Intuitionistic fuzzy induced ordered entropic weighted averaging operator for group decision making

Abstract

In the paper, we develop the intuitionistic fuzzy induced ordered entropic weighted averaging (IFIOEWA)operator for multiple attribute group decision making (MAGDM) problems. The new operator’s attribute values take the form of intuitionistic fuzzy numbers (IFNs) and the intuitionistic fuzzy entropy (IFE) is fully taken into account, after the intuitionistic fuzzy entropic weighted averaging (IFEWA)operator is proposed. Afterwards, the prime properties of the IFIOEWA operator are investigated under some operational laws of intuitionistic fuzzy numbers (IFNs). Moreover, a method based on the IFIOEWA operator for decision-making is presented. Eventually, a numerical example is constructed to reveal the availability and applicability of the raised operator.

Keywords

Group decision making induce ordered weighted averaging intuitionistic fuzzy numbers intuitionistic fuzzy entropy

1 Introduction

In the real world, we usually encounter uncertain information when we deal with multiple attributes group decision making (MAGDM) problems. Because of the restricting of time, lack of perceived professionalism, and individuals’ limited attention and capability of information processing, we can’t know everything about the aspects which are critical to rank the different alternatives. Many authors paid attention to measuring the fuzziness of indeterminacy information. In 1965, Zadeh [13, 14] first proposed a fuzzy method which used and cited entropy to measure the average uncertainty in a random experiment. Inspired by the concept of Shannon’s function, Luca and Termini [1] put forward the axioms of the fuzzy entropy, and defined the entropy of fuzzy sets (FSs). Yager [21] proposed the entropy of a FS by the distance between the FS and its complement, while Higashi and Klir [15] extended this concept based on a general class of fuzzy complements.

In 1986, Atanassov [12] proposed the definition of intuitionistic fuzzy set (IFS) taking account of a membership function and a nonmembership function. Comparing with Zadeh’s fuzzy set, IFS is more reasonable for handling the vague and imperfect information. The IFS emphasizes the importance of nonmembership function which is unprecedented in the history of the fuzzy set. Moreover, it is an extension of Zadeh’s fuzzy sets (FSs) that only assign to each element a membership function. Burille and Bustince [16] proposed the concept of the intuitionistic fuzzy entropy (IFE) to describe the degree of the fuzziness. Szmidt and Kacprzyk [3, 4] defined a kind of intuitionistic fuzzy entropy under non-probabilistic situation.

The most significant process of decision making is to aggregate the decision makers’ preferences by effective methods. Yager [18, 22] introduced the ordered weighted averaging (OWA) operator, which provides a parameterized family of aggregation operators that induce as special cases, such as the maximum, the minimum and the average. The OWA operator can also be extended to the induced OWA (IOWA) operator [19 , 24] when the order inducing variable is considered. The order inducing variable plays a key role in reordering the value of the argument. To copy with the fuzziness of the uncertain information, the IOWA operator is extended to the fuzzy IOWA operator [10] which shows favourable applications in the MAGDM problems. Because many given arguments are expressed by language, such as “bad”,“regular” and “good”, Xian [26] proposed the fuzzy language IOWA operator which shows more practical value than the previous operators, and so on [27–29 , 31].

In the practical application, for the vague and imperfect information, the MAGDM problems are complex and tough when decision makers try to deal with the fuzziness information and build a reasonable model. In such situations, an abundance of input arguments cannot represented by the exact numbers or pure fuzzy numbers, the intuitionistic fuzzy numbers considering the degree of the nonmembership is more general and favorable. The approaches based on IFNs have showed quite nice applications in the data statistics, financial prediction, and risk assessment [17]. As for the literatures on MAGDM problems, Xu et al. [35, 36] proposed many valuable methods for the circumstance interval-valued intuitionistic fuzzy set (IVIFS) environment and defined several arithmetic aggregation operators, then analyzed the prime properties of them. Wei [5 –7] introduced different kinds of induced geometric aggregation operators in view of IFS. Su et al. [34] investigated the intuitionistic fuzzy ordered weighted similarity measure for MAGDM and introduced some extensions in the MAGDM fields. Xian and Xue et al. [30] proposed the intuitionistic fuzzy linguistic induce OWA (IFLIOWA) operator for intuitionistic fuzzy linguistic MAGM, which shows more useful than the previous operators.

In recent years, many researches have paid attention to the combination of OWA, IOWA, WA, etc. in the same formulation [20, 25]. Since Torra [33] developed the weighted OWA (WOWA) oprator, and Xu and Da [37] introduced the hybrid averaging (HA) operator, the idea of hybrid weighted operator has triggered much attention and wide applications. Merigó [8] presented the ordered weighted averaging weighted averaging (OWAWA) operator which unifies OWA and WA in one formulation considering the degree of importance of the two elements. The coefficient of the OWAWA operator can give more importance to OWA or WA according to the decision makers’ interests on the problem analyzed and actual need. To overcome the complex reordering processes affected by different factors, such as the degree of optimism, time pressure and psychological aspects, Merigó et al. [9] also presented a method of business and economic decision-making problems by using the weighted average and the OWA operator in the same formulation and applied it to the framework for the entrepreneurial group decision support systems [11]. Zeng et al. [32] extended the induced OWAWA operator to accommodate the intuitionistic fuzzy situations and proposed the induced intuitionistic fuzzy ordered weighted averaging weighted average(I-IFOWAWA) operator.

According to the above literatures, we can see all kinds of the hybrid aggregation operators play a key role in copying with the information of the complex system for MAGDM problems. However, the weighting vector of the above operator, including I-IFOWAWA, is still decision makers’ subjective opinion which may be inaccurate due to the individuals’ limited attention and capability of information processing. Moreover, the intuitionistic fuzzy entropy, which is of a paramount importance to measure how much the mount of knowledge the uncertain information has, should be paid attention to hybrid aggregation operators. Inspired by Hwang and Yoon’s idea [2] which takes each variable’s entropy as weighted variable, we propose the intuitionistic fuzzy entropic weighted averaging (IFEWA) operator which takes each variable’s entropy as a part of weighted variable. The IFEWA operator can endow weightings to the arguments according to the mount of knowledge the arguments have. The greater value of intuitionistic fuzzy entropy an argument has, the less amount of knowledge it conveys, the less value of weighting it obtains. Then, the intuitionistic fuzzy induced ordered entropic weighted averaging (IFIOEWA) operator which unifies the IFEWA operator and the induced intuitionistic fuzzy ordered weighted averaging (I-IFOWA) operator is introduced. The IFIOEWA operator assimilates the reasonability of the intuitionistic fuzzy entropy and the I-IFOWA operator under the intuitionistic fuzzy circumstance. The IFIOEWA operator, which considers the substantive characteristic of IFN, is more reasonable than the I-IFOWAWA operator. That the new operator can endow larger weighting to the less fuzzier argument is superior to the I-IFOWAWA operator whose weighting is only a subjective evaluation. To our knowledge, there are few literatures towards the weighting based on the subjective and objective aspects. Therefore, the IFIOEWA operator is an applicable and valuable research subject, which should receive increasing attention.

In Section 2, some fundamental definitions are described, such as the intuitionistic fuzzy set, the intuitionistic fuzzy entropy, the IOWA, and I-IFOWA operators. In the following section, the IFEWA and IFIOEWA operators are introduced, and some main properties of the IFIOEWA operator are studied. Then, in Section 4, a method based on the IFIOEWA operator for MAGDM is constructed. In Section 5, an illustrative example of the IFIOEWA operator is developed. In the end, the prime conclusions of this paper are summarized briefly.

2 Preliminaries

We review some basic notions, such as IFS, IFE, the IOWA operator, and the I-IFOWA operator in this section.

2.1 The intuitionistic fuzzy sets

Atanassov [12] proposed the intuitionistic fuzzy numbers (A-IFNs), which are characterized as a measure of fuzziness or uncertain information in the FSs theory.

Definition 1. (Intuitionistic fuzzy set) Let a set X be an intuitionistic fuzzy set A in X is which defined as: $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in X},$ (1) where the functions μ_A (x) : X→ [0, 1] , x ∈ X →μ_A (x) ∈ [0, 1] ; ν_A (x) : X → [0, 1] , x ∈ X → ν_A (x)∈[0, 1] satisfy the condition: 0 ≤ μ_A (x) + ν_A (x) ≤1. And μ_A (x) , ν_A (x) denote the degree of membership and the degree of nonmembership of the element x ∈ X to the set A, respectively. In addition, for each IFS A ⊆ X, π_A (x) =1 - μ_A (x) - ν_A (x) is called the degree of indeterminacy of x to A, or called the degree of hesitancy of x to A.

We assume that $f_{\tilde{a}}^{l} (x), g_{\tilde{a}}^{l} (x), f_{\tilde{a}}^{r} (x), g_{\tilde{a}}^{r} (x)$ are the left and the right basis functions of the membership function and the nonmembership function respectively. According to concept of A-IFS, Xu and Yager [38] defined the ordered pair $(μ_{\tilde{a}}, ν_{\tilde{a}})$ as an intuitionistic fuzzy number (IFN). The values $μ_{\tilde{a}}$ and $ν_{\tilde{a}}$ respectively represent the maximum degrees of the membership and the nonmembership, such that $0 \leq μ_{\tilde{a}} \leq 1, 0 \leq ν_{\tilde{a}} \leq 1$ and $0 \leq μ_{\tilde{a}} + ν_{\tilde{a}} \leq 1$ .

Definition 2. Xu and Yager [38] proposed an effec-tive approach to compare two IFNs α₁, α₂ as follows. S (α) = μ_α - υ_α and H (α) = μ_α + υ_α are defined the score and accuracy degree of α. For any three IFNs α = (μ_α, υ_α) , α₁ = (μ_{α
₁}, υ_{α
₁}) , α₂ = (μ_{α
₂}, υ_{α
₂}), the following operational laws are valid. $\begin{matrix} α_{1} \oplus α_{2} = (μ_{α_{1}} + μ_{α_{2}} - μ_{α_{1}} \cdot μ_{α_{2}}, υ_{α_{1}} \cdot υ_{α_{2}}), \\ λ α = (1 - (1 - μ_{α})^{λ}, υ_{α}^{λ}) . \end{matrix}$ If S (α₁) < S (α₂), then α₁ ≺ α₂; If S (α₁) = S (α₂), then

(1) If H (α₁) < H (α₂), then α₁ ≺ α₂;

(2) If H (α₁) = H (α₂), then α₁ = α₂.

Example 1. Assume that α₁, α₂, …, α₅ are IFNsand α₁ = (0.55, 0.28) , α₂ = (0.63, 0.28) , α₃ = (0 .61, 0.27) , α₄ = (0.55, 0.26) , α₅ = (0.55, 0.36).

According to the above method of comparing, we calculate S (α₁) =0.27, S (α₂) =0.35, S (α₃) =0.34, S (α₄) =0.29, S (α₅) =0.19, then α₂ ≻ α₃ ≻ α₄ ≻ α₁ ≻ α₅ .

2.2 The intuitionistic fuzzy entropy

Many authors proposed many methods to measure the intuitionistic fuzzy entropy [3 , 16] based on distance, hesitation degree, probability. Szmidt and Kacprzyk [3] defined a kind of intuitionistic fuzzy entropy under non-probabilistic situation. In fact, it is a method to examine the distance from the A-IFS to being a crisp set.

Definition 3. (The intuitionistic fuzzy entropy) $\begin{matrix} E_{AIFS} (A) = \frac{d_{AIFS} (A, A_{near})}{d_{AIFS} (A, A_{far})} \\ = \frac{1}{n} \sum_{i = 1}^{n} \frac{\min {μ_{A} (x_{i}), ν_{A} (x_{i})} + π_{A} (x_{i})}{\max {μ_{A} (x_{i}), ν_{A} (x_{i})} + π_{A} (x_{i})} . \end{matrix}$ (2) where the d_AIFS (A, A_near) describes the distance from A to its nearest point A_near, similarly, the d_AIFS (A, A_far) describes the distance from A to its farthest point A_far in a three-dimensionalspace.

2.3 The IOWA operator

Yager and Filev [23] proposed the definition of the IOWA operator as follows.

Definition 4. (The IOWA operator) An IOWA operator of dimension n is a function φ_IOWA : Rⁿ× Rⁿ→ R, to which a weighting vector is associated W = (w₁, w₂, …, w_n)^T, such that w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , and it is defined to aggregate the set of second arguments of a list of n pairs {(u₁, p₁) , (u₂, p₂) , …, (u_n, p_n)} according to the following expression: $φ_{I - IFOWA} ((u_{1}, p_{1}), (u_{2}, p_{2}), \dots, (u_{n}, p_{n})) = \sum_{i = 1}^{n} w_{i} p_{σ (i)},$ (3) where σ : (1, 2, …, n) → (1, 2, …, n) is a permutation such that u_σ(i) ≥ u_σ(i+1), ∀ i = 1, 2, …, n - 1, that is, (u_σ(i), p_σ(i)) is the pair with u_σ(i) the ith highest value in the set {u₁, u₂, …, u_n}. p₁, p₂, …, p_n is induced by the ordering of the values u₁, u₂, …, u_n associated with them. Because of this use of the set of values u₁, u₂, …, u_n, Yager and Filev called them the values of an order inducing variable and p₁, p₂, …, p_n are the values of the argument variables.

2.4 The I-IFOWA operator

At present, the A-IFSs theory [12], for its excellent flexibility and agility in coping with fuzziness or imperfect data, has been widely applied in some domains, such as applications in the data statistics, financial prediction and risk assessment.

Definition 5. [32] Let Ψ be a set of intuition-istic fuzzy numbers. An I-IFOWA operator of dimension n is a function φ_I-IFOWA : Ψ ⁿ× Rⁿ→ Ψ, to which a weighting vector is associated W =(w₁, w₂, …, w_n)^T, such that w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , and it is defined to aggregate the set of second arguments of a list of n pairs {(u₁, p₁) , (u₂, p₂) , …, (u_n, p_n)} according to the following expression: $\begin{matrix} φ_{I - IFOWA} ((u_{1}, p_{1}), (u_{2}, p_{2}), \dots, (u_{n}, p_{n})) \\ = \sum_{i = 1}^{n} w_{i} p_{σ (i)}, \end{matrix}$ (4) where σ : (1, 2, …, n) → (1, 2, …, n) is a permutation such that u_σ(i) ≥ u_σ(i+1), ∀ i = 1, 2, …, n - 1, that is, (u_σ(i), p_σ(i)) is the pair with u_σ(i) the ith largest value in the set {u₁, u₂, …, u_n}. p₁, p₂, …, p_n are induced by the ordering of the values u₁, u₂, …, u_n associated with them.

3 Intuitionistic fuzzy induced ordered entropic weighted averaging (IFIOEWA) operator and its properties

In this section, the IFIOEWA operator is proposed after the intuitionistic fuzzy entropic weighted averaging (IFEWA) operator is presented. Afterwards, the prime properties of the IFIOEWA operator are investigated under the classical operational laws of IFNs.

3.1 The IFEWA operator

Inspired by [32], we presented the IFEWA operator after a deeply understanding of intuitionistic fuzzy entropy. The IFEWA operator extents the OWA operator in a way that each variable’s intuitionistic fuzzy entropy represented the weighted variable. The weighting of IFEWA is up to the mount of knowledge of each argument. The method of endowing weighting is objective and reasonable than traditional operators which only consider the decision makers’ subjective opinion in MAGDM.

Definition 6. (The IFEWA operator)An IFEWA operator of dimension n is a mapping Φ_IFEWA : Ψⁿ→ R, that has an associated weighting vector (e1, e2, ... , en) of dimension n and $\sum_{i = 1}^{n} e_{i} = 1, e_{i} \in [0, 1]$ , then: $Φ_{IFEWA} (a_{1}, a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} e_{i} a_{i},$ (5) where e_i is the intuitionistic fuzzy entropic weighted value of a_i. In addition, we can obtain the e_i such that: $e_{i} = \sum_{i = 1}^{n} \frac{1 - E_{i}}{\sum_{k = 1}^{n} (1 - E_{k})} a_{i},$ (6) where E_i is the corresponding intuitionistic fuzzy entropy of a_i. We can calculate the E_i by Definition 3.

Example 2. Assume that the available strategies in Table 1 [32] are evaluated by the intuitionistic fuzzy numbers.A_i is the possible best alternative that the decision makers may select. u_i represents the different environments of A_i in the future. We illuminate the application of the IFEWA operator as follows.

Firstly, analyze and determine the critical characteristics in the MAGDM problem, then decide the values of each attribute by intuitionistic fuzzy numbers, just as Table 1.

Secondly, calculate the entropy E_i of intuitionistic fuzzy set of each u_i by Definition 3. Then obtain the e_i by Equation (2). The results can be seen in Table 2.

Thirdly, calculate the results of the IFEWA operator by Definition 6 as follows: $A_{2} ≻ A_{3} ≻ A_{5} ≻ A_{1} ≻ A_{4} .$

Therefore, the most profitable alternative is A₂.

3.2 The IFIOEWA operator

Usually, because of the fuzziness and vagueness of the given information, it is difficult to represent it by real numbers. The intuitionistic fuzzy number is a useful technique to express the uncertain information. We present the IFIOEWA operator, which is a combination of the IFEWA operator and the I-IFOWA operator, considering the degree of relevance that each operator has in the case study. Because of individuals’ limited attention and capability of information processing, endowing the weighting of arguments by decision makers’ subjective opinion may be inaccurate. Moreover, to fully utilize the valuable of data, the intuitionistic fuzzy entropy, which measures the degree of fuzziness and how much the mount of knowledge the uncertain information has, is applied to presented operator. The decision makers can adjust the objective aspect: IFEWA and the subjective aspect: IFIOEWA by the parameter λ according to the actual need of decision-making.

Definition 7. (The IFIOEWA operator) An IFIOEWA operator of dimension n is a mapping Φ_IFIOEWA : Rⁿ× Ψⁿ→ Ψ that has a weighting vector is associated W = (w₁, w₂, …, w_n)^T, w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , such that, $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}), \dots, (s_{n}, a_{n})) \\ = λ \sum_{j = 1}^{n} w_{j} b_{j} + (1 - λ) \sum_{i = 1}^{n} e_{i} a_{i}, \end{matrix}$ (7) where b_j is a_i value of the IFIOEWA pair (s_i, a_i) having the jth largest s_i, s_i is the order inducing variable. Besides, λ ∈ [0, 1], argument a_i has an weight e_i which associates with the intuitionistic fuzzy entropy E_i. The operator can be defined as follows: $Φ_{IFIOEWA} ((s_{1}, a_{1}), \dots, (s_{n}, a_{n})) = \sum_{j = 1}^{n} w_{j}^{*} b_{j},$ (8) where $w_{j}^{*} = λ w_{j} + (1 - λ) e_{j}, e_{j}$ is the weight w_i ordered according to the jth largest of s_i. The IFIOEWA operator has the following properties.

Theorem 1. (Idempotency) Let ((s₁, a₁) , (s₂, a₂) , …, (s_n, a_n)) is the intuitionistic fuzzy vector and (s₁, s₂, …, s_n) is an order inducing variable. If a_i = a for all i = 1, 2, …, n, then $Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) = a .$ (9)

Proof. Since a_i = a for all, i = 1, 2, …, n, according to Definition 7, $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) \\ = w_{1}^{*} b_{1} + w_{2}^{*} b_{2} + \dots + w_{n}^{*} b_{n} \\ = w_{1}^{*} a + w_{2}^{*} a + \dots + w_{n}^{*} a \\ = a (w_{1}^{*} + w_{2}^{*} + \dots + w_{n}^{*}) \\ = a . \end{matrix}$

Theorem 2. (Commutativity) Assume that ${(s_{1}^{*}, a_{1}^{*}), (s_{2}^{*}, a_{2}^{*}), \dots, (s_{n}^{*}, a_{n}^{*})}$ is any permutation of theintuitionistic fuzzy vector {(s₁, a₁) , (s₂, a₂) , …, (s_n, a_n)} , then $\begin{matrix} Φ_{IFIOEWA} ((s_{1}^{*}, a_{1}^{*}), (s_{2}^{*}, a_{2}^{*}), \dots, (s_{n}^{*}, a_{n}^{*})) \\ = Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) . \end{matrix}$ (10)

Proof. Let $\begin{matrix} Φ_{IFIOEWA} ((s_{1}^{*}, a_{1}^{*}), \dots, (s_{n}^{*}, a_{n}^{*})) \\ = λ \sum_{j = 1}^{n} w_{j} b_{j} + (1 - λ) \sum_{i = 1}^{n} e_{i} a_{i}^{*}, \\ Φ_{IFIOEWA} ((s_{1}, a_{1}), \dots, (s_{n}, a_{n})) \\ = λ \sum_{j = 1}^{n} w_{j} b_{j} + (1 - λ) \sum_{i = 1}^{n} e_{i} a_{i} . \end{matrix}$ According to Definition 6, we obtain $Φ_{IFEWA} (a_{1}^{*}, a_{2}^{*}, \dots, a_{n}^{*}) = \sum_{i = 1}^{n} e_{i} a_{i}^{*} = \sum_{i = 1}^{n} e_{i} a_{i} .$ Therefore, $\begin{matrix} Φ_{IFIOEWA} ((s_{1}^{*}, a_{1}^{*}), (s_{2}^{*}, a_{2}^{*}), \dots, (s_{n}^{*}, a_{n}^{*})) \\ = Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) . \end{matrix}$

Theorem 3.(Monotonicity) Let ((s₁, α₁^*) , (s₂, α₂^*) , …, (s_n, α_n^*)) and ((s₁, α₁), (s₂, α₂) , …, (s_n, α_n)) are two intuitionistic fuzzy vectors, if $α_{i} ≺ α_{i}^{*}$ and the weighting vectors of a_i and $a_{i}^{*}$ are equal for all i (i = 1, 2, …, n) , then $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}^{*}), (s_{2}, a_{2}^{*}), \dots, (s_{n}, a_{n}^{*})) \\ \overset{≻}{_} Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) . \end{matrix}$ (11)Proof: Let $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}^{*}), (s_{2}, a_{2}^{*}), \dots, (s_{n}, a_{n}^{*})) = \sum_{j = 1}^{n} w_{j}^{*} {b_{j}}^{*} \\ Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) = \sum_{j = 1}^{n} w_{j}^{*} b_{j} . \end{matrix}$

Since $a_{i} ≺ a_{i}^{*}$ for all i (i = 1, 2, …, n), it follows that $b_{j} ≺ b_{j}^{*}, j = 1, 2, \dots, n,$ then $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}^{*}), (s_{2}, a_{2}^{*}), \dots, (s_{n}, a_{n}^{*})) \\ \overset{≻}{_} Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) . \end{matrix}$

Theorem 4. (Boundedness) Let $α_{m} = \min_{i} (a_{1},$ $a_{2}, \dots, a_{n}), α_{M} = \max_{i} (a_{1}, a_{2}, \dots, a_{n})$ , then $α_{m} ⪯ Φ_{IFIOEWA} ((s_{1}, a_{1}), (s_{2}, a_{2}), \dots, (s_{n}, a_{n})) ⪯ α_{M} .$ (12)

Proof. Given that α_m ⪯ a_i ⪯ α_M for all i (i = 1, 2, …, n), then α_m ⪯ b_j ⪯ α_M for all j (j = 1, 2, …, n), b_j is a_i value of the IFIOEWA pair (s_i, a_i) having the jth largest s_i, s_i is the order inducing variable. And $\sum_{i = 1}^{n} w_{i}^{*} = 1$ , using the Theorems 1-3 and Equation (8), then $\begin{matrix} Φ_{IFIOEWA} ((s_{1}, a_{1}), \dots, (s_{n}, a_{n})) & = & \sum_{j = 1}^{n} w_{j}^{*} b_{j} \\ \overset{≻}{_} & \sum_{j = 1}^{n} w_{j}^{*} α_{m} \\ = & α_{m} . \\ Φ_{IFIOEWA} ((s_{1}, a_{1}), \dots, (s_{n}, a_{n})) & = & \sum_{j = 1}^{n} w_{j}^{*} b_{j} \\ \overset{≻}{_} & \sum_{j = 1}^{n} w_{j}^{*} α_{M} \\ = & α_{M} . \end{matrix}$ Thus, α_m ⪯ Φ_IFIOEWA ((s₁, a₁) , (s₂, a₂) , …, (s_n, a_n)) ⪯ α_M .

Remark 1. If $w^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})^{T} = (1 / n,$ 1/n, …, 1/n)^T, then we obtain Φ_IFIOEWA (a₁, $a_{2}, \dots, a_{n}) = \sum_{i = 1}^{n} \frac{1}{n} b_{i} = \sum_{i = 1}^{n} \frac{1}{n} a_{i} .$

Remark 2. (1) If $w^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})^{T} = (1, 0, \dots, 0)^{T},$ then the IFIOEWA operator is red-uced to the $\max_{i} (a_{i})$ operator.

(2) If $w^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})^{T} = (0, 0, \dots, 1)^{T},$ then the IFIOEWA operator is reduced to the $\min_{i} (a_{i})$ operator.

(3) If $w^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{j}^{*}, \dots, w_{n}^{*})^{T} = (0, 0,$ …, 1, …, 0)^T, then the IFIOEWA operator is reduced to the b_j operator,where b_j is the jth largest of a_i (i = 1, 2, …, n) .

Remark 3. (1) If λ = 0, the IFIOEWA operator is reduced to the IFEWA operator.

(2) If λ = 1, the IFIOEWA operator is reduced to the I-IFOWA operator [32].

4 A new approach to decision making for intuitionistic fuzzy information with IFIOEWA operator

The IFIOEWA operator is practical in numerous domains, such as data analysis, venture investment and engineering application. Subsequently, a new method is developed in decision-making problems to select the most profitable investment strategies with intuitionistic fuzzy information.

Assume that A = {A₁, A₁, …, A_n} is a discrete set of alternatives (n< ∞), S = {S₁, S₂, …, S_n} be the set of attributes and S_j is represented by intuitionistic fuzzy numbers.

Step 1. Analyze and determine of the critical characteristics in the MAGDM problem, then obtain each attribute value by intuitionistic fuzzy numbers.

Step 2. Calculate the entropy E_i of intuitionistic fuzzy set of each S_i by Definition 3. Then obtain the e_i by Equation (6).

Step 3. Calculate the numerical results of the I-IFOWA operator and the IFEWA operator respectively.

Step 4. Utilize the given λ, I-IFOWA operator and IFEWA operator, then calculate results using Definition 7.

Step 5. Rank all the alternatives A₁, A₂, …, A_n and select the best one(s).

Step 6. End.

5 Numerical example

In this part, a numerical example is constructed to reveal the valid and applicability of the proposed operator in an investment strategy. We adapted the data of Zeng et al.’s paper [32] comparing with the outcome of the new operator that we proposed. Assume a real estate company intents to invest the most beneficial alternative(s), and they have five choices: A₁, A₂, A₃, A₄, A₅ which are presented by intuitionistic fuzzy numbers. With comprehensive understanding of the overall information, the decision makers have considered the characteristics of the five attributes:

(1) u₁ = Very bad. (2) u₂ = Bad. (3) u₃ = Regular. (4) u₄ = Good. (5) u₅ = Very good.

The decision makers give their opinion towards the possible conditions after the actual investigation, and the alternatives A_i are shown in Table 1.

In this issue, the decision makers assume that I-IFOWA’s weighting vector is: W = (0.1, 0.1, 0.2, 0.2, 0.4), and the order inducing variable S = (8, 4, 9, 3, 2). Then we obtain the results by different operators, such as IFWA, IFOWA, IFEWA, I-IFOWA, and IFIOEWA in Table 3.

With the above information, the outcomes of different operator for each A_i are shown in Table 4.

Besides, different values of coefficient λ may cause different ordering of A_i, which is shown in Table 4, by the IFIOEWA operator. Therefore, it is necessary to select the suitable value of coefficient λ according to the practical need.

According to the results, we can obtain the different order preferences if we use different operators or coefficients. Therefore, if the decision makers want to make the most reasonable choice, they should use the desirable operator or coefficient which is related to the real situation.

6 Conclusion

At present, to resolve MAGDM problems more effectively, many authors proposed an abundance of excellent methods when the arguments expressed by IFNs. After the definition of the IFEWA operator is proposed, we presented the IFIOEWA operator. The new operator’s attribute values take the form of intuitionistic fuzzy numbers and the intuitionistic fuzzy entropy is fully taken into account. The intuitionistic fuzzy entropy is fully considered in the new operator and it is a unification of the IFEWA operator and the I-IFOWA operator. The decision makers can flexibly determine the proportion of the two operators according to the degree of relevance that each operator has in the case study. Then,We have analyzed some of its main properties.

In this paper, we have also introduced a new method based on the IFIOEWA operator and the basic steps of the new approach is presented. The proposed operator can be applied to rank the alternatives in MAGDM problems, such as statistics, economics and engineering. It also expands the existing methods which aggregate uncertain information with the technique of IFNs. Because the presented operator not only inherits the feature of the I-IFOWA operator, but also deal with uncertain information with intuitionistic fuzzy entropy. Moreover, we can make decision more reasonably and precisely by adjusting the coefficient of the proposed operator.

Footnotes

Acknowledgments

The authors express their thankfulness to the editors and the anonymous reviewers for their constructive comments and suggestions. This work was supported by the Graduate Teaching Reform Research Program of Chongqing Municipal Education Commission (No. YJG143010), the Chongqing research and innovation project of graduate students (No. CYS15165), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (No. KJ120515 and No. KJ1400430), and Chongqing Basic and Frontier Research Project (No. cstc2014jcyjA1347, No. cstc2015jcyjA00034, and No. cstc2015jcyjB0269).

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