Similarity measures of intuitionistic fuzzy sets based on cosine function for the decision making of mechanical design schemes

1 Introduction

In conceptual design stage, mechanical design schemes should be given based on the available data and information, which are vague, imprecise and uncertain in nature. The decision-making process (evaluation process) in conceptual designs is one of these typical occasions, which frequently depends upon the method of treating uncertain data and information.

In the conceptual design, designers usually present many primary design schemes according to designers’ knowledge and experience. However, the subjective characteristics of the schemes are generally uncertain and need to be evaluated based on decision-makers’ insufficient knowledge and judgments. The nature of this vagueness and uncertainty is fuzzy rather than random, especially when subjective assessments are included in the decision-making process. The fuzzy theory proposed by Zahed [13] offers a useful tool to handle vague and uncertain data and information including the subjective characteristics of human nature in the decision making process. Fuzzy set allows the uncertainty of a set with a membership degree between 0 and 1. Later, Atanassov [10] extended a fuzzy set to an intuitionistic fuzzy set (IFS), which represents the uncertainty with respect to both membership degree and nonmembership degree and also implies its hesitancy degree (or called intuitionistic index). Then, Ban [1] investigated theory and applications of intuitionistic fuzzy measures, and then Atanassov et al. [12] further introduced measures and integrals of IFSs. Since a similarity measure is an important tool for determining the degree of similarity between two objects, it plays a main role in pattern recognition, medical diagnosis, decision making, clustering analysis, and so on. Therefore, various similarity measures between IFSs have been proposed in literature and applied to pattern recognition, medical diagnosis, and decision making. Li and Cheng [2] proposed some similarity measures of IFSs and applied them to pattern recognition. Liang and Shi [20] introduced several similarity measures of IFSs and investigated their relationships. Meanwhile, Mitchell [7] modified Li and Cheng’s similarity measures of IFSs from a statistical viewpoint. As a generalization of the Hamming and Euclidean distances between fuzzy sets, Szmidt and Kacprzyk [3–5] introduced the Hamming, Euclidean distances and similarity measures for IFSs. Hung and Yang [17] put forward the Hausdorff distance measure between IFSs and its several similarity measures between IFSs. Liu [8] presented some similarity measures between IFSs and between elements and applied them to pattern recognition. Also, Hung and Yang [18] proposed the similarity measures of IFSs induced by L _p metric. Xu and Xia [19] gave the geometric distance and similarity measures of IFSs and their application in group decision making. Whereas, Ye [9] presented the cosine similarity measure of IFSs as the vector representations of IFSs, which is the inner product of the two vectors divided by the product of their lengths, i.e., the cosine of the angle between two vectors, and then applied it to pattern recognition and medical diagnosis. Hung [11] introduced an intuitionistic fuzzy likelihood-based measurement and applied it to the medical diagnosis and bacteria classification problems in intuitionistic fuzzy setting. However, the cosine similarity measure of IFSs defined by Ye [9] implies some drawbacks in some cases [14]. To overcome the drawbacks, Shi and Ye [14] further improved the cosine similarity measure of IFSs (vague sets) defined in [9] by considering membership degrees, nonmembership degrees, and hesitancy degrees in IFSs as the vector representations of the three terms and applied it to the fault diagnosis of turbine. Also, Tian [15] developed the cotangent similarity measure of IFSs and applied it to medical diagnosis. Then, Rajarajeswari and Uma [16] further introduced the cotangent similarity measure of IFSs by considering membership, nonmembership and hesitation degrees in IFSs. Furthermore, Szmidt [6] discussed distances between IFSs in two ways: using the two term representation of IFSs (membership degrees and non-membership degrees are only taken into account) and the three term representation of IFSs (membership degrees, nonmembership degrees, and hesitation degrees are taken into account), and illustrated the usefulness of the three term distances in a measure for ranking the intuitionistic fuzzy alternatives. Expanding upon his previous work, he further introduced a family of similarity measures constructed by considering the three terms (membership, nonmembership and hesitation degrees) described in IFSs.

However, the trigonometric similarity measures, such as cosine and cotangent similarity measures, also play an important role in pattern recognition, medical diagnosis, fault diagnosis, and decision making. Then, the cosine similarity measures defined in vector space [9] have some drawbacks in some cases. For instance, they may produce undefined (unmeaningful) phenomena or some results calculated by the cosine similarity measures are unreasonable in some cases (details given in Sections 2). Since the similarity measures of IFSs using the cotangent function demonstrate their effectiveness and rationality [15, 16] and can overcome the aforementioned drawbacks, the cotangent similarity measures are very suitable for pattern recognition and decision making. Furthermore, the similarity measures based on the cotangent function show the simple algorithms by a comparison with the cosine similarity measures in vector space. Motivated by the similarity measures of the cotangent function for IFSs [15, 16], therefore, this paper wants to develop another form of similarity measures based on the cosine function to extend a family of trigonometric similarity measures under an IFS environment. Hence, this paper aims to propose two new similarity measures of IFSs based on the cosine function and to apply them to the decision-making problems of engineering alternatives under an intuitionistic fuzzy environment. To do so, the rest of this paper is organized as follows. Some basic concepts of IFSs and existing cosine and cotangent similarity measures of IFSs are briefly reviewed in Section 2. Section 3 introduces two new similarity measures between IFSs based on the cosine function and weighted cosine similarity measures between IFSs and investigates their properties. Section 4 gives the comparative analysis in a family of trigonometric similarity measures, such as cosine and cotangent similarity measures of IFSs, by several numerical examples to illustrate the effectiveness of the proposed cosine similarity measures. In Section 5, we develop a decision-making method using the weighted cosine similarity measures for selecting mechanical design schemes (alternatives) under an intuitionistic fuzzy environment. Section 6 provides a decision-making example on selecting mechanical design schemes to demonstrate the applications and efficiency of the proposed decision making method. Finally, the conclusions and future research are given in Section 7.

2 Some basic concepts of IFSs and existing cosine and cotangent similarity measures of IFSs

Atanassov [10] presented an IFS concept as an extension of the concept of a fuzzy set and gave the following definition.

Definition 1. [10] An IFS A in the universe of discourse X is defined as A ={ 〈 x,  μ _A  (x) ,  v _A  (x) 〉 |x ∈ X }, where μ _A  (x) : X → [0, 1] and v _A  (x) : X → [0, 1] are the membership degree and nonmembership degree, respectively, of the element x to the set A with the condition 0 ≤ μ _A  (x) + v _A  (x) ≤1 for x ∈ X.

Then π _A  (x) =1 - μ _A  (x) - v _A  (x) is called Atanassov’s intuitionistic index or a hesitancy degree of the element x in the set A. Obviously there is 0≤ π _A  (x) ≤1 for x ∈ X.

Assume that there are two IFSs A = {〈x,  μ _A  (x), v _A  (x) 〉|x ∈ X} and B ={ 〈 x,  μ _B  (x) ,  v _B  (x) 〉 |x ∈ X }. Then, the relations between IFSs A and B are defined as follows [10]:

(1) A ⊆ B if and only if μ _A  (x) ≤ μ _B  (x) and v _A  (x) ≥ v _B  (x) for any x ∈ X;

(2) A = B if and only if μ _A  (x) = μ _B  (x) and v _A  (x) = v _B  (x) for any x ∈ X.

Then, similarity measures have the following definition [15]:

Definition 2. [15] A real-valued function S: IFS (X) × IFS (X) → [0, 1] is called a similarity measure on IFS (X), if it satisfies the following axiomatic requirements:

(S1) 0 ≤ S (A,  B) ≤1;

(S2) S (A,  B) =1 if and only if A = B;

(S3) S (A,  B) = S (B,  A);

(S4) If A ⊆ B ⊆ C, then S (A,  C) ≤ S (A,  B) and S (A,  C) ≤ S (B,  C).

Assume that there are two IFSs A = {〈x _j , μ _A  (x _j ), v _A  (x _j ) 〉|x _j  ∈ X} and B = {〈x _j ,  μ _B  (x _j ), v _B  (x _j ) 〉|x _j  ∈ X} in the universe of discourse X = {x₁,  x₂, …,  x _n }. Then, based on the cosine similarity measure in vector space, Ye [9] proposed the following cosine similarity measure between IFSs A and B:C 1 ( A , B ) = 1 n ∑ j = 1 n μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) .(1)

However, one can find some drawbacks of the cosine similarity measure as follows:

(1) For two IFSs A and B, if μ _A  (x _j ) = v _A  (x _j ) =0 and/or μ _B  (x _j ) = v _B  (x _i ) =0 for any x _j in X (j = 1, 2, …,  n), the cosine similarity measure is undefined or unmeaningful. In this case, one cannot utilize it to calculate the cosine similarity measure between Aand B.

(2) If μ _A  (x _j ) =2μ _B  (x _j ) and v _A  (x _j ) =2v _B  (x _j ) or 2μ _A  (x _j ) = μ _B  (x _j ) and 2v _A  (x _j ) = v _B  (x _j ) for any x _j in X (j = 1,  2,   … ,  n), i.e. A ≠ B, the measure value of Equation (1) is equal to 1. This means that it only satisfies the necessary condition of the property (S2) in Definition 2, but not the sufficientcondition.

For example, if there are two IFSs A ={ 〈 x,  0.1,  0.2 〉 |x ∈ X } and B ={ 〈 x,  0.2,  0.4 〉 |x ∈ X } in X = {x}, Obviously, A ≠ B. Then, by using Equation (1), the calculating result is as follows: C 1 ( A , B ) = 0.1 × 0.2 + 0.2 × 0.4 0 . 1 2 + 0 . 2 2 · 0 . 2 2 + 0 . 4 2 = 1 .

In this case, therefore, it is unreasonable to apply it to pattern recognition and decision making.

In order to overcome aforementioned disadvantages, Shi and Ye [14] further presented the cosine similarity measure by considering membership degrees, nonmembership degrees, and hesitancy degrees in IFSs as the vector space of the three terms:C 2 ( A , B ) = 1 n ∑ j = 1 n ( μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) + π A ( x j ) π B ( x j ) ) ( μ A 2 ( x j ) + ν A 2 ( x j ) + π A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) + π B 2 ( x j ) ) .(2)

Thus, when μ _A  (x _j ) = v _A  (x _j ) =0 and/or μ _B  (x _j ) = v _B  (x _j ) =0 for any x _j in X = {x₁,  x₂, …,  x _n } (j = 1,  2, …, n), Equation (2) is meaningful and there is C₂ (A,  B) =1.

On the other hand, Tian [15] proposed a cotangent similarity measure between IFSs:CT 1 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ) ] ,(3)

where the symbol “∨” is the maximum operation.

When the three terms like membership degrees, non-membership degrees and hesitation degrees are considered in IFSs, Rajarajeswari and Uma [16] introduced the cotangent similarity measure of IFSs:CT 2 ( A , B ) = 1 n ∑ j = 1 n cot [ π 4 + π 4 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ∨ | π A ( x j ) - π A ( x j ) | ) ] .(4)

Again considering the similarity measure between A and B in the above example, we calculate the similarity measures between A and B by Equations (2–4), respectively, as follows: C 2 ( A , B ) = 0.1 × 0.2 + 0.2 × 0.4 + 0.7 × 0.4 0 . 1 2 + 0 . 2 2 + 0 . 7 2 · 0 . 2 2 + 0 . 4 2 + 0 . 4 2 , = 0.8619

CT 1 ( A , B ) = cot [ π + π ( | 0.1 - 0.2 | ∨ | 0.2 - 0.4 | ) 4 ] = 0.7265 , CT 2 ( A , B ) = cot [ π + π ( | 0.1 - 0.2 | ∨ | 0.2 - 0.4 | ∨ | 0.7 - 0.4 | ) 4 ] = 0.6128 .

Hence, Equations (2–4) can overcome the drawbacks of Equation (1) in some cases. Obviously, the trigonometric similarity measures (2–4) are superior to the cosine similarity measure (1).

Clearly, the cotangent similarity measures show the simple algorithms by a comparison with the cosine similarity measures in vector space. Motivated by the cotangent similarity measures of IFSs, we shall develop another form of similarity measures based on the cosine function in following section to extend a family of trigonometric similarity measures under an IFSenvironment.

For conveniently comparative analysis in the following decision-making example, we introduced the weighted cosine and cotangent similarity measures between IFSs A and B, respectively, as follows [9, 15, 16]:

WC 1 ( A , B ) = ∑ j = 1 n w j μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) μ A 2 ( x j ) + ν A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) ,(5) WC 2 ( A , B ) = ∑ j = 1 n w j ( μ A ( x j ) μ B ( x j ) + ν A ( x j ) ν B ( x j ) + π A ( x j ) π B ( x j ) ) ( μ A 2 ( x j ) + ν A 2 ( x j ) + π A 2 ( x j ) μ B 2 ( x j ) + ν B 2 ( x j ) + π B 2 ( x j ) ) ,(6) WCT 1 ( A , B ) = ∑ j = 1 n w j cot [ π 4 + π 4 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ) ] ,(7)

WCT 2 ( A , B ) = ∑ j = 1 n w j cot [ π 4 + π 4 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ∨ | π A ( x j ) - π A ( x j ) | ) ] ,(8)

where w _j (j = 1, 2, …, n) is the weight of an element x _j , w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1 and the symbol “∨” is the maximum operation.

Based on cosine function, the section proposes two cosine similarity measures between IFSs and investigates their properties.

Definition 3. Let A = {〈x _j ,  μ _A  (x _j ), v _A  (x _j ) 〉|x _j  ∈ X} and B = {〈x _j ,  μ _B  (x _j ) ,  v _B  (x _j ) 〉|x _j  ∈ X} be any two IFSs in X = {x₁,  x₂, …,  x _n }. Then, we define two cosine similarity measures between A and B, respectively, as follows:CS 1 ( A , B ) = 1 n ∑ j = 1 n cos [ π 2 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ∨ | π A ( x j ) - π A ( x j ) | ) ] ,(9)

CS 2 ( A , B ) = 1 n ∑ j = 1 n cos [ π 4 ( | μ A ( x j ) - μ B ( x j ) | + | ν A ( x j ) - ν B ( x j ) | + | π A ( x j ) - π A ( x j ) | ) ] ,(10)

where the symbol “∨” is the maximum operation. Then, the two cosine similarity measures satisfy the axiomatic requirements of similarity measures inDefinition 2.

Proposition 1. For two IFSs A and B in X = {x₁, x₂, …,  x _n }, the cosine similarity measure CS _k  (A,  B) (k = 1,  2) should satisfy the following properties(S1–S4):

0 ≤ CS _k  (A,  B) ≤1;

Proof:

Since the value of cosine function is within [0, 1], the similarity measure based on the cosine function also is within [0, 1]. Hence, there is 0 ≤ CS _k  (A,  B) ≤1 for k = 1, 2.

| μ A ( x j ) - μ B ( x j ) | ≤ | μ A ( x j ) - μ C ( x j ) | , | μ B ( x j ) - μ C ( x j ) | ≤ | μ A ( x j ) - μ C ( x j ) | , | ν A ( x j ) - ν B ( x j ) | ≤ | ν A ( x j ) - ν C ( x j ) | , | ν B ( x j ) - ν C ( x j ) | ≤ | ν A ( x j ) - ν C ( x j ) | , | π A ( x j ) - π B ( x j ) | ≤ | π A ( x j ) - π C ( x j ) | , | π B ( x j ) - π C ( x j ) | ≤ | π A ( x j ) - π C ( x j ) | .

Hence, CS _k  (A,  C) ≤ CS _k  (A,  B) and CS _k  (A,  C) ≤ CS _k  (B,  C) for k = 1,  2 as the cosine function is a decreasing function within the interval [0,  π/2].

Thus, the proofs of these properties are completed. □

Usually, one considers the weight of each element x _j for x _j  ∈ X. Assume that the weight of an element x _j is w _j (j = 1,  2,   … ,  n), w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1. Then we can introduce the following weighted cosine similarity measures between IFSs A and B:WCS 1 ( A , B ) = ∑ j = 1 n w j cos [ π 2 ( | μ A ( x j ) - μ B ( x j ) | ∨ | ν A ( x j ) - ν B ( x j ) | ∨ | π A ( x j ) - π A ( x j ) | ) ] ,(11)

WCS 2 ( A , B ) = ∑ j = 1 n w j cos [ π 4 ( | μ A ( x j ) - μ B ( x j ) | + | ν A ( x j ) - ν B ( x j ) | + | π A ( x j ) - π A ( x j ) | ) ] .(12)

where the symbol “∨” is the maximum operation. Especially when w _j  = 1/n for j = 1,  2, …,  n, Equations (11) and (12) reduce to Equations (9) and (10).

Obviously, the two weighted cosine similarity measures also satisfy the axiomatic requirements of similarity measures in Definition 2.

Proposition 2. For two IFSs A and B in X = {x₁,  x₂, …,  x _n }, the weighted cosine similarity measure WCS _k  (A,  B)   (k = 1,  2) should satisfy the following properties (S1–S4):

0 ≤ WCS _k  (A,  B) ≤1;

By similar proofs in Proposition 1, we can give the proofs of these properties (S1–S4), which are not repeated here.

In this section, the comparison of trigonometric similarity measures for IFSs is given by several numerical examples to demonstrate the effectiveness and rationality of the developed cosine similarity measures of IFSs.

To illustrate the effectiveness and rationality of the cosine similarity measures proposed herein, the comparisons of the trigonometric similarity measures (1)-(4), (9) and (10) are illustrated by several numerical examples in Table 1. These measure values between IFSs A and B are shown in Table 1.

For the results of the trigonometric similarity measures in Table 1, the cosine similarity measure C₁ (A,  B) cannot carry out our identification and has the unreasonable phenomena for Case 1 and Case 6, and also has the undefined phenomena for Case 2 and Case 3. Then, the cotangent similarity measure CT₁ (A,  B) cannot also carry out our identification between Case 1 and Case 5. The above results will get the decision maker into trouble in practical applications. However, the trigonometric similarity measures C₂ (A,  B), CT₂ (A,  B), CS₁ (A,  B) and CS₂ (A,  B) have stronger discrimination among them. Whereas, the measure values of the two cosine similarity measures CS₁ (A,  B) and CS₂ (A,  B) are identical. Obviously, the two cosine similarity measures proposed in this paper are effective and reasonable due to better discrimination. Furthermore, the trigonometric similarity measures constructed by considering all the three terms (membership degrees, nonmembership degrees and hesitation degrees) described in IFSs are more effective than the trigonometric similarity measures constructed by considering the two terms (membership degrees and nonmembership degrees) in IFSs.

5 Decision-making method using cosine similarity measures

In this section, the proposed similarity measures are used for the multiple attribute decision-making problems of mechanical design schemes with intuitionistic fuzzy information.

In conceptual design, usually there are various mechanical design schemes (alternatives) presented by designers. Assume that A = {A₁,  A₂, …,  A _m } is a set of alternatives. Then, the alternatives must satisfy the requirements of a set of attributes (criteria) C = {C₁,  C₂,   … C _n } by their assessments. Then we consider the weight w _j of the attribute C _j   (j = 1,  2, …,  n), entered by the decision-maker, with w _j  ∈ [0, 1] and ∑ j = 1 n w j = 1. In this case, the characteristic of the alternative A _i  (i = 1,  2, …,  m) on the attribute C _j   (j = 1,  2, …,  n) is expressed by the following IFS: A i = { 〈 C j , μ A i ( C j ) , v A i ( C j ) 〉 | C j ∈ C } ,

where 0 ≤ μ _{A i}  (C _j ) + v _{A i}  (C _j ) ≤1, μ _{A i}  (C _j ) ⩾0, v _{A i}  (C _j ) ⩾0, j = 1, 2, …, n, and i = 1,  2, …,  m. For convenience, a basic element 〈C _j , μ _{A i}  (C _j ) ,  v _{A i}  (C _j ) 〉 in an IFS A _i is denoted by a _ij  =〈 μ _ij ,  v _ij  〉, which is called an intuitionistic fuzzy value (IFV). Here, the IFV is usually obtained from the evaluated score to which the alternative A _i satisfies or does not satisfy the attribute C _j by means of a score law or appropriate membership functions in practical applications. Therefore, we can establish an intuitionistic fuzzy decision matrix D = (a _ij ) _m×n.

In multiple attribute decision-making environments, the concept of an ideal alternative has been used to help identify the best alternative in the decision set. Hence, we define the ideal alternative denoted by the following IFS: A * = { 〈 C j , a j * ( C j ) 〉 | C j ∈ C } = { 〈 C j , 1 , 0 〉 | C j ∈ C } ,

where an ideal IFV is denoted by a j * = 〈 μ j * , v j * 〉 = 〈 1 , 0 〉 for j = 1,  2,   … ,  n.

Then, by applying Equations (11) or (12) the weighted cosine similarity measure between an alternative A _i and the ideal alternative A^* is given byWCS 1 ( A * , A i ) = ∑ j = 1 n w j cos [ π ( | μ j * - μ ij | ∨ | ν j * - ν ij | ∨ | π j * - π ij | ) 2 ] ,(13)

orWCS 2 ( A * , A i ) = ∑ j = 1 n w j cos [ π ( | μ j * - μ ij | + | ν j * - ν ij | + | π j * - π ij | ) 4 ] ,(14)

which provides the global evaluation for each alternative regarding all attributes. For the similarity measure of Equation (13) or (14), the bigger the measure value WCS _k  (A^*,  A _i ) (k = 1,  2 ;  i = 1,  2,   … ,  m), the better the alternative A _i . According to the similarity measure value between the ideal alternative and each alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well.

This section provides a decision-making example on choosing mechanical design schemes (alternatives) for press machine to demonstrate the application and effectiveness of the proposed decision-making method.

In conceptual design, by considering the reducing mechanism and working mechanism of press machine, a set of four alternatives A = {A₁,  A₂,  A₃, A₄} is presented by specialists’ analyses and designers’ experiences, which are shown in Table 2. The chief designer (decision maker) must take a decision according to the four attributes: (1) C₁ is the manufacturing cost; (2) C₂ is the mechanical structure; (3) C₃ is the transmission effectiveness; (4) C₄ is the reliability. The weight vector of the four attributes is w = (0.3,  0.25,  0.25,  0.2)  ^T . The four possible alternatives of A _i   (i = 1,  2,  3,  4) are to be evaluated by the chief designer under the above four attributes according to “excellence”, which are represented by the form of IFVs, and then the intuitionistic fuzzy decision matrix D = (a _ij ) _4×4 is obtained asfollows:.

D = ( a ij ) 4 × 4 = [ 〈 0.9 , 0.1 〉 〈 0.92 , 0.05 〉 〈 0.9 , 0.1 〉 〈 0.7 , 0.2 〉 〈 0.9 , 0.1 〉 〈 0.97 , 0.0 〉 〈 0.85 , 0.1 〉 〈 0.8 , 0.1 〉 〈 0.8 , 0.1 〉 〈 0.65 , 0.3 〉 〈 0.8 , 0.1 〉 〈 0.7 , 0.2 〉 〈 0.7 , 0.2 〉 〈 0.8 , 0.15 〉 〈 0.75 , 0.2 〉 〈 0.9 , 0.1 〉 ] .

Then, we utilize the developed approach to obtain the most desirable alternative(s).

By using Equation (13) or (14), the similarity measures between an alternative A _i (i = 1, 2, 3, 4) and the ideal alternative A^* are shown in Table 3.

For comparative convenience, we also calculate the similarity measures between an alternative A _i (i = 1, 2, 3, 4) and the ideal alternative A^* by using Equations (5–8), which are also shown in Table 3.

In Table 3, we can see that results of all the ranking orders are identical. Therefore, A₂ is the optimal choice among all alternatives (design schemes). In fact, from intuitional viewpoint, the alternative A₂ should also satisfy practical requirements from the designers’ experience.

7 Conclusion

In this paper, we proposed another form of two similarity measures between two IFSs based on the cosine function and weighted cosine similarity measures between two IFSs by considering the degrees of membership, nonmembership and hesitancy in IFSs. The comparative analysis in a family of trigonometric similarity measures of IFSs demonstrated the effectiveness and rationality of the proposed two cosine similarity measures of IFSs. Then, the weighted cosine similarity measures were applied to decision-making problems of mechanical decision schemes (alternatives) under an intuitionistic fuzzy environment. Through the similarity measures between the ideal alternative and each alternative, we can determine the ranking order of all alternatives and the best one. Finally, a decision-making example on choosing mechanical design schemes was provided to demonstrate the applications and effectiveness of the developed approach. In fact, these cosine similarity measures can be also extended to interval-valued IFSs. In the future, we shall further apply the cosine similarity measures of IFSs and interval-valued IFSs to complex group decision making problems.