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

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

Abstract: In this paper, sine trigonometry operational laws (ST-OLs) have been extended to neutrosophic sets (NSs) and the operations and functionality of these laws are studied. Then, extending these ST-OLs to complex neutrosophic sets (CNSs) forms the core of this work. Some of the mathematical properties are proved based on ST-OLs. Fundamental operations and the distance measures between complex neutrosophic numbers (CNNs) based on the ST-OLs are discussed with numerical illustrations. Further the arithmetic and geometric aggregation operators are established and their properties are verified with numerical data. The general properties of the developed sine trigonometry weighted averaging/geometric aggregation operators for CNNs (ST-WAAO-CNN & ST-WGAO-CNN) are proved. A decision making technique based on these operators has been developed with the help of unsupervised criteria weighting approach called Entropy-ST-OLs-CNDM (complex neutrosophic decision making) method. A case study for material selection has been chosen to demonstrate the ST-OLs of CNDM method. To check the validity of the proposed method, entropy based complex neutrosophic CODAS approach with ST-OLs has been executed numerically and a comparative analysis with the discussion of their outcomes has been conducted. The proposed approach proves to be salient and effective for decision making with complex information.

Keywords: Complex neutrosophic sets (CNSs); sine trigonometric operational laws (ST-OLs); aggregation operator; entropy; CODAS; material selection; decision making

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

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.

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

Definition 2.1 [1] A fuzzy set

Definition 2.2 [2] An intuitionistic fuzzy set

Definition 2.3 [5] Let

Definition 2.4 [5] Let

Definition 2.5 Let

i.

ii.

Definition 2.6 Let

(1)

(2)

Also, the following conditions hold good.

(1) If

(2) If

(3) If

(4) If

(5) If

(6) If

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

From the above STOL of NNs, it is evident that the

Also,

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

Definition 3.2 Let

• Complement of

• Intersection of

• Union of

• Algebric sum of

• Algebric product of

• Scalar product of

• Power of

Definition 3.3 Let

i.

ii.

Definition 3.4 Let

1.

2.

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

4 Sine Trigonometry Operational Laws (STOLs) for Complex Neutrosophic Sets (CNSs)

First, we see some basic concepts of complex neutrosophic sets and their operations.

Definition 4.1 [18] A complex neutrosophic set

Example 4.1 Let

The STOLs have been introduced for complex neutrosophic sets and their boundary conditions are verified.

Definition 4.2 Let

Then, the sine trigonometric operational law of CNNs (ST-OLs-CNNs) is defined as follows:

From the above STOL of CNNs, it is evident that the

Also,

Example 4.2 Let

Then, the sine trigonometric operational law of CNNs (ST-OLs-CNNs) is also a CNN. We describe the function as follows:

Also, the modulus of ST-OLs of CNNs is listed below and it is observed that the sum of values is less than or equal to three.

Then we discuss the fundamental operations on sine trigonometric operational laws of CNNs and their properties.

Definition 4.3 [18] Let

Then the sine operational laws of

• Complement of

• Intersection of

• Union of

• Algebric sum of

• Scalar product of

• Algebric product of

• Power of

Definition 4.4 Let

Then, score and accuracy of

Next, the properties of sine trigonometric operational laws of CNNs are discussed.

Theorem 4.1 Let

•

•

Proof. The proofs are straightforward from the Definition 4.3.

Theorem 4.2 Let

(i).

(ii).

(iii).

(iv).

(v).

Proof. Let

Then, using Definition 4.3 the algebraic sum of two ST-OL-CNNs

(i). For any

Hence the property (i) is proved.

The proof of property (ii). is similar

(iii). For any

Hence the proof of property (iii).

Theorem 4.3 Let

Proof. For any two CNNs

Similarly, for indeterminacy and falsity membership functions.

Hence we get from the Definition 4.2 that

4.2 Subtraction of Two Sine Trigonometric CNNs

Definition 4.5 Let

The subtraction is defined as

Example 4.3 Let two sine trigonometric CNNs be

Then the subtraction is calculated as follows:

4.3 Distance Measure of ST-OL-CNNs

In this section, we discuss different types of distance measures of ST-OL-CNNs.

• Let

• When

• Similarly, when

Example 4.4 Let ST-OLs for two CNNs be given by

Then, the distance measures are

• Manhattan distance measure (Ma-D):

• Euclidean distance measure (ED):

• Minkowski distance measure (MD) when

Note: Here the subtraction of two ST-OL-CNNs is calculated using Eq. (6).

The Minkowski distance measures of ST-OL-CNNs satisfies the following properties:

(i).

(ii).

(iii).

(iv). If

5 Aggregation Operators for ST-OLs-CNSs

In this section, the weighted averaging and geometric aggregation operators are presented for ST-OLs-CNNs with numerical example.

5.1 Sine Trigonometry Weighted Averaging Aggregation Operator (ST-WAAO)

Definition 5.1 Let

Then the ST-WAAOs for CNNs is denoted by

where

Theorem 5.1 Let

Proof. The proof of Theorem 5.1 is examined by mathematical induction on n. For each P,

Step I. For

Using Definition 4.3, the algebraic sum of two ST-OL-CNNs

Step II. Suppose that Eq. (10) holds for

Step III. Next, we have to prove that Eq. (10) holds for

Hence the proof.

Example 5.1 Suppose that

Then absolute value of ST-WAAO-CNN is calculated as follows:

Next, we give some properties of the ST-WAAO-CNNs operator and establish that they preserve idempotency, boundedness, monotonically, and symmetry.

Theorem 5.2 Let

Proof. Let

Hence proved.

Theorem 5.3 Let

Then,

Proof. For any P,

This implies that

similarly,

Also, we have

similarly,

Then,

Theorem 5.4 Let

If

Proof. It follows from Theorem 5.3 and hence the proof is omitted.

Theorem 5.5 Let

Proof. The proof follows from Theorem 5.3.

5.2 Sine Trigonometry Weighted Geometric Aggregation Operator (ST-WGAO)

Definition 5.2 Let

Then the ST-WGAOs for CNNs is denoted by

where

Theorem 5.6 Let