Some New Trigonometric α-Order Fuzzy Entropies

In this paper, new α-order trigonometric and inverse trigonometric fuzzy entropies are proposed and the fuzzy entropy axiomatic requirements are satisfied for the new fuzzy entropies. A comparison of the new fuzzy entropies is done with several widely used fuzzy entropies in order to find the most fuzziness entropy. The results indicate that the new proposed α-order fuzzy entropy provides larger entropy value than those other fuzzy entropies which were defined in the paper.


Introduction
The fuzzy sets firstly defined by Zadeh (1965) to use for modeling non-statistical and vague information data (Chatterjee et al., 2017;Campion et al., 2018). Recently, the fuzzy sets become an interesting research topics in the applied sciences, e.g. engineering, image processing, medical sciences and so forth which are involving in using vague information (Fan and May, 2002;Hooda and Mishra, 2015;Dos et al., 2016). The fuzzy entropy plays an important role in the fuzzy set theory because it is a fuzziness measure of the fuzzy sets. Some authors used the fuzzy entropy in statistical inference, e.g. estimation and testing hypothesis concepts ( Zadeh (1968) proposed a fuzzy entropy based on the probabilistic framework. However, the fuzzy entropy based on the concept of the Shannon statistical entropy (Shannon, 1948) is defined by De Luca and Termini (1972). They provided an axiomatic framework to define new fuzzy entropy measures (FEM). Several studies employed these axioms to propose new fuzzy entropy. Bhandari and Pal (1993) provided a α-order type of FEM. There exists some new fuzzy αorder entropies proposed by Kapur (1997), Fan and Ma (2002) and Al-Talib and Al-Nasser (2018).
Fuzzy entropies based on the trigonometric functions are also considered by several authors e.g., Parkash et al. (2008) and Verma (2015). Hooda and Mishra (2015) defined several trigonometric fuzzy entropies based on sinus and cosine functions and along with studying their properties. In this paper, we generalized some α-order trigonometric fuzzy entropies and studied their properties. We found out the measure of the new α-order inverse trigonometric fuzzy entropy, based on arcsin function, is larger than that of the α-order fuzzy entropies and αorder trigonometric entropies for different values of α. This paper is organized as follows. In Section 2, some concepts and axioms of the fuzzy entropy and some fuzzy entropy measures, which are proposed by authors, are presented. We proposed several new α-order trigonometric and inverse trigonometric entropies in Section 3. The measurement comparison of the new fuzzy entropies with some widely used fuzzy entropies are done in Section 4 and Section 5 includes some conclusions.

Fuzzy Entropy Axioms
In this section, some concepts and axioms of the fuzzy entropy are presented. Let denote the fuzzy membership function of the finite set = { ; = 1,2, … , } where ( ) is the value of the membership function of the element form and, the fuzzy entropy of is denoted by ( ). Using the Shannon probabilistic entropy (1948), De Luca and Termini (1972) defined the fuzzy entropy for some constant equal to 1 ⁄ . They also provided following axiomatic requirements that are satisfied to propose new fuzzy entropy measure, as well. A fuzzy set * is called a sharpened version of the fuzzy set (or crisper than ) if for ∀ , ( ) * ≤ ( ) for 0 ≤ ( ) ≤ 0.5 and ( ) ≤ Furthermore, Kapur [11] proposed following -order FEM ]; ≠ 1, > 0. Some fuzzy entropy measures are suggested by authors, e.g., Al-Talib and Al-Nasser (2018) defined a -order entropy measure Some trigonometric fuzzy entropies are also proposed by Hood and Mishra (2015) as follows A -order inverse trigonometric fuzzy entropy is also defined Some authors proposed fuzzy entropy measures to be applicable in special cases, e.g., Hu and Yu (2004) and Gupta and Sheoran (2014).

New -order Trigonometric FEM
From mathematical point of views, trigonometric measures have important properties in the modeling of geometry applications. In this section, some new -order fuzzy entropy measures based on trigonometric functions are proposed as follows. We have 2 2 > 2 for ∀ > 2 and thus 2 1− 2 < 1. Therefore, the denominator of ( ) is negative. Moreover, However, 2 2 < 2 for ∀ < 2 and then 2 1− 2 < 1 and the ( ) denominator is positive.  Now, consider the second derivative of (1) as follows.
Thus, the FEM(1) attends its maximum at ( ) = 0.5. Figure 1 presents the graph of the first and second derivative of ( ) for different values of . As it is noticed from Figure  1, the second derivative of ( ) is negative at ( ) = 0.5. Therefore, the maximum value of ( ) happens at ( ) = 0.5 which is equal to 1. The fuzzy entropy (1) for various values of the membership ( ) for =0.2, 0.5, 1, 4 and 10 is given in Table 1. By taking a close look in Table 1, the FEM ( ) is increasing function of ( ) in the interval [0,0.5] and decreasing function in the interval [0.5,1] for diversity values of . Figure 2 and Table 1 indicate that the fuzzy entropy measure ( ) is a decreasing function of for values less than 4.09 and is an increasing function of for values larger than 4.09. Table 1. The value of ( ) for = . , . , , . : Plots of ( ) as a function of for different values of ( ) (left) and as a function of ( ) for different values of (right).

Resolutions:
As it is presented in the maximality axiom, the fuzzy entropy ( ) has a unique maximum at ( ) = 0.5 and thus, ( ) is a continuous concave function. Hence, ( ) monotonically increases f or ( ) ∈ [0,0.5] and monotonically decreases for ( ) ∈ [0.5,1]. Therefore, is a sharpened version of ( ) . Figure 2 illustrates the graph of ( ) for different values of . Therefore, we see that the maximum of FEM(2) will attend at ( ) = 0.5. For different values of , the graph of the first and second derivative of ( ) are presented in Figure  3. As Figure 3 indicates, the second derivative of ( ) for values =0.2, 0.5, 1, 4 and 10 is negative. Therefore, the maximum value of ( ) will be attended at ( ) = 0.5 which is equal to 1. Table 2 presents the numerical results of the measure (2) for different membership ( ) values by considering =0.2, 0.5, 1, 4 and 10. We can see form Table  2, the entropy measure (2) rises as a function of for every membership function μ A(x i ) . However, FEM H α CS ( ) increases for 0 ≤ μ A(x i ) ≤ 0.5 and declines for 0.5 ≤ μ A(x i ) ≤ 1 for any given α=0.2, 0.5, 1, 4 and 10.   denotes the sharpened version of ( ) . Figure 4 presents the graph of ( ) for different values of ( ) . Symmetry: By substituting 1 − ( ) instead of ( ) in Equation (2), we get (1 − ) = ( ).
Therefore, four requirement axioms are satisfied for FEM(2) and theorem 3.2 is proven. Now, the four measurement requirement axioms are to be investigated for FEM (3). We have 1 > 2 −1 for ∀ > 1 because, 2 > 2 then,  for different values of is presented in Figure 5.
Furthermore, the second derivative of (3.3) is given by Therefore, Thus, we can see the maximum value of the fuzzy entropy (3) is attended in ( ) = 0.5. Figure 5 indicates that the derivative of ( ) with respect to ( ) at value μ A(x i ) = 0.5 is negative for different values of . The maximum value of ( ) at ( ) = 0.5 is equal to 1. From Table 3 we found that the value of entropy =1.5 ( ) is increasing as increasing for 2.5 to 10 for the same membership value. However, the proposed entropy measure =1.5 ( ) as function of ( ) increases in the interval [0,0.5] and declines in the interval [0.5,1] at any given .  Figure 6 indicates, as a function of , the fuzzy entropy =1.5 ( ) has a unique minimum at = 3 for different values of the member function ( ) .  Symmetry: If we substitute 1 − ( ) instead of ( ) in FEM (3), we have (1 − ) = ( ). Therefore, the proof of theorem 3 is achieved.

A comparative measure of FEMs
In this section, the performance of the proposed fuzzy entropy measures is studied by a numerical comparison between the proposed fuzzy entropy measures and several defined measures which are defined by authors. Firstly, this comparison is done through the trigonometry fuzzy entropy. In the second step, the comparison is done with other fuzzy entropies which are defined by De Luca and Termini (1972), Bhandari and Pal (1993), Kapur (1997) and Al-Talib and Al-Naseer (2018). The results of the comparisons are given in Tables 4 and 5, respectively. Table 4 indicates that ( ) produces the greatest entropy measure through other trigonometry fuzzy entropies for values 1.5, 2.5, 4 and 10. Therefore, ( ) is most informative than that other trigonometry fuzzy entropies for different values. The results in Table 4 also indicate the following relationship  Table 5, the values of FEM ( ) are greater than the fuzzy entropy measures of other fuzzy entropies for = 1.5, 2.5, 4 and 10. Thus, ( ) provides the most informative fuzzy entropy compare with other fuzzy entropies that are presented in Table 5 for values of equal to 1.5, 2.5, 4 and 10. Table 5

Conclusions
In this paper, three new trigonometric fuzzy entropies of order α are proposed. We found that the four axiomatic requirements properties are satisfied with the new fuzzy entropies. The results of the preceding sections confirm that the FEM , α > 1 produces the greatest entropy value not only through some proposed α-order trigonometric fuzzy entropies but also, it produces the highest entropy value through some non-trigonometric fuzzy entropies which were suggested in previous studies. Another possible topic for future research is to use entropy of order in the fuzzy setting for multi criteria decision making problems (Adel Rastkhiz, 2019) which has an application in evaluating entrepreneurial opportunities.