Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.018267

ARTICLE

Sine Trigonometry Operational Laws for Complex Neutrosophic Sets and Their Aggregation Operators in Material Selection

D. Ajay1, J. Aldring1, G. Rajchakit2, P. Hammachukiattikul3 and N. Boonsatit4,*

1Sacred Heart College (Autonomous), Tamil Nadu, 635601, India
2Maejo University, Chiang Mai, 50290, Thailand
3Phuket Rajabhat University, Phuket, 83000, Thailand
4Rajamangala University of Technology Suvarnabhumi, Nonthaburi, 11000, Thailand
*Corresponding Author: N. Boonsatit. Email: nattakan.b@rmutsb.ac.th
Received: 11 July 2021; Accepted: 07 September 2021

1  Introduction

One of the complex problems in all types of industries/companies is making decisions based on ambiguous information. As a consequence, the concept of fuzzy set theory has been employed to deal with this type of situation or problem. Zadeh [1] first proposed the concept of fuzzy subsets in 1965. It has sparked numerous significant outcomes in the scientific community, which have been replicated in modern applications, particularly in decision-making and artificial intelligence. In addition, different researchers have extended the core idea of fuzzy sets to handle different types of uncertainty, and the extended notions of fuzzy sets include intuitionistic fuzzy sets (IFSs) [2], Pythagorean fuzzy sets (PFSs) [3], Fermatean fuzzy sets (FFSs) [4], Neutrosophic sets (NSs) [5], and Spherical fuzzy sets (SFSs) [6] among others and each of them addresses the problem of uncertainty in a unique way. Based on these notions, various multi criteria decision making (MCDM) approaches are available in the literature like weighted sum method (WSM), weighted product method (WPM), the multiple criteria optimization compromise solution (VIKOR), the technique for order performance by similarity to the ideal solution (TOPSIS), the Weighted Aggregates Sum Product Assessment (WASPAS), COmbinative Distance-Based ASsessment (CODAS), and the Evaluation Based on Distance from Average Solution (EDAS), etc., MCDM is a technique for selecting the best option from a set of alternatives and neutrosophic sets have found wide scope in such techniques. Edalatpanah established a new model of data envelopment analysis based on triangular neutrosophic numbers [7]. And also neutrosophic approach have been implemented to data envelopment analysis with undesirable outputs by Mao et al. [8]. A new ranking function of triangular neutrosophic number [9] and systems of neutrosophic linear equations are introduced by Edalatpanah [10].

Complex fuzzy sets (CFSs) are a significant research area of investigation in fuzzy logic. It was proposed by Ramot et al. [11,12]. CFSs use a complex membership function to handle uncertainty with periodicity which takes complex values within the complex unit circle. Clearly, the complex memership function M=μeiα of a CFS comprises two terms named amplitude term μ and phase term α which lie in the intervals [0,1] and [0,2π], respectively. The phase term accounts for the periodicity of the data and distinguishes CFSs from the traditional models of fuzzy set theory. CFSs and Complex Fuzzy Logic (CFL) have been utilized to develop accurate and efficient time series forecasting models [13,14], image processing [15], etc. The concept of CFSs, has been further extended to complex intuitionistic fuzzy sets (CIFSs) [16], complex Pythagorean fuzzy sets (CPFSs) [17], complex neutrosophic sets (CNSs) [18] and so on and also it is evident from the literature that notable research have been carried out in complex fuzzy sets scenario. Recently, Xu et al. [19] introduced an extended EDAS method with a single-valued complex neutrosophic set and its application in green supplier selection. Complex neutrosophic generalized dice similarity measures have been developed by Ali et al. [20]. Also, another MCDM model called a soft set based VIKOR is developed based on CNSs by Manna et al. [21]. Some of complex hybrid weighted averaging operators are introduced for decision making in [22]. Aggregating the fuzzy information plays an important role in decision theory and it is involved in majority of MCDM methods.

In addition operational laws also play a vital role in aggregation process. Also it is evident from the literature that various operational laws are available. Gou et al. [23] used new type of operational laws for IFSs. Then, Li et al. [24] introduced the logarithmic operational laws for IFSs in order to aggregate information. Later, Garg et al. [25] introduced a new logarithmic operational laws for single valued neutrosophic number which yields application in multi attribute decision making. Also, Ashraf et al. [26] used logarithmic hybrid aggregation operators for single valued neutrosophic sets. In continuation, Garg et al. [27] have presented some new exponential, logarithmic, and compensative exponential of logarithmic operational laws for complex intuitionistic fuzzy (CIF) numbers based on t-norm and co-norm. Garg also utilized the logarithmic operational laws for PFSs in [28]. Further, Nguyen et al. [29] developed exponential similarity measures for Pythagorean fuzzy sets and their application in pattern recognition. Haque et al. [30] utilized exponential operational laws for generalized SFSs. Another novel concept of neutrality operational laws have been introduced by Garg et al. [31] for q-rung orthopair fuzzy sets and Pythagorean fuzzy geometric aggregation operators [32]. Also, Garg extended the new exponential operation laws for q-rung orthopair fuzzy sets in [33].

Sine trigonometric operational laws were introduced by Garg [34]. The main advantage of sine trigonometric function is that it accounts for the periodicity and it is symmetric about the origin. Thus it satisfies the expectations of the decision-maker over the multi-time phase parameters. Garg [35] introduced a novel trigonometric operation based q-rung ortho pair fuzzy aggregation operators and he also extended these operational laws to Pythagorean fuzzy information [36]. Abdullah et al. [37] developed an approach of ST-OLs for picture fuzzy sets. Ashraf et al. [38] utilized the concept of single valued neutrosophic sine trigonometric aggregation operators for hydrogen power plant selection and further they implemented these operational laws for spherical fuzzy environment in [39]. MCDM methods namely TOPSIS and VIKOR have been developed based on ST-OLs by Qiyas et al. [40,41]. From the literature, it is clear that ST-OLs play predominant role in aggregation operators (AOs). Through this motivation and considering the advantage of ST-OLs, some new ST-OLs for CNSs must be established and their behaviour in complex scenario needs to be studied. Hence, this paper aims to modify sine trigonometric operational laws for complex neutrosophic sets and implement them in complex decision making method for material selection. So, the main objective of this paper can be described as follows:

(i). To present the ST-OLs for CNSs

(ii). To obtain some of the distance measure for complex neutrosophic sets based on ST-OLs

(iii). To develop an MCDM technique with the help of the proposed aggregation operators

(iv). To demonstrate an entropy technique based on ST-OLs for CNNs in order to attain complex weights of criteria

(v). To give an application of the proposed MCDM method in material selection in an industry

(vi). Finally, to present the validation of the developed method with existing CODAS approach

The organization of the paper is the following; We review the basic concept of NSs in the second section of the paper. In Section 3, we explore the operations of enhanced ST-OLs for NSs. These ST-OLs have been extended to CNSs in Section 4, including subtraction and distance measurement of ST-OLs for CNSs. In Section 5, we develop AOs and prove their properties for ST-OLs of CNSs. In Section 6, an MCDM approach has been explained in detailed steps with entropy technique for criteria weights. The proposed MCDM approach is used to provide an application for material selection in Section 7. The validation and discussion of the study is carried out in Section 8. Finally, the paper is concluded in Section 9 with direction for further research.

2  Preliminaries

Some of the basic concepts of neutrosophic sets (NSs) with their operations are discussed here.

Definition 2.1 [1] A fuzzy set F~ defined on a universe of discourse ℜ∗ has the form: F~={⟨ξF~(r˙)⟩|r˙∈ℜ∗}, where ξF~(r˙):ℜ∗→[0,1]. Here ξF~(r˙) denotes the membership function of each r˙.

Definition 2.2 [2] An intuitionistic fuzzy set IF~ is defined as a set of ordered pairs over a universal set ℜ∗ and is given by IF~={⟨(ξIF~(r˙),ψIF~(r˙))⟩|r˙∈ℜ∗}, where ξIF~(r˙):ℜ∗→[0,1],ψIF~(r˙):ℜ∗→[0,1] with the condition ξIF~(r˙)+ψIF~(r˙)≤1 for each element r˙∈ℜ∗. Here the membership and non-membership functions are denoted as ξIF~(r˙) and ψIF~(r˙), respectively.

Definition 2.3 [5] Let ℜ∗ be a universe of discourse or a non empty set. Any object in the neutrosophic set NF~ has the form NF~={⟨μNF~(r˙),σNF~(r˙),γNF00~(r˙)⟩|r˙∈ℜ∗}, where μNF~(r˙), σNF~(r˙) and γNF~(r˙) represent the degree of truth membership, the degree of indeterminacy and the degree of false membership respectively of each element r˙∈ℜ∗ to the set NF~ and it is defined as NF~={⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩|r˙∈ℜ∗}, where μNF~(r˙),σNF~(r˙),γNF~(r˙):ℜ∗→[0,1] such that 0≤μNF~(r˙)+σNF~(r˙)+γNF~(r˙)≤3.

Definition 2.4 [5] Let NF~=r˙:⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩ , NF1~=r˙:⟨μNF1~(r˙),σNF1~(r˙),γNF1~(r˙)⟩  and NF2~=r˙:⟨μNF2~(r˙),σNF2~(r˙),γNF2~(r˙)⟩ be three neutrosophic numbers (NNs) and let w¨ be any scalar. Then

Definition 2.5 Let NF1~=r˙:⟨μNF1~(r˙),σNF1~(r˙),γNF1~(r˙)⟩ and NF2~=r˙:⟨μNF2~(r˙),σNF2~(r˙),γNF2~(r˙)⟩ be two NNs. Then

i. NF1~⊆NF2~ if and only if μNF1~(r˙)≤μNF2~(r˙) , σNF1~(r˙)≥σNF2~(r˙) , γNF1~(r˙)≥γNF2~(r˙) for each (r˙)∈ℜ∗

ii. NF1~⊆NF2~ if and only if NF1~⊆NF2~ and NF1~⊇NF2~

Definition 2.6 Let NF~=r˙:⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩ , NF1~=r˙:⟨μNF1~(r˙),σNF1~(r˙),γNF1~(r˙)⟩ and NF2~=r˙:⟨μNF2~(r˙),σNF2~(r˙),γNF2~(r˙)⟩ be three NNs. Then the score and accuracy functions of NNs are defined as follows:

(1) Score(NF~)=μNF~(r˙)−σNF~(r˙)−γNF~(r˙)

(2)Accuracy(NF~)=μNF~(r˙)+σNF~(r˙)+γNF~(r˙)

Also, the following conditions hold good.

(1) If Score(NF1~)>Score(NF2~) then NF1~>NF2~

(2) If Score(NF1~)<Score(NF2~) then NF1~<NF2~

(3) If Score(NF1~)=Score(NF2~) then NF1~=NF2~

(4) If Accuracy(NF1~)>Accuracy(NF2~) then NF1~>NF2~

(5) If Accuracy(NF1~)<Accuracy(NF2~) then NF1~<NF2~

(6) If Accuracy(NF1~)=Accuracy(NF2~) then NF1~=NF2~

3  Sine Trigonometry Operational Law (STOL) for Neutrosophic Sets

First, the STOL [34] are applied to neutrosophic sets and the boundary conditions are verified.

Definition 3.1 Let the neutrosophic numbers (NNs) be NF~=r˙:⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩. Then, the sine trigonometric operational laws of NNs are defined as follows:

sin⁡(NF~)={⟨sin⁡(π2.μNF~(r˙)),sin2⁡(π2.σNF~(r˙)),2sin2⁡(π4.γNF~(r˙))⟩|r˙∈ℜ∗}(1)

From the above STOL of NNs, it is evident that the sin⁡(NF~) is also NNs. And it satisfies the following condition of neutrosophic set as the degree of truth, indeterminacy, and falsity of NS are defined, respectively

sin⁡(π2.μNF~(r˙)):ℜ∗→[0,1] such that

0≤sin⁡(π2.μNF~(r˙))≤1,

sin2⁡(π2.σNF~(r˙)):ℜ∗→[0,1] such that 0≤sin2⁡(π2.σNF~(r˙))≤1,

2sin2⁡(π4.γNF~(r˙)):ℜ∗→[0,1] such that 0≤2sin2⁡(π4.γNF~(r˙))≤1,

Also, 0≤⟨sin⁡(π2.μNF~(r˙))+sin2⁡(π2.σNF~(r˙))+2sin2⁡(π4.γNF~(r˙))⟩≤3. Therefore, STOL of NNs are also NNs, a fact which is also affirmed by Fig. 1.

Then we discuss the fundamental operations on sine trigonometric neutrosophic numbers (STNNs) and their properties.

Figure 1: This is a graph of STOLs of NNs

Definition 3.2 Let NF~=r˙:⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩, NF1~=r˙:⟨μNF1~(r˙),σNF1~(r˙),γNF1~(r˙)⟩ and NF2~=r˙:⟨μNF2~(r˙),σNF2~(r˙),γNF2~(r˙)⟩ be neutrosophic numbers (NNs) and let w¨ be any scalar. Then

•   Complement of sin⁡(NF)~:

•   Intersection of sin⁡(NF1)~ and sin⁡(NF2)~:

•   Union of sin⁡(NF1)~ and sin⁡(NF2)~:

•   Algebric sum of sin⁡(NF1)~ and sin⁡(NF2)~:

•   Algebric product of sin⁡(NF1)~ and sin⁡(NF2)~:

•   Scalar product of sin⁡(NF)~:

•   Power of sin⁡(NF)~:

Definition 3.3 Let sin⁡(NF1)~ and sin⁡(NF2)~ be two sine trigonometric neutrosophic numbers. Then

i.   sin⁡(NF1)~⊆sin⁡(NF2)~ if and only if sin⁡(π2.μNF1~(r˙))≤sin⁡(π2.μNF2~(r˙)), sin2⁡(π2.σNF1~(r˙))≥sin2⁡(π2.σNF2~(r˙)), 2sin2⁡(π4.γNF1~(r˙))≥2sin2⁡(π4.γNF2~(r˙)) for each (r˙)∈ℜ∗

ii.   sin⁡(NF1)~=sin⁡(NF2)~ if and only if sin⁡(NF1)~⊆sin⁡(NF2)~ and sin⁡(NF1)~⊇sin⁡(NF2)~

Definition 3.4 Let NF~=r˙:⟨μNF~(r˙),σNF~(r˙),γNF~(r˙)⟩ be a neutrosophic number. Then the score and accuracy functions of STOL of NNs are defined as follows:

1.    Score(sin⁡(NF~))=sin⁡(π2.μNF~(r˙))−sin2⁡(π2.σNF~(r˙))−2sin2⁡(π4.γNF~(r˙))

2.    Accuracy(sin⁡(NF~))=sin⁡(π2.μNF~(r˙))+sin2⁡(π2.σNF~(r˙))+2sin2⁡(π4.γNF~(r˙))

Here the score and accuracy range values of STOLs of NNs are discussed in pictorial representation in the following Figs. 2 and 3 when the membership values of μNF~(r˙), σNF~(r˙), γNF~(r˙) are equal.