Some ILOWA operators and their applications to group decision making with additive linguistic preference relations

Abstract

The aim of this paper is to present some cases of the induced linguistic ordered weighted averaging (ILOWA) operator which are very suitable to deal with group decision making (GDM) problems with additive linguistic preference relations. We propose the relative consensus degree ILOWA (RCD-ILOWA) operator which uses the relative consensus degree (RCD) variable as the order inducing variable, and the compatibility index ILOWA (CI-ILOWA) operator which uses the compatibility index (CI) variable as the order inducing variable. We also investigate some desired properties of RCD-ILOWA operator and the CI-ILOWA operator which provide theoretic basis for the application of the additive linguistic preference relations in GDM. A general formulation and a nonlinear model to determine experts’ weights and two approaches based on the RCD-ILOWA operator and the CI-ILOWA operator for GDM involving additive linguistic preference relations are developed respectively. A numerical example for comprehensive evaluation is given to illustrate the applications of the new operators.

Keywords

Group decision making additive linguistic preference relation ILOWA operator consistency compatibility

1 Introduction

Group decision making (GDM) consists of finding the best alternative(s) from a set of feasible alternatives [1]. To do this, experts have to express their opinions with preference relations by means of a set of evaluations over a set of alternatives, including multiplicative preference relations [13, 31], additive preference relations [2, 14], multiplicative linguistic preference relations [9 , 43], additive linguistic preference relations [7, 54], uncertain preference relations [4 , 59], uncertain linguistic preference relations [35 , 55], and intuitionistic preference relations [12 , 71].

In order to aggregate experts’ preference relations, Yager [62] introduced the ordered weighted averaging (OWA) operator, which provides a parameterized family of aggregation operators that includes the maximum, the minimum and the average criteria. Since its appearance, the OWA operator has been studied in a wide range of applications and extensions, including generalized aggregation operators [46 , 74], fuzzy aggregation operators [18], uncertain aggregation operators [75, 78], induced aggregation operators [23], fuzzy induced aggregation operators [24], linguistic aggregation operators [25 , 76], induced linguistic aggregation operators [26 , 77], uncertain linguistic aggregation operators [39 , 55], induced uncertain linguistic aggregation operators [27, 57], uncertain probabilistic aggregation operators [69], intuitionistic fuzzy aggregation operators [45 , 70], induced intuitionistic fuzzy aggregation operators[33 , 61], interval-valued intuitionistic fuzzy aggregation operators [68], and hesitant fuzzy aggregation operators [73]. In [64], Yager and Filev developed an extension of the OWA operator called the induced ordered weighted averaging (IOWA) operator in which the reordering step is not developed with the values of the arguments but their associated order inducing variables. In the last years, the IOWA operator has been receiving increasing attention [3 , 63], and a lot of extensions of induced aggregation operators have been developed, including extensions to uncertain environment [22, 44], fuzzy environment [24], linguistic environment [26, 56], uncertain linguistic environment [27, 57], intuitionistic fuzzy environment [33 , 61], and combination with other average operators [20 –22]. Another interesting extension of the OWA operator is the linguistic OWA (LOWA) [25, 51] that extends the OWA operator to the linguistic environment. Furthermore, Xu [56] proposed the induced linguistic OWA (ILOWA), which represents a linguistic version of the IOWA operator and generalizes a wide range of aggregation operators such as the IOWA operator and the LOWA operator. Note that a further generalization can be found in [27, 56].

The key features in order to use all kinds of preference relations in GDM are the consistency, consensus and the compatibility measure. The consistency is performed to ensure that experts are being neither random nor illogical in their pairwise comparisons. The lack of consistency in GDM with preference relations can lead to inconsistent conclusions [8]. Consensus is to ensure the maximum degree of agreement among the set of decision makers, which is based on the consistency measure. But consensus is generally to measure the deviation between the each preference relation and the group preference relation by using the distance measure. Moreover, compatibility also plays the role in ensuring that all the individual preference relations can be aggregated effectively. It can be seen as the extension of consensus measure in GDM problems, because the compatibility measure can be used to any two matrices rather than two preference relations. In GDM, the function of consensus measure is the same as the compatibility, the lack of consensus or acceptable compatibility can result in unsatisfied decision making with preference relations because of that there is significant difference among the preference relations provided by experts in GDM [6]. Recently, a lot of developments have appeared about consistency and compatibility in GDM with different types of preference relations, including consistency of multiplicative preference relation [32, 50], fuzzy preference relation [14 , 29], uncertain preference relation [10, 59], linguistic preference relation [8 , 72], uncertain linguistic preference relation [28], triangular fuzzy preference relation [17], incomplete preference relation [2, 48], and compatibility of uncertain preference relation [44, 53], uncertain linguistic preference relation [6], and intuitionistic fuzzy preference relation [15]. For example, Xu and Chen [60] defined some new concepts such as uncertain additive consistent preference relation, uncertain multiplicative consistent preference relation. Xu [59] developed two approaches to construct uncertain additive and multiplicative consistent preference relations. Dong, Xu and Li [8] presented a consistency index of linguistic preference relations and developed a consistency measure method based on the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Peng et al. [28] presented the consistency and consensus coinduced uncertain linguistic ordered weighted averaging operator to aggregate individual uncertain additive linguistic preference relation based on consistency and consensus measure. Peng, Wang and Gao [29] proposed a quantified SWOT decision analysis methodology to evaluate and analyze multiple preference structures based on consistency measure. In [32], Saaty and Vargas presented the compatibility to judge the difference between the two multiplicative preference relations. Xu [53] extended the compatibility degree to uncertain environment and developed the compatibility degree of two uncertain additive preference relations. Chen, Zhou and Han [6] proposed a new compatibility degree for the uncertain additive linguistic preference relations and utilized it to determine the optimal weights of experts in GDM. Therefore, it is necessary to investigate these issues.

The aim of this work is to present the relative consensus degree ILOWA (RCD-ILOWA) operator and the compatibility index ILOWA (CI-ILOWA) operator which are very suitable for GDM problems taking the form of additive linguistic preference relations. Firstly, we propose the RCD-ILOWA operator, which uses the relative consensus degree (RCD) variable as the order inducing variable. Then, an approach based on the RCD-ILOWA operator for GDM problems with additive linguistic preference relations is developed. Moreover, we present the compatibility index of additive linguistic preference relations and the CI-ILOWA operator which uses the compatibility index (CI) variable as the order inducing variable. Then a nonlinear model to determine experts’ weights based on the criterion of minimizing the compatibility index and an approach based on the CI-ILOWA operator for GDM problems with additive linguistic preference relations are investigated. Finally, the RCD-ILOWA operator and the CI-ILOWA operator are applied to group decision making with additive linguistic preference relations.

In order to do so, this paper is organized as follows. In Section 2, we briefly review some basic concepts. Section 3 presents the RCD-ILOWA operator and the CI-ILOWA operator, and then approaches based on the new operators for GDM problems with additive linguistic preference relations are developed. In Section 4, an illustrative example of the new approaches focusing on the evaluation of university. Finally, in Section 5 we summarize the main conclusions of the paper.

2 Preliminaries

In this section, we briefly review the additive linguistic preference relation, the OWA operator, the IOWA operator, the LOWA operator and the ILOWA operator.

2.1 Additive linguistic variable and operational laws

Let S = {s _α|α = - t, …, - 2, - 1, 0, 1, 2, …, t} be an additive linguistic label set with odd cardinality, which requires that the additive linguistic label set satisfies the following characteristics [6, 54]:

The additive linguistic label set S is ordered: if s _α, s _β ∈ S and α > β, then s _α > s _β.

There exists the negation operator: neg (s _α) = s _β such that α + β = 0, where s _α and s _β represent possible values for the linguistic variables and t is a positive integer.

The additive linguistic label set S is called the additive linguistic scale. For example, a set of nine labels S can be defined as: $\begin{matrix} S = {s_{- 4} = EL, s_{- 3} = VL, s_{- 2} = L, s_{- 1} = SL, \\ s_{0} = M, s_{1} = SH, s_{2} = H, s_{3} = VH, s_{4} = EH} . \end{matrix}$

Note that EL = Extremely low, VL = Very low, L = Low, SL = Slightly low, M = Medium, SH = Slightly high, H = High, VH = Very high, EH = Extremely high.

In order to preserve all the given information, we can extend the discrete additive linguistic label set S to a continuous additive linguistic label set $\tilde{S} = {s_{α} | α \in [- q, q]}$ , where q (q > t) is a sufficiently large positive integer. If s _α ∈ S, s _α ∈ S, we call s _α the original additive linguistic label, which is provided to evaluate alternatives by the experts, otherwise, we call s _α the virtual additive linguistic label, which can only appear in operations.

Let $s_{α}, s_{β} \in \tilde{S}$ and λ, μ ∈ [0, 1], then we have the following operational laws:

(1) s _α ⊕ s _β = s _α+β. (2) λs _α = s _λα.

(3) s _α ⊕ s _β = s _β ⊕ s _α. (4) λ (s _α ⊕ s _β) = λs _α ⊕ λs _β.

(5) λμs _α = λ (μs _α) = μ (λs _α).

2.2 Additive linguistic preference relation

In a GDM problem, let X = {x ₁, x ₂, …, x _n} be a finite set of alternatives. When a DM makes pairwise comparisons using the additive linguistic label set S, he/she can express his/her own opinion by an additive linguistic preference relation on X [54]. The additive linguistic preference relation can be defined asfollows:

Definition 1. An additive linguistic preference relation on the set X is denoted by a linguistic decision matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \subset X \times X$ , such that ${\tilde{a}}_{ij} \in S, {\tilde{a}}_{ij} \oplus {\tilde{a}}_{ji} = s_{0}, {\tilde{a}}_{ii} = s_{0}, \forall i, j = 1, 2, \dots, n,$ (1) where ${\tilde{a}}_{ij}$ represents the preference degree of the alternative x _i over x _j. Especially, ${\tilde{a}}_{ij} = s_{0}$ indicates that x _i is equivalent to x _j, ${\tilde{a}}_{ij} > s_{0}$ indicates that x _i is preferred to x _j, and ${\tilde{a}}_{ij} < s_{0}$ indicates that x _j is preferred to x _i.

A very crucial property of additive linguistic preference relation is the consistency [8]. It can be defined as follows:

Definition 2. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \subset X \times X$ be an additive linguistic preference relation, then $\tilde{A}$ is called a consistent additive linguistic preference relation, if ${\tilde{a}}_{ij} = {\tilde{a}}_{ik} \oplus {\tilde{a}}_{kj}, for all i, j, k .$ (2)

Note that throughout this paper, let M _n be the set of all n × n additive linguistic preference relations. For convenience, supposing that $s_{α} \in \tilde{S}$ , I (s _α) denotes the subscript of additive linguistic label s _α, then we have I (s _α) = α.

And more generally, Chen, Zhou and Han [6] developed the following theorem:

Theorem 1. If $s_{α_{i}} \in \tilde{S}$ and λ _i ∈ [0, 1] for i = 1, 2, …, n, then $I (\oplus_{i = 1}^{n} λ_{i} s_{α_{i}}) = \sum_{i = 1}^{n} λ_{i} I (s_{α_{i}}) .$ (3)

2.3 The OWA operator and the IOWA operator

The OWA operator [62] is an aggregation operator that provides a parameterized family of aggregation operators between the minimum and the maximum, which can be defined as follows:

Definition 3. An OWA operator of dimension n is a mapping OWA : R ⁿ → R that has an associated weighting vector w with $\sum_{j = 1}^{n} w_{j} = 1$ and w _j ∈ [0, 1], such that $OWA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j},$ (4) where b _j is the jth largest of a ₁, a ₂, …, a _n.

The OWA operator is monotonic, commutative, bounded and idempotent. In [64], Yager and Filev extended the OWA and obtain the IOWA operator. The main difference of the IOWA operator is that the reordering step is developed with order inducing variables u _i rather than the values of a _i.

Definition 4. An IOWA operator is a mapping IOWA : R ⁿ × R ⁿ → R that has an associated weighting vector W of dimension n with $\sum_{j = 1}^{n} w_{j} = 1$ and w _j ∈ [0, 1], such that:

$\begin{matrix} IOWA (〈 u_{1}, a_{1} 〉, 〈 u_{2}, a_{2} 〉, \dots, 〈 u_{n}, a_{n} 〉) \\ = \sum_{j = 1}^{n} w_{j} b_{j}, \end{matrix}$ (5) where b _j is the a _i value of the IOWA pair 〈u _i, a _i〉 having the jth largest u _i, u _i is the order inducing variable and a _i is the argument variable.

The IOWA operator is also monotonic, bounded, idempotent and commutative. For further properties and extensions on the IOWA, refer to [3 , 64]. For example, Chiclana et al. [3] proposed the induced ordered weighted geometric (IOWG) averaging operator, Merigó and Gil-Lafuente [23] extended the IOWA operator and obtained the induced generalized OWA (IGOWA) operator.

2.4 The LOWA operator and the ILOWA operator

In [51], Xu extended the OWA operator to the linguistic environment and obtained the linguistic OWA (LOWA) operator. It can be defined as follows:

Definition 5. A LOWA operator is a mapping $LOWA : {\tilde{S}}^{n} \to \tilde{S}$ that has an associated weighting vector W of dimension n with $\sum_{j = 1}^{n} w_{j} = 1$ and w _j ∈ [0, 1], such that: $LOWA (s_{α_{1}}, s_{α_{2}}, \dots, s_{α_{n}}) = \oplus_{j = 1}^{n} w_{j} s_{β_{j}},$ (6) where s _{β
_j} is the jth largest of the s _{α
_i}.

The induced LOWA (ILOWA) operator [56] is an extension of the OWA operator that uses linguistic information and inducing variables in the reordering of the arguments.

Definition 6. An ILOWA operator is defined as follows:

$\begin{matrix} ILOWA (〈 u_{1}, s_{α_{1}} 〉, \dots, 〈 u_{n}, s_{α_{n}} 〉) \\ = \oplus_{j = 1}^{n} w_{j} s_{β_{j}}, \end{matrix}$ (7) where W = (w ₁, w ₂, …, w _n) ^T is a weighting vector, such that $\sum_{j = 1}^{n} w_{j} = 1$ and w _j ∈ [0, 1], s _{β
_j} is the s _{α
_i} value of the ILOWA pair 〈u _i, s _{α
_i}〉 having the jth largest u _i, u _i is the order inducing variable and s _{α
_i} is the linguistic variable.

Especially, if u _i = s _{α
_i} for all i, then the ILOWA operator reduces to the LOWA operator. Note that the LOWA operator and the ILOWA operator are monotonic, bounded, idempotent and commutative. Note also that more information about LOWA operator and ILOWA operator can be found in [26 , 56].

3 Some ILOWA operators to aggregate additive linguistic preference relations

In [3], Chiclana et al. proposed the importance IOWG (I-IOWG) operator, the consistency IOWG (C-IOWG) operator and the preference IOWG (P-IOWG) operator. Wu et al. [44] presented the reliability induced continuous ordered weighted geometric (R-ICOWG) operator and the relative consensus degree induced continuous ordered weighted geometric (RCD-ICOWG) operator. And in [43], Wu, Cao and Zhang developed the compatibility index induced linguistic ordered weighted geometric (CI-ILOWG) operator and the importance induced linguistic ordered weightedgeometric (I-ILOWG) operator. However, all of these operators are somewhat unsuitable for dealing with additive decision making problems under linguistic environment.

In this section, we shall develop two special cases of ILOWA operator, including the relative consensus degree ILOWA (RCD-ILOWA) operator, which uses the relative consensus degree (RCD) variable as the order inducing variable, and the compatibility index ILOWA (CI-ILOWA) operator which uses the compatibility index (CI) variable as the order inducingvariable.

3.1 The RCD-ILOWA operator

Let E = {e ₁, e ₂, …, e _m} be a finite set of experts and ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n} \in M_{n}$ be the additive linguistic preference relation provided by expert e _k, k = 1, 2, …, m, then we can get the synthetic linguistic preference relation of ${\tilde{A}}^{(1)}, {\tilde{A}}^{(2)}, \dots, {\tilde{A}}^{(m)}$ asfollows:

Definition 7. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n} \in M_{n}$ for k = 1, 2, …, m. If

$\begin{matrix} {\tilde{a}}_{ij} & = & ILOWA (〈 u_{1}, {\tilde{a}}_{ij}^{(1)} 〉, \\ 〈 u_{2}, {\tilde{a}}_{ij}^{(2)} 〉, \dots, 〈 u_{m}, {\tilde{a}}_{ij}^{(m)} 〉) \\ = & \oplus_{k = 1}^{m} ω_{k} {\tilde{a}}_{ij}^{(σ (k))}, \end{matrix}$ (8) then the matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is called the synthetic linguistic preference relation of all experts, where ω = (ω ₁, ω ₂, …, ω _m) is the weighting vector of experts, which satisfies that ω _k ≥ 0 for all j = 1, 2, …, m and $\sum_{k = 1}^{m} ω_{k} = 1$ . (σ (1) , σ (2) , …, σ (m)) is a permutation of (1, 2, …, m) such that u _(σ(k-1)) ≥ u _(σ(k)).

Theorem 2. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ be additive linguistic preference relations provided by expert e _k for k = 1, 2, …, m, then the synthetic linguistic preference relation $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is also an additive linguistic preference relation.

Theorem 3. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n} \in M_{n}$ provided by expert e _k for k = 1, 2, …, m, and $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be the synthetic linguistic preference relation of ${\tilde{A}}^{(1)}, {\tilde{A}}^{(2)}, \dots, {\tilde{A}}^{(m)}$ . If ${\tilde{A}}^{(k)}$ is consistent for all k, then $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is also consistent.

In [19], Ma and Hu introduced the optimal transfer matrix as follows:

Definition 8. Let A = (a _ij) _n×n be an ordinary matrix, if $a_{ij} + a_{ji} = 0, a_{ii} = 0, for all i, j,$ (9) then A is called the antisymmetric matrix. If $a_{ij} + a_{ji} = 0, a_{ik} + a_{kj} = a_{ij}, for all i, j,$ (10) then A is called the transfer matrix.

Definition 9. Let A = (a _ij) _n×n be a transfer matrix. If there exists the minimization of the distance d between A = (a _ij) _n×n and C = (c _ij) _n×n, where $d = ∥ A - C ∥ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} (a_{ij} - c_{ij})^{2},$ (11) then C is called the optimal transfer matrix to A.

In [44], Wu et al. developed the following theorem:

Theorem 4. Let A = (a _ij) _n×n be an antisymmetric matrix. Then the optimal transfer matrix C = (c _ij) _n×n to A can be defined according to the following formula: $c_{ij} = \frac{1}{n} \sum_{k = 1}^{n} (a_{ik} - a_{jk}) .$ (12)

Definition 10. Let $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n}$ be the additive linguistic preference relation. $I (\tilde{B}) = (I ({\tilde{b}}_{ij}))_{n \times n}$ is called the subscript matrix of $\tilde{B}$ .

Theorem 5. If $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n}$ is the linguistic preference relation and $I (\tilde{B}) = (I ({\tilde{b}}_{ij}))_{n \times n}$ is the subscript matrix of $\tilde{B}$ , then $\tilde{B}$ is the additive linguistic preference relation if and only if $I (\tilde{B})$ is an antisymmetric matrix.

Definition 11. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ be additive linguistic preference relations provided by expert e _k for k = 1, 2, …, m. Then we call ${\tilde{C}}^{(k)} = ({\tilde{c}}_{ij}^{(k)})_{n \times n}$ the optimal transfer matrices to ${\tilde{A}}^{(k)}$ for all k, where $I ({\tilde{c}}_{ij}^{(k)}) = \frac{1}{n} \sum_{l = 1}^{n} (I ({\tilde{a}}_{il}^{(k)}) - I ({\tilde{a}}_{jl}^{(k)})), \forall i, j, k .$ (13)

$I ({\tilde{C}}^{(k)}) = (I ({\tilde{c}}_{ij}^{(k)}))_{n \times n}$ are subscript matrices of ${\tilde{C}}^{(k)}$ and $I ({\tilde{A}}^{(k)}) = (I ({\tilde{a}}_{ij}^{(k)}))_{n \times n}$ are the subscript matrices of ${\tilde{A}}^{(k)}$ , k = 1, 2, …, m.

Theorem 6. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ be additive linguistic preference relations provided by expert e _k for k = 1, 2, …, m, and $I ({\tilde{A}}^{(k)}) = (I ({\tilde{a}}_{ij}^{(k)}))_{n \times n}$ be the subscript matrices of ${\tilde{A}}^{(k)}$ , respectively. Then the optimal transfer matrices ${\tilde{C}}^{(k)} = ({\tilde{c}}_{ij}^{(k)})_{n \times n}$ are consistent additive linguistic preference relations.

Now, we can define the RCD-ILOWA operator as follows:

Definition 12. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ be additive linguistic preference relations provided by expert e _k and ${\tilde{C}}^{(k)} = ({\tilde{c}}_{ij}^{(k)})_{n \times n}$ be the optimal transfer matrices to ${\tilde{A}}^{(k)}$ for all k. The relative consensus degree of e _k is defined by using Equation (14) ${RCD}_{k} = \frac{cos θ_{k}}{\sum_{k = 1}^{m} cos θ_{k}}, k = 1, 2, \dots, m,$ (14) $cos θ_{k} = \frac{(vec (I ({\tilde{A}}^{(k)})), vec (I ({\tilde{C}}^{(k)})))}{∥ vec (I ({\tilde{A}}^{(k)})) ∥ \cdot ∥ vec (I ({\tilde{C}}^{(k)})) ∥},$ (15)

$vec (I ({\tilde{A}}^{(k)})) = (I ({\tilde{a}}_{11}^{(k)}), I ({\tilde{a}}_{21}^{(k)}), \dots, I ({\tilde{a}}_{n 1}^{(k)}); \dots;$ $I ({\tilde{a}}_{1 n}^{(k)})$ , $I ({\tilde{a}}_{2 n}^{(k)}), \dots, I ({\tilde{a}}_{nn}^{(k)}))^{T}$ , $vec (I ({\tilde{C}}^{(k)})) = (I ({\tilde{c}}_{11}^{(k)})$ , $I ({\tilde{c}}_{21}^{(k)}), \dots, I ({\tilde{c}}_{n 1}^{(k)}); \dots;$ $I ({\tilde{c}}_{1 n}^{(k)}), I ({\tilde{c}}_{2 n}^{(k)}), \dots, I ({\tilde{c}}_{nn}^{(k)}))^{T}$ , $I ({\tilde{C}}^{(k)}) = (I ({\tilde{c}}_{ij}^{(k)}))_{n \times n}$ are subscript matrices of ${\tilde{C}}^{(k)}$ and $I ({\tilde{A}}^{(k)}) = (I ({\tilde{a}}_{ij}^{(k)}))_{n \times n}$ are the subscript matrices of ${\tilde{A}}^{(k)}$ , k = 1, 2, …, m.

As we can see, the closer ${\tilde{A}}^{(k)}$ is to ${\tilde{C}}^{(k)}$ , the greater relative consensus degree RCD _k is, and thus bigger weight should be placed on the preference relation ${\tilde{A}}^{(k)}$ .

Definition 13. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ be additive linguistic preference relations provided by expert e _k k = 1, 2, …, m. The synthetic linguistic preference relation of all experts can be defined by the RCD-ILOWA operator as follows:

$\begin{matrix} \tilde{A} & = & RCD - ILOWA (〈 {RCD}_{1}, {\tilde{A}}^{(1)} 〉, \dots, \\ 〈 {RCD}_{m}, {\tilde{A}}^{(m)} 〉), \\ {\tilde{a}}_{ij} & = & RCD - ILOWA (〈 {RCD}_{1}, {\tilde{a}}_{ij}^{(1)} 〉, \dots, \\ 〈 {RCD}_{m}, {\tilde{a}}_{ij}^{(m)} 〉) \\ = & {RCD}_{σ (1)} {\tilde{a}}_{ij}^{(σ (1))} \oplus {RCD}_{σ (2)} {\tilde{a}}_{ij}^{(σ (2))} \oplus \dots \oplus \\ {RCD}_{σ (m)} {\tilde{a}}_{ij}^{(σ (m))} \\ = & \oplus_{k = 1}^{m} {RCD}_{σ (k)} {\tilde{a}}_{ij}^{(σ (k))} . \end{matrix}$ (16)

RCD _k is the relative consensus degree of e _k, k = 1, 2, …, m. (σ (1) , σ (2) , …, σ (m)) is a permutation of (1, 2, …, m) such that RCD _σ(k-1) ≥ RCD _σ(k).

3.2 An approach based on the RCD-ILOWA operator for GDM problems with additive linguistic preference relations

Consider a GDM problem. Let X = {x ₁, x ₂, …, x _n} be a set of finite alternatives and E = {e ₁, e ₂, …, e _m} be a finite set of experts. Each expert provides his/her own decision matrix ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ , which are additive linguistic preference relation provided by the expert e _k ∈ E. The process of new approach based on the RCD-ILOWA operator can be summarized asfollows:

Step 1: Utilize the model (14) to calculate the relative consensus degree of e _k: RCD _k.

Step 2: Utilize Equations (19) and (20) to calculate the synthetic linguistic preference relation $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ based on the RCD-ILOWA operator.

Step 3: Calculate the expected value ${\tilde{a}}_{i}$ of preference degree of the alternative x _i to all the alternative by the linguistic averaging (LA) operator [57]: ${\tilde{a}}_{i} = \oplus_{j = 1}^{n} {\tilde{a}}_{ij} / n, i = 1, 2, \dots, n .$ (17)

Step 4: Rank the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, n) in descending order.

Step 5: Rank all the alternatives x _i (i = 1, 2, …, n) and select the best one(s) in accordance with the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, n).

Step 6: End.

3.3 The CI-ILOWA operator

The compatibility degree is an important problem in GDM with preference relations. The lack of acceptable compatibility can result in unsatisfied decision making with preference relations because of that there is significant difference among the preference relations provided by experts in GDM [6]. In this section, we will propose a new compatibility index (CI) for the additive linguistic preference relations.

Definition 14. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n}$ be the additive linguistic preference relations. If ${\tilde{a}}_{ij} = {\tilde{b}}_{ij}$ for all i, j, then $\tilde{A}$ and $\tilde{B}$ are perfectly compatible.

It can be seen that only $\tilde{A}$ and $\tilde{B}$ were equal, they would be perfectly compatible. Then, based on Equation (1), $\tilde{A} = \tilde{B}$ if and only if ${\tilde{a}}_{ij} = {\tilde{b}}_{ij} \Leftrightarrow {\tilde{a}}_{ij} \oplus {\tilde{b}}_{ji} = s_{0} \Leftrightarrow I ({\tilde{a}}_{ij}) + I ({\tilde{b}}_{ji}) = 0$ .

Thus, the compatibility index can be defined asfollows:

Definition 15. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ be two additive linguistic preference relations, then $CI (\tilde{A}, \tilde{B}) = \frac{1}{2 {tn}^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} | I ({\tilde{a}}_{ij}) + I ({\tilde{b}}_{ji}) |,$ (18) is called the compatibility index (CI) of $\tilde{A}$ and $\tilde{B}$ .

It can be seen easily that the smaller the value of $CI (\tilde{A}, \tilde{B})$ , the closer the additive linguistic variables $\tilde{A}$ and $\tilde{B}$ will be. It is also can be obtained that in GDM, the function of compatibility index is the same as the consensus, because they are both applied to measure the deviation of two preference relation. In fact, compatibility can be also used to measure the deviation of any two matrices that they are not even consistency. But the relative consensus index is on the basis of the consistency, which results in that the relative consensus index is different from the compatibility index.

Based on Definition 14 and Definition 15, we can obtain the following theorems easily:

Theorem 7. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ and $\tilde{B} =$ $({\tilde{b}}_{ij})_{n \times n}$ ∈M _n, then $\tilde{A}$ and $\tilde{B}$ are perfectly compatible if and only if $CI (\tilde{A}, \tilde{B}) = 0$ .

Theorem 8. Assume that $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ , $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ and $\tilde{D} = ({\tilde{d}}_{ij})_{n \times n} \in M_{n}$ , then we have

Nonnegativity: $CI (\tilde{A}, \tilde{B}) \geq 0$ .

Reflexivity: $CI (\tilde{A}, \tilde{A}) = 0$ .

Commutativity: $CI (\tilde{A}, \tilde{B}) = CI (\tilde{B}, \tilde{A})$ .

Transitivity: If $CI (\tilde{A}, \tilde{B}) = 0$ and $CI (\tilde{B}, \tilde{D}) = 0$ , then $CI (\tilde{A}, \tilde{D}) = 0$ .

Triangle inequality: $CI (\tilde{A}, \tilde{D}) \leq CI (\tilde{A}, \tilde{B}) + CI (\tilde{B}, \tilde{D})$ .

Definition 16. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ . If $CI (\tilde{A}, \tilde{B}) \leq α,$ (19) then $\tilde{A}$ and $\tilde{B}$ are of acceptable compatibility, where α is the threshold of acceptable compatibility.

As illustrated in [6, 53], we can take α = 0.1 as the threshold of acceptable compatibility.

Now, we can develop the compatibility index ILOWA (CI-ILOWA) operator which uses the compatibility index (CI) variable as the order inducing variable in the ILOWA operator.

Definition 17. Let ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n} \in M_{n}$ provided by expert e _k, k = 1, 2, …, m, and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ . The synthetic linguistic preference relation of all experts can be defined by the CI-ILOWA operator as follows: $\tilde{A} = CI - ILOWA (〈 {CI}_{1}, {\tilde{A}}^{(1)} 〉, \dots, 〈 {CI}_{m}, {\tilde{A}}^{(m)} 〉),$ (20)

$\begin{matrix} {\tilde{a}}_{ij} & = & CI - ILOWA (〈 {CI}_{1}, {\tilde{a}}_{ij}^{(1)} 〉, \dots, 〈 {CI}_{m}, {\tilde{a}}_{ij}^{(m)} 〉) \\ = & ω_{1} {\tilde{a}}_{ij}^{(σ (1))} \oplus ω_{2} {\tilde{a}}_{ij}^{(σ (2))} \oplus \dots \oplus ω_{m} {\tilde{a}}_{ij}^{(σ (m))} \\ = & \oplus_{k = 1}^{m} ω_{k} {\tilde{a}}_{ij}^{(σ (k))} . \end{matrix}$ (21)

${CI}_{k} = CI ({\tilde{A}}^{(k)}, \tilde{B})$ is the compatibility index of ${\tilde{A}}^{(k)}$ and $\tilde{B}$ . (σ (1) , σ (2) , …, σ (m)) is a permutation of (1, 2, …, m) such that CI _σ(k-1) ≤ CI _σ(k). ω = (ω ₁, ω ₂, …, ω _m) is the weighting vector of experts, which satisfies that ω _k ≥ 0 for all j = 1, 2, …, m and $\sum_{k = 1}^{m} ω_{k} = 1$ .

It can be seen from Definition 1 and Theorem 2 that the synthetic linguistic preference relation of all experts $\tilde{A}$ defined by the CI-ILOWA operator is additive linguistic preference relation.

Theorem 9. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ for k = 1, 2, …, m, and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ . If $CI ({\tilde{A}}^{(k)}, \tilde{B}) \leq α$ , for all k = 1, 2, …, m, then $CI (\tilde{A}, \tilde{B}) \leq α,$ (22) where $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is the synthetic linguistic preference relation defined by the CI-ILOWA operator, and α is the threshold of acceptable compatibility.

Corollary 1. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ for k = 1, 2, …, m, and $\tilde{B} = ({\tilde{b}}_{ij})_{n \times n} \in M_{n}$ . If $CI ({\tilde{A}}^{(k)}, \tilde{B}) = 0$ , for all k = 1, 2, …, m, then $CI (\tilde{A}, \tilde{B}) = 0,$ (23) where $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is the synthetic linguistic preference relation defined by the CI-ILOWA operator.

3.4 To determine the weights of experts based on the CI-ILOWA operator with additive linguistic preference relations

It is evident that the less compatibility index of additive linguistic preference relations given by the expert e _k, the more reliable information provided by the expert e _k. Then, in order to determine the weights of experts, we can minimize the compatibility index of the synthetic linguistic preference relation $\tilde{A}$ and the ideal linguistic preference relation $\tilde{B}$ . For the convenience of calculation, we use the square sum instead of absolute values in compatibility index of $\tilde{A}$ and $\tilde{B}$ . Therefore, based on the proof of Theorem 9, the compatibility index of $\tilde{A}$ and $\tilde{B}$ can be rewritten as follows: $\begin{matrix} CI (\tilde{A}, \tilde{B}) = \frac{1}{2 {tn}^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {(I ({\tilde{a}}_{ij}) + I ({\tilde{b}}_{ji}))}^{2} \\ = \sum_{k_{1} = 1}^{m} \sum_{k_{2} = 1}^{m} ω_{k_{1}} ω_{k_{2}} (\frac{1}{2 {tn}^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (I ({\tilde{a}}_{ij}^{(σ (k_{1}))}) \\ + I ({\tilde{b}}_{ji})) (I ({\tilde{a}}_{ij}^{(σ (k_{2}))}) + I ({\tilde{b}}_{ji}))) \end{matrix}$

Let G = (g _{k
₁
k
₂}) _m×m, where

$\begin{matrix} g_{k_{1} k_{2}} = \frac{1}{2 {tn}^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (I ({\tilde{a}}_{ij}^{(σ (k_{1}))}) + I ({\tilde{b}}_{ji})) \\ \times (I ({\tilde{a}}_{ij}^{(σ (k_{2}))}) + I ({\tilde{b}}_{ji})), \forall k_{1}, k_{2} = 1, 2, \dots, m . \end{matrix}$ (24)

And let ω = (ω ₁, ω ₂, …, ω _m) ^T be the experts’ weighting vector. Then the compatibility index $CI (\tilde{A}, \tilde{B})$ can be regarded as the function of ω. Denoting $F (ω) = CI (\tilde{A}, \tilde{B})$ , we get $F (ω) = ω^{T} G ω .$ (25)

Therefore, we obtain the following optimal model to determine experts’ weights:

$\begin{matrix} Min F (ω) = ω^{T} G ω \\ s . t . {\begin{matrix} \sum_{k = 1}^{m} ω_{k} = 1, \\ ω_{k} \geq 0, k = 1, 2, \dots, m . (26) \end{matrix} \end{matrix}$

Letting R ^T = (1, 1, …, 1) _1×m, we have

$\begin{matrix} Min F (ω) = ω^{T} G ω \\ s . t . {\begin{matrix} R^{T} ω = 1, \\ ω \geq 0 . \end{matrix} \end{matrix}$ (26)

If the constraint ω ≥ 0 is not taken into account, Equation (27) can be viewed as follows: $\begin{matrix} Min F (ω) = ω^{T} G ω \\ s . t . R^{T} ω = 1 . \end{matrix}$ (27)

It follows that if the global optimal solution to model (28) ω ^* ≥ 0, then ω ^* is also the global optimal solution to model (27).

Theorem 10. If $\tilde{A}$ and $\tilde{B}$ are not perfectly compatible, then the optimal solution to Equation ( 28) is $ω^{*} = \frac{G^{- 1} R}{R^{T} G^{- 1} R} .$ (28)

Proof. Since $\tilde{A}$ and $\tilde{B}$ are not perfectly compatible, there exists i ₀, j ₀ ∈ {1, 2, …, n}, which satisfy ${\tilde{a}}_{i_{0} j_{0}} \neq {\tilde{b}}_{i_{0} j_{0}}$ , i.e., ${\tilde{a}}_{i_{0} j_{0}} \oplus {\tilde{b}}_{j_{0} i_{0}} \neq s_{0}$ , then we get $(I ({\tilde{a}}_{i_{0} j_{0}}) + I ({\tilde{b}}_{j_{0} i_{0}}))^{2} > 0$ . Thus, $F (ω) = ω^{T} G ω = \frac{1}{2 {tn}^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {(I ({\tilde{a}}_{ij}) + I ({\tilde{b}}_{ji}))}^{2} > 0 .$

By Equation (24), we get g _{k
₁
k
₂} = g _{k
₂
k
₁}, ∀k ₁, k ₂. Therefore, G = (g _{k
₁
k
₂}) _m×m is a nonsingular matrix. By constructing the Lagrange function corresponding to the model of Equation (28): $L (ω, λ) = ω^{T} G ω + λ (R^{T} ω - 1),$ (29) where λ is the Lagrange multiplier.

Taking the partial derivatives of Equation (30) with respect to ω and λ, and setting them to be equal to 0, we obtain that ∂L (ω, λ)/∂ω = 0, ∂L (ω, λ)/∂λ = 0.

Then we get $ω^{*} = \frac{G^{- 1} R}{R^{T} G^{- 1} R} .$

With the fact that ∂² L (ω, λ)/∂ω ² = 2G, which means that F (ω) is a strictly convex function, ω ^* is the unique optimal solution to Equation (28), which completes the proof of the Theorem.

3.5 An approach based on the CI-ILOWA operator for GDM problems with additive linguistic preference relations

Consider a GDM problem. Let X = {x ₁, x ₂, …, x _n} be a set of finite alternatives and E = {e ₁, e ₂, …, e _m} be a finite set of experts. Each expert provideshis/her own decision matrix ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ , which are additive linguistic preference relation provided by the expert e _k ∈ E. The process of new approach based on the CI-ILOWA operator can be summarized asfollows:

Step 1: Utilize the model (26) to calculate the compatibility index of e _k: $CI ({\tilde{A}}^{(k)}, \tilde{B})$ .

Step 2: Utilize the model Equation (31) to determine the optimal weights of experts: $ω^{*} = (ω_{1}^{*}, ω_{2}^{*}, \dots, ω_{m}^{*})^{T}$ .

Step 3: Utilize Equation (20) to obtain the synthetic linguistic preference relation $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ based on the CI-ILOWA operator.

Step 4: Calculate the expected value ${\tilde{a}}_{i}$ of preference degree of the alternative x _i to all the alternatives by the LA operator.

Step 5: Rank the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, n) in descending order.

Step 6: Rank all the alternatives x _i (i = 1, 2, …, n) and select the best one(s) in accordance with the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, n).

Step 7: End.

4 Illustrative example

In this section, we develop an approach for evaluating four schools of a university (adapted from [51]). Let X = {x ₁, x ₂, x ₃, x ₄, x ₅} be a finite set five schools to be evaluated. One main criterion used is research. The group of experts is constituted by three persons. Each expert e _k (k = 1, 2, 3) compares five schools with respect to the main criterion of research by using the following additive linguistic scales: $\begin{matrix} S = {s_{- 5} = EP, s_{- 4} = DP, s_{- 3} = SP, s_{- 2} = MP, \\ s_{- 1} = WP, s_{0} = EG, s_{1} = WG, s_{2} = MG, s_{3} = SG, \\ s_{4} = DG, s_{5} = EG} . \end{matrix}$

Note that EP = Extremely poor, DP = Demonstratedly poor, SP = Strongly poor, MP = Moderately poor, WP = Weakly poor, EG = Equally good, WG = Weakly good, MG = Moderately good, SG = Strongly good, DG = Demonstratedly good, EG = Extremely good.

Each expert gives his/her own additive linguistic preference relation ${\tilde{A}}^{(k)}$ (k = 1, 2, 3), respectively.

They are listed as follows: $\begin{matrix} {\tilde{A}}^{(1)} = (\begin{matrix} s_{0} & s_{1} & s_{- 2} & s_{3} & s_{- 1} \\ s_{- 1} & s_{0} & s_{4} & s_{1} & s_{- 2} \\ s_{2} & s_{- 4} & s_{0} & s_{2} & s_{3} \\ s_{- 3} & s_{- 1} & s_{- 2} & s_{0} & s_{4} \\ s_{1} & s_{2} & s_{- 3} & s_{- 4} & s_{0} \end{matrix}); \\ {\tilde{A}}^{(2)} = (\begin{matrix} s_{0} & s_{2} & s_{- 1} & s_{2} & s_{- 2} \\ s_{- 2} & s_{0} & s_{4} & s_{2} & s_{- 1} \\ s_{1} & s_{- 4} & s_{0} & s_{3} & s_{2} \\ s_{- 2} & s_{- 2} & s_{- 3} & s_{0} & s_{4} \\ s_{2} & s_{1} & s_{- 2} & s_{- 4} & s_{0} \end{matrix}); \\ {\tilde{A}}^{(3)} = (\begin{matrix} s_{0} & s_{3} & s_{- 3} & s_{1} & s_{- 3} \\ s_{- 3} & s_{0} & s_{3} & s_{2} & s_{- 2} \\ s_{3} & s_{- 3} & s_{0} & s_{3} & s_{1} \\ s_{- 1} & s_{- 2} & s_{- 3} & s_{0} & s_{4} \\ s_{3} & s_{2} & s_{- 1} & s_{- 4} & s_{0} \end{matrix}) . \end{matrix}$

With this information, we can use the RCD-ILOWA operator and the CI-ILOWA operator to get the ranking of the alternatives, respectively.

4.1 The approach based on the RCD-ILOWA operator

Step 1: Utilize the model (14) to calculate the relative consensus degree of e _k: ${RCD}_{1} = 0.3585, {RCD}_{2} = 0.3533, {RCD}_{3} = 0.2882 .$

Step 2: Utilize Equation (16) to obtain the synthetic linguistic preference relation $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ based on the RCD-ILOWA operator: $\tilde{A} = (\begin{matrix} s_{0} & s_{1.9297} & s_{- 1.9349} & s_{2.0703} & s_{- 1.9297} \\ s_{- 1.9297} & s_{0} & s_{3.7118} & s_{1.6415} & s_{- 1.6467} \\ s_{1.9349} & s_{- 3.7118} & s_{0} & s_{2.6415} & s_{2.0703} \\ s_{- 2.0703} & s_{- 1.6415} & s_{- 2.6415} & s_{0} & s_{4} \\ s_{1.9297} & s_{1.6467} & s_{- 2.0703} & s_{- 4} & s_{0} \end{matrix}) .$

Step 3: Calculate the expected value ${\tilde{a}}_{i}$ of preference degree of the alternative x _i to all the alternative by the linguistic averaging (LA) operator: $\begin{matrix} {\tilde{a}}_{1} = s_{0.0271}, {\tilde{a}}_{2} = s_{0.3554}, {\tilde{a}}_{3} = s_{0.5870}, \\ {\tilde{a}}_{4} = s_{- 0.4707}, {\tilde{a}}_{5} = s_{- 0.4988} . \end{matrix}$

Step 4: Rank the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, 5) in descending order: ${\tilde{a}}_{3} > {\tilde{a}}_{2} > {\tilde{a}}_{1} > {\tilde{a}}_{4} > {\tilde{a}}_{5} .$

Step 5: Rank all the alternatives x _i: $x_{3} ≻ x_{2} ≻ x_{1} ≻ x_{4} ≻ x_{5} .$

4.2 The approach based on the CI-ILOWA operator

Suppose that there is another decision maker who is a leading decision maker. The additive linguistic preference relation $\tilde{B}$ is given by the leading decision maker. It is listed as follows: $\tilde{B} = (\begin{matrix} s_{0} & s_{2} & s_{- 2} & s_{2} & s_{- 2} \\ s_{- 2} & s_{0} & s_{4} & s_{2} & s_{- 2} \\ s_{2} & s_{- 4} & s_{0} & s_{2} & s_{2} \\ s_{- 2} & s_{- 2} & s_{- 2} & s_{0} & s_{4} \\ s_{2} & s_{2} & s_{- 2} & s_{- 4} & s_{0} \end{matrix}) .$

Step 1: Utilize the model (18) to calculate the compatibility index of e _k: $CI ({\tilde{A}}^{(1)}, \tilde{B}) = 0.05$ , $CI ({\tilde{A}}^{(2)}, \tilde{B}) = 0.03, CI ({\tilde{A}}^{(3)}, \tilde{B}) = 0.07 .$

Step 2: Utilize the model Equation (26) to determine the optimal weights of experts: $ω^{*} = (0.2405, 0.4177, 0.3418)^{T} .$

Step 3: Utilize Equation (20) to obtain the synthetic linguistic preference relation $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ based on the CI-ILOWA operator: $\tilde{A} = (\begin{matrix} s_{0} & s_{1.9241} & s_{- 2.1013} & s_{2.0759} & s_{- 1.9241} \\ s_{- 1.9241} & s_{0} & s_{3.6582} & s_{1.5823} & s_{- 1.7595} \\ s_{2.1013} & s_{- 3.6582} & s_{0} & s_{2.5823} & s_{2.0759} \\ s_{- 2.0759} & s_{- 1.5823} & s_{- 2.5823} & s_{0} & s_{4} \\ s_{1.9241} & s_{1.7595} & s_{- 2.0759} & s_{- 4} & s_{0} \end{matrix}) .$

Step 4: Calculate the expected value ${\tilde{a}}_{i}$ of preference degree of the alternative x _i to all the alternative by the LA operator: $\begin{matrix} {\tilde{a}}_{1} = s_{- 0.0051}, {\tilde{a}}_{2} = s_{0.3114}, {\tilde{a}}_{3} = s_{0.6203}, \\ {\tilde{a}}_{4} = s_{- 0.4481}, {\tilde{a}}_{5} = s_{- 0.4785} . \end{matrix}$

Step 5: Rank the expected value ${\tilde{a}}_{i}$ (i = 1, 2, …, n) in descending order: ${\tilde{a}}_{3} > {\tilde{a}}_{2} > {\tilde{a}}_{1} > {\tilde{a}}_{4} > {\tilde{a}}_{5} .$

Step 6: Rank all the alternatives x _i: $x_{3} ≻ x_{2} ≻ x_{1} ≻ x_{4} ≻ x_{5} .$

As we can see, the decision of approach with CI-ILOWA operator is the same as the approach with RCD-ILOWA operator, which means that they can be applied to GDM problems efficiently. However, they are different in application to GDM. The approach based on the RCD-ILOWA operator is based on the consistency and consensus measure, in which the relative consensus degrees are obtained by constructing the optimal transfer matrices. The approach based on the CI-ILOWA operator is on the basis of the compatibility measure, in which compatibility index can be reached by using the ideal preference relation. In practical GDM problem, if the ideal preference relation is unknown, then decision maker can choose the approach with RCD-ILOWA, otherwise, CI-ILOWA is also a good choice.

5 Concluding remarks

In this paper, we have developed the RCD-ILOWA operator and the CI-ILOWA operator based on the consistency and compatibility of additive linguistic preference relations. We have investigated some desirable properties of the consistency and compatibility index of additive linguistic preference relations. We further have proposed an optimal model to determine the experts’ weights by minimizing the compatibility index in GDM and two approaches to GDM problem with additive linguistic preference relations based on the RCD-ILOWA operator and the CI-ILOWA operator. We have also applied the proposed approaches to GDM problem of evaluating university. In practical GDM problem, if the ideal preference relation is available, then the approach with CI-ILOWA operator is appreciate, otherwise, the approach with RCD-ILOWA operator can be used effectively.

Footnotes

Acknowledgments

The authors are very grateful to the Area Editor Professor J.M. Merigó and the anonymous referees for their insightful comments and suggestions. The work was supported by National Natural ScienceFoundation of China (Nos. 71301001, 71371011, 11426033), Project of Anhui Province for Excellent Young Talents in Universities, Higher School Specialized Research Fund for the Doctoral Program (No. 20123401110001), Anhui Provincial Natural Science Foundation (No. 1308085QG127), Humanity and Social Science Youth Foundation of Ministry of Education (No. 13YJC630092), Anhui Provincial Philosophy and Social Science Planning Youth Foundation (No. AHSKQ2014D13), Project of Anhui Province for Excellent Young Talents, The Doctoral Scientific Research Foundation of Anhui University.

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