Normal wiggly hesitant fuzzy TODIM approach for multiple attribute decision making

Abstract

A normal wiggly hesitant fuzzy set (NWHFS) is a powerful and useful tool to dig the potential indeterminacy of decision makers (DMs) in the process of expressing their preferences, which can be considered as an extended form of the traditional hesitant fuzzy set (HFS). The NWHFSs can not only retain the original hesitant fuzzy information completely, but also explore potential uncertainty of theses information. TODIM is an effective method to capture the psychological behavior based on prospect theory. Considering the advantages of NWHFS and TODIM method, in this paper, we define the distance measure of any two normal wiggly hesitant fuzzy elements (NWHFEs), and put forward an extended normal wiggly hesitant fuzzy TODIM (NWHF-TODIM) approach to handle multiple attribute decision making (MADM) problems with normal wiggly hesitant fuzzy (NWHF) information. Then we use the extended NWHF-TODIM method to rank alternatives and select an ideal one. Lastly, we compare it with two existing approaches to verify the rationality and validity of the proposed approach.

Keywords

Normal wiggly hesitant fuzzy sets multiple attribute decision making TODIM

1 Introduction

There are a lots of multiple-attribute decision making (MADM) in real life, and the more and more people are facing uncertain decision-making problems in all aspects of their lives, such as which bag is suitable for shopping today, which kind of fruit to buy, how to spend a wonderful time and so on. In the face of these decisions, it is particularly significant to help DMs in expressing their assessment information based on some natural contradictory criteria and to summarize all the information for the final ranking results of the alternatives. Thus, a series of multi-attribute decision-making models have been put forward and widely applied in practice. Since Zadeh [26] firstly put forward the concept of fuzzy sets (FSs) to describe the fuzziness, many efficient representative models had been extensively studied by scholars, such as interval-valued intuitionistic fuzzy sets (IVIFSs) [1], interval type-2 FSs (IT2FSs) [10], hesitant fuzzy sets (HFSs) [19], interval-valued hesitant fuzzy linguistic sets (IVHFLSs) [21], the single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) [17], Pythagorean fuzzy sets (PFSs) [16] and so on. The target of the above various types of fuzzy sets is to express the uncertain and complex evaluation information of DMs more effectively. In recent decades, for the sake of adapting to a variety of different application environment, these different forms of FSs have been recognized in academic research widely and have obtained rich research achievements. Among them, the conception of HFSs was defined by Torra [19], which is free to deal with uncertainties, allowing the DMs to describe their assessment information with a set of values that may belong to [0, 1]. Nevertheless, for DMs, the key problem in dealing with a complex real MADM is that how to determine a crisp value based on a given criterion to express their uncertainty and fuzziness. For instance, if DMs cannot determine which specific number should be given for an alternative under a specific attribute, he/she can give several numbers instead of a specific number to represent his/her evaluation information. Hence, compared with other extended forms of FSs, the HFSs has a wider range of applications and more practical significance.

However, in the context of many practical decision-making applications, DMs need to express their opinions in more complex forms than just some specific numerical values. In other words, DMs cannot represent all uncertain evaluation points only from some precise values. To put it another way, the existing representation model cannot contain all the hesitant information given by DMs in a centralized way, that is, DMs cannot give complete evaluation information by the existing models. Therefore, in order to cope with this kind of complex MADM problems more effectively and to mine the potential decision information from the original assessment in the form of HFSs, Ren et al. [18] firstly put forward the concept of the normal wiggly hesitant fuzzy sets (NWHFSs). This new extended form of HFSs shows that it is the more suitable and reasonable approach to obtain the reliable hesitant fuzzy information in the decision-making process by digging DMs’ potential uncertainty and fuzziness hidden behind their original assessments. It is worth emphasizing that the NWHFSs are based on the hypothesis that people’s uncertainty to any possible values can be treated as a normal wiggly range, and the range is in contact with the given original form of HFSs of DMs. The NWHFS is expressed by the real preference degree (RPD) and the normal wiggly range (NWR), which can not only retain the original preferences of DMs but also more effectively mine the deeper uncertain information. In addition, the outstanding advantage of NWHFS is that it can express potential uncertain information and help DMs obtain more reasonable and accurate preference information.

Decision-making methods play a crucial role in coping with the decision-making problems and have been widely used in multiple forms of practical applications in different fields, such as the VIKOR method [9 , 22], the MULTIMOORA method [4, 12], the ELECTRE method [3, 16], the TOPSIS method [14, 20] and the PROMETHEE method [28, 29]. Generally, in the process of the evaluation and ranking of alternatives, different experts have diverse views toward unknown information. Therefore, taking the risk attitudes into account is very significant for them to give the assessment. TODIM is a highly effective method to capture the psychological behavior based on prospect theory [5, 8], and now, various extension forms of the TODIM methods have been introduced and applied to MADM problems in a variety of different environments with different information forms. Qin et al. [15] proposed an extended TODIM method to solve the green supplier selection problems under the interval type-2 fuzzy environment. Ji et al. [7] put forward a projection-based TODIM method to solve the personnel-selection problems under multi-valued neutrosophic environments. Li et al. [9] proposed an extended TODIM method for multi-attribute risk decision making problem in emergency response. Zhang and Xu [27] put forward a hesitant fuzzy TODIM method to evaluate the sustainable water management efficiency. Zhu et al. [30] proposed an extended TODIM approach in which the dominance degree of failure modes is calculated by grey relational analysis to determine the risk priority of failure modes. The above studies make good use of the TODIM method to mine the psychological behavior of DMs to deal with the MADM problems. However, until now, there is very little research on the TODIM method under normal wiggly hesitant fuzzy environment. Moreover, NWHFS can explore potential and uncertain information behind the DMs’ feelings rather than directly given by someone.

Based on above analysis, the motivation of this paper is summarized as follows. NWHFSs can dig deeper uncertain information in line accordance with keeping the original hesitant fuzzy information, and the TODIM method can take the DMs’ psychological behavior into account. Obviously, it is necessary to extend the TODIM method to NWHFE to deal with the complex hesitant fuzzy information. Thus, considering the psychological behavior of DMs and maintaining the advantages of traditional TODIM method and NWHF information, we propose an extended NWHF-TODIM method to help DMs in selecting desirable products. The goals are to:

1) Define the distance measure of any two NWHFEs to enrich the theories of NWHFSs;

2) Propose a normal wiggly hesitant fuzzy TODIM method to cope with the MADM problems with NWHF information;

3) Make full use of the extended NWHF-TODIM method to rank alternatives and give an ideal choice;

4) Compare with two existing approaches to verify the rationality and validity of the approach we proposed.

In order to achieve these objectives, the rest of this paper is briefly described as follows. In Section 2, we briefly review some basic conceptions and introduce the traditional TODIM method. In Section 3, we put forward the distance measure of NWHFEs. In Section 4, we put forward a NWHF-TODIM method for coping with the MADM under NWHF environment. In Section 5, we give an example about the evaluation and ranking of high-level personnel training to show the implementation process, and to demonstrate the rationality and validity of our proposed method. The innovations of this paper are simply summarized and the conclusions are given in Section 6.

2 Preliminaries

In this section, to understand the main idea of this paper better, we briefly review some basic concepts, such as HFSs and NWHFSs, and their basic operational laws. By reviewing these basic concepts, it is helpful for readers to better understand the innovation and central idea of this paper. Then we will introduce the steps of the classical TODIM approach, which can help understand the extended form of the TODIM method.

2.1 Hesitant fuzzy sets

Definition 1 [19]. Let Γ be a reference set and a hesitant fuzzy set (HFS) HF on Γ can be defined by a function h_HF (τ) that returns a subset belonging to [0,1],which can be described as followed: $HF = {< τ, h_{HF} (τ) > | τ \in Γ}$ (1) where h_HF (τ) is a set of some computable values in [0,1], indicating the possible membership degrees of each element τ ∈ Γ to the set HF. As a matter of convenience, h_HF = h_HF (τ) is simplified as h_HF and is defined as a hesitant fuzzy element (HFE) [23] and Hcan represent the set of all HFEs on Γ.

In fact, even if DMs can precisely make their cognitive preference of an alternative by HFE, we cannot obtain all the uncertainty information hidden behind the original evaluation information given by DMs merely from several crisp values. That is to say, DMs cannot express all their concealed uncertain feelings by using fixed expression such as HFE when they cope with decision-making problems. For the purpose of mining deeper uncertain information, based on a hypothesis that human uncertainty is like a swinging clock within the bounds of a certain value and DMs’ actual degree of preference, Ren et al. [18] bring forward the concept of NWHFS, which includes not only the original information but also the potential uncertain information of HFSs.

2.2 Normal wiggly hesitant fuzzy sets

Definition 2 [18]. Let h ={ δ₁, δ₂, ·· · , δ_# h } be a HFE, and $\bar{h} = \frac{1}{# h} \sum_{i = 1}^{# h} δ_{i}$ be the average value of all values in h and $ɛ_{h} = \sqrt{\frac{1}{# h} \sum_{i = 1}^{# h} {(δ_{i} - \bar{h})}^{2}}$ be the standard deviation in h. The function $\tilde{g}$ represents the mapping from h to [0, ɛ_h], which is called the normal wiggly range of δ_i, and the function $\tilde{g} (δ_{i})$ can be described as followed: $\tilde{g} (δ_{i}) = ɛ_{h} e^{- \frac{(δ_{i} - \bar{h})^{2}}{2 ɛ_{h}^{2}}}$ (2) where # h denotes the association granularity of HFE h. Ren et al. [18] indicated that the function $\tilde{g} (δ_{i})$ is a similar normal function on the basis of the average and standard deviation of all values in h.

Definition 3 [18]. Let h ={ δ₁, δ₂, ·· · , δ_# h } be a HFE, and $\tilde{h} = {\tilde{δ} = δ_{i} / sum (h) | δ_{i} \in h}$ represent the standardized HFE, where $sum (h) = \sum_{i = 1}^{# h} δ_{i}$ is the sum of all values in h. The real preference degree rpd (h) of the DM in a HFE h can be estimated on the basis of the orness measure [25], which can be expressed in the following ways:

$rpd (h) = {\begin{matrix} \sum_{i = 1}^{# \tilde{h}} {\tilde{δ}}_{i} (\frac{# \tilde{h} - i}{# \tilde{h} - 1}), if mean (h) < 0.5; \\ 1 - \sum_{i = 1}^{# \tilde{h}} {\tilde{δ}}_{i} (\frac{# \tilde{h} - i}{# \tilde{h} - 1}), if mean (h) > 0.5; \\ 0.5, if mean (h) = 0.5; \end{matrix}$ (3)

where mean (h) is the average value of all values in h. And the degree of real preference can represent a DM uncertainty for each value in HFE.

Definition 4 [18]. Let Γ be a reference set and HF ={ (τ, h (τ)) |τ ∈ Γ } be a HFS on Γ. Then, the homologous normal wiggly hesitant fuzzy set (NWHFS) NWHF on Γ can be expressed in the following ways: $NWHF = {〈 τ, h (τ), ζ (h (τ)) 〉 | τ \in Γ}$ (4) where h (τ) is a hesitant fuzzy element in the hesitant fuzzy set HF. $ζ (h (τ)) = {δ_{1}^{⌣}, δ_{2}^{⌣}, \dots, δ_{# h (τ)}^{⌣}}$ , $δ_{i}^{⌣} = {μ_{i}^{L}, μ_{i}^{M}, μ_{i}^{U}} = {max (δ_{i} - \tilde{g} (δ_{i}), 0), (2 rpd (h) - 1) \tilde{g} (δ_{i}) + δ_{i}, min (δ_{i}) + \tilde{g} (δ_{i}), 1}$ , and δ_i is one of the values of h (τ), $\tilde{g} (δ_{i})$ represents the normal wiggly parameter of δ_i, and rpd (h) represents the real preference degree of h (τ). And ζ (h (τ)) is called a normal wiggly element (NWE). As a matter of convenience, the pairwise 〈h (τ) , ζ (h (τ))〉 is simplified as 〈h, ζ (h)〉 and can be called a normal wiggly hesitant fuzzy element (NWHFE).

The swing scope of every value in the HFE is the area of the triangle formed by the homologous number of triangles in the NWE. It is worth mentioning that ζ (h (τ)) is similar to a triangular fuzzy number, but maybe not an isosceles triangle.

Example 1: Given a reference set Γ = { τ₁, τ₂, τ₃ }. Then a HFS $\overset{⌣}{H}$ can be $\begin{matrix} \overset{⌣}{H} = & {〈 τ_{1}, (0.1, 0.2, 0.3) 〉, 〈 τ_{2}, (0.7, 0.8, 0.9) 〉, \\ 〈 τ_{3}, (0.4, 0.6, 0.9) 〉} \end{matrix}$

According to the definitions 3 and 4, a NWHFS ${NWHF}_{\overset{⌣}{H}}$ can be obtained:

${NWHF}_{\overset{⌣}{H}} = {\begin{matrix} 〈 τ_{1}, (0.1, 0.2, 0.3), {(0.0614, 0 . 0871, 0.1386), (0 . 1184, 0 . 1728, 0.2816), (0 . 2614, 0 . 2871, 0.3386)} 〉 \\ 〈 τ_{2}, (0.7, 0.8, 0.9), {(0.6614, 0.70 32, 0.7386), (0.7184, 0.8068, 0.8816), (0.8614, 0 . 9032, 0.9386)} 〉 \\ 〈 τ_{3}, (0.4, 0.6, 0.9), {(0 . 2922, 0 . 4284, 0.5078), (0 . 3972, 0 . 6534, 0.8028), (0 . 8115, 0 . 9233, 0.9885)} 〉 \end{matrix}}$

Taking the NWHFE 〈 (0.1, 0.2, 0.3) , { (0.0614, 0 . 0871, 0 . 1386) , (0 . 1184, 0 . 1728, 0 . 2816) , (0 . 2614, 0 . 2871, 0.3386) } 〉 as an example, (0.1, 0.2, 0.3) is the original hesitation fuzzy information, and { (0.0614, 0 . 0871, 0 . 1386) , (0 . 1184, 0 . 1728, 0 . 2816) , (0 . 2614, 0 . 2871, 0.3386)} is the potential uncertain information mined from the original hesitation fuzzy information (0.1, 0.2, 0.3). For easily understanding, we give the graphic form of the NWHFEs corresponding to HFEs (0.1, 0.2, 0.3) and (0.7, 0.8, 0.9) to elaborate the specific meaning of the NWHFS.

From Figs. 1 and 2, it’s not hard to find that the normal wiggly range ζ (h) of each value δ_i in HFE h ={ δ₁, δ₂, ·· · , δ_# h } is a triangle area. Among them, the abscissa coordinates of the bottom end points of each triangle region are denoted as $δ_{i} - \tilde{g} (δ_{i})$ and $δ_{i} + \tilde{g} (δ_{i})$ , that is to say, δ_i swings around the normal wiggly parameter $\tilde{g} (δ_{i})$ . As described by Ren et al. [18], human cognitive uncertainty is like the left and right swing in a range centered on one value. Therefore, in order to mine the potential uncertainty of the DM in the hesitant fuzzy evaluation, based on the mean value and standard deviation of values in HFE, the analogous normal function $\tilde{g}$ is constructed to obtain the normal wiggly parameter $\tilde{g} (δ_{i})$ of each value in HFE, and further obtain the left and right wiggly ranges $δ_{i} - \tilde{g} (δ_{i})$ and $δ_{i} + \tilde{g} (δ_{i})$ of each value, that is, the uncertainty wiggly range of the DM.

Fig. 1

The NWHFE of the corresponding HFE (0.1, 0.2, 0.3).

Fig. 2

The NWHFE of the corresponding HFE (0.7, 0.8, 0.9).

In addition, we think that the DM will tend to the smaller values (the average value is less than 0.5) when he/she uses (0.1, 0.2, 0.3) to give the evaluation, and the DM will tend to bigger values (the average value is greater than 0.5) when he/she uses (0.7, 0.8.0.9) to give the evaluation. Here, this potential preference information is reflected by the real preference degree function rpd (h). In Figs. 1 and 2, the abscissa coordinate of each triangle vertex is $(2 rpd (h) - 1) \tilde{g} (δ_{i}) + δ_{i}$ . Among them, the real preference degree rpd (h) of the DM determines the bias degree of the triangular vertex. Take Figs. 1 and 2 for example. Because the average value of (0.1, 0.2, 0.3) is less than 0.5, the vertices of all triangles in Fig. 1 deviate to the left. On the contrary, because the average value of (0.7, 0.8.0.9) is significantly greater than 0.5, the vertices of all triangles in Fig. 2 deviate to the right. Therefore, the NWHFE can accurately describe the potential preference of the DM in the evaluation process. Combined with the above analysis, through the analogous normal function $\tilde{g}$ and the real preference degree function rpd (h), the potential uncertain information of DMs hidden in the evaluation process can be mined.

From Definition 4 and Example 1, it’s easy to find that the calculated NWHFS not only retains the original hesitant fuzzy information, but also mines the potential uncertain information of DMs.

Definition 5 [18]. Let 〈h₁, ζ (h₁)〉 and 〈h₂, ζ (h₂)〉 be two different NWHFEs, then

$\begin{matrix} 1) 〈 h_{1}, ζ (h_{1}) 〉 \oplus 〈 h_{2}, ζ (h_{2}) 〉 \\ = 〈 \underset{δ_{1} \in h_{1}, δ_{2} \in h_{2}}{\cup} δ_{1} + δ_{2} - δ_{1} δ_{2}, \underset{δ_{1}^{⌣} \in ζ (h_{1}), δ_{2}^{⌣} \in ζ (h_{2})}{\cup} δ_{1}^{⌣} \oplus δ_{2}^{⌣} 〉 \end{matrix}$

$\begin{matrix} 2) 〈 h_{1}, ζ (h_{1}) 〉 \otimes 〈 h_{2}, ζ (h_{2}) 〉 \\ = 〈 \underset{δ_{1} \in h_{1}, δ_{2} \in h_{2}}{\cup} δ_{1} δ_{2}, \underset{δ_{1}^{⌣} \in ζ (h_{1}), δ_{2}^{⌣} \in ζ (h_{2})}{\cup} δ_{1}^{⌣} \otimes δ_{2}^{⌣} 〉 \end{matrix}$ $3) {(〈 h_{1}, ζ (h_{1}) 〉)}^{λ} = {\underset{δ_{1} \in h_{1}}{\cup} δ_{1}^{λ}, \underset{δ_{1}^{⌣} \in ζ (h_{1})}{\cup} δ_{1}^{⌣} λ}, λ > 0$ $\begin{matrix} 4) λ (〈 h_{1}, ζ (h_{1}) 〉) \\ = {\underset{δ_{1} \in h_{1}}{\cup} 1 - {(1 - δ_{1})}^{λ}, \underset{δ_{1}^{⌣} \in ζ (h_{1})}{\cup} λ δ_{1}^{⌣}}, λ > 0 \end{matrix}$

Definition 6 [18]. Suppose 〈h, ζ (h)〉 is a NWHFE, $\bar{h}$ is the average value and ɛ_h is the standard deviation value of all values in h. Then, the scoring function $ℕ (h)$ of a NWHFE 〈h, ζ (h)〉 is defined as:

$\begin{matrix} ℕ (h) = & S_{NW} (〈 h, ζ (h) 〉) = σ (\bar{h} - ɛ_{h}) \\ + (1 - σ) (\frac{1}{# h} \sum_{i = 1}^{# h} ({\bar{\overset{⌣}{δ}}}_{i} - ɛ_{δ_{i}^{⌣}})) \end{matrix}$ (5) where ${\bar{\overset{⌣}{δ}}}_{i} = \frac{μ_{i}^{L} + μ_{i}^{M} + μ_{i}^{U}}{3}$ and $ɛ_{δ_{i}^{⌣}} = \sqrt{(μ_{i}^{L})^{2} + (μ_{i}^{M})^{2} + (μ_{i}^{U})^{2} - μ_{i}^{L} μ_{i}^{M} - μ_{i}^{L} μ_{i}^{U} - μ_{i}^{M} μ_{i}^{U}}$ . σ ∈ (0, 1) can be considered as DMs’ confidence level to their original hesitant fuzzy information.

For ease of comparison and analysis, we set σ = 1/2 in this paper to mean that the DMs have a 50% confidence in its original hesitation ambiguity. In practical decision problems, the parameter σ should be confirmed in line accordance with the knowledge level of the decision support system on relevant problems and the ability to solve problems.

Definition 7 [18]. Let 〈h₁, ζ (h₁)〉 and 〈h₂, ζ (h₂)〉 be two diverse NWHFEs, then we call $ℕ (h_{1})$ and $ℕ (h_{2})$ are their scoring function values.

If $ℕ (h_{1}) > ℕ (h_{2})$ , then 〈h₁, ζ (h₁)〉 is prevail over 〈h₂, ζ (h₂)〉, which can be denoted as 〈h₁, ζ (h₁)〉 ≻ 〈 h₂, ζ (h₂) 〉. If $ℕ (h_{1}) = ℕ (h_{2})$ , then 〈h₁, ζ (h₁)〉 and 〈h₂, ζ (h₂)〉 are indistinguishable, which can be denoted as 〈h₁, ζ (h₁)〉 ∼ 〈 h₂, ζ (h₂) 〉. If $ℕ (h_{1}) < ℕ (h_{2})$ , the 〈h₁, ζ (h₁)〉 is inferior to 〈h₂, ζ (h₂)〉, which can be denoted as 〈h₁, ζ (h₁)〉 ≻ 〈 h₂, ζ (h₂) 〉.

2.3 The classical TODIM approach

The classical TODIM approach is to measure the dominance degree of each alternative to replace the others by establishing a multi-criteria value function φ_c (A_i, A_j) based on the theory of foreground. In addition, the TODIM approach can effectively handle the MCDM problems considering the psychological behavior of DM. An algorithm for the TODIM method is summarized as follows.

Suppose C_j (j = 1, 2, ⋯ n) is the referred criteria, whose weight vector is w = (w₁, w₂, ⋯ , w_n) ^T, with w_j ∈ [0, 1] (j = 1, 2, ⋯ , n) and $\sum_{j = 1}^{n} w_{j} = 1$ . For all alternatives A_i (i = 1, 2, ⋯ m), let H = (h_ij) _m×n be a decision matrix, where x_ij represents the criterion value given by the DMs for the alternative A_i with respect to the criterion C_j.

For convenience, suppose M = (1, 2, ⋯ m) and N = (1, 2, ⋯ n).

Step 1: Identify the decision matrix H = (h_ij) _m×n and normalize it into the matrix $\tilde{H} = {[{\tilde{h}}_{ij}]}_{m \times n}$ (i ∈ M, j ∈ N).

Step 2: Calculate the relative attribute weight w_jr for attribute C_j to reference attribute C_r, i.e., $w_{jr} = \frac{w_{j}}{w_{r}} (j \in N)$ (6) where w_r = max{ w_j|j ∈ N }.

Step 3: Compute the dominance degree of alternative A_i over alternative A_s (s ∈ M) according to attribute C_j,

$\begin{matrix} φ_{j} (A_{i}, A_{s}) \\ = {\begin{matrix} \sqrt{\frac{w_{jr} (h_{ij} - h_{sj})}{\sum_{j = 1}^{n} w_{jr}}}, if (h_{ij} - h_{sj}) > 0 \\ 0, if (h_{ij} - h_{sj}) = 0 \\ \frac{- 1}{θ} \sqrt{\frac{(\sum_{j = 1}^{n} w_{jr}) (h_{ij} - h_{sj})}{w_{jr}}}, if (h_{ij} - h_{sj}) < 0 \end{matrix} \end{matrix}$ (7)

where parameter θ denotes attenuation factor of the losses, which can be adjusted based on the specific problems, h_ij-h_sj represents the gain of the alternative A_i over the alternative A_s concerning C_j if (h_ij-h_sj) > 0, and the loss if (h_ij-h_sj) < 0.

Step 4: Compute the overall dominance degree of each alternative A_i over each alternative A_s by $ϑ (A_{i}, A_{s}) = \sum_{j = 1}^{n} φ_{j} (A_{i}, A_{s})$ (8)

Step 5: Compute the global value of the alternative A_i by normalizing the eventual matrix of dominance based on the following formula:

$\begin{matrix} ξ (A_{i}) = & \frac{\sum_{s = 1}^{m} ϑ (A_{i}, A_{s}) - min_{i} (\sum_{s = 1}^{m} ϑ (A_{i}, A_{s}))}{max_{i} (\sum_{s = 1}^{m} ϑ (A_{i}, A_{s})) - min_{i} (\sum_{s = 1}^{m} ϑ (A_{i}, A_{s}))}, \\ i \in M \end{matrix}$ (9)

Step 6: According to the global values of alternatives, rank all the alternatives. And the conclusion is that the better alternative A_i is on the basis of the bigger ξ (A_i) (i ∈ M).

3 Distance measure of normal wiggly hesitant fuzzy sets

Definition 8 [24]. It should be noted that different HFEs may have various cardinal numbers, and the values are generally out of orders, thus, for the sake of convenience, we can arrange them in any order. Given two HFEs h₁ and h₂, and | # h₁|, | # h₂| denotes the numbers of values in h₁ and h₂ respectively. in order to ensure the correct operation between NWHFEs, the shorter one should be added until h₁ and h₂ have the same length when | # h₁| ≠ | # h₂|, Based on the maximum terms h⁺ and minimum terms h^-, Zhu and Xu [24] introduced a method which could use the parameter λ to add different values to the HFE, as follows: $h^{*} = λ h^{+} + (1 - λ) h^{-}$ (10)

Where, λ (0 ≤ λ ≤ 1) is an optimized parameter to reflect the risk preference of DMs. Without losing generality, λ takes 1/2 here, i.e., h^* = 1/2 (h⁺ + h^-).

Definition 9 [18]. Let $ϒ_{1} = 〈 h_{1}, ζ (h_{1}) 〉 = 〈 {τ_{1}}, {ς_{1}^{⌣}} 〉$ and $ϒ_{2} = 〈 h_{2}, ζ (h_{2}) 〉 = 〈 {τ_{2}}, {ς_{2}^{⌣}} 〉$ be two given NWHFEs. If d (ϒ₁, ϒ₂) meets the following conditions:

(1) 0 ⩽ d (ϒ₁, ϒ₂) ⩽ 1,

(2) d (ϒ₁, ϒ₂) = 0 if and only if ϒ₁ = ϒ₂,

(3) d (ϒ₁, ϒ₂) = d (ϒ₂, ϒ₁),

(4) Let $ϒ_{3} = 〈 h_{3}, ζ (h_{3}) 〉 = 〈 {τ_{3}}, {δ_{3}^{⌣}} 〉$ be ∀ NWHFEs, then d (ϒ₁, ϒ₃) ⩽ d (ϒ₁, ϒ₂) + d (ϒ₂, ϒ₃).

Then, d (ϒ₁, ϒ₂) can be called distance measure (DME) between ϒ₁ and ϒ₂.

Definition 10. Let $ϒ_{a} = 〈 h_{a}, ζ (h_{a}) 〉 = 〈 {τ_{a}}, {δ_{a}^{⌣}} 〉$ and $ϒ_{b} = 〈 h_{b}, ζ (h_{b}) 〉 = 〈 {τ_{b}}, {δ_{b}^{⌣}} 〉$ be two given NWHFEs, where $h_{a} = {τ_{k}^{a} | k = 1, 2, \dots, | # h_{a} |}$ , $h_{b} = {τ_{k}^{b} | k = 1, 2, \dots, | # h_{b} |}$ , $ζ (h_{a}) = {δ_{k}^{⌣} a | k = 1, 2, \dots, | # h_{a} |} = {(μ_{k}^{La}, μ_{k}^{Ma}, μ_{k}^{Ua}) | k = 1, 2, \dots, | # h_{a} |}$ and $ζ (h_{b}) = {δ_{k}^{⌣} b | k = 1, 2, \dots, | # h_{b} |} = {(μ_{k}^{Lb}, μ_{k}^{Mb}, μ_{k}^{Ub}) | k = 1, 2, \dots, | # h_{b} |}$ .

Suppose that HFEs in h_a and h_b are arranged in an ascending order. Among them, | # h_a| = | # h_b| = H should be held. Otherwise,ϒ_j (j = a or b) with fewer cardinalities needs to be added to maintain the same length between ϒ_a and ϒ_b, and the normal wiggly range should be extended accordingly. Then, if we have various preference between the HFE and the NEW, we can put forward the distance measure by combing with preference coefficient ρ, which is listed as follows:

$\begin{matrix} d (ϒ_{a}, ϒ_{b}) = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | \\ + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) \end{matrix}$ (11) where ρ ∈ [0, 1] is considered as a measure of the rationality of the original fuzzy information given by DMs, and is called the confidence level [6]. For ease of comparison and analysis, in the calculation process of this paper, we choose a confidence level of 50%, that is, ρ = 0.5.

Proof. Firstly, we prove that Equation (11) satisfies the condition (1).

It is easy to observe that $0 ⩽ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) ⩽ 1$ and $\begin{matrix} 0 ⩽ (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | \\ + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) ⩽ 1 . \end{matrix}$

Then, $0 ⩽ ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) ⩽ ρ$ and $\begin{matrix} 0 ⩽ (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | \\ + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) ⩽ 1 - ρ . \end{matrix}$

Thus, $\begin{matrix} 0 ⩽ ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | \\ + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) ⩽ ρ + (1 - ρ) = 1 \\ 0 ⩽ d (ϒ_{a}, ϒ_{b}) ⩽ 1 . \end{matrix}$

Secondly, we prove that Equation (11) satisfies the condition (2).

If d (ϒ_a, ϒ_b) = 0,

Then, $ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) = 0$

Thus, $ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) = 0$ and $(1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) = 0$ ,

Thus, $\sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} | = 0$ and $(\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |)) = 0$ ,

Thus, $| τ_{k}^{a} - τ_{k}^{b} | = 0$ and $(| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |) = 0$ ,

Thus, $τ_{k}^{a} - τ_{k}^{b} = 0$ and $μ_{k}^{La} - μ_{k}^{Lb} = 0$ , $μ_{k}^{Ma} - μ_{k}^{Mb} = 0$ , $μ_{k}^{Ua} - μ_{k}^{Ub} = 0$ , ϒ₁ = ϒ₂.

If ϒ₁ = ϒ₂, $τ_{k}^{a} - τ_{k}^{b} = 0$ and $μ_{k}^{La} - μ_{k}^{Lb} = 0$ , $μ_{k}^{Ma} - μ_{k}^{Mb} = 0$ , $μ_{k}^{Ua} - μ_{k}^{Ub} = 0$ ,

Thus, $\sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} | = 0$ and $(\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |)) = 0$ ,

Thus, $ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) = 0$ , $d (ϒ_{a}, ϒ_{b}) = 0$

Thirdly, we prove that Equation (11) satisfies the condition (3). $\begin{matrix} d (ϒ_{b}, ϒ_{a}) = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{b} - τ_{k}^{a} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{Lb} - μ_{k}^{La} | \\ + | μ_{k}^{Mb} - μ_{k}^{Ma} | + | μ_{k}^{Ub} - μ_{k}^{Ua} |))) \end{matrix}$ $\begin{matrix} = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} \\ (| μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))) \\ = d (ϒ_{a}, ϒ_{b}) \end{matrix}$

Finally, we prove that Equation (11) satisfies the condition (4), i.e., d (ϒ_a, ϒ_c) ⩽ d (ϒ_a, ϒ_b) + d (ϒ_b, ϒ_c).

Let $ϒ_{c} = 〈 h_{c}, ζ (h_{c}) 〉 = 〈 {τ_{c}}, {δ_{c}^{⌣}} 〉$ , in which $h_{c} = {τ_{k}^{c} | k = 1, 2, \dots, | # h_{c} |}$ , $\begin{matrix} ζ (h_{c}) & = {δ_{k}^{⌣} c | k = 1, 2, \dots, | # h_{c} |} \\ = {(μ_{k}^{Lc}, μ_{k}^{Mc}, μ_{k}^{Uc}) | k = 1, 2, \dots, | # h_{c} |}, \end{matrix}$ then $\begin{matrix} d (ϒ_{a}, ϒ_{c}) = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{c} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lc} | \\ + | μ_{k}^{Ma} - μ_{k}^{Mc} | + | μ_{k}^{Ua} - μ_{k}^{Uc} |))), \end{matrix}$ $\begin{matrix} d (ϒ_{a}, ϒ_{b}) = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{a} - τ_{k}^{b} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{La} - μ_{k}^{Lb} | \\ + | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Ua} - μ_{k}^{Ub} |))), \end{matrix}$ $\begin{matrix} d (ϒ_{b}, ϒ_{c}) = ρ (\frac{1}{H} \sum_{k = 1}^{H} | τ_{k}^{b} - τ_{k}^{c} |) \\ + (1 - ρ) (\frac{1}{H} \sum_{k = 1}^{H} (\frac{1}{3} (| μ_{k}^{Lb} - μ_{k}^{Lc} | \\ + | μ_{k}^{Mb} - μ_{k}^{Mc} | + | μ_{k}^{Ub} - μ_{k}^{Uc} |))), \end{matrix}$

Since the mathematical theorem |a + c| ⩽ |a| + |c| holds, then | (a-b) + (b-c) | ⩽ |a-b| + |b-c|, i.e., |a-c| ⩽ |a-b| + |b-c|.

Hence, $| τ_{k}^{a} - τ_{k}^{c} | ⩽ | τ_{k}^{a} - τ_{k}^{b} | + | τ_{k}^{b} - τ_{k}^{c} |, | μ_{k}^{La} - μ_{k}^{Lc} | ⩽ | μ_{k}^{La} - μ_{k}^{Lb} | + | μ_{k}^{Lb} - μ_{k}^{Lc} |$ , $| μ_{k}^{Ma} - μ_{k}^{Mc} | ⩽ | μ_{k}^{Ma} - μ_{k}^{Mb} | + | μ_{k}^{Mb} - μ_{k}^{Mc} |$ and $| μ_{k}^{Ua} - μ_{k}^{Uc} | ⩽ | μ_{k}^{Ua} - μ_{k}^{Ub} | + | μ_{k}^{Ub} - μ_{k}^{Uc} |$ . Further, d (ϒ_a, ϒ_c) ⩽d (ϒ_a, ϒ_b) + d (ϒ_b, ϒ_c) is established.

The proof is finished.

Example 2: Given two NWHFSs $\begin{matrix} ϒ_{1} = 〈 {0.2, 0.5, 0.6}, {(0 . 1338, 0 . 1796, 0.2662), \\ (0 . 3426, 0 . 4516, 0.6574), (0 . 4949, 0 . 5677, 0.7051)} 〉 \\ ϒ_{2} = 〈 (0.4, 0.6, 0.9), {(0 . 2922, 0 . 4284, 0.5078), \\ (0 . 3972, 0 . 5466, 0.8028), (0 . 8115, 0 . 8767, 0.9885)} 〉 \end{matrix}$ and ρ = 0.5, so the distance between ϒ₁ and ϒ₁ is d (ϒ₁, ϒ₂) = 0.2029.

4 The extended TODIM under normal wiggly hesitant fuzzy environment

In this part, we will propose an extension of the TODIM method (NWHF-TODIM) for dealing with the MADM problems under normal wiggly hesitant fuzzy environment. This part mainly includes the following three aspects. Firstly, to understand the purpose of this article better, we will give a general overview of the following MADM problems in a normal wiggly hesitant fuzzy environment. Then, we will give the NWHF-TODIM method by extending the classical TODIM method to solve the MADM problem. Finally, we will provide the concrete steps for the NWHF-TODIM method.

4.1 The description of the MADM problems under normal wiggly hesitant fuzzy environment

Let A_i (i = 1, 2, ⋯ m) be a set which includes all the alternatives and C_j (j = 1, 2, ⋯ n) be a set including the criterions. The hesitant fuzzy MADM problems can be described by a hesitant fuzzy decision matrix expressed by H = (h_ij) _m×n, where h_ij is the rating of the alternative A_i (i = 1, 2, ⋯ m) according to the criteria C_j (j = 1, 2, ⋯ n) provided by the DMs, which is regarded as a hesitant fuzzy number.

By mining the underlying/deeper uncertainty message when HFSs is used by DMs to express evaluation information in the decision-making process, this paper extends the format of hesitation fuzzy matrix to general wiggly hesitation fuzzy decision matrix. In order to eliminate the influence of various physical dimensions and types, we normalize the hesitant fuzzy decision matrix H = (h_ij) _m×n and convert the value of the cost-type criterion into the value of the benefit-type criterion to obtain a homologous normalized matrix $\tilde{H} = {[{\tilde{h}}_{ij}]}_{m \times n}$ expressed by the following Equation (12)

${\tilde{h}}_{ij} = {\begin{matrix} h_{ij}, for benefit criterion C_{j} \\ {(h_{ij})}^{c}, for cost criterion C_{j} \end{matrix}, i \in M, j \in N$ (12)

Suppose the DMs give the evaluation values h_ij of each alternative A_j in regard to each attribute C_j, and summarize them in the decision matrix H = (h_ij) _m×n of Table 1. For the sake of digging the underlying fuzzy information given by DMs, based on Definitions 2 and 3, the normal wiggly hesitant fuzzy MADM problem can be expressed succinctly by the normal wiggly hesitant fuzzy decision matrix $\tilde{Λ} = {(η_{ij})}_{m \times n}$ in Table 2.

Table 1

Original hesitant decision matrix E

	C ₁	C ₂	⋯	C _n-1	C _n
A ₁	h ₁₁	h ₁₂	⋯	h _1,n-1	h _1n
A ₂	h ₂₁	h ₂₂	⋯	h _2,n-1	h _2n
A ₃	⋯	⋯	⋯	⋯	⋯
h _m-1,1	h _m-1,2	⋯	h _m-1,n-1	h _m-1,n
A ₄	h _m1	h _m2	⋯	h _m,n-1	h _mn

Table 2

The normal wiggly hesitant decision matrix $\tilde{Λ}$

	C ₁	C ₂	⋯	C _n-1	C _n
A ₁	$〈 {\tilde{h}}_{11}, ζ ({\tilde{h}}_{11}) 〉$	$〈 {\tilde{h}}_{12}, ζ ({\tilde{h}}_{12}) 〉$	⋯	$〈 {\tilde{h}}_{1, n - 1}, ζ ({\tilde{h}}_{1, n - 1}) 〉$	$〈 {\tilde{h}}_{1, n}, ζ ({\tilde{h}}_{1, n}) 〉$
A ₂	$〈 {\tilde{h}}_{21}, ζ ({\tilde{h}}_{21}) 〉$	$〈 {\tilde{h}}_{22}, ζ ({\tilde{h}}_{22}) 〉$	⋯	$〈 {\tilde{h}}_{2, n - 1}, ζ ({\tilde{h}}_{2, n - 1}) 〉$	$〈 {\tilde{h}}_{2, n}, ζ ({\tilde{h}}_{2, n}) 〉$
⋯	⋯	⋯	⋯	⋯	⋯
A _m-1	$〈 {\tilde{h}}_{m - 1, 1}, ζ ({\tilde{h}}_{m - 1, 1}) 〉$	$〈 {\tilde{h}}_{m - 1, 2}, ζ ({\tilde{h}}_{m - 1, 2}) 〉$	⋯	$〈 {\tilde{h}}_{m - 1, n - 1}, ζ ({\tilde{h}}_{m - 1, n - 1}) 〉$	$〈 {\tilde{h}}_{m - 1, n}, ζ ({\tilde{h}}_{m - 1, n}) 〉$
A _m	$〈 {\tilde{h}}_{m 1}, ζ ({\tilde{h}}_{m 1}) 〉$	$〈 {\tilde{h}}_{m 2}, ζ ({\tilde{h}}_{m 2}) 〉$	⋯	$〈 {\tilde{h}}_{m, n - 1}, ζ ({\tilde{h}}_{m, n - 1}) 〉$	$〈 {\tilde{h}}_{mn}, ζ ({\tilde{h}}_{mn}) 〉$

4.2 The normal wiggly hesitant fuzzy TODIM (NWHF-TODIM) approach

In the context of normal hesitant fuzzy information, the distance measure we proposed is combined with the traditional TODIM method to obtain the extended NWHF-TODIM approach in this paper. For the MADM problem, similar to the concrete process of the TODIM method, we should firstly use Equation (12) to normalize the original decision matrix. On the basis of prospect theory [8], we measure the dominance degree to which each option has an advantage over other options by establishing the foreground value function. For this purpose, we should define a reference standard and give the reference criterion. Afterwards, based on the score function $ℕ (η)$ , we could compare the size of the alternative ratings for each criterion denoted by the NWHFE. What’s more, similar to Equation (7), we can use the following expression to compute the optional gain and loss of the alternatives concerning the criterion C_j:

$φ_{j} (A_{i}, A_{s}) = {\begin{matrix} \sqrt{\frac{w_{jr} d (η_{ij}, η_{sj})}{\sum_{j = 1}^{n} w_{jr}}}, if (ℕ (η_{ij}) - ℕ (η_{sj})) > 0 \\ 0, if (ℕ (η_{ij}) - ℕ (η_{sj})) = 0 \\ \frac{- 1}{θ} \sqrt{\frac{(\sum_{j = 1}^{n} w_{jr}) d (η_{ij}, η_{sj})}{w_{jr}}}, if (ℕ (η_{ij}) - ℕ (η_{sj})) < 0 \end{matrix}$ (13) where the parameter θ represents the attenuation factor of the losses, d (η_ij, η_sj) denotes the distance between the NWHFEs η_ij and η_sj using Equation (11). It’s obvious to see that there are three situations in Equation (13): If $ℕ (η_{ij}) > ℕ (η_{sj})$ , then φ_j (A_i, A_s) represents a gain; If $ℕ (η_{ij}) = ℕ (η_{sj})$ , then φ_j (A_i, A_s) represents a nil; If $ℕ (η_{ij}) < ℕ (η_{sj})$ , then φ_j (A_i, A_s) denotes a loss.

The dominance of alternatives A_i over alternatives A_s can be given by polymerizing φ_j (A_i, A_s) with each criterion C_j as follows: $ϑ (A_{i}, A_{s}) = \sum_{j = 1}^{n} φ_{j} (A_{i}, A_{s})$ (14)

Then, we compute the overall prospect value of the alternative A_iby:

It’s obvious to see that 0 ⩽ ξ (A_i) ⩽ 1, and the alternative A_i is depending on the ξ (A_i). Hence, we can get the ranking order of all alternatives A_i on the basis of the selection of increasing order of overall outlook value of the alternative A_i, and choose the best alternative(s) from the set A.

According to the above analysis, the concrete steps for the NWHF-TODIM method is described as follows:

Step 1: Define the original decision matrix H = [h_ij] _m×n, and normalize it to get $\tilde{H} = {[{\tilde{h}}_{ij}]}_{m \times n}$ by Equation (12).

Step 2: Compute the relative attribute weight w_jr for attribute C_j to reference attribute C_r, i.e., $w_{jr} = \frac{w_{j}}{w_{r}} (j \in N)$ (16) where w_r = max{ w_j|j ∈ N }.

Step 3: Compute the NEW ζ (h (τ)) based on definition 4, and we can get the normal wiggly hesitant fuzzy decision matrix $\tilde{Λ} = {(η_{ij})}_{m \times n}$ . Then, the NWHFE $ℕ (η_{ij})$ can be determined.

Step 4: Compute the gain and loss of each alternative with respect to each criterion C_j using Equation (13).

Step 5: Compute the dominance of each alternative using Equation (14).

Step 6: Compute the overall prospect value of the alternative A_i using Equation (15).

Step 7: Get the ranking order of all alternatives based on the growing order of the overall prospect value of the alternative A_i and identify the expected alternatives in the alternative set A.

5 An illustrative example

In this part, we will take a decision-making problem related to the evaluation and ranking of high-level personnel training into account to show the implementation process, and demonstrate the rationality and validity of our proposed approach. In this paper, we use the proposed NWHF-TODIM approach to calculate this case and apply this case to other approaches to fully prove the rationality and effectiveness of our approach by comparing and analyzing the results.

5.1 Problem description

Since joining WTO, China has more demands and higher requirements for high-level talents. Accession to the WTO has brought opportunities for China’s economic development and the improvement of talent quality. At the same time, the international competition for talents has intensified. The competition for talents, especially high-level talents, has entered a white-hot state, which makes China face severe challenges. The establishment of high-level talent evaluation index system is beneficial to the survey of existing high-level talent resources and the construction of talent pool. Therefore, the cultivation of high-level talents is crucial. For the further purpose of improving the quality of high-level personnel training in China, the evaluation of high-level personnel training is carried out in four regions. The four regions could be expressed by A₁, A₂, A₃, A₄. There are many aspects in the high-level personnel training evaluation system, but we only consider the following five main aspects. These five main criteria are: Levels of knowledge (C₁), Basic quality(C₂), Ability level (C₃), Performance results (C₄), Mental model(C₅); Table 3 details the five standards. According to the comments of many High-level personnel training evaluation program, the weights of five criteria are given as w = (0.3, 0.2, 0.1, 0.1, 0.3) ^T. In fact, these criteria contain a number of quantitative attributes. However, the DMs give each criterion a thorough consideration based on the relevant attributes and express their preference information for each alternative concerning each criterion in the form of HFE. We gather all HFEs into a decision matrix, as shown in Table 4.

Table 3
The explanation of criteria

Criteria Explanation

C₁ Levels of knowledge The level of knowledge includes the level of education, qualifications and knowledge structure.

C₂ Basic quality Mental model includes ideological and moral character, professional ethics, personality quality, work style and cooperative spirit.

C₃ Ability level Ability level includes learning ability, strain ability, organization ability, coordination ability, innovation ability, decision-making ability and practical ability.

C₄ Performance results Achievements include work efficiency and work awards.

C₅ Mental model Mental model includes ideological and moral character, professional ethics, personality quality, work style and cooperative spirit.

Criteria	Explanation
C₁ Levels of knowledge	The level of knowledge includes the level of education, qualifications and knowledge structure.
C₂ Basic quality	Mental model includes ideological and moral character, professional ethics, personality quality, work style and cooperative spirit.
C₃ Ability level	Ability level includes learning ability, strain ability, organization ability, coordination ability, innovation ability, decision-making ability and practical ability.
C₄ Performance results	Achievements include work efficiency and work awards.
C₅ Mental model	Mental model includes ideological and moral character, professional ethics, personality quality, work style and cooperative spirit.

Table 4

The Original hesitant decision matrix H

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	{0.4,0.5,0.6}	{0.3,0.4,0.7}	{0.1,0.3}	{0.4,0.6}	{0.2,0.3,0.4}
A ₂	{0.5,0.7,0.8}	{0.2,0.3,0.4}	{0.2,0.4,0.5}	{0.5,0.6}	{0.1,0.4,0.7}
A ₃	{0.2,0.4,0.8}	{0.6,0.7,0.8}	{0.1,0.2,0.3}	{0.1,0.3,0.4}	{0.4,0.5}
A ₄	{0.6,0.7,0.8}	{0.3,0.4}	{0.4,0.5,0.6}	{0.2,0.3,0.6}	{0.5,0.7}

5.2 Decision steps

The specific steps are shown as follows:

Step 1: Define the original decision matrix H = [h_ij] _m×n, and normalize it to get $\tilde{H} = {[{\tilde{h}}_{ij}]}_{m \times n}$ shown in Table 5. Since all attributes are all benefit types, according to Equation (12), thus $\tilde{H} = {[{\tilde{h}}_{ij}]}_{m \times n} = H = {[h_{ij}]}_{m \times n}$ .

Table 5
The Original hesitant decision matrix $\tilde{H}$

C ₁ C ₂ C ₃ C ₄ C ₅

A ₁ {0.4,0.5,0.6} {0.3,0.4,0.7} {0.1,0.3} {0.4,0.6} {0.2,0.3,0.4}

A ₂ {0.5,0.7,0.8} {0.2,0.3,0.4} {0.2,0.4,0.5} {0.5,0.6} {0.1,0.4,0.7}

A ₃ {0.2,0.4,0.8} {0.6,0.7,0.8} {0.1,0.2,0.3} {0.1,0.3,0.4} {0.4,0.5}

A ₄ {0.6,0.7,0.8} {0.3,0.4} {0.4,0.5,0.6} {0.2,0.3,0.6} {0.5,0.7}

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	{0.4,0.5,0.6}	{0.3,0.4,0.7}	{0.1,0.3}	{0.4,0.6}	{0.2,0.3,0.4}
A ₂	{0.5,0.7,0.8}	{0.2,0.3,0.4}	{0.2,0.4,0.5}	{0.5,0.6}	{0.1,0.4,0.7}
A ₃	{0.2,0.4,0.8}	{0.6,0.7,0.8}	{0.1,0.2,0.3}	{0.1,0.3,0.4}	{0.4,0.5}
A ₄	{0.6,0.7,0.8}	{0.3,0.4}	{0.4,0.5,0.6}	{0.2,0.3,0.6}	{0.5,0.7}

Step 2: Compute the relative attribute weight w_jr for attribute C_j to reference attribute C_r. $\begin{matrix} w_{1 r} = 1, w_{2 r} = 0.6667, w_{3 r} = 0.3333, \\ w_{4 r} = 0.3333, w_{5 r} = 1 . \end{matrix}$

Step 3: Compute the NEW ζ (h (τ)) based on definition 4, and we can get the normal wiggly hesitant fuzzy decision matrix $\tilde{Λ} = {(η_{ij})}_{m \times n}$ and the NWHFE $ℕ (η_{ij})$ . $η_{11} = 〈 {0.4, 0.5, 0.6}, {(0 . 3614, 0 . 4000, 0.4386), (0 . 4184, 0 . 5000, 0.5816), (0 . 6000, 0 . 6386, 0.0819)} 〉$ $η_{12} = 〈 {0.3, 0.4, 0.7}, {(0 . 1949, 0 . 2700, 0.4051), (0.2426, 0 . 3550, 0.5574), (0 . 6338, 0 . 6811, 0.7662)} 〉$ $η_{13} = 〈 {0.1, 0.2, 0.3}, {(0.0614, 0.0817, 0.1386), (0.1184, 0.1728, 0.2816), (0.2614, 0.2871, 0.3386)} 〉$ $η_{14} = 〈 {0.4, 0.5, 0.6}, {(0.3614, 0.4000, 0.4386), (0.4184, 0.5000, 0.5816), (0 . 6000, 0 . 6386, 0.0819)} 〉$ $η_{15} = 〈 {0.2, 0.3, 0.4}, {(0 . 1614, 0 . 1914, 0.2386), (0 . 2184, 0 . 2819, 0.3816), (0 . 3614, 0 . 3914, 0.4386)} 〉$ $η_{21} = 〈 {0.5, 0.7, 0.8}, {(0 . 4489, 0 . 5077, 0.5511), (0 . 5797, 0 . 7181, 0.8203), (0 . 7296, 0.8106, 0.8704)} 〉$ $η_{22} = 〈 {0.2, 0.3, 0.4}, {(0 . 1614, 0 . 1914, 0.2386), (0 . 2184, 0 . 2819, 0.3816), (0 . 3614, 0 . 3914, 0.4386)} 〉$ $η_{23} = 〈 {0.2, 0.4, 0.5}, {(0 . 1489, 0 . 1861, 0.2511), (0 . 2797, 0 . 3672, 0.5203), (0 . 4296, 0 . 4808, 0.5704)} 〉$ $η_{24} = 〈 {0.5, 0.55, 0.6}, {(0 . 4807, 0 . 5012, 0.5193), (0 . 5092, 0 . 5525, 0.5908), (0 . 5807, 0 . 6012, 0.6193)} 〉$ $η_{25} = 〈 {0.1, 0.4, 0.7}, {(0 . 0000, 0 . 0421, 0.2157), (0 . 1551, 0 . 2775, 0.6449), (0 . 5843, 0 . 6421, 0.8157)} 〉$ $η_{31} = 〈 {0.2, 0.4, 0.8}, {(0 . 0591, 0 . 1396, 0.3409), (0 . 1593, 0 . 2968, 0.6407), (0 . 6979, 0 . 7562, 0.9021)} 〉$ $η_{32} = 〈 {0.6, 0.7, 0.8}, {(0 . 5614, 0.6037, 0.6386), (0 . 6184, 0.7078, 0.7816), (0.7614, 0.8037, 0.8386)} 〉$ $η_{33} = 〈 {0.1, 0.2, 0.3}, {(0 . 0614, 0 . 0871, 0.1386), (0 . 1184, 0 . 1728, 0.2816), (0 . 2614, 0 . 2871, 0.3386)} 〉$ $η_{34} = 〈 {0.1, 0.3, 0.4}, {(0 . 0489, 0 . 0808, 0.1511), (0 . 1797, 0 . 2549, 0.4203), (0 . 3296, 0 . 3736, 0.4704)} 〉$ $η_{35} = 〈 {0.4, 0.45, 0.5}, {(0 . 3807, 0 . 3986, 0.4193), (0 . 4092, 0 . 4470, 0.4908), (0 . 4807, 0 . 4986, 0.5193)} 〉$ $η_{41} = 〈 {0.6, 0.7, 0.8}, {(0 . 5614, 0 . 6037, 0.6386), (0 . 6184, 0 . 7078, 0.7816), (0 . 7614, 0 . 8037, 0.8386)} 〉$ $η_{42} = 〈 {0.3, 0.35, 0.4}, {(0 . 2807, 0 . 2982, 0.3193), (0 . 3092, 0 . 3461, 0.3908), (0 . 3807, 0 . 3982, 0.4193)} 〉$ $η_{43} = 〈 {0.4, 0.5, 0.6}, {(0 . 3614, 0.4000, 0.4386), (0.4184, 0.5000, 0.5816), (0 . 5614, 0 . 6000, 0.6386)} 〉$ $η_{44} = 〈 {0.2, 0.3, 0.6}, {(0 . 0949, 0 . 1618, 0.3051), (0 . 1426, 0 . 2428, 0.4574), (0 . 5338, 0 . 5759, 0.6662)} 〉$ $η_{45} = 〈 {0.5, 0.6, 0.7}, {(0 . 4614, 0 . 5043, 0.5386), (0 . 5184, 0 . 6091, 0.6816), (0 . 6614, 0 . 7043, 0.7386)} 〉$ $ℕ (η_{11}) = 0 . 0819 ℕ (η_{12}) = 0 . 0731 ℕ (η_{13}) = - 0 . 0588 ℕ (η_{14}) = 0 . 0819 ℕ (η_{15}) = - 0 . 0253$ $ℕ (η_{21}) = 0 . 2055 ℕ (η_{22}) = - 0 . 0253 ℕ (η_{23}) = 0 . 0132 ℕ (η_{24}) = 0 . 1138 ℕ (η_{25}) = 0 . 0615$ $ℕ (η_{31}) = 0 . 0896 ℕ (η_{32}) = 0 . 2308 ℕ (η_{33}) = - 0 . 0588 ℕ (η_{34}) = - 0 . 0290 ℕ (η_{35}) = 0 . 0498$ $ℕ (η_{41}) = 0 . 2308 ℕ (η_{42}) = - 0 . 0052 ℕ (η_{43}) = 0 . 0819 ℕ (η_{44}) = 0 . 0152 ℕ (η_{45}) = 0 . 1508$

Step 4: Add HFE for shorter ${\tilde{h}}_{ij}$ to get to the same length by Equation (24), and the results are shown in Table 6.

Table 6

The added hesitant decision matrix H

	C ₁	C ₂	C ₃	C ₄	C ₅
A ₁	{0.4,0.5,0.6}	{0.3,0.4,0.7}	{0.1,0.2,0.3}	{0.4,0.5,0.6}	{0.2,0.3,0.4}
A ₂	{0.5,0.7,0.8}	{0.2,0.3,0.4}	{0.2,0.4,0.5}	{0.5,0.55,0.6}	{0.1,0.4,0.7}
A ₃	{0.2,0.4,0.8}	{0.6,0.7,0.8}	{0.1,0.2,0.3}	{0.1,0.3,0.4}	{0.4,0.45,0.5}
A ₄	{0.6,0.7,0.8}	{0.3,0.35,0.4}	{0.4,0.5,0.6}	{0.2,0.3,0.6}	{0.5,0.6,0.7}

Step 5: When θ = 1, calculate the gain and loss of the alternatives over the others with respect to each criterion C_j using Equation (13). The results are shown in Table 7.

Table 7

Dominance degrees of each alternative over the others with respect to each criterion

C ₁	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	0.2250	0.2323	0.2455
A ₂	–07498	0.0000	–0.8506	0.1121
A ₃	–0.7743	0.2552	0.0000	0.2754
A ₄	–0.8182	0.3737	–0.9179	0.0000
C ₂	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	–0.9039	0.2188	–0.8122
A ₂	0.1808	0.0000	0.2838	0.1036
A ₃	–1.0941	–1.4192	0.0000	–1.3253
A ₄	0.1624	–0.5181	0.2651	0.0000
C ₃	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	0.1288	0.0000	0.1741
A ₂	–1.2882	0.0000	–1.2882	0.1170
A ₃	0.0000	0.1288	0.0000	0.1741
A ₄	–1.7405	–1.1705	–1.7405	0.0000
C ₄	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	0.0724	–1.5439	–1.1960
A ₂	–0.7239	0.0000	–1.6989	1.3981
A ₃	0.1544	0.1699	0.0000	0.1019
A ₄	0.1196	0.1398	–1.0191	0.0000
C ₅	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	0.2242	0.2133	0.3015
A ₂	–0.7472	0.0000	–0.8291	0.2580
A ₃	–0.7109	0.2487	0.0000	0.2131
A ₄	–1.0049	–0.8601	–0.7102	0.0000

Step 6: Calculate the dominance of the alternative A_i over the alternative A_s using Equation (14). The final results can be shown in Table 8, when θ = 1.

Table 8

Overall dominance degrees of each alternative over the others

	A ₁	A ₂	A ₃	A ₄
A ₁	0.0000	–0.2536	–0.8795	–1.2872
A ₂	–3.3284	0.0000	–4.3830	–0.8073
A ₃	–2.4249	–0.6165	0.0000	–0.5609
A ₄	–3.2816	–2.7825	–4.1226	0.0000

Step 7: Compute the overall prospect value ∂ of the alternative A_i using Equation (15)(θ = 1).

$\partial (A_{1}) = 0 . 0520, \partial (A_{2}) = 0 . 8518, \partial (A_{3}) = 0, \partial (A_{4}) = 1 .$

Step 8: Get the ranking order of all the alternatives in line with the growing order of the overall prospect value of the alternatives A_i, and we can get A₄ ≻ A₂ ≻ A₁ ≻ A₃. Apparently, A₄ is the most ideal one.

5.3 Parameter sensitivity analysis

Sensitivity analysis is performed by increasing and decreasing their values to modify the parameters (the attenuation factor of the losses) and recalculating the ordering of alternatives with various values of θ.

When computing the dominance degree of each alternative, we can see that the attenuation coefficient of loss θ plays a significant role, and the attenuation coefficient of loss is larger than that of gain. For the purpose of illustrating the influence of θ on the sorting result, the corresponding sorting result is obtained by changing the value θ from 0.1 to 30, as shown in Table 9 and Fig. 3.

Table 9
Sort the results of cases using different parameters θ

Alternatives θ = 0.1 θ = 0.5 θ = 1 θ = 1.58 θ = 1.59

∂ order ∂ order ∂ order ∂ order ∂ order

A ₁ 0.1489 3 0.1029 3 0.0520 3 0.0006 3 0.0000 4

A ₂ 0.9056 2 0.8800 2 0.8518 2 0.8233 2 0.8229 2

A ₃ 0 4 0 4 0 4 0 4 0.0002 3

A ₄ 1 1 1 1 1 1 1 1 1 1

Alternatives θ = 2 θ = 2.5 θ = 3 θ = 10 θ = 30

∂ order ∂ order ∂ order ∂ order ∂ order

A ₁ 0 4 0 4 0 4 0 4 0 4

A ₂ 0.8111 2 0.7990 2 0.7888 2 0.7259 2 0.6916 2

A ₃ 0.0314 3 0.0635 3 0.0906 3 0.2570 3 0.3478 3

A ₄ 1 1 1 1 1 1 1 1 1 1

Alternatives	θ = 0.1	θ = 0.5	θ = 1	θ = 1.58	θ = 1.59
A ₁	0.1489	3	0.1029	3	0.0520	3	0.0006	3	0.0000	4
A ₂	0.9056	2	0.8800	2	0.8518	2	0.8233	2	0.8229	2
A ₃	0	4	0	4	0	4	0	4	0.0002	3
A ₄	1	1	1	1	1	1	1	1	1	1
Alternatives	θ = 2	θ = 2.5	θ = 3	θ = 10	θ = 30
	∂	order	∂	order	∂	order	∂	order	∂	order
A ₁	0	4	0	4	0	4	0	4	0	4
A ₂	0.8111	2	0.7990	2	0.7888	2	0.7259	2	0.6916	2
A ₃	0.0314	3	0.0635	3	0.0906	3	0.2570	3	0.3478	3
A ₄	1	1	1	1	1	1	1	1	1	1

Fig. 3

Overall prospect values with different parameters θ.

From Table 9, it is not difficult to find that the sorting result of the alternatives may be changed with the change of variable value θ. That is to say, under the influence of different parameters θ, the orders are different. The same sorting result A₄ ≻ A₂ ≻ A₁ ≻ A₃ is get when θ ⩽ 1.58, but when θ ⩾ 1.59, the sorting result is changed to A₄ ≻ A₂ ≻ A₃ ≻ A₁. The cause leading to this difference is that the magnification degree of loss increases with the decrease of θ when θ ⩽ 1 and the extent to which the loss is reduced increases with the increase of θ when θ > 1. In this case, we can see that the degree of attenuation is too small to let the sorting inconsistent with the result when 1 ⩽ θ ⩽ 1.58, but when θ ⩾ 1.59, the degree of weakening is sufficient to change the rankings and the attenuation of losses make A₃ better than A₁. Therefore, it can be found from the change of alternative ranking results that the change of influence factor θ will have a significant impact on the alternative ranking results, which to some extent reflects that the risk preference of DMs has an important impact on the alternative ranking results.

5.4 Comparative analysis and discussions

Next, we will further analyze the rationality and validity of the extended NWHF-TODIM approach in the process of solving decision-making problems by comparing it with the two methods that are commonly used now. Up to now, there is no research on high-level personnel training evaluation problems under the normal wiggly hesitant fuzzy environment. Therefore, we can only compare and analyze existing approaches to this MADM problem in the context of hesitant fuzzy information., i.e., the HF-TOPSIS approach [24] and the HF-VIKOR approach [13].

5.4.1 Comparison with the HF-TOPSIS methods

The HF-TOPSIS approach proposed by Xu and Zhang [24] was applied to process the hesitant fuzzy information. Therefore, we can use it to deal with the hesitant decision matrix expressed in Table 5. The process can be reformulated shown as follows.

Step 1-2: Analogous to steps 1 and 2 in Section 5.2, we omit the result here.

Step 3: Count the PIS L⁺ and NIS L^- and obtain $\begin{matrix} L^{+} = {(0.6, 0.7, 0.8), (0.6, 0.7, 0.8), \\ (0.4, 0.5, 0.6), (0.5, 0.55, 0.6), (0.5, 0.6, 0.7)}, \\ L^{-} = {(0.2, 0.4, 0.6), (0.2, 0.3, 0.4), \\ (0.1, 0.2, 0.3), (0.1, 0.3, 0.4), (0.1, 0.3, 0.4)} . \end{matrix}$

Step 4: Count d (A_i, L⁺) and d (A_i, L^-), and obtain d_min (A_i, L⁺) and d_max (A_i, L^-) as follows: $\begin{matrix} d (A_{1}, L^{+}) = 0.1924, d (A_{2}, L^{+}) = 0.1282, \\ d (A_{3}, L^{+}) = 0.1357, d (A_{4}, L^{+}) = 0.0795, \end{matrix}$ $\begin{matrix} d (A_{1}, L^{-}) = 0.0664, d (A_{2}, L^{-}) = 0.1583, \\ d (A_{3}, L^{-}) = 0.1546, d (A_{4}, L^{-}) = 0.2145 . \end{matrix}$

And, d_min (A_i, L⁺) = 0.0795, d_max (A_i, L^-) = 0.2145 .

Step 5: Count the closeness coefficient CI (A_i) (i = 1, 2, 3, 4) and get $\begin{matrix} CI (A_{1}) = 0 . 2566, CI (A_{2}) = 0 . 5526, \\ CI (A_{3}) = 0 . 5326, CI (A_{4}) = 0 . 7295 . \end{matrix}$

Step 6: Get the ranking results by the values CI (A_i) (i = 1, 2, 3, 4), A₄ ≻ A₂ ≻ A₃ ≻ A₁.

5.4.2 Comparison with the HF-VIKOR methods

HF-VIKOR approach put forward by Liao and Xu [13] was applied to solve multi-attribute decision-making problems with hesitant preference information.

Step 1-2: Similar to steps 1 and 2 shown in Section 5.2, we omit the results too.

Step 3: Determine the PIS h^* and NIS h^-: $\begin{matrix} h^{*} = {(0.6, 0.7, 0.8), (0.6, 0.7, 0.8), (0.4, 0.5, 0.6), \\ (0.5, 0.55, 0.6), (0.5, 0.6, 0.7)}, \\ h^{-} = {(0.2, 0.4, 0.6), (0.2, 0.3, 0.4), (0.1, 0.2, 0.3), \\ (0.1, 0.3, 0.4), (0.1, 0.3, 0.4)} . \end{matrix}$

Step 4: Determine the S_i, R_i and Q_i (i = 1, 2, 3, 4) of alternatives, and get (v = 0.5)

$\begin{matrix} S_{1} = 0.7043, S_{2} = 0.4578, S_{3} = 0.5683, S_{4} = 0.2397, \\ R_{1} = 0.2700, R_{2} = 0.2000, R_{3} = 0.2333, R_{4} = 0.1750, \\ Q_{1} = 1, Q_{2} = 0.3663, Q_{3} = 0.6607, Q_{4} = 0 . \end{matrix}$

Step 5: Get the ranking results by the values S_i, R_i and Q_i (i = 1, 2, 3, 4) in increasing order. The results depicted in three ranking are shown in Table 10.

Table 10
The ranking and the compromise solutions

A ₁ A ₂ A ₃ A ₄ Ranking Compromise solutions

S 0.6967 0.4578 0.5683 0.2374 _SA₄ ≻ _SA₂ ≻ _SA₃ ≻ _SA₁ A ₄

R 0.2700 0.2000 0.2333 0.1727 _RA₄ ≻ _RA₂ ≻ _RA₃ ≻ _RA₁ A ₄

Q 1 0.3801 0.6717 0 _QA₄ ≻ _QA₂ ≻ _QA₃ ≻ _QA₁ A ₄

	A ₁	A ₂	A ₃	A ₄	Ranking	Compromise solutions
S	0.6967	0.4578	0.5683	0.2374	_SA₄ ≻ _SA₂ ≻ _SA₃ ≻ _SA₁	A ₄
R	0.2700	0.2000	0.2333	0.1727	_RA₄ ≻ _RA₂ ≻ _RA₃ ≻ _RA₁	A ₄
Q	1	0.3801	0.6717	0	_QA₄ ≻ _QA₂ ≻ _QA₃ ≻ _QA₁	A ₄

Step 6: Obtain the compromise solution(s). In accordance with the Q_i, the results can be shown as below:

As for the S, the ranking results are shown as: _SA₄ ≻ _SA₂ ≻ _SA₃ ≻ _SA₁.

As for the R, the ranking results are shown as: _RA₄ ≻ _RA₂ ≻ _RA₃ ≻ _RA₁.

As for the Q, the ranking results are shown as: _QA₄ ≻ _QA₂ ≻ _QA₃ ≻ _QA₁.

Since Q (A⁽²⁾) -Q (A⁽¹⁾) > 1/3, A₄ is the best solution.

5.4.3 Discussion

In this part, we mainly compare the extended NWHF-TODIM approach with two approaches just mentioned, and the results shown in Table 11.

Table 11
Ranking results of different methods

Methods Comprehensive values Ranking

NWHF-TODIM method (θ = 1.58) ∂ (A₁) = 0.0006, ∂ (A₂) = 0.8233, ∂ (A₃) = 0, ∂ (A₄) = 1. A₃ ≻ A₂ ≻ A₄ ≻ A₁

NWHF-TODIM method (θ = 1.59) ∂ (A₁) = 0, ∂ (A₂) = 0.8229, ∂ (A₃) = 0.0002, ∂ (A₄) = 1. A₄ ≻ A₂ ≻ A₃ ≻ A₁

HF-TOPSIS method [24] CI (A₁) = 0.2566, CI (A₂) = 0.5526, CI (A₃) = 0.5326, CI (A₄) = 0.7295. A₄ ≻ A₂ ≻ A₃ ≻ A₁

HF-VIKOR method [13] PLQ₁ = 1, PLQ₂ = 0.3603, PLQ₂ = 0.3603, PLQ₃ = 0.6607, PLQ₄ = 0. _QA₄ ≻ _QA₂ ≻ _QA₃ ≻ _QA₁

Methods	Comprehensive values	Ranking
NWHF-TODIM method (θ = 1.58)	∂ (A₁) = 0.0006, ∂ (A₂) = 0.8233, ∂ (A₃) = 0, ∂ (A₄) = 1.	A₃ ≻ A₂ ≻ A₄ ≻ A₁
NWHF-TODIM method (θ = 1.59)	∂ (A₁) = 0, ∂ (A₂) = 0.8229, ∂ (A₃) = 0.0002, ∂ (A₄) = 1.	A₄ ≻ A₂ ≻ A₃ ≻ A₁
HF-TOPSIS method [24]	CI (A₁) = 0.2566, CI (A₂) = 0.5526, CI (A₃) = 0.5326, CI (A₄) = 0.7295.	A₄ ≻ A₂ ≻ A₃ ≻ A₁
HF-VIKOR method [13]	PLQ₁ = 1, PLQ₂ = 0.3603, PLQ₂ = 0.3603, PLQ₃ = 0.6607, PLQ₄ = 0.	_QA₄ ≻ _QA₂ ≻ _QA₃ ≻ _QA₁

From Table 11, we can see that when θ = 1.6, the sorting results by HF-TOPSIS approach in [24] and HF-VIKOR approach in [13] are exactly same as that by the proposed NWHF-TODIM method, and are different when θ = 1.58 in this paper. What’s more, it’s not difficult to see that the sorting results by the approaches mentioned before are the same as those by the proposed method when θ ⩾ 1.59, and are different from that by the method we proposed when θ ⩽ 1.58 by combing Table 9 and Fig. 3. The same sorting order shows that the proposed NWHF-TODIM method is effective. What’s more different sorting results could better illustrate the advantages of method we put forward in this paper. The reasons for the results can be explained as follows.

The NWHF-TODIM approach we proposed has a distinct advantage in taking the psychological behavior of DMs into account by the parameter θ. The degree of loss aversion of DM decreases as its value θ increases. Namely, when the parameter value θ reaches1.58, the psychological behavior of DM can be ignored, i.e., we approximately think that the psychological behavior of DMs is not considered. Therefore, the NWHF-TODIM approach can produce the same sorting orders as the HF-TOPSIS and HF-VIKOR approaches in [13, 24] and the main reason is that they do not take the psychological behavior of DMs into account. It is obvious that this can illustrate the validity of the NWHF-TODIM approach. Furthermore, it is more reasonable and necessary to think about the DM’s psychological behavior in the actual decision-making process of practice because the DMs has bounded rationality, but the HF-TOPSIS and HF-VIKOR approaches [13, 24] don’t consider about this condition. In this paper, when θ ⩽ 1.58, the ranking result by the NWHF-TODIM method we proposed is changed to A₃ ≻ A₂ ≻ A₄ ≻ A₁. Obviously, it is more reasonable than two others, which is pretty vital for decision-making process.

Moreover, there is another advantages in the NWHF-TODIM, which can avoid the reverse order problem that may be caused by the existing methods [13, 24]. The occurrence of inversion depends largely on the choice of ideal solution in the two existing methods. For HF-TOPSIS method [24] or HF-VIKOR method [13], each method is based on the proximity to the ideal solution, and the ideal solution plays a crucial role in the calculation process of the two methods. For the purpose of better illustrating this advantage of the method presented in this paper, we have slightly changed the previous case by adding an alternative. The existing methods and our proposed NWHF-TODIM method are used to re-analyze the cases with five alternatives, and the original retained alternatives are sorted. The sorting results of the different methods are shown in Table 12.

Table 12

Ranking results of retained alternatives

Methods	Comprehensive values	Ranking
NWHF-TODIM method (θ = 1.59)	∂ (A₁) = 0, ∂ (A₂) = 0.2879, δ (A₃) = 0.0051, δ (A₄) = 0.3589.	A₄ ≻ A₂ ≻ A₃ ≻ A₁
HF-TOPSIS method [24]	CI (A₁) = 0.3149, CI (A₂) = 0.4947, CI (A₃) = 0.4724, CI (A₄) = 0.6315.	A₄ ≻ A₂ ≻ A₃ ≻ A₁
HF-VIKOR method [13]	PLQ₁ = 1, PLQ₂ = 0.4601, PLQ₃ = 0.3487, PLQ₄ = 0.	_QA₄ ≻ _QA₃ ≻ _QA₂ ≻ _QA₁

It can be easily found from Table 12 that after adding an alternative, the ranking results of the developed method and the original four schemes of NWHF-TODIM method [24] did not be changed. However, after the increase of alternative A₅, it can be seen from the final comprehensive value that the gap between alternative A₂ and alternative A₃ is almost reduced to 0 in HF-TOPSIS method [24]. From this analysis, it can be concluded that although this method does not generate inversions in this case, it is likely to cause inversions in other cases. What’s more, it is worth noting that the ranking results of HF-VIKOR method also changed significantly, and most importantly, the inversion problem was generated. The inversion of this approach can be illustrated by the fact that the order from alternative A₂ and alternative A₃ is changed after the addition of alternative A₅.

In line with the above analysis, we can draw the conclusion that the proposed NWHF-TODIM approach can not only adjust the loss avoidance behavior by adjusting the parameters θ, but also successfully avoid the reverse order and dig deeper level of uncertainty information. The advantages of the above two aspects strongly demonstrate the advantages of our proposed method over other existing methods [13, 24].

6 Conclusion

In this paper, in order to dig the DMs’ underlying uncertain information, by combining the traditional TODIM approach, the NWHF-TODIM is proposed under the NWHF environment. This new extended NWHFSs show that gaining DMs’ potential uncertainty and fuzziness hidden behind their original assessments in the process of decision-making is the more suitable and reasonable approach to obtain the reliable hesitant information. On this basis, a case is provided to demonstrate the validity and rationality of the proposed method. The main research results are shown as follows:

(1) Define the distance measure of any two NWHFEs for enriching the theories of NWHFSs.

(2) Put forward an extended normal wiggly hesitant fuzzy TODIM (NWHF-TODIM) approach to scope with MADM problems with normal wiggly information, which can dig the deeper potential uncertainty and fuzziness hidden behind the original information.

(3) Make a comparison of the proposed NWHF-TODIM approach with two existing approaches and further illustrate its obvious benefits, i.e., it takes DMs’ psychological behavior under risk into account, and void the revision orders of alternatives.

It is worth noting that there are still some deficiencies in this study, which deserve further study. This paper assumes that all attributes are unrelated, and if not, there are many information forms such as the Choquet integral, Bonferroni mean, Maclaurin Symmetric Mean can be applied to the approach to cope with the interrelationships between multiple attributes [31 –34]. In addition, we can also combine NWHFs with other approaches, such as the PROMETHEE method [35, 36] and the MULTIMOORA method [37], which consider the same kind of attribute relationship.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71771140, 71471172), (Project of cultural masters and “the four kinds of a batch” talents), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

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Normal wiggly hesitant fuzzy TODIM approach for multiple attribute decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy sets

4.1 The description of the MADM problems under normal wiggly hesitant fuzzy environment

5.1 Problem description

5.4.1 Comparison with the HF-TOPSIS methods

5.4.2 Comparison with the HF-VIKOR methods

Table 10 The ranking and the compromise solutions A 1 A 2 A 3 A 4 Ranking Compromise solutions S 0.6967 0.4578 0.5683 0.2374 S A4 ≻ S A2 ≻ S A3 ≻ S A1 A 4 R 0.2700 0.2000 0.2333 0.1727 R A4 ≻ R A2 ≻ R A3 ≻ R A1 A 4 Q 1 0.3801 0.6717 0 Q A4 ≻ Q A2 ≻ Q A3 ≻ Q A1 A 4

Footnotes

Acknowledgments

References