Similarity Measures between Intuitionistic Fuzzy Credibility Sets and Their Multicriteria Decision-Making Method for the Performance Evaluation of Industrial Robots

To enhance the credibility level/measure of an intuitionistic fuzzy set (IFS), this study proposes the notion of an intuitionistic fuzzy credibility set (IFCS) to express the hybrid information of a pair of a membership degree and a credibility degree and a pair of a nonmembership degree and a credibility degree. Next, we propose generalized distance and similarity measures between IFCSs and then further generalize the weighted generalized distance measure of IFCSs to the trigonometric function-based similarity measures of IFCSs, including the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure of IFCSs. (en, a multicriteria decision making (MCDM) method using the proposed similarity measures is developed in the environment of IFCSs. An illustrative example about the performance evaluation of industrial robots and comparative analysis are presented to indicate the applicability and efficiency of the developed method in the setting of IFCSs.


Introduction
A similarity measure is a critical mathematical tool for judging the degree of similarity between objects and plays a key role in decision making, clustering analysis, pattern recognition, and medical diagnosis. Since a fuzzy set [1] is only depicted by a membership degree, Atanassov [2] extended the fuzzy set to an intuitionistic fuzzy set (IFS) by considering both a membership degree and a nonmembership degree, which is much more practical and flexible than the traditional fuzzy set in handling vagueness and uncertainty problems. us, various similarity measures of IFSs have been presented in the current literature and mainly applied in decision making, medical diagnosis, clustering analysis, and pattern recognition. For instance, the Hamming and Euclidean distances and similarity measures of IFSs [3][4][5] and similarity measures of IFSs based on cotangent function [6] were proposed and utilized in medical diagnosis problems. en, some similarity measures of IFSs [7][8][9][10], similarity measures of IFSs based on the Hausdorff distance [11], distance and similarity measures of IFSs [12], and cosine measures of IFSs in vector space [13] were introduced and applied in pattern recognition. Similarity measures of IFSs based on Lp metric [14], similarity measures of IFSs based on cotangent function and the Hausdorff distance [15,16], the measure of inaccuracy of IFSs [17], similarity measures of IFSs based on cosine function [18], and the generalized Dice similarity measure of IFSs [19] were put forward and used in decision making problems. e similarity measures of IFSs and the Dice measure of IFSs also were utilized for clustering analysis [20][21][22]. As the extension of IFSs, the generalized Bonferroni mean operator of fuzzy number IFSs [23] and the generalized similarity measures and aggregation operators for Pythagorean fuzzy sets (PFSs) [24,25] were used for decision making problems.
Due to the uncertain knowledge and judgments of human cognitions in complicated decision making problems, the intuitionistic fuzzy evaluation value should be closed related to its measure/degree of credibility in the vagueness and uncertain environment to enhance the credibility level/degree of the intuitionistic fuzzy value. Clearly, IFSs and PFSs lack the credibility level/measure, which implies their insufficiency in the uncertain information measure and expression. To overcome their insufficiency and enhance their credibility level, the intuitionistic fuzzy concept should be combined with the degrees/levels of credibility to effectively express the hybrid information of a pair of a membership degree and a credibility degree and a pair of a nonmembership degree and a credibility degree. Hence, we need to propose a new intuitionistic fuzzy framework combining the credibility degrees of the membership and the nonmembership so as to enhance the credibility level of IFSs. Motivated by this new intuitionistic fuzzy framework with the credibility level, this study firstly proposes an intuitionistic fuzzy credibility set (IFCS) notion to make IFS more credible and generalizes distance and similarity measures between IFCSs and then further generalizes the weighted generalized distance measure of IFCSs to the trigonometric function-based similarity measures of IFCSs, including the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure of IFCSs. Next, a multicriteria decision making (MCDM) method using the proposed similarity measures is developed in the environment of IFCSs. Finally, an illustrative example about the performance evaluation of industrial robots and comparative analysis are presented to indicate the applicability and efficiency of the developed method.
In this original study, the novelty and advantages are summarized as follows: (a) e proposed IFCS enhances the credibility level of the intuitionistic fuzzy information with the help of the credibility degrees of a membership and a nonmembership.
(b) e proposed various similarity measures of IFCSs provide effective measure tools for MCDM problems with IFCS information.
(c) As the generalization/improvement of existing intuitionistic fuzzy MCDM methods, the developed MCDM method based on the proposed similarity measures not only strengthens the credibility level/ degree of existing intuitionistic fuzzy MCDM methods but also overcomes the insufficiency of existing intuitionistic fuzzy MCDM methods. e framework of this study is arranged as follows. Section 2 proposes a new IFCS notion and operational relations of intuitionistic fuzzy credibility numbers (IFCNs). Next, generalized distance and similarity measures between IFCSs are presented, and then the weighted generalized distance measure of IFCSs is further generalized to the trigonometric function-based similarity measures of IFCSs, including the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure of IFCSs in Section 3. In Section 4, an MCDM method using the proposed similarity measures is developed in the environment of IFCSs. In Section 5, the developed MCDM method is applied to an illustrative example about the performance evaluation of industrial robots to show its applicability and efficiency in the environment of IFCSs, and then the comparative analysis is given to demonstrate the main highlights of working in the environment of IFCSs. Conclusions and future work are included in Section 6.

Intuitionistic Fuzzy Credibility Sets (IFCSs)
To depict the hybrid information of a pair of a membership degree and a credibility degree and a pair of a nonmembership degree and a credibility degree, this section proposes the definition of an IFCS as the generalization of the IFS notion.
An IFS [2] in a universe set X is denoted by are the membership degree and nonmembership degree of When we considerate the credibility levels/measures of the membership degree and the nonmembership degree in IFS, we can give the definition of an IFCS below. Definition 1. Let X be a universe set. en, an IFCS in X is defined as where (t R (x), c t (x)): X ⟶ [0, 1] 2 is a pair of the membership degree t R (x) of the element x to R and the credibility degree For the convenient representation, the component < x, (t R (x), c t (x)), (f R (x), c f (x))> in R is simply denoted as r � <(t, c t ), (f, c f )>, which is named IFCN. Definition 2. Set r 1 � <(t 1 , c t1 ), (f 1 , c f1 )> and r 2 � <(t 2 , c t2 ), (f 2 , c f2 )> as two IFCNs and η > 0. en, their operational relations are given as follows: ), (f 1k , c f1k )> and r 2k � <(t 2k , c t2k ), (f 2k , c f2k )> are two IFCNs, and let λ ≥ 1 be any integer value. en, the generalized distance measure between R 1 and R 2 is defined as follows:

Similarity Measures Based on the Generalized Distance of IFCSs
Especially when λ � 1, 2, equation (2) reduces to the Hamming and Euclidean distances: Based on the properties of distance measures [12], the generalized distance measure of IFCSs implies the following proposition.

Proposition 1.
e generalized distance measure d λ (R 1 , R 2 ) for any integer value of λ ≥ 1 contains the following properties: Proof. It is obvious that d λ (R 1 , R 2 ) can satisfy the properties (d1) -(d3). us, we only verify the property (d4).
and c f3 ≤ c f2 ≤ c f1 . Hence, there are the following inequalities: Mathematical Problems in Engineering en, we give the following results: Hence, there also exist the following results: Consequently, there are d λ (R 1 , e verification is completed. When one considers the importance of r 1k and r 2k (k � 1, 2, . . ., n), the weight of r 1k and r 2k is specified as η k ∈ [0, 1] with n k�1 η k � 1. us, the weighted generalized distance measure between R 1 and R 2 for any integer value of λ ≥ 1 is presented as follows: Clearly, equation (7) also implies the following proposition.

Proposition 2.
e weighted generalized distance measure d wλ (R 1 , R 2 ) for any integer value of λ ≥ 1 contains the following properties: Similar to the proof of Proposition 1, we can easily verify the properties (d1)-(d4) in Proposition 2, which are not repeated.

Similarity Measures Based on the Generalized Distance of IFCSs.
Since there is the complementary relationship between the distance measure and the similarity measure [12], we can introduce the similarity measures of IFCSs.

Proposition 3.
e generalized distance-based similarity measure M λ (R 1 , R 2 ) for any integer value of λ ≥ 1 contains the following properties: Obviously, by equation (8) we can easily verify the above properties (M1)-(M4) regarding the complementary relationship between the similarity measure and the generalized distance measure. Hence, the proof is straightforward.
When one considers the importance of r 1k and r 2k (k � 1, 2, . . ., n), the weight of r 1k and r 2k is specified as η k ∈ [0, 1] with n k�1 η k � 1. us, the similarity measure based on the weighted generalized distance with any integer value of λ ≥ 1 is presented as the following formula: Obviously, the similarity measure M wλ (R 1 , R 2 ) also implies the following proposition.

Proposition 4.
e similarity measure M wλ (R 1 , R 2 ) for any integer value of λ ≥ 1 contains the following properties: Obviously, by equation (9), we can easily verify the properties (M1)-(M4) in Proposition 4 regarding the complementary relationship between the similarity measure and the distance measure. us, the proof is straightforward.
When we further generalize the weighted generalized distance measure of IFCSs to the trigonometric functionbased similarity measures, we can introduce the cosine, sine, Mathematical Problems in Engineering tangent, and cotangent similarity measures based on the weighted generalized distance measure for any integer value of λ ≥ 1, respectively, as follows: Obviously, the trigonometric function-based similarity measures also indicate the following proposition.
us, we complete the proof.

Illustrative Example about the Performance Evaluation of Industrial Robots.
In this section, the developed MCDM method is applied to the performance evaluation of industrial robots to show its applicability in the environment of IFCSs.
Suppose that there are four kinds of industrial robots (four alternatives), which are denoted by their set P � {P 1 , P 2 , P 3 , P 4 }. In the evaluation of the industrial robots, the four criteria that contain the principal performances include x 1 (dynamic accuracy), x 2 (dexterity), x 3 (working ability), and x 4 (communication and control capability). en, the weight vector for the four criteria is specified as η � (0.3, 0.25, 0.2, 0.25) by experts.
e evaluation values of four kinds of industrial robots over the four criteria are provided by the IFCN form that is composed of a pair of a membership degree and a credibility degree and a pair of a nonmembership degree and a credibility degree, and then all IFCNs can be constructed as the following decision matrix A � (r jk ) 4×4 :

Mathematical Problems in Engineering
From the decision matrix A, we first give the ideal solution (the ideal IFCS): en, based on equations (9)-(13) for λ � 1, 2, we obtain the values of the proposed similarity measures and decision results, which are shown in Tables 1 and 2 for the convenient  comparison. When λ � 1, all the ranking results in Table 1 are identical and the best one is P 4 , which show the effectiveness of the developed MCDM method based on the proposed similarity measures. When λ � 2, all the ranking orders in Table 2 are identical and the best one also is also P 4 , which also show the effectiveness of the developed MCDM method based on the proposed similarity measures. en, when λ � 1, 2, the ranking results between the Hamming distance-based similarity measures and the Euclidean distance-based similarity measures indicate some difference, but the best one is till P 4 . It is obvious that different values of λ may affect the ranking results, which are dependent on the preference value of λ chosen by experts/decision makers in actual applications.

Comparative Analysis.
To conveniently compare the proposed MCDM method with the intuitionistic fuzzy MCDM method for demonstrating the importance of the credibility measure in the decision making application, we ignore all measure values of the credibility and only keep intuitionistic fuzzy values in IFCNs. In this case, the decision matrix of IFCSs in the above example is reduced to the decision matrix of IFSs: Consequently us, equations (9)- (13) in the setting of IFSs are reduced to the following similarity measures between R j ′ (j � 1, 2, 3, 4) and R ′ * : Based on equations (18)- (22), for λ � 1, 2, we obtain the values of the similarity measures between R j ′ (j � 1, 2, 3, 4) and R ′ * and decision results, which are shown in Table 3.
In Table 3, all ranking orders and the best one regarding the special case of the illustrative example in the setting of IFSs are identical. en, the ranking orders based on the proposed MCDM method using the similarity measures of IFCSs in Table 2 are different from ones based on the MCDM method using the weighted similarity measures of IFSs in Table 3 when λ � 2. Hence, the credibility level/ measure can impact on the ranking of alternatives and indicate its importance in MCDM applications.
On the other hand, various decision making methods in the environments of IFSs [3-5, 11, 12, 14], fuzzy number IFSs [23], and PFSs [24,25] cannot handle the MCDM problems with IFCS information because IFSs, number IFSs, and PFSs lack their credibility measures in general. en, the proposed MCDM method based on the similarity measures of IFCSs can be used for the MCDM problems with IFCSs and make the decision results more credible and more reasonable in decision making applications. e main advantages of working in the environment of IFCSs are reflected as follows: (1) IFCS enhances the credibility measures of the membership degree and the nonmembership degree to make IFS more credible. is means that the proposed IFCS contains much more useful information than IFS.
(2) e MCDM method using the proposed similarity measures of IFCSs in this study strengthens the credibility and rationality of decision making results and demonstrates its superiority over existing intuitionistic fuzzy MDM methods without considering their credibility measures since the credibility degrees can affect ranking results of alternatives.

Conclusion
To enhance the credibility level/measure of both a membership degree and a nonmembership degree, this paper proposed a new IFCS notion, the generalized distance of IFCSs, and the generalized distance-based similarity measures and then further proposed the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure of IFCSs. Next, an MCDM method using the proposed similarity measures is developed in the environment of IFCSs. Finally, the developed MCDM method is applied to an illustrative example about the performance evaluation of industrial robots to show its applicability and efficiency in the environment of IFCSs. By comparative analysis, the main highlights of working in the environment of IFCSs were indicated as follows: (a) In decision making process, the information expression of IFCS can enhance the credibility level of the intuitionistic fuzzy evaluation information with the help of the credibility measure (b) e proposed similarity measures of IFCSs can provide new mathematical tools for the effective   modeling of MCDM problems with IFCS information (c) e developed MCDM method based on the proposed similarity measures can strengthen the credibility of decision results and extend existing intuitionistic fuzzy MCDM methods (d) In decision making process, the credibility degree can affect the ranking of alternatives, which reflects the importance In the future work, we should further extend new similarity measures between IFCSs and utilize them in pattern recognition, image processing, clustering analysis, medical diagnosis, and so on in the environment of IFCSs.

Data Availability
ere is no data used to support this study.