An extended multiple criteria decision making method based on neutrosophic hesitant fuzzy information

Abstract

Neutrosophic hesitant fuzzy set (NHFS) is an extension of neutrosophic set and the hesitant fuzzy set, which can easily express the uncertain, incomplete and inconsistent information, and the VIKOR (from Serbian: VIseKriterijumska Optimizacija I Kompromisno Resenje) method is an effective decision making tool which can select the optimal alternative by the maximizing “group utility” and minimizing “individual regret” with cognitive computation. In this paper, we firstly introduced some operational laws, comparison rules and the Hamming distance measure of neutrosophic hesitant fuzzy set, and described the traditional VIKOR method which only processes the crisp numbers. Then we extended the VIKOR method to process the Neutrosophic hesitant fuzzy information (NHFI), and proposed an extended VIKOR method for the multiple criteria decision making (MCDM) problems with NHFI, and an illustrative example shows the effectiveness of the proposed approach.

Keywords

Multiple neutrosophic hesitant fuzzy set VIKOR method multiple criteria decision making (MCDM)

1 Introduction

Decision making has been widely used in the politics, economic, military, management and the other fields. But in real decision making, the decision information is often inconsistent, incomplete and indeterminate and it is difficult or impossible to obtain the criteria values by the exact numbers. How to express the kind of information is a very important research topic. Since the fuzzy set (FS) was proposed [1], fuzzy MCDM problems have been widely researched. Because FS was only described by membership, it is not easy to denote some complicated uncertain information. For example, about a voting, there is a committee composed of ten members to do a selection for an issue, suppose five persons give the “support”, three “opposition”, and the others give up. Obviously, FS cannot fully express the polling information. Aimed at this deficiency, Atanassov [2, 3] defined the intuitionistic fuzzy set (IFS) and it consists of truth-membership and falsity-membership. The above example can be expressed by IFS (0.5, 0.3), i.e., its membership is 0.5 and non-membership is 0.3. However, IFSs can’t handle the inconsistent and indeterminate information yet, and the indeterminacy degree in IFSs is by default. In some complicated decision situations, IFS still has some limitations. For instance, when an expert is called to make a judgment for a statement, he/she may give the possibility of right is 0.4 and the false possibility is 0.5 and the uncertain possibility is 0.3 [4]. Obviously, this is a typical cognitive activity. However, on this occasion, IFS doesn’t cope with this problem. To handle this kind of information, Smarandache [5] proposed the neutrosophic set (NS) based on IFS. In NS, the truth-membership, false-membership and indeterminacy-membership are totally independent. To simplify NS and apply it to practical problems, Wang et al. [6] defined a single valued neutrosophic set (SVNS) with some examples. Ye [7, 8] defined the cross-entropy and the correlation coefficient of SVNS which was applied to the MCDM problems with SVNS.

On the other hand, FS has only one membership which will limit some decision making problems. In order to extend FS with multiple memberships, Torra and Narukawa [9], Torra [10] put forward the hesitant fuzzy sets (HFSs) which use several possible values instead of the single membership degree. Since then, in order to meet the needs of practical decision problems, many extended researches have been done, such as interval valued hesitant fuzzy sets (IVHFSs) by expressing the memberships with interval numbers [11], hesitant triangular fuzzy sets (HTFSs) in which the memberships are triangular fuzzy numbers [12], linguistic hesitant fuzzy sets (LHFSs) in which the different linguistic terms have the different HFSs [13], dual hesitant fuzzy sets (DHFSs) and dual interval hesitant fuzzy sets (DIHFSs) which have two sets of membership and non-membership [14, 15], hesitant interval-valued intuitionistic fuzzy sets by expressing the memberships with interval-valued IFSs [16]. Further, some aggregation operators for HTFSs, LHFSs and HIVIFSs were developed [12 , 16].

As mentioned above, the NS and HFS are extended in two directions based on FS, the HFS assigns the membership function to a set of some possible numbers, which is a good method to deal with uncertain information; however, it cannot process indeterminate or inconsistent information, while the NS can character uncertainty, incomplete and inconsistent information. Clearly, they all have the advantages and disadvantages. Based on this situation, the neutrosophic hesitant fuzzy set (NHFS) was proposed by Ye [17] by combining the HFS with NS, and it can easily express all kinds of fuzzy information. In NHFS, the memberships of truth, indeterminacy and falsity of NS were extended to a set of some possible values in interval [0, 1]. At the same time, some operators were developed for HNFSs.

In addition, the VIKOR method is an important decision tool to process the fuzzy MCDM problems because it can achieve the maximum “group utility” and minimum of “individual regret” and can consider two kinds of particular measures of “closeness” to the virtual ideal solution and the virtual negative ideal solution, simultaneously. Comparing with the other decision making methods, such as TOPSIS, TODIM and so on, VIKOR can give one compromise optimal choice or a group of choices with no differences based on the maximum “group utility” and minimum of “individual regret” while the other methods can only provide an optimal choice. However, because the traditional VIKOR method can only deal with the crisp numbers, some new extensions of VIKOR for the different fuzzy information have been studied. Liu and Wang [18] extended VIKOR to generalized interval-valued trapezoidal fuzzy numbers. Wu et al. [19] extended VIKOR to linguistic information. Liao et al. [20] extended VIKOR to Hesitant fuzzy linguistic information. Ghorabaee et al. [21] extended VIKOR to interval type-2 fuzzy sets. Zhang and Wei [22] extended VIKOR to deal with HFS. Liu and Wu [23] extended VIKOR to process the multi-granularity linguistic variables and apply it to the competency evaluation of human resources managers. Zhang and Liu [24] extended VIKOR to process the hybrid information, including crisp numbers, interval numbers, triangular fuzzy numbers, trapezoid fuzzy number s, linguistic variables, and so on. Du and Liu [25] extended VIKOR to deal with intuitionistic trapezoidal fuzzy numbers. However, until now, the extended VIKOR cannot process the neutrosophic hesitant fuzzy information, so it is useful and necessary to extend the VIKOR to neutrosophic hesitant fuzzy information.

In order to achieve the above purposes, the organization structure is shown as follows. In the next section, we introduce the single valued neutrosophic set, HFSs, NHFSs, and the traditional VIKOR method. In Section 3, we extend the traditional VIKOR method to the neutrosophic hesitant fuzzy information, and a MCDM approach is proposed. In Section 4, we give a numerical example to elaborate the effectiveness and feasibility of our approach. In Section 5, we give the main concluding remarks.

2 Preliminaries

2.1 The single valued neutrosophic set

Definition 1. [5, 26] Let X be a universe of discourse, with a generic element in X denoted by x. A single valued NS A in X is characterized by: $A = {x = (T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X},$ (1) where the functions T_A (x), I_A (x) and F_A (x) denote the truth-membership, the indeterminacy-membership and the falsity-membership of the element x ∈ X to the set A respectively. For each point x in X, we have T_A (x) , I_A (x) , F_A (x) ∈ [0, 1], and 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3.

For convenience, we can use x = (T_x, I_x, F_x) to denote an element x in SVNS, and the element x is called a single valued neutrosophic number (SVNN).

To compare two SVNNs, Smarandache and Vlâdâreanu [27] proposed the partial order relations for two SVNNs as follows.

Definition 2. [27] For two SVNNs x = (T₁, I₁, F₁) and y = (T₂, I₂, F₂), iff T₁ ≤ T₂, I₁ ≥ I₂, F₁ ≥ F₂, then x ≤ y.

Obviously, in practical applications, many cases can’t satisfy the above conditions. With respect to these, Ye [28] presented a comparison method based on the cosine similarity measure of a SVNN x = (T, I, F) to ideal solution (1,0,0), and offered the definition of the cosine similarity: $S (x) = \frac{T}{\sqrt{T^{2} + I^{2} + F^{2}}} .$ (2)

Definition 3. [28] Suppose x = (T_a, I_a, F_a) and y = (T_b, I_b, F_b) are two SVNNs, if S (x) ≤ S (y), then x ≤ y.

Definition 4. Let x = (T_a, I_a, F_a) and y = (T_b, I_b, F_b) be two SVNNs, then the normalized Hamming distance between two SVNNs x and y can be defined as follows: $d (x, y) = \frac{1}{3} (| T_{a} - T_{b} | + | I_{a} - I_{b} | + | F_{a} - F_{b} |) .$ (3)

2.2 The HFS

Definition 5. [10] Let X be a non-empty fixed set, a HFS A on X can be denoted by as follows $A = {〈 x, h_{A} (x) 〉 | x \in X}$ (4) where h_A (x) represents the some possible membership degrees of the element x ∈ X to A, and each value in the h_A (x) is in [0, 1].

For convenience, we call h_A (x) a hesitant fuzzy element (HFE), denoted by h, which reads h = {γ|γ ∈ h}.

Recently, Rodríguez et al. [29] give some new explanations about HFS and HFE. Please refer to it.

For any three HFEs h = {γ|γ ∈ h}, h₁ = {γ₁|γ₁ ∈ h₁} and h₂ = {γ₂|γ₂ ∈ h₂}, Torra [10] gave the definitions for some operational laws of HFEs as follows: $(1) h^{c} = \underset{γ \in h}{\cup} {1 - γ},$ (5) $(2) h_{1} \cup h_{2} = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}}{\cup} max {γ_{1}, γ_{2}} .$ (6) $(3) h_{1} \cap h_{2} = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}}{\cup} min {γ_{1}, γ_{2}} .$ (7)

Further, Xia and Xu [30] defined four operational rules about the HFEs h = {γ|γ ∈ h},

h₁ = {γ₁|γ₁ ∈ h₁} and h₂ = {γ₂|γ₂ ∈ h₂} as follows. $(1) h^{n} = \underset{γ \in h}{\cup} {γ^{n}} .$ (8) $(2) nh = \underset{γ \in h}{\cup} {1 - (- γ)^{n}},$ (9) $(3) h_{1} \oplus h_{2} = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}}{\cup} {γ_{1} + γ_{2} - γ_{1} γ_{2}},$ (10) $(4) h_{1} \otimes h_{2} = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}}{\cup} {γ_{1} γ_{2}} .$ (11)

Definition 6. Let h₁ and h₂ be two HFSs, then the hesitant normalized Hamming distance between two HFSs h₁ and h₂ can be defined as: $∥ h_{1} - h_{2} ∥ = \frac{1}{l} \sum_{j = 1}^{l} | h_{1 σ (j)} - h_{2 σ (j)} |,$ (12) where l (h) is the number of the elements in the HFS h and l = max {l (h₁) , l (h₂)}, and h_1σ(j) and h_2σ(j) express the jth element in two HFSs h₁ and h₂, respectively. In general, because l (h₁) ≠ l (h₂), i.e., the lengths of two HFSs h₁ and h₂ are not equal. In order to do the operations in Equation (12), we need to extend the shorter one to the length of longer one. How to extend the shorter one, Xia and Xu [30] thought that the best way is to add the same value in the shorter one until the lengths of two HFSs h₁ and h₂ are equal. Further, then they thought the decision makers with the different risk preferences maybe select the different values, optimists can select the maximum value, while pessimists may select the minimum value.

For example, h₁ = {0.3, 0.4, 0.5}, h₂ = {0.1, 0.6}. Because l (h₁) > l (h₂), we need to extend the HFS h₂ to the length of the HFS h₁. If the decision maker is an optimist, then the HFS h₂ will be extended to h₂ = {0.1, 0.6, 0.6}, and if the decision maker is an pessimist, then the HFS h₂ will be extended to h₂ = {0.1, 0.1, 0.6}. Of course, because the different values were adopted, and the distances between two HFSs h₁ and h₂ are different, however, Xia and Xu [30] thought this is reasonable because the decision makers with different risk preferences may give the different choices and make the different final decision. In this study, we only adopted the pessimistic selection (the other situations are similar to this study).

2.3 The NHFS

In this section, we will introduce the NHFS by combining NS with HFS.

Definition 7. [17] Let X be a non-empty fixed set, a NHFS on X is expressed by: $N = {〈 x, \tilde{t} (x), \tilde{i} (x), \tilde{f} (x) 〉 | x \in X},$ (13) where $\tilde{t} (x) = {\tilde{γ} | \tilde{γ} \in \tilde{t} (x)}$ , $\tilde{i} (x) = {\tilde{δ} | \tilde{δ} \in \tilde{i} (x)}$ , and $\tilde{f} (x) = {\tilde{η} | \tilde{η} \in \tilde{f} (x)}$ represents the possible truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees of the element x ∈ X to the set N, and they themselves are HFSs in interval [0, 1], and satisfies the conditions: $\tilde{γ} \in [0, 1], \tilde{δ} \in [0, 1], \tilde{η} \in [0, 1]$ and $0 \leq sup {\tilde{γ}}^{+} + sup {\tilde{δ}}^{+} + sup {\tilde{η}}^{+} \leq 3$ , where ${\tilde{γ}}^{+} = \cup_{\tilde{γ} \in \tilde{t} (x)} max {\tilde{γ}}$ , ${\tilde{δ}}^{+} = \cup_{\tilde{δ} \in \tilde{i} (x)} max {\tilde{δ}}$ , and ${\tilde{η}}^{+} = \cup_{\tilde{η} \in \tilde{f} (x)} max {\tilde{η}}$ for x ∈ X.

The $\tilde{n} = {\tilde{t} (x), \tilde{i} (x), \tilde{f} (x)}$ is called a neutrosophic hesitant fuzzy element (NHFE) and it is denoted by the symbol $\tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}}$ .

Then, some basic operations of NHFEs are defined as follows:

Definition 8. [17] Let ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ be two NHFEs in a non-empty fixed set X, then $(1) {\tilde{n}}_{1} \cup {\tilde{n}}_{2} = {{\tilde{t}}_{1} \cup {\tilde{t}}_{2}, {\tilde{i}}_{1} \cap {\tilde{i}}_{2}, {\tilde{f}}_{1} \cap {\tilde{f}}_{2}}$ (14) $(2) {\tilde{n}}_{1} \cap {\tilde{n}}_{2} = {{\tilde{t}}_{1} \cap {\tilde{t}}_{2}, {\tilde{i}}_{1} \cup {\tilde{i}}_{2}, {\tilde{f}}_{1} \cap {\tilde{f}}_{2}}$ (15)

Therefore, for two NHFEs ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ , ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ and a positive scale k > 0, the operations can be defined as follows: $\begin{matrix} (1) {\tilde{n}}_{1} \oplus {\tilde{n}}_{2} = {{\tilde{t}}_{1} \oplus {\tilde{t}}_{2}, {\tilde{i}}_{1} \otimes {\tilde{i}}_{2}, {\tilde{f}}_{1} \otimes {\tilde{f}}_{2}} = \\ \underset{\begin{matrix} {\tilde{γ}}_{2} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}, \\ {\tilde{γ}}_{2} \in {\tilde{t}}_{2}, {\tilde{δ}}_{2} \in {\tilde{i}}_{2}, {\tilde{η}}_{2} \in {\tilde{f}}_{2} \end{matrix}}{\cup} {γ_{1} + γ_{2} - γ_{1} γ_{2}, δ_{1} δ_{2}, η_{1} η_{2}} \end{matrix}$ (16)

$\begin{matrix} (2) {\tilde{n}}_{1} \otimes {\tilde{n}}_{2} = {{\tilde{t}}_{1} \otimes {\tilde{t}}_{2}, {\tilde{i}}_{1} \oplus {\tilde{i}}_{2}, {\tilde{f}}_{1} \oplus {\tilde{f}}_{2}} = \\ \underset{\begin{matrix} {\tilde{γ}}_{2} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}, \\ {\tilde{γ}}_{2} \in {\tilde{t}}_{2}, {\tilde{δ}}_{2} \in {\tilde{i}}_{2}, {\tilde{η}}_{2} \in {\tilde{f}}_{2} \end{matrix}}{\cup} {γ_{1} γ_{2}, δ_{1} + δ_{2} - δ_{1} δ_{2}, η_{1} + η_{2} - η_{1} η_{2}} \end{matrix}$ (17) $\begin{matrix} (3) k {\tilde{n}}_{1} = \underset{{\tilde{γ}}_{2} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}}{\cup} {1 - (1 - γ_{1})^{k}, δ_{1}^{k}, η_{1}^{k}} \end{matrix}$ (18) $\begin{matrix} (4) {\tilde{n}}_{1}^{k} = \underset{{\tilde{γ}}_{2} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}}{\cup} \\ {γ_{1}^{k}, 1 - (1 - δ_{1})^{k}, 1 - (1 - η_{1})^{k}} \end{matrix}$ (19)

Example 1. Let ${\tilde{n}}_{1} = 〈 {0.6}, {0.1, 0.2}, {0.2} 〉$ and ${\tilde{n}}_{2} = 〈 {0.5}, {0.3}, {0.2, 0.3} 〉$ be two NHFEs, and k = 2, then $\begin{matrix} (1) {\tilde{n}}_{1} \oplus {\tilde{n}}_{2} = 〈 {0.80}, {0.03, 0.06}, {0.04, 0.06} 〉 \\ (2) {\tilde{n}}_{1} \otimes {\tilde{n}}_{2} = 〈 {0.30}, {0.37, 0.44}, {0.36, 0.44} 〉 \\ (3) 2 . {\tilde{n}}_{1} = 〈 {0.84}, {0.01, 0.04}, {0.04} 〉 \\ (4) {\tilde{n}}_{1}^{2} = 〈 {0.36}, {0.19, 0.36}, {0.36} 〉 \end{matrix}$

Theorem 1. [17] Let ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ be two NHFEs in a non-empty fixed set X, and η, η₁, η₂ > 0, then we have $(1) {\tilde{n}}_{1} \oplus {\tilde{n}}_{2} = {\tilde{n}}_{2} \oplus {\tilde{n}}_{1};$ (20) $(2) {\tilde{n}}_{1} \otimes {\tilde{n}}_{2} = {\tilde{n}}_{2} \otimes {\tilde{n}}_{1};$ (21) $(3) η ({\tilde{n}}_{1} \oplus {\tilde{n}}_{2}) = η {\tilde{n}}_{1} \oplus η {\tilde{n}}_{2};$ (22) $(4) η_{1} {\tilde{n}}_{1} \oplus η_{2} {\tilde{n}}_{1} = (η_{1} + η_{2}) {\tilde{n}}_{1};$ (23) $(5) {\tilde{n}}_{1}^{η} \otimes {\tilde{n}}_{2}^{η} = ({\tilde{n}}_{2} \otimes {\tilde{n}}_{1})^{η};$ (24) $(6) {\tilde{n}}_{1}^{η_{1}} \otimes {\tilde{n}}_{1}^{η_{2}} = {\tilde{n}}_{1}^{η_{1} + η_{2}};$ (25)

Definition 9. [17] For an NHFE $\tilde{n}$ ,

$\begin{matrix} s (\tilde{n}) \\ = [\frac{1}{l} \sum_{i = 1}^{l} {\tilde{γ}}_{i} + \frac{1}{p} \sum_{i = 1}^{p} (1 - {\tilde{δ}}_{i}) + \frac{1}{q} \sum_{i = 1}^{q} (1 - {\tilde{η}}_{i})] / 3 \end{matrix}$ (26) is called the score function of the NHFE $\tilde{n}$ , where l, m, n are the numbers of the elements of the NHFEs $\tilde{γ}, \tilde{δ}, \tilde{η}$ , respectively. Obviously, $s (\tilde{n}) \in [0, 1]$ i.

Suppose ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ are any two NHFEs, the comparison method of NHFEs is expressed as follows [17]:

If $S ({\tilde{n}}_{1}) > S ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} > {\tilde{n}}_{2}$ ;

If $S ({\tilde{n}}_{1}) < S ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} < {\tilde{n}}_{2}$ ;

If $S ({\tilde{n}}_{1}) = S ({\tilde{n}}_{2})$ , then ${\tilde{n}}_{1} = {\tilde{n}}_{2}$ .

Example 2. Let ${\tilde{n}}_{1} = 〈 {0.6}, {0.1, 0.2}, {0.2} 〉$ and ${\tilde{n}}_{2} = 〈 {0.5}, {0.3}, {0.2, 0.3} 〉$ be two NHFEs, then $\begin{matrix} s ({\tilde{n}}_{1}) & = & [0.6 + \frac{1}{2} [(1 - 0.1) \\ + (1 - 0.2)] + (1 - 0.2)] / 3 = 0.75 \\ s ({\tilde{n}}_{2}) & = & [0.5 + [(1 - 0.3) \\ + \frac{1}{2} (1 - 0.2)] + (1 - 0.3)] / 3 = 0.65 \end{matrix}$

Because $S ({\tilde{n}}_{1}) > S ({\tilde{n}}_{2})$ , we can get ${\tilde{n}}_{1} > {\tilde{n}}_{2}$ .

Definition 10. Let ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ are any two NHFEs, then the normalized Hamming distance between ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ is defined as follows:

$\begin{matrix} d ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = | | {\tilde{n}}_{1} - {\tilde{n}}_{2} | | \\ = \frac{1}{3} (| {\tilde{γ}}_{1} - {\tilde{γ}}_{2} | + | {\tilde{δ}}_{1} - {\tilde{δ}}_{2} | + | {\tilde{η}}_{1} - {\tilde{η}}_{2} |) \\ = \frac{1}{3} (| \frac{1}{l} \sum_{j = 1}^{l} | {\tilde{γ}}_{1 σ (j)} - {\tilde{γ}}_{2 σ (j)} | | \\ + | \frac{1}{l} \sum_{j = 1}^{l} | {\tilde{δ}}_{1 σ (j)} - {\tilde{δ}}_{2 σ (j)} | | \\ + | \frac{1}{l} \sum_{j = 1}^{l} | {\tilde{η}}_{1 σ (j)} - {\tilde{η}}_{2 σ (j)} | |) \end{matrix}$ (27)

Example 3. Let ${\tilde{n}}_{1} = 〈 {0.6}, {0.1, 0.2}, {0.2} 〉$ and ${\tilde{n}}_{2} = 〈 {0.5}, {0.3}, {0.2, 0.3} 〉$ be two NHFEs, then $\begin{matrix} d ({\tilde{n}}_{1}, {\tilde{n}}_{2}) = | | {\tilde{n}}_{1} - {\tilde{n}}_{2} | | \\ = \frac{1}{3} (| | 0.6 - 0.5 | | + | \frac{1}{2} (| 0.1 - 0.3 | + | 0.2 - 0.3 |) | \\ | \frac{1}{2} (| 0.2 - 0.2 | + | 0.2 - 0.3 |) |) = 0.1 \end{matrix}$

2.4 VIKOR method

The VIKOR method, introduced for multi-criteria optimization problem, focuses on ranking the alternatives and selecting a compromise solution [31 –33]. The decision making problem, which can be solved by VIKOR, is expressed as follows.

Suppose there are m alternatives which are denoted as X₁, X₂, …, X_m, and there are n criteria which are denoted as A₁, A₂, …, A_m, the evaluation value of alternative X_i with respect to criterion A_j is expressed by x_ij. Suppose the $x_{j}^{*}$ and $x_{j}^{-}$ express the virtual positive ideal value and virtual negative ideal value under the criterion A_j, w = (w₁, w₂, … w_n) ^T is the criterion weight vector satisfying $w_{j} \in [0, 1], \sum_{j = 1}^{n} w_{j} = 1$ . The compromise ranking by VIKOR method is started with the form of L_p-metric [33].

$\begin{matrix} L_{pi} & = & {\sum_{j = 1}^{n} {[\frac{(f_{j}^{*} - f_{ij})}{(f_{j}^{*} - f_{j}^{-})}]}^{p}}^{1 / - p} \\ 1 & \leq & p \leq \infty; i = 1, 2, 3, \dots, m . \end{matrix}$ (28)

In the VIKOR method, the maximum group utility can be gotten by min S_i and minimum individual regret can be gotten by min R_i, where S_i = L_1,i, and R_i = L_∞,i.

The steps of the VIKOR method can be described as following:

Step 1: Computing the virtual positive ideal $x_{j}^{*}$ and the virtual negative ideal $x_{j}^{-}$ values under the criterion A_j, we have $x_{j}^{*} = max_{i} x_{ij}, x_{j}^{-} = min_{i} x_{ij}$ (29)

Step 2: Computing the group utility value S_i and individual regret value R_i; i = 1, 2, …, m, as follows. $S_{i} = \sum_{j = 1}^{n} w_{j} (x_{j}^{*} - x_{ij}) / (x_{j}^{*} - x_{j}^{-}),$ (30) $R_{i} = max_{j} w_{j} (x_{j}^{*} - x_{ij}) / (x_{j}^{*} - x_{j}^{-})$ (31)

Step 3: Computing the values Q_i: i = 1, 2, …, m, according to the following formulas:

$Q_{i} = \frac{v (S_{i} - S^{*})}{(S^{-} - S^{*})} + \frac{(1 - v) (R_{i} - R^{*})}{(R^{-} - R^{*})}$ (32) where $S^{*} = min_{i} S_{i},$ $S^{-} = max_{i} S_{i}$ , $R^{*} = min_{i} R_{i},$ $R^{-} = max_{i} R_{i}$ , v is the balance parameter which can balance the group utility and individual regret. When v > 0.5, considering “the maximum group utility” is more than “minimum individual regret”, and when v < 0.5, considering “minimum individual regret” is more than “the maximum group utility”. In this study, we suppose that v = 0.5, the “minimum individual regret” and “the maximum group utility” are the same important.

Step 4: Ranking the alternatives. According to the values S, R and Q, we will get three ranking results, and then we can obtain the compromise solution.

Step 5: Obtaining a compromise solution X⁽¹⁾, which is in the first position of all ranking alternatives produced by the value Q (i.e., the alternative is with minimum value Q) if it meets the following two conditions:

Condition 1: Acceptable advantage: $Q (X^{(2)}) - Q (X^{(1)}) \geq \frac{1}{m - 1}$ , where Q (X⁽²⁾) is the Q value in the second position of all ranking alternatives produced by the value Q, and m is the number of alternatives.

Condition 2: Acceptable stability. Alternative X⁽¹⁾ must also be in the first position of all ranking alternatives produced by the value by S or/and R.

If one of above two conditions is not met, then we will get a collection of compromise alternatives and not one compromise solution.

If condition 2 is not met, then we can get that alternatives X⁽¹⁾ and X⁽²⁾ should be compromise solutions.

If condition 1 is not met, then the maximum M can be got by the formula Q (X^(M)) - Q (X⁽¹⁾) < MQ, $MQ = \frac{1}{m - 1}$ , and we can get the alternatives X⁽¹⁾, X⁽²⁾, …, X^(M) are compromise solutions.

Based on the above analysis, we know that the best solution is the one with the minimum Q value when the conditions 1 and 2 are met, and when one of two conditions is not met, the compromise solutions may be have more than one.

The VIKOR method is an effective tool for solving the MCDM problems, and it can get one compromise solution or a collection of compromise solutions according to some conditions based on the maximum “group utility” and minimum “individual regret”.

3 VIKOR method for decision making problem with neutrosophic hesitant fuzzy numbers

In real decision making, because decision problems are complex, sometimes, it is difficult or impossible to obtain the criteria values by the exact numbers, however, the NHFS is a very useful tool to process the uncertain MCDM problems in which each criteria can be expressed by the neutrosophic hesitant fuzzy numbers [17]. The VIKOR is very effective method to solve the decision making problems, however, the traditional VIKOR is only suitable for crisp numbers, and then it has been extended to deal with the different fuzzy information [22 –25]. Until now, it has not been used to process the neutrosophic hesitant fuzzy information. So, in this paper, we will extend the VIKOR method to solve MADM problem with the neutrosophic hesitant fuzzy information (NHFI).

To do this, we firstly describe the decision making problem.

For a multiple criteria decision making problem, let X = {X₁, X₂, …, X_m} be a collection of m alternatives, A = {A₁, A₂, …, A_m} be a collection of n criteria, which weight vector is w = (w₁, w₂, … w_n) ^T satisfying $w_{j} \in [0, 1], \sum_{j = 1}^{n} w_{j} = 1$ . Suppose that ${\tilde{n}}_{ij} = ({\tilde{t}}_{ij}, {\tilde{i}}_{ij}, {\tilde{f}}_{ij})$ is the evaluation value of the alternative X_i with respect to the criteria A_j which is expressed by the NHFI, where ${\tilde{t}}_{ij} = {{\tilde{γ}}_{ij} | {\tilde{γ}}_{ij} \in {\tilde{t}}_{ij}}$ , ${\tilde{i}}_{ij} = {{\tilde{δ}}_{ij} | {\tilde{δ}}_{ij} \in {\tilde{i}}_{ij}}$ and ${\tilde{f}}_{ij} = {{\tilde{η}}_{ij} | {\tilde{η}}_{ij} \in {\tilde{f}}_{ij}}$ are three collections in which each is hesitant fuzzy numbers (HFNs) in interval [0, 1], which represent the possible truth-membership, indeterminacy-membership, and falsity-membership of NS, and satisfy following limits: $\tilde{γ} \in [0, 1], \tilde{δ} \in [0, 1], \tilde{η} \in [0, 1]$ , and $0 \leq sup {\tilde{γ}}^{+} + sup {\tilde{δ}}^{+} + sup {\tilde{η}}^{+} \leq 3$ , where ${\tilde{γ}}^{+} = \underset{{\tilde{γ}}_{ij} \in {\tilde{t}}_{ij}}{\cup} max {r_{ij}}$ , ${\tilde{δ}}^{+} = \underset{{\tilde{δ}}_{ij} \in {\tilde{i}}_{jj}}{\cup} max {δ_{ij}}$ and ${\tilde{η}}^{+} = \underset{{\tilde{η}}_{ij} \in {\tilde{f}}_{ij}}{\cup} max {η_{ij}}$ . The decision matrix denoted by the neutrosophic hesitant fuzzy numbers are listed in Table 1, and the goal of this MCDM problem is to rank the alternatives.

Table 1
Decision making matrix with the neutrosophic hesitant fuzzy information

A ₁ A ₂ ⋯ A _n

X ₁ ${\tilde{n}}_{11}$ ${\tilde{n}}_{12}$ ⋯ ${\tilde{n}}_{1 n}$

X ₂ ${\tilde{n}}_{21}$ ${\tilde{n}}_{22}$ ⋯ ${\tilde{n}}_{2 n}$

⋯ ⋯ ⋯ ⋯ ⋯

X _m ${\tilde{n}}_{m 1}$ ${\tilde{n}}_{m 2}$ ⋯ ${\tilde{n}}_{mn}$

	A ₁	A ₂	⋯	A _n
X ₁	${\tilde{n}}_{11}$	${\tilde{n}}_{12}$	⋯	${\tilde{n}}_{1 n}$
X ₂	${\tilde{n}}_{21}$	${\tilde{n}}_{22}$	⋯	${\tilde{n}}_{2 n}$
⋯	⋯	⋯	⋯	⋯
X _m	${\tilde{n}}_{m 1}$	${\tilde{n}}_{m 2}$	⋯	${\tilde{n}}_{mn}$

In this study, we think the weight information is known. If the weight information is unknown, we can use the AHP method or some objective weight determination methods to obtain them.

The procedures of the proposed method as follows:

Step 1: Normalize the decision matrix.

In MAGDM problems, there are two types in criteria, that is, benefit criteria and cost criteria. To maintain consistency of the criteria, we usually transform the cost criteria into benefit criteria.

For the cost criteria, the normalization can be done by the following formula

$\begin{matrix} ({\tilde{n}}_{ij}^{k}) \\ = {\cup_{\tilde{γ} \in {\tilde{t}}_{ij}} {1 - \tilde{γ}}, \cup_{\tilde{δ} \in {\tilde{i}}_{ij}} {1 - \tilde{δ}}, \cup_{\tilde{η} \in {\tilde{f}}_{ij}} {1 - \tilde{η}}} \end{matrix}$ (33)

Step 2: Obtain the virtual positive ideal solution and the virtual negative ideal solution. There are two methods to determine them.

(1) According to the partial order relation, we have the virtual positive ideal solution (PIS): $X^{*} = {{\tilde{n}}_{1}^{*}, \dots, {\tilde{n}}_{n}^{*}}$ (34) where

$\begin{matrix} {\tilde{n}}_{j}^{*} & = & {\max_{i} \underset{\tilde{γ} \in {\tilde{t}}_{ij}}{\cup} {\tilde{γ}}, \min_{i} \underset{\tilde{δ} \in {\tilde{i}}_{ij}}{\cup} {\tilde{δ}}, \min_{i} \underset{\tilde{η} \in {\tilde{f}}_{ij}}{\cup} {\tilde{η}}}, \\ j & = & 1, 2, \dots, n \end{matrix}$ (35) the virtual negative ideal solution (NIS) $X^{-} = {{\tilde{n}}_{1}^{-}, \dots, {\tilde{n}}_{n}^{-}}$ (36) where

$\begin{matrix} {\tilde{n}}_{j}^{-} & = & {\min_{i} \underset{\tilde{γ} \in {\tilde{t}}_{ij}}{\cup} {\tilde{γ}}, \max_{i} \underset{\tilde{δ} \in {\tilde{i}}_{ij}}{\cup} {\tilde{δ}}, \max_{i} \underset{\tilde{η} \in {\tilde{f}}_{ij}}{\cup} {\tilde{η}}}, \\ j & = & 1, 2, \dots, n \end{matrix}$ (37)

(2) According to score function, we have $X^{*} = {{\tilde{n}}_{1}^{*}, \dots, {\tilde{n}}_{n}^{*}},$ where ${\tilde{n}}_{j}^{*} = max {S ({\tilde{n}}_{1 j}), \dots, S ({\tilde{n}}_{mj})}, j = 1, 2, \dots, n$ (38) $X^{-} = {{\tilde{n}}_{1}^{-}, \dots, {\tilde{n}}_{n}^{-}},$ where ${\tilde{n}}_{j}^{-} = min {S ({\tilde{n}}_{1 j}), \dots, S ({\tilde{n}}_{mj})}, j = 1, 2, \dots, n$ (39)

Where S (.) is the score function of neutrosophic hesitant fuzzy number which is defined by Equations (26).

Step 3: Compute S_i and R_i, and we have $S_{i} = \frac{\sum_{j = 1}^{n} w_{j} | | {\tilde{n}}_{j}^{*} - {\tilde{n}}_{ij} | |}{| | {\tilde{n}}_{j}^{*} - {\tilde{n}}_{j}^{-} | |}, i = 1, 2, \dots, m,$ (40) $R_{i} = \frac{max_{j} w_{j} | | {\tilde{n}}_{j}^{*} - {\tilde{n}}_{ij} | |}{| | {\tilde{n}}_{j}^{*} - {\tilde{n}}_{j}^{-} | |}, i = 1, 2, \dots, m$ (41)

Where $∥ {\tilde{n}}_{1} - {\tilde{n}}_{2} ∥$ is the distance between two neutrosophic hesitant fuzzy numbers ${\tilde{n}}_{1}$ and ${\tilde{n}}_{2}$ , which is defined by Equation (27).

Step 4: Compute the values Q_i, and we have $Q_{i} = \frac{v (S_{i} - S^{*})}{(S^{-} - S^{*})} + \frac{(1 - v) (R_{i} - R^{*})}{(R^{-} - R^{*})}$ (42) where $S^{*} = min_{i} S_{i},$ $S^{-} = max_{i} S_{i}$ , $R^{*} = min_{i} R_{i},$ $R^{-} = max_{i} R_{i}$ , v is the balance parameter which can balance the group utility and individual regret, here suppose that v = 0.5, it shows that the “minimum individual regret” and “the maximum group utility” are the same important.

Step 5: Same as the step 4 of Section 2.

Step 6: Same as the step 5 of Section 2.

4 A numerical example

We cited an example from [34] which is a invest selection problem. One investment company intends to invest for one enterprise from the following four candidate enterprises, and they are marked by X_i (i = 1, 2, 3.4), and are measured by three criteria: (1) A₁ (the enterprise’s anti-risk ability); (2) A₂ (the enterprise’s growth ability); (3) A₃ (enterprise’s environmental impact) (suppose it is cost type), and the evaluation values are denoted by NHFNs and their weight is w = (0.35, 0.25, 0.4) ^T. The decision matrix R is listed in the Table 2. Then give the ranking the alternatives.

Table 2
The neutrosophic hesitant fuzzy decision matrix

A ₁ A ₂ A ₃

X ₁ {{0.4, 0.5}, {0.2}, {0.3}} {{0.4 },{0.2,0.3},{0.3}} {{0.2},{0.2},{0.5}}

X ₂ {{0.6},{0.1,0.2},{0.2}} {{0.6},{0.1},{0.2}} {{0.5},{0.2},{0.1,0.2}}

X ₃ {{0.3,0.4},{0.2},{0.3}} {{0.5},{0.2},{0.3}} {{0.5},{0.2,0.3},{0.2}}

X ₄ {{0.7},{0.1,0.2},{0.1}} {{0.6},{0.1},{0.2}} {{0.6},{0.3},{0.2}}

	A ₁	A ₂	A ₃
X ₁	{{0.4, 0.5}, {0.2}, {0.3}}	{{0.4 },{0.2,0.3},{0.3}}	{{0.2},{0.2},{0.5}}
X ₂	{{0.6},{0.1,0.2},{0.2}}	{{0.6},{0.1},{0.2}}	{{0.5},{0.2},{0.1,0.2}}
X ₃	{{0.3,0.4},{0.2},{0.3}}	{{0.5},{0.2},{0.3}}	{{0.5},{0.2,0.3},{0.2}}
X ₄	{{0.7},{0.1,0.2},{0.1}}	{{0.6},{0.1},{0.2}}	{{0.6},{0.3},{0.2}}

Obviously, this example is a MADM problem, and its attribute values take the form of NHFNs. In order to obtain the best enterprise to invest, we can use the proposed method (i.e., the extended VIKOR method) to rank the all candidate enterprises, and then select the best one. Based on the extended VIKOR method proposed in section 3, we can give the decision making steps shown as follows.

4.1 The evaluation steps from the proposed method

Step 1: Normalize the decision matrix.

Considering all the criteria should be uniform types, the cost type ${\tilde{n}}_{1}$ should be transformed into benefit type, and then we obtain the normalized NHFNS decision matrix ${\tilde{R}}^{k} = (β_{ij}^{k})_{4 \times 3}$ by Equation (33) shown in Table 3.

Table 3
The normalized neutrosophic hesitant fuzzy decision matrix

A ₁ A ₂ A ₃

X ₁ {{0.4, 0.5}, {0.2}, {0.3}} {{0.4 },{0.2,0.3},{0.3}} {{0.8},{0.8},{0.5}}

X ₂ {{0.6},{0.1,0.2},{0.2}} {{0.6},{0.1},{0.2}} {{0.5},{0.8},{0.9,0.8}}

X ₃ {{0.3,0.4},{0.2},{0.3}} {{0.5},{0.2},{0.3}} {{0.5},{0.8,0.7},{0.8}}

X ₄ {{0.7},{0.1,0.2},{0.1}} {{0.6},{0.1},{0.2}} {{0.6},{0.7},{0.8}}

	A ₁	A ₂	A ₃
X ₁	{{0.4, 0.5}, {0.2}, {0.3}}	{{0.4 },{0.2,0.3},{0.3}}	{{0.8},{0.8},{0.5}}
X ₂	{{0.6},{0.1,0.2},{0.2}}	{{0.6},{0.1},{0.2}}	{{0.5},{0.8},{0.9,0.8}}
X ₃	{{0.3,0.4},{0.2},{0.3}}	{{0.5},{0.2},{0.3}}	{{0.5},{0.8,0.7},{0.8}}
X ₄	{{0.7},{0.1,0.2},{0.1}}	{{0.6},{0.1},{0.2}}	{{0.6},{0.7},{0.8}}

Step 2: Obtain the virtual positive ideal solution and the virtual negative ideal solution by Equations (34–37), we can get $\begin{matrix} X^{*} & = & {{\tilde{n}}_{1}^{*}, {\tilde{n}}_{2}^{*}, {\tilde{n}}_{3}^{*}} = {{{0.7}, {0.1}, {0.1}}, \\ {{0.6}, {0.1}, {0.2}}, {{0.8}, {0.7}, {0.5}}} \end{matrix}$ $\begin{matrix} X^{-} & = & {{\tilde{n}}_{1}^{-}, {\tilde{n}}_{2}^{-}, {\tilde{n}}_{3}^{-}} = {{{0.3}, {0.2}, {0.3}}, \\ {{0.4}, {0.3}, {0.3}}, {{0.5}, {0.8}, {0.9}}} \end{matrix}$

Step 3: Compute S_i and R_i by Equations (40) and (41), and we have $\begin{matrix} S_{1} & = & \frac{w_{1} ∥ n_{1}^{*} - n_{11} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥} + \frac{w_{2} ∥ n_{2}^{*} - n_{12} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥} \\ + \frac{w_{3} ∥ n_{3}^{*} - n_{13} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥} = 0.450 \\ S_{2} & = & \frac{w_{1} ∥ n_{1}^{*} - n_{21} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥} + \frac{w_{2} ∥ n_{2}^{*} - n_{22} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥} \\ + \frac{w_{3} ∥ n_{3}^{*} - n_{23} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥} = 0.587 \\ S_{3} & = & \frac{w_{1} ∥ n_{1}^{*} - n_{31} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥} + \frac{w_{2} ∥ n_{2}^{*} - n_{32} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥} \\ + \frac{w_{3} ∥ n_{3}^{*} - n_{33} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥} = 0.783 \\ S_{4} & = & \frac{w_{1} ∥ n_{1}^{*} - n_{41} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥} + \frac{w_{2} ∥ n_{2}^{*} - n_{42} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥} \\ + \frac{w_{3} ∥ n_{3}^{*} - n_{43} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥} = 0.184 \\ R_{1} & = & max_{3} {\frac{w_{1} ∥ n_{1}^{*} - n_{11} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥}, \frac{w_{2} ∥ n_{2}^{*} - n_{12} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥}, \\ \frac{w_{3} ∥ n_{3}^{*} - n_{13} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥}} = 0.225 \\ R_{2} & = & max_{3} {\frac{w_{1} ∥ n_{1}^{*} - n_{21} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥}, \frac{w_{2} ∥ n_{2}^{*} - n_{22} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥}, \\ \frac{w_{3} ∥ n_{3}^{*} - n_{23} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥}} = 0.516 \\ R_{3} & = & max_{3} {\frac{w_{1} ∥ n_{1}^{*} - n_{31} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥}, \frac{w_{2} ∥ n_{2}^{*} - n_{32} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥}, \\ \frac{w_{3} ∥ n_{3}^{*} - n_{33} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥}} = 0.447 \\ R_{4} & = & max_{3} {\frac{w_{1} ∥ n_{1}^{*} - n_{41} ∥}{∥ n_{1}^{*} - n_{1}^{-} ∥}, \frac{w_{2} ∥ n_{2}^{*} - n_{42} ∥}{∥ n_{2}^{*} - n_{2}^{-} ∥}, \\ \frac{w_{3} ∥ n_{3}^{*} - n_{43} ∥}{∥ n_{3}^{*} - n_{3}^{-} ∥}} = 0.167 \end{matrix}$

Step 4: Compute the values Q_i (i = 1, 2, 3, 4) by Equation (42) (suppose v = 0.5), we have $Q_{1} = 0.305, Q_{2} = 0.836, Q_{3} = 0.901, Q_{4} = 0$

Step 5: Rank the alternatives.

We firstly gave the ranking results by the values S, R and Q, the smaller the values S, R and Q are, the better the alternatives are. The results are listed in Table 4.

Table 4

The ranking and the compromise solutions

	X ₁	X ₂	X ₃	X ₄	Ranking	Compromise solutions
S	0.450	0.587	0.783	0.184	A₄ ≻ A₁ ≻ A₂ ≻ A₃	X ₄
R	0.225	0.516	0.447	0.167	A₄ ≻ A₁ ≻ A₃ ≻ A₂	X ₄
Q (v = 0.5)	0.305	0.836	0.901	0	A₄ ≻ A₁ ≻ A₂ ≻ A₃	X ₄
Compromise						X₄, X₁
solutions

Step 6: Obtain the compromise ranking results. Firstly, we rank the alternatives by Q in decreasing order, the alternative with first position is X₄ with Q (X₄) =0, and X₁ is the alternative with second position with Q (X₁) =0.305. As MQ = 1/(m-1) = 1/(4-1) = 0.333, so $Q (X_{1}) - Q (X_{4}) = 0.305 < 0.333$

Which is not satisfied $Q (X_{1}) - Q (X_{4}) \geq \frac{1}{4 - 1}$ , but alternative X₄ is the best ranked by S and R, which satisfies the condition 2. By computing, we get: $\begin{matrix} Q (X_{3}) - Q (X_{4}) = 0.901 > 0.333 \\ Q (X_{2}) - Q (X_{4}) = 0.836 > 0.333 \\ Q (X_{1}) - Q (X_{4}) = 0.305 < 0.333 \end{matrix}$ so X₄, X₁ are both compromise solutions.

4.2 Comparison analysis

In order to verify the feasibility and effectiveness of the proposed approach, a comparison analysis with multi-valued neutrosophic TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) method introduced by Wang and Li [34] is given for this illustrative example.

With regard to the method in Wang and Li [34], the multi-valued neutrosophic number is defined, and the traditional TODIM method is extended to the neutrosophic environment. In this new method, a reference criterion is selected first and then built the value function based on the Hamming distance between multi-valued neutrosophic numbers. Its decision-making steps are shown as below:

Step 1: Select the highest weight criterion as the reference criterion, so select A₃ as the reference criterion.

Step 2: Calculate the degree of alternative X_i superior to alternative X_j, and the results are shown in the Table 5.

Table 5
The degree of priority among alternatives [34]

X ₁ X ₂ X ₃ X ₄

X ₁ 0 –0.893 0.236 –1.358

X ₂ 0.539 0 0.272 –0.765

X ₃ 0.927 –1.406 0 –1.955

X ₄ –0.41 0.248 0.498 0

	X ₁	X ₂	X ₃	X ₄
X ₁	0	–0.893	0.236	–1.358
X ₂	0.539	0	0.272	–0.765
X ₃	0.927	–1.406	0	–1.955
X ₄	–0.41	0.248	0.498	0

Step 3: Calculate the comprehensive ranking value of X_i, and get ξ₁ = 0.492, ξ₂ = 0.702, ξ₃ = 0, ξ₄ = 1.

Step 4: Rank all the alternatives X_i (i = 1, 2, 3, 4) based on the values ξ_i. The bigger ξ_i, the better alternative X_i (i = 1, 2, 3, 4), we can get X₄ ≻ X₂ ≻ X₁ ≻ X₃.

Clearly, the ranking has a little difference; however, the best alternative is the same as X₄. The advantage of the proposed method is that it can select the optimal alternative by the maximum “group utility” and minimum “individual regret” and the advantage of Wang and Li’ method [34] is that it can take the bounded rationality of decision makers into account. Because the ranking principle is different, it is reasonable for not completely same ranking results. In this example, these two methods produced the same best and worst alternatives, and this can show the validity of the proposed method.

In addition, the proposed method can provide the compromise optimal group, i.e., X₄ and X₁, while TODIM method [34] only gives the optimal alternative X₄.

5 Conclusion

NHFS is the generalization of neutrosophic set and the hesitant fuzzy set. Some operational laws, comparison rules of NHFS and the Hamming distance between two NHFNs are defined. For the MCDM problems with NHFS, the traditional VIKOR method is extended, and an approach is given based on the maximum “group utility” and minimum “individual regret”. So this method can provide a compromise best solution or a set of compromise solutions. Compared with the other MCDM methods, the advantages of the proposed method are that it can simultaneously consider the “group utility’ and “individual regret” and can give a compromise best solution or a set of compromise solutions. In the future, we will research some extended VIKOR for new fuzzy sets, such as multi-valued neutrosophic set, uncertain linguistic neutrosophic set, single valued neutrosophic graphs [35, 36], intuitionistic hesitant fuzzy set [37] and so on. In addition, in this research, we didn’t consider the case of uncertain weights, this needs to do further study in the future, and at the same time, we need research the applications of the proposed method in real decision making environment, such as performance evaluation, supply chain selection, and so on.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province, National Soft Science Project of China (No. 2014GXQ4D192), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), Shandong Provincial Social Science Planning Project (No. 15BGLJ06), The teaching reform research project of undergraduate colleges and Universities in Shandong province (No. 2015Z057). The authors also would like to express appreciations to the anonymous reviewers and Editors for their very helpful comments that improved this paper.

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An extended multiple criteria decision making method based on neutrosophic hesitant fuzzy information

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The single valued neutrosophic set

Table 1 Decision making matrix with the neutrosophic hesitant fuzzy information A 1 A 2 ⋯ A n X 1 n ˜ 11 n ˜ 12 ⋯ n ˜ 1 n X 2 n ˜ 21 n ˜ 22 ⋯ n ˜ 2 n ⋯ ⋯ ⋯ ⋯ ⋯ X m n ˜ m 1 n ˜ m 2 ⋯ n ˜ mn

Table 5 The degree of priority among alternatives [34] X 1 X 2 X 3 X 4 X 1 0 –0.893 0.236 –1.358 X 2 0.539 0 0.272 –0.765 X 3 0.927 –1.406 0 –1.955 X 4 –0.41 0.248 0.498 0

Footnotes

Acknowledgments

References

Table 5
The degree of priority among alternatives [34]

X ₁ X ₂ X ₃ X ₄

X ₁ 0 –0.893 0.236 –1.358

X ₂ 0.539 0 0.272 –0.765

X ₃ 0.927 –1.406 0 –1.955

X ₄ –0.41 0.248 0.498 0