The interval complex single-valued neutrosophic fuzzy set (ICSVNFS) and hesitant fuzzy set (HFS) are two important different generalizations of the fuzzy set (FS) to cope with unreliable and unpredictable information in the real world. The ICSVNHFS is characterized by interval complex-valued membership, interval complex-valued abstinence, and interval complex-valued non-membership grades, whose ranges are restricted to a unit disc in a complex plane instead of real numbers. In this paper, the notions of interval-valued complex single-valued neutrosophic hesitant fuzzy sets (IVCSVNHFSs) are initiated and also their operational laws are described with examples. Further, based on IVCSVNHFSs, we develop the notions of the interval-valued complex single-valued neutrosophic fuzzy generalized weighted (IVCSVNHFGW) operator, interval-valued complex single-valued neutrosophic fuzzy generalized ordered weighted (IVCSVNHFGOW) operator, and interval-valued complex single-valued neutrosophic fuzzy generalized hybrid weighted (IVCSVNHFGHW) operator to cope with decision information and also study their properties. Further, a new multi-attribute group decision making (MAGDM) problem is initiated based on the proposed operators. Finally, we provide some numerical examples to illustrate the reliability and superiority of the proposed methods by comparison with other existing methods.
The Intuitionistic fuzzy set (IFS) was proposed by Atanassov [1] as an extension of the notion of a fuzzy set (FS) [2]. IFS comprising two functions so-called membership and non-membership grades whose sum is limited to the unit interval [0, 1]. Therefore, they are more reliable and practical when addressing fuzziness and uncertainty than the FS. In some circumstances, the truth grade, falsity grade and hesitant grade of an object in the IFS may not be a specific number. So the IFS is generalized into the interval-valued intuitionistic fuzzy set (IVIFS), proposed by Atanassov and Gargov [3]. The characteristic of the IVIFS is that the membership and non-membership grades are intervals instead of exact numeric values. The concepts of the IFS and IVIFS are applied in many fields like distance measures [4–6], entropy measures [7, 8], similarity measures [9–11], and MAGDM problems [12–14]. To handle such situations where people are hesitant shown their performance regarding objects in a decision-making process, Aoroa [15] proposed the hesitant fuzzy set (HFS) as a generalization of the FS. The HFS comprising only membership functions whose ranges are a finite subset of the unit interval [0, 1]. As for the present, the HFS has been widely applied in solving MAGDM problems [16, 17], neural networks [18], medical diagnosis [19], color region extraction [20], market prediction [21], and so forth.
The complex intuitionistic fuzzy set (CIFS) was initiated by Alkouri and Salleh [22]. It is proven to be an important tool to cope with uncertain and unpredictable information in real-life problems. The CIFS is a generalization of the complex fuzzy set (CFS) [23] by including the non-membership degree in the environment of the CFS. The difference between the CIFS and IFS is that the range of membership and non-membership grades in the CIFS is restricted to the unit disc instead of real numbers belonging to the unit interval [0, 1]. Further, complex Atanassov’s intuitionistic fuzzy relations are also proposed by Alkouri and Salleh [24]. Kumar and Bajaj [25] developed the notion of complex intuitionistic fuzzy soft sets with distance measures and entropies. Rani and Grag [26, 27] founded complex intuitionistic fuzzy power aggregation operators and distance measures for the CIFS and their applications in MAGDM problems. The generalized complex intuitionistic fuzzy aggregation operators and robust correlation coefficient for the CIFS and their applications in MAGDM problem were proposed by Grag and Rani [28, 29].
Whenever the IFS and CIFS cannot describe effectively when a decision maker provides (0.5,0.4,0.7) for truth value, for abstinence value, and for a false value. For handling such types of situations, Smarandache [30] initiated the concept of the neutrosophic set (NS) as an extension of the FS. The NS is characterized by membership, abstinence, and non-membership degrees belonging to [0-, 1+] with a condition: the sum of membership, abstinence and non-membership degrees is limited to [0-, 3+]. However, without specific description, the NS is difficult to apply to real-life situations. Therefore, the single-valued neutrosophic set (SVNS) was founded by Haibin et al. [31]. The NS and SVNS are applied in many areas such as distance measures [32–34], entropy measures [35–37], operators [38–41], and MAGDM problems [42, 43]. Recently, Ali and Smarandache [44] pioneered the idea of the complex neutrosophic set (CNS). The CNS is the combination of the CFS and NS, which are two different tools to cope with unreliable and difficult information in real-life problems. Basically, the CFS is comprising of two dimension-information in a single set. Further, Ali et al. [45] developed interval complex neutrosophic sets (ICNSs). Singh et al. [46] initiated complex neutrosophic concept lattice and its applications to air quality analysis. The complex neutrosophic soft expert set and its application in MAGDM were proposed by Ashraf and Hassan [47]. Therefore, the aim of this paper is to initiate the notion of interval-valued complex neutrosophic hesitant fuzzy sets and its properties. Our study is more general than the existing work. The comparison between our proposed research with existing methods is discussed below:
If the non-membership grade is zero, then the interval-valued complex single valued neutrosophic hesitant fuzzy set (IVCSVNHFS) is reduced into an interval-valued complex intuitionistic hesitant fuzzy set (IVCIHFS).
If the non-membership and abstinence grades are zero, then the IVCSVNHFS is reduced into an interval-valued complex hesitant fuzzy set (IVCHFS).
If the membership, non-membership, and abstinence grades are singleton sets, then the IVCSVNHFS is reduced into an interval-valued complex single valued neutrosophic set (IVCSVNS).
If the membership, non-membership, and abstinence grades are singleton sets and the non-membership grade is zero, then the IVCSVNHFS is reduced into an interval-valued complex intuitionistic fuzzy set (IVCIFS).
If the membership, non-membership, and abstinence grades are singleton sets and the non-membership and abstinence grades are zero, then the IVCSVNHFS is reduced into an interval-valued complex fuzzy set (IVCFS).
The existing works and our proposed work are also explained with the help of diagram, depicted as in Fig. 1.
Flowchart of fuzzy sets and their generalizations.
It is to see that the flow chart extended from the fuzzy sets to the interval-valued fuzzy sets, complex fuzzy sets, intuitionistic fuzzy sets, interval-valued complex neutrosophic hesitant fuzzy sets, and more other concepts (Fig. 1). In a flowchart, we discuss the fuzzy sets and their generalizations. The above discussions show the proposed methods in this paper have advantages. In this paper, we propose a novel approach of interval-valued complex single-valued neutrosophic hesitant fuzzy sets and their generalized hybrid averaging operators in the environment of MAGDM problems. When a decision maker provides interval-valued complex single-valued neutrosophic hesitant fuzzy kind of information, the existing methods, such as IVCIHFS, IVCHFS, IVCSVNS, IVCIFS, and IVCFS, cannot describe them effectively. As mentioned above, the complex-valued hesitancy is the most common problem in decision making, for which the complex hesitant fuzzy set (CHFS) can be considered as a suitable means allowing several possible degrees for an element to a set. However, in a CHFS, there is only one complex-valued truth-membership hesitant function, and it cannot express this problem with a few different values assigned by the complex-valued truth-membership degree, complex-valued indeterminacy-membership degree, and complex-valued falsity membership degree, due to doubts of decision makers. Thus, in this situation, it can represent only one kind of complex-valued hesitancy information and cannot express three kinds of complex-valued hesitancy information. A complex single-valued neutrosophic set (CSVNS) is an instance of a complex neutrosophic set which provides an additional possibility to represent uncertain, imprecise, incomplete, and inconsistent information that exists in the real world. It seems to be more suitable for dealing with indeterminate information and inconsistent information. However, it can only express a complex-value truth-membership degree, a complex-valued indeterminacy-membership degree, and a complex-valued falsity-membership degree, whereas it cannot represent this problem with a few different values assigned by complex-valued truth-membership degree, complex-valued indeterminacy-membership degree, and complex-valued falsity-membership degree, respectively, due to doubts of decision makers. In such a situation, the aforementioned algorithms based on complex
neutrosophic sets or CHFSs 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 propose the IVCSVNHFS which is more powerful and more general than existing methods. The framework of existing methods are taken from Ref. [48]. The comparison between existing methods and our proposed methods are discussed in Table 1.
Comparisons between existing methods and proposed methods
The aim of existing works
The aim of this paper
To initiate the notion of interval-valued single-valued neutrosophic hesitant fuzzy sets and describe their basic properties, which cannot be described using two dimension-information in a single set.
To initiate the notion of interval-valued complex single-valued neutrosophic hesitant fuzzy sets and describe their basic properties, which effectively describe the existing type of information.
To initiate the notion of interval-valued single-valued neutrosophic hesitant fuzzy generalized weighted averaging operator, interval-valued single-valued neutrosophic hesitant fuzzy generalized ordered weighted averaging operator, and interval-valued single-valued neutrosophic hesitant fuzzy generalized hybrid weighted averaging operator and describe their basic properties, which cannot hold in the environment of complex fuzzy information.
To initiate the notion of interval-valued complex single-valued neutrosophic hesitant fuzzy generalized weighted averaging operator, interval-valued complex single-valued neutrosophic hesitant fuzzy generalized ordered weighted averaging operator, and interval-valued complex single-valued neutrosophic hesitant fuzzy generalized hybrid weighted averaging operator and describe their basic properties, which is the generalization of existing methods.
To initiate a new MAGDM problem based on existing methods.
To initiate a new MAGDM problem based on our proposed methods.
The comparison between our proposed method and existing methods cannot be calculated.
To initiate the comparison between proposed method and existing methods to show the superiority and effectiveness of our methods.
The rest of the paper is follow as: in Section 2, we review some basic notions of fuzzy sets (FSs), complex fuzzy sets (CFSs), complex neutrosophic sets (CNSs), interval-valued single valued neutrosophic sets (IVSVNSs), hesitant fuzzy sets (HFSs), and their properties. In Section 3, the notions of interval-valued complex single-valued neutrosophic hesitant fuzzy sets (IVCSVNHFSs) are initiated and their operational laws also are described with examples. In Section 4, based on IVCSVNHFSs, we develop the notions of the interval-valued complex single-valued neutrosophic fuzzy generalized weighted (IVCSVNHFGW) operator, interval-valued complex single-valued neutrosophic fuzzy generalized ordered weighted (IVCSVNHFGOW) operator, and interval-valued complex single-valued neutrosophic fuzzy generalized hybrid weighted (IVCSVNHFGHW) operator to cope with decision information and also study their properties. Section 5 initiates a new multi-attribute group decision making (MAGDM) problem based on the proposed methods and provides some numerical examples to illustrate the reliability and superiority of the proposed methods by comparison with other existing methods. The conclusion of this paper is discussed in the last section.
Preliminaries
We will review some fundamental concepts of the FS, CFS, CNS, IVSVNS, and HFS, and their properties.
Definition 1: [2] The FS is denoted and defined by:
where T (x) represents the degree of membership, which satisfies the following conditions: 0≤ T (x)≤ 1 .
Definition 2: [23] The CFS is denoted and defined by:
where represents the degree of complex-valued membership, satisfying the following conditions: and 0 ≤ θ η′ ≤ 2 ℼ
Definition 3: [44] The CNFS is denoted and defined by:
where , and represents the complex-valued membership, complex-valued abstinence, and complex-valued non-membership degrees, respectively, which satisfy the following conditions: 0 ≤ η′ + F′ + ξ′ ≤ 3 and 0 ≤ θη′ + θF′ + θξ′ ≤ 6 ℼ, where 0 ≤ η′, F′, ξ′ ≤ 1 and 0 ≤ θη′, θF′, θξ′ ≤ 2 ℼ. Further, ᶇ = {T(x), ζ(x), Υ(x)} represents the complex neutrosophic fuzzy number (CNFN), simply we write ᶇ = (T, ζ, Υ) = (η′eiθη′, F′eiθF′, ξ′eiθξ′).
Definition 4: [31] The SVNS is denoted and defined by:
where T(x), ζ(x) and Υ(x) represents the membership, abstinence, and non-membership degrees, which are finite subsets of [0,1] and satisfy the following conditions: 0 ≤ TU(x) + ζU(x) + ΥU(x) ≤ 3 and 0 ≤ TU(x) + ζU(x) + ΥU(x) ≤ 1. Further, ᶇ = {T(x), ζ(x), Υ(x)} represents the single-valued neutrosophic number (SVNN), simply we write ᶇ = {T, ζ, Υ}.
Definition 5: [37] The IVNS is denoted and defined by:
where T(x) = [TL(x), TU(x)], ζ(x) = [ζL(x), ζU(x)] and Υ(x) = [ΥL(x), ΥU(x)] represents the interval-valued membership, interval-valued abstinence, and interval-valued non-membership degrees, which are finite subsets of [0,1] and satisfy the following conditions: 0+ ≤ TU(x) + ζU(x) + ΥU(x) ≤ 3+ and 0+ ≤ TU(x) + ζU(x) + ΥU(x) ≤ 1+. Further, ᶇ = {T(x), ζ(x), Υ(x)} represents the interval-valued neutrosophic number (IVNN), simply we write ᶇ = (T, ζ, Υ)
Definition 6: [15] The HFS is denoted and defined by:
E={(x, hE(x)): x ∈ X, where hE(x) is a finit subset of [0, 1]}
where h = hE (x) is called a hesitant fuzzy element (HFE) [15].
Definition 7: [15] Let h, h1 and h2 be three HFEs with Ɣ > 0. Then
In this section, we introduce the notion of IVCSVNHFSs and their basic properties.
Definition 8: The IVCSVNHFS is represented and described by:
where T(x) = {η = [η′L, η′U]ei2ℼ[θη′L, θη′U]/η ∈ T(x)}, ζ(x) = {F = [F′L, F′U]ei2ℼ[θF′L, θF′U]/F ∈ ζ(x)} and Υ(x) = {ξ = [ξ′L, ξ′U]ei2ℼ[θξ′L, θξ′U]/ξ ∈ Υ(x)} represents the complex-valued membership, complex-valued abstinence, and complex-valued non-membership degrees, which are interval values belonging to unite disc in the complex plane, respectively, and satisfy these limits: 0 ≤ max(η′U) + max(F′U) + max(ξ′U) ≤ 3 and 0 ≤ max(θη′U) + max(θF′U) + max(θξ′U) ≤ 3, where 0 ≤ η′U, F′U, ξ′U) ≤ 1 0 ≤ θη′U, θF′U, θξ′U) ≤ 1 and lower values are less than or equal to upper values for all degrees. Further, ᶇ = {T(x), ζ(x), Υ(x)} represents the interval-valued complex single-valued neutrosophic hesitant fuzzy number (IVCSVNHFN), simply we write ᶇ = (T, ζ, Υ) = .
Example 1: and
are two IVCSVNHFNs.
Definition 9: Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs. Then
1.
2.
Definition 10: Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs with Ɣ > 0. Then
Theorem 1: Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs with Ɣ, Ɣ1, Ɣ2 > 0. Then
1. ᶇ1 ⊕ ᶇ2 = ᶇ2 ⊕ ᶇ1;
2. ᶇ1 ⊗ ᶇ2 = ᶇ2 ⊗ ᶇ1;
3. ɣ (ᶇ1 ⊕ ᶇ2) = ɣᶇ2 ⊕ ɣᶇ1;
4.
5. ɣ1 ᶇ1 ⊕ ɣ2 ᶇ2 = (ɣ1 + ɣ2)ᶇ1;
6.
Proof: We examine the part (1), part (3), and part (5). The other parts are straightforward.
1. Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs with Ɣ, Ɣ1, Ɣ2 > 0. Then
ᶇ1 ⊕ ᶇ2 =
∪η1′ ∈ T1, F1′ ∈ ζ1, ξ1′ ∈ Υ1,
η2′ ∈ T2, F2′ ∈ ζ2, ξ2′ ∈ Υ2
2. Straightforward (omitted).
3. Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs. Then ɣ1(ᶇ1 ⊕ ᶇ2)
= ɣ1∪η1′ ∈ T1, F1′ ∈ ζ1, ξ1′ ∈ Υ1,
η2′ ∈ T2, F2′ ∈ ζ2, ξ2′ ∈ Υ2
= ∪η1′ ∈ T1, F1′ ∈ ζ1, ξ1′ ∈ Υ1,
η2′ ∈ T2, F2′ ∈ ζ2, ξ2′ ∈ Υ2
4. Straightforward (omitted).
5. Let ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) be two IVCSVNHFNs. Then
= ɣ1ᶇ1 ⊕ ɣ2ᶇ1
Hence, (ɣ1 + ɣ2)ᶇ1 = ɣ1ᶇ1 ⊕ ɣ2ᶇ1.
6. Straightforward (omitted).
Definition 11: Let be an IVCSVNHFN. Then, the score function and the accuracy function are calculated as follows:
and
respectively.
Let us consider two IVCSVNHFNs ᶇ1 = (T1, ζ1, Υ1) and ᶇ2 = (T2, ζ2, Υ2) . Then,
If
If
If Ş(ᶇ1) = Ş(ᶇ2), then
If
If
If
Some aggregation operators based on interval-valued complex single-valued neutrosophic hesitant fuzzy sets
In this section, we propose the notions of the interval-valued complex single-valued neutrosophic hesitant fuzzy geometricweighted averaging (IVCSVNHFGWA) operator, interval-valued complex single-valued neutrosophic hesitant fuzzy geometric ordered weighted averaging (IVCSVNHFGOWA) operator, and interval-valued complex single-valued neutrosophic hesitant fuzzy geometric hybrid weighted averaging (IVCSVNHFGHWA) operator to aggregate the interval-valued complex single-valued neutrosophic hesitant fuzzy numbers (IVCSVNHFNs) effectively. Throughout this paper, the finite universal set is represented by X and the weight vector is denoted and defined by τ = (τ1, τ2, .., τᶇ)T, τi ∈ [0, 1], where
The IVCSVNHFGWA Operator
In this sub-section, we propose the notion of the IVCSVNHFGWA operator and its properties.
Definition 12: The IVCSVNHFGWA operator is given by: IVCSVNHFGWA : Ωᶇ → Ω by IVCSVNHFGWA (ᶇ1, ᶇ1, .., ᶇᶇ) = where Ω represents the family of all IVCSVNHFNs with Ɣ > 0. The IVCSVNHFN is of the form (i = 1, 2, .., ᶇ). Using the operational laws of our work, we want to prove some results as follows.
Theorem 2: Let (i = 1, 2, .., ᶇ) be the IVCSVNHFNs with Ɣ > 0. Then
IVCSVNHFGWA (ᶇ1, ᶇ2, . . , ᶇᶇ)
Proof:
First we will examine that
For the above equality, we can prove it by mathematical induction.
1. If ᶇ = 1, then
It is true for ᶇ = 1.
2. If ᶇ = k, let us assume that the above equality is right, then
Then, we check the above equality for ᶇ = k + 1. We get
and
It is true also for ᶇ = k + 1. So it is true for all ᶇ.
3. From a step (2), we get
Hence the result has been proven.
Next, we prove the idempotency property for IVCSVNHFSs.
Theorem 3: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with Ɣ > 0. Then IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) = ᶇ
Proof: If , then IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) =
Hence the result has been proven. In the following, we prove the monotonicity property for IVCSVNHFSs.
Theorem 4: Let (i = 1, 2, .., ᶇ) "be IVCSVNHFNs with ɣ > 0. If ᶇi ≥ ᶇj′. Then IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≥ IVCSVNHFGWA (ᶇ1′, ᶇ2′, .., ᶇᶇ′)
Proof: Assume that ᶇi ≥ ᶇi′ which means that and for all i. Similarly, for lower part, we get and for all i. Then, firstly we prove for membership grades such that
[(1 - (ηi′L) ɣ) τi, (1 - (ηi′U) ɣ) τi] ei2-Pi[(1-(θηi′L)ɣ)τi,(1-(θηi′U)ɣ)τi] ≤ [(1 - (ηi′L′) ɣ) τi, (1 - (ηi′U′) ɣ) τi]
Similarly, for falsity and non-membership grades, we get
and
Hence we combine the above equations such that
So, IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≥ IVCSVNHFGWA (ᶇ1′, ᶇ2′, .., ᶇᶇ′). Hence the result has been proven.
We want to prove the boundedness property for IVCSVNHFSs.
Theorem 5: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then min(ᶇ1, ᶇ2, .., ᶇᶇ) ≤ IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ max(ᶇ1, ᶇ2, .., ᶇᶇ)
That is min (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ IVCSVNHFGWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ max(ᶇ1, ᶇ2, .., ᶇᶇ) Hence the result has been proven.
The IVCSVNHFGOWA operator
In this sub-section, we propose the notion of the IVCSVNHFGOWA operator and its properties.
Definition 13: The IVCSVNHFGOWA operator is given by: IVCSVNHFGOWA: Ωᶇ → Ω so that IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ) = where Ω represents the family of all IVCSVNHFNs with Ɣ > 0 and ᶇo(i) means that the provided IVCSVNHFNs are ranked in an ascending or descending order, i.e., ᶇo(i) ≤ ᶇo(i-1) . The IVCSVNHFN is of the form (i = 1, 2, .., ᶇ).
Using the operational laws of our work, we want to prove the following some results.
Theorem 6: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ)
Proof: The proof of this result is similar to Theorem 2 (omitted). We will prove the idempotency property for IVCSVNHFSs.
Theorem 7: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ) = ᶇ
Proof: The proof of this result is similar to Theorem 3.
Next, we will prove the monotonicity property for IVCSVNHFSs.
Theorem 8: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. If ᶇ_i ≥ ᶇ_j'. Then IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≥ IVCSVNHFGOWA (ᶇ1′, ᶇ2′, .., ᶇᶇ′)
Proof: The proof of this result is similar to Theorem 4.
We will prove the boundedness property for IVCSVNHFSs.
Theorem 9: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then min(ᶇ1, ᶇ2, .., ᶇᶇ) ≤ IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ max(ᶇ1, ᶇ2, .., ᶇᶇ)
That is min(ᶇ1, ᶇ2, .., ᶇᶇ) ≤ IVCSVNHFGOWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ max(ᶇ1, ᶇ2, .., ᶇᶇ)
Hence the result has been proven.
The IVCSVNHFGHWA operator
In this sub-section, we propose the notion of the IVCSVNHFGHWA operator and its properties.
Definition 14: The IVCSVNHFGHWA operator is given by: IVCSVNHFGHWA :Ωᶇ → Ω so that IVCSVNHFGHWA (ᶇ1, ᶇ2, .., ᶇᶇ) = where Ω represents the family of all IVCSVNHFNs with Ɣ > 0 and ᶇo(i) = ᶇώi ᶇi means that the provided IVCSVNHFNs is ranked in an ascending or descending order, i.e., ᶇo(i) ≤ ᶇo(i-1). The IVCSVNHFN is of the form (i = 1, 2, .., ᶇ), where
Using the operational laws of our work, we want to prove the following some results.
Theorem 10: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then IVCSVNHFGHWA (ᶇ1, ᶇ2, .., ᶇᶇ)
Proof: The proof of this result is similar to Theorem 2.
We will prove the idempotency, monotonicity, and boundedness properties for IVCSVNHFSs.
Theorem 11: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. Then IVCSVNHFGHWA (ᶇ1, ᶇ2, .., ᶇᶇ) = ᶇ
Proof: The proof of this result is similar to Theorem 3 (omitted).
Theorem 12: Let (i = 1, 2, .., ᶇ) be IVCSVNHFNs with ɣ > 0. If ᶇi ≥ ᶇj′. Then
Proof: Obviously, we have α ≤ IVCSVNHFGHWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ β
Then, we get
Therefore
That is, min (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ IVCSVNHFGHWA (ᶇ1, ᶇ2, .., ᶇᶇ) ≤ max(ᶇ1, ᶇ2, .., ᶇᶇ). Hence the result has been proven.
Special cases
We will discuss some special cases of our study, which can be proven to be more general than existing works. The special cases are discussed for three ideas, which are as follows:
Remark 1: We will discuss special cases for the weighted averaging operator.
1. If Ɣ → 0, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy weighted geometric operator as follows:
2. If Ɣ = 1, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy weighted averaging operator as follows:
3. If Ɣ = 2, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy weighted quadratic averaging operator as follows:
Remark 2: We will discuss special cases for the weighted geometric operator.
1. If Ɣ → 0, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy ordered weighted geometric operator as follows:
2. If Ɣ = 1, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy ordered weighted averaging operator as follows:
3. If Ɣ = 2, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy ordered weighted quadratic averaging operator as follows:
Remark 3: We will discuss special cases for the hybrid weighted geometric operator.
1. If Ɣ → 0, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy hybrid weighted geometric (IVCSVNHFHWG) operator as follows:
2. If Ɣ = 1, then the our proposed operator is reduced into the intrerval-valued complex single-valued neutrosophic hesitant fuzzy hybrid weighted averaging (IVCSVNHFHWA) operator as follows:
3. If Ɣ = 2, then the our proposed operator is reduced into the interval-valued complex single-valued neutrosophic hesitant fuzzy hybrid weighted quadratic averaging (CNHFHWQA) operator as follows:
MAGDM method based on interval-valued complex single-valued neutrosophic hesitant fuzzy sets
In this section, we will consider the developed operators for the interval-valued complex single-valued
neutrosophic hesitant fuzzy numbers (IVCSVNHFNs) to the multi-attribute group decision making (MAGDM)
problems. We are taking the attributes values from the interval-valued complex single-valued neutrosophic
hesitant fuzzy information. For the MAGDM problem, the set of alternatives and the set of attributes are denoted
and defined by: X = {x1, x2, x3, .., xm} and C = {c1, c2, c3, .., cn}, where τ = {τ1, τ2, τ3, .., τn}T is the weight vector, . Let us consider that ᶇij = (Tij, ζij, Υij) be IVCSVNHFNs for attribute cj and alternative Ai such that IVCSVNHFNs are of the form
Then the steps of algorithm is follow as:
Using the def. 14 to solve the IVCSVNHFNs.
Using the def. 11 to solve the IVCSVNHFNs.
Rank the all alternative and choose the best one.
End.
A Numerical example
In this subsection, we will discuss an example to illustrate the application of our proposed methods. The example is taken from ref. [48] proposed by Liu and Shi.
Let us consider the investment company which wants to invest with another company. There are four possible alternatives with respect to four applicants, denoted by Ai (i = 1, 2, 3, 4). In this application we consider the three attributes, which are denoted and defined by: C1: Risk Index; C2: Growth Index; C3: Environmental Impact Index.The above three attributes are of the form IVCSVNHFNs. We consider the weighted vector as τ = { 0.35, 0.45, 0.20 } T. Then, we get Table 2. This application is discussed for three different ideas as follows:
Point 1: We consider the original form of the weight vector τ = { 0.35, 0.45, 0.20 } T and solve the application for finding the best alternative.
Point 2: We consider the ascending form of the weight vector τ = { 0.20, 0.35, 0.45 } T and solve the application for finding the best alternative.
Point 3: We consider the descending form of the weight vector τ = { 0.45, 0.35, 0.20 } T and solve the application for finding the best alternative.
In many research papers, the authors used the point 1, and examined the results. But in this paper, we will check the above three possibility and examine the results and find the same or different results.
Then we compare the results and find the difference between the above three points. Before some researchers used point 2 or point 3, but no one used all points and compared the results. In this paper, we use the all point and discuss the results to find the best one.
We will study the application of point 1, then the steps of the algorithm is follow as:
1. Using the def. 14 to solve Table 2, which contains IVCSVNHFNs. Suppose the positional weighted vector with constant is denoted and defined as and lambda = 2 such that
2. Using the def. 11 to find the sore function of IVCSVNHFNs, we get
3. Rank the all alternatives and choose the best one.
Hence the best alternative is A3 when we consider the point 1.
Similarly, we consider the point 2 and solve the application. Then we get the final result such that.
Rank the all alternatives and choose the best one.
Hence the best alternative is A3. when we consider the point 2.
In last, we consider the point 3 and solve the application. Then we get the final result such that
Rank the all alternative and choose the best one.
Hence the best alternative is A3 when we consider the point 3. The ranking values in Tables 3, 4 and 5 are given the same results. The best choice for investment is the same.
Using score functions to examine the best one
Alternatives
Score functions
Positions of alternatives in decending order
A1
Ş ; (A1) = 0.48
Third
A2
Ş ; (A2) = 0.51
Second
A3
Ş ; (A3) = 0.52
First
A4
Ş ; (A4) = 0.46
Fourth
Using score functions to examine the best one
Alternatives
Score functions
Positions of alternatives in decending order
A1
Ş ; (A1) = 0.50
Fourth
A2
Ş ; (A2) = 0.54
Second
A3
Ş ; (A3) = 0.55
First
A4
Ş ; (A4) = 0.51
Third
Using score functions to examine the best one
Alternatives
Score function
Position of alternatives in decending order
A1
Ş ; (A1) = 0.501
Fourth
A2
Ş ; (A2) = 0.539
Second
A3
Ş ; (A3) = 0.541
First
A4
Ş ; (A4) = 0.503
Third
4. End.
Therefore we use different methods to discuss the above three points and examine the best alternative, which is A3. Whenever, if we discuss any point in above, the best alternative is provided as A3. So there is no problem, if we discuss any point. When we discuss any point above, which cannot be effected on the best alternative but provide to change the last results. Therefore, there is no effect when we discuss ascending or descending order.
Advantages and comparative study
In this sub-section, we will verify that the superiority and effectiveness of the proposed methods by comparison with some existing methods. First we will show the influence of the parameter Ɣ, we will take the different value Ɣ in step 1 to rank the alternatives. The ranking results are shown in Table 6. It is to see from Table 6 that the ranking results are identical for the different values of the parameter Ɣ in IVCSVNHFHWA operator.
Here we will discuss two cases for the parameters Ɣ.
If 0.01 ≤ Ɣ ≤ 5, then the best alternative is A3.
If 0.01 ≤ Ɣ ≤ 5, then the required results are the same for different values of the parameter. The performance of the parameter Ɣ is discussed in Table 6.
Different results for different values of the parameters γ
The sensitivity analysis of the proposed method is follow as:
To investigate the variation trends of the scores and the rankings of the alternatives with the change in the values of the attitudinal character parameter Ɣ from 0 to 5, an analysis has been conducted by the proposed operator, IVCSVNHFHWA, and their corresponding results are summarized in Table 6. These results show that the decision makers can choose the values according to their preferences. Thus, the management meaning of Ɣ is that the decision makers’ different preferences have effects on the score values of alternatives, which lead to the different optimal alternatives. For different values of the parameter, the best alternative is A3.
We compare our proposed work with existing methods such as:
If we will ignore the imaginary part of the membership, abstinence and non-membership grades and using the extended brackets in IVCSVCNHFSs, then the IVCSVCNHFS is reduced into the INHFS proposed by Liu and Shi [48]. The example of INHFS is discussed in Table 7.
If we will ignore the imaginary part of the membership, abstinence, non-membership grades and interval and using the extended brackets in IVCSVCNHFSs, then the IVCSVCNHFS is reduced into the NHFS proposed by Liu and Zhang [50]. The example of NHFS is discussed in Table 8.
If we will ignore the interval-valued, single-valued, imaginary parts and abstinence degree in the proposed methods, then the proposed method is reduced into the intuitionistic hesitant fuzzy set (IHFS) proposed by Peng et al. [51]. The example of IHFS is discussed in Table 9.
If we will ignore the interval-valued and the imaginary parts in IVSVNHFSs, then the IVSVNHF is reduced into the SVNHFS proposed by Ye [49]. The example of SVNHFS is discussed in Table 10.
If we will ignore the interval-valued, single-valued, imaginary parts, abstinence degree and non-membership degree in IVSVNHFSs, then the IVSVNHFS is reduced into the hesitant fuzzy set (HFS) proposed by Torra [15]. The example of the HFS is follow as:
If we will ignore the interval-valued, single-valued and considered the singleton set in IVSVNHFSs, then the IVSVNHFS is reduced into the CNS proposed by Ali and Smarandache [44]. The example of the CNS is follow as:
Interval neutrosophic hesitant fuzzy sets
Alternatives
C1
C2
C3
A1
A2
A3
Neutrosophic hesitant fuzzy sets
Alternatives
C1
C2
C3
A1
A2
A3
Intuitionistic hesitant fuzzy sets
Alternatives
C1
C2
C3
A1
{{ 0.4, 0.3, 0.5 } , { 0.4, 0.9 }}
{{ 0.2, 0.6 } , { 0.8, 0.9 }}
{{ 0.3, 0.5 } , { 0.4 }}
A2
{{ 0.4, 0.35 } , { 0.9 }}
{{ 0.2, 0.8, 0.5, 0.6 } , { 0.8, 0.9 }}
{{ 0.5 } , { 0.22, 0.9 }}
A3
{{ 0.3, 0.5 } , { 0.4 }}
{{ 0.2, 0.8, 0.5, 0.6 } , { 0.6, 0.8, 0.8, 0.9 }}
{{ 0.45, 0.6 } , { 0.22, 0.9 }}
Single-valued neutrosophic hesiotant fuzzy sets
Alternatives
C1
C2
C3
A1
A2
A3
Our proposed methods have failed many other concepts, but we will not discuss in here. We will work on these concepts in a future. The characteristic comparison of the proposed methods and existing methods is follow as:
Clearly, the above all points are special cases of our proposed work. So the proposed methods are more reliable and more general than existing methods. In Table 13, the notation “Yes” is used for agree and the notation “No” describes that the corresponding methods are fails. In Table 13 we compare our proposed work with existing works. When a decision maker is provided wits such kinds of information like IVCSVNHFS, the all existing works (see Table 4) cannot effectively deal with the above information. Therefore, our proposed methods are more superior and more reliable than existing methods.
Hesitant fuzzy sets
Alternatives
C1
C2
C3
A1
{0.4, 0.3, 0.5}
{0.6, 0.7, 0.3}
{0.3, 0.5}
A2
{0.4, 0.35}
{0.4, 0.35, 0.78, 0.87}
{0.35}
A3
{0.4, 0.6}
{0.4, 0.56, 0.6}
{0.6}
Complex neutrosophic sets
Alternatives
C1
C2
C3
A1
A2
A3
The characteristic comparison of different approaches
Methods
Generalized operators based on t-norm and t-conorm
Contain interval-valued kinds of information’s
Ability to capture information using complex numbers
Ability to handle two dimensional information
Flexible according to decision maker’s preferences
The ICSVNFS and HFS are two different extensions of the FS to cope with unreliable and unpredictable information in the real world. The ICSVNHFS is characterized by the interval complex-valued membership, interval complex-valued abstinence, and interval complex-valued non-membership grades, whose ranges are restricted to unit disc in a complex plane instead of real numbers. In this paper, the notion of IVCSVNHFSs is initiated and its operational laws are discussed with examples. Further, based on IVCSVNHFSs, we develop the notions of IVCSVNHFGW operator, IVCSVNHFGOW operator, and IVCSVNHFGHW operator to cope with uncertain and unpredictable information in real life problems. Some special cases of the proposed approaches and their describable properties are also discussed in detail. Further, we use the IVCSVNHFGHWA operator to develop an approach to solve the IVCSVNHF MAGDM problems. Finally, we provide some numerical examples to illustrate the reliability and superiority of the proposed methods and deliver the sensitivity analysis and compare them with other existing methods.
In the future, we will concentrate on developing linear programming methods [57] to solve heterogeneous multi-attribute decision problems [58–60] with IVCSVNHFSs.
Author Contributions
Funding
Conflicts of interest
The authors declare no conflict of interest.
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