Intuitionistic multiplicative preference relation and its application in group decision making

1 Introduction

Preference relation is one of the most crucial tools used by experts in decision making problems, because it can express their information about the alternatives or criteria. In the past decades, a plenty of efforts have been done about preference relations. Among these efforts, the fuzzy preference relations [15] and the multiplicative preference relations [18] are two types of preference relations which are widely studied by many researchers. In [22, 26], some methods to obtain the priorities of preference relations were proposed. The consistency, a vital property of preference relations, has drawn much attention in the decision making field and many related researches have been accomplished in [2, 10]. Considerable works have been done in [7, 27] for studying the consensus of a group of preference relations. The above two preference relations are different. The fuzzy preference relations apply the 0.1–0.9 scale, which is a symmetrical distribution around 0.5 to express the experts’ preference information about the alternatives or criteria. However, the multiplicative preference relations use the Saaty’s 1–9 scale, which is a non-symmetrical distribution around 1.

Due to the increasing complexity of the socio-economic environment and the lack of knowledge or data about the problem domain, decision makers have great difficulties in using the exact values to express their preference information about the alternatives or criteria. In order to deal with such cases, the experts provided the interval-valued fuzzy preference relations [17, 30] and the interval-valued multiplicative preference relations [3,8,24, 3,8,24] by using the interval-valued numbers to express their preference information.

The basic element in the above preference relations only provides the degree that an alternative is prior to the other. However, decision makers may also need to provide the degree that an alternative is not prior to the other in some practical problems. To overcome this issue, in [1, 32] the intuitionistic fuzzy preference relation consisted of the membership functions and the non-membership functions was defined based on the intuitionistic fuzzy set [11], which have more advantages in dealing with the inevitably imprecise or not totally reliable judgment than the usual fuzzy set [12].

It is noted that the interval-valued fuzzy preference relation and the intuitionistic fuzzy preference relation use the 0.1–0.9 scale, which is a balanced scale to express the preference information. But in real life, the information are usually distributed asymmetrically, and the law of diminishing marginal utility is the common phenomenon in economics even in our dailylife.

For above mentioned reasons, Xia et al. [13] used the 1–9 scale instead of the 0.1–0.9 scale in the intuitionistic fuzzy preference relation and first gave the concept of intuitionistic multiplicative preference relation and proposed some corresponding aggregation techniques, which can avoid some disadvantages of the intuitionistic fuzzy aggregation operators. Based on the above theory, Xia and Xu [14] proposed some extended operations about the intuitionistic multiplicative information, and then gave a relevant method to deal with the group decision making based on intuitionistic multiplicative preference relations. However, the operational laws given in [13] and [14] can not guarantee closedness. In order to overcome this problem, some new operational laws of intuitionistic multiplicative numbers are developed in this paper. Based on these operational laws, some operators are proposed to aggregate the intuitionistic multiplicative preference information, and properties of the operator are investigated. Finally, these results are applied to group decision making problems based on intuitionistic multiplicative preference information.

The remainder of the paper is organized as follows: Section 2 describes some basic concepts regarding the intuitionistic multiplicative set and intuitionistic multiplicative preference relations. In Section 3, some new operational laws of intuitionistic multiplicative numbers are introduced. Based on these operational laws, Section 4 provides some aggregation operators and discusses some properties of aggregation operators. Section 5 uses the new aggregation operators to deal with the group decision making under intuitionistic multiplicative preference relations. At last, a short conclusion is given in Section 6.

2 Preliminaries

Xia et al. [13] first gave the concept of intuitionistic multiplicative preference relation using the 1-9 scale to replace the 0.1–0.9 scale of an intuituionistic fuzzy preference relation to express the preference information. Some basic definitions of intuitionistic multiplicative set and intuitionistic multiplicative preference relations are given as follows:

Definition 2.1. Let X be a fixed set. An intuitionistic multiplicative set (IMS) is defined as: D = { < x , ρ ( x ) , σ ( x ) | x ∈ X } ,(1) which assigns each element x a membership information ρ (x) and a non-membership information σ (x), with the conditions: 0 < ρ ( x ) σ ( x ) ≤ 1 , 1 q ≤ ρ ( x ) , σ ( x ) ≤ q ,(2) where q > 1.

For convenience, let the pair (ρ (x) , σ (x)) be an intuitionistic multiplicative number (IMN) and M be the set of all IMNs.

Definition 2.2. [14] Let X = {x₁, x₂, …, x _n } be n alternatives. Then, the intuitionistic multiplicative preference relation is defined as A = (α _ij ) _n×n, where α _ij  = (ρ _{α ij} , σ _{α ij} ) is an intuitionistic multiplicative number (IMN), ρ _{α ij} can be considered as the intensity degree that x _i is preferred to x _j , σ _{α ij} can be considered as the intensity degree that x _i is not preferred to x _j , and both of them should satisfy the following conditions ρ α ij = σ α ji , σ α ij = ρ α ji , and 0 < ρ α ij σ α ij ≤ 1 , 1 q ≤ ρ α ij , σ α ij ≤ q , q > 1 .

If q = 9, then Definition 2.1 and Definition 2.2 reduce to Definition 2 and Definition 3 given in [13], respectively.

It can be seen that the basic element of an intuitionistic multiplicative preference relation is the IMN. To compare two IMNs, Xia et al. [13] gave the comparison laws as follows.

Definition 2.3. [13] For an IMN α = (ρ _α, σ _α), s (α) = ρ _α/σ _α is called the score function of α, and h (α) = ρ _ασ _α the accuracy function of α. The comparison laws are given as follows:

If s (α₁) > s (α₂), then α₁ > α₂;

In [13] and [14], Xia and Xu proposed the following operational laws of IMNs.

Definition 2.4. [13] Let α₁ = (ρ _α1, σ _α1), α₂ = (ρ _α2, σ _α2) and α = (ρ _α, σ _α) be three IMNs, and λ > 0.Then 1 ) α 1 ⊕ α 2 = ( ( 1 + 2 ρ α 1 ) ( 1 + 2 ρ α 2 ) - 1 2 , 2 σ α 1 σ α 2 ( 2 + σ α 1 ) ( 2 + σ α 2 ) - σ α 1 σ α 2 ) ; 2 ) α 1 ⊗ α 2 = ( 2 ρ α 1 ρ α 2 ( 2 + ρ α 1 ) ( 2 + ρ α 2 ) - ρ α 1 ρ α 2 , ( 1 + 2 σ α 1 ) ( 1 + 2 σ α 2 ) - 1 2 ) ; 3 ) λ α = ( ( 1 + 2 ρ α ) λ - 1 2 , 2 σ α λ ( 2 + σ α ) λ - σ α λ ) ; 4 ) α λ = ( 2 ρ α λ ( 2 + ρ α ) λ - ρ α λ , ( 1 + 2 σ α ) λ - 1 2 ) .

Definition 2.5. [14] Let α₁ = (ρ _α1, σ _α1), α₂ = (ρ _α2, σ _α2) and α = (ρ _α, σ _α) be three IMNs, λ > 0 and γ > 0. Then, 1 ) α 1 ⊕ α 2 = ( ( 1 + γ ρ α 1 ) ( 1 + γ ρ α 2 ) - 1 γ , γ σ α 1 σ α 2 ( γ + σ α 1 ) ( γ + σ α 2 ) - σ α 1 σ α 2 ) ; 2 ) α 1 ⊗ α 2 = ( γ ρ α 1 ρ α 2 ( γ + ρ α 1 ) ( γ + ρ α 2 ) - ρ α 1 ρ α 2 , ( 1 + γ σ α 1 ) ( 1 + γ σ α 2 ) - 1 γ ) ; 3 ) λ α = ( ( 1 + γ ρ α ) λ - 1 γ , γ σ α λ ( γ + σ α ) λ - σ α λ ) ; 4 ) α λ = ( γ ρ α λ ( γ + ρ α ) λ - ρ α λ , ( 1 + γ σ α ) λ - 1 γ ) .

However, Definition 2.4 and Definition 2.5 can not guarantee closedness of operation. For example, let α 1 = ( 7 , 1 8 ) and α 2 = ( 6 , 1 7 ) be two IMNs with q = 9. Then α 1 ⊕ α 2 = ( 97 , 1 127 ) and α 1 ⊕ α 2 = ( 42 γ + 13 , 1 56 γ + 15 ) are obtained by Definition 2.4 and Definition 2.5, respectively. It can be seen that α₁ ⊕ α₂ is not an IMN with q = 9. In order to overcome this shortcoming, some new operational laws of IMNs are presented in next section.

The new operational laws of IMNs are expressed as follows:

Definition 3.1. Let α₁ = (ρ _α1, σ _α1), α₂ = (ρ _α2, σ _α2) and α = (ρ _α, σ _α) be three IMNs, and λ > 0. Then 1 ) α 1 ⊕ α 2 = ( q log q ρ α 1 ρ α 2 - log q ρ α 1 log q ρ α 2 + 1 2 , q log q σ α 1 σ α 2 + log q σ α 1 log q σ α 2 - 1 2 ) ; 2 ) α 1 ⊗ α 2 = ( q log q ρ α 1 ρ α 2 + log q ρ α 1 log q ρ α 2 - 1 2 , q log q σ α 1 σ α 2 - log q σ α 1 log q σ α 2 + 1 2 ) ; 3 ) λ α = ( q 1 - 2 ( 1 - log q ρ α 2 ) λ , q 2 ( 1 + log q σ α 2 ) λ - 1 ) ; 4 ) α λ = ( q 2 ( 1 + log q ρ α 2 ) λ - 1 , q 1 - 2 ( 1 - log q σ α 2 ) λ ) .

Theorem 3.1. Let α₁ = (ρ _α1, σ _α1), α₂ = (ρ _α2, σ _α2) and α = (ρ _α, σ _α) be three IMNs. Then α₁ ⊕ α₂, α₁ ⊗ α₂, λα, and α ^λ  (λ > 0) are also IMNs.

Proof. Sine 1 q ≤ ρ α i, σ _{α i}  ≤ q (i = 1, 2), then 0 ≤ log q q ρ α i ≤ 2 and 0 ≤ log  _q qσ _{α i}  ≤ 2 (i = 1, 2), which imply that - 1 ≤ 2 - log q q ρ α 1 · log q q ρ α 2 2 ≤ 1 ,(3)- 1 ≤ log q q σ α 1 · log q q σ α 2 - 2 2 ≤ 1 .(4)

From Equations (3) and (4), one has 1 q ≤ ρ α 1 ⊕ α 2 = q log q ρ α 1 ρ α 2 - log q ρ α 1 · log q ρ α 2 + 1 2 = q 2 - log q q ρ α 1 · log q q ρ α 2 2 ≤ q , 1 q ≤ σ α 1 ⊕ α 2 = q log q σ α 1 σ α 2 + log q σ α 1 · log q σ α 2 - 1 2 = q log q q σ α 1 · log q q σ α 2 - 2 2 ≤ q ,

By Definition 2.1, for any IMN α = (ρ _α, σ _α), one has log q q σ α + log q ρ α q = log q σ α ρ α ≤ 0 .

Thus, one has 0 ≤ log q q σ α ≤ log q q ρ α .(5)

Using Equation (5), one has 0 ≤ ρ α 1 ⊕ α 2 · σ α 1 ⊕ α 2 = q 2 - log q q ρ α 1 · log q q ρ α 2 2 · q log q q σ α 1 · log q q σ α 2 - 2 2 = q log q q σ α 1 · log q q σ α 2 - log q q ρ α 1 · log q q ρ α 2 2 ≤ 1 .

Therefore, α₁ ⊕ α₂ is an IMN. Similarly, it can also be obtained that α₁ ⊗ α₂ is an IMN.

Since 1 q ≤ ρ α, σ _α ≤ q, then 0 ≤ log q q ρ α ≤ 2 and 0 ≤ log  _q qσ _α ≤ 2. For λ > 0, one has - 1 ≤ 1 - 2 ( log q q ρ α 2 ) λ ≤ 1 ,(6) and - 1 ≤ 2 ( log q q σ α 2 ) λ - 1 ≤ 1 .(7)

Using Equations (6) and (7), one has 1 q ≤ ρ λ α = q 1 - 2 ( 1 - log q ρ α 2 ) λ = q 1 - 2 ( log q q ρ α 2 ) λ ≤ q ,

1 q ≤ σ λ α = q 2 ( 1 + log q σ α 2 ) λ - 1 = q 2 ( log q q σ α 2 ) λ - 1 ≤ q ,

By Equation (5), one has ρ λ α · σ λ α = q 1 - 2 ( 1 - log q ρ α 2 ) λ · q 2 ( 1 + log q σ α 2 ) λ - 1 = q 2 ( log q q σ α 2 ) λ - 2 ( log q q ρ α 2 ) λ ≤ 1 .

Therefore, λα is an IMN. Similarly, it can also be obtained that α ^λ is an IMN. This completes the proof.

In this section, some aggregation operators are proposed based on the operational laws defined in Definition 3.1 to aggregate the intuitionistic multiplicative information.

Definition 4.1. Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs. An intuitionistic multiplicative weighted averaging (IMWA) operator is a mapping M ⁿ  → M, such that IMWA ( α 1 , α 2 , … , α n ) = oplus i = 1 n ( ω i α i ) ,(8) where ω = (ω₁, ω₂, …, ω _n )  ^T is the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1.

Particularly, if ω 1 = ω 2 = ⋯ = ω n = 1 n, then the IMWA operator reduces to the intuitionistic multiplicative averaging (IMA) operator: IMA ( α 1 , α 2 , … , α n ) = 1 n oplus i = 1 n α i .(9)

Definition 4.2. Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs. An intuitionistic multiplicative weighted geometric (IMWG)operator is a mapping M ⁿ  → M, such that IMWG ( α 1 , α 2 , … , α n ) = ⨂ i = 1 n α i ω i ,(10) where ω = (ω₁, ω₂, …, ω _n )  ^T is the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1.

In the case where ω 1 = ω 2 = ⋯ = ω n = 1 n, the IMWG operator reduces to the intuitionistic multiplicative geometric (IMG) operator: IMG ( α 1 , α 2 , … , α n ) = ⨂ i = 1 n α i 1 n .(11)

Based on Definition 3.1 and Definition 4.1, the following theorems can be obtained.

Theorem 4.1.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and ω = (ω₁, ω₂, …, ω _n )  ^T be the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1. Then

IMWA ( α 1 , α 2 , … , α n ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 )(12)

Proof. By using mathematical induction on n: for n = 2, one has IMWA ( α 1 , α 2 ) = ω 1 α 1 ⊕ ω 2 α 2 = ( q 1 - 2 ( 1 - log q ρ α 1 2 ) ω 1 , q 2 ( 1 + log q σ α 1 2 ) ω 1 - 1 ) ⊕ ( q 1 - 2 ( 1 - log q ρ α 2 2 ) ω 2 , q 2 ( 1 + log q σ α 2 2 ) ω 2 - 1 ) = ( q 1 - 2 ( 1 - log q ρ α 1 2 ) ω 1 ( 1 - log q ρ α 2 2 ) ω 2 , q 2 ( 1 + log q σ α 1 2 ) ω 1 ( 1 + log q σ α 2 2 ) ω 2 - 1 )

Suppose Equation (12) holds for n = k, that is IMWA ( α 1 , α 2 , … , α k ) = oplus i = 1 k ω i α i = ( q 1 - 2 ∏ i = 1 k ( 1 - log q ρ α i 2 ) ω i , q 2 ∏ i = 1 k ( 1 + log q σ α i 2 ) ω i - 1 ) , then, when n = k + 1, by the operational laws inDefinition 3.1, one has IMWA ( α 1 , α 2 , … , α k + 1 ) = oplus i = 1 k + 1 ω i α i = ( q 1 - 2 ∏ i = 1 k ( 1 - log q ρ α i 2 ) ω i , q 2 ∏ i = 1 k ( 1 + log q σ α i 2 ) ω i - 1 ) ⊕ ( q 1 - 2 ( 1 - log q ρ α k + 1 2 ) ω k + 1 , q 2 ( 1 + log q σ α k + 1 2 ) ω k + 1 - 1 ) = ( q 1 - 2 ∏ i = 1 k ( 1 - log q ρ α i 2 ) ω i ( 1 - log q ρ α k + 1 2 ) ω k + 1 , q 2 ∏ i = 1 k ( 1 + log q σ α i 2 ) ω i ( 1 + log q σ α k + 1 2 ) ω k + 1 ) = ( q 1 - 2 ∏ i = 1 k + 1 ( 1 - log q ρ α i 2 ) ω i , q 2 ∏ i = 1 k + 1 ( 1 + log q σ α i 2 ) ω i - 1 ) , that is, Equation (12) holds for n = k + 1. Therefore, Equation (12) holds for all n, which completes the proof of the theorem.

In the same way, the following theorem is easily obtained.

Theorem 4.2.Let α _i  = (ρ _{α i} , σ _{α i} ) be a collection of IMNs, and ω = (ω₁, ω₂, …, ω _n )  ^T be the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1. Then

IMWG ( α 1 , α 2 , … , α n ) = ( q 2 ∏ i = 1 n ( 1 + log q ρ α i 2 ) ω i - 1 , q 1 - 2 ∏ i = 1 n ( 1 - log q σ α i 2 ) ω i ) .(13)

Now, taking the IMWA operator as an example, some desirable properties of the developed operators are investigated.

Property 4.1.Let α _i  (i = 1, 2, …, n) be a collection of IMNs. If all α _i are equal, i.e., α₁ = α₂ = ⋯ = α _n  = α = (ρ _α, σ _α). Then IMWA ( α 1 , α 2 , … , α n ) = oplus i = 1 n ω i α = α ,(14)which is called the idempotency.

Proof. By Definition 3.1, one has IMWA ( α 1 , α 2 , … , α n ) = oplus i = 1 n ω i α = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α 2 ) ω i , q 2 ∏ i = 1 n ( 1 + log q σ α 2 ) ω i - 1 ) = ( q log q ρ α , q log q σ α ) = ( ρ α , σ α ) = α .

The proof is completed.

Property 4.2.Let α _i  = (ρ _{α i} , σ _{α i} ) and α i * = ( ρ α i * , σ α i * ) ( i = 1 , 2 , … , n ) be two collections of IMNs. If ρ α i ≤ ρ α i * and σ α i ≥ σ α i *, for all i. Then

IMWA ( α 1 , α 2 , … , α n ) ≤ IMWA ( α 1 * , α 2 * , … , α n * )(15) which is called the monotonicity.

Proof. Since ρ α i ≤ ρ α i * and σ α i ≥ σ α i *, for all i,one has q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ≤ q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i * 2 ) ω i , and q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 ≥ q 2 ∏ i = 1 n ( 1 + log q σ α i * 2 ) ω i - 1 .

Therefore, by Definition 2.3, one has IMWA ( α 1 , α 2 , … , α n ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 ) ≤ ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i * 2 ) ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i * 2 ) ω i - 1 ) = IMWA ( α 1 * , α 2 * , … , α n * ) .

The proof is completed.

Based on the monotonicity, the following property can be obtained.

Property 4.3.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, α - = ( min i { ρ α i } , max i { σ α i } ), and α + = ( max i { ρ α i } , min i { σ α i } ). Then α - ≤ IMWA ( α 1 , α 2 , … , α n ) ≤ α + .(16)which is called the boundedness.

Property 4.4.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and β = (ρ _β , σ _β ) be an IMN,then

IMWA ( α 1 ⊕ β , α 2 ⊕ β , … , α n ⊕ β ) = IMWA ( α 1 , α 2 , … , α 2 ) ⊕ β .(17)

Proof. By Definition 3.1, one has α i ⊕ β = ( q log q ρ α i ρ β - log q ρ α i log q ρ β + 1 2 , q log q σ α i σ β + log q σ α i log q σ β - 1 2 ) .

Thus IMWA ( α 1 ⊕ β , α 2 ⊕ β , … , α n ⊕ β ) = ( q 1 - 2 ∏ i = 1 n ( 1 + log q ρ α i log q ρ β - log q ρ α i ρ β 4 ) ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i log q σ β + log q σ α i σ β 4 ) ω i - 1 ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ( 1 - log q ρ α β 2 ) , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i ( 1 + log q σ α β 2 ) - 1 ) . IMWA ( α 1 , α 2 , … , α 2 ) ⊕ β = ( q 2 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ( 1 - log q ρ β ) 2 , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i ( 1 + log q σ β ) - 2 2 ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ( 1 - log q ρ α β 2 ) , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i ( 1 + log q σ α β 2 ) - 1 ) .

The proof is completed.

Property 4.5.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs. If r > 0, then

IMWA ( r α 1 , r α 2 , … , r α n ) = r IMWA ( α 1 , α 2 , … , α 2 ) .(18)

Proof. According to Definition 3.1 and Theorem 4.1, one has IMWA ( r α 1 , r α 2 , … , r α n ) = ( q 1 - 2 ∏ i = 1 n ( 1 - 1 + 2 ( 1 - log q ρ α i 2 ) r 2 ) ω i , q 2 ∏ i = 1 n ( 1 + 2 ( 1 + log q σ α i 2 ) r - 1 2 ) ω i - 1 ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) r ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) r ω i - 1 ) , and r IMWA ( α 1 , α 2 , … , α n ) = ( q 1 - 2 ( 1 - 1 + 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i 2 ) r , q 2 ( 1 + 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 2 ) r - 1 ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) r ω i , q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) r ω i - 1 ) ,

This completes the proof.

According to Properties 4.4 and 4.5, it is easy to get the property 4.6.

Property 4.6.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and β = (ρ _β , σ _β ) be an IMN. Ifr > 0, then

IMWA ( r α 1 ⊕ β , r α 2 ⊕ β , ⋯ , r α n ⊕ β ) = r IMWA ( α 1 , α 2 , … , α n ) ⊕ β(19)

Property 4.7.Let α _i  = (ρ _{α i} , σ _{α i} ) and β _i  = (ρ _{β i} , σ _{β i} ) (i = 1, 2, …, n) be two collections of IMNs. Then,

IMWA ( α 1 ⊕ β 1 , α 2 ⊕ β 2 , … , α n ⊕ β n ) = IMWA ( α 1 α 2 , … , α n ) ⊕ IMWA ( β 1 β 2 , … , β n ) .(20)

Proof. Similar to the proof of Property 4.4, the proof can be completed.

Property 4.8.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and ω = (ω₁, ω₂, …, ω _n )  ^T be the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1. Then

IMWA ( α 1 α 2 , … , α n ) ≥ ( ∏ i = 1 n ρ α i ω i , ∏ i = 1 n σ α i ω i ) ,(21) and the equality holds if and only if α₁ = α₂ = ⋯ = α _n .

Proof. Suppose ρ _{α i}  < q, for all i, then log q q ρ α i > 0, one has 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i = ∏ i = 1 n ( log q q ρ α i ) ω i = e ∑ i = 1 n ω i ln log q q ρ α i .

According to the definition of the convex function, one has e ∑ i = 1 n ω i ln log q q ρ α i ≤ ∑ i = 1 n ω i e ln log q q ρ α i = ∑ i = 1 n ω i log q q ρ α i = 1 - ∑ i = 1 n log q ρ α i ω i , and the equality holds if and only if ρ _α1 = ρ _α2 = ⋯ = ρ _{α n} .

Then, one has q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ≥ q ∑ i = 1 n log q ρ α i ω i = ∏ i = 1 n ρ α i ω i , and the equality holds if and only if ρ _α1 = ρ _α2 = ⋯ = ρ _{α n} .

Suppose that there exists i ∈ {1, 2, …, n} such that ρ _{α i}  = q, one has q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i = q ≥ ∏ i = 1 n ρ α i ω i , and the equality holds if and only if ρ _α1 = ρ _α2 = ⋯ = ρ _{α n} .

Similarly, the following inequality can be obtained q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 ≤ ∏ i = 1 n σ α i ω i , and the equality holds if and only if σ _α1 = σ _α2 = ⋯ = σ _{α n} .

By Theorem 4.1 and Property 4.2, one has

IMWA ( α 1 α 2 , … , α n ) ≥ ( ∏ i = 1 n ρ α i ω i , ∏ i = 1 n σ α i ω i ) , and the equality holds if and only if α₁ = α₂ = ⋯ = α _n . This completes the proof.

Property 4.9.Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and ω = (ω₁, ω₂, …, ω _n )  ^T be the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1. Then

IMWA ( α 1 α 2 , … , α n ) ≥ IMWG ( α 1 α 2 , … , α n ) ,(22) and the equality holds if and only if α₁ = α₂ = ⋯ = α _n .

Proof. Similar to the proof of Property 4.8, one has IMWG ( α 1 α 2 , … , α n ) ≤ ( ∏ i = 1 n ρ α i ω i , ∏ i = 1 n σ α i ω i ) , and the equality holds if and only if α₁ = α₂ = ⋯ = α _n .

By Equation (21), one has IMWA ( α 1 α 2 , … , α n ) ≥ IMWG ( α 1 α 2 , … , α n ) , and the equality holds if and only if α₁ = α₂ = ⋯ = α _n . This completes the proof.

Property 4.10. Let α _i  = (ρ _{α i} , σ _{α i} ) (i = 1, 2, …, n) be a collection of IMNs, and ω = (ω₁, ω₂, …, ω _n )  ^T be the weighting vector of α _i with ω _i  ∈ [0, 1] and ∑ i = 1 n ω i = 1. Then 1 ) IMWA ( α 1 c α 2 c , … , α n c ) = ( IMWG ( α 1 α 2 , … , α n ) ) c(23)2 ) ( IMWA ( α 1 α 2 , … , α n ) ) c = IMWG ( α 1 c α 2 c , … , α n c ) .(24) where α i c = ( σ α i , ρ α i ) ( i = 1 , 2 , … , n ) denotes the complement of α _i .

Proof. From Theorem 4.1 and Theorem 4.2, one has IMWA ( α 1 c α 2 c , … , α n c ) = ( q 1 - 2 ∏ i = 1 n ( 1 - log q σ α i 2 ) ω i , q 2 ∏ i = 1 n ( 1 + log q ρ α i 2 ) ω i - 1 ) = ( IMWG ( α 1 α 2 , … , α n ) ) c , ( IMWA ( α 1 α 2 , … , α n ) ) c = ( q 2 ∏ i = 1 n ( 1 + log q σ α i 2 ) ω i - 1 , q 1 - 2 ∏ i = 1 n ( 1 - log q ρ α i 2 ) ω i ) = IMWG ( α 1 c α 2 c , … , α n c ) .

This completes the proof.

Let X = {x₁, x₂, …, x _n } (n ≥ 2) be a set of alternatives in a decision making problem, and E = {e₁, e₂, …, e _p } be a set of p decision makers. The decision maker e _k  (k = 1, 2, …, p) uses the Saaty’s 1-9 scale to provide his/her preference about the alternatives x _i and x _j in the set X, denoted by IMN α ij ( k ) = ( ρ α ij ( k ) , σ α ij ( k ) ), with the conditions ρ _{α ij}  = σ _{α ji} , σ _{α ij}  = ρ _{α ji} , ρ _{α ij} σ _{α ij}  ≤ 1, 1 q ≤ ρ α ij ≤ q, 1 q ≤ σ α ij ≤ q and q > 1.ρ α ij ( k ) indicates the intensity degree which the alternative x _i is preferred to x _j . σ α ij ( k ) indicates the intensity degree which the alternatives x _i is not preferred to x _j . If the decision maker e _k provides all the preferences for n alternatives, then the intuitionistic multiplicative fuzzy preference relation A ( k ) = ( α ij ( k ) ) n × n can be obtained. To get the best alternative, a detailed procedure is given as follows:

Step 1. Utilize the IMWA or IMWG operator to aggregate the preference values α ij ( k ) into the preference values α i ( k ) of the alternative x _i for the decision maker e _k .

Step 2. Utilize intuitionistic multiplicative weighted averaging (IMWA) operator α i = IMWA ( α i ( 1 ) , α i ( 2 ) , … , α i ( p ) ) = oplus i = 1 p ( λ k α i ( k ) ) to aggregate the decision maker’s preference values α i ( k ) ( i = 1 , 2 , … , n ; k = 1 , 2 , … , p ) into the group ones α _i , where λ _k are the weights of the decision maker e _k  (k = 1, 2, …, p).

Step 3. Calculate the score function s (α _i ) and the accuracy function h (α _i ) of α _i by Definition 2.3.

Step 4. Rank s (α _i ) and h (α _i ) (i = 1, 2, …, n) to get the priority of alternatives x _i .

Next, an example is used to illustrate the developed method.

Example 5.1. Suppose that there is a group decision making problem involving the evaluation of four branch offices X = {x₁, x₂, x₃, x₄} in a company. An expert group is formed which consists of four decision makers e _k  (k = 1, 2, 3, 4) (whose weight vector is λ = (0.2, 0.1, 0.3, 0.4)) from each strategic decision area. These decision makers e _k  (k = 1, 2, 3, 4) provide their intuitionistic multiplicative preference relations A ( k ) = ( α ij ( k ) ) ( k = 1 , 2 , 3 , 4 ) ( here q = 9) over alternatives x _i  (i = 1, 2, 3, 4), respectively, as follows: A ( 1 ) = ( ( 1 , 1 ) ( 2 , 2 5 ) ( 1 4 , 5 3 ) ( 1 2 , 2 ) ( 2 5 , 2 ) ( 1 , 1 ) ( 2 , 1 3 ) ( 8 7 , 1 3 ) ( 5 3 , 1 4 ) ( 1 3 , 2 ) ( 1 , 1 ) ( 5 , 1 7 ) ( 2 , 1 2 ) ( 1 3 , 8 7 ) ( 1 7 , 5 ) ( 1 , 1 ) ) A ( 2 ) = ( ( 1 , 1 ) ( 1 2 , 6 5 ) ( 1 , 2 3 ) ( 1 7 , 3 ) ( 6 5 , 1 2 ) ( 1 , 1 ) ( 2 , 1 5 ) ( 1 4 , 2 3 ) ( 2 3 , 1 ) ( 1 5 , 2 ) ( 1 , 1 ) ( 3 4 , 1 ) ( 3 , 1 7 ) ( 2 3 , 1 4 ) ( 1 , 3 4 ) ( 1 , 1 ) ) A ( 3 ) = ( ( 1 , 1 ) ( 1 3 , 3 2 ) ( 1 , 3 5 ) ( 1 3 , 3 ) ( 3 2 , 1 3 ) ( 1 , 1 ) ( 2 , 2 5 ) ( 1 , 6 7 ) ( 3 5 , 1 ) ( 2 5 , 2 ) ( 1 , 1 ) ( 1 3 , 2 ) ( 3 , 1 3 ) ( 6 7 , 1 ) ( 2 , 1 3 ) ( 1 , 1 ) ) A ( 4 ) = ( ( 1 , 1 ) ( 2 7 , 3 ) ( 2 5 , 2 ) ( 5 4 , 1 2 ) ( 3 , 2 7 ) ( 1 , 1 ) ( 4 , 1 6 ) ( 3 2 , 1 3 ) ( 2 , 2 5 ) ( 1 6 , 4 ) ( 1 , 1 ) ( 3 , 1 4 ) ( 1 2 , 5 4 ) ( 1 3 , 3 2 ) ( 1 4 , 3 ) ( 1 , 1 ) )