Grey relational projection method for multiple attribute decision making with interval-valued dual hesitant fuzzy information

Abstract

This paper investigates the multiple attribute decision making (MADM) problems in which the attribute values take the form of interval-valued dual hesitant fuzzy elements (IVDHFEs) and the attribute weights are unknown. Firstly, motivated by the distance definitions of the hesitant fuzzy elements and dual hesitant fuzzy elements, the distance definitions of the interval-valued dual hesitant fuzzy elements are proposed. Then, the grey relational projection method based on the distance measure between the interval-valued dual hesitant fuzzy elements is proposed. To get the attribute weights, the nonlinear optimization model with the minimum total deviation of the grey relational coefficient is constructed. Furthermore, a novel interval-valued dual hesitant fuzzy MADM method is proposed based on the grey relational projection. Finally, a numerical example is given to demonstrate the effectiveness and practicality of the developed method.

Keywords

Dual hesitant fuzzy set interval-valued dual hesitant fuzzy set grey relational projection multiple attribute decision making

1 Introduction

Since Zadeh [1] proposed the fuzzy set which can describe complex and uncertain phenomenon, fuzzy set theory has been widely used in edge detection [2, 3], pattern recognition [4 –6], image reconstruction [7, 8] and decision making [9 –13] and so on. As an extended form of fuzzy set, the hesitant fuzzy set was proposed by Torra and Narukawa [14, 15], which permits the membership degree having a set of possible values on [0,1]. The hesitant fuzzy set can be more objective and effective in describing people’s hesitant attitude toward things than traditional fuzzy set and it has become one of the hot topics of scholars [16 –22]. However, while using the hesitant fuzzy theory, only the element values in the membership function are used to express the degree of certainty of the property, the importance of the uncertainty is ignored. In which case, Zhu et al. [23] proposed the dual hesitant fuzzy set, which permits both the membership degree and the non-membership degree having a set of possible values on [0,1]. The dual hesitant fuzzy set not only takes into account the membership degree for expressing certainty but also considers the non-membership degree for expressing uncertainty. The dual hesitant fuzzy set can describe the fuzzy phenomenon more comprehensively, which accords with the objective reality, so it has been used by more scholars to study MADM problems.

Wang et al. [24] extended the WA, WG, OWA, OWG, HA and HG operators to the dual hesitant fuzzy environment and proposed a series of dual hesitant fuzzy aggregation operators, then applied the DHFWA operator and the DHFWG operator to a potential evaluation of emerging technology commercialization. Wang et al. [25] proposed the induced dual hesitant fuzzy Hamacher ordered weighted geometric (IDHFHOWG) operator and applied the IDHFHOWG operator to the performance evaluation in customs service management. Yu and Li [26] developed the generalized dual hesitant fuzzy weighted averaging (GDHFWA) operator, the generalized dual hesitant fuzzy ordered weighted averaging (GDHFOWA) operator and the generalized dual hesitant fuzzy hybrid averaging (GDHFHA) operator and applied the GDHFWA operator to teaching quality assessment. Ju et al. [27] derived some new dual hesitant fuzzy aggregation operators according to the Choquet integral, proposed an approach to dual hesitant fuzzy multiple attribute decision making based on the dual hesitant fuzzy Choquet ordered average (DHFCOA) operator. Similar to [24], Lu and Wei [28] extended the WA, WG, OWA, OWG, HA and HG operators to the dual hesitant fuzzy uncertain linguistic environment and proposed a series of dual hesitant fuzzy uncertain linguistic aggregation operators, then an MADM problem was used to show the application of the aggregation operators. Su et al. [29] studied some Hamming distance, Euclidean distance and Hausdorff distance for dual hesitant fuzzy sets, and proposed the similarity measures for dual hesitant fuzzy sets, then the results of pattern recognition were presented according to the proposed distances. Singh [30] presented the geometric distance of dual hesitant fuzzy elements, and two similarity measures based on the set-theoretic approach and the matching functions were proposed, then the proposed measures were applied in dual hesitant fuzzy multiple attribute decision making. Ren and Wei [31] defined a correctional score function of DHFE, on which, the dice similarity measure of DHFSs was proposed and this measure was applied in MADM problem with prioritization relationship. Ye [32] proposed the correlation coefficient between dual hesitant fuzzy sets, and the weighted correlation coefficient was used to solve the investment alternatives. Tyagi [33] derived an extended correlation coefficient between dual hesitant fuzzy sets and the measure method was applied in determining the correlation coefficient between different parameters of the water in four different lakes. Chen et al. [34] used the correlation coefficient to solve the MADM problem with dual hesitant fuzzy information, where the attribute weight was incompletely known or completely unknown.

Sometimes due to insufficiency in available information, it is hard to express the dual hesitant fuzzy information with a crisp number. Ju et al. [35] proposed the interval-valued dual hesitant fuzzy set, which permits both the membership degrees and the non-membership degrees having a set of possible interval values, and they extended the WA, OWA, HA operators to interval-valued dual hesitant fuzzy environment and developed some interval-valued dual hesitant fuzzy aggregation operators, then applied the IVDHFWA operator to the invest option. Zhang et al. [36] explored the Einstein operations under interval-valued dual hesitant fuzzy environment and proposed a series of interval-valued dual hesitant fuzzy aggregation operators, then a MADM problem is used to show the application of the aggregation operators. Farhadinia [37] proposed the correlation coefficient of interval-valued dual hesitant fuzzy sets, then a medical diagnosis practical example was given to illustrate the utilization of the proposed measure.

Summarily, through the above investigation, the research on the MADM problems with dual hesitant fuzzy information is still in infancy, especially the research on those with interval-valued dual hesitant fuzzy information is rarer. At present, the main research topics focuses on aggregation operator, distance and similarity measure and correlation coefficient. On the one hand, grey relational analysis is an important part of grey system theory, which has been widely applied in decision science fields, such as the real numbers, interval numbers and linguistic labels decision making [38], 2-tuple linguistic decision making [39], intuitionistic fuzzy decision making [40, 41], and hesitant fuzzy decision making [42]. On the other hand, the projection measure is one of the most broadly applied methods for dealing with decision making problems.The projection measure has been extended to the fuzzy environment [43], intuitionistic fuzzy environment [44], intuitionistic trapezoidal fuzzy environment [45], and hesitant fuzzy environment [46]. As one of the projection methods, the grey relational projection method combines grey system theory and vector projection principle, which can comprehensively analyze the relationship among the attribute, reflect the influence of the whole index space and avoid the unidirectional deviation [45 , 50].

In this paper,we extend the grey relational projection method to the interval-valued dual hesitant fuzzy environment, which is a new extension of the projection measure of intuitionistic fuzzy sets and hesitant fuzzy sets. Therefore, the principal purposes of this paper are summarized as follows: (1) to develop a grey relational projection measure of the interval-valued dual hesitant fuzzy set; (2) to introduce an optimization model with the minimum total deviation of the grey relational coefficient to determine the attribute weights; (3) to develop an interval-valued dual hesitant fuzzy multi-attribute decision making method based on the grey relational projection.

The rest of this paper is organized as follows: In Section 2, we presents some distance definitions of the HFEs, IVHFEs and DHFEs. In Section 3, we proposes the distance definitions of the IVDHFEs. Section 4 denotes to study the interval-valued dual hesitant fuzzy MADM based on the grey relational projection. In Section 5, we presents a numerical example to demonstrate the effectiveness and practicality of the proposed method and also discusses the advantages of the proposed method over the existing methods. In Section 6, we briefly conclude the paper.

2 Preliminaries

2.1 Distance measures for HFEs

Definition 1. [14, 15] Let X be a reference set, a hesitant fuzzy set (HFS) A on the set X is defined in terms of a function that when applied to X returns a subset of [0,1], the mathematical symbol is as: $A = {〈 x, h_{A} (x) 〉 | x \in X}$ where h_A (x) is a set of some values in [0,1], denoting the possible membership degrees of the element x ∈ X to the set A, For convenience, h = h_A (x) is called the hesitant fuzzy element (HFE).

Several common distance measures for two HFEs h₁ and h₂ are given as [48]:

The hesitant normalized Hamming distance: $d_{hnh}^{*} (h_{1}, h_{2}) = \frac{1}{l} \sum_{i = 1}^{l} | h_{1 σ (i)} - h_{2 σ (i)} |$ (1)

The hesitant normalized Euclidean distance: $d_{hne}^{*} (h_{1}, h_{2}) = {(\frac{1}{l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{2})}^{1 / 2}$ (2)

The generalized hesitant normalized distance: $d_{ghn}^{*} (h_{1}, h_{2}) = {(\frac{1}{l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ})}^{1 / λ}$ (3)

The generalized hesitant normalized Hausdorff distance: $d_{ghnh}^{*} (h_{1}, h_{2}) = {(max_{i} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ})}^{1 / λ}$ (4)

The generalized hybrid hesitant normalized distance:

$\begin{matrix} d_{ghhn}^{*} (h_{1}, h_{2}) \\ = [\frac{1}{2} (\frac{1}{l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ} \\ + {max_{i} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ})]}^{1 / λ} \end{matrix}$ (5) where h_1σ(i), h_2σ(i) are the ith largest value in h₁, h₂, and l = max{ l₁, l₂ }, where l₁, l₂ are the number of values in h₁, h₂,respectively. Sometimes, l₁ and l₂ are not equal, we adopt the method proposed by [49] to make them equivalent by adding some values to the shorter HFE.

Definition 2. [48] The above distance measures between h₁ and h₂ satisfy the following properties:

0≤ d^∗ (h₁, h₂) ≤ 1 ;

d^∗ (h₁, h₂) = 0 if and only if h₁ = h₂ ;

d^∗ (h₁, h₂) = d^∗ (h₂, h₁) .

2.2 Distance measures for DHFEs

Definition 3. [23] Let X be a fixed set, a dual hesitant fuzzy set (DHFS) D on the set X is defined as $D = {〈 x, h (x), g (x) 〉 | x \in X}$ where h (x) =∪ _γ∈h(x) { γ } and g (x) =∪ _η∈g(x) { η } are two sets of some values in [0,1], denoting the possible membership degrees and non-membership degrees of the element x ∈ X to the set D, respectively, with the conditions: $γ, η \in [0, 1], 0 \leq γ^{+} + η^{+} \leq 1 .$ where γ∈ h (x) , η ∈ g (x) , γ⁺ ∈ ∪ _γ∈h(x) max { γ }, η⁺∈ ∪ _η∈g(x) max { η }. For convenience, the pair d (x) ={ h (x) , g (x) } is called the dual hesitant fuzzy element (DHFE), denoted as d ={ h, g }.

Based on the distance measures for DHFSs and their properties proposed by Su et al. [29], the distance measures for DHFEs d₁ = {h₁, g₁} and d₂ = {h₂, g₂} are defined as:

The dual hesitant normalized Hamming distance:

$\begin{matrix} d_{dhnh}^{*} (d_{1}, d_{2}) \\ = \frac{1}{2} (\frac{1}{l} \sum_{i = 1}^{l} | h_{1 σ (i)} - h_{2 σ (i)} | \\ + \frac{1}{k} \sum_{i = 1}^{k} | g_{1 σ (i)} - g_{2 σ (i)} |) \end{matrix}$ (6)

The dual hesitant normalized Euclidean distance:

$\begin{matrix} d_{dhne}^{*} (d_{1}, d_{2}) \\ = [\frac{1}{2} (\frac{1}{l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{2} \\ + {\frac{1}{k} \sum_{i = 1}^{k} {| g_{1 σ (i)} - g_{2 σ (i)} |}^{2})]}^{1 / 2} \end{matrix}$ (7)

With the generalization of the Equations (6) and (7), we can obtain the generalized dual hesitant normalized distance:

$\begin{matrix} d_{gdhn}^{*} (d_{1}, d_{2}) \\ = [\frac{1}{2} (\frac{1}{l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ} \\ + {\frac{1}{k} \sum_{i = 1}^{k} {| g_{1 σ (i)} - g_{2 σ (i)} |}^{λ})]}^{1 / λ} \end{matrix}$ (8)

Based on the standard of Hausdorff distance measure, the generalized dual hesitant normalized Hausdorff distance is expressed as:

$\begin{matrix} d_{gdhnh}^{*} (d_{1}, d_{2}) \\ = [max {max_{i} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ}, \\ {max_{i} {| g_{1 σ (i)} - g_{2 σ (i)} |}^{λ}}]}^{1 / λ} \end{matrix}$ (9)

With the combination of the Equations (8) and (9), a generalized hybrid dual hesitant normalized distance can be defined as follows:

$\begin{matrix} d_{ghdhn}^{*} (d_{1}, d_{2}) \\ = [\frac{1}{2} (\frac{1}{2 l} \sum_{i = 1}^{l} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ} \\ + \frac{1}{2 k} \sum_{i = 1}^{k} {| g_{1 σ (i)} - g_{2 σ (i)} |}^{λ} \\ + max {max_{i} {| h_{1 σ (i)} - h_{2 σ (i)} |}^{λ}, \\ {max_{i} {| g_{1 σ (i)} - g_{2 σ (i)} |}^{λ}})]}^{1 / λ} \end{matrix}$ (10) where h_1σ(i) and h_2σ(i) are the ith largest value in h₁ and h₂, g_1σ(i) and g_1σ(i) are the ith largest value in g₁ and g₂, respectively. Here, l = max{ l₁, l₂ } and k = max{ k₁, k₂ }, where l₁, l₂ are the number of values in h₁, h₂ and k₁, k₂ are the number of values in g₁, g₂, respectively. When l₁ ≠ l₂ or k₁ ≠ k₂, we adopt the method proposed by [29] to make them equivalent by adding some values to the shorter DHFE.

Definition 4. The distance measures d^∗ (d₁, d₂) between d₁ and d₂ satisfy the following properties:

0≤ d^∗ (d₁, d₂) ≤ 1 ;

d^∗ (d₁, d₂) = 0 if and only if d₁ = d₂ ;

d^∗ (d₁, d₂) = d^∗ (d₂, d₁) .

3 Distance measures for IVDHFEs

Definition 5. [35] Let X be a fixed set, an interval-valued dual hesitant fuzzy set (IVDHFS) $\tilde{D}$ on the set X is defined as $D = {〈 x, \tilde{h} (x), \tilde{g} (x) 〉 | x \in X}$ where $\tilde{h} (x) = \cup_{[γ^{L}, γ^{U}] \in \tilde{h} (x)} {[γ^{L}, γ^{U}]}$ and $\tilde{g} (x) = \cup_{[η^{L}, η^{U}] \in \tilde{g} (x)} {[η^{L}, η^{U}]}$ are two sets of some interval values in [0, 1], denoting the possible membership degrees and non-membership degrees of the element x ∈ X to the set $\tilde{D}$ , respectively, with the conditions: $\begin{matrix} [γ^{L}, γ^{U}], [η^{L}, η^{U}] \\ \subset [0, 1], 0 \leq {(γ^{U})}^{+} + {(η^{U})}^{+} \leq 1 . \end{matrix}$ where $[γ^{L}, γ^{U}] \in \tilde{h} (x), [η^{L}, η^{U}] \in \tilde{g} (x), {(γ^{U})}^{+} \in {\tilde{h}}^{+} (x) = \cup_{[γ^{L}, γ^{U}] \in \tilde{h} (x)} max {γ^{U}}, {(η^{U})}^{+} \in {\tilde{g}}^{+} (x) = \cup_{[η^{L}, η^{U}] \in \tilde{g} (x)}$ max{ η^U }. For convenience, the pair $\tilde{d} (x) = {\tilde{h} (x), \tilde{g} (x)}$ is called the interval-valued dual hesitant fuzzy element (IVDHFE), denoted as $\tilde{d} = {\tilde{h}, \tilde{g}}$ . Especially, when γ^L = γ^U, η^L = η^U, the interval-valued dual hesitant fuzzy set reduces to the dual hesitant fuzzy set.

Similar to the definition of the distance measures for HFEs and DHFEs, we define the distance measures for IVDHFEs ${\tilde{d}}_{1} = {{\tilde{h}}_{1}, {\tilde{g}}_{1}}$ and ${\tilde{d}}_{2} = {{\tilde{h}}_{2}, {\tilde{g}}_{2}}$ .

We should do some preparing work before calculating the distance between ${\tilde{d}}_{1}$ and ${\tilde{d}}_{2}$ . Let l₁, l₂ be the number of values in ${\tilde{h}}_{1}$ , ${\tilde{h}}_{2}$ , and k₁, k₂ be the number of values in ${\tilde{g}}_{1}$ , ${\tilde{g}}_{2}$ , respectively. In most cases, l₁ ≠ l₂ or k₁ ≠ k₂, we can make them equivalent by adding some values to the shorter IVDHFE. In terms of pessimistic principles,the smallest value will be added. For example, if l₁ < l₂, then we should add the minimum value in ${\tilde{h}}_{1}$ until ${\tilde{h}}_{1}$ has the same length as ${\tilde{h}}_{2}$ .

The distance measures for IVDHFEs ${\tilde{d}}_{1} = {{\tilde{h}}_{1}, {\tilde{g}}_{1}}$ and ${\tilde{d}}_{2} = {{\tilde{h}}_{2}, {\tilde{g}}_{2}}$ are defined as follows:

The interval-valued dual hesitant normalized Hamming distance:

$\begin{matrix} d_{ivdhnh}^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \\ = \frac{1}{4} [\frac{1}{l} \sum_{i = 1}^{l} (| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} | + | {\tilde{h}}_{1 σ (i)}^{U} \\ - {\tilde{h}}_{2 σ (i)}^{U} |) + \frac{1}{k} \sum_{i = 1}^{k} (| {\tilde{g}}_{1 σ (i)}^{L} - {\tilde{g}}_{2 σ (i)}^{L} | \\ + | {\tilde{g}}_{1 σ (i)}^{U} - {\tilde{g}}_{2 σ (i)}^{U} |)] \end{matrix}$ (11)

The interval-valued dual hesitant normalized Euclidean distance:

$\begin{matrix} d_{ivdhne}^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \\ = \frac{1}{4} [\frac{1}{l} \sum_{i = 1}^{l} ({| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} |}^{2} + | {\tilde{h}}_{1 σ (i)}^{U} \\ - {{\tilde{h}}_{2 σ (i)}^{U} |}^{2}) + \frac{1}{k} \sum_{i = 1}^{k} ({| {\tilde{g}}_{1 σ (i)}^{L} - {\tilde{g}}_{2 σ (i)}^{L} |}^{2} \\ + {{| {\tilde{g}}_{1 σ (i)}^{U} - {\tilde{g}}_{2 σ (i)}^{U} |}^{2})]}^{1 / 2} \end{matrix}$ (12)

With the generalization of the Equations (11) and (12), we can obtain the generalized interval-valued dual hesitant normalized distance:

$\begin{matrix} d_{givdhn}^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \\ = \frac{1}{4} [\frac{1}{l} \sum_{i = 1}^{l} ({| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} |}^{λ} + | {\tilde{h}}_{1 σ (i)}^{U} \\ - {{\tilde{h}}_{2 σ (i)}^{U} |}^{λ}) + \frac{1}{k} \sum_{i = 1}^{k} ({| {\tilde{g}}_{1 σ (i)}^{L} - {\tilde{g}}_{2 σ (i)}^{L} |}^{λ} \\ + {{| {\tilde{g}}_{1 σ (i)}^{U} - {\tilde{g}}_{2 σ (i)}^{U} |}^{λ})]}^{1 / λ} \end{matrix}$ (13)

Based on the standard of Hausdorff distance measure, the generalized interval-valued dual hesitant normalized Hausdorff distance is expressed as:

$\begin{matrix} d_{givdhnh}^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \\ = [max {max_{i} ({| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} |}^{λ}, \\ {| {\tilde{h}}_{1 σ (i)}^{U} - {\tilde{h}}_{2 σ (i)}^{U} |}^{λ}), max_{i} (| {\tilde{g}}_{1 σ (i)}^{L} \\ - {{{\tilde{g}}_{2 σ (i)}^{L} |}^{λ}, {| {\tilde{g}}_{1 σ (i)}^{U} - {\tilde{g}}_{2 σ (i)}^{U} |}^{λ})}]}^{1 / λ} \end{matrix}$ (14)

With the combination of the Equations (13) and (14), a generalized hybrid interval-valued dual hesitant normalized distance can be defined as:

$\begin{matrix} d_{ghivdhn}^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \\ = [\frac{1}{2} (\frac{1}{4} (\frac{1}{l} \sum_{i = 1}^{l} ({| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} |}^{λ} + | {\tilde{h}}_{1 σ (i)}^{U} \\ - {\tilde{h}}_{2 σ (i)}^{U} |) + \frac{1}{k} \sum_{i = 1}^{k} ({| {\tilde{g}}_{1 σ (i)}^{L} - {\tilde{g}}_{2 σ (i)}^{L} |}^{λ} + | {\tilde{g}}_{1 σ (i)}^{U} \\ - {{\tilde{g}}_{2 σ (i)}^{U} |}^{λ})) + max {max_{i} ({| {\tilde{h}}_{1 σ (i)}^{L} - {\tilde{h}}_{2 σ (i)}^{L} |}^{λ}, \\ {| {\tilde{h}}_{1 σ (i)}^{U} - {\tilde{h}}_{2 σ (i)}^{U} |}^{λ}), max_{i} ({| {\tilde{g}}_{1 σ (i)}^{L} - {\tilde{g}}_{2 σ (i)}^{L} |}^{λ}, \\ {{| {\tilde{g}}_{1 σ (i)}^{U} - {\tilde{g}}_{2 σ (i)}^{U} |}^{λ})})]}^{1 / λ} \end{matrix}$ (15) where ${\tilde{h}}_{1 σ (i)} = [{\tilde{h}}_{1 σ (i)}^{L}, {\tilde{h}}_{1 σ (i)}^{U}]$ and ${\tilde{h}}_{2 σ (i)} = [{\tilde{h}}_{2 σ (i)}^{L}, {\tilde{h}}_{2 σ (i)}^{U}]$ are the ith largest value in ${\tilde{h}}_{1}$ and ${\tilde{h}}_{2}$ , ${\tilde{g}}_{1 σ (i)} = [{\tilde{g}}_{1 σ (i)}^{L}, {\tilde{g}}_{1 σ (i)}^{U}]$ and ${\tilde{g}}_{2 σ (i)} = [{\tilde{g}}_{2 σ (i)}^{L}, {\tilde{g}}_{2 σ (i)}^{U}]$ are the ith largest value in ${\tilde{g}}_{1}$ and ${\tilde{g}}_{2}$ , Here, l = max{ l₁, l₂ } and k = max{ k₁, k₂ } where l₁, l₂ are the number of values in ${\tilde{h}}_{1}, {\tilde{h}}_{2}$ and k₁, k₂ are the number of values in ${\tilde{g}}_{1}, {\tilde{g}}_{2}$ , respectively.

Definition 6. Let ${\tilde{d}}_{1}$ and ${\tilde{d}}_{2}$ be two IVDHFEs, $d^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2})$ be the distance measure between ${\tilde{d}}_{1}$ and $\tilde{d}$ which satisfies the following properties:

$0 \leq d^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) \leq 1;$

$d^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) = 0 if and only if {\tilde{d}}_{1} = {\tilde{d}}_{2};$

$d^{*} ({\tilde{d}}_{1}, {\tilde{d}}_{2}) = d^{*} ({\tilde{d}}_{2}, {\tilde{d}}_{1}) .$

4 Proposed method for MADM

Let S = (A, C, W) be the interval-valued dual hesitant fuzzy MADM system, A = {A₁, A₂,. . . , A_m} be m alternatives; C = {C₁, C₂,. . . , C_n} be n attributes, W = (w₁, w₂,. . . , w_n) ^T be the weight vector of the attributes, which satisfies $w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1$ . The attribute value of alternative A_i (i = 1, 2,. . . , m) on the attribute C_j (j = 1, 2,. . . , n) is the IVDHFE ${\hat{d}}_{ij} = {{\hat{h}}_{ij}, {\hat{g}}_{ij}}$ , and the interval-valued dual hesitant fuzzy decision matrix $\hat{D} = {[{\hat{d}}_{ij}]}_{m \times n}$ is constructed, and the purpose is to rank the alternatives.

4.1 Normalized decision matrix

We need to translate the decision matrix $\hat{D} = {[{\hat{d}}_{ij}]}_{m \times n}$ into the normalized decision matrix $\tilde{D} = {[{\tilde{d}}_{ij}]}_{m \times n}$ . There are usually two types of attributes, namely, benefit attributes and cost attributes, the normalized ${\tilde{d}}_{ij}$ can be calculated as follows: ${\tilde{d}}_{ij} = {\begin{matrix} {{\hat{h}}_{ij}, {\hat{g}}_{ij}}, C_{j} \in benefit attributes \\ {{\hat{g}}_{ij}, {\hat{h}}_{ij}}, C_{j} \in cost attributes \end{matrix}$

4.2 Grey relational projection

The grey relational projection method is a multi-attribute system decision method. In this paper, we extend the grey relational projection into interval-valued dual hesitant fuzzy environment. The process is shown as follows:

The interval-valued dual hesitant fuzzy ideal solution

For a normalized interval-valued dual hesitant fuzzy decision making matrix $\tilde{D}$ , the interval-valued dual hesitant fuzzy positive ideal solution(IVDHFPIS) and the interval-valued dual hesitant fuzzy negative ideal solution (IVDHFNIS) are expressed as: $\begin{matrix} {\tilde{d}}^{+} & = & {{\tilde{d}}_{1}^{+}, {\tilde{d}}_{2}^{+}, . . ., {\tilde{d}}_{n}^{+}} \\ = & {{{\tilde{h}}_{1}^{+}, {\tilde{g}}_{1}^{+}}, {{\tilde{h}}_{2}^{+}, {\tilde{g}}_{2}^{+}}, . . ., {{\tilde{h}}_{n}^{+}, {\tilde{g}}_{n}^{+}}} \end{matrix}$ (16) $\begin{matrix} {\tilde{d}}^{-} & = & {{\tilde{d}}_{1}^{-}, {\tilde{d}}_{2}^{-}, . . ., {\tilde{d}}_{n}^{-}} \\ = & {{{\tilde{h}}_{1}^{-}, {\tilde{g}}_{1}^{-}}, {{\tilde{h}}_{2}^{-}, {\tilde{g}}_{2}^{-}}, . . ., {{\tilde{h}}_{n}^{-}, {\tilde{g}}_{n}^{-}}} \end{matrix}$ (17) where $\begin{matrix} {\tilde{h}}_{j}^{+} & = & {[max_{i} {\tilde{h}}_{ij σ (1)}^{L}, max_{i} {\tilde{h}}_{ij σ (1)}^{U}], [max_{i} {\tilde{h}}_{ij σ (2)}^{L}, \\ max_{i} {\tilde{h}}_{ij σ (2)}^{U}], . . ., [max_{i} {\tilde{h}}_{ij σ (n)}^{L}, max_{i} {\tilde{h}}_{ij σ (n)}^{U}]} \\ {\tilde{g}}_{j}^{+} & = & {[min_{i} {\tilde{g}}_{ij σ (1)}^{L}, min_{i} {\tilde{g}}_{ij σ (1)}^{U}], [min_{i} {\tilde{g}}_{ij σ (2)}^{L}, \\ min_{i} {\tilde{g}}_{ij σ (2)}^{U}], . . ., [min_{i} {\tilde{g}}_{ij σ (n)}^{L}, min_{i} {\tilde{g}}_{ij σ (n)}^{U}]} \\ {\tilde{h}}_{j}^{-} & = & {[min_{i} {\tilde{h}}_{ij σ (1)}^{L}, min_{i} {\tilde{h}}_{ij σ (1)}^{U}], [min_{i} {\tilde{h}}_{ij σ (2)}^{L}, \\ min_{i} {\tilde{h}}_{ij σ (2)}^{U}], . . ., [min_{i} {\tilde{h}}_{ij σ (n)}^{L}, min_{i} {\tilde{h}}_{ij σ (n)}^{U}]} \\ {\tilde{g}}_{j}^{-} & = & {[max_{i} {\tilde{g}}_{ij σ (1)}^{L}, max_{i} {\tilde{g}}_{ij σ (1)}^{U}], [max_{i} {\tilde{g}}_{ij σ (2)}^{L}, \\ max_{i} {\tilde{g}}_{ij σ (2)}^{U}], . . ., [max_{i} {\tilde{g}}_{ij σ (n)}^{L}, max_{i} {\tilde{g}}_{ij σ (n)}^{U}]} \end{matrix}$

The grey relational coefficient

The grey relational coefficient of each alternative from the IVDHFPIS is formulated as: ${\tilde{r}}_{ij}^{+} = \frac{min_{i} min_{j} {\tilde{D}}_{ij}^{+} + θ max_{i} max_{j} {\tilde{D}}_{ij}^{+}}{{\tilde{D}}_{ij}^{+} + θ max_{i} max_{j} {\tilde{D}}_{ij}^{+}},$ (18) where θ ∈ [0, 1] represents the resolution coefficient which is given by the decision makers, and ${\tilde{D}}_{ij}^{+}$ is the generalized hybrid dual hesitant normalized distance between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{+}$ , and

$\begin{matrix} {\tilde{D}}_{ij}^{+} \\ = [\frac{1}{2} (\frac{1}{4} (\frac{1}{l} \sum_{s = 1}^{l} {| {\tilde{h}}_{ij σ (s)}^{L} - {\tilde{h}}_{j σ (s)}^{L +} |}^{λ} + | {\tilde{h}}_{ij σ (s)}^{U} \\ - {{\tilde{h}}_{j σ (s)}^{U +} |}^{λ}) + \frac{1}{k} \sum_{s = 1}^{k} ({| {\tilde{g}}_{ij σ (s)}^{L} - {\tilde{g}}_{j σ (s)}^{L +} |}^{λ} \\ + {| {\tilde{g}}_{ij σ (s)}^{U} - {\tilde{g}}_{j σ (s)}^{U +} |}^{λ})) + max {max_{s} (| {\tilde{h}}_{ij σ (s)}^{L} \\ - {{\tilde{h}}_{j σ (s)}^{L +} |}^{λ}, {| {\tilde{h}}_{ij σ (s)}^{U} - {\tilde{h}}_{j σ (s)}^{U +} |}^{λ}), max_{s} (| {\tilde{g}}_{ij σ (s)}^{L} \\ - {{\tilde{g}}_{j σ (s)}^{L +} |}^{λ}, {{| {\tilde{g}}_{ij σ (s)}^{U} - {\tilde{g}}_{j σ (s)}^{U +} |}^{λ})})]}^{1 / λ} \end{matrix}$ (19)

Similarly, the grey relational coefficient of each alternative from the IVDHFNIS is formulated as: ${\tilde{r}}_{ij}^{-} = \frac{min_{i} min_{j} {\tilde{D}}_{ij}^{-} + θ max_{i} max_{j} {\tilde{D}}_{ij}^{-}}{{\tilde{D}}_{ij}^{-} + θ max_{i} max_{j} {\tilde{D}}_{ij}^{-}},$ (20) where θ ∈ [0, 1] represents the resolution coefficient which is given by the decision makers, and ${\tilde{D}}_{ij}^{-}$ is the generalized hybrid dual hesitant normalized distance between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{-}$ , and

$\begin{matrix} {\tilde{D}}_{ij}^{-} \\ = [\frac{1}{2} (\frac{1}{4} (\frac{1}{l} \sum_{s = 1}^{l} {| {\tilde{h}}_{ij σ (s)}^{L} - {\tilde{h}}_{j σ (s)}^{L -} |}^{λ} + | {\tilde{h}}_{ij σ (s)}^{U} \\ - {{\tilde{h}}_{j σ (s)}^{U -} |}^{λ}) + \frac{1}{k} \sum_{s = 1}^{k} ({| {\tilde{g}}_{ij σ (s)}^{L} - {\tilde{g}}_{j σ (s)}^{L -} |}^{λ} \\ + {| {\tilde{g}}_{ij σ (s)}^{U} - {\tilde{g}}_{j σ (s)}^{U -} |}^{λ})) + max {max_{s} (| {\tilde{h}}_{ij σ (s)}^{L} \\ - {{\tilde{h}}_{j σ (s)}^{L -} |}^{λ}, {| {\tilde{h}}_{ij σ (s)}^{U} - {\tilde{h}}_{j σ (s)}^{U -} |}^{λ}), max_{s} (| {\tilde{g}}_{ij σ (s)}^{L} \\ - {{\tilde{g}}_{j σ (s)}^{L -} |}^{λ}, {{| {\tilde{g}}_{ij σ (s)}^{U} - {\tilde{g}}_{j σ (s)}^{U -} |}^{λ})})]}^{1 / λ} \end{matrix}$ (21)

Then we can construct the grey relational coefficient matrix ${\tilde{r}}^{+} = {[{\tilde{r}}_{ij}^{+}]}_{m \times n}$ and ${\tilde{r}}^{-} = {[{\tilde{r}}_{ij}^{-}]}_{m \times n}$ . That is, $\begin{matrix} {\tilde{r}}^{+} & = & [\begin{matrix} {\tilde{r}}_{11}^{+} & {\tilde{r}}_{12}^{+} & \dots & {\tilde{r}}_{1 n}^{+} \\ {\tilde{r}}_{21}^{+} & {\tilde{r}}_{22}^{+} & \dots & {\tilde{r}}_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{r}}_{m 1}^{+} & {\tilde{r}}_{m 2}^{+} & \dots & {\tilde{r}}_{mn}^{+} \end{matrix}] \\ {\tilde{r}}^{-} & = & [\begin{matrix} {\tilde{r}}_{11}^{-} & {\tilde{r}}_{12}^{-} & \dots & {\tilde{r}}_{1 n}^{-} \\ {\tilde{r}}_{21}^{-} & {\tilde{r}}_{22}^{-} & \dots & {\tilde{r}}_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{r}}_{m 1}^{-} & {\tilde{r}}_{m 2}^{-} & \dots & {\tilde{r}}_{mn}^{-} \end{matrix}] \end{matrix}$

The grey relational coefficient between IVDHFPIS and IVDHFPIS, and that between IVDHFNIS and IVDHFNIS are shown as: ${\tilde{r}}_{0}^{+} = ({\tilde{r}}_{01}^{+}, {\tilde{r}}_{02}^{+}, . . ., {\tilde{r}}_{0 n}^{+}) = \underset{n}{\underset{︸}{1, 1, . . ., 1}},$ (22) ${\tilde{r}}_{0}^{-} = ({\tilde{r}}_{01}^{-}, {\tilde{r}}_{02}^{-}, . . ., {\tilde{r}}_{0 n}^{-}) = \underset{n}{\underset{︸}{1, 1, . . ., 1}} .$ (23)

The weighted grey relational coefficient matrices ${\tilde{R}}^{+} = {[w_{j} {\tilde{r}}_{ij}^{+}]}_{m \times n}$ and ${\tilde{R}}^{-} = {[w_{j} {\tilde{r}}_{ij}^{-}]}_{m \times n}$ are given as: $\begin{matrix} {\tilde{R}}^{+} & = & [\begin{matrix} w_{1} {\tilde{r}}_{11}^{+} & w_{2} {\tilde{r}}_{12}^{+} & \dots & w_{n} {\tilde{r}}_{1 n}^{+} \\ w_{1} {\tilde{r}}_{21}^{+} & w_{2} {\tilde{r}}_{22}^{+} & \dots & w_{n} {\tilde{r}}_{2 n}^{+} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{1} {\tilde{r}}_{m 1}^{+} & w_{2} {\tilde{r}}_{m 2}^{+} & \dots & w_{n} {\tilde{r}}_{mn}^{+} \end{matrix}] \\ {\tilde{R}}^{-} & = & [\begin{matrix} w_{1} {\tilde{r}}_{11}^{-} & w_{2} {\tilde{r}}_{12}^{-} & \dots & w_{n} {\tilde{r}}_{1 n}^{-} \\ w_{1} {\tilde{r}}_{21}^{-} & w_{2} {\tilde{r}}_{22}^{-} & \dots & w_{n} {\tilde{r}}_{2 n}^{-} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{1} {\tilde{r}}_{m 1}^{-} & w_{2} {\tilde{r}}_{m 2}^{-} & \dots & w_{n} {\tilde{r}}_{mn}^{-} \end{matrix}] \end{matrix}$

The weighted grey relational coefficient between IVDHFPIS and IVDHFPIS, and that between IVDHFNIS and IVDHFNIS are shown as: $\begin{matrix} {\tilde{R}}_{0}^{+} & = & (w_{1} {\tilde{r}}_{01}^{+}, w_{2} {\tilde{r}}_{02}^{+}, . . ., w_{n} {\tilde{r}}_{0 n}^{+}) \\ = & (w_{1}, w_{2}, . . ., w_{n}) \end{matrix}$ (24) $\begin{matrix} {\tilde{R}}_{0}^{-} & = & (w_{1} {\tilde{r}}_{01}^{-}, w_{2} {\tilde{r}}_{02}^{-}, . . ., w_{n} {\tilde{r}}_{0 n}^{-}) \\ = & (w_{1}, w_{2}, . . ., w_{n}) \end{matrix}$ (25)

The grey relational projection

Suppose ${\tilde{R}}_{0}^{+} = (w_{1}, w_{2}, . . ., w_{n})$ and ${\tilde{R}}_{0}^{-} = (w_{1}, w_{2}, . . ., w_{n})$ are two vectors. Referring to the projection formula [43, 44], and the weighted grey relational coefficient matrix ${\tilde{R}}^{+} = {[w_{j} {\tilde{r}}_{ij}^{+}]}_{m \times n}$ and ${\tilde{R}}^{-} = {[w_{j} {\tilde{r}}_{ij}^{-}]}_{m \times n}$ , we can get the weighted grey relational projection of alternative A_i onto IVDHFPIS or IVDHFNIS as shown: $\begin{matrix} {\tilde{p}}_{i}^{+} & = & | {\tilde{R}}_{i}^{+} | \cdot cos ({\tilde{R}}_{i}, {\tilde{R}}_{0}^{+}) \\ = & \sqrt{\sum_{j = 1}^{n} {(w_{j} {\tilde{r}}_{ij}^{+})}^{2}} \cdot \frac{\sum_{j - 1}^{n} (w_{j} {\tilde{r}}_{ij}^{+} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} {(w_{j} {\tilde{r}}_{ij}^{+})}^{2}} \cdot \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ = & \frac{\sum_{j = 1}^{n} (w_{j} {\tilde{r}}_{ij}^{+} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{w}}_{j} {\tilde{r}}_{ij}^{+} \end{matrix}$ (26) $\begin{matrix} {\tilde{p}}_{i}^{-} & = & | {\tilde{R}}_{i}^{-} | \cdot cos ({\tilde{R}}_{i}, {\tilde{R}}_{0}^{-}) \\ = & \sqrt{\sum_{j = 1}^{n} {(w_{j} {\tilde{r}}_{ij}^{-})}^{2}} \cdot \frac{\sum_{j = 1}^{n} (w_{j} {\tilde{r}}_{ij}^{-} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} {(w_{j} {\tilde{r}}_{ij}^{-})}^{2}} \cdot \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ = & \frac{\sum_{j = 1}^{n} (w_{j} {\tilde{r}}_{ij}^{-} \cdot w_{j})}{\sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} = \sum_{j = 1}^{n} {\bar{w}}_{j} {\tilde{r}}_{ij}^{-} \end{matrix}$ (27)

where ${\bar{w}}_{j} = w_{j}^{2} / \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}$ .

For alternative A_i, the bigger the value of ${\tilde{p}}_{i}^{+}$ is, the closer it is to IVDHFPIS, whereas the smaller the value of ${\tilde{p}}_{i}^{-}$ is, the closer it is to IVDHFNIS, and vice versa. Thus, in order to comprehensively aggregate the grey relational projection values and obtain the optimal alternative, we construct the relative closeness formula which refers to the topsis method.

${\tilde{C}}_{i} = \frac{{\tilde{p}}_{i}^{+}}{{\tilde{p}}_{i}^{+} + {\tilde{p}}_{i}^{-}} = \frac{\sum_{j = 1}^{n} {\bar{w}}_{j} {\tilde{r}}_{ij}^{+}}{\sum_{j = 1}^{n} {\bar{w}}_{j} {\tilde{r}}_{ij}^{+} + \sum_{j = 1}^{n} {\bar{w}}_{j} {\tilde{r}}_{ij}^{-}} .$ (28)

Obviously, the larger ${\tilde{C}}_{i}$ is, the better A_i is, and vice versa.

4.3 Determine the attribute weight

In the multi-attribute decision problems, in some cases, the attribute weights are known, sometimes, the attribute weights are unknown in some cases. There are several methods to determine the attribute weights, such as information entropy method [51, 52], minimum deviation method [53, 54]. In this paper, we construct the nonlinear optimization model with the minimum total deviation of the grey relational coefficient to determine the attribute weights, which can avoid the interference of subjective human factors.

Consider the weight of each attribute, under attribute C_j, the grey relational coefficient deviation between the alternative A_i and IVDHFPIS is $(1 - {\tilde{r}}_{ij}^{+})$ . To eliminate the impact of symbolic factors, the deviation sums take the form of squared sums and we get the integrated grey relational coefficient weighted deviation sum between the alternative A_i and IVDHFPIS as $\sum_{j = 1}^{n} {[(1 - {\tilde{r}}_{ij}^{+}) w_{j}]}^{2}$ . Furthermore,we have the integrated grey relational coefficient weighted deviation sum of all the alternatives as $C (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {[(1 - {\tilde{r}}_{ij}^{+}) w_{j}]}^{2},$ where W = (w₁, w₂,. . . , w_n) ^T is the vector of the attribute weights, which satisfies $w_{j} \geq 0, \sum_{j = 1}^{n} w_{j} = 1$ . The resulting weight vector should minimize the integrated grey relational coefficient weighted deviation sum of all the alternatives, thus we construct the following objective function: $min C (w) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {[(1 - {\tilde{r}}_{ij}^{+}) w_{j}]}^{2}$ (29) $s . t . {\begin{matrix} \sum_{j = 1}^{n} w_{j} = 1 \\ 0 \leq w_{j} \leq 1 \end{matrix}$

Then construct the Lagrangian function:

$\begin{matrix} L (w, λ) \\ = \sum_{i = 1}^{m} \sum_{j = 1}^{n} {[(1 - {\tilde{r}}_{ij}^{+}) w_{j}]}^{2} \\ + 2 λ (\sum_{j = 1}^{n} w_{j} - 1) \end{matrix}$ (30)

According to the necessary conditions for the existence of extreme value, we take the partial derivative of w_j and λ, respectively, ${\begin{matrix} \frac{\partial L}{\partial w_{j}} = 2 \sum_{i = 1}^{m} {(1 - {\tilde{r}}_{ij}^{+})}^{2} w_{j} + 2 λ = 0, \\ \frac{\partial L}{\partial λ} = \sum_{j = 1}^{n} w_{j} - 1 = 0 . \end{matrix}$ (31) Then get ${\begin{matrix} λ = - {[\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} {(1 - {\tilde{r}}_{ij}^{+})}^{2})}^{- 1}]}^{- 1}, \\ \begin{matrix} w_{j} = {[\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} {(1 - {\tilde{r}}_{ij}^{+})}^{2})}^{- 1}]}^{- 1} \\ \times {[\sum_{i = 1}^{m} {(1 - {\tilde{r}}_{ij}^{+})}^{2}]}^{- 1} . \end{matrix} \end{matrix}$ (32)

4.4 The procedure of the MADM method

In sum, the algorithm for MADM with interval-valued dual hesitant fuzzy information, using the grey relational projection method, is given as follows:

Step 1. Establish the decision matrix $\hat{D}$ according to the interval-valued dual hesitant fuzzy information given by decision maker, then translate it into the normalized decision matrix $\tilde{D}$ .

Step 2. Determine the interval-valued dual hesitant fuzzy positive ideal solution (IVDHFPIS) and the interval-valued dual hesitant fuzzy negative ideal solution (IVDHFNIS) shown in Equations (16) and (17).

Step 3. Calculate the generalized hybrid dual hesitant normalized distance between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{+}$ , and that between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{-}$ , shown in Equations (19) and (21).

Step 5. Determine the attribute weight vector W = (w₁, w₂,. . . , w_n) ^T by Equation (32).

Step 6. Calculate the grey relational projection of each alternative onto the IVDHFPIS ${\tilde{p}}_{i}^{+}$ , and the grey relational projection of each alternative onto the IVDHFNIS ${\tilde{p}}_{i}^{-}$ , shown in Equations (26) and (27).

Step 7. Construct the relative closeness of each alternative ${\tilde{C}}_{i}$ shown in Equation (28).

Step 8. Rank all the alternatives according to the relative closeness, then choose the best one.

5 Illustrative example

5.1 Example

In this section, a practical example is used to illustrate the professional exercise of the proposed method, which refers to the one given by Ju et al. [35] and Zhang et al. [36]. There are four alternatives A₁, A₂, A₃, A₄ and three attributes C₁, C₂, C₃ in the MADM problem. We suppose attribute weights are unknown. Among the attributes, C₁ and C₃ are the benefit attributes, C₂ is the cost attribute. The four possible alternatives A_i (i = 1, 2, 3, 4) are evaluated by using the IVDHFEs under the three attributes C_j (j = 1, 2, 3), and the decision matrix $\hat{D}$ is shown as follow: $\begin{matrix} \hat{D} & = & [\begin{matrix} \begin{matrix} {{[0.2, 0.3], [0.2, 0.4]}, \\ {[0.3, 0.4], [0.4, 0.5], [0.5, 0.6]}} \end{matrix} \\ \begin{matrix} {{[0.3, 0.4]}, \\ {[0.4, 0.5], [0.5, 0.6]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.3], [0.3, 0.5]}, \\ {[0.3, 0.5], [0.4, 0.5]}} \end{matrix} \\ \begin{matrix} {{[0.1, 0.2]}, \\ {[0.3, 0.4], [0.4, 0.6], [0.5, 0.8]}} \end{matrix} \end{matrix} \\ \to & \begin{matrix} \begin{matrix} {{[0.3, 0.5], [0.4, 0.5]}, \\ {[0.2, 0.3], [0.3, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.3, 0.4], [0.2, 0.4]}, \\ {[0.2, 0.3], [0.4, 0.5]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.4], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.1, 0.3], [0.2, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.4, 0.5], [0.5, 0.6]}, \\ {[0.2, 0.3], [0.2, 0.4]}} \end{matrix} \end{matrix} \\ \to & \begin{matrix} \begin{matrix} {{[0.1, 0.3], [0.2, 0.3]}, \\ {[0.3, 0.5], [0.4, 0.5], [0.4, 0.6]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.4], [0.3, 0.4]}, \\ {[0.2, 0.3], [0.4, 0.5], [0.5, 0.6]}} \end{matrix} \\ \begin{matrix} {{[0.1, 0.3], [0.2, 0.3]}, \\ {[0.4, 0.5], [0.5, 0.6], [0.6, 0.7]}} \end{matrix} \\ \begin{matrix} {{[0.1, 0.2], [0.1, 0.3]}, \\ {[0.4, 0.6], [0.6, 0.7]}} \end{matrix} \end{matrix}] \end{matrix}$

The goal is to obtain the ranking order of the alternatives.

Step 1. Standardize decision matrix, we have $\begin{matrix} \tilde{D} & = & [\begin{matrix} \begin{matrix} {{[0.3, 0.4], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.2, 0.3], [0.2, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.4, 0.5], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.3, 0.4], [0.3, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.3, 0.5], [0.3, 0.5], [0.4, 0.5]}, \\ {[0.2, 0.3], [0.3, 0.5]}} \end{matrix} \\ \begin{matrix} {{[0.3, 0.4], [0.4, 0.6], [0.5, 0.8]}, \\ {[0.1, 0.2], [0.1, 0.2]}} \end{matrix} \end{matrix} \\ \to & \begin{matrix} \begin{matrix} {{[0.3, 0.5], [0.3, 0.5], [0.4, 0.5]}, \\ {[0.2, 0.3], [0.3, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.4], [0.2, 0.4], [0.3, 0.4]}, \\ {[0.2, 0.3], [0.4, 0.5]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.4], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.1, 0.3], [0.2, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.4, 0.5], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.2, 0.3], [0.2, 0.4]}} \end{matrix} \end{matrix} \\ \to & \begin{matrix} \begin{matrix} {{[0.3, 0.5], [0.4, 0.5], [0.4, 0.6]}, \\ {[0.1, 0.3], [0.2, 0.3]}} \end{matrix} \\ \begin{matrix} {{[0.2, 0.3], [0.4, 0.5], [0.5, 0.6]}, \\ {[0.2, 0.4], [0.3, 0.4]}} \end{matrix} \\ \begin{matrix} {{[0.4, 0.5], [0.5, 0.6], [0.6, 0.7]}, \\ {[0.1, 0.3], [0.2, 0.3]}} \end{matrix} \\ \begin{matrix} {{[0.4, 0.6], [0.4, 0.6], [0.6, 0.7]}, \\ {[0.1, 0.2], [0.1, 0.3]}} \end{matrix} \end{matrix}] \end{matrix}$

Step 2. Calculate the interval-valued dual hesitant fuzzy positive ideal solution and the interval-valued dual hesitant fuzzy negative ideal solution. According to Equations (16) and (17), we get $\begin{matrix} {\tilde{d}}^{+} & = & {({[0.4, 0.5], [0.4, 0, 6], [0.5, 0.8]}, \\ {[0.1, 0.2], [0.1, 0.2]}), ({[0.4, 0.5], \\ [0.4, 0.5], [0.5, 0.6]}, {[0.1, 0.3], \\ [0.2, 0.4]}), ({[0.4, 0.6], [0.5, 0.6], \\ [0.6, 0.7]}, {[0.1, 0.2], [0.1, 0.3]})} \\ {\tilde{d}}^{-} & = & {({[0.3, 0.4], [0.3, 0.5], [0.4, 0.5]}, \\ {[0.3, 0.4], [0.3, 0.5]}), ({[0.2, 0.4], \\ [0.2, 0.4], [0.3, 0.4]}, {[0.2, 0.3], \\ [0.4, 0.5]}), ({[0.2, 0.3], [0.4, 0.5], \\ [0.4, 0.6]}, {[0.2, 0.4], [0.3, 0.4]})} \end{matrix}$

Step 3. Calculate the distance between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{+}$ , and that between ${\tilde{d}}_{ij}$ and ${\tilde{d}}_{j}^{-}$ , from Equations (19) and (21). Let λ = 2, we have $\begin{matrix} {\tilde{D}}^{+} & = & [\begin{matrix} 0.1652 & 0.0890 & 0.1581 \\ 0.1791 & 0.1768 & 0.2415 \\ 0.2445 & 0.1486 & 0.0816 \\ 0.0764 & 0.0750 & 0.0736 \end{matrix}] \\ {\tilde{D}}^{-} & = & [\begin{matrix} 0.0935 & 0.0935 & 0.1568 \\ 0.0878 & 0.0000 & 0.0736 \\ 0.0816 & 0.1708 & 0.1696 \\ 0.2512 & 0.1750 & 0.2432 \end{matrix}] \end{matrix}$

Step 4. Calculate the grey relational coefficient of each alternative from the IVDHFPIS ${\tilde{r}}_{ij}^{+}$ , and the grey relational coefficient of each alternative from the IVDHFNIS ${\tilde{r}}_{ij}^{-}$ , shown in Equations (18) and (20). Let θ = 0.5, we obtain $\begin{matrix} {\tilde{r}}^{+} & = & [\begin{matrix} 0.6813 & 0.9272 & 0.6986 \\ 0.6499 & 0.6550 & 0.5384 \\ 0.5340 & 0.7231 & 0.9605 \\ 0.9860 & 0.9929 & 1.0000 \end{matrix}] \\ {\tilde{r}}^{-} & = & [\begin{matrix} 0.5731 & 0.5731 & 0.4448 \\ 0.5886 & 1.0000 & 0.6305 \\ 0.6060 & 0.4238 & 0.4255 \\ 0.3333 & 0.4178 & 0.3405 \end{matrix}] \end{matrix}$

Step 5. Calculate the attribute weights. According to Equation (32), we get $W = {(0.2155, 0.4731, 0.3114)}^{T}$

Step 6. Calculate the weighted grey relational projection of each alternative onto the IVDHFPIS ${\tilde{p}}_{i}^{+}$ , and the weighted grey relational projection of each alternative onto the IVDHFNIS ${\tilde{p}}_{i}^{-}$ . According to Equations (26) and (27) we obtain $\begin{matrix} {\tilde{p}}_{1}^{+} & = & 0.5065, {\tilde{p}}_{2}^{+} = 0.3779, {\tilde{p}}_{3}^{+} = 0.4617, \\ {\tilde{p}}_{4}^{+} & = & 0.6023, {\tilde{p}}_{1}^{-} = 0.3268, {\tilde{p}}_{2}^{-} = 0.5154, \\ {\tilde{p}}_{3}^{-} & = & 0.2711, {\tilde{p}}_{4}^{-} = 0.2344 . \end{matrix}$

Step 7. Calculate the relative closeness. According to Equation (28), we have $\begin{matrix} {\tilde{C}}_{1} & = & 0.6078, {\tilde{C}}_{2} = 0.4230, {\tilde{C}}_{3} = 0.6301, \\ {\tilde{C}}_{4} & = & 0.7199 . \end{matrix}$

Step 8. Rank the alternatives $A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} .$

The above calculation process is based on that the attribute weights are unknown. For comparative analysis, if we use the attribute weights given in [35, 36], that is W = (0.35, 0.25, 0.40) ^T, then we have:

Step 6*. Calculate the weighted grey relational projection of each alternative onto the IVDHFPIS ${\tilde{p}}_{i}^{+}$ , and the weighted grey relational projection of each alternative onto the IVDHFNIS ${\tilde{p}}_{i}^{-}$ . According to Equations (26) and (27), we obtain $\begin{matrix} {\tilde{p}}_{1}^{+} & = & 0.4311, {\tilde{p}}_{2}^{+} = 0.3519, {\tilde{p}}_{3}^{+} = 0.4500, \\ {\tilde{p}}_{4}^{+} & = & 0.5837, {\tilde{p}}_{1}^{-} = 0.3017, {\tilde{p}}_{2}^{-} = 0.4009, \\ {\tilde{p}}_{3}^{-} & = & 0.2874, {\tilde{p}}_{4}^{-} = 0.2067 . \end{matrix}$

Step 7*. Calculate the relative closeness. According to Equation (28), we have $\begin{matrix} {\tilde{C}}_{1} & = & 0.5883, {\tilde{C}}_{2} = 0.4675, {\tilde{C}}_{3} = 0.6102, \\ {\tilde{C}}_{4} & = & 0.7385 . \end{matrix}$

Step 8*. Rank the alternatives $A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} .$

The results are the same as those in [35] and [36], so the proposed method in this paper is effective.

We notice that in the process of calculating the relative closeness, there are two parameters: λ and θ. To get the impact of the two parameters on the relative closeness, let λ or θ be fixed, the trends of the relative closeness of four alternatives are shown in Figs. 1–4. Furthermore, Figs. 5–8 present the relative closeness for the four alternatives when the parameters λ and θ change simultaneously.

Fig.1

Relative closeness with respect to θ when λ = 2.

Fig.2

Relative closeness with respect to θ when λ = 5.

Fig.3

Relative closeness with respect to λ when θ = 0.2.

Fig.4

Relative closeness with respect to λ when θ = 0.5.

Fig.5

Relative closeness for A₁ (λ ∈ (0, 10] , θ ∈ (0, 1]).

Fig.6

Relative closeness for A₂ (λ ∈ (0, 10] , θ ∈ (0, 1]).

Fig.7

Relative closeness for A₃ (λ ∈ (0, 10] , θ ∈ (0, 1]).

Fig.8

Relative closeness for A₄ (λ ∈ (0, 10] , θ ∈ (0, 1]).

According to the above figures, we can observe that when λ is fixed, the value of θ has obvious effect on relative closeness. However, when θ is fixed, the value of λ has no obvious effect on relative closeness. Therefore, if the decision maker is risk-seeking, let θ be a smaller value, but if the decision maker is risk-averse, let θ be a larger value.

5.2 Advantages of the proposed method

Now we summarize the advantages of our proposed method compared with the other existed methods.

In contrast to the dual hesitant fuzzy multi-attribute decision making methods studied in [32, 33] and [29, 30], the method proposed in this paper does not take the magnitude of the angle cosine (correlation coefficient in [32, 33]) or distance measure in [29, 30] as the criterion of alternative ranking. The grey relational projection method combines the correlation coefficient and the distance, which can reflect the close degree between the alternative and the ideal solution.

Compared with the general projection method, the grey relational projection method integrates the grey relation analysis into the projection, which can integrate the influence of the whole attribute space and avoid the unidirectional deviation. That is, it can avoid the deviations caused by comparing the single attribute value (IVDHFE) of each alternative, so that can comprehensively reflect the influence of the whole attribute space.

The weight is determined based on the evaluation information itself, which can reduce the influence of the lack of information, and make it closer to the real intention of decision makers. Also, the determination of the parameters in this paper can reflect the risk preference of decision makers.

As we know, the interval-valued dual hesitant fuzzy set is a generalization of the dual hesitant fuzzy set, the hesitant fuzzy set and the intuitionistic fuzzy set. Therefore, the grey relational projection method proposed in this paper can be used to solve not only multi-attribute decision making with IVDHFSs but also the MADM with the dual hesitant fuzzy set, the hesitant fuzzy set and the intuitionistic fuzzy set.

6 Conclusion

In this paper, a novel grey relational projection method was proposed for MADM, in which the attribute values take the form of interval-dual hesitant fuzzy elements and attribute weights are unknown. Firstly, the distance measures of the interval-dual hesitant fuzzy elements were introduced. Then, the grey relational projection method based on the distance measure was proposed, and the nonlinear optimization model with the minimum total deviation of the grey relational coefficient was constructed to determine attribute weights. Furthermore, this method is also suitable for solving the MADM. Finally, the effectiveness and feasibility of the proposed method is illustrated by an example. The proposed method is clear and simple, and is easy to be operated, which provides a new idea to solve the fuzzy multi-attribute decision making problem. In the future, we will apply the proposed method to some practical problems, such as pattern recognition,boundary detection, image reconstruction. We will also be committed to the expansion of the algorithm for interval-valued dual hesitant fuzzy multi-attribute decision making.

Footnotes

Acknowledgments

The authors would like to thank the support from the National Natural Science Foundation of China (Nos. 71671159, 71301139), the Humanity and Social Science Foundation of Ministry of Education of China (No. 16YJC630106), the Natural Science Foundation of Hebei Province (No. G2016203236), the China Postdoctoral Science Foundation (No. 2016M600196), and the project Funded by Hebei Education Department (Nos. BJ2016063, BJ2017029).

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