Linguistic Intuitionistic Fuzzy Group Decision Making Based on Aggregation Operators

Abstract

This paper researches decision making with linguistic intuitionistic fuzzy variables. Several linguistic intuitionistic fuzzy operations are first defined. Then, several linguistic intuitionistic fuzzy aggregation operators are provided, including the linguistic intuitionistic fuzzy hybrid weighted arithmetical averaging operator, and the linguistic intuitionistic fuzzy hybrid weighted geometric mean operator. Considering the interactive characteristics between the weights of elements in a set, several linguistic intuitionistic fuzzy Shapley aggregation operators are presented, including the linguistic intuitionistic fuzzy hybrid Shapley arithmetical averaging operator, and the linguistic intuitionistic fuzzy hybrid Shapley geometric mean operator. To ensure the application reasonably, several desirable properties are discussed. When the weighting information is incompletely known, models for the optimal fuzzy and additive measures are constructed. After that, an approach to multi-criteria group decision making with linguistic intuitionistic fuzzy information is performed. Finally, a practical example about evaluating different types of engines is provided to illustrate the developed procedure.

1 Introduction

In modern decision-making problems, uncertain and fuzzy information always exists. Considering this situation, researchers introduced Zadeh’s fuzzy sets [41] into decision making and developed fuzzy decision-making theory. Atanassov [1] noted that Zadeh’s fuzzy sets can only denote the preferred information of decision makers (DMs). When the DMs want to offer their preferred and non-preferred judgments simultaneously, Zadeh’s fuzzy sets are helpless. Thus, Atanassov [1] further introduced the concept of intuitionistic fuzzy sets (IFSs) that are composed by a membership degree, a non-membership degree, and a hesitancy degree. After the original work of Atanassov [1], many decision-making methods with intuitionistic fuzzy information are proposed, such as methods based on intuitionistic fuzzy entropies [3, 30], methods based on intuitionistic fuzzy similarity measures [14, 20], and methods based on aggregation operators [17, 43]. Furthermore, Bao et al. [4] developed an intuitionistic fuzzy decision-making method based on prospect theory and evidential reasoning, and Krishankumar et al. [13] proposed an intuitionistic fuzzy PROMETHEE method. The applications of intuitionistic fuzzy decision making can be seen in the literature [5, 28, 38]. To denote the uncertain preferred and non-preferred judgments rather than exact ones, Atanassov and Gargov [2] further introduced the concept of interval-valued intuitionistic fuzzy sets that use intervals in [0, 1] to denote the uncertain preferred and non-preferred recognitions of the DMs, respectively. Because interval-valued intuitionistic fuzzy sets endow the DMs with more flexibility, many researchers devoted themselves to studying interval-valued intuitionistic fuzzy decision making. Taking interval-valued intuitionistic fuzzy decision making based on aggregation operators for example, interval-valued intuitionistic fuzzy aggregation operators can be classified into five types following the adopted operational laws: the interval-valued intuitionistic fuzzy Archimedean aggregation operator [35], the interval-valued intuitionistic fuzzy Frank aggregation operator [44], the interval-valued intuitionistic fuzzy Hamacher aggregation operator [15], the interval-valued intuitionistic fuzzy Einstein aggregation operator [36], and the interval-valued intuitionistic fuzzy Algebraic aggregation operator [16, 21, 37]. Note that the Einstein and Algebraic aggregation operators are two special cases of the Hamacher aggregation operators.

Although (interval-valued) intuitionistic fuzzy sets are efficient to denote the preferred and non-preferred information of the DMs, they only permit the DMs to use real numbers or intervals in [0, 1] to express the quantitative judgments. However, in many situations, the DMs may want to give their qualitative preferences; namely, they use linguistic variables to express their judgments, such as “good”, “bad”, or “fair”. Zadeh [42] first noted this issue and introduced the concept of linguistic variables. To facilitate the application, Herrera et al. [10] presented the concept of linguistic term sets to denote the possible values of linguistic variables. With the development of decision making with linguistic information, several extending forms based on different points of view are proposed, such as hesitant fuzzy linguistic term sets [29], interval intuitionistic uncertain linguistic fuzzy sets [18], linguistic hesitant fuzzy sets [22], uncertain linguistic hesitant fuzzy sets [23], and interval-valued intuitionistic uncertain linguistic sets [24]. To avoid information loss, there are two linguistic computing models: the 2-tuple linguistic representation model [6, 11] and the continuous linguistic representation model [39]. Dong et al. [9] proved the equivalence of these two representation models. Most of all researches about decision making with linguistic information are based on these two models. For instance, decision making based on 2-tuple linguistic representation model is studied in the literature [7, 25, 40], while decision making using the continuous linguistic representation model is discussed in the literature [26, 46, 47].

However, all of the above-mentioned linguistic variables cannot denote the preferred and non-preferred qualitative judgments of the DMs. To address this problem, Chen et al. [8] presented the concept of linguistic intuitionistic fuzzy variables (LIFVs) and defined several basic operations. Then, the authors defined two aggregation operators: the linguistic intuitionistic fuzzy weighted geometric (LIFWG) operator and the linguistic intuitionistic fuzzy hybrid geometric (LIFHG) operator, by which a group decision-making method with linguistic intuitionistic fuzzy information is presented. Garg and Kumar [10] introduced the set pair analysis (SPA) theory into LIFVs and proposed the linguistic connection number (LCN). Then, the authors developed an approach for decision making with LIFVs using the linguistic connection number ordered weighted geometric (LCNOWG) operator and the linguistic connection number hybrid geometric (LCNHG) operator. Furthermore, Liu and Liu [19] introduced a method for decision making with LIFVs using the scaled prioritized linguistic intuitionistic fuzzy weighted operator. Notably, this method needs to distinguish the categories of attributes. Moreover, Zhang et al. [45] offered an extended outranking approach for decision making with LIFVs.

After reviewing researches about decision making with LIFVs, we find that there are several limitations. The linguistic intuitionistic fuzzy hybrid operators [8, 10] do not satisfy idempotency and boundary, and methods in [8, 10] are based on the assumption that the weighting information is completely known. While methods in [19, 45] did not research group decision making. Furthermore, all of these methods are based on the assumption that there is no interaction between the weights of elements in a set. To avoid these issues, this paper continues to study linguistic intuitionistic fuzzy decision making based on aggregation operators. To do this, several linguistic intuitionistic fuzzy operational laws are defined. To calculate comprehensive LIFVs, several linguistic intuitionistic fuzzy aggregation operators are offered. Meanwhile, several linguistic intuitionistic fuzzy Shapley aggregation operators are provided to cope with the situations where there are interactive characteristics. Furthermore, models for determining the optimal fuzzy and additive measures are constructed to address decision making with incomplete weighting information. The organization is offered as follows:

Section 2 first reviews several basic concepts, including linguistic variables, LIFVs, and a ranking order. Then, it defines several linguistic intuitionistic fuzzy operational laws. Section 3 contains two parts. The first part defines several linguistic intuitionistic fuzzy aggregation operators based on additive measures, and the second part provides several linguistic intuitionistic fuzzy Shapley aggregation operators to reflect the interactions between the weights of elements. Section 4 builds several programming models to determine fuzzy and additive measures on the DM set, on the criteria set and on their ordered sets, respectively. Section 5 gives a group decision-making method with linguistic intuitionistic fuzzy information. Meanwhile, a practical example is offered to show the application of the new method.

2 Preliminary

To denote the qualitative judgments of the DMs rather than quantitative ones, linguistic variables introduced by Zadeh [42] are useful, which permit the DMs to apply linguistic variables to express their judgments, such as “fast”, “slow”, or “fair”. To facilitate the application of linguistic variables, Herrera and Martinez [11] introduced linguistic term sets to denote linguistic variables. For example, a linguistic term set may be denoted as S = {s_i | i = 0, 1,…, 2t}, where t is a positive integer. The linguistic term s_i represents a possible value for a linguistic variable. Furthermore, Herrera and Martinez [11] defined the following four properties of linguistic terms: (1) The set is ordered: s_i > s_j, if i > j; (2) Maximum operator: max(s_i, s_j) = s_i, if s_i ≥ s_j; (3) Minimum operator: min(s_i, s_j) = s_i, if s_i ≤ s_j; and (4) A negation operator: neg(s_i) = s_j such that j = 2t−i.

For example, the linguistic term set S may be expressed as S = {s₀: extremely bad, s₁: very bad, s₂: bad, s₃: fair, s₄: good, s₅: very good, s₆: extremely good}. To preserve information, Xu [39] extended the discrete linguistic term set S to the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. For any $s_{\alpha } \in S_{c}$, if $s_{\alpha } \in S$, then it is called an original linguistic term. Otherwise, it is called a virtual linguistic term.

Let $s_{\alpha }$ and $s_{\beta }$ be any two linguistic variables, then several of their operational laws are defined as follows [39]:

• (1) $s_{\alpha } \oplus s_{\beta } = s_{\alpha + \beta }$;

• (2) $s_{\alpha } { \ominus }s_{\beta } = s_{\alpha - \beta }$;

• (3) $\lambda s_{\alpha } = s_{\lambda \alpha } ,\,\lambda \in [0,1]$;

• (4) $\lambda \left( {s_{\alpha } \oplus s_{\beta } } \right) = \lambda s_{\alpha } \oplus \lambda s_{\beta } ,\,\lambda \in [0,1]$;

• (5) $\left( {\lambda_{1} + \lambda_{2} } \right)s_{\alpha } = \lambda_{1} s_{\alpha } \oplus \lambda_{2} s_{\alpha } ,\,\lambda_{1} ,\lambda_{2} \in [0,1]$.

To denote the preferred and non-preferred qualitative judgments of the DMs, Chen et al. [8] introduced linguistic intuitionistic fuzzy variables (LIFVs) in a similar way as intuitionistic fuzzy variables [1].

Definition 1

[8] A LIFV on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$ is expressed as $\tilde{s} = \left( {s_{\alpha } ,s_{\beta } } \right)$, where $s_{\alpha }$ and $s_{\beta }$ are the preferred and non-preferred qualitative degrees, respectively, and $s_{\alpha } \oplus s_{\beta } \le s_{2t}$.

Considering the order relationship between LIFVs, Zhang et al. [45] offered the following ranking order

Definition 2

[45] Let $\tilde{s}_{1} = \left( {s_{{\alpha_{1} }} ,s_{{\beta_{1} }} } \right)$ and $\tilde{s}_{2} = \left( {s_{{\alpha_{2} }} ,s_{{\beta_{2} }} } \right)$ be any two LIFVs on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, their order relationship is defined as follows:

• (1) $\tilde{s}_{1} \ge \tilde{s}_{2}$ if $s_{{\alpha_{1} }} \ge s_{{\alpha_{2} }} \wedge s_{{\beta_{1} }} \le s_{{\beta_{2} }} \wedge s_{{2t - \alpha_{1} - \beta_{1} }} \le s_{{2t - \alpha_{2} - \beta_{2} }}$;

• (2) $\tilde{s}_{1} = \tilde{s}_{2}$ if $\tilde{s}_{1} \ge \tilde{s}_{2} \wedge \tilde{s}_{1} \le \tilde{s}_{2}$.

Next, we define several linguistic intuitionistic fuzzy operational laws to calculate the comprehensive LIFVs.

Definition 3

Let $\tilde{s}_{1} = \left( {s_{{\alpha_{1} }} ,s_{{\beta_{1} }} } \right)$ and $\tilde{s}_{2} = \left( {s_{{\alpha_{2} }} ,s_{{\beta_{2} }} } \right)$ be any two LIFVs on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, their operational laws are defined as follows:

• (1) $\lambda \tilde{s}_{1} = \left( {s_{{\lambda \alpha_{1} }} ,s_{{\lambda \beta_{1} }} } \right),\,\lambda \in [0,1]$;

• (2) $\tilde{s}_{1}^{\lambda } = \left( {s_{{\alpha_{1}^{\lambda } }} ,s_{{\beta_{1}^{\lambda } }} } \right),\,\lambda \in [0,1]$;

• (3) $\lambda_{1} \tilde{s}_{1} \oplus \lambda_{2} \tilde{s}_{2} = \left( {s_{{\lambda_{1} \alpha_{1} + \lambda_{2} \alpha_{2} }} ,s_{{\lambda_{1} \beta_{1} + \lambda_{2} \beta_{2} }} } \right),\,\lambda_{1} ,\lambda_{2} \in [0,1] \wedge \lambda_{1} + \lambda_{2} \le 1$;

• (4) $\tilde{s}_{1}^{{\lambda_{1} }} \otimes \tilde{s}_{2}^{{\lambda_{2} }} = \left( {s_{{\alpha_{1}^{{\lambda_{1} }} \times \alpha_{2}^{{\lambda_{2} }} }} ,s_{{\beta_{1}^{{\lambda_{1} }} \times \beta_{2}^{{\lambda_{2} }} }} } \right),\,\lambda_{1} ,\lambda_{2} \in [0,1] \wedge \lambda_{1} + \lambda_{2} \le 1$.

From Definition 3, we can check that their results are still LIFVs in $S_{c}$.

Definition 2 shows that Zhang’s ranking method for LIFVs needs the preferred, non-preferred, and hesitant linguistic values of LIFVs to satisfy the defined relationship simultaneously. This ranking method in fact compares three-dimensional vectors formed by LIFVs. Thus, many situations are incomparable. For example, let $\tilde{s}_{1} = \left( {s_{4} ,s_{5} } \right)$ and $\tilde{s}_{2} = \left( {s_{3} ,s_{4} } \right)$ be two LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,8]\}$. Because $s_{4} > s_{3}$ and $s_{5} > s_{4}$, we cannot derive their order relationship following Zhang’s ranking method. However, we can easily check that $\tilde{s}_{1}$ is better than $\tilde{s}_{2}$. Considering this situation, we applied the ranking order introduced in the literature [8]:

Definition 4

[8] Let $\tilde{s} = \left( {s_{\alpha } ,s_{\beta } } \right)$ be a LIFV on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the score function is defined as: $S(\tilde{s}) = \alpha - \beta$, and the accuracy function is defined as: $A(\tilde{s}) = \alpha + \beta$.

Let $\tilde{s}_{1} = \left( {s_{{\alpha_{1} }} ,s_{{\beta_{1} }} } \right)$ and $\tilde{s}_{2} = \left( {s_{{\alpha_{2} }} ,s_{{\beta_{2} }} } \right)$ be any two LIFVs on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, their order relationship is defined as follows:

• If $S(\tilde{s}_{1} ) \ge S(\tilde{s}_{2} )$, then $\tilde{s}_{1} \ge \tilde{s}_{2}$;

• If $S(\tilde{s}_{1} ) = S(\tilde{s}_{2} )$, then $\left\{ {\begin{array}{*{20}c} {A(\tilde{s}_{1} ) > A(\tilde{s}_{2} ),\tilde{s}_{1} > \tilde{s}_{2} } \\ {A(\tilde{s}_{1} ) = A(\tilde{s}_{2} ),\tilde{s}_{1} = \tilde{s}_{2} } \\ \end{array} } \right.$.

Property 1

Let$\tilde{s}_{1} = \left( {s_{{\alpha_{1} }} ,s_{{\beta_{1} }} } \right)$and$\tilde{s}_{2} = \left( {s_{{\alpha_{2} }} ,s_{{\beta_{2} }} } \right)$be any two LIFVs defined on the continuous linguistic term set$S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then,$\tilde{s}_{1} = \tilde{s}_{2}$if and only if$s_{{\alpha_{1} }} = s_{{\alpha_{2} }}$and$s_{{\beta_{1} }} = s_{{\beta_{2} }}$.

Proof

When $\tilde{s}_{1} = \tilde{s}_{2}$, we have $S(\tilde{s}_{1} ) = S(\tilde{s}_{2} )$ and $A(\tilde{s}_{1} ) = A(\tilde{s}_{2} )$, namely, $\alpha_{1} - \beta_{1} = \alpha_{2} - \beta_{2}$ and $\alpha_{1} + \beta_{1} = \alpha_{2} + \beta_{2}$. Thus, $\alpha_{1} = \alpha_{2}$ and $\beta_{1} = \beta_{2}$. On the other hand, if $\alpha_{1} = \alpha_{2}$ and $\beta_{1} = \beta_{2}$, we easily derive $\tilde{s}_{1} = \tilde{s}_{2}$. □

3 Several Linguistic Intuitionistic Fuzzy Aggregation Operators

On the basis of the defined linguistic intuitionistic fuzzy operations, this section introduces several linguistic intuitionistic fuzzy aggregation operators. Following the adopted weighting information, we classify them into two types. The first type uses additive measures, while the second type applies fuzzy measures.

3.1 Linguistic Intuitionistic Fuzzy Aggregation Operators Based on Additive Measures

Definition 5

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the linguistic intuitionistic fuzzy weighted arithmetical averaging (LIFWAA) operator is defined as:

$${\text{LIFWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{i = 1}^{n} \omega_{i} \tilde{s}_{i} = \left( {s_{{\sum\nolimits_{i = 1}^{n} {\omega_{i} } \alpha_{i} }} ,s_{{\sum\nolimits_{i = 1}^{n} {\omega_{i} } \beta_{i} }} } \right)$$

(1)

and the linguistic intuitionistic fuzzy weighted geometric mean (LIFWGM) operator is defined as:

$${\text{LIFWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{i = 1}^{n} \tilde{s}_{i}^{{\omega_{i} }} = \left( {s_{{\varPi_{i = 1}^{n} \alpha_{i}^{{\omega_{i} }} }} ,s_{{\varPi_{i = 1}^{n} \beta_{i}^{{\omega_{i} }} }} } \right)$$

(2)

where $\omega = (\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} )$ is a weighting vector on $\left\{ {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right\}$ such that $\sum\nolimits_{i = 1}^{n} {\omega_{i} } = 1$ and $\omega_{i} \ge 0$ for all i = 1, 2, …, n.

Considering the importance of the ordered positions to reduce the influence of extreme values, the linguistic intuitionistic fuzzy ordered weighted arithmetical averaging (LIFOWAA) operator and the linguistic intuitionistic fuzzy ordered weighted geometric mean (LIFOWGM) operator are defined as follows:

Definition 6

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the LIFOWAA operator is defined as:

$${\text{LIFOWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{j = 1}^{n} w_{j} \tilde{s}_{(j)} = \left( {s_{{\sum\nolimits_{j = 1}^{n} {w_{j} } \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {w_{j} } \beta_{(j)} }} } \right)$$

(3)

and the LIFOWGM operator is defined as:

$${\text{LIFOWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{j = 1}^{n} \tilde{s}_{(j)}^{{w_{j} }} = \left( {s_{{\varPi_{j = 1}^{n} \alpha_{(j)}^{{w_{j} }} }} ,s_{{\varPi_{j = 1}^{n} \beta_{(j)}^{{w_{j} }} }} } \right)$$

(4)

where $( \cdot )$ is a permutation on N = {1, 2, …, n} such that $\tilde{s}_{(1)} \le \text{ }\tilde{s}_{(2)} \le \cdots \le \tilde{s}_{(n)}$, and $w = (w_{1} ,w_{2} , \ldots ,w_{n} )$ is a weighting vector on the ordered position set N such that $\sum\nolimits_{j = 1}^{n} {w_{j} } = 1$ and $w_{j} \ge 0$ for all j = 1, 2, …, n.

Definitions 5 and 6 show that the LIFWAA and LIFWGM operators only give the importance of LIFVs, while the LIFOWAA and LIFOWGM operators only consider the weights of the ordered positions. To show these two aspects simultaneously, we further offer the linguistic intuitionistic fuzzy hybrid weighted arithmetical averaging (LIFHWAA) operator and the linguistic intuitionistic fuzzy hybrid weighted geometric mean (LIFHWGM) operator as follows:

Definition 7

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the LIFHWAA operator is defined as:

$${\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{j = 1}^{n} \frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}\tilde{s}_{(j)} = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta_{(j)} }} } \right)$$

(5)

and the LIFHWGM operator is defined as:

$${\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{j = 1}^{n} \tilde{s}_{(j)}^{{\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}}} = \left( {s_{{\varPi_{j = 1}^{n} \alpha_{(j)}^{{\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}}} }} ,s_{{\varPi_{j = 1}^{n} \beta_{(j)}^{{\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}}} }} } \right)$$

(6)

where $( \cdot )$ is a permutation on N such that $\omega_{(1)} \tilde{s}_{(1)} \le \text{ }\omega_{(2)} \tilde{s}_{(2)} \le \cdots \le \omega_{(n)} \tilde{s}_{(n)}$, and the other notations as shown in Definitions 5 and 6.

Theorem 1

Let$\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set$S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, their aggregated values using the LIFHWAA and LIFHWGM operators are still IVIFVs.

Proof

Following Eqs. (5) and (6), we can easily derive the conclusions. □

Next, we discuss several desirable properties of the above-defined aggregation operators.

Theorem 2

Let$\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$and$\tilde{t}_{i} = \left( {s_{{\mu_{i} }} ,s_{{v_{i} }} } \right)$, i = 1, 2, …, n, be any two collections of LIFVs defined on the continuous linguistic term set$S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$.

• (1) Commutativity.Let$\tilde{s}'_{i} = \left( {s_{{\alpha '_{i} }} ,s_{{\beta '_{i} }} } \right)$be a permutation of$\tilde{s}_{i}$, i = 1, 2, …, n, then

  $$\begin{aligned} {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = {\text{LIFHWAA}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} ) \\ {\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = {\text{LIFHWGM}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} ) \\ \end{aligned}$$

• (2) Idempotency.If all LIFVs$\tilde{s}_{i}$, i = 1, 2, …, n, are equal, i.e.,$\tilde{s}_{i} = \tilde{s} = \left( {s_{\alpha } ,s_{\beta } } \right)$for all i, then

  $$\begin{aligned} {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \tilde{s} \\ {\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \tilde{s} \\ \end{aligned}$$

• (3) Comonotonicity.If$\tilde{s}_{i}$and$\tilde{t}_{i}$are comonotonic, namely,$\tilde{s}_{(1)} \le \tilde{s}_{(2)} \le \cdots \le \tilde{s}_{(n)}$if and only if$\tilde{t}_{(1)} \le \tilde{t}_{(2)} \le \ldots \le \tilde{t}_{(n)}$for all i, where$( \cdot )$is a permutation on N = {1, 2, …, n} such that$\tilde{s}_{(j)}$and$\tilde{t}_{(j)}$are the jth least values of$\tilde{s}_{i}$and$\tilde{t}_{i}$, i = 1, 2, …, n, respectively. Then,

  $$\begin{aligned} {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & \le {\text{LIFHWAA}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} ) \\ {\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & \le {\text{LIFHWGM}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} ) \\ \end{aligned}$$

• (4) Boundary.Let$\tilde{s}^{ - } = \left( {s_{{\mathop {\hbox{min} }\limits_{i \in N} \alpha_{i} }} ,s_{{\mathop {\hbox{max} }\limits_{i \in N} \beta_{i} }} } \right)$and$\tilde{s}^{ + } = \left( {s_{{\mathop {\hbox{max} }\limits_{i \in N} \alpha_{i} }} ,s_{{\mathop {\hbox{min} }\limits_{i \in N} \beta_{i} }} } \right)$, then

  $$\tilde{\alpha }^{ - } \le {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{\alpha }^{ + } \,and\,\tilde{\alpha }^{ - } \le {\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{\alpha }^{ + }$$

  where N = {1, 2, …, n}.

Proof

For (1): We have

$$\begin{aligned} {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta_{(j)} }} } \right) = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega '_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega '_{(j)} } }}} \alpha '_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega '_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega '_{(j)} } }}} \beta '_{(j)} }} } \right) \\ & = {\text{LIFHWAA}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} ). \\ \end{aligned}$$

Similarly, we have ${\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = {\text{LIFHWGM}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} )$.

For (2): We have

$${\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta_{(j)} }} } \right) = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta }} } \right) = \left( {s_{\alpha } ,s_{\beta } } \right) = \tilde{s}$$

and

$${\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \left( {s_{{\Pi_{j = 1}^{n} \alpha_{(j)}^{{\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}}} }} ,s_{{\Pi_{j = 1}^{n} \beta_{(j)}^{{\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}}} }} } \right) = \left( {s_{{\alpha^{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} }} }} ,s_{{\beta^{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} }} }} } \right) = \left( {s_{\alpha } ,s_{\beta } } \right) = \tilde{s}$$

For (3): From $\tilde{s}_{(i)} \le \tilde{s}_{(j)}$ and $\tilde{t}_{(i)} \le \tilde{t}_{(j)}$ for any i, j = 1, 2, …, n, we obtain $\omega_{(i)} \tilde{s}_{(i)} \le \omega_{(j)} \tilde{s}_{(j)} \Leftrightarrow \omega_{(i)} \tilde{t}_{(i)} \le \omega_{(j)} \tilde{t}_{(j)}$ and $\omega_{(i)} \tilde{s}_{(i)} \ge \omega_{(j)} \tilde{s}_{(j)} \Leftrightarrow \omega_{(i)} \tilde{t}_{(i)} \ge \omega_{(j)} \tilde{t}_{(j)}$. Taking the first case for example, if $\omega_{(i)} \tilde{s}_{(i)} \le \omega_{(j)} \tilde{s}_{(j)}$, we have

$$LS(\omega_{(i)} \tilde{s}_{(i)} ) \le LS(\omega_{(j)} \tilde{s}_{(j)} )\, {\text{or}}\,LS(\omega_{(i)} \tilde{s}_{(i)} ) = LS(\omega_{(j)} \tilde{s}_{(j)} )\,{\text{and}}\,LA(\omega_{(j)} \tilde{s}_{(j)} ) \le LA(\omega_{(j)} \tilde{s}_{(j)} )$$

by which we derive $s_{{\omega_{(i)} (\alpha_{(i)} - \beta_{(i)} )}} \le s_{{\omega_{(j)} (\alpha_{(j)} - \beta_{(j)} )}}$ or $s_{{\omega_{(i)} (\alpha_{(i)} - \beta_{(i)} )}} = s_{{\omega_{(j)} (\alpha_{(j)} - \beta_{(j)} )}}$ and $s_{{\omega_{(i)} (\alpha_{(i)} + \beta_{(i)} )}} \le s_{{\omega_{(j)} (\alpha_{(j)} + \beta_{(j)} )}}$. Thus,

$$s_{{\alpha_{(i)} - \beta_{(i)} }} \le s_{{\alpha_{(j)} - \beta_{(j)} }} \,{\text{or}}\,s_{{\alpha_{(i)} - \beta_{(i)} }} = s_{{\alpha_{(j)} - \beta_{(j)} }} \,{\text{and}}\,s_{{\alpha_{(i)} + \beta_{(i)} }} \le s_{{\alpha_{(j)} + \beta_{(j)} }} .$$

Following the order relationship of $\tilde{s}_{i}$ and $\tilde{t}_{i}$, i = 1, 2, …, n, we have $\omega_{(i)} \tilde{s}_{(i)} \le \omega_{(j)} \tilde{s}_{(j)} \Leftrightarrow \omega_{(i)} \tilde{t}_{(i)} \le \omega_{(j)} \tilde{t}_{(j)}$. Following Eq. (5), we obtain

$${\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta_{(j)} }} } \right) \le \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \mu_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} v_{(j)} }} } \right) = {\text{LIFHWAA}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} )$$

Furthermore, we have ${\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le {\text{LIFHWGM}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} )$.

For (4): From $\tilde{s}^{ - } \le \tilde{s}_{i} \le \tilde{s}^{ + }$ for any i = 1, 2, …, n, we have

$$\begin{aligned} {\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \beta_{(j)} }} } \right) \ge \left( {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \mathop {\hbox{min} }\limits_{j \in N} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{w_{j} \omega_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{(j)} } }}} \mathop {\hbox{min} }\limits_{j \in N} \beta_{(j)} }} } \right) \\ & = \left( {s_{{\mathop {\hbox{min} }\limits_{j \in N} \alpha_{(j)} }} ,s_{{\mathop {\hbox{min} }\limits_{j \in N} \beta_{(j)} }} } \right) = \left( {s_{{\mathop {\hbox{min} }\limits_{j \in N} \alpha_{j} }} ,s_{{\mathop {\hbox{min} }\limits_{j \in N} \beta_{j} }} } \right) = \tilde{s}^{ - } \\ \end{aligned}$$

Similarly, we obtain ${\text{LIFHWAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{s}^{ + }$. □

We can further have $\tilde{\alpha }^{ - } \le {\text{LIFHWGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{\alpha }^{ + }$ as the proof for the LIFHWAA operator.

Remark 1

Because the LIFWAA, LIFWGM, LIFOWAA, and LIFOWGM operators can be seen as special cases of the LIFHWAA and LIFHWGM operators, they all satisfy the properties listed in Theorem 2.

3.2 Linguistic Intuitionistic Fuzzy Aggregation Operators Based on Fuzzy Measures

The aggregation operators defined in Sect. 3.1 are based on the assumption that the weights of elements in a set are independent. However, this conclusion does not hold in many situations [23, 24, 35]. Considering the interactions among the weights of elements, this subsection defines several linguistic intuitionistic fuzzy Shapley aggregation operators based on fuzzy measures. First, we recall the concept of fuzzy measures.

Definition 8

[31] A fuzzy measure μ on the finite set $X = \{ x_{1} ,x_{2} , \ldots ,x_{n} \}$ is a set function μ: P(X) → [0, 1] satisfying

• (1) μ (ϕ) = 0, μ (X) = 1,

• (2) If A, B ∈ P(X) and A ⊆ B, then μ (A) ≤ μ (B),

where P(X) is the power set of X.

In cooperative game theory, the Shapley function [32] is one of the most important payoff indices that satisfies several desirable properties, such as efficiency, dummy, null, symmetry, and linear property. When we restrict the Shapley function on fuzzy measures, the following expression is obtained:

$$Sh_{{x_{i} }} (\mu ,X )= \sum\limits_{{S \subseteq N\backslash x_{i} }} {\frac{(n - s - 1)!s!}{n!}} (\mu (S \cup x_{i} ) - \mu (S))\quad \forall x_{i} \in X$$

(7)

where $\mu$ is a fuzzy measure on $X = \{ x_{1} ,x_{2} , \ldots ,x_{n} \}$, s, t, and n are the cardinalities of S, T, and X, respectively.

Property 2

[27] Let$\mu$be a fuzzy measure on$X = \{ x_{1} ,x_{2} , \ldots ,x_{n} \}$, and let$Sh$be the Shapley function for the fuzzy measure μ on the set X. Then, we have$\sum\nolimits_{i = 1}^{n} {Sh_{{x_{i} }} (\mu ,X )} = 1$and$Sh_{{x_{i} }} (\mu ,X )\ge 0$for any$x_{i} \in X$.

Property 2 shows that the Shapley function Sh is a weighting vector for the fuzzy measure μ. Using the Shapley function shown in Eq. (7), the linguistic intuitionistic fuzzy Shapley arithmetical averaging (LIFSAA) operator and the linguistic intuitionistic fuzzy Shapley geometric mean (LIFSGM) operator are defined as follows:

Definition 9

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the LIFSAA operator is defined as:

$${\text{LIFSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{i = 1}^{n} Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})\tilde{s}_{i} = \left( {s_{{\sum\nolimits_{i = 1}^{n} {Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})} \alpha_{i} }} ,s_{{\sum\nolimits_{i = 1}^{n} {Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})} \beta_{i} }} } \right)$$

(8)

and the LIFSGM operator is defined as:

$${\text{LIFSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{i = 1}^{n} \tilde{s}_{i}^{{Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})}} = \left( {s_{{\varPi_{i = 1}^{n} \alpha_{i}^{{Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})}} }} ,s_{{\varPi_{i = 1}^{n} \beta_{i}^{{Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})}} }} } \right)$$

(9)

where $Sh_{{\tilde{s}_{i} }} (\mu ,\tilde{S})$ is the Shapley value of the LIFV $\tilde{s}_{i}$, and μ is a fuzzy measure on the set $\tilde{S} = \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right)$.

Similar to the LIFSAA and LIFSGM operators, the linguistic intuitionistic fuzzy ordered Shapley arithmetical averaging (LIFOSAA) operator and the linguistic intuitionistic fuzzy ordered Shapley geometric mean (LIFOSGM) operator are defined as follows:

Definition 10

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the LIFOSAA operator is defined as:

$${\text{LIFOSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{j = 1}^{n} Sh_{j} (v,N)\tilde{s}_{(j)} = \left( {s_{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (v,N)} \alpha_{(j)} }} ,s_{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (v,N)} \beta_{(j)} }} } \right)$$

(10)

and the LIFOSGM operator is defined as:

$${\text{LIFOSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{i = 1}^{n} \tilde{s}_{(j)}^{{Sh_{j} (v,N)}} = \left( {s_{{\varPi_{i = 1}^{n} \alpha_{(j)}^{{Sh_{j} (v,N)}} }} ,s_{{\varPi_{i = 1}^{n} \beta_{(j)}^{{Sh_{j} (v,N)}} }} } \right)$$

(11)

where $( \cdot )$ is a permutation on N = {1, 2, …, n} such that $\tilde{s}_{(1)} \le \text{ }\tilde{s}_{(2)} \le \cdots \le \tilde{s}_{(n)}$, and $Sh_{j} (v,N)$ is the Shapley value of the jth ordered position, and v is a fuzzy measure on the set N.

Considering the importance of these two aspects, we further define the linguistic intuitionistic fuzzy hybrid Shapley arithmetical averaging (LIFHSAA) operator and the linguistic intuitionistic fuzzy hybrid Shapley geometric mean (LIFHSGM) operator.

Definition 11

Let $\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set $S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, the LIFHSAA operator is defined as:

$${\text{LIFHSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \oplus_{j = 1}^{n} \frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}\tilde{s}_{(j)} = \left( {s_{{\frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}\alpha_{(j)} }} ,s_{{\frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}\beta_{(j)} }} } \right)$$

(12)

and the LIFHSGM operator is defined as:

$${\text{LIFHSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) = \otimes_{i = 1}^{n} \tilde{s}_{(j)}^{{\frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}}} = \left( {s_{{\Pi_{i = 1}^{n} \alpha_{(j)}^{{\frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}}} }} ,s_{{\Pi_{i = 1}^{n} \beta_{(j)}^{{\frac{{Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})}}{{\sum\nolimits_{j = 1}^{n} {Sh_{j} (\mu ,N)Sh_{{\tilde{s}_{(j)} }} (\mu ,\tilde{S})} }}}} }} } \right)$$

(13)

where $( \cdot )$ is a permutation on N such that $Sh_{{\tilde{s}_{(1)} }} (\mu ,\tilde{S})\tilde{s}_{(1)} \le \text{ }Sh_{{\tilde{s}_{(2)} }} (\mu ,\tilde{S})\tilde{s}_{(2)} \le \cdots \le Sh_{{\tilde{s}_{(n)} }} (\mu ,\tilde{S})\tilde{s}_{(n)}$, and the other notations as shown in Definitions 9 and 10.

Theorem 3

Let$\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$, i = 1, 2, …, n, be a set of LIFVs defined on the continuous linguistic term set$S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$. Then, their aggregated values using the LIFHSAA and LIFHSGM operators are still IVIFVs.

Theorem 4

Let$\tilde{s}_{i} = \left( {s_{{\alpha_{i} }} ,s_{{\beta_{i} }} } \right)$and$\tilde{t}_{i} = \left( {s_{{\mu_{i} }} ,s_{{v_{i} }} } \right)$, i = 1, 2, …, n, be any two collections of LIFVs defined on the continuous linguistic term set$S_{c} = \{ s_{\alpha } |\alpha \in [0,2t]\}$.

• (1) Commutativity.Let$\tilde{s}'_{i} = \left( {s_{{\alpha '_{i} }} ,s_{{\beta '_{i} }} } \right)$be a permutation of$\tilde{s}_{i}$, i = 1, 2, …, n, then

  $$\begin{aligned} {\text{LIFHSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = {\text{LIFHSAA}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} ) \\ {\text{LIFHSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = {\text{LIFHSGM}}(\tilde{s}'_{1} ,\tilde{s}'_{2} , \ldots ,\tilde{s}'_{n} ) \\ \end{aligned}$$

• (2) Idempotency.If all LIFVs$\tilde{s}_{i}$, i = 1, 2, …, n, are equal, i.e.,$\tilde{s}_{i} = \tilde{s} = \left( {s_{\alpha } ,s_{\beta } } \right)$for all i, then

  $$\begin{aligned} {\text{LIFHSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \tilde{\alpha } \\ {\text{LIFHSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & = \tilde{\alpha } \\ \end{aligned}$$

• (3) Comonotonicity.If$\tilde{s}_{i}$and$\tilde{t}_{i}$are comonotonic, namely,$\tilde{s}_{(1)} \le \tilde{s}_{(2)} \le \cdots \le \tilde{s}_{(n)}$if and only if$\tilde{t}_{(1)} \le \tilde{t}_{(2)} \le \cdots \le \tilde{t}_{(n)}$for all i, where$( \cdot )$is a permutation on N = {1, 2, …, n} such that$\tilde{s}_{(j)}$and$\tilde{t}_{(j)}$are the jth least values of$\tilde{s}_{i}$and$\tilde{t}_{i}$, i = 1, 2, …, n, respectively. Then,

  $$\begin{aligned} {\text{LIFHSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & \le {\text{LIFHSAA}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} ) \\ {\text{LIFHSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) & \le {\text{LIFHSGM}}(\tilde{t}_{1} ,\tilde{t}_{2} , \ldots ,\tilde{t}_{n} ) \\ \end{aligned}$$

• (4) Boundary.Let$\tilde{s}^{ - } = \left( {s_{{\mathop {\hbox{min} }\limits_{i \in N} \alpha_{i} }} ,s_{{\mathop {\hbox{max} }\limits_{i \in N} \beta_{i} }} } \right)$and$\tilde{s}^{ + } = \left( {s_{{\mathop {\hbox{max} }\limits_{i \in N} \alpha_{i} }} ,s_{{\mathop {\hbox{min} }\limits_{i \in N} \beta_{i} }} } \right)$, then

  $$\tilde{\alpha }^{ - } \le {\text{LIFHSAA}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{\alpha }^{ + } \, and\,\tilde{\alpha }^{ - } \le {\text{LIFHSGM}}(\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} ) \le \tilde{\alpha }^{ + }$$

  where N = {1, 2, …, n}.

Proof

From Theorem 2 and Property 2, we can easily derive the conclusions. □

4 Programming Models for the Optimal Fuzzy and Additive Measures

In decision-making problems, the weighting information may not be completely known. This section builds several programming models to determine the optimal fuzzy and additive measures on the DM set, on the criteria set, and on their ordered position sets, respectively.

Considering a group decision-making problem, without loss of generality, suppose that there are m alternatives A = {a₁, a₂,…, a_m}, which are evaluated by q DMs E = {e₁, e₂,…, e_q} following n criteria C = {c₁, c₂,…, c_n}. Let $\tilde{S}^{k} = \left( {\tilde{s}_{ij}^{k} } \right)_{m \times n}$ be the individual linguistic intuitionistic fuzzy decision matrix (LIFDM) offered by the DM e_k, k = 1, 2, …, q, where $\tilde{s}_{ij}^{k} = \left( {s_{{\alpha_{ij}^{k} }} ,s_{{\beta_{ij}^{k} }} } \right)$ is the LIFV offered by the DM e_k for the alternative a_i∈A with respect to the criterion c_j∈ C.

First, we introduce the concept of correlation coefficient decision matrices as follows, which is then used to build model for determining the optimal fuzzy and additive measures on the DM set.

Definition 12

Let $\tilde{S}^{k} = \left( {\tilde{s}_{ij}^{k} } \right)_{m \times n}$ and $\tilde{S}^{l} = \left( {\tilde{s}_{ij}^{l} } \right)_{m \times n}$ be any two LIFDMs. Then, their correlation coefficient decision matrix (CCDM)$C^{kl} = (c_{ij}^{kl} )_{m \times n}$ is defined as follows:

$$c_{ij}^{kl} = \frac{{c(\tilde{s}_{ij}^{k} ,\tilde{s}_{ij}^{l} )}}{{\sqrt {e(\tilde{s}_{ij}^{k} ) \times e(\tilde{s}_{ij}^{l} )} }}$$

(14)

for all i = 1, 2, …, m, j = 1, 2, …, n, where $c(\tilde{s}_{ij}^{k} ,\tilde{s}_{ij}^{l} ) = \frac{1}{2}(\alpha_{ij}^{k} \alpha_{ij}^{l} + \beta_{ij}^{k} \beta_{ij}^{l} )$, $e(\tilde{s}_{ij}^{k} ) = \frac{{(\alpha_{ij}^{k} )^{2} + (\beta_{ij}^{k} )^{2} }}{2}$ and $e(\tilde{s}_{ij}^{k} ) = \frac{{(\alpha_{ij}^{k} )^{2} + (\beta_{ij}^{k} )^{2} }}{2}$.

Following the correlation coefficient decision matrix defined in Eq. (14), we build the following programming model to determine the weights of the DMs:

$$\begin{aligned} \phi^{*} & = \hbox{max} \sum\limits_{k = 1}^{q} {\left( {\sum\limits_{l = 1,l \ne k}^{q} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {c_{ij}^{kl} } } } } \right)Sh_{{e_{k} }} (\mu_{E} ,E)} \\ & \quad s.t.\left\{ \begin{array}{l} \mu_{E} (S) \le \mu_{E} (T){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall S,T \subseteq E \wedge S \subseteq T \hfill \\ \mu_{E} (e_{k} ) \in H_{{e_{k} }} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,q \hfill \\ \mu_{E} (\emptyset ) = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{E} (E) = 1 \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-1)

where $H_{{e_{k} }}$ is the known weighting information of the DM e_k, and $Sh_{{e_{k} }} (\mu_{E} ,E)$ is the Shapley value of the DM e_k for the fuzzy measure μ_E on the DM set E.

Let $\tilde{S}^{{Sh_{{e_{k} }} ,k}} = \left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,k}} } \right)_{m \times n}$ be the individual weighted LIFDM (WLIFDM) offered by the DM e_k, where $\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,k}} = \left( {s_{{Sh_{{e_{k} }} (\mu_{E} ,E) \times \alpha_{ij}^{k} }} ,s_{{Sh_{{e_{k} }} (\mu_{E} ,E) \times \beta_{ij}^{k} }} } \right)$ for all i = 1, 2, …, m, and all j = 1, 2, …, n, and $Sh_{{e_{k} }} (\mu ,E)$ is the Shapley value of the DM e_k determined by model (M-1). Furthermore, let $d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,k}} } \right) = Sh_{{e_{k} }} (\mu_{E} ,E) \times \alpha_{ij}^{k} - Sh_{{e_{k} }} (\mu_{E} ,E) \times \beta_{ij}^{k}$ for all k = 1, 2, …, q, which is the Shapley weighted score of the LIFV $\tilde{s}_{ij}^{k}$.

For each pair of (i, j), we reorder $d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,k}} } \right)$ for k = 1, 2, …, q in an increasing order, denoted by $d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(k)}} } \right)$, k = 1, 2, …, q, where $d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(1)}} } \right) \le d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(2)}} } \right) \le \cdots \le d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(q)}} } \right)$ for all i = 1, 2, …, m, and all j = 1, 2, …, n.

To reduce the influence of extreme evaluation values offered by the DMs, we construct the following programming model to determine the fuzzy measure $\mu_{{N_{E} }}$ on the ordered position set N_E = {1, 2, …, q}:

$$\begin{aligned} \varphi^{*} & = \hbox{min} \left( {\left( {\sum\limits_{k = \hbox{min} (q) + 1}^{q} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(k)}} } \right)} } }}{m \times n}} \right)Sh_{k} (\mu_{{N_{E} }} ,N_{E} )} } \right) - \left( {\sum\limits_{k = 1}^{\hbox{min} (q)} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(k)}} } \right)} } }}{m \times n}} \right)Sh_{k} (\mu_{{N_{E} }} ,N_{E} )} } \right)} \right) \\ & \quad s.t.\left\{ \begin{array}{l} \mu_{{N_{E} }} (S) \le \mu_{{N_{E} }} (T){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall S \subseteq T \subseteq N_{E} {\kern 1pt} \hfill \\ \mu_{{N_{E} }} (k) \in W_{k} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,q{\kern 1pt} \hfill \\ \mu_{{N_{E} }} (k) \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,q \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-2)

where ${\text{mid(}}q ) { = }\left\{ {\begin{array}{*{20}c} {\tfrac{q}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{even}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}{\kern 1pt} {\kern 1pt} } \\ {\tfrac{q + 1}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\text{odd}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}} \\ \end{array} } \right.$, $W_{k}$ is the known weighting information of the kth ordered position, and $Sh_{k} (\mu_{{N_{E} }} ,N_{E} )$ is the Shapley value of the kth ordered position for the fuzzy measure $\mu_{{N_{E} }}$ on the ordered position set N_E.

Note that when there is no interactive, models (M-1) and (M-2) reduce to the following models for the additive weighting vectors on the DM set E and on the ordered position set N_E:

$$\begin{aligned} f^{*} & = \hbox{max} \sum\limits_{k = 1}^{q} {\left( {\sum\limits_{l = 1,l \ne k}^{q} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {c_{ij}^{kl} } } } } \right)\omega_{{e_{k} }} } \\ & \quad s.t.\left\{ \begin{array}{l} \sum\nolimits_{k = 1}^{q} {\omega_{{e_{k} }} = 1} \hfill \\ \omega_{{e_{k} }} \in H_{{e_{k} }} \wedge \omega_{{e_{k} }} \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,q \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-3)

and

$$\begin{aligned} h^{*} & = \hbox{min} \left( {\left( {\sum\limits_{k = \hbox{min} (q) + 1}^{q} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(k)}} } \right)} } }}{m \times n}} \right)w_{k} } } \right) - \left( {\sum\limits_{k = 1}^{\hbox{min} (q)} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {\sum\nolimits_{j = 1}^{n} {d\left( {\tilde{s}_{ij}^{{Sh_{{e_{k} }} ,(k)}} } \right)} } }}{m \times n}} \right)w_{k} } } \right)} \right) \\ & \quad s.t.\left\{ \begin{array}{l} \sum\nolimits_{k = 1}^{q} {w_{k} = 1} \hfill \\ w_{k} \in W_{k} \wedge w_{k} \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,q \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-4)

where $\omega_{E} = \left( {\omega_{{e_{1} }} ,\omega_{{e_{2} }} , \ldots ,\omega_{{e_{q} }} } \right)$ is the additive weighting vector on the DM set E, $w_{{N_{E} }} = \left( {w_{1} ,w_{2} , \ldots ,w_{q} } \right)$ is the additive weighting vector on the ordered set N_E and the other notations as shown in models (M-1) and (M-2).

Let $\tilde{S} = \left( {\tilde{s}_{ij} } \right)_{m \times n}$ be the comprehensive linguistic intuitionistic fuzzy decision matrix (CLIFDM), and let $\tilde{s}^{ - } = (s_{0} ,s_{2t} )$ and $\tilde{s}^{ + } = (s_{2t} ,s_{0} )$. We define the following correlation coefficient for the LIFV $\tilde{s}_{ij}$:

$$c_{ij}^{ - } = \frac{{c(\tilde{s}_{ij} ,\tilde{s}^{ - } )}}{{\sqrt {e(\tilde{s}_{ij} ) \times e(\tilde{s}^{ - } )} }}\, {\text{and}}\,c_{ij}^{ + } = \frac{{c(\tilde{s}_{ij} ,\tilde{s}^{ + } )}}{{\sqrt {e(\tilde{s}_{ij} ) \times e(\tilde{s}^{ + } )} }}$$

(15)

where i = 1, 2, …, m, and j = 1, 2, …, n.

To determine the optimal fuzzy measure μ_C on the criteria set C, we establish the following programming model:

$$\begin{aligned} \psi^{*} & = \hbox{max} \sum\limits_{j = 1}^{n} {\sum\limits_{i = 1}^{m} {\frac{{c_{ij}^{ + } }}{{c_{ij}^{ - } + c_{ij}^{ + } }}Sh_{{c_{j} }} (\mu_{C} ,C)} } \\ & \quad s.t.\left\{ \begin{array}{l} \mu_{C} (S) \le \mu_{C} (T){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall S,T \subseteq C \wedge S \subseteq T \hfill \\ \mu_{C} (c_{j} ) \in H_{{c_{j} }} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n \hfill \\ \mu_{C} (\emptyset ) = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mu_{C} (C) = 1 \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-5)

where $H_{{c_{j} }}$ is the known weighting information of the criterion c_j, and $Sh_{{c_{j} }} (\mu_{C} ,C)$ is its Shapley value for the fuzzy measure μ_C on the criteria set C.

Let $\tilde{S}^{Sh} = \left( {\tilde{s}_{ij}^{Sh} } \right)_{m \times n}$ be the weighted comprehensive linguistic intuitionistic fuzzy decision matrix (WCLIFDM), and let $d\left( {\tilde{s}_{ij}^{Sh} } \right) = Sh_{{c_{j} }} (\mu_{C} ,C) \times \alpha_{ij} - Sh_{{c_{j} }} (\mu_{C} ,C) \times \beta_{ij}$ for all each pair of (i, j), which is the Shapley weighted score of the LIFV $\tilde{s}_{ij}$. For each i = 1, 2, …, m, we reorder $d\left( {\tilde{s}_{ij}^{Sh} } \right)$ in an increasing order for all j = 1, 2, …, n, where $d\left( {\tilde{s}_{i(1)}^{Sh} } \right) \le d\left( {\tilde{s}_{i(2)}^{Sh} } \right) \le \cdots \le d\left( {\tilde{s}_{i(n)}^{Sh} } \right)$ for all i = 1, 2, …, m.

To decrease the influence of extreme evaluation values, we construct the following programming model to determine the fuzzy measure on the ordered position set N_C = {1, 2, …, n}:

$$\begin{aligned} \zeta^{*} & = \hbox{min} \left( {\left( {\sum\limits_{j = \hbox{min} (n) + 1}^{n} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {d\left( {\tilde{s}_{i(j)}^{Sh} } \right)} }}{m}} \right)Sh_{j} (\mu_{{N_{C} }} ,N_{C} )} } \right) - \left( {\sum\limits_{j = 1}^{\hbox{min} (n)} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {d\left( {\tilde{s}_{i(j)}^{Sh} } \right)} }}{m}} \right)Sh_{j} (\mu_{{N_{C} }} ,N_{C} )} } \right)} \right) \\ & \quad s.t.\left\{ \begin{array}{l} \mu_{{N_{C} }} (S) \le \mu_{{N_{C} }} (T){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \forall S \subseteq T \subseteq N_{C} {\kern 1pt} \hfill \\ \mu_{{N_{C} }} (j) \in W_{j} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n{\kern 1pt} \hfill \\ \mu_{{N_{C} }} (j) \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-6)

where ${\text{mid(}}n ) { = }\left\{ {\begin{array}{*{20}c} {\tfrac{n}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{even}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}{\kern 1pt} {\kern 1pt} } \\ {\tfrac{n + 1}{2}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\kern 1pt} {\text{an}}{\kern 1pt} {\kern 1pt} {\text{odd}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{number}}} \\ \end{array} } \right.$, and $W_{j}$ is the known weighting information of the jth ordered position, and $Sh_{j} (\mu_{{N_{C} }} ,N_{C} )$ is the Shapley value of the jth ordered position for the fuzzy measure $\mu_{{N_{C} }}$ on the ordered position set N_C.

Similar to the additive measures on the DM set E and on the ordered position set N_E, when the importance of criteria and the ordered positions have no interaction, models (M-5) and (M-6) reduce to the following programming models:

$$\begin{aligned} p^{*} & = \hbox{max} \sum\limits_{j = 1}^{n} {\sum\limits_{i = 1}^{m} {\frac{{c_{ij}^{ + } }}{{c_{ij}^{ - } + c_{ij}^{ + } }}\omega_{{c_{j} }} } } \\ & \quad s.t.\left\{ \begin{array}{l} \sum\nolimits_{j = 1}^{n} {\omega_{{c_{j} }} = 1} \hfill \\ \omega_{{c_{j} }} \in H_{{c_{j} }} \wedge \omega_{{c_{j} }} \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-7)

and

$$\begin{aligned} g^{*} & = \hbox{min} \left( {\left( {\sum\limits_{j = \hbox{min} (n) + 1}^{n} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {d\left( {\tilde{s}_{i(j)}^{Sh} } \right)} }}{m}} \right)w_{j} } } \right) - \left( {\sum\limits_{j = 1}^{\hbox{min} (n)} {\left( {\frac{{\sum\nolimits_{i = 1}^{m} {d\left( {\tilde{s}_{i(j)}^{Sh} } \right)} }}{m}} \right)w_{j} } } \right)} \right) \\ & \quad s.t.\left\{ \begin{array}{l} \sum\nolimits_{j = 1}^{n} {w_{j} = 1} \hfill \\ w_{j} \in W_{j} \wedge w_{j} \ge 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \ldots ,n \hfill \\ \end{array} \right. \\ \end{aligned}$$

(M-8)

where $\omega_{C} = \left( {\omega_{{c_{1} }} ,\omega_{{c_{2} }} , \ldots ,\omega_{{c_{n} }} } \right)$ is the additive weighting vector on the criteria set C, $w_{{N_{C} }} = \left( {w_{1} ,w_{2} , \ldots ,w_{n} } \right)$ is the additive weighting vector on the ordered set N_C, and the other notations as shown in models (M-5) and (M-6).

5 A Group Decision-Making Method with Linguistic Intuitionistic Fuzzy Information

This section contains two parts. This first part introduces a group decision-making algorithm based on the defined aggregation operators and the built programming models for the optimal fuzzy and additive measures. The second part selects a practical group decision-making problem to show the specific application of the offered procedure.

5.1 An algorithm

• Step 1: Let $\tilde{S}^{k} = \left( {\tilde{s}_{ij}^{k} } \right)_{m \times n}$ be the individual LIFDM offered by the DM e_k, k = 1, 2, …, q. Model (M-1) or model (M-3) is used to determine the optimal fuzzy or additive measure on the DM set E = {e₁, e₂,…, e_q};

• Step 2: Model (M-2) or model (M-4) is adopted to calculate the optimal fuzzy or additive measure on the ordered position set N_E = {1, 2, …, q};

• Step 3: The LIFHSAA and LIFHSGM operators or the LIFHWAA and LIFHWGM operators are utilized to calculate the comprehensive LIFDM;

• Step 4: Model (M-5) or model (M-7) is adopted to calculate the optimal fuzzy or additive measure on the criteria set C = {c₁, c₂,…, c_n};

• Step 5: Model (M-6) or model (M-8) is used to calculate the optimal fuzzy or additive measure on the ordered position set N_C = {1, 2, …, n};

• Step 6: The LIFHSAA and LIFHSGM operators or the LIFHWAA and LIFHWGM operators is adopted to calculate the comprehensive LIFVs $\tilde{s}_{i} = (s_{{\alpha_{i} }} ,s_{{\beta_{i} }} )$, i = 1, 2, …, m;

• Step 7: The score and accuracy functions are utilized to calculate the score and accuracy values of the comprehensive LIFVs $\tilde{s}_{i}$, i = 1, 2, …, m;

• Step 8: We rank objects x_i, i = 1, 2, …, m, based on the order relationship of $\tilde{s}_{i}$, i = 1, 2, …, m.

5.2 A Case Study

We consider the decision-making problem of assessing engines (adapted from Ref. [12]). There are four brands of engines (alternatives) A = {A₁, A₂, A₃, A₄} that are assessed using the linguistic term set S = {s₀: extremely poor, s₁: very poor, s₂: poor, s₃: slight poor, s₄: indifferent, s₅: slight good, s₆: good, s₇: very good, s₈: extremely good} with respect to four criteria: C₁: responsiveness, C₂: fuel economy, C₃: vibration, and C₄: starting. The individual LIFDMs offered by four DMs E = {e₁, e₂, e₃, e₄} are listed as shown in Tables 1, 2, 3, and 4.

Table 1 Individual LIFDM $\tilde{S}^{1}$ offered by the DM e₁

Table 2 Individual LIFDM $\tilde{S}^{2}$ offered by the DM e₂

Table 3 Individual LIFDM $\tilde{S}^{3}$ offered by the DM e₃

Table 4 Individual LIFDM $\tilde{S}^{4}$ offered by the DM e₄

The known weighting information of the experts is defined as:

$$\omega_{{e_{1} }} = [0.1,0.3],\omega_{{e_{2} }} = [0.3,0.5],\omega_{{e_{3} }} = [0.1,0.2],\omega_{{e_{4} }} = [0.2,0.4],$$

and the known weighting information of the criteria is offered as:

$$\omega_{{c_{1} }} = [0.1,0.2],\omega_{{c_{2} }} = [0.3,0.4],\omega_{{c_{3} }} = [0.2,0.3],\omega_{{c_{4} }} = [0.1,0.2]$$

Furthermore, the known weighting information of the ordered positions is offered as:

$$w_{1} = [0.1,0.3],w_{2} = [0.2,0.4],w_{3} = [0.2,0.4],w_{4} = [0.1,0.3].$$

To rank these four brands of engines, the following procedure is needed:

Step 1: Following model (M-1), the optimal fuzzy measure on the DM set E is

$$\begin{aligned} \mu_{E} (e_{1} ) = \mu_{E} (e_{3} ) & = \mu_{E} (e_{4} ) = \mu_{E} (e_{1} ,e_{4} ) = \mu_{E} (e_{3} ,e_{4} ) = 0.2,\,\mu_{E} (e_{2} ) = \mu_{E} (e_{2} ,e_{4} ) \\ & = \mu_{E} (e_{1} ,e_{2} ,e_{4} ) = 0.3,\mu_{E} (e_{1} ,e_{3} ) = \mu_{E} (e_{2} ,e_{3} ) = \mu_{E} (e_{1} ,e_{2} ,e_{3} ) \\ & = \mu_{E} (e_{1} ,e_{3} ,e_{4} ) = \mu_{E} (e_{2} ,e_{3} ,e_{4} ) = \mu_{E} (E) = 1. \\ \end{aligned}$$

Following Eq. (7), the Shapley values of the DMs are

$$Sh_{{e_{1} }} (\mu_{E} ,E) = 0. 1 8 3 ,Sh_{{e_{2} }} (\mu_{E} ,E) = 0. 2 3 3 ,Sh_{{e_{3} }} (\mu_{E} ,E) = 0. 5 3 3 ,Sh_{{e_{4} }} (\mu_{E} ,E) = 0.0 5.$$

Step 2: On the basis of model (M-2), the optimal fuzzy measure on the ordered position set N_E is

$$\begin{aligned} \mu_{{N_{E} }} (1) & = \mu_{{N_{E} }} (4) = \mu_{{N_{E} }} (1,4) = 0.1,\mu_{{N_{E} }} (3) = \mu_{{N_{E} }} (1,3) \\ & = \mu_{{N_{E} }} (3,4) = \mu_{{N_{E} }} (1,3,4) = 0.2,\mu_{{N_{E} }} (2) = \mu_{{N_{E} }} (2,4) = 0.4,\mu_{{N_{E} }} (1,2) \\ & = \mu_{{N_{E} }} (2,3) = \mu_{{N_{E} }} (1,2,3) = \mu_{{N_{E} }} (1,2,4) = \mu_{{N_{E} }} (2,3,4) = \mu_{{N_{E} }} (N_{E} ) = 1. \\ \end{aligned}$$

Following Eq. (7), the Shapley values of the ordered positions are

$$Sh_{1} (\mu_{{N_{E} }} ,N_{E} ) = 0. 1 2 5 ,Sh_{2} (\mu_{{N_{E} }} ,N_{E} ) = 0. 6 7 5 ,Sh_{3} (\mu_{{N_{E} }} ,N_{E} ) = 0. 1 7 5 ,Sh_{4} (\mu_{{N_{E} }} ,N_{E} ) = 0.0 2 5.$$

Step 3: Using the LIFHSAA operator, the comprehensive LIFDM is shown in Table 5.

Table 5 Comprehensive LIFDM $\tilde{S}$

Step 4: Model (M-5) is used to calculate the optimal fuzzy measure on the criteria set C, where

$$\begin{aligned} \mu_{C} (c_{1} ) & = 0. 1,\mu_{C} (c_{3} ) = \mu_{C} (c_{4} ) = \mu_{C} (c_{1} ,c_{2} ) = \mu_{C} (c_{1} ,c_{3} ) \\ & = 0.2,\mu_{C} (c_{2} ) = \mu_{C} (c_{2} ,c_{3} ) = \mu_{C} (c_{1} ,c_{2} ,c_{3} ) = 0.3,\mu_{C} (c_{1} ,c_{4} ) \\ & = \mu_{C} (c_{2} ,c_{4} ) = \mu_{C} (c_{3} ,c_{4} ) = \mu_{C} (c_{1} ,c_{2} ,c_{4} )\mu_{C} (c_{1} ,c_{3} ,c_{4} ) \\ & = \mu_{C} (c_{2} ,c_{3} ,c_{4} ) = \mu_{C} (C) = 1. \\ \end{aligned}$$

Following Eq. (7), the Shapley values of the criteria are

$$Sh_{{c_{1} }} (\mu_{C} ,C) = 0. 0 8 3 { ,}Sh_{{c_{2} }} (\mu_{C} ,C) = 0. 1 6 7 ,Sh_{{c_{3} }} (\mu_{C} ,C) = 0. 1 3 3 { ,}Sh_{{c_{4} }} (\mu_{C} ,C) = 0. 6 1 7.$$

Step 5: Model (M-6) is adopted to calculate the optimal fuzzy measure on the ordered position set N_C, where

$$\begin{aligned} \mu_{{N_{C} }} (4) & = 0.1,\mu_{{N_{C} }} (1) = \mu_{{N_{C} }} (3) = \mu_{{N_{C} }} (1,3) = \mu_{{N_{C} }} (1,4) = \mu_{{N_{C} }} (3,4) \\ & = \mu_{{N_{C} }} (1,3,4) = 0.2,\mu_{{N_{E} }} (2) = \mu_{{N_{C} }} (2,3) = \mu_{{N_{C} }} (2,4) = \mu_{{N_{C} }} (2,3,4) \\ & = 0.4,\mu_{{N_{C} }} (1,2) = \mu_{{N_{C} }} (1,2,3) = \mu_{{N_{C} }} (1,2,4) = \mu_{{N_{C} }} (N_{C} ) = 1. \\ \end{aligned}$$

Following Eq. (7), the Shapley values of the ordered positions are

$$Sh_{1} (\mu_{{N_{C} }} ,N_{C} ) = 0. 3 5 8 ,Sh_{2} (\mu_{{N_{C} }} ,N_{C} ) = 0. 5 5 8 { ,}\,Sh_{3} (\mu_{{N_{C} }} ,N_{C} ) = 0. 0 5 8 ,Sh_{4} (\mu_{{N_{C} }} ,N_{C} ) = 0. 0 2 5.$$

Step 6: Again using the LIFHSAA operator, the comprehensive LIFVs are

$$\tilde{s}_{1} = (s_{ 2. 9 3 4 7} ,s_{ 4. 2 9 1} ),\tilde{s}_{2} = (s_{ 3. 4 2 4 3} ,s_{ 3. 6 2 7 1} ),\tilde{s}_{3} = (s_{ 4. 5 5 2 5} ,s_{ 2. 5 1 6} ),\tilde{s}_{4} = (s_{ 3. 6 6 4 1} ,s_{ 3. 4 4 0 2} ).$$

Step 7: Following the comprehensive LIFVs, the linguistic scores are

$$S(\tilde{s}_{1} ) = - 1. 3 5 6 3,S(\tilde{s}_{2} ) = - 0. 2 0 2 7,S(\tilde{s}_{3} ) = 2. 0 3 6 5,S(\tilde{s}_{4} ) = 0. 2 2 3 9.$$

Thus, the ranking order of objects is $A_{3} \succ A_{4} \succ A_{2} \succ A_{1}$, and the third brand of engines is the best.

In this example, ranking values and orders obtained from different aggregation operators are derived as shown in Table 6.

Table 6 Ranking values and orders obtained from different aggregation operators

Table 6 shows that different ranking orders are obtained for different aggregation operators. However, all ranking orders show that the third brand of engines is the best. In practical decision-making examples, we suggest the DMs to apply the aggregation operators based on fuzzy measures. However, when there is an explanation that the importance of elements in a set is independent. It is sufficient to use the aggregation operators based on additive measures. Because the hybrid aggregation operators consider more weighting information than the arithmetical averaging and geometric mean operators, we recommend the DMs to adopt the hybrid aggregation operators to calculate the comprehensive ranking values of alternatives.

In this example, when Chen et al’s method [8] is adopted, the ranking scores of objects are

$$S(\tilde{s}_{1} ) = 4. 0 7 6 4,S(\tilde{s}_{2} ) = 7.8481,S(\tilde{s}_{3} ) = 7. 8 3 5 4,S(\tilde{s}_{4} ) = 1. 4 0 6 6,$$

by which the ranking is $x_{2} \succ x_{3} \succ x_{1} \succ x_{4}$.

On the other hand, when Chen et al’s method [8] based on the linguistic intuitionistic fuzzy weighted geometric (LIFWG) operator and the linguistic intuitionistic fuzzy hybrid geometric (LIFHG) operator is used, the ranking scores of objects are

$$S(\tilde{s}_{1} ) = - 0. 4 0 8 3,S(\tilde{s}_{2} ) = 0.4380,S(\tilde{s}_{3} ) = 1. 4 8 7 0,S(\tilde{s}_{4} ) = - 0. 1 6 5 4,$$

by which the ranking is $x_{3} \succ x_{2} \succ x_{4} \succ x_{1}$.

Furthermore, when Garg and Kumar’s method [10] is applied, the ranking scores of objects are

$$S(\tilde{s}_{1} ) = 4. 0 9 2 7,S(\tilde{s}_{2} ) = 4. 2 5 5 0,S(\tilde{s}_{3} ) = 4. 4 9 1 3,S(\tilde{s}_{4} ) = 4. 1 3 5 9,$$

by which the ranking is $x_{3} \succ x_{2} \succ x_{4} \succ x_{1}$.

Except for Chen et al’s method [8] based on the linguistic intuitionistic fuzzy arithmetical weighted geometric (LIFAW) operator and the linguistic intuitionistic fuzzy hybrid arithmetical weighted (LIFHAW) operator, all other methods show that the third brand of engines is the best. However, their ranking orders and ranking values are different.

Remark 2

Methods in [19, 45] cannot be applied in this example, which did not consider group decision-making situation. Notably, Liu and Liu’s method [19] needs to divide attributes into different categories. The hybrid linguistic intuitionistic fuzzy aggregation operators in [8, 10] do not satisfy idempotency and boundary, which is caused by the balancing coefficient. Furthermore, methods in [8, 10] are based on the assumption that the weight information is completely known. Moreover, none of them can address the situation where there are interactive characteristics.

6 Conclusions

To denote the qualitative preferred and non-preferred information of the DMs rather than quantitative ones, this paper applied LIFVs [8] to develop a linguistic intuitionistic fuzzy group decision-making method. The main contributions include: (1) several new linguistic intuitionistic fuzzy operations are defined; (2) two types of aggregation operators are defined that satisfy several desirable properties; (3) models for determining the optimal fuzzy and additive measures on the DM set, on the criteria set and on their ordered position sets are built, respectively; and (4) a practical group decision-making problem about evaluating engines is provided to show the specific application of the developed theoretical results.

This paper focuses on linguistic intuitionistic fuzzy group decision making based on aggregation operators. In future, we shall continue to research linguistic intuitionistic fuzzy group decision making using other measures, including distance measure, entropy, and similarity measure. Furthermore, we shall study the application of linguistic intuitionistic fuzzy decision making in some other fields, including engineering project management, medical recommendation, software quality assurance management, and selecting cooperative partner. Notably, most of the current researches for decision making with fuzzy information are restricted to concrete problems rather than in the setting of dynamical systems. Thus, we will pay more attention to research methods for dynamical decision making in a similar way as the literature [33, 34].