Intuitionistic fuzzy multi-criteria decision making with application to job hunting: A comparative perspective

Abstract

The existing intuitionistic fuzzy information aggregation methods based on power mean cannot work well when the membership or non-membership degree of the intuitionistic fuzzy numbers (IFNs) reduced to zero. For solving the considered problem effectively, a group of new power intuitionistic fuzzy aggregation operators are proposed in this paper. We focus on comparisons with the existing power mean based intuitionistic fuzzy information aggregation approaches on numerical studies. Through comparative studies, we found some similarities between the operators proposed in this paper and the existing ones as well as a few differences. We use the proposed aggregation operators to multi-criteria decision making (MCDM) problem combined with IFNs. A real case about the job hunting problem of the contemporary college students is addressed based on the proposed method. The research of this paper indicates that the proposed MCDM is effective.

Keywords

Intuitionistic fuzzy set power mean aggregation operator MCDM

1 Introduction

Voting is one of the common techniques and widely exists in the social, economic, political and other activities. In the process of voting, three scenarios often appear at the same time, such as vote in favor, against and abstentions. How to describe the voting results in mathematical language is very important for the development of various decision making events. The biggest characteristic of vote is the vote in favor, against and abstentions coexist during the whole process. Coincidentally, the intuitionistic fuzzy set (IFS) theory first emanated from Atanassov [1] is a very appropriate mathematic technique for the description of the problem resembles the vote. The best feature of IFS is it uses the membership, non-membership and hesitant degree to describe the fuzziness of the subjective world [25]. Because of the features of IFS, it has been successfully applied to various areas, such as engineering, decision making and so on [33 –37].

In the field of IFS theory, MCDM has attracted many attentions from researchers [9]. MCDM problems related two issues including the performance presentation of each criterion and the aggregation methods used to aggregate these performances [28]. Typically, intuitionistic fuzzy MCDM problems can be divided into two categories: weight known and weight unknown [18]. However, aggregation operator is the key technique to any kind of intuitionistic fuzzy MCDM problems.

Intuitionistic fuzzy information aggregation operator is the hotspot in the study of IFS theory and it can be divided into two types, namely, independent and non-independent aggregation operators. Since there is an obvious connection between each other in actual use, the non-independent aggregation operators for IFNs are with wider application range. The power average (PA) operator first proposed by Yager [30] is a very useful technique to aggregate the correlated data. Combined the well-known OWA operator, the ordered PA (POWA) operator are also proposed by Yager [30] to fusion the arguments when their weights are depended on the support obtained from other arguments. Another two modified power operator is the power geometric (PG) operator and ordered PG (POWG) operator proposed by Xu and Yager [28]. The above four operators have been studied and extended to the fuzzy environment [26], triangular fuzzy environment [19], hesitant fuzzy environment [41] and linguistic environment [20].

Some power aggregation operators for IFNs are proposed which are mainly based on the extension of the existing power operators [24]. For example, the intuitionistic fuzzy power weighted average (IFPWA) operator and ordered IFPWA (IFPOWA) operator are the extensions of PA and POWA operators respectively. While the IFPWG operator and ordered IFPWG (IFPOWG) operator are the extensions of PG and POWG operators respectively. Inspired by the GOWA operator [31], Zhou et al. [42] extended the PA and POWA operators and proposed the generalized PA (GPA) and generalized POWA (GPOWA) operators, respectively. Furthermore, the GPA and GPOWA operators are extended for aggregating the IFNs. Zhang [41] improved the research of Zhou et al. [42] and introduced the weighted GIFPA operator (WGIFPA). From the point of geometric mean, Zhang [41] proposed the generalized intuitionistic fuzzy power geometric averaging (GIFPGA) operator, weighted GIFPGA (WGIFPGA) operator and ordered WGIFPGA (GIFPOWGA) operator.

Though some good progress has already been achieved regarding the power intuitionistic fuzzy aggregation operators, but more needs to be done for each aspect. However, the most fundamental is that all the existing intuitionistic fuzzy power operators are based on the operations proposed by Xu and Yager [27] which is cited hundreds of times. Nevertheless, these operations should be improved since they cannot deal with some special cases effectively, especially when the membership or the non-membership degrees of the aggregated IFNs reduced to zero [4, 5]. Therefore, the existing power aggregation operators for IFNs should be improved which is the main focus of this paper.

The following Section introduces the concept of IFS, the two kinds operations of IFNs, distance measures between IFNs. New developed power aggregation operators for IFNs are introduced in Section 3 and the comparison studies are presented in this section. Section 4 extends the results of Section 3 to interval-valued IFNs (IIFNs). Section 5 presents the MCDM method based on the proposed new power operator with IFNs, two numerical examples are provided to illustrate the differences with other existing methods. Finally, Section 6 summarizes thepaper.

2 Some basic concepts

On the basis of fuzzy set theory, Atanassov [1] first proposed the IFS theory which is a very useful technique for processing fuzziness and vagueness. At present, the IFS theory has been intensively studied and achieved some significant findings [16, 38].

Definition 1. [1] Suppose there an objective set and expressed as X, the IFS A on the objective set X is given as follows: $A = {< x, μ_{A} (x), v_{A} (x), h_{A} (x) > | x \in X}$ (1) where μ_A (x), v_A (x) and h_A (x) are the membership, non-membership and hesitant functions respectively.

IFN is the basic element of the IFS theory. To make the expression easier, Xu [21] expressed an IFN as α = 〈 μ, v 〉 and μ, v, μ + v ∈ [0, 1]. For three given IFNs α =〈 μ, v 〉, α₁ =〈 μ₁, v₁ 〉 and α₂ =〈 μ₂, v₂ 〉, the operations were defined as follows [21]: α₁ ⊕ α₂ = (μ₁ + μ₂ - μ₁μ₂, v₁v₂), α₁ ⊗ α₂ = (μ₁μ₂, v₁ + v₂ - v₁v₂), λα = (1 - (1 - μ) ^λ, v^λ) and α^λ = (μ^λ, 1 - (1 - v) ^λ) where λ > 0 is parameter. He et al. [4, 5] point out that what these operations mean in practice is debatable and introduced some modified operations to improve the existing research results.

$\begin{matrix} α_{1} \oplus α_{2} & = & 〈 1 - (1 - μ_{1}) (1 - μ_{2}), \\ (1 - μ_{1}) (1 - μ_{2}) \\ - (1 - (v_{1} + μ_{1})) (1 - (v_{2} + μ_{2})) 〉 \end{matrix}$ (2)

$\begin{matrix} λ α & = & 〈 1 - (1 - μ)^{λ}, (1 - μ)^{λ} - (1 - (v + μ))^{λ} 〉 \end{matrix}$ (3) $\begin{matrix} α_{1} \otimes α_{2} & = & 〈 (1 - v_{1}) (1 - v_{2}) \\ - (1 - (v_{1} + μ_{1})) (1 - (v_{2} + μ_{2})), \\ 1 - (1 - v_{1}) (1 - v_{2}) 〉 \end{matrix}$ (4) $\begin{matrix} α^{λ} & = & 〈 (1 - v)^{λ} - (1 - (v + μ))^{λ}, 1 - (1 - v)^{λ} 〉 \end{matrix}$ (5)

For two given IFNs α₁ =〈 μ₁, v₁ 〉 and α₂ =〈 μ₂, v₂ 〉, Xu and Yager [27] proposed a very simple approach to ranking the two IFNs mainly through two steps. 1) Calculate the score functions s (α₁) = μ₁ - v₁, s (α₂) = μ₂ - v₂ and accuracy functions: h (α₁) = μ₁ + v₁, h (α₂) = μ₂ + v₂. 2)s (α₁) > s (α₂) means α₁ is bigger than α₂; s (α₁) = s (α₂) and h (α₁) > h (α₂) means α₁ is bigger than α₂; s (α₁) = s (α₂) and h (α₁) = h (α₂) means α₁ is equal to α₂.

The great advantage of the power average (PA) operator is it can reflect the relationship between the aggregated data effectively. Yager [30] first proposed the concept of PA. Based on the PA operator, the power geometric (PG) operator was proposed by Xu and Yager [29].

3 Intuitionistic fuzzy interactive aggregation operators based on power mean

3.1 Power intuitionistic fuzzy interactive aggregation operators

The PA and PG operators have been extended for IFNs and some corresponding operators have been proposed. In this section, we propose some new power aggregation operators for IFNs and the comparison studies with the existing ones are presented in detail.

Definition 2. For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them is w = (w₁, w₂, …, w_n). Then the intuitionistic fuzzy power interactive weighted average (IFPIWA) operator is defined as:

$\begin{matrix} IFPIWA (α_{1}, α_{2}, \dots, α_{n}) \\ = \frac{(w_{1} (1 + T (α_{1}))) α_{1} \oplus \cdot \cdot \cdot \oplus (w_{n} (1 + T (α_{n}))) α_{n}}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} \end{matrix}$ (6) where $T (α_{j}) = \sum_{i = 1, i \neq j}^{n} w_{i} sup (α_{i}, α_{j}), j = 1, 2, \dots, n$ (7)

Sup (α_i, α_j) = 1 - d (α_i, α_j) is the support of α_i obtained from α_j, d is a distance measure. There are a lot of distance measures for IFNs [7, 40]. Szmidt and Kacprzyk [15] proposed a series of distance measures for IFNs, such as intuitionistic fuzzy Hamming distance and intuitionistic fuzzy Euclidean distance. In order to solve the problem of information loss, Zhang and Yu [40] proposed two kinds of new intuitionistic fuzzy distance measures.

Definition 3. [23] Let α₁ =〈 μ₁, v₁ 〉 and α₂ =〈 μ₂, v₂ 〉 be two IFNs,

$\begin{matrix} d (α_{1}, α_{2}) & = & \sqrt{\frac{1}{2} ({(μ_{1} - μ_{2})}^{2} + {(v_{1} - v_{2})}^{2})}, \\ d^{'} (α_{1}, α_{2}) & = & \frac{1}{2} (| μ_{1} - μ_{2} | + | v_{1} - v_{2} |) \end{matrix}$ (8)

Then d (α₁, α₂) is called the intuitionistic fuzzy Euclidean distance between IFNs α₁ and α₂. d′ (α₁, α₂) is called the intuitionistic fuzzy Hamming distance between IFNs α₁ and α₂ [11].

It should be noted that different distance measures adopted for intuitionistic fuzzy power operators, different aggregated results could be achieved. The main reason is the weight vector of the IFNs to be aggregated is highly correlated with intuitionistic fuzzy power operator. Using different distance measures, different power operators generalized and different aggregated results produced. Since the sensitivity analysis of the intuitionistic fuzzy power operator is not the focus of this paper, we just use Euclidean distance measure in the following studies.

Based on the Equations (2–5) defined by He et al. [4, 5], we can obtain the Theorem 1 as follows.

Theorem 1.For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them isw = (w₁, w₂, …, w_n). Based on the IFPIWA operator, the aggregated result was:

$\begin{matrix} IFPIWA (α_{1}, α_{2}, \dots, α_{n}) \\ = (1 - \prod_{j = 1}^{n} (1 - μ_{j})^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}, \\ \prod_{j = 1}^{n} (1 - μ_{j})^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} \\ - \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}) \end{matrix}$ (9)

Example 1. Suppose α₁ = 〈 0.3, 0.4 〉 , α₂ = 〈0.5, 0.2〉 and α₃ = 〈 0.7, 0.1 〉 be three IFNs, and w = (0.35, 0.52, 0.13) ^T be the standardized importance degree of them. In the following, we use the IFPIWA operator to aggregate the three IFNs.

First, we compute the distances between the three IFNs: $\begin{matrix} d (α_{1}, α_{2}) \\ = \sqrt{\frac{1}{2} ({(0.3 - 0.5)}^{2} + {(0.4 - 0.2)}^{2})} = 0 . 2000 \\ d (α_{1}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.3 - 0.7)}^{2} + {(0.4 - 0.1)}^{2})} = 0 . 3536 \\ d (α_{2}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.5 - 0.7)}^{2} + {(0.2 - 0.1)}^{2})} = 0 . 1581 \end{matrix}$

Therefore, $\begin{matrix} sup (α_{1}, α_{2}) = 0 . 8000 sup (α_{1}, α_{3}) = 0 . 6464 \\ sup (α_{2}, α_{3}) = 0 . 8419 \\ T (d_{1}) = 0 . 5000 T (d_{2}) = 0 . 3894 \\ T (d_{3}) = 0 . 6640 \end{matrix}$ and $\begin{matrix} \frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 3587 \\ \frac{w_{2} (1 + T (α_{2}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 4936 \\ \frac{w_{3} (1 + T (α_{3}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 1478 \end{matrix}$

Then,

IFPIWA (α₁, α₂, α₃) =〈 0 . 4769, 0 . 2406 〉, and its score is 0.2363.

If we use the IFPWA operator proposed by Xu [24] to aggregate the three IFNs, the result is

IFPWA (α₁, α₂, α₃) =〈 0 . 4769, 0 . 2315 〉, and its score is 0.2454.

Furthermore, if the IFN α₃ changed to $α_{3}^{'} = 〈 0.7, 0.0 〉$ , then the aggregated results may completely change.

IFPIWA $(α_{1}, α_{2}, α_{3}^{'}) = 〈 0 . 4761, 0 . 2329 〉$ , and its score is 0.2522.

If we use the IFPWA operator proposed by Xu [24] to aggregate the three IFNs, the result is

IFPWA $(α_{1}, α_{2}, α_{3}^{'}) = 〈 0 . 4761, 0 . 0 〉$ , and its score is 0.4769.

When the IFN α₃ changed to $α_{3}^{'} = 〈 0.7, 0.0 〉$ , the aggregated results have been changed completely. The aggregated results based on IFPWA operator is 〈0 . 4761, 0 . 0〉. Obviously, the non-membership degree is zero which is totally decided by the non-membership degree of IFN $α_{3}^{'}$ and take no account of the non-membership degrees of IFNs α₁ and α₂. Of course, when there is no non-membership degrees of IFNs takes the value of zero, the results based on IFPWA and IFPIWA operators are similar.

Definition 4. For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them is w = (w₁, w₂, …, w_n). Then the intuitionistic fuzzy power interactive weighted geometric (IFPIWG) operator is defined as:

$\begin{matrix} IFPIWG (α_{1}, α_{2}, \dots, α_{n}) \\ = (α_{1})^{\frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} \otimes \cdot \cdot \cdot \otimes (α_{n})^{\frac{w_{n} (1 + T (α_{n}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} \end{matrix}$ (10)

Based on the Theorem 1, we have Corollary 1 as follows:

Corollary 1.For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them isw = (w₁, w₂, …, w_n). Based on the IFPIWG operator, the aggregated result was:

$\begin{matrix} IFPIWG (α_{1}, α_{2}, . . ., α_{n}) \\ = (\prod_{j = 1}^{n} (1 - v_{j})^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} \\ - \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}, \\ 1 - \prod_{j = 1}^{n} (1 - v_{j})^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}) \end{matrix}$ (11)

Example 2. Suppose α₁ = 〈 0.7, 0.2 〉 , α₂ = 〈0.8, 0.1〉 and α₃ = 〈 0.0, 0.8 〉 be three IFNs, and w = (0.27, 0.48, 0.25) ^T be the standardized importance degree of them. In the following, we use the IFPIWG operator to aggregate the three IFNs.

The distances between the three IFNs were given as follows: $\begin{matrix} d (α_{1}, α_{2}) \\ = \sqrt{\frac{1}{2} ({(0.7 - 0.8)}^{2} + {(0.2 - 0.1)}^{2})} = 0 . 1000 \end{matrix}$

$\begin{matrix} d (α_{1}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.7 - 0.0)}^{2} + {(0.2 - 0.8)}^{2})} = 0 . 6519 \\ d (α_{2}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.8 - 0.0)}^{2} + {(0.1 - 0.8)}^{2})} = 0 . 7517 \end{matrix}$

Therefore, $\begin{matrix} sup (α_{1}, α_{2}) = 0 . 9000 sup (α_{1}, α_{3}) = 0 . 3481 \\ sup (α_{2}, α_{3}) = 0 . 2483 \\ T (d_{1}) = 0 . 5190 T (d_{2}) = 0 . 3051 \\ T (d_{3}) = 0 . 2132 \end{matrix}$ and $\begin{matrix} \frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 3061 \\ \frac{w_{2} (1 + T (α_{2}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 4675 \\ \frac{w_{3} (1 + T (α_{3}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 2264 \end{matrix}$

IFPIWG (α₁, α₂, α₃) =〈 0 . 5006, 0 . 3824 〉, and its score is 0.1182.

If we use the IFPWG operator proposed by Xu [24] to aggregate the three IFNs, the result is

IFPWG (α₁, α₂, α₃) =〈 0 . 0, 0 . 3824 〉, and its score is –0.3824.

Obviously, the membership degree of the aggregated IFN is zero which is totally decided by the membership degree of IFN α₃ and take no account of the membership degrees of IFNs α₁ and α₂.

3.2 Generalized power intuitionistic fuzzy interactive aggregation operators

In Section 3.1, the IFPIWA and IFPIWG operators are proposed which are the good complements of the existing power intuitionistic fuzzy aggregation operators proposed by Xu [24]. In this Section, we intend to generalize the IFPIWA and IFPIWG operators by introducing the GOWA operator [31].

Definition 5. For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them is w = (w₁, w₂, …, w_n). Then the generalized intuitionistic fuzzy power interactive weighted average (GIFPIWA) operator is defined as:

$\begin{matrix} GIFPIWA (α_{1}, α_{2}, \dots, α_{n}) \\ = (\frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} α_{1}^{λ} \oplus \cdot \cdot \cdot \oplus \\ {\frac{w_{n} (1 + T (α_{n}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} α_{n}^{λ})}^{1 / λ} \end{matrix}$ (12)

Corollary 2.For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them isw = (w₁, w₂, …, w_n). Based on the GIFPIWA operator, the aggregated result was:

$\begin{matrix} GIFPIWA (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 {(1 - \prod_{j = 1}^{n} {(1 - (1 - v_{j})^{λ} + (1 - (μ_{j} + v_{j}))^{λ})}^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} + \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{λ \frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}})}^{1 / λ} \\ - \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}), \\ 1 - {(1 - \prod_{j = 1}^{n} {(1 - (1 - v_{j})^{λ} + (1 - (μ_{j} + v_{j}))^{λ})}^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} + \prod_{j = 1}^{n} {(1 - (μ_{j} + v_{j}))}^{λ \frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}})}^{1 / λ} 〉 \end{matrix}$ (13)

Example 3. Suppose α₁ = 〈 0.7, 0.2 〉 , α₂ = 〈0.5, 0.1〉 and α₃ = 〈 0.2, 0.0 〉 be three IFNs, and w = (0.12, 0.46, 0.42) ^T be the standardized importance degree of them. Based on the GIFPIWA operator, the aggregated results can be obtained.

First, the distances between the three IFNs can be calculated as follows. $\begin{matrix} d (α_{1}, α_{2}) \\ = \sqrt{\frac{1}{2} ({(0.7 - 0.5)}^{2} + {(0.2 - 0.1)}^{2})} = 0 . 1581 \\ d (α_{1}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.7 - 0.2)}^{2} + {(0.2 - 0.0)}^{2})} = 0 . 3808 \\ d (α_{2}, α_{3}) \\ = \sqrt{\frac{1}{2} ({(0.5 - 0.2)}^{2} + {(0.1 - 0.0)}^{2})} = 0 . 2236 \end{matrix}$

Therefore, $\begin{matrix} sup (α_{1}, α_{2}) = 0 . 8419 sup (α_{1}, α_{3}) = 0 . 6192 \\ sup (α_{2}, α_{3}) = 0 . 7764 \\ T (d_{1}) = 0 . 6473 T (d_{2}) = 0 . 4271 \\ T (d_{3}) = 0 . 4314 \end{matrix}$ and $\begin{matrix} \frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 1358 \\ \frac{w_{2} (1 + T (α_{2}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 4511 \\ \frac{w_{3} (1 + T (α_{3}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 4131 \end{matrix}$

Then,

when λ = 1, GIFPIWA (α₁, α₂, α₃) = 〈0 . 4335, 0 . 1252〉 and its score is 0.3083.

when λ = 2, GIFPIWA (α₁, α₂, α₃) = 〈0 . 4203, 0 . 1385〉 and its score is 0.2818.

when λ = 5, GIFPIWA (α₁, α₂, α₃) = 〈0 . 4655, 0 . 0933〉 and its score is 0.3722.

when λ = 10, GIFPIWA (α₁, α₂, α₃) = 〈0 . 5206, 0 . 0382〉 and its score is 0.4824.

If we use the GIFPWA operator proposed by Zhou et al. [42] to aggregate the three IFNs, the results can be obtained as follows when the parameter λ assigned to specific numbers. When λ = 1, GIFPWA (α₁, α₂, α₃) = 〈 0 . 4335, 0 . 0 〉 and its score is 0.4335.

When λ = 2, GIFPWA (α₁, α₂, α₃) = 〈 0 . 4603, 0 . 0 〉 and its score is 0.4603.

When λ = 5, GIFPWA (α₁, α₂, α₃) = 〈 0 . 5218, 0 . 0 〉 and its score is 0.5218.

When λ = 10, GIFPWA (α₁, α₂, α₃) = 〈 0 . 5802, 0 . 0 〉 and its score is 0.5802.

Figure 1 shows the different results based on GIFPWA and GIFPIWA operators.

Definition 6. For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them is w = (w₁, w₂, …, w_n). Then the generalized intuitionistic fuzzy power interactive weighted geometric (GIFPIWG) operator is defined as:

$\begin{matrix} GIFPIWG (α_{1}, α_{2}, \dots, α_{n}) \\ = \frac{1}{λ} ({(λ α_{1})}^{\frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} \otimes \cdot \cdot \cdot \otimes \\ {(λ α_{n})}^{\frac{w_{n} (1 + T (α_{n}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}) \end{matrix}$ (14)

Corollary 3.For a given set of IFNs (α₁, α₂, …, α_n), the standardized importance degree of them isw = (w₁, w₂, …, w_n). Based on the GIFPIWAG operator, the aggregated result was: $\begin{matrix} GIFPIWG (α_{1}, α_{2}, \dots, α_{n}) \\ = 〈 1 - {(1 - \prod_{j = 1}^{n} {(1 - (1 - μ_{j})^{λ} + (1 - (μ_{j} + v_{j}))^{λ})}^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} + \prod_{j = 1}^{n} {(1 - (μ_{j} + v_{j}))}^{λ \frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}})}^{1 / λ}, \\ {(1 - \prod_{j = 1}^{n} {(1 - (1 - μ_{j})^{λ} + (1 - (μ_{j} + v_{j}))^{λ})}^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}} + \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{λ \frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}})}^{1 / λ} \\ - \prod_{j = 1}^{n} (1 - (μ_{j} + v_{j}))^{\frac{w_{j} (1 + T (α_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))}}) 〉 \end{matrix}$ (15)

Example 4. Here we use the data of example 3. Based on the GIFPIWG operator, the results can be obtained as follows.

When λ = 1, GIFPIWG (α₁, α₂, α₃) = 〈0.4839, 0.0749〉 and its score is 0.4090.

When λ = 2, GIFPIWG (α₁, α₂, α₃) = 〈0.5029, 0.0559〉 and its score is 0.4470.

When λ = 5, GIFPIWG (α₁, α₂, α₃) = 〈0.5159, 0.0429〉 and its score is 0.4731.

When λ = 10, GIFPIWG (α₁, α₂, α₃) = 〈0.5182, 0.0406〉 and its score is 0.4777.

Figure 2 shows the different results based on GIFPIWA and GIFPIWG operators.

4 Interval-valued intuitionistic fuzzy interactive aggregation operatorsbased on power mean

In this Section, we intend to extend the IFPIWA and IFPIWG operators to interval IFS [2]. The operations for IIFNs have been defined by Xu [22] and cited widely [10 , 37]. However, Yu [39] pointed out that the operations are not quite perfect and introduced some new operations for IIFNs as follows.

Definition 7. Suppose there are three IIFNs $\tilde{α} = ([a, b], [c, d])$ ${\tilde{α}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{α}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ , the operations were defined as:

$\begin{matrix} {\tilde{α}}_{1} \oplus {\tilde{α}}_{2} \\ = 〈 [1 - (1 - a_{1}) (1 - a_{2}), 1 - (1 - b_{1}) (1 - b_{2})], \\ [(1 - a_{1}) (1 - a_{2}) - (1 - (a_{1} + c_{1})) (1 - (a_{2} + c_{2})), \\ (1 - b_{1}) (1 - b_{2}) - (1 - (b_{1} + d_{1})) (1 - (b_{2} + d_{2})) 〉 \end{matrix}$ (16) $\begin{matrix} {\tilde{α}}_{1} \otimes {\tilde{α}}_{2} \\ = 〈 [(1 - c_{1}) (1 - c_{2}) - (1 - (a_{1} + c_{1})) (1 - (a_{2} + c_{2})), \\ (1 - d_{1}) (1 - d_{2}) - (1 - (b_{1} + d_{1})) (1 - (b_{2} + d_{2})), \\ [1 - (1 - c_{1}) (1 - c_{2}), 1 - (1 - d_{1}) (1 - d_{2})] 〉 \end{matrix}$ (17) $\begin{matrix} λ \tilde{α} = 〈 [1 - (1 - a)^{λ}, 1 - (1 - b)^{λ}], \\ [(1 - a)^{λ} - (1 - (a + c))^{λ}, (1 - b)^{λ} - (1 - (b + d))^{λ}] 〉 \end{matrix}$ (18) $\begin{matrix} (\tilde{α})^{λ} = 〈 [(1 - c)^{λ} - (1 - (a + c))^{λ}, \\ (1 - d)^{λ} - (1 - (b + d))^{λ}], \\ [1 - (1 - c)^{λ}, 1 - (1 - d)^{λ}] 〉 \end{matrix}$ (19)

Definition 8. For a given set of IIFNs $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ , the standardized importance degree of them is w = (w₁, w₂, …, w_n). Then the interval-valued intuitionistic fuzzy power interactive weighted average (IIFPIWA) operator and interval-valued intuitionistic fuzzy power interactive weighted geometric (IIFPIWG) operator were defined as: $\begin{matrix} IIFPIWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, . . ., {\tilde{α}}_{n}) \\ = \frac{(w_{1} (1 + T ({\tilde{α}}_{1}))) {\tilde{α}}_{1} \oplus \cdot \cdot \cdot \oplus (w_{n} (1 + T ({\tilde{α}}_{n}))) {\tilde{α}}_{n}}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))} \end{matrix}$ (20) $\begin{matrix} IIFPIWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = ({\tilde{α}}_{1})^{\frac{w_{1} (1 + T ({\tilde{α}}_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} \otimes \cdot \cdot \cdot \otimes ({\tilde{α}}_{n})^{\frac{w_{n} (1 + T ({\tilde{α}}_{n}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} \end{matrix}$ (21) where $T ({\tilde{α}}_{j}) = \sum_{i = 1, i \neq j}^{n} w_{i} sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}), j = 1, 2, \dots, n$ (22)

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) = 1 - d ({\tilde{α}}_{i}, {\tilde{α}}_{j})$ is the support of ${\tilde{α}}_{i}$ obtained from ${\tilde{α}}_{j}$ , d is a distance measure.

Corollary 4.For a given set of IIFNs $({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n})$ , the standardized importance degree of them isw = (w₁, w₂, …, w_n). Based on the IIFPIWA and IIFPIWG operators, the aggregated results were:

$\begin{matrix} IIFPIWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = ([1 - \prod_{j = 1}^{n} (1 - a_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}, 1 - \prod_{j = 1}^{n} (1 - b_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}], \\ [\prod_{j = 1}^{n} (1 - a_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} - \prod_{j = 1}^{n} (1 - (a_{j} + c_{j}))^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}, \prod_{j = 1}^{n} (1 - b_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} \\ - \prod_{j = 1}^{n} (1 - (b_{j} + d_{j}))^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}]) \end{matrix}$ (23)

$\begin{matrix} IIFPIWG ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ = ([\prod_{j = 1}^{n} (1 - c_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} - \prod_{j = 1}^{n} (1 - (a_{j} + c_{j}))^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}, \prod_{j = 1}^{n} (1 - d_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}} \\ - \prod_{j = 1}^{n} (1 - (b_{j} + d_{j}))^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}], \\ [1 - \prod_{j = 1}^{n} (1 - c_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}, 1 - \prod_{j = 1}^{n} (1 - d_{j})^{\frac{w_{j} (1 + T ({\tilde{α}}_{j}))}{\sum_{j = 1}^{n} w_{j} (1 + T ({\tilde{α}}_{j}))}}]) \end{matrix}$ (24)

5 Job hunting under intuitionistic fuzzy environment

5.1 Problem description

The research of Super and Hall (1978) have shown that the first job out of school is critical to a person’s whole career. A dozen years ago, job hunting is not received enough attentions from Chinese university graduates, since there are few graduates in China and they are arranged to work after graduation [6]. However, it began to change significantly in 2003; the number of graduates increased dramatically and the unemployment has become an inevitable topic for graduates [8, 12]. As a result, job hunting is important choices for college students on the roads they take in their lives [13]. To a certain extent, their choices for the career directly decide their futures. Search a good job to develop own talent and get fully utility is the dream for all college students. As for them, Job hunting is necessary one to up and is the start point for their lives.

Job hunting is so serious, thus college students should take full consideration during the whole process. It always involves many factors for college students’ job hunting. For example, Li Ming is a college graduate and he is trying to find his first job. After send his resumes through internet, email and so on, Li Ming received several interview invitations and he Selects 5 of them. Due to the energy and time limitation, he decides to choose 1 company’ interview invitation. The problem is which company he should choose? After full consideration, He decided to make a decision through the comparison through below aspects.

The first attribute (C₁): The first aspect is the personal development. Personal development is always the top priority during the career selecting process; it includes many factors, Such as promotion chance, training opportunity, the implementation of personal ability and so on. Since it is the first job for Li Ming and it may in relation to his future occupation career, this aspect appears particularly important.

The second attribute (C₂): Second aspect is the work location. Location selection always involves many influence factors. For one thing we need to consider natural factors like climatic factors, People in the South cannot adapt to the northern cold, northern people may not be able to suit for the southern coastal city’ wet. If you do not adjust to the selected location, you may need to search a work again. For another thing, many other influence factors will also play important influences, For example, interpersonal relationships. If you have a lot of social relations can be used in a city, such as relatives and friends, classmates, parents, colleagues superior in there, these social relations can come in handy when necessary, at least temporary accommodation.

The third attribute (C₃): Compensation is the third aspect need to consider during the decision making process. Compensation is not only referring to wage, it also includes various benefits, such as social and commercial insurance, annual leave, health check-up and so on. Actually, benefits provided by company can improve one’s life quality in a certain extent. Besides, we need to combine the local price levels to be the judge of the wage level, it is always complicated.

The forth attribute (C₄): The last but not the least factor is the work environment. A job with competitive compensation in an ideal location is always fascinating, but unpleasant work environment may destroy the beauty and bring bad mood, pressures and so on. Working environment including soft and hard environment Soft environment always refer to the working atmosphere, group sprint and so on. While hard environment means to the objective external conditions such as office layout.

Since the four attributes are difficult to accurately describe, IFS theory is adopted to depict the four attributes. During the evaluation process, we assume the weight of the four attributes are (0.2, 0.3, 0.3, 0.2)^T. The performance of the five potential jobs is described in Table 1.

5.2 Job selection based on GIFIPWA operator

In order to find the most suitable work from the five alternative jobs, the GIFIPWA operator was adopted. The aggregated IFNs can be obtained and were shown in Tables 2 and 3. The ranking results of the five jobs based on GIFPIWA operator was shown by Fig. 3. The detailed calculation process were omitted in this section and just present the calculate process for Job1.

Let a₁ =〈 0.2, 0.7 〉, a₂ =〈 0.3, 0.6 〉, a₃ =〈 0.4, 0.2 〉 and a₄ =〈 0.3, 0.4 〉

Then, the distance measure between the four IFNs can be calculated as follows. $\begin{matrix} d_{12} & = & 0 . 1000, d_{13} = 0 . 3808, d_{14} = 0 . 2236, \\ d_{23} & = & 0 . 2915, d_{24} = 0 . 1414, d_{34} = 0 . 1581 \end{matrix}$ and $\begin{matrix} sup_{12} & = & 0 . 9000, sup_{13} = 0 . 6192, sup_{14} = 0 . 7764, \\ sup_{23} & = & 0 . 7085, sup_{24} = 0 . 8586, sup_{34} = 0 . 8419 \end{matrix}$

Therefore, $\begin{matrix} \frac{w_{1} (1 + T (α_{1}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 2044 \\ \frac{w_{2} (1 + T (α_{2}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 2978 \\ \frac{w_{3} (1 + T (α_{3}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 2864 \\ \frac{w_{4} (1 + T (α_{4}))}{\sum_{j = 1}^{n} w_{j} (1 + T (α_{j}))} = 0 . 2113 \end{matrix}$

We can get,

When λ = 1, GIFPIWA (α₁, α₂, α₃, α₄) = 〈0.3117, 0.5007〉 and its score is –0.1890.

When λ = 2, GIFPIWA (α₁, α₂, α₃, α₄) = 〈0.3678, 0.4446〉 and its score is –0.0768.

When λ = 5, GIFPIWA (α₁, α₂, α₃, α₄) = 〈0.4686, 0.3438〉 and its score is 0.1248.

When λ = 10, GIFPIWA (α₁, α₂, α₃, α₄) = 〈0.5238, 0.2885〉 and its score is 0.2353.

Systematic comparison with GIFPWA operator proposed by Zhou et al.

In the following, we use the GIFPWA operator to select the best Job from the five alternatives. The aggregated IFNs and corresponding scores and ranking results are shown in Tables 4 and 5 in details. Figure 4 illustrates the ranking results of the five jobs when The values of λ are in [0, 4].

Remarks.

The final results based on GIFPWA and GIFPIWA are different. From Fig. 3 we can find out that the Job5 is the most inappropriate working while the Fig. 4 shows that the job5 is the second behind Job3 among five Jobs. Table 6 shows the detail performance of the five Jobs as follows.

From Table 6, we know that the performances of the four attributes are <0.1, 0.8>, <0.2, 0.7>, <0.2, 0.7 >and <0.1, 0.0>. By comparison with performance of the remaining four Jobs, we can know that the Job5 performs worst on each different attribute. Therefore, we have no reason to believe that the Job5 can rank second among five Jobs. In other words, the ranking results based on GIFPIWA are more feasible.

From Table 4 we find that the non-membership degree of four attribute for Job5 are all zero which seems very strange. This phenomenon is due to the fourth attribute value which has the zero non-membership degree. Therefore, the GIFPWA operator proposed by Zhou et al. [42] is not applicable to aggregate IFNs for Job5. The GIFPIWA operator proposed in this paper can help ease the problem.

6 Concluding remarks and future works

Power mean is an effective tool for dealing with the information aggregation problem, especially in the condition that the aggregated arguments are correlated. However, existing intuitionistic fuzzy power mean operators cannot work well in some circumstances. In this paper, we introduced some new power based aggregation operators for IFNs to improve the existing research. The academic Contribution of this stud can be summarized: (1) As discussed in Sections 3 and 4, the proposed operators can be used to deal with the situation where the membership or the non-membership of the IFNs takes the value of zero. (2) The new proposed operators can effectively consider the relationship between the IFNs being aggregated. For the above reason, the proposed operators are more practical in practical applications. (3) Job hunting problem for college students have been investigated based on proposed operators. Furthermore, the systematic comparison with other operator is provided to show the advantage of our proposed operators in solving the job hunting problem.

The further works of this paper are two folds: (1) The proposed operators can be extended to the hesitant fuzzy environment, interval-valued hesitant fuzzy environment and dual hesitant fuzzy environment. (2) The research results of this paper can also be applied to other decision making problem, such as sustainable energy planning, engineering technology, supplier selection and so on.

Footnotes

Acknowledgments

This work has been supported by China National Natural Science Foundation (No. 71301142), Zhejiang Science & Technology Plan of China (2015C33024), Zhejiang Provincial Natural Science Foundation of China (No. LQ13G010004), Project Funded by China Postdoctoral Science Foundation (No. 2014M550353) and the National Education Information Technology Research (No. 146242069).

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