Some interactive uncertain linguistic aggregation operators based on Shapley function and their application

Abstract

In this paper, two uncertain linguistic aggregation operators named the induced uncertain linguistic hybrid Shapley arithmetical averaging (IULHSAA) operator and the induced uncertain linguistic hybrid Shapley geometric mean (IULHSGM) operator are defined. These operators do not only globally consider the importance of the elements and their ordered positions but overall reflect the correlations between them, respectively. Furthermore, to reduce the complexity of solving a fuzzy measure, the induced uncertain linguistic hybrid 2-additive Shapley arithmetical averaging (IULHASAA) operator and the induced uncertain linguistic hybrid 2-additive Shapley geometric mean (IULHASGM) operator are defined. Moreover, if the weight information of experts and attributes is incompletely known, models for the optimal fuzzy measures are established. Meanwhile, an approach to multi-attribute group decision making under uncertain linguistic environment is developed, which considers the correlative characteristics between experts as well as between attributes. Finally, a numerical example is provided to illustrate the developed procedure.

1 Introduction

At present, many aggregation operators have been developed to aggregate information, which can be briefly classified into the following two categories: one is the quantitative aggregation operators; the other is the qualitative aggregation operators. For the former: according to the definition domain of aggregation operators, it can be divided into the real number aggregation operators (Yager 1988; Yager and Xu 2006; Wu et al. 2009; Ji et al. 2021; Xian et al. 2020), the fuzzy aggregation operators (Lin and Jiang 2014; Casanovas et al. 2015; Yi et al. 2021; Gong et al. 2022) and the intuitionistic fuzzy aggregation operators (Meng et al. 2015; Jana et al. 2020; Seikh and Mandal 2021; Kumar and Chen 2022). For the latter: we usually use the linguistic variables to denote the qualitative preference relations (Herrera and Herrera-Viedma 2000a; Herrera et al. 2009). There are two main kinds of linguistic aggregation operators. One is the deterministic linguistic aggregation operators (Herrera and Herrera-Viedma 1997; Herrera-Viedma et al. 2006; Meng et al. 2016a; Wu and Chen 2007; Wei 2011; Xu and Huang 2008), and the other is the uncertain linguistic aggregation operators (Cheng et al. 2017; Meng et al. 2016b, 2020; Wei 2009; Zhao et al. 2021).

Under many conditions, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in linguistic terms. In recent years, linguistic aggregation operators are receiving more and more attentions, and many linguistic aggregation operators are presented, such as the linguistic weighted averaging (LWA) operator (Herrera and Herrera-Viedma 1997), the linguistic ordered weighted averaging operator (LOWA) (Herrera et al. 1996), the linguistic weighted arithmetic averaging (LWAA) operator (Wu and Chen 2007), and the linguistic hybrid geometric averaging (LHGA) operator (Xu 2004a). To generate the linguistic recommendations according to the majority of the evaluation judgments provided by different visitors, Herrera-Viedma et al. (2006) introduced the majority guided linguistic IOWA (MLIOWA) operator and the weighted MLIOWA operator. With the emergence of 2-tuple linguistic variables, some 2-tuple linguistic aggregation operators are proposed, such as the 2-tuple ordered weighted averaging (TOWA) operator (Herrera and Herrera-Viedma 2000b), the 2-tuple ordered weighted geometric (TOWG) operator (Jiang and Fan 2003), the extended 2-tuple ordered weighted averaging (ET-OWA) operator (Zhang and Fan 2006), the 2-tuple weighted geometric averaging (TWGA) operator (Xu and Huang 2008), the 2-tuple ordered weighted geometric averaging (TOWGA) operator (Xu and Huang 2008), the induced generalized 2-tuple ordered weighted average (IG-2TOWA) operator (Wei 2011). On the aggregation of uncertain linguistic information, there are also some aggregation operators, such as the uncertain linguistic ordered weighted averaging (ULOWA) operator (Xu 2004b), the uncertain linguistic hybrid aggregation (ULHA) operator (Xu 2004b), the uncertain 2-tuple linguistic Muirhead mean (UL2-tupleMM) operator and the uncertain 2-tuple linguistic weighted Muirhead mean (UL2-tupleWMM) operator (Liu et al. 2019a, b), the induced uncertain linguistic OWA (LULOWA) operator (Jin et al. 2020), the uncertain linguistic hybrid geometric mean (ULHGM) operator (Wei 2009), the uncertain linguistic hybrid harmonic mean (ULHHM) operator (Park et al. 2011). In addition, Liu et al. (2019a, b) proposed the cloud weighted Maclaurin symmetric mean (CWMSM) operator.

All these linguistic aggregation operators are based on the assumption that the elements in a set are independent, i.e., they only consider the independent situation of the importance of individual elements. However, in many practical situations, the elements are usually correlative (Grabisch 1995; Meng et al. 2015; Tan 2011; Xu 2010; Xu and Xia 2011; Dumnić et al. 2022), which means that it is unreasonable to aggregate the values of elements using additive measures. To reflect the interdependent characteristics between elements, many aggregation operators are presented in the setting of intuitionistic fuzzy sets (Tan 2011; Xu 2010; Xu and Xia 2011; Meng and Chen 2016a, b; Jia and Wang 2022; Keikha et al. 2021), and there is only one linguistic aggregation operator (Tan et al. 2011) that considers the correlative characteristics between elements. Considering the advantages of uncertain linguistic evaluation information in representing the subjective cognitive uncertainty of decision makers, it is necessary to develop the aggregation operators of uncertain linguistic variables in the case of interactions.

Based on the operational laws of uncertain linguistic variables (Xu 2004b, 2006) and the Shapley function (Shapley 1953), we introduce some new interactive uncertain linguistic aggregation operators in this paper. The main contributions are listed as below:

• (i) The induced uncertain linguistic hybrid Shapley arithmetical averaging (IULHSAA) operator and the induced uncertain linguistic hybrid Shapley geometric mean (IULHSGM) operator are defined, which do not only globally consider the importance of elements and their ordered positions but also overall reflect the correlations between them, respectively;

• (ii) To reduce the complexity of solving a fuzzy measure, the induced uncertain linguistic hybrid 2-additive Shapley arithmetical averaging (IULHASAA) operator and the induced uncertain linguistic hybrid 2-additive Shapley geometric mean (IULHASGM) operator are further defined;

• (iii) If the information about the weights of experts, attributes and their ordered positions is partly known, models for the optimal fuzzy measures are established. Based on the introduced aggregation operator and models for the optimal fuzzy measures, an approach to milt-attribute group decision making under uncertain linguistic environment is developed.

This paper is organized as follows: In Sect. 2, some basic concepts related to uncertain linguistic variables, uncertain linguistic hybrid aggregation operators, fuzzy measure and the Shapley function are briefly reviewed. In Sect. 3, based on the Shapley function, the IULHSAA and IULHSGM operators are defined. Meanwhile, several desirable properties are studied. Furthermore, the IULHASAA and IULHASGM operators are presented. In Sect. 4, models for the optimal fuzzy measures are first built. Then, an approach to multi-attribute group decision making under uncertain linguistic environment is developed, which considers the correlative characteristics between experts and attributes. In Sect. 5, a practical application of the developed approach to the problem of evaluating air-conditioning systems for a municipal library is provided. The conclusion is made in the last section.

2 Preliminaries

2.1 Some concepts about uncertain linguistic variables

The linguistic approach is an approximate technique, which represents qualitative aspects as linguistic values by means of linguistic variables. Let S = {s_i | i = 1, 2,…, t} be a linguistic term set with odd cardinality. Any label s_i represents a possible value for a linguistic variable, and it should satisfy the following characteristics (Herrera and Herrera-Viedma 2000b):

• (i) The set is ordered: s_i > s_j, if i > j;

• (ii) Max operator: max(s_i, s_j) = s_i, if i ≥ j;

• (iii) Min operator: min(s_i, s_j) = s_i, if i ≤ j.

For example, S can be defined as:

S = {s₁: extremely poor, s₂: very poor, s₃: poor, s₄: slightly poor, s₅: fair, s₆: slightly good, s₇:good, s₈:very good, s₉: extremely good}.

To preserve all the given information, Xu (2004b) extended the discrete linguistic term set S to a continuous linguistic term set $\overline{S} = \left\{ {s_{\alpha } |s_{1} \le s_{\alpha } \le s_{t} ,\alpha \in [1,t]} \right\}$, whose elements also meet all the characteristics above. If $s_{\alpha } \in S$, then it is called the original linguistic term; otherwise, it is called the virtual linguistic term (Xu 2004b).

Definition 1

(Xu 2004b) Let $\tilde{s} = [s_{\alpha } ,s_{\beta } ]$, where $s_{\alpha } ,s_{\beta } \in \overline{S},$ $s_{\alpha }$ and $s_{\beta }$ are the lower and upper limitations, respectively, then $\tilde{s}$ is called an uncertain linguistic variable.

Let $\tilde{S}$ be the set of all uncertain linguistic variables, and $\tilde{s} = [s_{\alpha } ,s_{\beta } ],$ $\tilde{s}_{1} = [s_{{\alpha_{1} }} ,s_{{\beta_{1} }} ]$ and $\tilde{s}_{2} = [s_{{\alpha_{2} }} ,s_{{\beta_{2} }} ]$ be three uncertain linguistic variables. Some of their operational laws are defined as (Xu 2004b, 2006):

• (i) $\tilde{s}_{1} \oplus \tilde{s}_{2} = \left[ {s_{{\alpha_{1} + \alpha_{2} }} ,s_{{\beta_{1} + \beta_{2} }} } \right];$

• (ii) $\tilde{s}_{1} \otimes \tilde{s}_{2} = \left[ {s_{{\alpha_{1} \alpha_{2} }} ,s_{{\beta_{1} \beta_{2} }} } \right];$

• (iii) $\lambda \tilde{s} = [s_{\lambda \alpha } ,s_{\lambda \beta } ]$, $\lambda \in [0,1];$

• (iv) $\tilde{s}^{\lambda } = \left[ {s_{{\alpha^{\lambda } }} ,s_{{\beta^{\lambda } }} } \right]$, $\lambda \in [0,1];$

• (v) $\tilde{s}_{1} \oplus \tilde{s}_{2} = \tilde{s}_{2} \oplus \tilde{s}_{1} ;$

• (vi) $\tilde{s}_{1} \otimes \tilde{s}_{2} = \tilde{s}_{2} \otimes \tilde{s}_{1} ;$

• (vii) $\lambda \left( {\tilde{s}_{1} \oplus \tilde{s}_{2} } \right) = \lambda \tilde{s}_{1} \oplus \lambda \tilde{s}_{2}$, $\lambda \in [0,1];$

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

• (ix) $\left( {\lambda_{1} + \lambda_{2} } \right)\tilde{s} = \lambda_{1} \tilde{s} \oplus \lambda_{2} \tilde{s}$, $\lambda_{1} ,\lambda_{2} \in [0,1];$

• (x) $\tilde{s}^{{\lambda_{1} + \lambda_{2} }} = \tilde{s}^{{\lambda_{1} }} \otimes \tilde{s}^{{\lambda_{2} }}$, $\lambda_{1} ,\lambda_{2} \in [0,1].$

Based on the operational laws of uncertain linguistic variables, Xu (2004b) defined the uncertain linguistic weighted averaging (ULWA) operator and the uncertain linguistic ordered weighted averaging (ULOWA) operator and pointed out that the ULWA operator weights the uncertain linguistic arguments while the ULOWA operator weights the ordered positions of the uncertain linguistic arguments instead of weight the arguments themselves. To solve this drawback, Xu (2004b) further proposed the uncertain linguistic hybrid aggregation (ULHA) operator.

Definition 2

(Xu 2004b) An uncertain linguistic hybrid aggregation (ULHA) operator is a mapping ULHA: $\tilde{S}^{n} \to \tilde{S}$, which has an associated weight vector $w = (w_{1} ,w_{2} , \ldots ,w_{n} )^{T}$ such that $w_{j} \in [0,1]$ and $\sum\nolimits_{j = 1}^{n} {w_{j} = 1}$, denoted by

$${\text{ULHA}}_{w.\omega } \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} w_{i} \tilde{\varepsilon }_{j} ,$$

(1)

where $\tilde{\varepsilon }_{j}$ is the jth largest value of the weighted arguments $n\omega_{j} \tilde{s}_{j}$$(j = 1,2, \ldots ,n)$, $\omega = (\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} )^{T}$ is the weight vector on $\tilde{A} = \{ \tilde{s}_{i} \}_{i = 1, \ldots ,n}$ (i = 1,2,…,n) with $\omega_{i} > 0$ and $\sum\nolimits_{i = 1}^{n} {\omega_{i} = 1}$, and n is the balancing coefficient.

Based on the uncertain linguistic geometric mean (ULGM) operator and the uncertain linguistic ordered weighted geometric (ULOWG) operator (Xu 2006), Wei (2009) defined the following uncertain linguistic hybrid geometric mean (ULHGM) operator in a similar way to the ULHA operator.

Definition 3

(Wei 2009) An uncertain linguistic hybrid geometric mean (ULHGM) operator is a mapping ULHGM: $\tilde{S}^{n} \to \tilde{S}$, which has an associated weight vector $w = (w_{1} ,w_{2} , \ldots ,w_{n} )^{T}$ such that $w_{j} \in [0,1]$ and $\sum\nolimits_{j = 1}^{n} {w_{j} = 1}$, denoted by

$${\text{ULHGM}}_{w.\omega } \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{\varepsilon }_{j}^{{w_{i} }} ,$$

(2)

where $\tilde{\varepsilon }_{j}$ is the jth largest value of the weighted arguments $\tilde{s}_{j}^{{n\omega_{j} }}$$(j = 1,2, \ldots ,n)$, $\omega = (\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} )^{T}$ is the weight vector on $\tilde{A} = \{ \tilde{s}_{i} \}_{i = 1, \ldots ,n}$ (i = 1,2,…,n) with $\omega_{i} > 0$ and $\sum\nolimits_{i = 1}^{n} {\omega_{i} = 1}$, and n is the balancing coefficient.

From Eqs. (1) and (2), we know that the ULHA and ULHGM operators do not satisfy boundary and idempotent, which are desirable properties for aggregating a finite collection of arguments. Furthermore, these two operators are based on the assumption that elements in a set are independent.

Definition 4

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 two uncertain linguistic variables, then the degree of possibility $\tilde{s}_{1} \ge \tilde{s}_{2}$ is defined as

$$p\left( {\tilde{s}_{1} \ge \tilde{s}_{2} } \right) = \left\{ {\begin{array}{*{20}c} 1 & {\alpha_{1} \ge \beta_{2} } \\ {1 - \frac{{(\beta_{2} - \alpha_{1} )^{2} }}{{2d(\tilde{s}_{1} )d(\tilde{s}_{2} )}}} & {\alpha_{2} \le \alpha_{1} < \beta_{2} \le \beta_{1} } \\ {\frac{{2\beta_{1} - (\beta_{2} + \alpha_{2} )}}{{2d(\tilde{s}_{1} )}}} & {\alpha_{1} < \alpha_{2} < \beta_{2} \le \beta_{1} } \\ \end{array} } \right.,$$

(3)

where $d(\tilde{s}_{1} ) = \beta_{1} - \alpha_{1}$ and $d(\tilde{s}_{2} ) = \beta_{2} - \alpha_{2} .$

From Definition 4, one can easily get the following results:

• (i) $0 \le p(\tilde{s}_{1} \ge \tilde{s}_{2} ) \le 1$, $0 \le p(\tilde{s}_{2} \ge \tilde{s}_{1} ) \le 1;$

• (ii) $p(\tilde{s}_{1} \ge \tilde{s}_{2} ) + p(\tilde{s}_{2} \ge \tilde{s}_{1} ) = 1$. Especially, $p(\tilde{s}_{1} \ge \tilde{s}_{1} ) = p(\tilde{s}_{2} \ge \tilde{s}_{2} ) = 0.5.$

2.2 Fuzzy measures and the Shapley function

As an effective tool to measure the importance of elements and the correlations between them, fuzzy measures (Sugeno 1974) have been deeply studied by many researchers (Dubois and Prade 1988; Grabisch 1997; Miranda et al. 2002; Sugeno 1974) and successfully used in many different fields, especially in game theory and decision making.

Definition 5

(Sugeno 1974) A fuzzy measure on finite set N is a set function μ: P(N) → [0, 1] satisfying.

• (i) μ ($\emptyset$) = 0, μ (N) = 1,

• (ii) If A, B ∈ P(N) and A ⊆ B, then μ (A) ≤ μ (B),

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

In the multi-attribute decision making, $\mu (A)$ can be viewed as the importance of the attribute set A. Thus, in addition to the usual weights on attribute set taken separately, weight on any combination in attribute set is also defined.

The Shapley function (Shapley 1953) as an important solution concept in game theory has been deeply discussed by many scholars that is based on several reasonable axioms, expressed by

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

(4)

where $\mu$ is the fuzzy measure on finite set N, and s and n denote the cardinalities of S and N, respectively.

From Eq. (4), it shows that the Shapley function is an expect value of the globally marginal contribution between the element i and any coalition in N\i.

3 New uncertain linguistic hybrid aggregation operators

3.1 The IULHSAA and IULHSGM operators

This section defines two uncertain linguistic hybrid aggregation operators called the induced uncertain linguistic hybrid Shapley arithmetical averaging (IULHSAA) operator and the induced uncertain linguistic hybrid Shapley geometric mean (IULHSGM) operator, which overall consider the correlations between elements.

Based on the Shapley function (Shapley 1953), the induced ordered weighted averaging (IOWA) operator (Chiclana et al. 2007) and the ULHA operator (Xu 2004b), we define the IULHSAA operator as follows:

Definition 6

An induced uncertain linguistic hybrid Shapley arithmetical averaging (IULHSAA) operator is a mapping IULHSAA: $\tilde{S}^{n} \to \tilde{S}$ defined on the set of second arguments of two tuples $< u_{1} ,\tilde{s}_{1} > ,$ $< u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} >$ with a set of order-inducing variables u_i (i = 1, 2, …, n), denoted by

$${\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \frac{{\mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\tilde{\varepsilon }_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }},$$

(5)

where (⋅) is a permutation on u_i (i = 1, 2, …, n) such that $u_{(j)}$ being the jth largest value of u_i (i = 1, 2, …, n),$\tilde{\varepsilon }_{i}$ is the Shapley weighted argument $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})\tilde{s}_{i}$ with $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})$ being the Shapley value for the fuzzy measure v on $\tilde{A} = \{ \tilde{s}_{i} \}_{i = 1,\ldots,n}$ for $\tilde{s}_{i}$(i = 1, 2, …, n), and $\varphi_{j} (\mu ,N)$ is the Shapley value for the fuzzy measure $\mu$ on ordered set N = {1, 2, …, n} for the jth position.

Remark 1

If $\mu$ and v are both additive, then the IULHSAA operator reduces to the induced uncertain linguistic hybrid weighted averaging (IULHWA) operator.

$${\text{IULHWA}}_{w,\omega } \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \frac{{\mathop \oplus \limits_{j = 1}^{n} w_{j} \tilde{\varepsilon }_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{{\tilde{s}_{(j)} }} } }},$$

(6)

where $w_{j} = \mu (j)$ and $\omega_{{\tilde{s}_{j} }} = v(\tilde{s}_{j} )$ for each j = 1, 2, …, n.

Remark 2

If u_i = u_j for all i, j = 1, 2, …, n with i $\ne$ j, then the IULHSAA operator reduces to the uncertain linguistic hybrid ordered Shapley arithmetical averaging (ULHOSAA) operator.

$${\text{ULHOSAA}}_{\mu ,v} \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \frac{{\mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\tilde{\varepsilon }_{j} }}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{j} }} (v,\tilde{A})} }},$$

(7)

where $\tilde{\varepsilon }_{j}$ is the jth largest value of the Shapley weighted argument $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})\tilde{s}_{i}$, i = 1, 2, …, n.

Remark 3

If $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A}) = \frac{1}{n}$, i = 1, 2, …, n, then the IULHSAA operator reduces to the induced uncertain linguistic ordered Shapley arithmetical averaging (IULOSAA) operator.

$${\text{IULOSAA}}_{\mu } \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\tilde{s}_{(j)} ,$$

(8)

where $u_{(j)}$ is the jth largest value of u_i (i = 1,2,…,n).

Remark 4

If $\varphi_{i} (v,N) = \frac{1}{n}$, i = 1, 2, …, n, then the IULHSAA operator reduces to the uncertain linguistic Shapley arithmetical averaging (ULSAA) operator.

$${\text{ULSAA}}_{v} \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \mathop \oplus \limits_{j = 1}^{n} \varphi_{{\tilde{s}_{j} }} (v,\tilde{A})\tilde{s}_{j} .$$

(9)

Theorem 1

Let $\tilde{s}_{i} = [s_{{\alpha_{i} }} ,s_{{\beta_{i} }} ]$, i = 1, 2, …, n, be a collection of uncertain linguistic variables in $\tilde{S},$ and $\mu$ and v be a fuzzy measure on N = {1, 2, …, n} and $\tilde{A} = \{ \tilde{s}_{i} \}_{i \in N} ,$ respectively. Then, their collective value using the IULHSAA operator is also an uncertain linguistic variable in $\tilde{S}.$

Proof

From Definition 5 and Eq. (4), it has $\varphi_{i} (\mu ,N),\varphi_{{\tilde{s}_{i} }} (v,\tilde{A}) \ge 0$ for any i ∈ N, and $\sum\nolimits_{i = 1}^{n} \varphi_{i} (\mu ,N) = \sum\nolimits_{i = 1}^{n} {\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})} = 1.$ By the operational laws of uncertain linguistic variables and Eq. (5), we have

$$\begin{aligned} & {\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) \\ & = \frac{{\mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\tilde{\varepsilon }_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }} \\ & = \frac{{\mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\left[ {s_{{\alpha_{(j)} \varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})}} ,s_{{\beta_{(j)} \varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})}} } \right]}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }} \\ & = \mathop \oplus \limits_{j = 1}^{n} \left[ {s_{{\alpha_{(j)} \varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\frac{{\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}}} ,s_{{\beta_{(j)} \varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\frac{{\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}}} } \right] \\ & = \left[ {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\alpha_{(j)} } }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\beta_{(j)} } }} } \right]. \\ \end{aligned}$$

(10)

From $\alpha_{(j)} \le \beta_{(j)}$ for each $j \in N$, we get that $\left[ {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\alpha_{(j)} } }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{s_{\left( j \right)} }} (v,\tilde{A})} }}\beta_{(j)} } }} } \right]$ is an uncertain linguistic variable.

Because

$$\mathop {\min }\limits_{1 \le j \le n} \alpha_{j} \le \sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\alpha_{(j)} } \le \mathop {\max }\limits_{1 \le j \le n} \alpha_{j} ,$$

$$\mathop {\min }\limits_{1 \le j \le n} \beta_{j} \le \sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\beta_{(j)} } \le \mathop {\max }\limits_{1 \le j \le n} \beta_{j} ,$$

we get $\left[ {s_{{\mathop {\min }\limits_{1 \le j \le n} \alpha_{j} }} ,s_{{\mathop {\min }\limits_{1 \le j \le n} \beta_{j} }} } \right] \le \left[ {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\alpha_{(j)} } }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\beta_{(j)} } }} } \right]$$\le [s_{{\mathop {\max }\limits_{1 \le j \le n} \alpha_{j} }} ,s_{{\mathop {\max }\limits_{1 \le j \le n} \beta_{j} }} ]$.

Namely,$\left[ {s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\alpha_{(j)} } }} ,s_{{\sum\nolimits_{j = 1}^{n} {\frac{{\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}\beta_{(j)} } }} } \right]$ is an uncertain linguistic variable in $\tilde{S}$.

Based on the Shapley function (Shapley 1953), the induced uncertain linguistic ordered weighted geometric (IULOWG) operator (Xu 2006) and the ULHGM operator (Wei 2009), we define the following IULHSGM operator in a similar way to the IULHSAA operator:

Definition 7

An induced uncertain linguistic hybrid Shapley geometric mean (IULHSGM) operator is a mapping IULHSGM: $\tilde{S}^{n} \to \tilde{S}$ defined on the set of second arguments of two tuples $< u_{1} ,\tilde{s}_{1} > ,$ $< u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} >$ with a set of order-inducing variables u_i, i = 1, 2, …, n, denoted by

$${\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{\varepsilon }_{(j)}^{{\frac{{\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}}} ,$$

(11)

where (⋅) is a permutation on u_i, i = 1, 2, …, n, such that $u_{(j)}$ being the jth largest value of u_i, i = 1, 2, …, n,$\tilde{\varepsilon }_{i}$ is the Shapley weighted argument $\tilde{s}_{i}^{{\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})}}$ with $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})$ being the Shapley value for the fuzzy measure v on $\tilde{A} = \{ \tilde{s}_{i} \}_{i = 1,\ldots,n}$ for $\tilde{s}_{i}$, i = 1, 2, …, n, and $\varphi_{j} (\mu ,N)$ is the Shapley value for the fuzzy measure $\mu$ on ordered set N = {1, 2, …, n} for the jth position.

Remark 5

If $\mu$ and v are both additive, then the IULHSGM operator reduces to the induced uncertain linguistic hybrid geometric mean (IULHGM) operator.

$${\text{IULHGM}}_{w,\omega } \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{\varepsilon }_{(j)}^{{\frac{{\omega_{{\tilde{s}_{(j)} }} }}{{\sum\nolimits_{j = 1}^{n} {w_{j} \omega_{{\tilde{s}_{(j)} }} } }}}} ,$$

(12)

where $w_{j} = \mu (j)$ and $\omega_{{\tilde{s}_{j} }} = v(\tilde{s}_{j} )$ for each j = 1, 2, …, n.

Remark 6

If u_i = u_j for all i, j = 1, 2, …, n with i $\ne$ j, then the IULHSGM operator reduces to the uncertain linguistic hybrid ordered Shapley geometric mean (ULHOSGM) operator.

$${\text{ULHOSGM}}_{\mu ,v} \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{\varepsilon }_{j}^{{\frac{{\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{j} }} (v,\tilde{A})} }}}} ,$$

(13)

where $\tilde{\varepsilon }_{j}$ is the jth largest value of the Shapley weighted argument $\tilde{s}_{i}^{{\varphi_{{\tilde{s}_{i} }} (v,\tilde{A})}}$, i = 1, 2, …, n.

Remark 7

If $\varphi_{{\tilde{s}_{i} }} (v,\tilde{A}) = \frac{1}{n}$, i = 1, 2, …, n, then the IULHSGM operator reduces to the induced uncertain linguistic ordered Shapley geometric mean (IULOSGM) operator.

$${\text{IULOSGM}}_{\mu } \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{s}_{(j)}^{{\varphi_{j} (\mu ,N)}} ,$$

(14)

where $u_{(j)}$ is the jth largest value of u_i, i = 1, 2, …, n.

Remark 8

If $\varphi_{i} (v,N) = \frac{1}{n}$, i = 1, 2, …, n, then the IULHSGM operator reduces to the uncertain linguistic Shapley geometric mean (ULSGM) operator.

$${\text{ULSGM}}_{v} \left( {\tilde{s}_{1} ,\tilde{s}_{2} , \ldots ,\tilde{s}_{n} } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{s}_{j}^{{\varphi_{{\tilde{s}_{j} }} (v,\tilde{A})}} .$$

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Theorem 2

Let $\tilde{s}_{i} = [s_{{\alpha_{i} }} ,s_{{\beta_{i} }} ],$ i = 1, 2, …, n , be a collection of uncertain linguistic variables in $\tilde{S},$ and $\mu$ and v be a fuzzy measure on N = {1, 2, …, n} and $\tilde{A} = \{ \tilde{s}_{i} \}_{i \in N} ,$ respectively. Then, their collective value using the IULHSGM operator is also an uncertain linguistic variable in $\tilde{S}.$

Proof

From the operational laws of uncertain linguistic variables and Theorem 1, it is not difficult to get the conclusion.

Next, let us investigate some desirable properties of the IULHSAA and IULHSGM operators, which are necessary for us to apply the operators to solve real decision-making problems.

Theorem 3

Let $\tilde{s}_{i} = [s_{{\alpha_{i} }} ,s_{{\beta_{i} }} ]$ and $\tilde{t}_{i} = [t_{{\tau_{i} }} ,t_{{\theta_{i} }} ],$ i = 1, 2, …, n , be two collections of uncertain linguistic variables in $\tilde{S},$ and $\mu$ and v be a fuzzy measure on N = {1, 2, …, n} and $\tilde{A} = \{ \tilde{s}_{i} \}_{i \in N} ,$ respectively.

• (i) Commutativity. Let $\tilde{s}^{\prime}_{i} = [s^{\prime}_{{\alpha_{i} }} ,s^{\prime}_{{\beta_{i} }} ]$, i = 1, 2, …, n, be a permutation of $\tilde{s}_{i}$, then

  $${\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = {\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}^{\prime}_{1} > , \ldots , < u_{n} ,\tilde{s}^{\prime}_{n} > } \right),$$
  $${\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = {\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}^{\prime}_{1} > , \ldots , < u_{n} ,\tilde{s}^{\prime}_{n} > } \right).$$

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

  $${\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \tilde{s},$$
  $${\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \tilde{s}.$$

• (iii) 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 $\tilde{t}_{(1)} \le \tilde{t}_{(2)} \le \cdots \le \tilde{t}_{(n)}$, and $\tilde{s}_{(i)} \le \tilde{t}_{(i)}$ for all i, where $( \cdot )$ is a permutation on N such that $u_{(j)}$ being the jth largest value of u_i, i = 1, 2, …, n. Then,

  $${\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) \le {\text{IULHSAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{t}_{1} > , \ldots , < u_{n} ,\tilde{t}_{n} > } \right),$$
  $${\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) \le {\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{t}_{1} > , \ldots , < u_{n} ,\tilde{t}_{n} > } \right).$$

• (iv) Boundary. Let $\tilde{s}^{ - } = \left[ {s_{{\mathop {\min }\limits_{1 \le i \le n} \alpha_{i} }} ,s_{{\mathop {\min }\limits_{1 \le i \le n} \beta_{i} }} } \right]$ and $\tilde{s}^{ + } = \left[ {s_{{\mathop {\max }\limits_{1 \le i \le n} \alpha_{i} }} ,s_{{\mathop {\max }\limits_{1 \le i \le n} \beta_{i} }} } \right]$, then

  $$\tilde{s}^{ - } \le {\text{IULHSA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) \le \tilde{s}^{ + } ,$$
  $$\tilde{s}^{ - } \le {\text{IULHSGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) \le \tilde{s}^{ + } .$$

Proof

From the expression of the Shapley value, it is easy to see that $\{ \varphi_{i} (\mu ,N)\}_{i \in N}$ is a weight vector. By Definitions 6 and 7, it is not difficult to get the conclusion.

3.2 A special case

Because fuzzy measures are defined on the power set, it makes the problem exponentially complex. To reflect the interactions between elements and reduce the complexity of solving a fuzzy measure, we further define the induced uncertain linguistic hybrid 2-additive Shapley arithmetical averaging (IULHASAA) operator and the induced uncertain linguistic hybrid 2-additive Shapley geometric mean (IULHASGM) operator using 2-additive measures (Grabisch 1997), which greatly reduces the complexity of solving a fuzzy measure.

Let $f:\{ 0,1\} \to {\mathbb{R}}$ be a pseudo-Boolean function. Grabisch (1997) noted that any fuzzy measure $\mu$ can be seen as a particular case of pseudo-Boolean function, and put under a multilinear polynomial in n variables:

$$\mu (A) = \sum\limits_{T \subseteq N} {\left[ {a_{T} \mathop \Pi \limits_{i \in T} y_{i} } \right]} \quad \forall A \subseteq N,$$

(16)

where $a_{T} \in {\mathbb{R}}$, $y = (y_{1} ,y_{2} , \ldots ,y_{n} ) \in \{ 0,1\}^{n}$, and $y_{i} = 1$ if and only if $i \in A.$

The set of coefficients $a_{T}$(T ⊆ N) in fact corresponds to the Möbius transform, denoted by $a_{T} =$$\sum\limits_{S \subseteq T} {( - 1)^{|T\backslash S|} \mu (S)}$. Since the transform is inversible, $\mu$ can be recovered from $a_{T}$ by $\mu (A) =$$\sum\limits_{B \subseteq A} {a_{B} } .$

Definition 8

(Grabisch 1997) A fuzzy measure $\mu$ defined on N is said to be k-additive if its corresponding pseudo-Boolean function is a multilinear polynomial of degree k, i.e., $a_{T} = 0$ for all T such that t > k, and there exists at least one subset T of k elements such that $a_{T} \ne 0.$

When k = 1, the k-additive measure $\mu$ reduces to an additive measure; When k = n, the k-additive measure $\mu$ is a fuzzy measure as usual. Especially, when k = 2, by Eq. (16) we get a 2-additive measure. For a 2-additive measure $\mu$, one can easily get (Grabisch 1997), for any S $\subseteq N$ with $\left| {\text{S}} \right| \ge 2,$

$$\mu (S) = \sum\limits_{i = 1}^{n} {a_{i} x_{i} } + \sum\limits_{{\{ i,j\} \subseteq N}} {a_{ij} x_{i} x_{j} } = \sum\limits_{i \in S} {a_{i} } + \sum\limits_{{\{ i,j\} \subseteq S}} {a_{ij} } = \sum\limits_{{\{ i,j\} \subseteq S}} {\mu (i,j)} - (|S| - 2)\sum\limits_{i \in S} {\mu (i),}$$

(17)

where $\mu (i) = a_{i}$ and $\mu (i,j) = a_{i} + a_{j} + a_{ij} .$

From Eq. (17), we know that it only needs ${{n(n + 1)} \mathord{\left/ {\vphantom {{n(n + 1)} 2}} \right. \kern-\nulldelimiterspace} 2}$ coefficients to determine a 2-additive measure on a set with n elements.

Theorem 4

(Grabisch 1997) Let $\mu$ be fuzzy measure on N, then $\mu$ is a 2-additive measure if and only if there exist coefficients μ(i) and μ(i, j) for all i, j $\in$ N that satisfy the following conditions:

• (i) $\mu (i) \ge 0\;\forall i \in N,$

• (ii) $\sum\limits_{{\{ i,j\} \subseteq N}} {\mu (i,j)} - (|N| - 2)\sum\limits_{i \in N} {\mu (i)} = 1,$

• (iii) $\sum\limits_{i \subseteq S\backslash j} {\left( {\mu (i,j) - \mu (i)} \right) \ge } (|S| - 2)\mu (j)$ $\forall S \in N$ s.t. $j \in S$ and $\left| S \right| \ge 2.$

Theorem 5

(Meng et al. 2015) Let $\mu$ be a 2-additive measure, then the Shapley function given in Eq. (4) with respect to the 2-additive measure $\mu$ can be expressed by

$$\varphi_{i} (\mu ,N) = \frac{3 - n}{2}\mu_{i} + \sum\limits_{j \in N\backslash i} \frac{1}{2} (\mu_{ij} - \mu_{j} )\quad \forall i \in N,$$

(18)

where $\mu_{i} = \mu (i)$ and $\mu_{ij} = \mu (i,j)$ for all i, j $\in$ N.

Similar to the IULHSAA operator, the induced uncertain linguistic hybrid 2-additive Shapley arithmetical averaging (IULHASAA) operator is defined as:

$${\text{IULHASAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \frac{{\mathop \oplus \limits_{j = 1}^{n} \varphi_{j} (\mu ,N)\tilde{\varepsilon }_{(j)} }}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}.$$

(19)

Similar to the IULHSGM operator, the induced uncertain linguistic hybrid 2-additive Shapley geometric mean (IULHASGM) operator is written as:

$${\text{IULHASGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{1} > , < u_{2} ,\tilde{s}_{2} > , \ldots , < u_{n} ,\tilde{s}_{n} > } \right) = \mathop \otimes \limits_{j = 1}^{n} \tilde{\varepsilon }_{(j)}^{{\frac{{\varphi_{j} (\mu ,N)}}{{\sum\nolimits_{j = 1}^{n} {\varphi_{j} (\mu ,N)\varphi_{{\tilde{s}_{(j)} }} (v,\tilde{A})} }}}} ,$$

(20)

where $\mu$ and v are, respectively, a 2-additive measure on N = {1,2,…,n} and $\tilde{A} = \{ \tilde{s}_{i} \}_{i \in N} .$

3.3 Comparison

The differences between the new defined linguistic operator and the linguistic weighted averaging operator (Herrera and Herrera-Viedma 1997; Wu and Chen 2007; Xu and Huang 2008) are: the former considers the importance of elements and ordered positions, while the latter only gives the importance of elements. Furthermore, the former reflects the interactions between elements, while the latter is based on the assumption that the elements are independent.

The differences between the new defined linguistic operator and the linguistic ordered weighted averaging operator (Herrera et al. 1996; Herrera and Herrera-Viedma 2000b; Herrera-Viedma et al. 2006; Jiang and Fan 2003; Wei 2011; Xu and Huang 2008; Xu 2004b; Zhang and Fan 2006) are similar to that between the new defined linguistic operator and the linguistic weighted averaging operator. The main difference is: the former considers the importance of elements and ordered positions, while the latter only gives the importance of ordered positions. It is worth pointing out that the linguistic Choquet geometric operator (Tan et al. 2011) gives the importance of ordered positions and reflects the interactions between them. However, it neither considers the importance of elements nor reflects the interactions between them.

The differences between the new defined linguistic operator and the linguistic hybrid weighted averaging operator (Park et al. 2011; Wei 2009; Xu 2004a, b) are: the former reflects interactions between elements, while the latter is based on the assumption that elements are independent. Furthermore, the former satisfies some desirable properties such as commutativity, idempotency, comonotonicity, and boundary, while the latter does not satisfy the properties idempotency and boundary.

The main drawback of the new defined linguistic operator is: the calculation seems to be a little complex than other linguistic operators, which is caused by fuzzy measures.

4 An approach to uncertain linguistic multi-attribute group decision making

Considering a multi-attribute group decision making problem under uncertain linguistic environment, in which the experts and attributes are correlative, respectively. Let A = {a₁, a₂, …, a_m} be the alternative set, C = {c₁, c₂, …, c_n} be the attribute set, and E = {e₁, e₂, …, e_q} be the expert set. Assume that $S^{k} = \left( {\tilde{s}_{ij}^{k} } \right)_{m \times n}$ is the uncertain linguistic decision matrix, where $\tilde{s}_{ij}^{k} = \left[ {s_{{\alpha_{ij}^{k} }} ,s_{{\beta_{ij}^{k} }} } \right]$ is the uncertain linguistic variable given by the expert e_k for the alternative a_i ∈ A with respect to the attribute c_j ∈ C.

When the weight vectors on the expert set, the attribute set and their ordered sets are completely known, we can use the associated operator to develop a method to multi-attribute group decision making under uncertain linguistic environment. Due to the complexity and uncertainty of real world decision-making problems and the inherent subjective nature of human thinking, the information about the weight vectors is usually partly known. In this situation, our primary work is to get their weight vectors, which is an important research topic in decision making.

4.1 Models for the optimal fuzzy measures

For each attribute c_j (j ∈ N = {1, 2, …, n}), let

$$d_{j}^{kl} = \sum\limits_{i = 1}^{m} {\left| {\alpha_{ij}^{k} - \alpha_{ij}^{l} | + |\beta_{ij}^{k} - \beta_{ij}^{l} } \right|}$$

(21)

for all k, l ∈ Q = {1, 2, …, q}.

Because every expert’s knowledge, skill and experience are different with respect to the different attributes, it is unreasonable to endow an expert with the same weight for the different attributes. Let $d_{j}^{k} = \sum\limits_{l = 1}^{q} {d_{j}^{kl} }$ for any k ∈ Q, the following model for the optimal fuzzy measure on expert set E with respect to the attribute c_j (j ∈ N) is established:

$$\min \sum\limits_{k = 1}^{q} {d_{j}^{k} } \varphi_{{e_{k} }} (v,E)$$

$$s.t.\left\{ \begin{aligned} & v(E) = 1 \\ & v(S) \le v(T)\;{\kern 1pt} \forall S,\;T \subseteq E{\kern 1pt} \;S \subseteq T \\ & v(e_{k} ) \in W_{{e_{k} }}^{j} ,\;v(e_{k} ) \ge 0\;\forall e_{k} \in E \\ \end{aligned} \right.,$$

(22)

where v is the fuzzy measure on the expert set E, $W_{{e_{k} }}^{j}$ is the known weight information of the expert e_k with respect to the attribute c_j, and $\varphi_{{e_{k} }} (v,E)$ is the associated Shapley value.

Solving the above model, the optimal fuzzy measure on the expert set E with respect to the attribute c_j (j ∈ N) is obtained. If v is a 2-additive measure, then the following model is built:

$$\min \sum\limits_{k = 1}^{q} {\frac{{d_{j}^{k} }}{2}} \left( {(3 - n)v(e_{k} ) + \sum\limits_{{e_{l} \subseteq E\backslash e_{k} }} {\left( {v(e_{k} ,e_{l} ) - v(e_{l} )} \right)} } \right)$$

$$s.t.\left\{ \begin{aligned} & \sum\limits_{{e_{l} \subseteq S\backslash e_{k} }} {\left( {v(e_{l} ,e_{k} ) - v(e_{l} )} \right) \ge } (s - 2)v(e_{k} ),\;\forall S \subseteq E,\;\forall e_{k} \in S,\;s \ge 2 \\ & \sum\limits_{{\{ e_{k} ,e_{l} \} \subseteq E}} {v(e_{l} ,e_{k} )} - (q - 2)\sum\limits_{{e_{l} \in E}} {v(e_{l} )} = 1 \\ & v(e_{k} ) \in W_{{e_{k} }}^{j} ,v(e_{k} ) \ge 0,\;k = 1,2,\ldots,q \\ \end{aligned} \right..$$

(23)

Now, let us consider model for the optimal fuzzy measure on the ordered set N. Let

$$d_{ij}^{k} = \sum\limits_{l = 1}^{q} {|\alpha_{ij}^{k} - \alpha_{ij}^{l} | + |\beta_{ij}^{k} - \beta_{ij}^{l} |} .$$

(24)

Reorder $d_{ij}^{k}$(k ∈ Q) in decreasing order $d_{ij}^{(1)} \ge d_{ij}^{(2)} \ge \cdots \ge d_{ij}^{(q)}$. Based on the distance deviation method, the following model for the optimal fuzzy measure on the ordered set Q for each j ∈ N is established:

$$\min \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\sum\limits_{k = 1}^{q} {d_{ij}^{(k)} } \varphi_{(k)} (\mu ,Q)} }$$

$$s.t.\left\{ \begin{aligned} & \mu (Q) = 1 \\ & \mu (S) \le \mu (T)\;\forall S,T \subseteq Q\;S \subseteq T \\ & \mu (k) \in W_{k} ,\mu (k) \ge 0{\kern 1pt} \;\forall k \in Q \\ \end{aligned} \right.,$$

(25)

where µ is the fuzzy measure on the ordered set Q, $W_{k}$ is the known weight information of the kth position, and $\varphi_{(k)} (\mu ,Q)$ is the associated Shapley value.

Solving the above model, the optimal fuzzy measure on the ordered set Q with respect to each j ∈ N is obtained. If µ is a 2-additive measure, then the following model is built:

$$\min \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\sum\limits_{k = 1}^{q} {\frac{{d_{ij}^{(k)} }}{2}} \left( {(3 - n)\mu ((k)) + \sum\limits_{l \subseteq Q\backslash (k)} {\left( {\mu ((k),l) - \mu (l)} \right)} } \right)} }$$

$$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{l \subseteq S\backslash k} {\left( {\mu (k,l) - \mu (l)} \right) \ge } (s - 2)\mu (k),{\kern 1pt} \;\forall S \subseteq Q,\;\forall k \in S,{\kern 1pt} \;s \ge 2} \\ {\sum\limits_{{\{ l,k\} \subseteq Q}} {\mu (k,l)} - (q - 2)\sum\limits_{l \in Q} {\mu (l)} = 1} \\ {\mu (k) \in W_{k}^{j} ,\mu (k) \ge 0,\;k = 1,2,\ldots,q} \\ \end{array} } \right..$$

(26)

Next, let us consider models for the optimal fuzzy measures on the attribute set C and their ordered set N. Let $S = \left( {\tilde{s}_{ij} } \right)_{m \times n}$ be the comprehensive uncertain linguistic decision matrix, where $\tilde{s}_{ij} = [s_{{\alpha_{ij} }} ,s_{{\beta_{ij} }} ]$ is an uncertain linguistic variable.

Because the optimal fuzzy measure on the attribute set C should make the comprehensive uncertain linguistic values of alternatives as big as possible, and all alternatives are non-inferior, the following models for the optimal fuzzy measures on attribute set C are built:

$$\max \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\alpha_{ij} } \varphi_{{c_{j} }} (v,C)}$$

$$s.t.\left\{ \begin{aligned} & v(C) = 1 \\ & v(S) \le v(T)\;\forall S,T \subseteq C\;S \subseteq T \\ & v(c_{j} ) \in W_{{c_{j} }} ,\;v(c_{j} ) \ge 0\;\forall c_{j} \in C \\ \end{aligned} \right.$$

(27)

and

$$\max \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\beta_{ij} } \varphi_{{c_{j} }} (v,C)}$$

$$s.t.\left\{ \begin{aligned} & v(C) = 1 \\ & v(S) \le v(T)\;\forall S,T \subseteq C\;S \subseteq T \\ & v(c_{j} ) \in W_{{c_{j} }} ,v(c_{j} ) \ge 0\;\forall c_{j} \in C \\ \end{aligned} \right..$$

(28)

Because models (27) and (28) have the same constraints, they can be combined to formulate the following linear programming:

$$\max \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {(\alpha_{ij} + \beta_{ij} )} \varphi_{{c_{j} }} (v,C)}$$

$$s.t.\left\{ \begin{aligned} & v(C) = 1 \\ & v(S) \le v(T)\;\forall S,T \subseteq C\;S \subseteq T \\ & v(c_{j} ) \in W_{{c_{j} }} ,v(c_{j} ) \ge 0\;\forall c_{j} \in C \\ \end{aligned} \right..$$

(29)

Solving model (29), the optimal fuzzy measure on the attribute set C is obtained. If v is a 2-additive measure, then the following model is built:

$$\max \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\frac{{\alpha_{ij} + \beta_{ij} }}{2}} \left( {(3 - n)v(c_{j} ) + \sum\limits_{{c_{i} \subseteq C\backslash c_{j} }} {\left( {v(c_{j} ,c_{i} ) - v(c_{i} )} \right)} } \right)}$$

$$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{{c_{i} \subseteq S\backslash c_{j} }} {\left( {v(c_{i} ,c_{j} ) - v(c_{i} )} \right) \ge } (s - 2)v(c_{j} ),\;\forall S \subseteq C,\;\forall c_{j} \in S,\;s \ge 2} \\ {\sum\limits_{{\{ c_{i} ,c_{j} \} \subseteq C}} {v(c_{i} ,c_{j} )} - (n - 2)\sum\limits_{{c_{i} \in C}} {v(c_{i} )} = 1} \\ {v(c_{j} ) \in W_{{c_{j} }} ,v(c_{j} ) \ge 0,\;j = 1,2,\ldots,n} \\ \end{array} } \right..$$

(30)

Similar to model for the optimal fuzzy measure on the ordered set Q, the following model for the optimal fuzzy measure on the ordered set N is established:

$$\min \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {d_{i(j)} } \varphi_{(j)} (\mu ,N)}$$

$$s.t.\left\{ \begin{aligned} & \mu (N) = 1 \\ & \mu (S) \le \mu (T)\;\forall S,T \subseteq N\;S \subseteq T \\ & \mu (j) \in W_{j} ,\;\mu (j) \ge 0{\kern 1pt} \;\forall j \in N \\ \end{aligned} \right.,$$

(31)

where $d_{i(1)} \ge d_{i(2)} \cdots. \ge d_{i(n)}$ with $d_{ij} = |\alpha_{ij} - \frac{1}{m}\sum\limits_{i = 1}^{m} {\alpha_{ij} } | + |\beta_{ij} - \frac{1}{m}\sum\limits_{i = 1}^{m} {\beta_{ij} } |$ for each i = 1, 2, …, m.

Solving model (31), the optimal fuzzy measure on ordered set N is obtained. If µ is a 2-additive measure, then the following model is built.

$$\min \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\frac{{d_{i(j)} }}{2}\left( {(3 - n)\mu ((j)) + \sum\limits_{i \subseteq N\backslash (j)} {\left( {\mu ((j),i) - \mu (i)} \right)} } \right)} }$$

$$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{j \subseteq S\backslash i} {\left( {\mu (i,j) - \mu (j)} \right) \ge } (s - 2)\mu (i),{\kern 1pt} \;\forall S \subseteq N,\;{\kern 1pt} \forall i \in S,\;{\kern 1pt} s \ge 2} \\ {\sum\limits_{{\{ i,j\} \subseteq N}} {\mu (i,j)} - (n - 2)\sum\limits_{j \in N} {\mu (j)} = 1} \\ {\mu (j) \in W_{j} ,\;\mu (j) \ge 0,\;j = 1,2,\ldots,n} \\ \end{array} } \right..$$

(32)

4.2 A new algorithm

This section develops an approach to multi-attribute group decision making under uncertain linguistic environment. The main decision procedure can be described as follows:

• Step 1: Assume that the uncertain linguistic variable $\tilde{s}_{ij}^{k} = [s_{{\alpha_{ij}^{k} }} ,s_{{\beta_{ij}^{k} }} ]$ is the important degree of the alternative a_i (i = 1,2,…,m) with respect to the attribute c_j (j = 1, 2, …, n) given by the expert e_k (k = 1, 2,…,q), then the uncertain linguistic decision matrix $S^{k} = \left( {\tilde{s}_{ij}^{k} } \right)_{m \times n}$ is obtained.

• Step 2: Utilize model (23) to calculate the optimal 2-additive measure on experts set E for each attribute c_j (j = 1, 2, …, n).

• Step 3: Utilize model (26) to calculate the optimal 2-additive measure on ordered set Q for each j = 1, 2, …, n.

• Step 4: Use the IULHASAA operator or the IULHASGM operator to get the comprehensive uncertain linguistic decision matrix $D = \left( {\tilde{s}_{ij} } \right)_{m \times n}$ with

  $$\tilde{s}_{ij} = [s_{{\alpha_{ij} }} ,s_{{\beta_{ij} }} ] = {\text{IULHASA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{ij}^{1} > , < u_{2} ,\tilde{s}_{ij}^{2} > , \ldots , < u_{q} ,\tilde{s}_{ij}^{q} > } \right)$$

  or

  $$\tilde{s}_{ij} = [s_{{\alpha_{ij} }} ,s_{{\beta_{ij} }} ] = {\text{IULHASGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{ij}^{1} > , < u_{2} ,\tilde{s}_{ij}^{2} > , \ldots , < u_{q} ,\tilde{s}_{ij}^{q} > } \right)$$

  for all i, j = 1, 2, …, n.

• Step 5: Utilize model (30) to calculate the optimal 2-additive measure on attribute set C.

• Step 6: Adopt model (32) to calculate the optimal 2-additive measure on ordered set N.

• Step 7: Again use the IULHASAA operator or the IULHASGM operator to get the comprehensive uncertain linguistic value of the alternative a_i, i = 1, 2, …, m, where

  $$\tilde{s}_{i} = {\text{IULHASA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{ij} > , < u_{2} ,\tilde{s}_{ij} > , \ldots , < u_{n} ,\tilde{s}_{ij} > } \right)$$

  or

  $$\tilde{s}_{i} = {\text{IULHASGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{ij} > , < u_{2} ,\tilde{s}_{ij} > , \ldots , < u_{n} ,\tilde{s}_{ij} > } \right).$$

• Step 8: Compare each uncertain linguistic value $\tilde{s}_{i}$ with all $\tilde{s}_{i}$, i = 1, 2, …, m, by Definition 4. For simplicity, let $p_{ij} = p(\tilde{s}_{i} \ge \tilde{s}_{j} )$, then it develops a complementary matrix as $P = (p_{ij} )_{m \times m}$, where $p_{ij} + p_{ji} = 1,p_{ii} = 0.5$, i = 1, 2, …, m; j = 1, 2, …, n. Summing up all elements in each line of matrix P, we have $p_{i} = \sum\nolimits_{j = 1}^{n} {p_{ij} }$, i = 1, 2, …, m. We rank $\tilde{s}_{i}$, i = 1, 2, …, m, in descending order in accordance with the value $p_{i}$ (Wei 2009; Xu 2004b).

• Step 9: Rank all the alternatives a_i, i = 1, 2, …, m, and then select the best one(s) in accordance with $p_{i}$, i = 1, 2, …, m.

• Step 10: End.

5 An illustrative example

In this section, we use a multi-attribute group decision making problem of determining which kind of air-conditioning systems should be installed in a library to illustrate the proposed approach.

A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine which kind of air-conditioning systems should be installed in the library. The contractor offers five feasible alternatives, which might be adapted to the physical structure of the library. The offered air-conditioning system must take a decision according to four attributes: (i) c₁ is performance; (ii) c₂ is maintainability; (iii) c₃ is flexibility; (iv) c₄ is safety. Let S = {s₁: very bad; s₂: bad; s₃: a little bad; s₄: fair; s₅: a little good; s₆: good; s₇: very good} be the predefined linguistic term set. In order to take into account the requirements of different stakeholders and obtain objective and comprehensive evaluation results, three experts with different professional backgrounds participate in the decision-making, one is an expert in air-conditioning system technology, one is an expert in engineering project management, and the other is an expert in municipal library operation management. The five possible alternatives a_j (j = 1, 2, 3, 4,5) are to be evaluated using the uncertain linguistic variables by three decision makers under the above four attributes, and construct, respectively, the decision matrices as listed in the following:

$$D^{1} = \left( {\begin{array}{*{20}ll} {[s_{5} ,s_{7} ]} & {[s_{4} ,s_{5} ]} & {[s_{2} ,s_{4} ]} & {[s_{3} ,s_{4} ]} \\ {[s_{3} ,s_{4} ]} & {[s_{1} ,s_{3} ]} & {[s_{5} ,s_{6} ]} & {[s_{2} ,s_{3} ]} \\ {[s_{2} ,s_{4} ]} & {[s_{3} ,s_{4} ]} & {[s_{1} ,s_{3} ]} & {[s_{3} ,s_{5} ]} \\ {[s_{4} ,s_{5} ]} & {[s_{3} ,s_{5} ]} & {[s_{6} ,s_{7} ]} & {[s_{2} ,s_{3} ]} \\ {[s_{2} ,s_{3} ]} & {[s_{4} ,s_{6} ]} & {[s_{4} ,s_{5} ]} & {[s_{3} ,s_{4} ]} \\ \end{array} } \right),$$

$$D^{2} = \left( {\begin{array}{*{20}ll} {[s_{3} ,s_{5} ]} & {[s_{2} ,s_{4} ]} & {[s_{1} ,s_{2} ]} & {[s_{3} ,s_{5} ]} \\ {[s_{4} ,s_{5} ]} & {[s_{2} ,s_{3} ]} & {[s_{2} ,s_{3} ]} & {[s_{4} ,s_{6} ]} \\ {[s_{1} ,s_{2} ]} & {[s_{3} ,s_{5} ]} & {[s_{1} ,s_{2} ]} & {[s_{2} ,s_{3} ]} \\ {[s_{3} ,s_{5} ]} & {[s_{2} ,s_{4} ]} & {[s_{2} ,s_{4} ]} & {[s_{1} ,s_{3} ]} \\ {[s_{1} ,s_{3} ]} & {[s_{4} ,s_{5} ]} & {[s_{5} ,s_{6} ]} & {[s_{4} ,s_{6} ]} \\ \end{array} } \right),$$

$$D^{3} = \left( {\begin{array}{*{20}ll} {[s_{2} ,s_{3} ]} & {[s_{3} ,s_{4} ]} & {[s_{1} ,s_{3} ]} & {[s_{2} ,s_{3} ]} \\ {[s_{3} ,s_{5} ]} & {[s_{1} ,s_{3} ]} & {[s_{3} ,s_{5} ]} & {[s_{2} ,s_{4} ]} \\ {[s_{1} ,s_{3} ]} & {[s_{4} ,s_{5} ]} & {[s_{2} ,s_{3} ]} & {[s_{4} ,s_{5} ]} \\ {[s_{2} ,s_{3} ]} & {[s_{3} ,s_{4} ]} & {[s_{4} ,s_{5} ]} & {[s_{1} ,s_{2} ]} \\ {[s_{4} ,s_{5} ]} & {[s_{3} ,s_{4} ]} & {[s_{3} ,s_{4} ]} & {[s_{2} ,s_{4} ]} \\ \end{array} } \right).$$

Assume that the weight vectors on the expert set with respect to these four attributes are given as:

$$W_{{e_{1} }} = \left( {[0.2,0.4],\;[0.3,0.5],\;[0.2,0.4],\;[0.3,0.5]} \right),$$

$$W_{{e_{2} }} = \left( {[0.1,0.3],\;[0.2,0.3],\;[0.3,0.4],\;[0.1,0.3]} \right),$$

$$W_{{e_{3} }} = \left( {[0.2,0.4],\;[0.2,0.4],\;[0.1,0.3],\;[0.2,0.3]} \right),$$

and the weight vector on ordered set Q = {1,2,3} is defined as:

$$W_{Q} = \left( {[0.2,0.3],\;[0.3,0.5],\;[0.4,0.6]} \right).$$

Furthermore, the weight vector on attribute set is defined as $W_{C} = \left( {[0.1,0.3],\;[0.1,0.2],\;[0.2,0.4],\;[0.3,0.5]} \right)$, and the weight vector on ordered set N = {1, 2, 3, 4} is defined as: $W_{N} = \left( {[0.1,0.2],\;[0.2,0.3],\;[0.3,0.4],\;[0.4,0.5]} \right)$. Based on the IULHASAA operator, to get the best alternative(s), the following steps are involved.

• Step 1: According to Eq. (18) and model (23), the following linear programming for the optimal 2-additive measure on the expert set E for the attribute c₁ is established:

  $$\min - 27.5\left( {v^{1} (e_{1} ) - v^{1} (e_{2} ,e_{3} )} \right) - 30\left( {v^{1} (e_{2} ) - v^{1} (e_{1} ,e_{3} )} \right) - 26.5\left( {v^{1} (e_{3} ) - v^{1} (e_{1} ,e_{2} )} \right)$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{{e_{l} \subseteq S\backslash e_{k} }} {\left( {v^{1} (e_{l} ,e_{k} ) - v^{1} (e_{l} )} \right) \ge } (s - 2)v^{1} (e_{k} ),{\kern 1pt} \;\forall S \subseteq \{ e_{1} ,e_{2} ,e_{3} \} ,{\kern 1pt} \;\forall e_{k} \in S,\;s \ge 2} \\ {v^{1} (e_{1} ,e_{2} ) + v^{1} (e_{1} ,e_{3} ) + v^{1} (e_{2} ,e_{3} ) - v^{1} (e_{1} ) - v^{1} (e_{2} ) - v^{1} (e_{3} ) = 1} \\ {v^{1} (e_{1} ) \in [0.2,0.4],v^{1} (e_{2} ) \in [0.1,0.3],v^{1} (e_{3} ) \in [0.2,0.4]} \\ \end{array} } \right..$$

  Solving it; we obtain the following optimal 2-additive measure

  $$v^{1} (e_{1} ) = v^{1} (e_{3} ) = v^{1} (e_{1} ,e_{3} ) = 0.2,v^{1} (e_{2} ) = 0.3,v^{1} (e_{1} ,e_{2} ) = 1,v^{1} (e_{2} ,e_{3} ) = 0.5.$$

  Similar to the calculation of the optimal 2-additive measure on expert set E for the attribute c₁, the optimal 2-additive measures on expert set E for the attributes c_j, j = 2, 3, 4, are obtained as follows:

  $$v^{2} (e_{1} ) = 0.3,v^{2} (e_{2} ) = 0.29,v^{2} (e_{3} ) = 0.35,v^{2} (e_{1} ,e_{2} ) = 0.45,v^{2} (e_{1} ,e_{3} ) = 0.49,v^{2} (e_{2} ,e_{3} ) = 1.$$
  $$v^{3} (e_{1} ) = 0.26,v^{3} (e_{2} ) = v^{3} (e_{3} ) = v^{3} (e_{1} ,e_{2} ) = 0.3,v^{3} (e_{1} ,e_{3} ) = 1,v^{3} (e_{2} ,e_{3} ) = 0.56.$$
  $$v^{4} (e_{1} ) = 0.5,v^{4} (e_{2} ) = 0.1,v^{4} (e_{3} ) = v^{4} (e_{2} ,e_{3} ) = 0.2,v^{4} (e_{1} ,e_{2} ) = 0.6,v^{4} (e_{1} ,e_{3} ) = 1.$$

• Step 2: According to Eq. (18) and model (26), the following model for the optimal 2-additive measure on ordered set Q is established:

  $$\min - 49(\mu (1) - \mu (2,3)) - 56(\mu (2) - \mu (1,2)) - 63(\mu (3) - \mu (1,2))$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{j \subseteq S\backslash i} {\left( {\mu (i,j) - \mu (j)} \right) \ge } (s - 2)\mu (i),{\kern 1pt} \;\forall S \subseteq \{ 1,2,3\} ,\;\forall i \in S,\;s \ge 2} \\ {\mu (1,2) + \mu (1,3) + \mu (2,3) - \mu (1) - \mu (2) - \mu (3) = 1} \\ {\mu (1) \in [0.2,0.3],\mu (2) \in [0.3,0.5],\mu (3) \in [0.4,0.6]{\kern 1pt} } \\ \end{array} } \right..$$

  Solving it, we obtain the following optimal 2-additive measure:

  $$\mu (1) = 0.2,\;\mu (2) = \mu (1,2) = 0.3,\;\mu (3) = 0.6,\;\mu (1,3) = 0.8,\;\mu (2,3) = 1.$$

• Step 3: Let $u_{(k)} = d_{ij}^{(k)}$(k = 1, 2, 3), according to the IULHASAA operator, we get the comprehensive uncertain linguistic values $\tilde{s}_{ij} = [s_{{\alpha_{ij} }} ,s_{{\beta_{ij} }} ]$ (i = 1, 2, 3, 4, 5; j = 1, 2, 3, 4), e.g., i = j = 1,

  $$\begin{aligned} \tilde{s}_{11} & = {\text{IULHASAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{11}^{1} > , < u_{2} ,\tilde{s}_{11}^{2} > , < u_{3} ,\tilde{s}_{11}^{3} > } \right) \\ & = \frac{{\varphi_{1} (\mu^{1} ,Q)\varphi_{{e_{2}^{1} }} (v^{1} ,E)\tilde{s}_{11}^{2} \oplus \varphi_{2} (\mu^{1} ,Q)\varphi_{{e_{1}^{1} }} (v^{1} ,E)\tilde{s}_{11}^{1} \oplus \varphi_{3} (\mu^{1} ,Q)\varphi_{{e_{3}^{1} }} (v^{1} ,E)\tilde{s}_{11}^{3} }}{{\varphi_{1} (\mu^{1} ,Q)\varphi_{{e_{2}^{1} }} (v^{1} ,E) + \varphi_{2} (\mu^{1} ,Q)\varphi_{{e_{1}^{1} }} (v^{1} ,E) + \varphi_{3} (\mu^{1} ,Q)\varphi_{{e_{3}^{1} }} (v^{1} ,E)}} \\ & = \frac{{0.1 \times 0.55[s_{3} ,s_{5} ] \oplus 0.25 \times 0.35[s_{5} ,s_{7} ] \oplus 0.65 \times 0.1[s_{2} ,s_{3} ]}}{0.1 \times 0.55 + 0.25 \times 0.35 + 0.65 \times 0.1} \\ & = \left[ {s_{3.53} ,s_{5.22} } \right]. \\ \end{aligned}$$

  Similar to the calculation of $\tilde{s}_{11}$, the comprehensive uncertain linguistic matrix $D = \left( {\tilde{s}_{ij} } \right)_{5 \times 4}$ is obtained as follows:

  $$D = \left( {\begin{array}{*{20}ll} {[s_{3.53} ,s_{5.22} ]} & {[s_{2.99} ,s_{4.40} ]} & {[s_{1.14} ,s_{3.19} ]} & {[s_{2.57} ,s_{3.75} ]} \\ {[s_{3.27} ,s_{4.58} ]} & {[s_{1.16} ,s_{3.00} ]} & {[s_{3.33} ,s_{4.54} ]} & {[s_{2.38} ,s_{4.00} ]} \\ {[s_{1.15} ,s_{2.57} ]} & {[s_{3.19} ,s_{4.60} ]} & {[s_{1.21} ,s_{2.59} ]} & {[s_{3.25} ,s_{4.62} ]} \\ {[s_{3.11} ,s_{4.37} ]} & {[s_{2.59} ,s_{4.40} ]} & {[s_{3.51} ,s_{4.89} ]} & {[s_{1.38} ,s_{2.57} ]} \\ {[s_{2.36} ,s_{3.63} ]} & {[s_{3.55} ,s_{4.94} ]} & {[s_{4.20} ,s_{5.20} ]} & {[s_{2.75} ,s_{4.38} ]} \\ \end{array} } \right).$$

• Step 4: According to Eq. (18), model (18) and the comprehensive uncertain linguistic matrix $D = \left( {\tilde{s}_{ij} } \right)_{m \times n}$, the following linear programming for the optimal 2-additive measure on attribute set C is established:

  $$\begin{aligned} & \max - 66.69\left( {v(c_{1} ) + v(c_{2} ) + v(c_{3} ) + v(c_{4} )} \right) + 33.99v(c_{1} ,c_{2} ) + 33.5v(c_{1} ,c_{3} ) + 32.36v(c_{1} ,c_{4} ) \\ & \quad + 34.33v(c_{2} ,c_{3} ) + 33.19v(c_{2} ,c_{4} ) + 32.7v(c_{3} ,c_{4} ) \\ \end{aligned}$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{{c_{j} \subseteq S\backslash c_{i} }} {\left( {v(c_{i} ,c_{j} ) - v(c_{j} )} \right) \ge } (s - 2)v(c_{i} ),\;\forall S \subseteq \{ c_{1} ,c_{2} ,c_{3} ,c_{4} \} ,\;\forall c_{i} \in S,\;s \ge 2} \\ {v(c_{1} ,c_{2} ) + v(c_{1} ,c_{3} ) + v(c_{1} ,c_{4} ) + v(c_{2} ,c_{3} ) + v(c_{2} ,c_{4} ) + v(c_{3} ,c_{4} ) - 2(v(c_{1} ) + v(c_{2} ) + v(c_{3} ) + v(c_{4} )) = 1} \\ {v(c_{1} ) \in [0.1,0.3],v(c_{2} ) \in [0.1,0.2],v(c_{3} ) \in [0.2,0.4],v(c_{4} ) \in [0.3,0.5]} \\ \end{array} } \right..$$

  Solving it, we obtain the following optimal 2-additive measure

  $$v(c_{1} ) = 0.15,v(c_{2} ) = v(c_{3} ) = 0.2,v(c_{4} ) = v(c_{1} ,c_{4} ) = 0.3,v(c_{2} ,c_{3} ) = 1,$$
  $$v(c_{1} ,c_{2} ) = v(c_{1} ,c_{3} ) = v(c_{2} ,c_{4} ) = v(c_{3} ,c_{4} ) = 0.35.$$

• Step 5: According to Eq. (18), model (18) and the comprehensive uncertain linguistic matrix $D = \left( {\tilde{s}_{ij} } \right)_{m \times n}$, the following model for the optimal 2-additive measure on ordered set N = {1, 2, 3, 4} is established:

  $$\begin{aligned} & \min - 16.24\left( {\mu (1) + \mu (2) + \mu (3) + \mu (4)} \right) + 11.35\mu (1,2) + 9.79\mu (1,3) + 7.83\mu (1,4) \\ {\kern 1pt} & \quad + 8.41\mu (2,3) + 6.45\mu (2,4) + 4.89\mu (3,4) \\ \end{aligned}$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{j \subseteq S\backslash i} {\left( {\mu (i,j) - \mu (j)} \right) \ge } (s - 2)\mu (i),\;\forall S \subseteq \{ 1,2,3,4\} ,\;\forall i \in S,\;s \ge 2} \\ \begin{gathered} \mu (1,2) + \mu (1,3) + \mu (1,4) + \mu (2,3) + \mu (2,4) + \mu (3,4) - 2(\mu (1) + \hfill \\ \mu (2) + \mu (3) + \mu (4)) = 1 \hfill \\ \end{gathered} \\ {\mu (1) \in [0.1,0.2],\mu (2) \in [0.2,0.3],\mu (3) \in [0.3,0.4],\mu (4) \in [0.4,0.5]} \\ \end{array} } \right..$$

  Solving it, we obtain the following optimal 2-additive measure

  $$\mu (1) = 0.1,\;\mu (2) = 0.2,\;\mu (3) = 0.3,\;\mu (4) = 0.5,\;\mu (1,2) = 0.25,\;\mu (1,3) = \mu (2,3) = 0.35,$$
  $$\mu (1,4) = 0.55,\;\mu (2,4) = 0.7,\;\mu (3,4) = 1.$$

• Step 6: Let $u_{(j)} = d_{i(j)}$ (j = 1, 2, 3, 4) for each i = 1, 2, 3, 4, 5. According to the IULHASAA operator, we obtain the comprehensive uncertain linguistic values $\tilde{s}_{i} = [s_{{\alpha_{i} }} ,s_{{\beta_{i} }} ]$ of alternatives a_i (i = 1, 2, 3, 4, 5), e.g., i = 1,

  $$\begin{aligned} \tilde{s}_{1} & = {\text{IULHASAA}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{11} > , < u_{2} ,\tilde{s}_{12} > , < u_{3} ,\tilde{s}_{13} > , < u_{4} ,\tilde{s}_{14} > } \right) \\ & = \frac{{\varphi_{1} (\mu ,N)\varphi_{{c_{2} }} (v,C)\tilde{s}_{12} \oplus \varphi_{2} (\mu ,N)\varphi_{{c_{3} }} (v,C)\tilde{s}_{13} \oplus \varphi_{3} (\mu ,N)\varphi_{{c_{1} }} (v,C)\tilde{s}_{11} \oplus \varphi_{4} (\mu ,N)\varphi_{{c_{4} }} (v,C)\tilde{s}_{14} }}{{\varphi_{1} (\mu ,N)\varphi_{{c_{2} }} (v,C) + \varphi_{2} (\mu ,N)\varphi_{{c_{3} }} (v,C) + \varphi_{3} (\mu ,N)\varphi_{{c_{1} }} (v,C) + \varphi_{4} (\mu ,N)\varphi_{{c_{4} }} (v,C)}} \\ & = \frac{{0.025 \times 0.425[s_{2.99} ,s_{4.40} ] \oplus 0.1 \times 0.425[s_{1.14} ,s_{3.19} ] \oplus 0.3 \times 0.075[s_{3.53} ,s_{5.22} ] \oplus 0.575 \times 0.075[s_{2.57} ,s_{3.75} ]}}{0.025 \times 0.425 + 0.1 \times 0.425 + 0.3 \times 0.075 + 0.575 \times 0.075} \\ & = \left[ {s_{2.27,} s_{3.89} } \right]. \\ \end{aligned}$$

  Similar to the calculation of $\tilde{s}_{1}$, the comprehensive uncertain linguistic values $\tilde{s}_{i} = [s_{{\alpha_{i} }} ,s_{{\beta_{i} }} ]$ of alternatives a_i (i = 2, 3, 4, 5) are obtained as follows:

  $$\tilde{s}_{2} = [s_{2.19} ,s_{3.80} ],\tilde{s}_{3} = [s_{1.75} ,s_{3.15} ],\tilde{s}_{4} = [s_{2.33} ,s_{3.77} ],\tilde{s}_{5} = [s_{3.03} ,s_{4.38} ].$$

• Step 7: From Definition 4, the following complementary matrix is obtained

  $$P = \left( {\begin{array}{*{20}ll} {0.5000} & {0.5505} & {0.8314} & {0.5183} & {0.1700} \\ {0.4495} & {0.5000} & {0.7982} & {0.4664} & {0.1380} \\ {0.1686} & {0.2018} & {0.5000} & {0.1652} & {0.0038} \\ {0.4817} & {0.5336} & {0.8348} & {0.5000} & {0.8570} \\ {0.8300} & {0.8620} & {0.9962} & {0.1430} & {0.5000} \\ \end{array} } \right).$$

  Summing up all elements in each line of matrix P, we have

  $$p_{1} = 2.5702,\;p_{2} = 2.3521,\;p_{3} = 1.0394,\;p_{4} = 3.2071,\;p_{5} = 3.3312.$$

  According to p_i (i = 1, 2, 3, 4, 5), we have $\tilde{s}_{5} > \tilde{s}_{4} > \tilde{s}_{1} > \tilde{s}_{2} > \tilde{s}_{3} .$

• Step 8: Rank all the alternatives a_i (i = 1, 2, 3, 4, 5) in accordance with $\tilde{s}_{i}$: $a_{5} > a_{4} > a_{1} > a_{2} > a_{3} .$ Thus, the best alternative is a₅.

In the above example, if the IULHASGM operator is used to make decisions, then the main steps are given as follows:

• Step 1′–Step 2′: See Step 1–Step 2.

• Step 3′: Let $u_{(k)} = d_{ij}^{(k)}$(k = 1, 2, 3), according to the IULHASGM operator, we get the comprehensive uncertain linguistic values $\tilde{s}^{\prime}_{ij} = [s^{\prime}_{{\alpha_{ij} }} ,s^{\prime}_{{\beta_{ij} }} ]$ (i = 1, 2, 3, 4, 5; j = 1, 2, 3, 4), e.g., i = j = 1,

  $$\begin{aligned} \tilde{s}^{\prime}_{11} & = {\text{IULHASGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}_{11}^{1} > , < u_{2} ,\tilde{s}_{11}^{2} > , < u_{3} ,\tilde{s}_{11}^{3} > } \right) \\ & = \tilde{s}_{11}^{{2}{\frac{{\varphi_{1} {(}\mu^{1} ,Q{)}\varphi_{{e_{2}^{1} }} {(}v^{1} ,E{)}}}{{\varphi_{1} {(}\mu^{11} ,Q{)}\varphi_{{e_{2}^{1} }} {(}v^{1} ,E{)} + \varphi_{2} {(}\mu^{1} ,Q{)}\varphi_{{e_{1}^{1} }} {(}v^{1} ,E{)} + \varphi_{3} {(}\mu^{1} ,Q{)}\varphi_{{e_{3}^{1} }} {(}v^{1} ,E{)}}}}} \otimes \tilde{s}_{11}^{{1}{\frac{{\varphi_{2} {(}\mu^{1} ,Q{)}\varphi_{{e_{1}^{1} }} {(}v^{1} ,E{)}}}{{\varphi_{1} {(}\mu^{1} ,Q{)}\varphi_{{e_{2}^{1} }} {(}v^{1} ,E{)} + \varphi_{2} {(}\mu^{1} ,Q{)}\varphi_{{e_{1}^{1} }} {(}v^{1} ,E{)} + \varphi_{3} {(}\mu^{1} ,Q{)}\varphi_{{e_{3}^{1} }} {(}v^{1} ,E{)}}}}} \\ & \quad \otimes \tilde{s}_{11}^{{3}{\frac{{\varphi_{3} {(}\mu^{1} ,Q{)}\varphi_{{e_{3}^{1} }} {(}v^{1} ,E{)}}}{{\varphi_{1} {(}\mu^{1} ,Q{)}\varphi_{{e_{2}^{1} }} {(}v^{1} ,E{)} + \varphi_{2} {(}\mu^{1} ,Q{)}\varphi_{{e_{1}^{1} }} {(}v^{1} ,E{)} + \varphi_{3} {(}\mu^{1} ,Q{)}\varphi_{{e_{3}^{1} }} {(}v^{1} ,E{)}}}}} \\ & = [s_{3} ,s_{5} ]^{{\frac{0.1 \times 0.55}{{0.1 \times 0.55 + 0.25 \times 0.35 + 0.65 \times 0.1}}}} \otimes [s_{5} ,s_{7} ]^{{\frac{0.25 \times 0.35}{{0.1 \times 0.55 + 0.25 \times 0.35 + 0.65 \times 0.1}}}} \otimes [s_{2} ,s_{3} ]^{{\frac{0.65 \times 0.1}{{0.1 \times 0.55 + 0.25 \times 0.35 + 0.65 \times 0.1}}}} \\ & = \left[ {s_{{{3}.{28},}} s_{{{4}.{91}}} } \right]. \\ \end{aligned}$$

  Similar to the calculation of $\tilde{s}^{\prime}_{11}$, the comprehensive uncertain linguistic matrix $D^{\prime} = \left( {\tilde{s}^{\prime}_{ij} } \right)_{5 \times 4}$ is obtained as follows:

  $$D^{\prime} = \left( {\begin{array}{*{20}ll} {[s_{3.28} ,s_{4.91} ]} & {[s_{2.85} ,s_{4.38} ]} & {[s_{1.10} ,s_{2.68} ]} & {[s_{2.52} ,s_{3.68} ]} \\ {[s_{3.24} ,s_{4.55} ]} & {[s_{1.12} ,s_{3.78} ]} & {[s_{3.07} ,s_{4.33} ]} & {[s_{2.28} ,s_{3.87} ]} \\ {[s_{1.11} ,s_{2.89} ]} & {[s_{3.16} ,s_{4.57} ]} & {[s_{1.16} ,s_{2.54} ]} & {[s_{3.15} ,s_{4.54} ]} \\ {[s_{2.98} ,s_{4.26} ]} & {[s_{2.54} ,s_{4.38} ]} & {[s_{3.49} ,s_{5.17} ]} & {[s_{1.30} ,s_{2.52} ]} \\ {[s_{2.07} ,s_{3.52} ]} & {[s_{3.51} ,s_{4.86} ]} & {[s_{4.13} ,s_{5.14} ]} & {[s_{2.66} ,s_{4.32} ]} \\ \end{array} } \right).$$

• Step 4′: According to Eq. (18), model (18) and the comprehensive uncertain linguistic matrix $D^{\prime} = \left( {\tilde{s}^{\prime}_{ij} } \right)_{m \times n}$, the following linear programming for the optimal 2-additive measure on attribute set C is built:

  $$\begin{aligned} & \max - 65.99\left( {v^{\prime}(c_{1} ) + v^{\prime}(c_{2} ) + v^{\prime}(c_{3} ) + v^{\prime}(c_{4} )} \right) + 33.67v^{\prime}(c_{1} ,c_{2} ) + 33.04v^{\prime}(c_{1} ,c_{3} ) + 32v^{\prime}(c_{1} ,c_{4} ) \\ {\kern 1pt} & \quad + 33.99v^{\prime}(c_{2} ,c_{3} ) + 32.95v^{\prime}(c_{2} ,c_{4} ) + 32.32v^{\prime}(c_{3} ,c_{4} ) \\ \end{aligned}$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{{c_{j} \subseteq S\backslash c_{i} }} {\left( {v^{\prime}(c_{i} ,c_{j} ) - v^{\prime}(c_{j} )} \right) \ge } (s - 2)v^{\prime}(c_{i} ),\;\forall S \subseteq \{ c_{1} ,c_{2} ,c_{3} ,c_{4} \} ,\;\forall c_{i} \in S,\;s \ge 2} \\ {v^{\prime}(c_{1} ,c_{2} ) + v^{\prime}(c_{1} ,c_{3} ) + v^{\prime}(c_{1} ,c_{4} ) + v^{\prime}(c_{2} ,c_{3} ) + v^{\prime}(c_{2} ,c_{4} ) + v^{\prime}(c_{3} ,c_{4} ) - 2\left( {v^{\prime}(c_{1} ) + v^{\prime}(c_{2} ) + v^{\prime}(c_{3} ) + v^{\prime}(c_{4} )} \right) = 1} \\ {v^{\prime}(c_{1} ) \in [0.1,0.3],v^{\prime}(c_{2} ) \in [0.1,0.2],v^{\prime}(c_{3} ) \in [0.2,0.4],v^{\prime}(c_{4} ) \in [0.3,0.5]} \\ \end{array} } \right..$$

  Solving it, we get $v^{\prime}(S) = v(S)$ for any S ⊆ C.

• Step 5′: According to Eq. (18), model (18) and the comprehensive uncertain linguistic matrix $D^{\prime} = \left( {\tilde{s}^{\prime}_{ij} } \right)_{m \times n}$, the following model for the optimal 2-additive measure on ordered set N = {1, 2, 3, 4} is built:

  $$\begin{gathered} \min - 16.32\left( {\mu ^{\prime}(1) + \mu ^{\prime}(2) + \mu ^{\prime}(3) + \mu ^{\prime}(4)} \right) + 11.37\mu ^{\prime}(1,2) + 9.88\mu ^{\prime}(1,3) + 7.88\mu ^{\prime}(1,4) \hfill \\ {\kern 1pt} + 8.44\mu ^{\prime}(2,3) + 6.43\mu ^{\prime}(2,4) + 4.94\mu ^{\prime}(3,4) \hfill \\ \end{gathered}$$
  $$s.t.\left\{ {\begin{array}{*{20}ll} {\sum\limits_{j \subseteq S\backslash i} {\left( {\mu ^{\prime}(i,j) - \mu ^{\prime}(j)} \right) \ge (s - 2)\mu ^{\prime}(i),\;\forall S \subseteq \{ 1,2,3,4\} ,\;\forall i \in S,\;s \ge 2} } \\ {\mu ^{\prime}(1,2) + \mu ^{\prime}(1,3) + \mu ^{\prime}(1,4) + \mu ^{\prime}(2,3) + \mu ^{\prime}(2,4) + \mu ^{\prime}(3,4) - 2(\mu ^{\prime}(1) + \mu ^{\prime}(2) + \mu ^{\prime}(3) + \mu ^{\prime}(4)) = 1} \\ {\mu ^{\prime}(1) \in [0.1,0.2],\;\mu ^{\prime}(2) \in [0.2,0.3],\;\mu ^{\prime}(3) \in [0.3,0.4],\;\mu ^{\prime}(4) \in [0.4,0.5]} \\ \end{array} } \right..$$

• Solving it, we get $\mu ^{\prime}(S) = \mu (S)$ for any S ⊆ N.

• Step 6′: Let $u_{(j)} = d_{i(j)}$(j = 1,2,3,4) for each i = 1, 2, 3, 4, 5. Using the IULHASGM operator, we get the comprehensive uncertain linguistic values $\tilde{s}^{\prime}_{i} = [s^{\prime}_{{\alpha_{i} }} ,s^{\prime}_{{\beta_{i} }} ]$ of alternatives a_i, i = 1, 2, 3, 4, 5, e.g., i = 1,

  $$\begin{aligned} \tilde{s}_{1} & = {\text{IULHASGM}}_{\mu ,v} \left( { < u_{1} ,\tilde{s}^{\prime}_{11} > , < u_{2} ,\tilde{s}^{\prime}_{12} > , < u_{3} ,\tilde{s}^{\prime}_{13} > , < u_{4} ,\tilde{s}^{\prime}_{14} > } \right) \\ & = \tilde{{s}^{\prime}}_{12}^{{\frac{{\varphi_{1} (\mu ^{\prime},N)\varphi_{{c_{2} }} (v^{\prime},C)}}{{\varphi_{1} (\mu ^{\prime},N)\varphi_{{c_{2} }} (v^{\prime},C) + \varphi_{2} (\mu ^{\prime},N)\varphi_{c3} (v^{\prime},C) + \varphi_{3} (\mu ^{\prime},N)\varphi_{{c_{1} }} (v^{\prime},C) + \varphi_{4} (\mu ^{\prime},N)\varphi_{c4} (v^{\prime},C)}}}} \\ & \quad \otimes \tilde{{s}^{\prime}}_{13}^{{\frac{{\varphi_{2} (\mu ^{\prime},N)\varphi_{c3} (v^{\prime},C)}}{{\varphi_{1} (\mu ^{\prime},N)\varphi_{{c_{2} }} (v^{\prime},C) + \varphi_{2} (\mu ^{\prime},N)\varphi_{{c_{3} }} (v^{\prime},C) + \varphi_{3} (\mu ^{\prime},N)\varphi_{{c_{1} }} (v^{\prime},C) + \varphi_{4} (\mu ^{\prime},N)\varphi_{c4} (v^{\prime},C)}}}} \\ & \quad \otimes \tilde{{s}^{\prime}}_{11}^{{\frac{{\varphi_{3} (\mu ^{\prime},N)\varphi_{{c_{1} }} (v^{\prime},C)}}{{\varphi_{1} (\mu ^{\prime},N)\varphi_{{c_{2} }} (v^{\prime},C) + \varphi_{2} (\mu ^{\prime},N)\varphi_{{c_{3} }} (v^{\prime},C) + \varphi_{3} (\mu ^{\prime},N)\varphi_{{c_{1} }} (v^{\prime},C) + \varphi_{4} (\mu ^{\prime},N)\varphi_{{c_{4} }} (v^{\prime},C)}}}} \\ & \quad \otimes \tilde{{s}^{\prime}}_{14}^{{\frac{{\varphi_{4} (\mu ^{\prime},N)\varphi_{{c_{4} }} (v^{\prime},C)}}{{\varphi_{1} (\mu ^{\prime},N)\varphi_{c2} (v^{\prime},C) + \varphi_{2} (\mu ^{\prime},N)\varphi_{c3} (v^{\prime},C) + \varphi_{3} (\mu ^{\prime},N)\varphi_{{c_{1} }} (v^{\prime},C) + \varphi_{4} (\mu ^{\prime},N)\varphi_{{c_{4} }} (v^{\prime},C)}}}} \\ & = [s_{2.85} ,s_{4.38} ]^{{\frac{0.025 \times 0.425}{{0.025 \times 0.425 + 0.1 \times 0.425 + 0.3 \times 0.075 + 0.575 \times 0.075}}}} \otimes [s_{1.10} ,s_{2.68} ]^{{\frac{0.1 \times 0.425}{{0.025 \times 0.425 + 0.1 \times 0.425 + 0.3 \times 0.075 + 0.575 \times 0.075}}}} \\ & \quad \otimes [s_{3.28} ,s_{4.91} ]^{{\frac{0.3 \times 0.075}{{0.025 \times 0.425 + 0.1 \times 0.425 + 0.3 \times 0.075 + 0.575 \times 0.075}}}} \otimes [s_{2.52} ,s_{3.68} ]^{{\frac{0.575 \times 0.075}{{0.025 \times 0.425 + 0.1 \times 0.425 + 0.3 \times 0.075 + 0.575 \times 0.075}}}} \\ & = \left[ {s_{1.99} ,s_{3.57} } \right]. \\ \end{aligned}$$

  Similar to the calculation of $\tilde{s}^{\prime}_{1}$, the comprehensive uncertain linguistic values $\tilde{s}^{\prime}_{i} = [s^{\prime}_{{\alpha_{i} }} ,s^{\prime}_{{\beta_{i} }} ]$ of alternatives a_i (i = 2, 3, 4, 5) are obtained as follows:

  $$\tilde{s}^{\prime}_{2} = [s_{1.94} ,s_{4.00} ],\;\tilde{s}^{\prime}_{3} = [s_{1.51} ,s_{3.13} ],\;\tilde{s}^{\prime}_{4} = [s_{2.11} ,s_{3.61} ],\;\tilde{s}^{\prime}_{5} = [s_{2.79} ,s_{4.25} ]$$

• Step 7′: From Definition 4, the following complementary matrix is obtained.

  $$P^{\prime} = \left( {\begin{array}{*{20}ll} {0.5000} & {0.4079} & {0.7452} & {0.4472} & {0.1319} \\ {0.5921} & {0.5000} & {0.7871} & {0.5513} & {0.2434} \\ {0.2548} & {0.2129} & {0.5000} & {0.2135} & {0.0249} \\ {0.5528} & {0.4487} & {0.7865} & {0.5000} & {0.1553} \\ {0.8681} & {0.7566} & {0.9751} & {0.8447} & {0.5000} \\ \end{array} } \right).$$

  Summing up all elements in each line of matrix P′, we have

  $$p^{\prime}_{1} = 2.2322,\;p^{\prime}_{2} = 2.674,\;p^{\prime}_{3} = 1.2061,\;p^{\prime}_{4} = 2.4432,\;p^{\prime}_{5} = 3.9445.$$

  According to p′_i (i = 1, 2, 3, 4, 5), we derive $\tilde{s}^{\prime}_{5} > \tilde{s}^{\prime}_{2} > \tilde{s}^{\prime}_{4} > \tilde{s}^{\prime}_{1} > \tilde{s}^{\prime}_{3} .$

• Step 8′: Rank all the alternatives a_i (i = 1, 2, 3, 4, 5) in accordance with $\tilde{s}_{i}$: $a_{5} > a_{2} > a_{4} > a_{1} > a_{3} .$ Thus, the best alternative is still a₅.

Based on the IULHASAA and IULHASGM operators, the different ranking results are obtained. However, the best choice is both the alternative a₅.

For the convenience of comparison, the ranking results with respect to the ULHA operator (Xu 2004b), the ULHGM operator (Wei 2009) and the ULHHM operator (Park et al. 2011) are obtained as shown in Table 1.

Table 1 The ranking orders with different methods

It can be found from Table 1 that although the ranking results of alternatives obtained by these methods are different, except for the method in (Park et al. 2011), all the other methods get the best alternative a₅ and the worst alternative a₃. The main reason for the difference between the aggregation operator in (Park et al. 2011) and the other ones is that this method does not comprehensively consider the importance weight and ordered positions weight information. It should be noted that all the decision-making results are based on the optimal 2-additive measures obtained by the programming models constructed in this paper. Compared with the other aggregation operators, the IULHASAA and IULHASGM operators proposed in this paper can not only globally consider the importance of the elements and their ordered positions but overall reflect the correlations between them, respectively.

From the provided example, we know that the different ranking results and the different optimal alternatives may be obtained using the different aggregation operators, and thus, the decision maker can properly select the desirable alternative according to his interest and the actual needs. As we noted, the IULHASAA and IULHASGM operators are based on fuzzy measures that consider the interactions between elements. When there is no special explanation that elements in a set are independent, we suggest that the experts adopt the aggregation operator based on fuzzy measures.

6 Conclusion

Based on the Shapley function, we have researched some uncertain linguistic aggregation operators called the induced uncertain linguistic hybrid Shapley arithmetical averaging (IULHSAA) operator and the induced uncertain linguistic hybrid Shapley geometric mean (IULHSGM) operator, which consider the correlative characteristics between elements. To reduce the complexity of solving a fuzzy measure, we further define the induced uncertain linguistic hybrid 2-additive Shapley arithmetical averaging (IULHASAA) operator and the induced uncertain linguistic hybrid 2-additive Shapley geometric mean (IULHASGM) operator. These operators do not only globally consider the importance of the elements and their ordered positions but overall reflect the correlations between them and their ordered positions. When the information about the weight vectors is partly known, models for the optimal fuzzy measures are, respectively, built. As a series of development, an approach to uncertain linguistic multi-attribute group decision making with incomplete weighting information and correlative conditions is developed, and an example is provided to illustrate the practicality and validity of the proposed procedure.

This paper discusses the group decision making method based on the aggregation and ranking of evaluation information. In order to meet different decision situations, it is necessary to further explore some other group decision making methods, such as TODIM method (Wu et al. 2022), and MULTIMOORA method (Qin and Ma 2022). In the process of group decision making, it is necessary to study the group consensus problem in order to ensure the decision effect (Wu et al. 2021). In addition, with the wide range of information and the large number of decision makers, we can further study the large-scale group decision making problems based on social network analysis and minimum cost consensus models (Qin et al. 2022).