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

Authors: Jun Ye; Shigui Du; Rui Yong
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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. Then, 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.


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## Body

## 1. 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. Thus, 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 [3–5] and similarity measures of IFSs based on cotangent function [6] were proposed and utilized in medical diagnosis problems. Then, some similarity measures of IFSs [7–10], 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. The similarity measures of IFSs and the Dice measure of IFSs also were utilized for clustering analysis [20–22]. 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)
The 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)
The 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.The framework of this study is arranged as follows. Section2 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.


## 2. 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 B = {<x, tB(x), fB(x)>|x ∈ X}, where tB(x): X ⟶ [0, 1] and fB(x): X ⟶ [0, 1] are the membership degree and nonmembership degree of the element x to B with 0 ≤ tB(x) + fB(x) ≤ 1 for x ∈ X.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.
LetX be a universe set. Then, an IFCS in X is defined as(1)R=x,tRx,ctx,fRx,cfx|x∈X,where (tR(x), ct(x)): X ⟶ [0, 1]2 is a pair of the membership degree tR(x) of the element x to R and the credibility degree ct(x) to tR(x) and (fR(x), cf(x)): X ⟶ [0, 1]2 is a pair of the nonmembership degree fR(x) of the element x to R and the credibility degree cf(x) to fR(x) for x ∈ X, along with the condition tR(x), ct(x), tf(x), cf(x) ∈ [0, 1] and 0 ≤ tR(x) + fR(x) ≤ 1.
For the convenient representation, the component <x, (tR(x), ct(x)), (fR(x), cf(x))> in R is simply denoted as r = <(t, ct), (f, cf)>, which is named IFCN.Definition 2.
Setr1 = <(t1, ct1), (f1, cf1)> and r2 = <(t2, ct2), (f2, cf2)> as two IFCNs and η > 0. Then, their operational relations are given as follows:(1)
r1 ⊇ r2 ⇔ t1 ≥ t2, ct1 ≥ ct2, f1 ≤ f2, and cf1 ≤ cf2(2)
r1 = r2 ⇔ r1 ⊇ r2 and r2 ⊇ r1(3)
r1 ∪ r2 = <(t1 ∨ t2, ct1 ∨ ct2), (f1 ∧ f2, cf1 ∧ cf2)>(4)
r1 ∩ r2 = <(t1 ∧ t2, ct1 ∧ ct2), (f1 ∨ f2, cf1 ∨ cf2)>(5)
(r1)c = <(f1, cf1), (t1, ct1)> (the complement of r1)(6)
r1 ⊕ r2 = <(t1 + t2 − t1t2, ct1 + ct2 − ct1ct2), (f1f2, cf1cf2)>(7)
r1 ⊗ r2 = <(t1t2, ct1ct2), (f1 + f2 − f1f2, cf1 + cf2 − cf1cf2)>(8)
ηr1 = <(1 − (1 − t1)η, 1 − (1 − ct1)η), ((f1)η, (cf1)η)>(9)
(r1)η = <((t1)η, (ct1)η), (1 − (1 − f1)η, 1 − (1 − cf1)η)>


## 3. Similarity Measures Based on the Generalized Distance of IFCSs

### 3.1. Generalized Distance Measure of IFCSs

For two IFCSs, we give the definition of the generalized distance measure of IFCSs below.Definition 3.
SetR1 = {r11, r12, …, r1n} and R2 = {r21, r22, …, r2n} as two IFCSs, where r1k = <(t1k, ct1k), (f1k, cf1k)> and r2k = <(t2k, ct2k), (f2k, cf2k)> are two IFCNs, and let λ ≥ 1 be any integer value. Then, the generalized distance measure between R1 and R2 is defined as follows:(2)dλR1,R2=1212n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ+12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ.
Especially whenλ = 1, 2, equation (2) reduces to the Hamming and Euclidean distances:(3)d1R1,R2=14n∑k=1nt1k−t2k+f1k−f2k+14n∑k=1nct1k−ct2k+cf1k−cf2k,d2R1,R2=1212n∑k=1nt1k−t2k2+f1k−f2k21/2+12n∑k=1nct1k−ct2k2+cf1k−cf2k21/2.
Based on the properties of distance measures [12], the generalized distance measure of IFCSs implies the following proposition.Proposition 1.
The generalized distance measuredλ(R1, R2) for any integer value of λ ≥ 1 contains the following properties:(d1)dλ(R1, R2) ∈ [0, 1](d2)dλ(R1, R2) = 0 iff R1 = R2(d3)dλ(R1, R2) = dλ(R2, R1)(d4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then dλ(R1, R3) ≥ dλ(R1, R2) and dλ(R1, R3) ≥ dλ(R2, R3)Proof.
It is obvious thatdλ(R1, R2) can satisfy the properties (d1) – (d3). Thus, we only verify the property (d4).
IfR1 ⊆ R2 ⊆ R3, then t3 ≥ t2 ≥ t1, ct3 ≥ ct2 ≥ ct1, f3 ≤ f2 ≤ f1, and cf3 ≤ cf2 ≤ cf1. Hence, there are the following inequalities:(4)t1k−t2kλ≤t1k−t3kλ,ct1k−ct2kλ≤ct1k−ct3kλ,f1k−f2kλ≤f1k−f3kλ,cf1k−cf2kλ≤cf1k−cf3kλ,t2k−t3kλ≤t1k−t3kλ,ct2k−ct3kλ≤ct1k−ct3kλ,f2k−f3kλ≤f1k−f3kλ,cf2k−cf3kλ≤cf1k−cf3kλ.
Then, we give the following results:(5)12n∑k=1nt1k−t2kλ+f1k−f2kλ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ,12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ,12n∑k=1nt2k−t3kλ+f2k−f3kλ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ,12n∑k=1nct2k−ct3kλ+cf2k−cf3kλ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ.
Hence, there also exist the following results:(6)12n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ1/λ,12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ1/λ,12n∑k=1nt2k−t3kλ+f2k−f3kλ1/λ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ1/λ,12n∑k=1nct2k−ct3kλ+cf2k−cf3kλ1/λ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ1/λ.
Consequently, there aredλ(R1, R3) ≥ dλ(R1, R2) and dλ(R1, R3) ≥ dλ(R2, R3).
The verification is completed.
When one considers the importance ofr1k and r2k (k = 1, 2, …, n), the weight of r1k and r2k is specified as ηk ∈ [0, 1] with ∑k=1nηk=1. Thus, the weighted generalized distance measure between R1 and R2 for any integer value of λ ≥ 1 is presented as follows:(7)dwλR1,R2=1212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.
Clearly, equation (7) also implies the following proposition.Proposition 2.
The weighted generalized distance measuredwλ(R1, R2) for any integer value of λ ≥ 1 contains the following properties:(d1)dwλ(R1, R2) ∈ [0, 1](d2)dwλ(R1, R2) = 0 iff R1 = R2(d3)dwλ(R1, R2) = dwλ(R2, R1)(d4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then dwλ(R1, R3) ≥ dwλ(R1, R2) and dwλ(R1, R3) ≥ dwλ(R2, R3)
Similar to the proof of Proposition1, we can easily verify the properties (d1)–(d4) in Proposition 2, which are not repeated.


### 3.2. 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.SetR1 = {r11, r12, …, r1n} and R2 = {r21, r22, …, r2n} as two IFCSs, where r1k = <(t1k, ct1k), (f1k, cf1k)> and r2k = <(t2k, ct2k), (f2k, cf2k)> are two IFCNs, and let λ ≥ 1 be any integer value. Then, we propose the similarity measure based on the generalized distance measure between R1 and R2:(8)MλR1,R2=1−dλR1,R2=1−1212n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ+12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ.Corresponding to the complementary relationship between the distance measure and the similarity measure, the generalized distance-based similarity measureMλ(R1, R2) implies the following proposition.Proposition 3.
The generalized distance-based similarity measureMλ(R1, R2) for any integer value of λ≥1 contains the following properties:(M1)Mλ(R1, R2) ∈ [0, 1](M2)Mλ(R1, R2) = 1 iff R1 = R2(M3)Mλ(R1, R2) = Mλ(R2, R1)(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then Mλ(R1, R3) ≤ Mλ(R1, R2) and Mλ(R1, R3) ≤ Mλ(R2, R3)
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 ofr1k and r2k (k = 1, 2, …, n), the weight of r1k and r2k is specified as ηk ∈ [0, 1] with ∑k=1nηk=1. Thus, the similarity measure based on the weighted generalized distance with any integer value of λ ≥ 1 is presented as the following formula:(9)MwλR1,R2=1−dwλR1,R2=1−1212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.Obviously, the similarity measureMwλR1,R2 also implies the following proposition.Proposition 4.
The similarity measureMwλR1,R2 for any integer value of λ ≥ 1 contains the following properties:(M1)MwλR1,R2 ∈ [0, 1](M2)MwλR1,R2  = 1 iff R1 = R2(M3)MwλR1,R2  =  MwλR2,R1(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then MwλR1,R3 ≤ MwλR1,R2 and MwλR1,R3 ≤ MwλR2,R3
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. Thus, the proof is straightforward.
When we further generalize the weighted generalized distance measure of IFCSs to the trigonometric function-based similarity measures, we can introduce the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure for any integer value ofλ ≥ 1, respectively, as follows:(10)CwλR1,R2=cosπ2dwλR1,R2=cosπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(11)SwλR1,R2=1−sinπ2dwλR1,R2=1−sinπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(12)TwλR1,R2=1−tanπ4dwλR1,R2=1−tanπ812∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(13)CTwλR1,R2=cotπ4+π4dwλR1,R2=cotπ4+π812∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.
Obviously, the trigonometric function-based similarity measures also indicate the following proposition.Proposition 5.
The trigonometric function-based similarity measuresCwλ(R1, R2), Swλ(R1, R2), Twλ(R1, R2), and CTwλ(R1, R2) for any integer value of λ ≥ 1 contain the following properties:(M1)Cwλ(R1, R2) ∈ [0, 1], Swλ(R1, R2) ∈ [0, 1], Twλ(R1, R2) ∈ [0, 1], and CTwλ(R1, R2) ∈ [0, 1](M2)Cwλ(R1, R2) = Swλ(R1, R2) = Twλ(R1, R2) = CTwλ(R1, R2) = 1 iff R1 = R2(M3)Cwλ(R1, R2) = Cwλ(R2, R1), Swλ(R1, R2) = Swλ(R2, R1), Twλ(R1, R2) = Twλ(R2, R1), and CTwλ(R1, R2) = CTwλ(R2, R1)(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then Cwλ(R1, R3) ≤ Cwλ(R1, R2) and Cwλ(R1, R3) ≤ Cwλ(R2, R3); Swλ(R1, R3) ≤ Swλ(R1, R2) and Swλ(R1, R3) ≤ Mwλ(R2, R3); Twλ(R1, R3) ≤ Twλ(R1, R2) and Twλ(R1, R3) ≤ Twλ(R2, R3); and CTwλ(R1, R3) ≤ CTwλ(R1, R2) and CTwλ(R1, R3) ≤ CTwλ(R2, R3)Proof.
It is obvious thatCwλ(R1, R2), Swλ(R1, R2), Twλ(R1, R2), and CTwλ(R1, R2) can satisfy the properties (M1)–(M3). Thus, we only verify the property (M4).
Since cos(x) for x ∈ [0, π/2] is a decreasing function with 1 ≥ cos(x) ≥ 0, there are Cwλ(R1, R3) ≤ Cwλ(R1, R2) and Cwλ(R1, R3) ≤ Cwλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (10).
Since sin(x) for x ∈ [0, π/2] is an increasing function with 0 ≤ sin(x) ≤ 1, there are Swλ(R1, R3) ≤ Swλ(R1, R2) and Swλ(R1, R3) ≤ Swλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (11).
Since tan(x) for x ∈ [0, π/4] is an increasing function with 0 ≤ tan(x) ≤ 1, there are Twλ(R1, R3) ≤ Twλ(R1, R2) and Twλ(R1, R3) ≤ Twλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (12).
Since cot(x) for x ∈ [π/4, π/2] is a decreasing function with 1 ≥ cot(x) ≥ 0, there are CTwλ(R1, R3) ≤ CTwλ(R1, R2) and CTwλ(R1, R3) ≤ CTwλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (13).
Thus, we complete the proof.


## 3.1. Generalized Distance Measure of IFCSs

For two IFCSs, we give the definition of the generalized distance measure of IFCSs below.Definition 3.
SetR1 = {r11, r12, …, r1n} and R2 = {r21, r22, …, r2n} as two IFCSs, where r1k = <(t1k, ct1k), (f1k, cf1k)> and r2k = <(t2k, ct2k), (f2k, cf2k)> are two IFCNs, and let λ ≥ 1 be any integer value. Then, the generalized distance measure between R1 and R2 is defined as follows:(2)dλR1,R2=1212n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ+12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ.
Especially whenλ = 1, 2, equation (2) reduces to the Hamming and Euclidean distances:(3)d1R1,R2=14n∑k=1nt1k−t2k+f1k−f2k+14n∑k=1nct1k−ct2k+cf1k−cf2k,d2R1,R2=1212n∑k=1nt1k−t2k2+f1k−f2k21/2+12n∑k=1nct1k−ct2k2+cf1k−cf2k21/2.
Based on the properties of distance measures [12], the generalized distance measure of IFCSs implies the following proposition.Proposition 1.
The generalized distance measuredλ(R1, R2) for any integer value of λ ≥ 1 contains the following properties:(d1)dλ(R1, R2) ∈ [0, 1](d2)dλ(R1, R2) = 0 iff R1 = R2(d3)dλ(R1, R2) = dλ(R2, R1)(d4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then dλ(R1, R3) ≥ dλ(R1, R2) and dλ(R1, R3) ≥ dλ(R2, R3)Proof.
It is obvious thatdλ(R1, R2) can satisfy the properties (d1) – (d3). Thus, we only verify the property (d4).
IfR1 ⊆ R2 ⊆ R3, then t3 ≥ t2 ≥ t1, ct3 ≥ ct2 ≥ ct1, f3 ≤ f2 ≤ f1, and cf3 ≤ cf2 ≤ cf1. Hence, there are the following inequalities:(4)t1k−t2kλ≤t1k−t3kλ,ct1k−ct2kλ≤ct1k−ct3kλ,f1k−f2kλ≤f1k−f3kλ,cf1k−cf2kλ≤cf1k−cf3kλ,t2k−t3kλ≤t1k−t3kλ,ct2k−ct3kλ≤ct1k−ct3kλ,f2k−f3kλ≤f1k−f3kλ,cf2k−cf3kλ≤cf1k−cf3kλ.
Then, we give the following results:(5)12n∑k=1nt1k−t2kλ+f1k−f2kλ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ,12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ,12n∑k=1nt2k−t3kλ+f2k−f3kλ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ,12n∑k=1nct2k−ct3kλ+cf2k−cf3kλ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ.
Hence, there also exist the following results:(6)12n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ1/λ,12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ1/λ,12n∑k=1nt2k−t3kλ+f2k−f3kλ1/λ≤12n∑k=1nt1k−t3kλ+f1k−f3kλ1/λ,12n∑k=1nct2k−ct3kλ+cf2k−cf3kλ1/λ≤12n∑k=1nct1k−ct3kλ+cf1k−cf3kλ1/λ.
Consequently, there aredλ(R1, R3) ≥ dλ(R1, R2) and dλ(R1, R3) ≥ dλ(R2, R3).
The verification is completed.
When one considers the importance ofr1k and r2k (k = 1, 2, …, n), the weight of r1k and r2k is specified as ηk ∈ [0, 1] with ∑k=1nηk=1. Thus, the weighted generalized distance measure between R1 and R2 for any integer value of λ ≥ 1 is presented as follows:(7)dwλR1,R2=1212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.
Clearly, equation (7) also implies the following proposition.Proposition 2.
The weighted generalized distance measuredwλ(R1, R2) for any integer value of λ ≥ 1 contains the following properties:(d1)dwλ(R1, R2) ∈ [0, 1](d2)dwλ(R1, R2) = 0 iff R1 = R2(d3)dwλ(R1, R2) = dwλ(R2, R1)(d4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then dwλ(R1, R3) ≥ dwλ(R1, R2) and dwλ(R1, R3) ≥ dwλ(R2, R3)
Similar to the proof of Proposition1, we can easily verify the properties (d1)–(d4) in Proposition 2, which are not repeated.


## 3.2. 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.SetR1 = {r11, r12, …, r1n} and R2 = {r21, r22, …, r2n} as two IFCSs, where r1k = <(t1k, ct1k), (f1k, cf1k)> and r2k = <(t2k, ct2k), (f2k, cf2k)> are two IFCNs, and let λ ≥ 1 be any integer value. Then, we propose the similarity measure based on the generalized distance measure between R1 and R2:(8)MλR1,R2=1−dλR1,R2=1−1212n∑k=1nt1k−t2kλ+f1k−f2kλ1/λ+12n∑k=1nct1k−ct2kλ+cf1k−cf2kλ1/λ.Corresponding to the complementary relationship between the distance measure and the similarity measure, the generalized distance-based similarity measureMλ(R1, R2) implies the following proposition.Proposition 3.
The generalized distance-based similarity measureMλ(R1, R2) for any integer value of λ≥1 contains the following properties:(M1)Mλ(R1, R2) ∈ [0, 1](M2)Mλ(R1, R2) = 1 iff R1 = R2(M3)Mλ(R1, R2) = Mλ(R2, R1)(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then Mλ(R1, R3) ≤ Mλ(R1, R2) and Mλ(R1, R3) ≤ Mλ(R2, R3)
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 ofr1k and r2k (k = 1, 2, …, n), the weight of r1k and r2k is specified as ηk ∈ [0, 1] with ∑k=1nηk=1. Thus, the similarity measure based on the weighted generalized distance with any integer value of λ ≥ 1 is presented as the following formula:(9)MwλR1,R2=1−dwλR1,R2=1−1212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.Obviously, the similarity measureMwλR1,R2 also implies the following proposition.Proposition 4.
The similarity measureMwλR1,R2 for any integer value of λ ≥ 1 contains the following properties:(M1)MwλR1,R2 ∈ [0, 1](M2)MwλR1,R2  = 1 iff R1 = R2(M3)MwλR1,R2  =  MwλR2,R1(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then MwλR1,R3 ≤ MwλR1,R2 and MwλR1,R3 ≤ MwλR2,R3
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. Thus, the proof is straightforward.
When we further generalize the weighted generalized distance measure of IFCSs to the trigonometric function-based similarity measures, we can introduce the cosine, sine, tangent, and cotangent similarity measures based on the weighted generalized distance measure for any integer value ofλ ≥ 1, respectively, as follows:(10)CwλR1,R2=cosπ2dwλR1,R2=cosπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(11)SwλR1,R2=1−sinπ2dwλR1,R2=1−sinπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(12)TwλR1,R2=1−tanπ4dwλR1,R2=1−tanπ812∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ,(13)CTwλR1,R2=cotπ4+π4dwλR1,R2=cotπ4+π812∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ+12∑k=1nηkct1k−ct2kλ+cf1k−cf2kλ1/λ.
Obviously, the trigonometric function-based similarity measures also indicate the following proposition.Proposition 5.
The trigonometric function-based similarity measuresCwλ(R1, R2), Swλ(R1, R2), Twλ(R1, R2), and CTwλ(R1, R2) for any integer value of λ ≥ 1 contain the following properties:(M1)Cwλ(R1, R2) ∈ [0, 1], Swλ(R1, R2) ∈ [0, 1], Twλ(R1, R2) ∈ [0, 1], and CTwλ(R1, R2) ∈ [0, 1](M2)Cwλ(R1, R2) = Swλ(R1, R2) = Twλ(R1, R2) = CTwλ(R1, R2) = 1 iff R1 = R2(M3)Cwλ(R1, R2) = Cwλ(R2, R1), Swλ(R1, R2) = Swλ(R2, R1), Twλ(R1, R2) = Twλ(R2, R1), and CTwλ(R1, R2) = CTwλ(R2, R1)(M4) IfR1 ⊆ R2 ⊆ R3 for an IFCS R3, then Cwλ(R1, R3) ≤ Cwλ(R1, R2) and Cwλ(R1, R3) ≤ Cwλ(R2, R3); Swλ(R1, R3) ≤ Swλ(R1, R2) and Swλ(R1, R3) ≤ Mwλ(R2, R3); Twλ(R1, R3) ≤ Twλ(R1, R2) and Twλ(R1, R3) ≤ Twλ(R2, R3); and CTwλ(R1, R3) ≤ CTwλ(R1, R2) and CTwλ(R1, R3) ≤ CTwλ(R2, R3)Proof.
It is obvious thatCwλ(R1, R2), Swλ(R1, R2), Twλ(R1, R2), and CTwλ(R1, R2) can satisfy the properties (M1)–(M3). Thus, we only verify the property (M4).
Since cos(x) for x ∈ [0, π/2] is a decreasing function with 1 ≥ cos(x) ≥ 0, there are Cwλ(R1, R3) ≤ Cwλ(R1, R2) and Cwλ(R1, R3) ≤ Cwλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (10).
Since sin(x) for x ∈ [0, π/2] is an increasing function with 0 ≤ sin(x) ≤ 1, there are Swλ(R1, R3) ≤ Swλ(R1, R2) and Swλ(R1, R3) ≤ Swλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (11).
Since tan(x) for x ∈ [0, π/4] is an increasing function with 0 ≤ tan(x) ≤ 1, there are Twλ(R1, R3) ≤ Twλ(R1, R2) and Twλ(R1, R3) ≤ Twλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (12).
Since cot(x) for x ∈ [π/4, π/2] is a decreasing function with 1 ≥ cot(x) ≥ 0, there are CTwλ(R1, R3) ≤ CTwλ(R1, R2) and CTwλ(R1, R3) ≤ CTwλ(R2, R3) based on dwλ(R1, R3) ≥ dwλ(R1, R2), dwλ(R1, R3) ≥ dwλ(R2, R3), and equation (13).
Thus, we complete the proof.


## 4. MCDM Method Based on the Proposed Similarity Measures of IFCSs

In this section, an MCDM method is developed by using the proposed similarity measures of IFCSs to choose the best alternative in a set of finite alternatives.Assume the collection ofm alternatives be denoted by P = {P1, P2, …, Pm} in an MCDM problem, which is assessed by the set of n criteria X = {x1, x2, …, xn}, along with the specific weight vector η = (η1, η2, …, ηn). Then, the decision makers give the evaluation value of each alternative over the criteria 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. Then, all IFCNs can be constructed as their decision matrix A = (rjk)m×n, where rjk = <(tjk, ctjk), (fjk, cfjk)> (j = 1, 2, …, m; k = 1, 2, …, n) are IFCNs and forms the IFCS Rj = {rj1, rj2, …, rjn} for Pj.Thus, the MCDM algorithm using one of the proposed similarity measures is demonstrated as follows:Step 1. From the decision matrix A = (rjk)m×n, we obtain the ideal solution (the ideal IFCS) by the following form:(14)R∗=t1∗,ct1∗,f1∗,cf1∗,t2∗,ct2∗,f2∗,cf2∗,…,tn∗,ctn∗,fn∗,cfn∗maxjtj1,maxjctj1,minjfj1,minjcfj1,maxjtj2,maxjctj2,minjfj2,minjcfj2,…,maxjtjn,maxjctjn,minjfjn,minjcfjnStep 2. Based on one of equations (9)–(13), we obtain the values of similarity measures between Rj andStep 3. The alternatives are ranked, and the best one is chosenStep 4. End


## 5. Illustrative Example and Comparative Analysis

### 5.1. 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 setP = {P1, P2, P3, P4}. In the evaluation of the industrial robots, the four criteria that contain the principal performances include x1 (dynamic accuracy), x2 (dexterity), x3 (working ability), and x4 (communication and control capability). Then, the weight vector for the four criteria is specified as η = (0.3, 0.25, 0.2, 0.25) by experts. The 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 = (rjk)4×4:(15)A=R1R2R3R4=0.65,0.85,0.3,0.70.72,0.8,0.25,0.750.6,0.78,0.35,0.70.6,0.88,0.3,0.760.7,0.7,0.2,0.80.65,0.7,0.3,0.850.86,0.8,0.1,0.80.8,0.8,0.1,0.80.6,0.8,0.3,0.850.75,0.8,0.2,0.70.85,0.7,0.12,0.750.7,0.8,0.2,0.860.8,0.8,0.1,0.80.9,0.8,0.1,0.80.7,0.8,0.25,0.80.7,0.75,0.2,0.8.From the decision matrixA, we first give the ideal solution (the ideal IFCS):(16)R∗=0.8,0.85,0.1,0.7,0.9,0.8,0.1,0.7,0.86,0.8,0.1,0.7,0.8,0.88,0.1,0.76.Then, 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.Table 1
The values of the proposed similarity measures and decision results forλ = 1.
Similarity measureMeasure valueRankingThe best oneMw1Rj,R∗0.8985, 0.9100, 0.9066, 0.9389P4 > P2 > P3 > P1P4Cw1Rj,R∗0.9873, 0.9900, 0.9893, 0.9954P4 > P2 > P3 > P1P4Sw1Rj,R∗0.8412, 0.8591, 0.8539, 0.9041P4 > P2 > P3 > P1P4Tw1Rj,R∗0.9201, 0.9292, 0.9265, 0.9520P4 > P2 > P3 > P1P4CTw1Rj,R∗0.8520, 0.8678, 0.8631, 0.9083P4 > P2 > P3 > P1P4Table 2
The values of the proposed similarity measures and decision results forλ = 2.
Similarity measureMeasure valueRankingThe best oneMw2Rj,R∗0.8916, 0.8848, 0.8898, 0.9171P4 > P1 > P3 > P2P4Cw2Rj,R∗0.9855, 0.9837, 0.9851, 0.9915P4 > P1 > P3 > P2P4Sw2Rj,R∗0.8305, 0.8201, 0.8278, 0.8702P4 > P1 > P3 > P2P4Tw2Rj,R∗0.9146, 0.9093, 0.9132, 0.9348P4 > P1 > P3 > P2P4CTw2Rj,R∗0.8427, 0.8337, 0.8403, 0.8776P4 > P1 > P3 > P2P4Whenλ = 1, all the ranking results in Table 1 are identical and the best one is P4, 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 P4, which also show the effectiveness of the developed MCDM method based on the proposed similarity measures. Then, 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 P4. 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.


### 5.2. 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:(17)A′=R1′R2′R3′R4′=0.65,0.30.72,0.250.6,0.350.6,0.30.7,0.20.65,0.30.86,0.10.8,0.10.6,0.30.75,0.20.85,0.120.7,0.20.8,0.10.9,0.10.7,0.250.7,0.2.Consequently, the ideal solution (the ideal IFS) isR′∗=0.8,0.1,0.9,0.1,0.86,0.1,0.8,0.1.Thus, equations (9)–(13) in the setting of IFSs are reduced to the following similarity measures between Rj′j=1,2,3,4 and R′∗:(18)Mwλ′Rj′,R′∗=1−12∑k=1nηktjk−tk∗λ+fjk−fk∗λ1/λ,(19)Cwλ′Rj′,R′∗=cosπ212∑k=1nηkt1k−tk∗λ+f1k−fk∗λ1/λ,(20)Swλ′Rj′,R′∗=1−sinπ212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ,(21)Twλ′Rj′,R′∗=1−tanπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ,(22)CTwλ′Rj′,R′∗=cotπ4+π412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ.Based on equations (18)–(22), for λ = 1, 2, we obtain the values of the similarity measures between Rj′j=1,2,3,4 and R′∗ and decision results, which are shown in Table 3.Table 3
The values of the similarity measures betweenRj′ and R′∗ and decision results.
Similarity measureMeasure value forλ = 1Measure value forλ = 2RankingThe best oneMwλ′Rj′,R′∗0.8053, 0.9137, 0.8808, 0.94400.8019, 0.8743, 0.8636, 0.9145P4 > P2 > P3 > P1P4Cwλ′Rj′,R′∗0.9536, 0.9908, 0.9825, 0.99610.9520, 0.9806, 0.9771, 0.9910P4 > P2 > P3 > P1P4Swλ′Rj′,R′∗0.6988, 0.8649, 0.8138, 0.91210.6938, 0.8038, 0.7873, 0.8661P4 > P2 > P3 > P1P4Twλ′Rj′,R′∗0.8458, 0.9322, 0.9061, 0.95600.8431, 0.9009, 0.8924, 0.9327P4 > P2 > P3 > P1P4CTwλ′Rj′,R′∗0.7329, 0.8729, 0.8283, 0.91570.7288, 0.8197, 0.8058, 0.8740P4 > P2 > P3 > P1P4In Table3, all ranking orders and the best one regarding the special case of the illustrative example in the setting of IFSs are identical. Then, 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. Then, 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.The 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. This means that the proposed IFCS contains much more useful information than IFS.(2)
The 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.


## 5.1. 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 setP = {P1, P2, P3, P4}. In the evaluation of the industrial robots, the four criteria that contain the principal performances include x1 (dynamic accuracy), x2 (dexterity), x3 (working ability), and x4 (communication and control capability). Then, the weight vector for the four criteria is specified as η = (0.3, 0.25, 0.2, 0.25) by experts. The 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 = (rjk)4×4:(15)A=R1R2R3R4=0.65,0.85,0.3,0.70.72,0.8,0.25,0.750.6,0.78,0.35,0.70.6,0.88,0.3,0.760.7,0.7,0.2,0.80.65,0.7,0.3,0.850.86,0.8,0.1,0.80.8,0.8,0.1,0.80.6,0.8,0.3,0.850.75,0.8,0.2,0.70.85,0.7,0.12,0.750.7,0.8,0.2,0.860.8,0.8,0.1,0.80.9,0.8,0.1,0.80.7,0.8,0.25,0.80.7,0.75,0.2,0.8.From the decision matrixA, we first give the ideal solution (the ideal IFCS):(16)R∗=0.8,0.85,0.1,0.7,0.9,0.8,0.1,0.7,0.86,0.8,0.1,0.7,0.8,0.88,0.1,0.76.Then, 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.Table 1
The values of the proposed similarity measures and decision results forλ = 1.
Similarity measureMeasure valueRankingThe best oneMw1Rj,R∗0.8985, 0.9100, 0.9066, 0.9389P4 > P2 > P3 > P1P4Cw1Rj,R∗0.9873, 0.9900, 0.9893, 0.9954P4 > P2 > P3 > P1P4Sw1Rj,R∗0.8412, 0.8591, 0.8539, 0.9041P4 > P2 > P3 > P1P4Tw1Rj,R∗0.9201, 0.9292, 0.9265, 0.9520P4 > P2 > P3 > P1P4CTw1Rj,R∗0.8520, 0.8678, 0.8631, 0.9083P4 > P2 > P3 > P1P4Table 2
The values of the proposed similarity measures and decision results forλ = 2.
Similarity measureMeasure valueRankingThe best oneMw2Rj,R∗0.8916, 0.8848, 0.8898, 0.9171P4 > P1 > P3 > P2P4Cw2Rj,R∗0.9855, 0.9837, 0.9851, 0.9915P4 > P1 > P3 > P2P4Sw2Rj,R∗0.8305, 0.8201, 0.8278, 0.8702P4 > P1 > P3 > P2P4Tw2Rj,R∗0.9146, 0.9093, 0.9132, 0.9348P4 > P1 > P3 > P2P4CTw2Rj,R∗0.8427, 0.8337, 0.8403, 0.8776P4 > P1 > P3 > P2P4Whenλ = 1, all the ranking results in Table 1 are identical and the best one is P4, 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 P4, which also show the effectiveness of the developed MCDM method based on the proposed similarity measures. Then, 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 P4. 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.


## 5.2. 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:(17)A′=R1′R2′R3′R4′=0.65,0.30.72,0.250.6,0.350.6,0.30.7,0.20.65,0.30.86,0.10.8,0.10.6,0.30.75,0.20.85,0.120.7,0.20.8,0.10.9,0.10.7,0.250.7,0.2.Consequently, the ideal solution (the ideal IFS) isR′∗=0.8,0.1,0.9,0.1,0.86,0.1,0.8,0.1.Thus, equations (9)–(13) in the setting of IFSs are reduced to the following similarity measures between Rj′j=1,2,3,4 and R′∗:(18)Mwλ′Rj′,R′∗=1−12∑k=1nηktjk−tk∗λ+fjk−fk∗λ1/λ,(19)Cwλ′Rj′,R′∗=cosπ212∑k=1nηkt1k−tk∗λ+f1k−fk∗λ1/λ,(20)Swλ′Rj′,R′∗=1−sinπ212∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ,(21)Twλ′Rj′,R′∗=1−tanπ412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ,(22)CTwλ′Rj′,R′∗=cotπ4+π412∑k=1nηkt1k−t2kλ+f1k−f2kλ1/λ.Based on equations (18)–(22), for λ = 1, 2, we obtain the values of the similarity measures between Rj′j=1,2,3,4 and R′∗ and decision results, which are shown in Table 3.Table 3
The values of the similarity measures betweenRj′ and R′∗ and decision results.
Similarity measureMeasure value forλ = 1Measure value forλ = 2RankingThe best oneMwλ′Rj′,R′∗0.8053, 0.9137, 0.8808, 0.94400.8019, 0.8743, 0.8636, 0.9145P4 > P2 > P3 > P1P4Cwλ′Rj′,R′∗0.9536, 0.9908, 0.9825, 0.99610.9520, 0.9806, 0.9771, 0.9910P4 > P2 > P3 > P1P4Swλ′Rj′,R′∗0.6988, 0.8649, 0.8138, 0.91210.6938, 0.8038, 0.7873, 0.8661P4 > P2 > P3 > P1P4Twλ′Rj′,R′∗0.8458, 0.9322, 0.9061, 0.95600.8431, 0.9009, 0.8924, 0.9327P4 > P2 > P3 > P1P4CTwλ′Rj′,R′∗0.7329, 0.8729, 0.8283, 0.91570.7288, 0.8197, 0.8058, 0.8740P4 > P2 > P3 > P1P4In Table3, all ranking orders and the best one regarding the special case of the illustrative example in the setting of IFSs are identical. Then, 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. Then, 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.The 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. This means that the proposed IFCS contains much more useful information than IFS.(2)
The 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.


## 6. 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)
The proposed similarity measures of IFCSs can provide new mathematical tools for the effective modeling of MCDM problems with IFCS information(c)
The 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 importanceIn 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.



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*Source: 6630898-2021-01-19.xml*