Multiple-attribute Decision-Making Method under a Single-Valued Neutrosophic Hesitant Fuzzy Environment

1 Introduction

The neutrosophic set [2] is a powerful general formal framework that generalizes the concept of the classic set, fuzzy set, interval-valued fuzzy set, intuitionistic fuzzy set, interval-valued intuitionistic fuzzy set, paraconsistent set, dialetheist set, paradoxist set, and tautological set. In the neutrosophic set, indeterminacy is quantified explicitly, and truth membership, indeterminacy membership, and falsity membership are independent. However, the neutrosophic set generalizes the above-mentioned sets from a philosophical point of view, and its functions T_A(x), I_A(x), and F_A(x) are real standard or non-standard subsets of ]^–0, 1⁺[, i.e., T_A(x): X →]^–0, 1⁺[, I_A(x): X →]^–0, 1⁺[, and F_A(x): X →]^–0, 1⁺[. Thus, it will be difficult to apply in real scientific and engineering areas [5, 6]. Therefore, Wang et al. [5, 6] proposed the concepts of an interval neutrosophic set (INS) and a single-valued neutrosophic set (SVNS), which are the subclasses of a neutrosophic set, and provided the set-theoretic operators and various properties of SVNSs and INSs. SVNSs and INSs provide an additional possibility to represent uncertain, imprecise, incomplete, and inconsistent information that exists in the real world. They would be more suitable to handle indeterminate information and inconsistent information. Recently, Ye [13] presented the correlation coefficient of SVNSs based on the extension of the correlation of intuitionistic fuzzy sets and proved that the cosine similarity measure is a special case of the correlation coefficient of SVNSs, and then applied it to single-valued neutrosophic decision-making problems. Then, Ye [14] presented another form of correlation coefficient between SVNSs and applied it to multiple-attribute decision making under a single-valued neutrosophic environment. Ye [15] proposed a single-valued neutrosophic cross-entropy measure and applied it to multicriteria decision-making problems with single-valued neutrosophic information. Ye [16] also introduced the Hamming and Euclidean distances between INSs and the similarity measures, and then applied them to decision-making problems in an interval neutrosophic setting. Furthermore, Ye [17] presented the concept of a simplified neutrosophic set (SNS), which is a subclass of the neutrosophic set and includes an SVNS and an INS, and defined some operations of SNSs. Then, he developed a simplified neutrosophic weighted averaging (SNWA) operator, a simplified neutrosophic weighted geometric (SNWG) operator, and a multicriteria decision-making method using the SNWA and SNWG operators and the cosine measure of SNSs under a simplified neutrosophic environment.

In fuzzy decision-making problems, however, decision makers sometimes find it difficult to determine the membership of an element to a set, and in some circumstances they cause this difficult problem of giving a few different values due to doubt. To deal with such cases, Torra and Narukawa [3] and Torra [4] presented the concept of a hesitant fuzzy set (HFS) as another extension of the fuzzy set. Xu and Xia [9, 10] defined some similarity measures, distance and correlation measures of HFSs, and applied them to multicriteria decision making. Then, Xia and Xu [8] developed a series of aggregation operators for hesitant fuzzy information and applied them in solving decision-making problems. Xu and Xia [11] introduced hesitant fuzzy entropy and cross-entropy and their use in multiple-attribute decision making. Zhu et al. [19] introduced hesitant fuzzy geometric Bonferroni means and applied the proposed aggregation operators to multicriteria decision making. Wei [7] also introduced hesitant fuzzy prioritized operators and their application to multiple-attribute decision making. Chen et al. [1] generalized the concept of HFS to that of interval-valued HFS (IVHFS) in which the membership degrees of an element to a given set are not exactly defined but denoted by several possible interval values, and gave systematic aggregation operators to aggregate interval-valued hesitant fuzzy information. They then developed an approach to group decision making based on interval-valued hesitant preference relations to consider the differences of opinions between individual decision makers. Xu et al. [12] investigated the aggregation of hesitancy fuzzy information, proposed several series of aggregation operators, and discussed their connections. Then, they applied the Choquet integral to obtain the weights of criteria and established a group decision-making method under a hesitant fuzzy environment.

However, in some situations, decision makers sometimes cause this difficult problem of assigning a few different values for satisfied and unsatisfied degrees. Thus, the HFS and the IVHFS are difficult to use for such a decision-making problem. To handle such cases, Zhu et al. [20] originally introduced a dual HFS (DHFS) as a generalization of HFSs, which encompasses fuzzy sets, intuitionistic fuzzy sets, HFSs, and fuzzy multisets as special cases [20]. The DHFS consists of two parts – the membership hesitancy function and the non-membership hesitancy function – and can handle two kinds of hesitancy in this situation. Then, Ye [18] proposed a correlation coefficient of DHFSs and applied it to multiple-attribute decision-making problems under a dual hesitant fuzzy environment.

As mentioned above, hesitancy is the most common problem in decision making, for which HFS can be considered as a suitable means allowing several possible degrees for an element to a set. However, in an HFS, there is only one truth-membership hesitant function, and it cannot express this problem with a few different values assigned by truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees, due to doubts of decision makers. Thus, in this situation, it can represent only one kind of hesitancy information and cannot express three kinds of hesitancy information. An SVNS is an instance of a neutrosophic set that provides an additional possibility to represent uncertain, imprecise, incomplete, and inconsistent information that exists in the real world. It would be more suitable to handle indeterminate information and inconsistent information. However, it can only express a truth-membership degree, an indeterminacy-membership degree, and a falsity-membership degree, and it cannot represent this problem with a few different values assigned by truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees, respectively, due to doubts of decision makers. In such a situation, the aforementioned algorithms based on neutrosophic sets or HFSs and their decision-making methods are difficult to use for such a decision-making problem with three kinds of hesitancy information that exists in the real world. To handle this case, we need to introduce a concept of a single-valued neutrosophic HFS (SVNHFS), which is a combination of HFS and SVNS, and some algorithms for SVNHFSs. The SVNHFS consists of three parts – the truth-membership hesitancy function, indeterminacy-membership hesitancy function, and falsity-membership hesitancy function – and can express three kinds of hesitancy information in this situation. Therefore, the purposes of this article are (i) to propose an SVNHFS based on the combination of SVNS and HFS, (ii) to introduce the basic operational relations and cosine measure function of SVNHFSs, (iii) to develop a single-valued neutrosophic hesitant fuzzy weighted averaging (SVNHFWA) operator and a single-valued neutrosophic hesitant fuzzy weighted geometric (SVNHFWG) operator and investigate their properties, and (iv) to establish a multiple-attribute decision making method based on the SVNHFWA and SVNHFWG operators and the cosine measure under a single-valued neutrosophic hesitant fuzzy environment.

The rest of the article is organized as follows. Section 2 briefly describes some concepts of neutrosophic sets, SVNSs, HFSs, and DHFSs. Section 3 proposes the concept of SVNHFSs and defines the corresponding basic operations and cosine measure for SVNHFSs. In Section 4, we develop the SVNHFWA and SVNHFWG operators and investigate their properties. Section 5 establishes a decision-making approach based on the SVNHFWA and SVNHFWG operators and the cosine measure. An illustrative example validating our approach is presented and discussed in Section 6. Section 7 contains the conclusion and future research direction.

2 Preliminaries

3 Single-Valued Neutrosophic Hesitant Fuzzy Set

HFSs can reflect the original information given by the decision makers as much as possible, and more values are obtained from the decision makers or experts, which belonging to [0, 1] may be assigned to easily express hesitant judgments. Therefore, in this section, the concept of an SVNHFS is presented on the basis of the combination of SVNSs and HFSs as a further generalization of the concept of SVNSs and HFSs, which is defined as follows.

Definition 12. Let X be a fixed set; an SVNHFS on X is defined as

N = {〈x,t˜(x),i˜(x),f˜(x)〉 | x ∈ X},

in which t˜(x),i˜(x), and f˜(x) are three sets of some values in [0, 1], denoting the possible truth-membership hesitant degrees, indeterminacy-membership hesitant degrees, and falsity-membership hesitant degrees of the element x ∈ X to the set N, respectively, with the conditions 0 ≤ δ, γ, η ≤ 1 and 0 ≤ γ⁺ + δ⁺ + η⁺ ≤ 3, where γ ∈ t˜(x),δ ∈ i˜(x),η ∈ f˜(x),γ+ ∈ t˜+(x) = ∪γ ∈ t˜(x)max{γ},δ+ ∈ i˜+(x) = ∪δ ∈ i˜(x)max{δ}, and η+ ∈ f˜+(x) = ∪η ∈ f˜(x)max{η} for x ∈ X.

For convenience, the three tuple n(x) = {t˜(x),i˜(x),f˜(x)} is called a single-valued neutrosophic hesitant fuzzy element (SVNHFE) or a triple hesitant fuzzy element, which is denoted by the simplified symbol n = {t˜,i˜,f˜}.

From Definition 12, we can see that the SVNHFS consists of three parts, i.e., the truth-membership hesitancy function, the indeterminacy-membership hesitancy function, and the falsity-membership hesitancy function, supporting a more exemplary and flexible access to assign values for each element in the domain, and can handle three kinds of hesitancy in this situation. Thus, the existing sets, including fuzzy sets, intuitionistic fuzzy sets, SVNSs, HFSs, and DHFSs, can be regarded as special cases of SVNHFSs.

Then, we can define the union and intersection of SVNHFEs as follows.

Definition 13. Let n₁ and n₂ be two SVNHFEs in a fixed set X; then, their union and intersection are defined, respectively, by

1. n1∪n2 = {t˜ ∈ (t˜1 ∪ t˜2) | t˜ ≥ max(t˜1−,t˜2−),i˜ ∈ (i˜1 ∩ i˜2) | i˜ ≤ min(i˜1+,i˜2+),f˜ ∈ (f˜1 ∩ f˜2) | f˜ ≤ min(f˜1+,f˜2+)}.

2. n1 ∩ n2 = {t˜ ∈ (t˜1 ∩ h˜2) | t˜ ≤ min(t˜1+,t˜2+),i˜ ∈ (i˜1 ∪ i˜2) | i˜ ≥ max(i˜1−,i˜2−),f˜ ∈ (f˜1 ∪ f˜2) | f˜ ≥ max(f˜1−,f˜2−)}.

Therefore, for two SVNHFEs n₁ and n₂, we can give the following basic operations:

1. n1 ⊕ n2 = {t˜1 ⊕ t˜2,i˜1 ⊗ i˜2,f˜1 ⊗ f˜2} = ∪γ1 ∈ t˜1,δ1 ∈ i˜1,η1 ∈ f˜1,γ2 ∈ t˜2,δ2 ∈ i˜2,η2 ∈ f˜2{{γ1 + γ2 − γ1γ2},{δ1δ2},{η1η2}}.

2. n1 ⊗ n2 = {t˜1 ⊗ t˜2,i˜1 ⊕ i˜2,f˜1 ⊕ f˜2}=∪γ1 ∈ t˜1,δ1 ∈ i˜1,η1 ∈ f˜1,γ2 ∈ t˜2,δ2 ∈ i˜2,η2 ∈ f˜2{{γ1γ2},{δ1 + δ2 − δ1δ2},{η1 + η2 − η1η2}}.

3. λn1 = ∪γ1 ∈ t˜1,δ1 ∈ i˜1,η1 ∈ f˜1{{1 − (1 − γ1)λ},{δ1λ},{η1λ}},λ > 0.

4. n1λ = ∪γ1 ∈ t˜1,δ1 ∈ i˜1,η1 ∈ f˜1{{γ1λ},{1 − (1 − δ1)λ},{1 − (1 − η1)λ}},λ > 0.

On the basis of the cosine measure of SVNSs [13, 17], we give the following cosine measure between SVNHFEs.

Definition 14. Let n1 = {t˜1,i˜1,f˜1} and n2 = {t˜2,i˜2,f˜2} be any two SVNHFEs; thus, the cosine measure between n₁ and n₂ is defined by

where l_i, p_i, and q_i for i = 1, 2 are the numbers of the elements in t˜i,i˜i,f˜i for i = 1, 2, respectively, and cos(n₁, n₂) ∈ [0, 1].

To compare the SVNHFEs, we give the following comparative laws based on the cosine measure.

Let n1 = {t˜1,i˜1,f˜1} and n2 = {t˜2,i˜2,f˜2} be any two SVNHFEs; thus, the cosine measure between n_i (i = 1, 2) and the ideal element n^* = <1, 0, 0> is obtained by applying Eq. (1) as follows:

where l_i, p_i, and q_i for i = 1, 2 are the numbers of the elements in t˜i,i˜i,f˜i for i = 1, 2, respectively, and cos(n_i, n^*) ∈ [0, 1] for i = 1, 2. Then, there are the following comparative laws based on the cosine measure:

1. If cos(n₁, n^*) > cos(n₂, n^*), then n₁ is superior to n₂, denoted by n₁≿n₂.

2. If cos(n₁, n^*) = cos(n₂, n^*), then n₁ is equivalent to n₂, denoted by n₁∼n₂.

4 Weighted Aggregation Operators for SVNHFEs

On the basis of the basic operations of SVNHFEs in Section 3, this section proposes the following two weighted aggregation operators for SVNHFEs as a further generalization of the weighted aggregation operators of HFSs and SNSs [8, 17], and investigates their properties as the weighted aggregation operators are more basic operators in information aggregation and decision making.

5 Decision-Making Method Based on the SVNHFWA and SVNHFWG Operators

In this section, we apply the SVNHFWA and SVNHFWG operators to multiple-attribute decision-making problems with single-valued neutrosophic hesitant fuzzy information.

For a multiple-attribute decision-making problem under a single-valued neutrosophic hesitant fuzzy environment, let A = {A₁, A₂, …, A_m} be a discrete set of alternatives and C = {C₁, C₂, …, C_k} be a discrete set of attributes. When the decision makers are required to evaluate the alternative A_i (i = 1, 2, …, m) under the attribute C_j (j = 1, 2, …, k), they may assign a set of several possible values to each of truth-membership degrees, indeterminacy-membership degrees, and falsity-membership degrees to which an alternative A_i (i = 1, 2, …, m) satisfies and/or dissatisfies an attribute C_j (j = 1, 2,…, k), and then these evaluated values can be expressed as an SVNHFE nij = (t˜ij,i˜ij,f˜ij) (i = 1, 2,…, m; j = 1, 2,…, k). Thus, we can elicit a single-valued neutrosophic hesitant fuzzy decision matrix D = (n_ij)_m×k, where n_ij (i = 1, 2, …, m; j = 1, 2, …, k) is in the form of SVNHFEs.

The weight vector of attributes for the different importance of each attribute is given as w = (w₁, w₂,…, w_k)^T, where w_j≥ 0, j = 1, 2, …, k, and ∑j = 1kwj = 1. Then, we use the SVNHFWA or SVNHFWG operator and the cosine measure to develop an approach for multiple-attribute decision-making problems with single-valued neutrosophic hesitant fuzzy information, which can be described as follows:

Step 1. Aggregate all SVNHFEs of n_ij (i = 1, 2, …, m; j = 1, 2, …, k) by using the SVNHFWA operator to derive the collective SVNHFE n_i (i = 1, 2, …, m) for an alternative A_i (i = 1, 2, …, m):

or by using the SVNHFWG operator:

Step 2. Calculate the measure values of the collective SVNHFE n_i (i = 1, 2, …, m) and the ideal element n^* = <1, 0, 0> by Eq. (2).

Step 3. Rank the alternatives and select the best one(s) in accordance with the measure values.

Step 4. End.

6 Illustrative Example

An illustrative example about investment alternatives for a multiple-attribute decision-making problem adapted from Ye [13, 18] is used as the demonstration of the applications of the proposed decision-making method under a single-valued neutrosophic hesitant fuzzy environment. There is an investment company that wants to invest a sum of money in the best option available. There is a panel with four possible alternatives in which to invest the money: (i) A₁ is a car company; (ii) A₂ is a food company; (iii) A₃ is a computer company; and (iv) A₄ is an arms company. The investment company must make a decision according to the following three attributes: (i) C₁ is the risk; (ii) C₂ is the growth; and (iii) C₃ is the environmental impact. The attribute weight vector is given as w = (0.35, 0.25, 0.4)^T. The four possible alternatives, A_i (i = 1, 2, 3, 4), are to be evaluated using the single-valued neutrosophic hesitant fuzzy information by some decision makers or experts under three attributes, C_j (j = 1, 2, 3).