Generalized Parametric Intuitionistic Fuzzy Measures Based on Trigonometric Functions for Improved Decision-Making Problem

Information theory is the study of collecting, storing, and sharing digital information. It is a nexus of disciplines such as statistics, computer science, statistical mechanics, and probability theory. This study pertains to intuitionistic fuzzy sets theory, which is a substantial component of fuzzy set theory. Nonetheless, the motive of the study is to find vague information intuitionistic fuzzy entropy measures. The authors are extend the parametric intuitionistic fuzzy entropy measures by using trigonometric functions and investigate the difference between proposed study & existing entropy measures. Furthermore, discuss the significance analysis and authenticity of the proposed study. It concludes that the proposed measure could be a good perspective for decision-making problems. Using a suitable illustration, the applicability of the proposed study has been demonstrated. Depict the graph of proposed and existing entropy measure together with their average measure. Additionally, these estimations enhance the study of information theory and produce superior information.


INTRoDUCTIoN
The communication genrally exist with transferring of information from one source to other.According to the information theory, the communication process is a crucial aspect of information collection, as it is the source of authentic information.Shannon (Shannon, 1948) established probabilistic entropy for continuous and discrete probability distributions.Analogues to the concept of probability theory, (L.A. Zadeh, 1965) describe the fuzzy sets (FSs) theory and introduced fuzzy entropy on fuzzy sets.Afterwards, (K.T. Atanassov, 1986) invented the intuitionistic fuzzy sets (IFSs) theory, which raised the study of information theory, it facilitates the investigation of information theory.Atanassov and Gargov (1989) developed the study of interval valued intuitionistic fuzzy sets (IVFSs) which is the extension of intuitionistic fuzzy sets.In this present study, the authors are analyze the non-membership function beyond the membership function and construct the measures to determine the uncertain information.It attained the considering degree values between zero and one, corresponding to every point of the universe of discourse.The intuitionistic fuzzy set theory is an extensive study of (L.A. Zadeh, 1965) fuzzy set theory.Under this environment, IFSs entropy deals with the study of uncertain information.Burillo & Bustince (1996) reveal the concept of IFSs entropy measures for uncertainty and found the relationship between IFSs and IVFSs.Gau and Buehrer (1993) discussed the idea of vague sets, which is based on the notion of fuzzy set theory.Szmidt & Kacprzyk (2001) drew out the postulates De Luca and Termini (1971) for IFSs entropy measure and developed a non-probabilistic IFSs entropy measure based on the geometrical interpretation of IFSs.Here, few complexity is arise in the present communication, therefore, some new entropies were created.This overcomes the past study for finding the relevant results of uncertainty.On the basis of the pioneering work of (A. De Luca and S. Termini, 1971), (Vlachos & Sergiadis, 2007) generalised the IFS entropy measure.Subsequently, the several authors overgeneralize the different types of entropies in the IFSs environment.Zhang & Jiang (2008) additionally enhanced (A. De Luca and S. Termini, 1971) entropy on the IFSs environment.On this concept, Verma and Sharma (2014) developed the IFSs entropy measure based on (N.R. Pal & Pal, 1989) fuzzy entropy.Afterwards, numours authors are suggested several kinds of entropy for IFSs and having the applicability with their relevance.Numerous authers were introduced the different types of IF entropy measure and they have investigated some shortcoming related to to developed study.Wei et al. (2012) and Wang et al. (2012) suggested an IF entropy measure employing the cosine function and cotangent function, respectively.
IFSs have advantages in a variety of real-world applications.Yager (2000) discusses the pattern recognination as well as logical reasoning problem.By using some different aggregations, the numerous authors are addressed the problem of MCDM problem but most of these are unabled to provide the best decisions.In the past few decades, a substantial amount of literature has proposed various decision-making strategies.The following authors (Garg, 2016;He & He, 2016;R. Verma & Sharma, 2014;Xu, 2007) are elaborated the MCDM problem in different ways with their relavent field.MCDM problem is and ranking based process, it determined by quantitative and qualitative assessment of some finite sets from some autonomous and mutual conditions.Intuitionistic fuzzy entropy measure is more effective for measuring both fuzziness and intuitionism, whereas fuzzy measures are only effective for measuring fuzziness.The similarity and divergence measures are novel paramount concept of fuzzy sets.These information measures are widely applicable for the decision making problem.The various authors (Dengfeng & Chuntian, 2002;Li et al., 2012;Liang & Shi, 2003;E. Szmidt & Kacprzyk, 2005;Eulalia Szmidt & Kacprzyk, 2001) have developed IFSs similarity and weight similarity information measures.They proposed the idea for locating uncertain information, which has been implemented as a problem involving decision-making and pattern recognition.Wang et al. (Wang & Wang, 2012) provide a comparative analysis of IFS similarity measures.In addition, Dilshad et al (Ansari et al., 2018) and Kaushik et al. (Kaushik et al., 2015) has generalized the intuitionistic fuzzy divergence measures, which have been widely used for edge detection problem.Lin (Lin, 1991) created a novel IFS divergence measure based on the difference between two IFSs.The various authors (W.L. Hung & Yang, 2008;Mao et al., 2013;Mishra et al., 2019;Srivastava & Maheshwari, 2016;Vlachos & Sergiadis, 2007;Zhang & Jiang, 2008) have extensively developed IFSs divergence measures and implemented their application in different fields.On the basis of existing literature review, the numours authors are introduced IFSs divergence measure to which deals with the study about discriminate information.IFSs divergence measures are the most significant source for MCDM problem, pattern recognition, image processing and medical diagnosis.Numerous strategies for solving MADM problems necessitate the definition of quantifiable weights for aspects.Because this evaluation of attribute weights could change the order of the options, it is important to use an aspect weighting method correctly.The weight in MADM can be categorised as either subjective or objective.Subjective weights may represent personal judgment or perception, based on personal choice knowledge of attributes provided by decision maker, questionnaires, or trade-off investigation.
In this paper, we proposed two new generalized parametric IFSs entropy measures, and these are the generalized variant of (Ye, 2010).The proposed study is associated with trigonometric functions of sine and cosine along with ± -parameter.Also, discussed the authentication and validation of the proposed study along with an illustration.It may provide a useful perspective on the decision-making problem.Researchers from around the world have developed some novel IFSs measures that offer some different ideas for solving decision-making problem.It is inferred from literature screening that the parametric approach needs to construct for finding reliable and flexible decisions from fresh perspectives to promote decision making.The study included a numerical example for analysis the multicriteria decision-making problem.It describe the decision making problem for selecting the best AC manufacturing company and in order to determine which company is the most profitable and productive.Finally, depict the graph of the proposed study and their comparison between existing measures.

Some Fe and IFe Measures
To illustrate some essential definitions corresponding to the proposed study with their relevant functions and perceptions.

Zadeh (1965): A non-empty set A
Ú is said to be fuzzy set on some universe of discourse X = { t t t t , 0 1 is the membership function, each value is between 0 and 1, i.e. 0 ≤ µ τ ( ) → + .It satisfies all following axioms of (A. De Luca and S. Termini, 1971).
The numerous authors describe fuzzy measures that correspond to probabilistic entropy measures in relation to fuzzy sets.
De Luca and Termini (1971) suggest the FSs entropy measure analogue to (Shannon, 1948) probabilistic entropy as (1) Pal & Pal (1989) introduced the exponential FSs entropy measure corresponding to itself generalized probabilistic exponential entropy as Corresponding to (Rényi, 1961) probabilistic entropy measure, (Nikhil R Pal & Bhandari, 1993) suggested the fuzzy entropy measure of α-order as Analogue to (N.R. Pal & Pal, 1989) exponential entropy measure, (Raj Mishra et al., 2016) developed the fuzzy logarithmic measures are given by Verma and Sharma (2011) developed fuzzy exponential measure corresponding to (Kvalseth, 2000) probabilistic entropy measure with order α On the basis of Verma and Sharma (2011) fuzzy entropy, Joshi and Kumar (2017) introduced a novel ideas of fuzzy entropy measure with α,β-parameters is given by Gupta et al. (2015) extended the idea of Pal and Pal (1989) from fuzzy entropy measure to fuzzy parametric entropy measure is given

Definition
Atanassov (1986): A non-empty finite set A is said to be intuitionistic fuzzy set on some universe of discourse , is given by In this study, we obtained the intuitionistic fuzzy measures are defined in the sense of  E : IFS(X) → R + .The present measure satisfies all of axioms of Szmidt and Kacprzyk (2001).It provides a valid entropy measure which could be a good perspective for the ensuing study.

Z1: 
In this present section, we will examine the existing study on parametric and non-parametric IFSs measures.The classification of the existing study is described in various forms with their relevant function.

Definition
(Set operation on IFSs): Let the family of all IFSs are denoted by IFSs(X) on the universe of discourse X.Let A, B є IFSs(X) is given by A = τ µ τ ν τ τ , , : Some ordinary set operations and relations are defined as follows ( These relations and operations are holds for each IFS.

GeNeRALIZeD PARAMeTRIC INTUITIoNISTIC FUZZy MeASUReS
In this section, we proposed two generalized parametric intuitionistic fuzzy measures on the basis of Ye (2010) effective IFSs entropy measures.The study will determine the generalization of trigonometric IFS measures.
Let the universe of discourse , have an intuitionistic fuzzy set A. Then, each IFS element provides a specific degree of membership and non-membership.Li et al. (2003) determined the relationship between converting IFSs to FSs based on the values of membership, non-membership, and hesitancy.Let A Ú represent an FS with the membership function m In this outline, two parameterized intuitionistic fuzzy measures with parameter α were obtained.The parameter can modify the structure of the present study so by introducing the parameter we obtained two new generalized parametric measure of IFSs are given by Theorem 3.1 The proposed measures,  E a 1 (A) and  E a 2 .(A) with α-parameters are valid entropy measures for IFSs.
Proof.For prove the result, it is necessary to satisfy some axioms Z1 to Z2. ( ) ( ) is minimum or zero if and only if A is crisp set.

Z1.   E A and E A
Let it suppose that, A is crisp set, then set A must have the elements either 0 or 1.

Put, µ τ ν τ
Now, it is sufficint to prove that A is a crisp set.Therefore, It is only possible when ψ τ Thus, it holds Z1.
) is maximum or one if and only if A is most fuzzy set.
First, suppose that A is most fuzzy set, i.e. m A ( where, And, Then, Thus, Equation ( 29) and (30) will holds if, ψ τ Also, differentiating again partially ( 24) and (25) w.r.t ψ τ For, 1 21 Therefore, the given function ) is concave nature and having maximum value at ψ τ Hence, proved axiom Z2 Z3.In this part, it is necessary to prove that ( 16) and ( 17) satisfy property Z3.Now, It is essential to prove that the given function.Sce, Both functions are increasing w.r.t.r and decreasing w.r.t.t.Now, Therefore, corresponding critical points of f 1 and f 2 are Similarly, Therefore, the present functions f 1 and f 2 are increasing if r ≤ t and decreasing if r ≥ t.

Z4.
Let A be an IFS and A c is its complement, It is held for representation of complement of intuitionistic fuzzy set, then the above equations ( 16) and ( 17), Hence, equations ( 16) and ( 17) are valid parametric intuitionistic fuzzy entropy measures on given parameter α.
Theorem 3.2 Consider two IFSs R and S, then Proof.Consider the universe of discourse are separately divided into two sets such that, In, X ' , Now, obtaining the equations ( 16) and ( 17), we have For 1.21 < α ≤ 1.42, adding equations ( 35) and ( 37), we have Similarly, the results will hold for X b ' .Also, for 1 21 1 42 .
. < ≤ a , adding equations ( 40) and ( 42), we have Similarly, the results will hold for X b ' .Hence, it proved the results.

NUMeRICAL ILLUSTRATIoN
, n ¼ } be the universe of discourse and R = { τ µ τ ν τ , , A A ( ) ( ) : t є X} be the IFS on X.We obtained that IFS on X, which follow as; On the universe of discourse X, the concentration of an intuitionistic fuzzy set A is represented by CON (R) & DIL(R) it followed as The following authors (Burillo & Bustince, 1996;Wen Liang Hung & Yang, 2006;Eulalia Szmidt & Kacprzyk, 2001;Zeng & Li, 2006;Zhang & Jiang, 2008) relates the study to intuitionistic fuzzy set theory.They suggested several IFSs measures that did not satisfy the general sequencing order for the corresponding sets.Furthermore, the proposed entropy measure satisfied the desire sequence of the order.The authors will classify the values of proposed and existing entropy measures to the related sets in given Table 1.
According to the above tables, we analysis that the various values of entropy occurred for various IFSs with their distinct parameter.The behaviour and reliability of the suggested and existing entropy  3 shows that no existing entropy measure achieves an excellent and valid result and all these do not satisfying the general sequencing order from their corresponding sets.However, the proposed IFSs entropy measures are producing incredibly useful and impressive results.In addition, we will discuss the dominance of proposed and existing entropy measures with the help of an illustration example.
In the present study, we obtained here the proposed entropy measures performing under their conditions.The proposed entropy measurements were found to be effective in the current study under the given circumstances.According to the above figures, fig.

THe APPLICABILITy oF PRoPoSeD MeASURe IN MCDM PRoBLeM
MCDM is a useful technique for providing a valuable decision, which conduces to choose the best substitute from the different one.In this scenario, there are some various alternatives H = { H 1 , H 2 , H 3 ,..., H m } to be studied over the distinct criteria K = { K 1 , K 2 , , K 3 ..., K n } on the IFS nature, and it figures out to detect the best alternative among them.The present study will resolve the MCDM problem with the help of the decision matrix along with some corresponding steps.
Step.1 To make a decision-making matrix First, we placed the value of the choices of the decision-maker with each alternative H i there corresponding each criterion K j in the given decision matrix as, where , m ij is the degree of H i which is desirable for K j and n ij is the degree of degree of H i which do not make it desirable for K j .Consequently, , m ij & n ij lies between 0 and 1, also Based on the values of membership & non-membership the decision matrix from (Garg et al., 2017) is obtained as: . Step.2 Normalized the decision matrix In the present study, the criteria should be distinct; if two or more criteria are identical, then we will make them distinct by converting its profit type F to expense G by normalised formulation: Therefore, the resultant decision matrix will be D m n x ( r ij ).
Step.3 Determine the total value of alternatives termine the aggregate values of an alternative H i using proposed entropy measures with the help of their concerning attribute K j is r ij having the overall value is r i .
Step.4 Determine grading of the alternatives Remark the grade of all alternatives corresponding based on their aggregated values as obtained in step 3. We can determine the most appropriable alternative for this grade value.And the highest grade rating will identify the most reliable option Step.5 Performed sensitivity analysis From this perspective, decision-makers will undertake sensitivity analysis with their respective parameters.

ILLUSTRATIVe eXAMPLe
In this study, we have discuss the MCDM problem with the help of a numerical example.Through the above discussion, we were able to categorise a broad strategy for resolving decision-making problems.There are four companies that make air conditioning (AC) for commercial purposes.All production company manufactures the different kinds of air-conditioning (AC).Here, we will compare four AC manufacturing companies in order to determine which company is the most profitable and productive.In addition, the comparative analysis of each production company is motivated by the aim to increase corporate profits in the future year.Additionally, we desire to formulate a problem for recommending the maximum yield of an A.C. manufacturing company.From the above discussion, we have taken four alternatives, which are as follows; H 1 : Voltas A.C. company; H 2 : Samsung A.C. company; H 3 : Daikin AC company; and H 4 : Hitachi AC company.According to the above present four alternatives H i ( i = 1, 2, 3, 4), for applying the intuitionistic fuzzy decision matrix, D( x , i = 1, 2, 3, 4 & j = 1, 2, 3, 4 the experts has been provided some perception values.Following is the layout of the appropriate perception values in an intuitionistic fuzzy matrix: ., . Since the introduced matrix is already in normalized form, we need not be manipulated to the given data.Therefore, all the given values of the decision matrix are in presentable order.We have apply the proposed IFSs entropy measures on the given matrix for some fixed α = 1.2.Thus, the desire values r ij ( i = 1, 2, 3, 4) relates to i th alternative, then the aggregate required values r i analogue to H i ( i .( ) = 0.6865.The occurrence of the pattern H 3 ˃ H 2 ˃ H 4 ˃ H 1 in the alternatives has led to the conclude that H 3 is the optimal choice.As a result, the AC manufacturer business H 3 making the most profit will be the most efficient, producing more products and achieving more profits.Therefore, from a customer's perspective, Hitachi will be the best AC manufacturer for commercial uses.

SeNSITIVe ANALySIS
The findings of the study, the entropy measures are more affected by given parameter α, these having a monotone nature with their respective parameter.There will be obtained the different values of alternative with respect to the different parameters.It indicates that entropy values will decrease and increase when the values of α-parameter value is concurrently increased and decreased.From a variety of choices, H 3 has the most desirable characteristics, whereas H 1 has the least.

CoMPARISoN oF DeVeLoPeD MeASURe To oTHeRS
Table .1 reveals that a number of existing IFSs entropy measures have identical decisions, as illtrated by illustrative cases (R. Verma & Sharma, 2013;Rajkumar Verma & Sharma, 2014;Vlachos & Sergiadis, 2007;Wei et al., 2012;Ye, 2010;Zhang & Jiang, 2008).The present entropy measures have satisfied the classed sequence in arranging order, thus the proposed study is more practical for deciding the decision-making problem.Furthermore, the proposed measures could turn out to be a breakthrough point in the domain of decision making.
The proposed measure also makes the same decision as existing measures, hence it determined that H 3 is the best alternative among the offered viewpoints and H 1 is the worst.From the above study, we analyse the measures that cannot have various attributes from the given alternatives; otherwise, they will be not valid.

Particular Case
In addition to the study's findings, we admitted a limitation, i.e., ± = 1 .The proposed entropy measures will turn into another entropy measures, as suggested by (Ye 2010).

DISCUSSIoN
1.The present study has been developed the idea of novel generalized parametric intuitionistic fuzzy entropy measures, it consist of two different parametric measures along with trigonometric functions.It consist of trigonometric function with some parameters that make to difference between present study and existing study.2. The authors are extended the fuzzy trigonometric measures of (Ye, 2010) in the form of parametric intuitionistic fuzzy entropy measures.The highlights of the developed parametric study have been created by the setting of membership and non-membership along with a -parameter.3. The authenticity and validity of proposed study is discuss with a numerical example and depict the classification of them.And analysis the comparison between existing study and proposed entropy measures.Depict the graph of proposed study are presented to which shows that how to behave of the present study is coincide with existing study.4. It concludes that the proposed measure could be a good perspective for decision-making problems.
Using a suitable illustration, the applicability of the proposed study has been demonstrated.

( 1 )
If A ⊑ B and B ⊑ A if and only if A = B.

Now
Figure 1.The classification of proposed entropy measure  E a 2 on corresponding sets

Figure 3 .
Figure 3.The classification of existing entropy measures on corresponding sets