Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators

Abstract

On the basis of Frank operators, the operational rules of intuitionistic fuzzy numbers are redefined, then the intuitionistic fuzzy Frank power average operator (IFFPA), intuitionistic fuzzy Frank power weighted average operator (IFFPWA) and intuitionistic fuzzy Frank power ordered weighted average operator (IFFPOWA) are proposed by combining Frank operations and power aggregation operators. At the same time, some desirable properties of these operators, such as idempotency, commutativity and boundedness, are studied, and some special cases are analyzed. Furthermore, with respect to multiple attribute group decision making (MAGDM) problems in which attribute values take the form of the intuitionistic fuzzy numbers, two group decision making methods based on IFFPWA and IFFPOWA operators are developed. Finally, an illustrative example is given to verify the proposed methods and to demonstrate their practicality and effectiveness.

Keywords

Intuitionistic fuzzy number Frank operator power average operator multiple attribute group decision making

1 Introduction

Group decision theory and methods have wide application requirements in the political, economic, military, science, culture and other aspects. Because of fuzziness and complexity of decision making problems, for group decision making problems, decision makers are difficult to evaluate the alternatives by using real numbers, and in many cases, especially for some qualitative evaluation, it is more reasonable and natural to evaluate alternatives by utilizing linguistic information such as good, fair, poor or fuzzy concept such as slightly, obviously, mightily [7 , 38]. Since Zadeh [39] proposed the fuzzy set (FS) theory in 1965, it has gotten a rapid development and extensive application in fuzzy multi-attribute decision making problems [7]. Further, on the basis of the FS, the intuitionistic fuzzy set (IFS), which was developed by Atanassov [1, 2], has also been rapidly extended and obtained a wide range of attentions in multi-attribute decision making (MADM) and multi-attribute group decision making (MAGDM) [11–13 , 33]. Later, Atanassov and Gargov [5], Atanassov [3] further introduced the interval-valued intuitionistic fuzzy set (IVIFS),and Xu [30], Wang [22] developed the decision-making approaches based on IVIFS. Shu, Cheng and Chang [20] defined the intuitionistic triangular fuzzy number, and Zhang and Liu [40] defined the triangular intuitionistic fuzzy number, and they proposed the relevant decision making methods respectively. Wang [23] gave the conception of intuitionistic trapezoidal fuzzy number and interval intuitionistic trapezoidal fuzzy number, then some decision making methods based on the intuitionistic triangular fuzzy number had been proposed [14 , 16–18, 21]. Furthermore, Wang and Li [24] proposed intuitionistic linguistic sets, intuitionistic linguistic numbers, intuitionistic two-semantics and the Hamming distance between two intuitionistic two-semantics, and ranked the alternatives by calculating the comprehensive membership degree to the ideal solution for each alternative.

The information aggregation operators have become an important research topic and obtained a wide range of research results, which are still receiving more and more concerns [29 , 42]. Yager [35] developed ordered weighted averaging (OWA) operator, which weighted the inputs according to their ranking position; Xu [31], Xu and Yager [33] established the arithmetic weighted aggregation operators and geometric weighted aggregation operators based on intuitionistic fuzzy information; Zhao [41] proposed the generalized aggregation operators with respect to intuitionistic fuzzy information, which included the generalized intuitionistic fuzzy weighted aggregation (GIFWA) operator, the generalized intuitionistic fuzzy ordered weighted aggregation (GIFOWA) operator and the generalized intuitionistic fuzzy hybrid aggregation (GIFHA) operator, and proved that the arithmetic aggregation operators and geometric aggregation operators are the special cases of the generalized aggregation operators. In addition, because there are the relationships between the values being aggregated in some cases, Yager [36] proposed a power-average (PA) operator and a power OWA (POWA) operator, which weighting vectors depend on the input data and allow values being fused to support and to reinforce each other. Xu and Yager [34] developed a power-geometric (PG) operator, a power-ordered-geometric (POG) operator and a power-ordered-weighted-geometric (POWG) operator, and discussed some properties of these operators. Then, an uncertain PG (UPG) operator and its weighted form, and an uncertain power-ordered-weighted-geometric (UPOWG) operator are proposed to aggregate the input arguments which take the form of interval values, and the approaches to group decision making are developed based on these operators.

All above aggregation operators are based on the algebraic operational rules and the keys of the algebraic operations are algebraic product and algebraic sum, which are a special case of operations that can be chosen to model the intersection and union of intuitionistic fuzzy sets [28, 37]. In general, a general t-norm and t-conorm can be utilized to shape the intersection and union of intuitionistic fuzzy sets. As the only one type of t-norm which meets the compatibility, Frank operators involve a parameter, which can make the information fusion process more flexible and robust [6, 10]. However, the researches on the applications of Frank operators are rather rare, especially in information aggregation and decision making. Thus, motivated by the aforementioned analysis, we established new operational rules of intuitionistic fuzzy numbers on the basis of Frank operators, and prove their characteristics. Based on new operational laws and the ideas of power aggregation (PA) operator proposed by Yager [36], we propose intuitionistic fuzzy Frank power average (IFFPA) operator, intuitionistic fuzzy Frank power weighted average (IFFPWA) operator and intuitionistic fuzzy Frank power ordered weighted average (IFFPOWA) operator, respectively. At the same time, their properties and some special cases are explored. Further, we give two decision making methods for MAGDM problems.

2 Preliminaries

2.1 Intuitionistic fuzzy set (IFS)

Definition 1. [1] Let X be a universe of discourse, with a generic element in X denoted by x. An intuitionistic fuzzy set (IFS) A in X is given by $A = {< x, u_{A} (x), v_{A} (x) > x \in X}$ (1) where u _A (x) is the degree of membership value, and v _A (x) is the degree of non-membership value. For each point x in X, we have u _A (x) , v _A (x) ∈ [0, 1] and 0 ≤ u _A (x) + v _A (x) ≤1.

For each IFS A in X, let π (x) =1 - u _A (x) - v _A (x), ∀x ∈ X, and we call π (x) the indeterminacy degree of the element x to the set A [1, 2]. It can be easily proved that 0 ≤ π (x) ≤1, ∀x ∈ X.

To the given element x, the pair (u _A (x) , v _A (x)) is called an intuitionistic fuzzy number (IFN) [32]. For convenience, we use α = (u _α, v _α) to represent IFN, which meets u _α ∈ [0, 1], v _α ∈ [0, 1] and 0 ≤ u _α + v _α ≤ 1.

Definition 2. [4] Let α = (u _α, v _α) be an IFN, then the expected value of α is defined as follows: $E (α) = \frac{μ_{α} + 1 - ν_{α}}{2}$ (2)

Definition 3. Let α = (u _α, v _α) be an IFN, then the score function of α can be described as follows: $S (α) = (μ_{α} - ν_{α}) \times E (α)$ (3) where E (α) is the expected value of the IFN α.

Definition 4. Let α = (u _α, v _α) be an IFN, then the accuracy function of α can be described as follows: $H (α) = (μ_{α} + ν_{α}) \times E (α)$ (4) where E (α) is the expected value of the IFN α.

Definition 5. If α = (u _α, v _α) and β = (u _β, v _β) are any two IFNs, then

(1) If E (α) > E (β), then α > β

(2) If E (α) = E (β), then,

If S (α) > S (β), then α > β

(3) If S (α) = S (β), then

If H (α) > H (β), then α > β;

If H (α) = H (β), then α = β.

Definition 6. Let α = (u _α, v _α) and β = (u _β, v _β) be any two IFNs, then the normalized Hamming distance between α and β is defined as follows: $d (α, β) = \frac{| μ_{α} - μ_{β} | + | ν_{α} - ν_{β} |}{2}$ (5)

2.2 Frank operations

Frank operations involve the Frank product and Frank sum, which are special cases of t-norms and t-conorms, respectively.

Definition 7. Frank product ⊗_F is a t-norm and Frank sum ⊕_F is a t-conorm, they are defined as below [10]:

$\begin{matrix} a \oplus_{F} b = 1 - {log}_{λ} (1 + \frac{(λ^{1 - a} - 1) (λ^{1 - b} - 1)}{λ - 1}) \\ λ > 1, \forall (a, b) \in [0, 1]^{2} \\ a \otimes_{F} b = {log}_{λ} (1 + \frac{(λ^{a} - 1) (λ^{b} - 1)}{λ - 1}) \end{matrix}$ (6) $λ > 1, \forall (a, b) \in [0, 1]^{2}$ (7)

It has two interesting conclusions shown as follows [26]:

(1) If λ → 1, then a ⊕ _F b → a + b - ab, a ⊗ _F b → ab, the Frank product and Frank sum are reduced to the probability product and probability sum;

(2) If λ→ ∞, then a ⊕ _F b → min (a + b, 1), a ⊗ _F b → max (0, a + b - 1), the Frank product and Frank sum are reduced to the Lukasiewicz product and Lukasiewicz sum.

2.3 The power operators

Definition 8. [36] A power average (PA) operator, which is proposed by Yager [36], can be defined as follows: $PA (a_{1}, a_{2}, \dots, a_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (a_{i})) a_{i}}{\sum_{i = 1}^{n} (1 + T (a_{i}))}$ (8)

where $T (a_{i}) = {\sum_{j = 1}}_{j \neq i}^{n} Sup (a_{i}, a_{j}),$ and Sup (a _i, a _j) expresses the support for a _i from a _j, which must meet the following conditions: $(1) Sup (a_{i}, a_{j}) \in [0, 1];$ (9) $(2) Sup (a_{i}, a_{j}) = Sup (a_{j}, a_{i});$ (10) $(3) Sup (a_{i}, a_{j}) \geq Sup (a_{k}, a_{l}), if | a_{i} - a_{j} | < | a_{k} - a_{l} |$ (11) PA operator considers the relationship between the values being aggregated, and its weighting vector depends on the input arguments and makes values being integrated support and reinforces each other.

Definition 9. [36] A power ordered weighted average (POWA) operator, also defined by Yager [36], is shown as follows. $POWA (a_{1}, a_{2,} \dots, a_{n}) = \sum_{i = 1}^{n} u_{i} a_{σ (i)}$ (12) where $u_{i} = g (\frac{R_{i}}{TV}) - g (\frac{R_{i - 1}}{TV}), R_{i} = \sum_{j = 1}^{i} V_{σ (j)}$ g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function which meets the following properties: (1) g (0) =0; (2) g (1) =1; (3) g (x) ≥ g (y), if x > y. $TV = \sum_{i = 1}^{n} V_{σ (i)}, V_{σ (i)} = 1 + T (a_{σ (i)})$ (13)

T (a _σ(i)) indicates the support of the ith largest argument a _σ(i) by all the other arguments, and it can be calculated by the following formula. $T (a_{σ (i)}) = \sum_{\begin{array}{l} i, j = 1 \\ i, \neq j \end{array}}^{n} Sup (a_{σ (i)}, a_{σ (j)})$ (14)

Sup (a _σ(i), a _σ(j)) is used to express the support of jth largest argument a _σ(j) for the ith largest argument a _σ(i) in the formula (14).

When g (x) = x, POWA operator degrades into PA operator.

3 Frank operations of intuitionistic fuzzy numbers

In this section, we shall establish some essential algorithms of intuitionistic fuzzy numbers based on Frank operations.

Based on the T-norm and T-conorm, a generalized union and a generalized intersection of intuitionistic fuzzy sets were introduced by Deschrijver and Kerre [8].

Definition 10. [8] Let A and B be any two intuitionistic fuzzy sets, then, the generalized intersection and union are defined as follows:

$\begin{matrix} A \cap_{T, T^{*}} B & = {< x, T (u_{A} (x), u_{B} (x)), \\ T^{*} (v_{A} (x), v_{B} (x)) > x \in X} \end{matrix}$ (15)

$\begin{matrix} A \cup_{T, T^{*}} B & = {< x, T^{*} (u_{A} (x), u_{B} (x)), \\ T (v_{A} (x), v_{B} (x)) > x \in X} \end{matrix}$ (16) where T denotes a T-norm and T ^* a T-conorm.

Based on the Definition 10 and Frank product and Frank sum, we can establish the Frank operational laws with respect to IFNs.

Let α = (u _α, v _α) and β = (u _β, v _β) be any two IFNs and λ > 1, then the operational laws based on Frank product and Frank sum are defined as follows. $\begin{matrix} (1) α \oplus_{F} β & = (1 - {log}_{λ} (1 + \frac{(λ^{1 - μ_{α}} - 1) (λ^{1 - μ_{β}} - 1)}{λ - 1}), \\ {log}_{λ} (1 + \frac{(λ^{ν_{α}} - 1) (λ^{ν_{β}} - 1)}{λ - 1})) \end{matrix}$ (17) $\begin{matrix} (2) α \otimes_{F} β & = ({log}_{λ} (1 + \frac{(λ^{μ_{α}} - 1) (λ^{μ_{β}} - 1)}{λ - 1}), \\ 1 - {log}_{λ} (1 + \frac{(λ^{1 - ν_{α}} - 1) (λ^{1 - ν_{β}} - 1)}{λ - 1})) \end{matrix}$ (18) $\begin{matrix} (3) k \cdot_{F} α & = (1 - {log}_{λ} (1 + \frac{(λ^{1 - μ_{α}} - 1)^{k}}{(λ - 1)^{k - 1}}), \\ {log}_{λ} (1 + \frac{(λ^{ν_{α}} - 1)^{k}}{(λ - 1)^{k - 1}})), k > 0 \end{matrix}$ (19) $\begin{matrix} (4) α^{\land Fk} & = ({log}_{λ} (1 + \frac{(λ^{μ_{α}} - 1)^{k}}{(λ - 1)^{k - 1}}), \\ 1 - {log}_{λ} (1 + \frac{(λ^{1 - ν_{α}} - 1)^{k}}{(λ - 1)^{k - 1}})), k > 0 \end{matrix}$ (20)

Theorem 1. Let α = (u _α, v _α) and β = (u _β, v _β) be any two IFNs and λ > 1, then $(1) α \oplus_{F} β = β \oplus_{F} α$ (21) $(2) α \otimes_{F} β = β \otimes_{F} α$ (22) $(3) k \cdot_{F} (α \oplus_{F} β) = k \cdot_{F} α \oplus_{F} k \cdot_{F} β, k > 0$ (23) $(4) k_{1} \cdot_{F} α \oplus_{F} k_{2} \cdot_{F} α = (k_{1} + k_{2}) \cdot_{F} α, k_{1}, k_{2} > 0$ (24) $(5) α^{\land_{F^{k 1}}} \otimes_{F} α^{\land_{{Fk}_{2}}} = α^{\land_{F (k_{1} + k_{2})}}, k_{1}, k_{2} > 0$ (25) $(6) α^{\land_{F^{k 1}}} \otimes_{F} β^{\land_{F^{k}}} = (α \otimes_{F} β)^{\land_{F^{k}}}, k > 0$ (26)

4 Intuitionistic fuzzy Frank power aggregation operators

Based on the operational rules of IFNs with respect to Frank operations in Section 3, and the advantages of power aggregation operator, we will introduce and analyze three aggregation operators in the following, which are intuitionistic fuzzy Frank power average operator, intuitionistic fuzzy Frank power weighted average operator and intuitionistic fuzzy Frank power ordered weighted average operator, respectively.

Definition 11. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs, and IFFPA : Ω ⁿ → Ω, if $IFFPA (a_{1}, a_{2}, \dots, a_{n}) = \frac{{\oplus_{F}}_{j = 1}^{n} ((1 + T (a_{j})) \cdot_{F} a_{j})}{\sum_{j = 1}^{n} (1 + T (a_{j}))}$ (27) where Ω is the set of all IFNs, and $T (a_{j}) = \sum_{\begin{array}{l} i, j = 1 \\ i, \neq j \end{array}}^{n} Sup (a_{j}, a_{i})$ , then IFFPA is called the intuitionistic fuzzy Frank power average operator.

Theorem 2. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs and λ > 1, then, the result aggregated from Definition 11 is still an IFN, and even $IFFPA (a_{1}, a_{2}, \dots, a_{n}) = (\begin{matrix} 1 - \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{1 - μ_{j}} - 1)^{ζ_{j}}), \\ \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{ν_{j}} - 1)^{ζ_{j}}) \end{matrix})$ (28) where ζ _j (j = 1, 2, ⋯ , n) is a collection of integrated weights such that $ζ_{j} = \frac{1 + T (a_{j})}{\sum_{j = 1}^{n} (1 + T (a_{j}))} .$

We can use mathematical induction to prove this theorem, and it is omitted here because of the constraints of space.

The IFFPA operator has the following properties: commutativity, idempotency and boundedness.

In Definition 11, we assumed that all of the objects (a ₁, a ₂, ⋯ , a _n) being aggregated were equally important. However, in many cases, the degrees of importance are not equal; thus, we should assign different weights for different objects. So, we establish intuitionistic fuzzy Frank power weighted average operator in the following.

Definition 12. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs, and IFFPWA : Ω ⁿ → Ω, if $IFFPWA (a_{1}, a_{2}, \dots, a_{n}) = \frac{{\oplus_{F}}_{j = 1}^{n} (ω_{j} (1 + T (a_{j})) \cdot_{F} a_{j})}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{j}))}$ (29) where Ω is the set of all IFNs, $T (a_{j}) = \sum_{\begin{array}{l} i, j = 1 \\ i, \neq j \end{array}}^{n} Sup (a_{j}, a_{i})$ , and ω = (ω ₁, ω ₂, ⋯ , ω _n) ^T is the weight vector of a _j (j = 1, 2 ⋯ , n), $ω_{j} \in [0, 1], \sum_{j = 1}^{n} ω_{j} = 1$ , then IFFPWA is called the intuitionistic fuzzy Frank power weighted average operator.

Specially, when $ω = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T}$ , IFFPWA operator will be reduced to IFFPA operator.

Theorem 3. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs, λ > 1, and ω = (ω ₁, ω ₂, ⋯ , ω _n) ^T is the weight vector of a _j (j = 1, 2 ⋯ , n), $ω_{j} \in [0, 1], \sum_{j = 1}^{n} ω_{j} = 1$ , then, the result aggregated from Definition 12 is still an IFN, and even $IFFPWA (a_{1}, a_{2}, \dots, a_{n}) = (\begin{matrix} 1 - \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{1 - μ_{j}} - 1)^{δ_{j}}), \\ \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{ν_{j}} - 1)^{δ_{j}}) \end{matrix})$ (30) where δ _j (j = 1, 2, ⋯ , n) is a collection of integrated weights such that $δ_{j} = \frac{ω_{j} (1 + T (a_{j}))}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{j}))} .$ The proof of this theorem is similar with Theorem 2, it’s omitted here.

Some properties of the IFFPWA operator are shown as follows.

Theorem 4. Suppose Sup (a _i, a _j) = k, for all i ≠ j, then $\begin{matrix} IFFPWA (a_{1}, a_{2}, \dots, a_{n}) & = \frac{{\oplus_{F}}_{j = 1}^{n} (ω_{j} (1 + T (a_{j})) \cdot_{F} a_{j})}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{j}))} \\ = {\oplus_{F}}_{j = 1}^{n} (ω_{j} \cdot_{F} a_{j}) \end{matrix}$

This manifests that when all the supports are equal, the IFFPWA operator becomes an intuitionistic fuzzy Frank weighted average (IFFWA) operator.

Limited to the space, the proof of theorem 4 is omitted here.

Besides, IFFPWA operator has the idempotency and boundedness but not the commutativity property.

In the following, we will discuss some special cases of IFFPWA operator.

(1) When λ → 1 $\begin{matrix} {IFFPWA}_{λ \to 1} (a_{1}, a_{2}, \dots, a_{n}) \\ = (1 - \prod_{j = 1}^{n} (1 - μ_{j})^{δ_{j}}, \prod_{j = 1}^{n} ν_{j}^{δ_{j}}) \end{matrix}$ (31) where $δ_{j} = \frac{ω_{j} (1 + T (a_{j}))}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{j}))} .$

(2) When λ→ ∞ ${IFFPWA}_{λ \to \infty} (a_{1}, a_{2}, \dots, a_{n}) = (\sum_{j = 1}^{n} δ_{j} μ_{j}, \sum_{j = 1}^{n} δ_{j} ν_{j})$ (32) where $δ_{j} = \frac{ω_{j} (1 + T (a_{j}))}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{j}))} .$

The proofs of the above two formulas can refer to the reference [19].

However, we often need to consider the ordered position weights of the input arguments in many actual decision-making problems. For example, in order to prevent the condition that there exists partiality, spite or misjudgment and so on among the jury, the highest and lowest scores are always removed when calculating the final result in many singing contests. Therefore, on the basis of the POWA operator, we develop an intuitionistic fuzzy Frank power ordered weighted average operator to solve above cases.

Definition 13. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs, and IFFPOWA : Ω ⁿ → Ω, if $IFFPOWA (a_{1}, a_{2}, \dots, a_{n}) = {\oplus_{F}}_{j = 1}^{n} (ξ_{j} \cdot_{F} a_{σ (j)})$ (33) where Ω is the set of all IFNs, and a _σ(j) is the jth largest of the IFNs a _j (j = 1, 2 ⋯ , n). ξ _j (j = 1, 2, ⋯ , n) are the integrated weights such that $ξ_{j} = g (\frac{R_{j}}{TV}) - g (\frac{R_{j - 1}}{TV}), R_{j} = \sum_{i = 1}^{j} V_{σ (i)}$ g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function which meets the following properties: (1) g (0) =0; (2) g (1) =1; (3) g (x) ≥ g (y), if x > y. $TV = \sum_{j = 1}^{n} V_{σ (j)}, V_{σ (j)} = 1 + T (a_{σ (j)})$ (34)

T (a _σ(j)) indicates the support of the jth largest argument a _σ(j) by all the other arguments which can be calculated as follows: $T (a_{σ (j)}) = \sum_{\begin{array}{l} k = 1 \\ k \neq j \end{array}}^{n} Sup (a_{σ (j)}, a_{σ (k)})$ (35)

Then IFFPOWA is called the intuitionistic fuzzy Frank power ordered weighted average operator.

Theorem 5. Let a _j = (u _j, v _j) (j = 1, 2 ⋯ , n) be a collection of the IFNs and λ > 1, then, the result aggregated from Definition 13 is still an IFN, and even $IFFPOWA (a_{1}, a_{2}, \dots, a_{n}) = (\begin{matrix} 1 - \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{1 - μ_{j}} - 1)^{ξ_{j}}), \\ \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{ν_{j}} - 1)^{ξ_{j}}) \end{matrix})$ (36)

The proof of this theorem is similar with theorem 2, it’s omitted here.

The IFFPOWA operator has the following properties.

(1) Theorem 6. When g (x) = x, IFFPOWA operator becomes the IFFPA operator.

(2) Theorem 7. Suppose Sup (a _i, a _j) = k, for all i ≠ j, and g (x) = x, then $IFFPOWA (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{n} {\oplus_{F}}_{j = 1}^{n} a_{j}$

This manifests that when all the supports are equal, the IFFPOWA operator becomes an intuitionistic fuzzy Frank average (IFFA) operator.

Limited to the space, the proofs of the above two theorems are omitted here.

Besides, IFFPOWA operator has the commutativity, idempotency and boundedness properties.

In the following, we will also discuss some special cases of the IFFPOWA operator.

(1) When λ → 1

$\begin{matrix} {IFFPOWA}_{λ \to 1} (a_{1}, a_{2}, \dots, a_{n}) \\ = (1 - \prod_{j = 1}^{n} (1 - μ_{j})^{ξ_{j}}, \prod_{j = 1}^{n} ν_{j}^{ξ_{j}}) \end{matrix}$ (37)

(2) When λ→ ∞

$\begin{matrix} {IFFPOWA}_{λ \to \infty} (a_{1}, a_{2}, \dots, a_{n}) \\ = (\sum_{j = 1}^{n} ξ_{j} μ_{j}, \sum_{j = 1}^{n} ξ_{j} ν_{j}) \end{matrix}$ (38)

Clearly, both the IFFPWA and IFFPOWA operators can consider the given arguments and their relationships; the difference between two operators is that the IFFPWA operator emphasizes their own importance of these arguments, while the IFFPOWA operator stresses their ordered position importance.

Usually, in real group decision making problems, some decision-makers may provide unfair evaluation values to some objects. If the evaluation value given by a decision maker is more similar to all the other values provided by the other decision makers, the bigger weight will be assigned; on the contrary, the smaller weight will be assigned. Thus, these operators can reduce the influence of outlier arguments. So, the decision results based on these operators are more reasonable and reliable.

5 Multiple attribute group decision making methods based on IFFPWA and IFFPOWA operators

Considering the MAGDM problems based on intuitionistic fuzzy information: let X ={ X ₁, X ₂, ⋯ , X _m } be the set of alternatives, and G ={ G ₁, G ₂, ⋯ , G _n } be the set of attributes. ω _j is the weight of the attribute G _j (j = 1, 2, ⋯ , n), where 0 ≤ ω _j ≤ 1 (j = 1, 2, ⋯ , n), $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that D ={ D ₁, D ₂, ⋯ , D _p } is the set of decision makers, and γ _k (k = 1, 2, ⋯ , p) is a weight of decision maker D _γ with 0 ≤ γ _k ≤ 1 (k = 1, 2, ⋯ , p), $\sum_{k = 1}^{p} γ_{k} = 1$ . Suppose that $A^{k} = {[a_{ij}^{k}]}_{m \times n}$ is the decision matrix, where $a_{ij}^{k} = (μ_{ij}^{k}, ν_{ij}^{k})$ takes the form of the IFN and $u_{ij}^{k}, v_{ij}^{k} \in [0, 1]$ , $0 \leq u_{ij}^{k} + v_{ij}^{k} \leq 1$ , which means that the decision maker D _k evaluates the attribute G _j with respect to the alternative X _i. Then, we can rank the order of the alternatives based on the given information.

Because the IFFPWA and IFFPOWA operators have different emphasis, we apply them to MAGDM problems based on intuitionistic fuzzy information in the following, respectively.

5.1 The method based on the IFFPWA operator with the known weight information

The method involves the following steps:

Step 1. Calculate the supports. $\begin{matrix} Sup (a_{ij}^{k}, a_{ij}^{t}) = 1 - d (a_{ij}^{k}, a_{ij}^{t}), \\ k, t = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (39)

which satisfies the support conditions expressed by formulas (9)–(11), where $d (a_{ij}^{k}, a_{ij}^{t})$ is the distance between two IFNs $a_{ij}^{k}$ and $a_{ij}^{t}$ , which is defined by formula (5).

Step 2. Calculate $T (a_{ij}^{k})$ . $\begin{matrix} T (a_{ij}^{k}) = \sum_{\begin{array}{l} t = 1 \\ t \neq k \end{array}}^{p} Sup (a_{ij}^{k}, a_{ij}^{t}) \\ k = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (40)

Step 3. Calculate the weight $δ_{ij}^{k}$ associated with the IFN $a_{ij}^{k}$ .

$\begin{matrix} δ_{ij}^{k} = \frac{γ_{k} (1 + T (a_{ij}^{k}))}{\sum_{k = 1}^{p} γ_{k} (1 + T (a_{ij}^{k}))} k = 1, 2, \dots, p; \\ i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (41)

Step 4. Aggregate the assessment information of each decision maker by the IFFPWA operator. $\begin{matrix} a_{ij} = IFFPWA (a_{ij}^{1}, a_{ij}^{2}, \dots, a_{ij}^{p}) = (μ_{ij}, ν_{ij}) \\ = (\begin{matrix} 1 - \log_{λ} (1 + \prod_{k = 1}^{p} (λ^{1 - μ_{ij}^{k}} - 1)^{δ_{ij}^{k}}), \\ \log_{λ} (1 + \prod_{k = 1}^{p} (λ^{ν_{ij}^{k}} - 1)^{δ_{ij}^{k}}) \end{matrix}) \end{matrix}$ (42) where i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n.

Step 5. Calculate T (a _ij)

$\begin{matrix} T (a_{ij}) = \sum_{\begin{array}{l} h = 1 \\ h \neq j \end{array}}^{n} Sup (a_{ij}, a_{ih}) i = 1, 2, \dots, \\ m; j = 1, 2, \dots, n \end{matrix}$ (43)

Step 6. Calculate the weight δ _ij associated with the IFN a _ij.

$\begin{matrix} δ_{ij} = \frac{ω_{j} (1 + T (a_{ij}))}{\sum_{j = 1}^{n} ω_{j} (1 + T (a_{ij}))} i = 1, 2, \dots, \\ m; j = 1, 2, \dots, n \end{matrix}$ (44)

Step 7. Calculate the comprehensive evaluation value of each alternative.

$\begin{matrix} a_{i} = IFFPWA (a_{i 1}, a_{i 2}, \dots, a_{in}) = (μ_{i}, ν_{i}) \\ = (1 - \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{1 - μ_{ij}} - 1)^{δ_{ij}}), \\ \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{ν_{ij}} - 1)^{δ_{ij}})) \end{matrix}$ (45) where i = 1, 2, ⋯ , m.

Step 8. Rank a _i (i = 1, 2, ⋯ , m) in descending order according to the comparison method of IFNs described in Definition 5.

Step 9. End.

5.2 The method based on the IFFPOWA operator with the unknown weight information

The method involves the following steps:

Step 1. Calculate the supports. $\begin{matrix} Sup (a_{ij}^{σ (k)}, a_{ij}^{σ (t)}) = 1 - d (a_{ij}^{σ (k)}, a_{ij}^{σ (t)}), \\ k, t = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (46) which satisfies the support conditions expressed by formulas (9)–(11), where $d (a_{ij}^{σ (k)}, a_{ij}^{σ (t)})$ is the distance between two IFNs $a_{ij}^{σ (k)}$ and $a_{ij}^{σ (t)}$ , which is defined by formula (5).

Step 2. Calculate $T (a_{ij}^{σ (k)})$ . $\begin{matrix} T (a_{ij}^{σ (k)}) = \sum_{\begin{array}{l} t = 1 \\ t \neq j \end{array}}^{p} Sup (a_{ij}^{σ (k)}, a_{ij}^{σ (t)}) \\ k = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (47)

Step 3. Calculate the weight $ξ_{ij}^{k}$ associated with the IFN $a_{ij}^{σ (k)}$ . $\begin{matrix} ξ_{ij}^{k} = g (\frac{R_{ij}^{k}}{{TV}_{ij}}) - g (\frac{R_{ij}^{k - 1}}{{TV}_{ij}}), R_{ij}^{k} = \sum_{t = 1}^{k} V_{ij}^{σ (t)}, \\ {TV}_{ij} = \sum_{t = 1}^{p} V_{ij}^{σ (t)}, V_{ij}^{σ (t)} = 1 + T (a_{ij}^{σ (t)}) \\ k = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (48) where g : [0, 1] → [0, 1] is a basic unit-interval monotonic (BUM) function which meets three conditions described in definition 13.

Step 4. Aggregate the assessment information of each decision maker by the IFFPOWA operator. $\begin{matrix} a_{ij} & = IFFPOWA (a_{ij}^{1}, a_{ij}^{2}, \dots, a_{ij}^{p}) = (μ_{ij}, ν_{ij}) \\ = (\begin{matrix} 1 - \log_{λ} (1 + \prod_{k = 1}^{p} (λ^{1 - μ_{ij}^{σ (k)}} - 1)^{ξ_{ij}^{k}}), \\ \log_{λ} (1 + \prod_{k = 1}^{p} (λ^{ν_{ij}^{σ (k)}} - 1)^{ξ_{ij}^{k}}) \end{matrix}) \end{matrix}$ (49) where i = 1, 2, ⋯ , m ; j = 1, 2, ⋯ , n.

Step 5. Calculate T (a _i,σ(j)) (note: T (a _ij) can be expressed as T (a _i,j))

$\begin{matrix} T (a_{i, σ (j)}) & = \sum_{\begin{array}{l} h = 1 \\ h \neq j \end{array}}^{n} Sup (a_{i, σ (j)}, a_{i, σ (h)}) i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n \end{matrix}$ (50)

Step 6. Calculate the weight ξ _ij associated with the IFN a _i,σ(j). $\begin{matrix} ξ_{ij} = g (\frac{R_{i, j}}{{TV}_{i}}) - g (\frac{R_{i, j - 1}}{{TV}_{i}}), R_{i, j} = \sum_{h = 1}^{j} V_{i, σ (h)}, \\ {TV}_{i} = \sum_{h = 1}^{n} V_{i, σ (h)}, V_{i, σ (h)} = 1 + T (a_{i, σ (h)}) \\ k = 1, 2, \dots, p; i = 1, 2, \dots, m; j = 1, 2, \dots, n \end{matrix}$ (51)

Step 7. Calculate the comprehensive evaluation value of each alternative.

$\begin{matrix} a_{i} & = IFFPOWA (a_{i 1}, a_{i 2}, \dots, a_{in}) = (μ_{i}, ν_{i}) \\ = (1 - \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{1 - μ_{i, σ (j)}} - 1)^{ξ_{ij}}), \\ \log_{λ} (1 + \prod_{j = 1}^{n} (λ^{ν_{i, σ (j)}} - 1)^{ξ_{ij}})) \end{matrix}$ (52) where i = 1, 2, ⋯ , m.

Step 8. Rank a _i (i = 1, 2, ⋯ , m) in descending order according to the comparison method of IFNs described in Definition 5.

Step 9. End.

6 A numerical example

In order to demonstrate the application of the proposed method, we will cite an example about the investment for five possible companies {X ₁, X ₂, X ₃, X ₄ X ₅} (adapted from [27]). There are three decision makers D _k (k = 1, 2, 3) with weight vector γ = (0.35, 0.40, 0.25) ^T, and four attributes (suppose that their weight vector is ω = (0.2, 0.1, 0.3, 0.4) ^T) are shown as follows: the risk analysis (G ₁), the growth analysis (G ₂), the social-political impact analysis (G ₃) and the environmental impact analysis (G ₄).

The decision makers D _k (k = 1, 2, 3) evaluate the companies X _i (i = 1, 2, 3, 4, 5) with respect to the attributes G _j (j = 1, 2, 3, 4) by using the IFNs and construct three following decision matrices $A^{k} = {[a_{ij}^{k}]}_{5 \times 4} (k = 1, 2, 3)$ as listed in Tables 1–3, where $a_{ij}^{k}$ can be expressed as $(μ_{ij}^{k}, ν_{ij}^{k})$ .

6.1 The evaluation steps

The steps are shown as follows:

(1) Calculate the supports $Sup (a_{ij}^{k}, a_{ij}^{t})$ k, t = 1, 2, 3 i, j = 1, 2, 3, 4 by formula (39) (for simplicity, $Sup (a_{ij}^{k}, a_{ij}^{t})$ is written to $S_{ij}^{kt}$ for short).

For example, according to the formula (39), we can calculate $S_{11}^{12}$ $\begin{matrix} S_{11}^{12} = 1 - d (a_{11}^{1}, a_{11}^{2}) = 1 - \frac{| μ_{11}^{1} - μ_{11}^{2} | + | ν_{11}^{1} - ν_{11}^{2} |}{2} \\ = 1 - \frac{| 0.5 - 0.4 | + | 0.4 - 0.5 |}{2} = 0.9 \end{matrix}$ (53)

Limited to the space, the values of the other $S_{ij}^{kt}$ are not to be shown in detail.

(2) Calculate $T (a_{ij}^{k})$ k = 1, 2, 3; i, j = 1, 2, 3, 4 by formula (40).

As an example, according to the formula (40), we can calculate $\begin{matrix} T (a_{ij}^{1}) = \sum_{\begin{array}{l} t = 1 \\ t \neq 1 \end{array}}^{3} Sup (a_{11}^{1}, a_{11}^{t}) = Sup (a_{11}^{1}, a_{11}^{2}) + Sup (a_{11}^{1}, a_{11}^{3}) \\ = S_{11}^{12} + S_{11}^{13} = 0.9 + 0.85 = 1.75 \end{matrix}$

Limited to the space, the calculations of the other $T (a_{ij}^{k})$ are omitted here.

(3) Calculate the weights $δ_{ij}^{k}$ (k = 1, 2, 3; i, j = 1, 2, 3, 4) by formula (41)

For example, we can calculate the weight $δ_{11}^{1}$ , and its calculative process is shown as follows. $\begin{matrix} \begin{matrix} δ_{11}^{1} = \frac{γ_{1} (1 + T (a_{11}^{1}))}{\sum_{k = 1}^{3} γ_{k} (1 + T (a_{11}^{k}))} \\ = \frac{0.35 \times (1 + 1.75)}{0.35 \times (1 + 1.75) + 0.4 \times (1 + 1.75) + 0.25 \times (1 + 1.7)} \\ = 0.352 \end{matrix} \end{matrix}$

Limited to the space, the calculations of the other $ζ_{ij}^{k}$ are omitted here.

(4) Aggregate the assessment information of each decision maker by the IFFPWA operator in formula (42), suppose λ = 2.

As an example, we can calculate the weight $δ_{11}^{1}$ , and its calculative process is shown as follows. $\begin{matrix} a_{11} & = IFFPWA (a_{11}^{1}, a_{11}^{2}, a_{11}^{3}) = (μ_{11}, ν_{11}) \\ = (1 - \log_{2} (1 + \prod_{k = 1}^{3} (2^{1 - μ_{11}^{k}} - 1)^{δ_{11}^{k}}), \\ \log_{2} (1 + \prod_{k = 1}^{3} (2^{ν_{11}^{k}} - 1)^{δ_{11}^{k}})) \\ = (0.437, 0.371) \end{matrix}$

So we can get an aggregated matrix A based on three decision matrices given by decision makers, which is shown as follows: $A = [\begin{matrix} (0.437, 0.371) & (0.542, 0.231) & (0.413, 0.427) & (0.467, 0.300) \\ (0.579, 0.337) & (0.617, 0.256) & (0.600, 0.235) & (0.668, 0.235) \\ (0.437, 0.438) & (0.418, 0.438) & (0.476, 0.295) & (0.342, 0.395) \\ (0.628, 0.296) & (0.660, 0.221) & (0.376, 0.335) & (0.542, 0.200) \\ (0.539, 0.271) & (0.587, 0.239) & (0.400, 0.332) & (0.590, 0.325) \end{matrix}]$

(5) Calculate T (a _ij) i, j = 1, 2, 3, 4 by formula (43), we can get $\begin{matrix} T (a_{11}) = 2.787, T (a_{12}) = 2.642, T (a_{13}) = 2.707, T (a_{14}) = 2.787 \\ T (a_{21}) = 2.784, T (a_{22}) = 2.886, T (a_{23}) = 2.886, T (a_{24}) = 2.835 \\ T (a_{31}) = 2.831, T (a_{32}) = 2.831, T (a_{33}) = 2.691, T (a_{34}) = 2.756 \\ T (a_{41}) = 2.710, T (a_{42}) = 2.679, T (a_{43}) = 2.505, T (a_{44}) = 2.689 \\ T (a_{51}) = 2.807, T (a_{52}) = 2.774, T (a_{53}) = 2.661, T (a_{54}) = 2.805 \end{matrix}$ (6) Calculate the weights δ _ij i, j = 1, 2, 3, 4 by formula (44), we can get $\begin{matrix} δ_{11} = 0.202, δ_{12} = 0.097, δ_{13} = 0.297, δ_{14} = 0.404 \\ δ_{21} = 0.197, δ_{22} = 0.101, δ_{23} = 0.303, δ_{24} = 0.399 \\ δ_{31} = 0.204, δ_{32} = 0.102, δ_{33} = 0.295, δ_{34} = 0.400 \\ δ_{41} = 0.204, δ_{42} = 0.101, δ_{43} = 0.289, δ_{44} = 0.406 \\ δ_{51} = 0.203, δ_{52} = 0.100, δ_{53} = 0.292, δ_{54} = 0.405 \end{matrix}$

(7) Calculate the comprehensive evaluation value of each alternative by formula (45) (suppose λ = 2), we can get $\begin{matrix} a_{1} = (0.454, 0.340) \\ a_{2} = (0.626, 0.255) \\ a_{3} = (0.411, 0.375) \\ a_{4} = (0.533, 0.255) \\ a_{5} = (0.529, 0.306) \end{matrix}$

(8) Calculate expectations about each a _i (i = 1, 2, 3, 4) based on formula (2), we can get $\begin{matrix} E (a_{1}) = 0.557, E (a_{2}) = 0.686, E (a_{3}) = 0.518, \\ E (a_{4}) = 0.639, E (a_{5}) = 0.612 \end{matrix}$

(9) Rank the alternatives

According to the comparison method described in Definition 5, the ranking is X ₂ ≻ X ₄ ≻ X ₅ ≻ X ₁ ≻ X ₃, and the best alternative is X ₂.

6.2 Discussion

In order to demonstrate the influence of the parameter λ on decision making result of this example, we use the different value λ in steps (4) and (7) to rank the alternatives. The ranking results are shown in Table 4.

Obviously, although the integrated result may be different for the different value λ in IFFPWA operator in Table 4, the ordering of the alternatives is still X ₂ ≻ X ₄ ≻ X ₅ ≻ X ₁ ≻ X ₃ and the best alternative is also X ₂. In general, decision makers can assign different value to the parameter with respect to their preferences. By further analysis, it is easy to find that the expectations obtained by the IFFPWA operator become smaller as the parameter λ increases for the same integrated elements.

In addition, in order to further verify the effective of the proposed method, we also utilized this method to solve the illustrate example developed by Wei [27], and we can get the same ranking result. However, in this paper, the proposed method has a variety of weighted average modes with the change of parameter λ, in other words, the method proposed in this paper is more flexible because of parameter λ.

7 Conclusion

On the one hand, Frank operators are able to make the process of information fusion more flexible and robust because of their containing a parameter, on the other hand, PA operator considers the relationship between the values being aggregated. So, in this paper, we proposed Frank operations of IFSs, and established some aggregation operators with respect to IFNs by combining the Frank operations with Power aggregation operators, such as intuitionistic fuzzy Frank power average operator (IFFPA), intuitionistic fuzzy Frank power weighted average operator (IFFPWA) and intuitionistic fuzzy Frank power ordered weighted average operator (IFFPOWA). At the same time, some properties and special cases of these operators are discussed. Furthermore, on the basis of the IFFPWA and IFFPOWA operators, we propose two approaches to multiple attribute group decision making with IFNs. The prominent characteristic of the developed approaches is that they can take all the relationships of input arguments into account, and can reflect the preference of the decision-makers, and the proposed methods are more scientific to do decision making.

In further research, it is necessary and meaningful to study some applications of these operators such as the supplier select, data mining, performance evaluation and cluster analysis and so on. In addition, considering that Frank operator has the advantage of compatibility among all types of triangle norms, we also study some new aggregation operators based on Frank operator, such as the intuitionistic fuzzy Frank power geometric operator (IFFPG), the intuitionistic fuzzy Frank power ordered geometric operator (IFFPOG), the intuitionistic fuzzy Frank power ordered weighted geometric operator (IFFPOWG), etc.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), Shandong Provincial Social Science Planning Project (No. 13BGLJ10), the Natural Science Foundation of Shandong Province (No. ZR2011FM036), and Graduate education innovation projects in Shandong Province (SDYY12065).

References

Atanassov

1986

Intuitionistic fuzzy sets

Fuzzy Sets and Systems 20 87 96

Atanassov

1989