In particular, when \( \gamma = 1 \), Hamacher t-norm and t-conorm will reduce to

$$ T\left( {a,b} \right) = a \otimes b = ab $$
(6)
$$ T*\left( {a,b} \right) = a \oplus b = a + b - ab $$
(7)

which are their algebraic counterparts; and when \( \gamma = 2 \), Eqs. (4) and (5) will reduce to

$$ T\left( {a,b} \right) = a \otimes b = \frac{ab}{{1 + \left( {1 - a} \right)\left( {1 - b} \right)}} $$
(8)
$$ T*\left( {a,b} \right) = a \oplus b = \frac{a + b}{1 + ab} $$
(9)

which are referred to as the Einstein t-norm and t-conorm, respectively [50, 51].

2.3 Hamacher operations of bipolar fuzzy numbers

Let \( \tilde{b}_{1} = \left( {\mu_{1}^{ + } ,\nu_{1}^{ - } } \right) \), \( \tilde{b}_{2} = \left( {\mu_{2}^{ + } ,\nu_{2}^{ - } } \right) \) and \( \tilde{b} = \left( {\mu_{{}}^{ + } ,\nu_{{}}^{ - } } \right) \) denote BFNs. We define the following basic Hamacher operators of BFNs with \( \gamma > 0 \).

$$ \tilde{b}_{1} \oplus \tilde{b}_{2} = \left( {\frac{{\mu_{1}^{ + } + \mu_{2}^{ + } - \mu_{1}^{ + } \mu_{2}^{ + } - (1 - \gamma )\mu_{1}^{ + } \mu_{2}^{ + } }}{{1 - (1 - \gamma )\mu_{1}^{ + } \mu_{2}^{ + } }},} \right.\left. {\frac{{ - \nu_{1}^{ - } \nu_{2}^{ - } }}{{\gamma + (1 - \gamma )\left( {\nu_{1}^{ - } + \nu_{2}^{ - } - \nu_{1}^{ - } \nu_{2}^{ - } } \right)}}} \right) $$
(10)
$$ \tilde{b}_{1} \otimes \tilde{b}_{2} = \left( {\frac{{\mu_{1}^{ - } \mu_{2}^{ - } }}{{\gamma + (1 - \gamma )\left( {\mu_{1}^{ - } + \mu_{2}^{ - } - \mu_{1}^{ - } \mu_{2}^{ - } } \right)}},} \right.\left. {\frac{{\nu_{1}^{ - } + \nu_{2}^{ - } - \nu_{1}^{ - } \nu_{2}^{ - } - (1 - \gamma )\nu_{1}^{ - } \nu_{2}^{ - } }}{{1 - (1 - \gamma )\nu_{1}^{ - } \nu_{2}^{ - } }}} \right) $$
(11)
$$ \lambda \tilde{b} = \left( {\frac{{\left( {1 + \left( {\gamma - 1} \right)\mu^{ + } } \right)^{\lambda } - \left( {1 - \mu^{ + } } \right)^{\lambda } }}{{\left( {1 + \left( {\gamma - 1} \right)\mu^{ + } } \right)^{\lambda } + \left( {\gamma - 1} \right)\left( {1 - \mu^{ + } } \right)^{\lambda } }}} \right., \left. {\frac{{{ - }\gamma \left| {\nu^{ - } } \right|^{\lambda } }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu^{ - } } \right)} \right)^{\lambda } + \left( {\gamma - 1} \right)\left| {\nu^{ - } } \right|^{\lambda } }}} \right),\;\lambda > 0 $$
(12)
$$ \left( {\tilde{b}} \right)^{\lambda } = \left( {\frac{{\gamma \left( {\mu^{ + } } \right)^{\lambda } }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 - \mu^{ + } } \right)} \right)^{\lambda } + \left( {\gamma - 1} \right)\left( {\mu^{ + } } \right)^{\lambda } }},} \right. \left. {\;\;\;\;\;\;\;\;\;\;\; - \frac{{\left( {1 + \left( {\gamma - 1} \right)\left| {\nu^{ - } } \right|} \right)^{\lambda } - \left( {1 + \nu^{ - } } \right)^{\lambda } }}{{\left( {1 + \left( {\gamma - 1} \right)\left| {\nu^{ - } } \right|} \right)^{\lambda } + \left( {\gamma - 1} \right)\left( {1 + \nu^{ - } } \right)^{\lambda } }}} \right),\;\lambda > 0$$
(13)

3 Bipolar Fuzzy Hamacher Aggregation Operators

3.1 Bipolar Fuzzy Hamacher Arithmetic Aggregation Operators

Let \( \tilde{b}_{j} = \left( {\mu_{j}^{ + } ,\nu_{j}^{ - } } \right)\left( {j = 1,2, \ldots ,n} \right) \) be a collection of BFNs. We next establish bipolar fuzzy Hamacher arithmetic aggregation operators.

Definition 5

The bipolar fuzzy Hamacher weighted average (BFHWA)operator is

$$ {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\omega_{j} \tilde{b}_{j} } \right) $$
(14)

where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) denotes the weight vector associated with \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \), and \( \omega_{j} > 0 \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \), \( \gamma > 0 \).

Theorem 2 can be shown by its definition.

Theorem 2

The BFHWA operator returns a BFN with

$$ {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \nolimits_{j = 1}^{n} \left( {\omega_{j} \tilde{b}_{j} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{n} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{j}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right) $$
(15)

Theorem 2 can be proved by Mathematical induction shown as follows:

Proof

  1. (1)

    When \( n = 1 \), \( \therefore w_{1} = 1 \), for the left side of the (15), \( {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \tilde{b}_{1} = \left( {\mu_{1}^{ + } ,\nu_{1}^{ - } } \right) \) and for the right side of the (15), we have

    $$ \left( {\frac{{1 + \left( {\gamma - 1} \right)\mu_{1}^{ + } - \left( {1 - \mu_{1}^{ + } } \right)}}{{\left( {1 + \left( {\gamma - 1} \right)\mu_{1}^{ + } } \right) + \left( {\gamma - 1} \right)\left( {1 - \mu_{1}^{ + } } \right)}},} \right.\left. {\frac{{ - \gamma \left| {\nu_{1}^{ - } } \right|}}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{1}^{ - } } \right)} \right) + \left( {\gamma - 1} \right)\left| {\nu_{1}^{ - } } \right|}}} \right) = \left( {\mu_{1}^{ + } ,\nu_{1}^{ - } } \right) $$

    Therefore, (15) holds for \( n = 1 \).

  2. (2)

    Assume that (15) holds for \( n = k \), we have

    $$ {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{k} } \right) = \mathop \oplus \nolimits_{j = 1}^{k} \left( {\omega_{j} \tilde{b}_{j} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{k} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{k} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{j}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right) $$

    When \( n = k + 1 \),

    $${\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{k} ,\tilde{b}_{k + 1} } \right) = \mathop \oplus \nolimits_{j = 1}^{k} \left( {\omega_{j} \tilde{b}_{j} } \right) \oplus \left( {\omega_{k + 1} \tilde{b}_{k + 1} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{k} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{k} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{k} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{j}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right) \oplus \left( {\frac{{\left( {1 + \left( {\gamma - 1} \right)\mu_{k + 1}^{ + } } \right)^{{\omega_{k + 1} }} - \left( {1 - \mu_{k + 1}^{ + } } \right)^{{\omega_{k + 1} }} }}{{\left( {1 + \left( {\gamma - 1} \right)\mu_{k + 1}^{ + } } \right)^{{\omega_{k + 1} }} + \left( {\gamma - 1} \right)\left( {1 - \mu_{k + 1}^{ + } } \right)^{{\omega_{k + 1} }} }},} \right.\left. {\frac{{ - \gamma \left| {\nu_{{^{k + 1} }}^{ - } } \right|^{{\omega_{k + 1} }} }}{{\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{k + 1}^{ - } } \right)} \right)^{{\omega_{k + 1} }} + \left( {\gamma - 1} \right)\left| {\nu_{k + 1}^{ - } } \right|^{{\omega_{k + 1} }} }}} \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{k + 1} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{k + 1} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{k + 1} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k + 1} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{k + 1} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{k + 1} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{j}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{k + 1} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right) $$

    Therefore, \( n = k + 1 \), (15) holds.

  3. (3)

    According to steps (1) and (2), we have get (15) holds for any \( n \).

It can be easily proven that the BFHWA operator possesses three properties stated below.

Theorem 3

(Idempotency) If all \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) are equal, i.e., \( \tilde{b}_{j} = \tilde{b} \) for all \( j \) , then

$$ {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \tilde{b} $$
(16)

Theorem 4

(Boundedness) Let \( \tilde{b}^{ - } = \mathop {\hbox{min} }\limits_{j} \tilde{b}_{j} ,\quad \tilde{b}^{ + } = \mathop {\hbox{max} }\limits_{j} \tilde{b}_{j} \)

Then

$$ \tilde{b}^{ - } \le {\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le \tilde{b}^{ + } $$
(17)

Theorem 5

(Monotonicity) Let \( \tilde{b}_{j} = \left( {\mu_{j}^{ + } ,\nu_{j}^{ - } } \right)\left( {j = 1,2, \ldots ,n} \right) \) and \( \tilde{b}_{j}^{{\prime }} = \left( {\mu_{j}^{{{\prime } + }} ,\nu_{j}^{{{\prime } - }} } \right)\left( {j = 1,2, \ldots ,n} \right) \) be two set of BFNs. If \( \mu_{j}^{ + } \le \mu_{j}^{{{\prime } + }} \) and \( \nu_{j}^{ - } \le \nu_{j}^{{{\prime - }}} \) , for all \( j \) , then

$$ {\text{BFHWA}}_{\omega } (\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} ) \le {\text{BFHWA}}_{\omega } (\tilde{b}_{1}^{{\prime }} ,\tilde{b}_{2}^{{\prime }} , \ldots ,\tilde{b}_{n}^{{\prime }} ) $$
(18)

We subsequently discuss two special cases of the BFHWA operator.

  • If \( \gamma = 1 \), the BFHWA operator is equivalent to the bipolar fuzzy weighted average (BFWA)operator [50]:

    $$ {\text{BFWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\omega_{j} \tilde{b}_{j} } \right) = \left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } , - \prod\limits_{j = 1}^{n} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } } \right) $$
    (19)

  • If \( \gamma = 2 \), the BFHWA operator coincides with the bipolar fuzzy Einstein weighted average (BFEWA) operator:

    $$ {\text{BFEWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\omega_{j} \tilde{b}_{j} } \right) \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \mu_{j}^{ + } } \right)^{{\omega_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},\frac{{ - 2\prod\nolimits_{j = 1}^{n} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right)$$
    (20)

Example 1

Let \( \tilde{b}_{1} = \left( {0.6, - 0.3} \right),\tilde{b}_{2} = \left( {0.5, - 0.4} \right),\tilde{b}_{3} = \left( {0.7, - 0.2} \right),\tilde{b}_{4} = \left( {0.2, - 0.3} \right) \) be three bipolar fuzzy numbers, \( \omega = \left( {0.2,0.1,0.3,0.4} \right) \) is the weight vector of \( \tilde{b}_{j} \left( {j = 1,2,3,4} \right) \). Suppose that \( \gamma = 3 \), then

$${\text{BFHWA}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} ,\tilde{b}_{3} ,\tilde{b}_{4} } \right) = \mathop \oplus \limits_{j = 1}^{4} \left( {\omega_{j} \tilde{b}_{j} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{4} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{4} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{4} {\left( {1 + \left( {\gamma - 1} \right)\mu_{j}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{4} {\left( {1 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } } }},} \right. \left. {\;\;\;\frac{{ - \gamma \prod\nolimits_{j = 1}^{4} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{4} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{j}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{4} {\left| {\nu_{j}^{ - } } \right|^{{\omega_{j} }} } }}} \right) = \left( {\frac{ \left( {1 + \left( {3 - 1} \right)0.6} \right)^{0.2} \times \left( {1 + \left( {3 - 1} \right)0.5} \right)^{0.1} \times \left( {1 + \left( {3 - 1} \right)0.7} \right)^{0.3} \times \left( {1 + \left( {3 - 1} \right)0.2} \right)^{0.3} - \left( {1 - 0.6} \right)^{0.2} \times \left( {1 - 0.5} \right)^{0.1} \times \left( {1 - 0.7} \right)^{0.3} \times \left( {1 - 0.2} \right)^{0.4} }{ \left( {1 + \left( {3 - 1} \right)0.6} \right)^{0.2} \times \left( {1 + \left( {3 - 1} \right)0.5} \right)^{0.1} \times \left( {1 + \left( {3 - 1} \right)0.7} \right)^{0.3} \times \left( {1 + \left( {3 - 1} \right)0.2} \right)^{0.3}+ \left( {3 - 1} \right) \times \left( {1 - 0.6} \right)^{0.2} \times \left( {1 - 0.5} \right)^{0.1} \times \left( {1 - 0.7} \right)^{0.3} \times \left( {1 - 0.2} \right)^{0.4} }} \right., \left. {\frac{{ - 3 \times \left| { - 0.3} \right|^{0.2} \times \left| { - 0.4} \right|^{0.1} \left| { - 0.2} \right|^{0.3} \left| { - 0.3} \right|^{0.4} }}{ \left( {1 + \left( {3 - 1} \right)\left( {1 - 0.3} \right)} \right)^{0.2} \times \left( {1 + \left( {3 - 1} \right)\left( {1 - 0.4} \right)} \right)^{0.1} \times \left( {1 + \left( {3 - 1} \right)\left( {1 - 0.2} \right)} \right)^{0.3} \times \left( {1 + \left( {3 - 1} \right)\left( {1 - 0.3} \right)} \right)^{0.4} + \left( {3 - 1} \right) \times \left| { - 0.3} \right|^{0.2} \times \left| { - 0.4} \right|^{0.1} \left| { - 0.2} \right|^{0.3} \left| { - 0.3} \right|^{0.4} }} \right) = \left( {0.48, - 0.20} \right) $$

Definition 6

The bipolar fuzzy Hamacher ordered weighted average (BFHOWA)operator is defined as

$$ {\text{BFHOWA}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \tilde{b}_{\sigma \left( j \right)} } \right) $$
(21)

where \( \left( {\sigma \left( 1 \right),\sigma \left( 2 \right), \ldots ,\sigma \left( n \right)} \right) \) is a permutation of \( \left( {1,2, \ldots ,n} \right) \), such that \( \tilde{b}_{{\sigma \left( {j - 1} \right)}} \ge \tilde{b}_{\sigma \left( j \right)} \) for all \( j = 2, \ldots ,n \), and \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)^{T} \) is the aggregation-associated weight vector such that \( w_{j} \in \left[ {0,1} \right] \) and \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \gamma > 0 \).

Theorem 6 is straightforward.

Theorem 6

The bipolar fuzzy Hamacher ordered weighted average (BFHOWA) operator is evaluated as

$$ {\text{BFHOWA}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \tilde{b}_{\sigma \left( j \right)} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{n} {\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)} \right)^{{w_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}} \right) $$
(22)

where \( \left( {\sigma \left( 1 \right),\sigma \left( 2 \right), \ldots ,\sigma \left( n \right)} \right) \) is a permutation of \( \left( {1,2, \ldots ,n} \right) \), such that \( \tilde{b}_{{\sigma \left( {j - 1} \right)}} \ge \tilde{b}_{\sigma \left( j \right)} \) for all \( j = 2, \ldots ,n \), and \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)^{T} \) is the aggregation-associated weight vector such that \( w_{j} \in \left[ {0,1} \right] \) and \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \gamma > 0 \).

We can prove that the BFHOWA operator has the following properties.

Theorem 7

(Idempotency) If all \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) are equal, i.e., \( \tilde{b}_{j} = \tilde{b} \) for all \( j \) , then

$$ {\text{BFHOWA}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \tilde{b} $$
(23)

Theorem 8

(Boundedness) Let \( \tilde{b}^{ - } = \mathop {\hbox{min} }\limits_{j} \tilde{b}_{j} ,\quad \tilde{b}^{ + } = \mathop {\hbox{max} }\limits_{j} \tilde{b}_{j} , \)

Then

$$ \tilde{b}^{ - } \le {\text{BFHOWA}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le \tilde{b}^{ + } $$
(24)

Theorem 9

(Monotonicity) \( \tilde{b}_{j} = \left( {\mu_{j}^{ + } ,\nu_{j}^{ - } } \right)\left( {j = 1,2, \ldots ,n} \right) \) and \( \tilde{b}_{j}^{{\prime }} = \left( {\mu_{j}^{{{\prime } + }} ,\nu_{j}^{{{\prime } - }} } \right)\left( {j = 1,2, \ldots ,n} \right) \) be two set of BFNs. If \( \mu_{j}^{ + } \le \mu_{j}^{{{\prime } + }} \) and \( \nu_{j}^{ - } \le \nu_{j}^{{{\prime } - }} \) , for all \( j \) , then

$$ {\text{BFHOWA}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le {\text{BFHOWA}}_{w} \left( {\tilde{b}_{1}^{{\prime }} ,\tilde{b}_{2}^{{\prime }} , \ldots ,\tilde{b}_{n}^{{\prime }} } \right) $$
(25)

When the parameter \( \gamma \) takes value 1 or 2, the BFHOWA operator reduces to the following operators.

  • If \( \gamma = 1 \), the BFHOWA operator reduces to the bipolar fuzzy ordered weighted average (BFOWA) operator:

    $$ {\text{BFOWA}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \tilde{b}_{\sigma \left( j \right)} } \right) = \left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } , - \prod\limits_{j = 1}^{n} {\left| {\nu_{\sigma \left( j \right)}^{ + } } \right|^{{w_{j} }} } } \right) $$
    (26)

  • If \( \gamma = 2 \), the BFHOWA operator reduces to the bipolar fuzzy Einstein ordered weighted average (BFEOWA) operator:

    $$ {\text{BFEOWA}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \nolimits_{j = 1}^{n} \left( {w_{j} \tilde{b}_{\sigma \left( j \right)} } \right) \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }},\frac{{ - 2\prod\nolimits_{j = 1}^{n} {\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}} \right) $$
    (27)

Definitions 5 and 6 suggest that the BFHWA operator and the BFHOWA operator weigh the Bipolar fuzzy arguments and the ordered positions of the bipolar fuzzy arguments, respectively. A bipolar fuzzy Hamacher hybrid average (BFHHA) operator is proposed below to combine the characteristics of the BFHWA operator and the BFHOWA operator together.

Definition 7

A bipolar fuzzy Hamacher hybrid average (BFHHA) operator is defined as follows:

$$ {\text{BFHHA}}_{w,\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \dot{\tilde{b}}_{\sigma \left( j \right)} } \right) $$
(28)

where \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) is the associated weighting vector, with \( w_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \dot{\tilde{b}}_{\sigma \left( j \right)} \) is the j-th largest element of the bipolar fuzzy arguments \( \dot{\tilde{b}}_{j} \left( {\dot{\tilde{b}}_{j} = \left( {n\omega_{j} } \right)\tilde{b}_{j} ,\quad j = 1,2, \ldots ,n} \right) \),\( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right) \) is the weighting vector of bipolar fuzzy arguments \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \),with \( \omega_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \), and \( n \) is the balancing coefficient, \( \gamma > 0 \).

Note that the BFHHA operator reduces to the BFHWA operator if \( w = \left( {1/n,1/n, \ldots ,1/n} \right)^{T} \), and the (BFHOWA) operator if \( \omega = (1/n,1/n, \ldots ,1/n) \).

The next theorem holds for the BFHHA operator.

Theorem 10

The BFHHA operator returns a BFN, and

$$ {\text{BFHHA}}_{w,\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \dot{\tilde{b}}_{\sigma \left( j \right)} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{n} {\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)} \right)^{{w_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}} \right) $$
(29)

where \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) is the associated weighting vector, with \( w_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \dot{\tilde{b}}_{\sigma \left( j \right)} \) is the j-th largest element of the Bipolar fuzzy arguments \( \dot{\tilde{b}}_{j} \left( {\dot{\tilde{b}}_{j} = \left( {n\omega_{j} } \right)\tilde{b}_{j} ,\quad j = 1,2, \ldots ,n} \right) \),\( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right) \) is the weighting vector of Bipolar fuzzy arguments \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) ,with \( \omega_{j} \in \left[ {0,1} \right] \),\( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \) , and \( n \) is the balancing coefficient, \( \gamma > 0 \).

We now discuss two special cases of the BFHHA operator.

  • If \( \gamma = 1 \), the BFHHA operator reduces to the bipolar fuzzy hybrid average (BFHA) operator as follows:

    $$ {\text{BFHWA}}_{\omega ,w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \dot{\tilde{b}}_{\sigma \left( j \right)} } \right) = \left( {1 - \prod\limits_{j = 1}^{n} {\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } , - \prod\limits_{j = 1}^{n} {\left| {\dot{\nu }_{\sigma \left( j \right)}^{ + } } \right|^{{w_{j} }} } } \right) $$
    (30)

  • If \( \gamma = 2 \), the BFHHA operator reduces to the bipolar fuzzy Einstein hybrid average (BFEHA) operator as follows:

    $$ {\text{BFEHWA}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {w_{j} \dot{\tilde{b}}_{\sigma \left( j \right)} } \right) \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } } }},\frac{{ - 2\prod\nolimits_{j = 1}^{n} {\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 + \dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|^{{w_{j} }} } }}} \right) $$
    (31)

3.2 Bipolar Fuzzy Hamacher Geometric Aggregation Operators

Applying the bipolar fuzzy Hamacher arithmetic aggregation operators and the concept of geometric mean [52,53,54,55,56,57], we can define bipolar fuzzy Hamacher geometric aggregation operators.

Definition 8

The bipolar fuzzy Hamacher weighted geometric (BFHWG) operator is defined as

$$ {\text{BFHWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{j} } \right)^{{\omega_{j} }} $$
(32)

where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) is the weight vector of \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) with \( \omega_{j} > 0 \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \), \( \gamma > 0 \).

By definition 8, we can prove the following theorem by mathematical induction.

Theorem 11

The BFHWG operator returns a BFN, and

$$ {\text{BFHWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{j} } \right)^{{\omega_{j} }} = \left( {\frac{{\gamma \prod\nolimits_{j = 1}^{n} {\left( {\mu_{j}^{ + } } \right)^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 - \mu_{j}^{ + } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {\mu_{j}^{ + } } \right)^{{\omega_{j} }} } }},} \right.\left. { - \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{j}^{ - } } \right|} \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{j}^{ - } } \right|} \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } } }}} \right) $$
(33)

where \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) is the weight vector of \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) with \( \omega_{j} > 0 \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \), \( \gamma > 0 \).

Similar to Theorem 2, it is easy to prove Theorem 11. We omit the proof.

It is easy to shown that the BFHWG operator exhibits the following properties.

Theorem 12

(Idempotency) If all \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) are equal, i.e., \( \tilde{b}_{j} = \tilde{b} \) for all \( j \) , then

$$ {\text{BFHWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \tilde{b} $$
(34)

Theorem 13

(Boundedness) Let \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) be a collection of BFNs, and

$$ \tilde{b}^{ - } = \mathop {\hbox{min} }\limits_{j} \tilde{b}_{j} ,\quad \tilde{b}^{ + } = \mathop {\hbox{max} }\limits_{j} \tilde{b}_{j} $$

Then

$$ \tilde{b}^{ - } \le {\text{BFHWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le \tilde{b}^{ + } $$
(35)

Theorem 14

(Monotonicity) \( \tilde{b}_{j} = \left( {\mu_{j}^{ + } ,\nu_{j}^{ - } } \right)\left( {j = 1,2, \ldots ,n} \right) \) and \( \tilde{b}_{j}^{{\prime }} = \left( {\mu_{j}^{{{\prime } + }} ,\nu_{j}^{{{\prime } - }} } \right)\left( {j = 1,2, \ldots ,n} \right) \) be two set of BFNs. If \( \mu_{j}^{ + } \le \mu_{j}^{{{\prime } + }} \) and \( \nu_{j}^{ - } \le \nu_{j}^{{{\prime } - }} \) , for all \( j \) , then

$$ {\text{BFHWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \cdots ,\tilde{b}_{n} } \right) \le {\text{BFHWG}}_{\omega } \left( {\tilde{b}^{\prime}_{1} ,\tilde{b}^{\prime}_{2} , \cdots ,\tilde{b}^{\prime}_{n} } \right) $$
(36)

Next we present two special cases of the BFHWG operator.

  • If \( \gamma = 1 \), BFHWG operator reduces to the bipolar fuzzy weighted geometric (BFWG) operator:

    $$ {\text{BFWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{j} } \right)^{{\omega_{j} }} = \left( {\prod\limits_{j = 1}^{n} {\left( {\mu_{j}^{ + } } \right)^{{\omega_{j} }} } , - 1 + \prod\limits_{j = 1}^{n} {\left( {1 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } } \right) $$
    (37)

  • If \( \gamma = 2 \), BFHWG operator reduces to the bipolar fuzzy Einstein weighted geometric (BFEWG) operator:

    $$ {\text{BFEWG}}_{\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {\mu_{j}^{ + } } \right)^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - \mu_{j}^{ + } } \right)^{{\omega_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left( {\mu_{j}^{ + } } \right)^{{\omega_{j} }} } }}, - \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\nu_{j}^{ - } } \right|} \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\nu_{j}^{ - } } \right|} \right)^{{\omega_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{j}^{ - } } \right)^{{\omega_{j} }} } } }}} \right) $$
    (38)

Definition 9

The bipolar fuzzy Hamacher ordered weighted geometric (BFHOWG)operator is defined as

$$ {\text{BFHOWG}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{\sigma \left( j \right)} } \right)^{{w_{j} }} $$
(39)

where \( \left( {\sigma \left( 1 \right),\sigma \left( 2 \right), \ldots ,\sigma \left( n \right)} \right) \) is a permutation of \( \left( {1,2, \ldots ,n} \right) \), such that \( \tilde{b}_{{\sigma \left( {j - 1} \right)}} \ge \tilde{b}_{\sigma \left( j \right)} \) for all \( j = 2, \ldots ,n \), and \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)^{T} \) is the aggregation-associated weight vector such that \( w_{j} \in \left[ {0,1} \right] \) and \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \gamma > 0 \).

The following theorem is valid for the BFHOWG operator.

Theorem 15

The BFHOWG operator returns a BFN, and

$$ {\text{BFHOWG}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{\sigma \left( j \right)} } \right)^{{w_{j} }} = \left( {\frac{{\gamma \prod\nolimits_{j = 1}^{n} {\left( {\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 - \mu_{\sigma \left( j \right)}^{ + } } \right)} \right)^{{w_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }},} \right. - \left. {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}} \right) $$
(40)

where \( \left( {\sigma \left( 1 \right),\sigma \left( 2 \right), \ldots ,\sigma \left( n \right)} \right) \) is a permutation of \( \left( {1,2, \ldots ,n} \right) \) , such that \( \tilde{b}_{{\sigma \left( {j - 1} \right)}} \ge \tilde{b}_{\sigma \left( j \right)} \) for all \( j = 2, \ldots ,n \) , and \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)^{T} \) is the aggregation-associated weight vector such that \( w_{j} \in \left[ {0,1} \right] \) and \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \gamma > 0 \).

The following properties of the BFHOWG operator can be proven.

Theorem 16

(Idempotency) If all \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) are equal, i.e., \( \tilde{b}_{j} = \tilde{b} \) for all \( j \) , then

$$ {\text{BFHOWG}}_{\text{w}} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \tilde{b} $$
(41)

Theorem 17

(Boundedness) Let

$$ \tilde{b}^{ - } = \mathop {\hbox{min} }\limits_{j} \tilde{b}_{j} ,\quad \tilde{b}^{ + } = \mathop {\hbox{max} }\limits_{j} \tilde{b}_{j} $$

Then

$$ \tilde{b}^{ - } \le {\text{BFHOWG}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le \tilde{b}^{ + } $$
(42)

Theorem 18

(Monotonicity) \( \tilde{b}_{j} = \left( {\mu_{j}^{ + } ,\nu_{j}^{ - } } \right)\left( {j = 1,2, \ldots ,n} \right) \) and \( \tilde{b}_{j}^{{\prime }} = \left( {\mu_{j}^{{{\prime } + }} ,\nu_{j}^{{{\prime } - }} } \right)\left( {j = 1,2, \ldots ,n} \right) \) be two set of BFNs. If \( \mu_{j}^{ + } \le \mu_{j}^{{{\prime } + }} \) and \( \nu_{j}^{ - } \le \nu_{j}^{{{\prime } - }} \) , for all \( j \) , then

$$ {\text{BFHOWG}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \le {\text{BFHOWG}}_{w} \left( {\tilde{b}_{1}^{{\prime }} ,\tilde{b}_{2}^{{\prime }} , \ldots ,\tilde{b}_{n}^{{\prime }} } \right) $$
(43)

Two special cases of the BFHOWG operator are discussed as follows.

  • If \( \gamma = 1 \), the BFHOWG operator coincides with the bipolar fuzzy ordered weighted geometric (BFOWG) operator:

    $$ {\text{BFOWG}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \cdots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{b}_{j} } \right)^{{w_{j} }} = \left( {\prod\limits_{j = 1}^{n} {\left( {\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } , - 1 + \prod\limits_{j = 1}^{n} {\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } \right) $$
    (44)

  • If \( \gamma = 2 \), the BFHOWG operator is equivalent to the bipolar fuzzy Einstein ordered weighted geometric (BFEOWG) operator as follows:

    $$ {\text{BFEOWG}}_{w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - \mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left( {\mu_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }}, - \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\nu_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\nu_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}} \right) $$
    (45)

Definitions 8 and 9 imply that the BFHWG operator and the BFHOWG operator target, respectively, the bipolar fuzzy argument itself and the ordered positions of the bipolar fuzzy arguments. To mix the features of these two operators together, we propose the bipolar fuzzy Hamacher hybrid geometric (BFHHG) operator below.

Definition 10

The bipolar fuzzy Hamacher hybrid geometric (BFHHG) operator is defined as

$$ {\text{BFHHG}}_{w,\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\dot{\tilde{b}}_{\sigma \left( j \right)} } \right)^{{w_{j} }} $$
(46)

where \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) is the associated weighting vector, with \( w_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \dot{\tilde{b}}_{\sigma \left( j \right)} \) is the j-th largest element of the bipolar fuzzy arguments \( \dot{\tilde{b}}_{j} \left( {\dot{\tilde{b}}_{j} = \left( {\tilde{b}_{j} } \right)^{{n\omega_{j} }} ,\quad j = 1,2, \ldots ,n} \right) \),\( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right) \) is the weighting vector of Bipolar fuzzy arguments \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \),with \( \omega_{j} \in \left[ {0,1} \right] \),\( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \), and \( n \) is the balancing coefficient, \( \gamma > 0 \).

By definition, BFHHG becomes the BFHWG operator when \( w = \left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}, \ldots ,{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right)^{T} \), and the BFHOWG operator when if \( \omega = \left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n},{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}, \ldots ,{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-0pt} n}} \right) \).

Theorem 19 is a natural conclusion.

Theorem 19

The BFHHG operator returns a BFN, and

$$ {\text{BFHHG}}_{w,\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\dot{\tilde{b}}_{\sigma \left( j \right)} } \right)^{{w_{j} }} = \left( {\frac{{\gamma \prod\nolimits_{j = 1}^{n} {\left( {\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)} \right)^{{w_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }},} \right.\left. {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1{ + }\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1{ + }\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}} \right) $$
(47)

where \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) is the associated weighting vector, with \( w_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \), \( \dot{\tilde{b}}_{\sigma \left( j \right)} \) is the j-th largest element of the bipolar fuzzy arguments \( \dot{\tilde{b}}_{j} \left( {\dot{\tilde{b}}_{j} = \left( {\tilde{b}_{j} } \right)^{{n\omega_{j} }} ,\quad j = 1,2, \ldots ,n} \right) \), \( \omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right) \) is the weighting vector of Bipolar fuzzy arguments \( \tilde{b}_{j} \left( {j = 1,2, \ldots ,n} \right) \) , with \( \omega_{j} \in \left[ {0,1} \right] \), \( \sum\nolimits_{j = 1}^{n} {\omega_{j} } = 1 \) , and \( n \) is the balancing coefficient, \( \gamma > 0 \).

It is easy to derive special cases of the BFHHG operator when the parameter γ takes particular values.

  • If \( \gamma = 1 \), the BFHHG operator reduces to the bipolar fuzzy hybrid geometric (BFHG) operator:

    $$ {\text{BFHG}}_{\omega ,w} \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\dot{\tilde{b}}_{j} } \right)^{{w_{j} }} = \left( {\prod\limits_{j = 1}^{n} {\left( {\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } , - 1{ + }\prod\limits_{j = 1}^{n} {\left( {1{ + }\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } \right) $$
    (48)

  • If \( \gamma = 2 \), the BFHHG operator reduces to the bipolar fuzzy Einstein hybrid geometric (BFEHG) operator:

    $$ {\text{BFEHG}}_{w,\omega } \left( {\tilde{b}_{1} ,\tilde{b}_{2} , \ldots ,\tilde{b}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\dot{\tilde{b}}_{\sigma \left( j \right)} } \right)^{{w_{j} }} = \left( {\frac{{2\prod\nolimits_{j = 1}^{n} {\left( {\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {2 - \dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } + \prod\nolimits_{j = 1}^{n} {\left( {\dot{\mu }_{\sigma \left( j \right)}^{ + } } \right)^{{w_{j} }} } }},\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1{ + }\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left| {\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right|} \right)^{{w_{j} }} + \prod\nolimits_{j = 1}^{n} {\left( {1{ + }\dot{\nu }_{\sigma \left( j \right)}^{ - } } \right)^{{w_{j} }} } } }}} \right) $$
    (49)

4 Models for Multiple Attribute Decision Making with Bipolar Fuzzy Information

We next apply the bipolar Hamacher aggregation operators developed in the previous section to solve MADM problems with Bipolar fuzzy information.

The following assumptions or notations are used to represent the MADM problem for potential evaluation of emerging technology commercialization with bipolar fuzzy information. Denote a discrete set of alternatives by \( A = \left\{ {A_{1} ,A_{2} , \ldots ,A_{m} } \right\} \) and the set of attributes by \( G = \left\{ {G_{1} ,G_{2} , \ldots ,G_{n} } \right\} \). Let \( w = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right) \) be the weight vector of attributes, where \( w_{j} \ge 0 \), \( j = 1,2, \ldots ,n \), \( \sum\nolimits_{j = 1}^{n} {w_{j} } = 1 \). Suppose that \( \tilde{R} = \left( {\tilde{r}_{ij} } \right)_{m \times n} = \left( {\mu_{ij}^{ + } ,\nu_{ij}^{ - } } \right)_{m \times n} \) is the bipolar fuzzy decision matrix, where \( \mu_{ij}^{ + } \) and \( \nu_{ij}^{ - } \) indicate, respectively, the positive degree and negative degree assessed by the decision maker that the alternative \( A_{i} \) satisfies the attribute \( G_{j} \), \( \mu_{ij}^{ + } \in \left[ {0,1} \right] \), \( \nu_{ij}^{ - } \in \left[ { - 1,0} \right] \), \( i = 1,2, \ldots ,m \), \( j = 1,2, \ldots ,n \).

The process of utilizing the BFHWA (or BFHWG) operator to solve an MADM problem is presented below.

  • Step 1 Applying the BFHWA operator to process the information in matrix \( \tilde{R} \), derive the overall values \( \tilde{r}_{i} \left( {i = 1,2, \ldots ,m} \right) \) of the alternative \( A_{i} \).

    $$ \tilde{r}_{i} = \left( {\mu_{i}^{ + } ,\nu_{i}^{ - } } \right){\text{ = BFHWA}}_{\omega } \left( {\tilde{r}_{i1} ,\tilde{r}_{i2} , \ldots ,\tilde{r}_{in} } \right) = \mathop \oplus \limits_{j = 1}^{n} \left( {\omega_{j} \tilde{r}_{ij} } \right) = \left( {\frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{ij}^{ + } } \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{ij}^{ + } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\mu_{ij}^{ + } } \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 - \mu_{ij}^{ + } } \right)^{{\omega_{j} }} } } }},} \right.\left. {\frac{{ - \gamma \prod\nolimits_{j = 1}^{n} {\left| {\nu_{ij}^{ - } } \right|^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 + \nu_{ij}^{ - } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left| {\nu_{ij}^{ - } } \right|^{{\omega_{j} }} } }}} \right) $$
    (50)

    If the BFHWG operator is chosen instead, we have

    $$ \tilde{r}_{i} = \left( {\mu_{i}^{ + } ,\nu_{i}^{ - } } \right){\text{ = BFHWG}}_{\omega } \left( {\tilde{r}_{i1} ,\tilde{r}_{i2} , \ldots ,\tilde{r}_{in} } \right) = \mathop \otimes \limits_{j = 1}^{n} \left( {\tilde{r}_{ij} } \right)^{{\omega_{j} }} = \left( {\frac{{\gamma \prod\nolimits_{j = 1}^{n} {\left( {\mu_{ij}^{ + } } \right)^{{\omega_{j} }} } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left( {1 - \mu_{ij}^{ + } } \right)} \right)^{{\omega_{j} }} } + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {\mu_{ij}^{ + } } \right)^{{\omega_{j} }} } }},} \right.\left. { - \frac{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{ij}^{ - } } \right|} \right)^{{\omega_{j} }} - \prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{ij}^{ - } } \right)^{{\omega_{j} }} } } }}{{\prod\nolimits_{j = 1}^{n} {\left( {1 + \left( {\gamma - 1} \right)\left| {\nu_{ij}^{ - } } \right|} \right)^{{\omega_{j} }} + \left( {\gamma - 1} \right)\prod\nolimits_{j = 1}^{n} {\left( {1 + \nu_{ij}^{ - } } \right)^{{\omega_{j} }} } } }}} \right) $$
    (51)

  • Step 2 Calculate the scores \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \ldots ,m} \right) \).

  • Step 3 Rank all the alternatives \( A_{i} \left( {i = 1,2, \ldots ,m} \right) \) in terms of \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \ldots ,m} \right) \). If there is no difference between two scores \( S\left( {\tilde{r}_{i} } \right) \) and \( S\left( {\tilde{r}_{j} } \right) \), then calculate the accuracy degrees \( H\left( {\tilde{r}_{i} } \right) \) and \( H\left( {\tilde{r}_{j} } \right) \) to rank the alternatives \( A_{i} \) and \( A_{j} \).

  • Step 4 Select the best alternative(s).

5 Numerical example

In this section, we utilize a practical multiple attribute decision-making problems to illustrate the application of the developed approaches. Suppose an organization plans to implement enterprise resource planning (ERP) system (adapted from [58]). The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all possible information about ERP vendors and systems, project term chooses five potential ERP systems \( A_{i} \left( {i = 1,2, \ldots ,5} \right) \) as candidates. The company employs some external professional organizations (or experts) to aid this decision making. The project team selects four attributes to evaluate the alternatives: (1) function and technology G1, (2) strategic fitness G2, (3) vendor’s ability G3; (4) vendor’s reputation G4. The five possible ERP systems \( A_{i} \left( {i = 1,2, \ldots ,5} \right) \) are to be evaluated using the bipolar fuzzy numbers by the decision makers under the above four attributes [whose weighting vector is \( \omega = \left( {0.2,0.1,0.3,0.4} \right) \)]. The ratings are presented in the following matrix.

$$ \tilde{R} = \left[ {\begin{array}{*{20}c} {\left( {0.7, - 0.3} \right)} & {\left( {0.2, - 0.8} \right)} & {\left( {0.2, - 0.3} \right)} & {\left( {0.8, - 0.2} \right)} \\ {\left( {0.6, - 0.4} \right)} & {\left( {0.3, - 0.5} \right)} & {\left( {0.5, - 0.4} \right)} & {\left( {0.4, - 0.6} \right)} \\ {\left( {0.4, - 0.2} \right)} & {\left( {0.4, - 0.1} \right)} & {\left( {0.3, - 0.5} \right)} & {\left( {0.5, - 0.4} \right)} \\ {\left( {0.2, - 0.6} \right)} & {\left( {0.3, - 0.7} \right)} & {\left( {0.4, - 0.2} \right)} & {\left( {0.5, - 0.3} \right)} \\ {\left( {0.7, - 0.3} \right)} & {\left( {0.5, - 0.4} \right)} & {\left( {0.5, - 0.4} \right)} & {\left( {0.4, - 0.4} \right)} \\ \end{array} } \right] $$

In the following, in order to select the most desirable ERP systems, we utilize the BFHWA operator (BFHWG) operator to develop an approach to multiple attribute decision-making problems with bipolar fuzzy information, which can be described as following.

  • Step 1 Let \( \gamma = 3 \). Apply the BFHWA operator to compute the overall preference values \( \tilde{r}_{i} \) of the ERP systems \( A_{i} \left( {i = 1,2,3,4,5} \right) \)

    $$ \tilde{r}_{1} = \left( {0.588, - 0.202} \right),\tilde{r}_{2} = \left( {0.464, - 0.292} \right),\tilde{r}_{3} = \left( {0.412, - 0.223} \right) \tilde{r}_{4} = \left( {0.393, - 0.227} \right),\tilde{r}_{5} = \left( {0.508, - 0.251} \right) $$

  • Step 2 Calculate the scores \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \ldots ,5} \right) \) of the overall bipolar fuzzy numbers \( \tilde{r}_{i} \;\left( {i = 1,2, \ldots ,5} \right) \)

    $$ S\left( {\tilde{r}_{1} } \right) = 0.693,S\left( {\tilde{r}_{2} } \right) = 0.586,S\left( {\tilde{r}_{3} } \right) = 0.594 S\left( {\tilde{r}_{4} } \right) = 0.583,S\left( {\tilde{r}_{5} } \right) = 0.628 $$

  • Step 3 Rank all the ERP systems \( A_{i} \left( {i = 1,2,3,4,5} \right) \) in accordance with the scores \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \cdots ,5} \right) \) of the overall bipolar fuzzy numbers:\( A_{1} \succ A_{5} \succ A_{3} \succ A_{2} \succ A_{4} \).

  • Step 4 \( A_{1} \) is chosen as the most desirable ERP system.

If the BFHWG operator is applied instead, the problem can be solved in a similar way.

  • Step 1 Let \( \gamma = 3 \). Aggregate all BFNs via the BFHWG operator to derive the overall BFNs \( \tilde{r}_{i} \;\left( {i = 1,2, \ldots ,5} \right) \) of the ERP system \( A_{i} \):

    $$ \tilde{r}_{1} = \left( {0.490,{ - }0.325} \right),\tilde{r}_{2} = \left( {0.455,{ - }0.494} \right),\tilde{r}_{3} = \left( {0.404,{ - }0.363} \right) \tilde{r}_{4} = \left( {0.376,{ - }0.381} \right),\tilde{r}_{5} = \left( {0.495,{ - }0.380} \right) $$

  • Step 2 Calculate the scores \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \ldots ,5} \right) \) of the overall BFNs \( \tilde{r}_{i} \;\left( {i = 1,2, \ldots ,5} \right) \) of the ERP system \( A_{i} \):\( S\left( {\tilde{r}_{1} } \right) = 0.583,S\left( {\tilde{r}_{2} } \right) = 0.480,S\left( {\tilde{r}_{3} } \right) = 0.521 S\left( {\tilde{r}_{4} } \right) = 0.497,S\left( {\tilde{r}_{5} } \right) = 0.557 \).

  • Step 3 Rank all the ERP systems \( A_{i} \left( {i = 1,2,3,4,5} \right) \) in accordance with the scores \( S\left( {\tilde{r}_{i} } \right)\;\left( {i = 1,2, \ldots ,5} \right) \):\( A_{1} \succ A_{5} \succ A_{3} \succ A_{4} \succ A_{2} \).

  • Step 4 Return \( A_{1} \) as the most desirable ERP system.

From the above analysis, it is easily seen that although the overall rating values of the alternatives are different by using two operators, respectively, the ranking orders of the alternatives are slightly different. However, the most desirable ERP system is \( A_{1} \).

6 Conclusion

The existing aggregation operators for BFNs are based on the traditional arithmetic and geometric operators. In this paper, we have investigated the multiple attribute decision-making problems with bipolar fuzzy information. Motivated by the Hamacher operations, we have proposed some bipolar fuzzy Hamacher aggregating operators: bipolar fuzzy Hamacher weighted average (BFHWA) operator, bipolar fuzzy Hamacher ordered weighted average (BFHOWA) operator, bipolar fuzzy Hamacher hybrid average (BFHHA) operator, bipolar fuzzy Hamacher weighted geometric (BFHWG) operator, bipolar fuzzy Hamacher ordered weighted geometric (BFHOWG) operator, bipolar fuzzy Hamacher hybrid geometric (BFHHG) operator. We investigate the characteristics and special cases of these operators. Then, we have utilized these operators to develop some approaches to solve the multiple attribute decision-making problems with bipolar fuzzy information. Finally, a practical example for enterprise resource planning (ERP) system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness. In our future study, we shall extend the proposed models to other domains and other applications [59,60,61,62,63,64].