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.

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 [

Complex fuzzy sets (CFSs) are a significant research area of investigation in fuzzy logic. It was proposed by Ramot et al. [

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. [

Sine trigonometric operational laws were introduced by Garg [

(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

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

i.

ii.

(1)

(2)

Also, the following conditions hold good.

(1) If

(2) If

(3) If

(4) If

(5) If

(6) If

First, the STOL [

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.

Complement of

Intersection of

Union of

Algebric sum of

Algebric product of

Scalar product of

Power of

Here the score and accuracy range values of STOLs of NNs are discussed in pictorial representation in the following

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

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

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,

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.

Then the sine operational laws of

• Complement of

• Intersection of

• Union of

• Algebric sum of

• Scalar product of

• Algebric product of

• Power of

Then, score and accuracy of

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

(i).

(ii).

(iii).

(iv).

(v).

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).

Similarly, for indeterminacy and falsity membership functions.

Hence we get from the Definition 4.2 that

The subtraction is defined as

Then the subtraction is calculated as follows:

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

Let

When

Similarly, when

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

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

(i).

(ii).

(iii).

(iv). If

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

Then the ST-WAAOs for CNNs is denoted by

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

Hence the proof.

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.

Hence proved.

Then,

This implies that

Also, we have

Then,

If

Then the ST-WGAOs for CNNs is denoted by

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