A novel bidirectional projection measures of circular intuitionistic fuzzy sets and its application to multiple attribute group decision-making problems

1.   Introduction

Multi-attribute group decision-making (MAGDM) ^[1,2,3] is an important area of decision-making research. For example, Xu et al.^[1] used the interval-valued intuitionistic fuzzy set to deal with the group attribute decision-making problem. Akram et al.^[2,3] applied the Pythagorean fuzzy set and the VIKOR method to the group multi-attribute decision-making problem. Liu et al.^[4] applied the fuzzy linguistic term set to the group multi-attribute decision-making problem. The multi-attribute group decision-making process is typically divided into three steps: collecting experts' views on the different attributes of the alternatives, aggregating expert opinions, and selecting the best alternative. In the decision-making processes involving experts, the decision information about alternatives is usually uncertain or fuzzy due to the increasing complexity of the socio-economic environment and the inherent fuzziness in human cognition. In 1965, an American computer and cybernetics expert, Zadeh, proposed the concept of fuzzy sets ^[5] to solve fuzzy phenomena; ordinary fuzzy sets use a single value for membership. However, many researchers think that single-valued fuzzy sets make it challenging to represent fuzzy information accurately. Thus, many extended fuzzy sets have been proposed. For example, the interval fuzzy set ^[6] proposed by Guinness involves the replacement of membership from exact values to interval values. Atanassov introduced the concept of intuitionistic fuzzy sets (IFS) ^[7] in 1986, which considers both membership and non-membership for fuzzy sets. Non-membership represents fuzziness from the opposite side of things. Intuitionistic fuzzy sets can be more flexible than ordinary fuzzy sets to represent the fuzziness of information. Subsequently, intuitionistic fuzzy set theory has been used in various fields of research ^[8,9,10], such as in the fields of decision analysis, pattern recognition, clustering analysis, etc. For example, Demir ^[8] applied it in the multi-attribute decision-making (MADM), specifically for the security management of smart cities. Moslem ^[9] also applied it in the MADM model. Zeng et al.^[10] applied the intuitionistic fuzzy sets to the pattern recognition problem. Kuo et al. ^[11] applied intuitionistic fuzzy sets for clustering analysis. It is Rahman ^[12] and Iqbal who applied intuitionistic fuzzy sets to the selection and optimization problem of railway systems.

However, to ensure the accuracy of the decision, it is often necessary to have several experts working together to provide a decision evaluation. Experts often struggle to give precise and consistent membership degree and non-membership degree in practical applications of intuitionistic fuzzy sets. In 1999, Atanassov extended the intuitionistic fuzzy set to an interval-valued intuitionistic fuzzy set (IVIFS) ^[13], which extension of membership degree and non-membership degree with exact values to interval values. The theoretical study of applying intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets to MADM problems has produced many results, and the main research focuses on score functions, distance measures, similarity measures, etc. Chen et al. ^[14] first proposed a score function on intuitionistic fuzzy numbers. Sahin et al. ^[15] proposed a new score function based on considering the hesitancy degree of interval-valued intuitionistic fuzzy sets for IVIFS. Kumar ^[16] converts IVIFVs into connection numbers (CNs), and develops a score function for ranking CNs. A large number of research results ^[17,18,19] have been presented on the distance between intuitionistic fuzzy sets. Szmidt and Kacprzyk ^[17] analogized vectors using the three parameters of the membership degree, non-membership degree, and hesitation degree of intuitionistic fuzzy sets, and proposed a distance formula for intuitionistic fuzzy sets. Zhang et al. ^[18] proposed a new distance measure for intuitionistic fuzzy sets, which significantly overcomes the problem of information loss. A similarity measure ^[19] is defined by using the weighted average distance between the intuitionistic fuzzy set and the two most fuzzy sets.

In addition to the aforementioned extensions of intuitionistic fuzzy sets, when the sum of membership degree and non-membership degree exceeds 1 in practical problems, the intuitionistic fuzzy sets will become inapplicable. To address this issue, Yager ^[20] proposed Pythagorean fuzzy sets, which makes the sum of the squares of the membership degree and the non-membership degree less than 1. It expands the value range of the membership degree and non-membership degree in the intuitionistic fuzzy set, and can provide decision-makers with a broader space for information expression. Subsequently, Yager extended the Pythagorean fuzzy set to the q-rung orthopair fuzzy set ^[21], this extension ensures that the sum of the q-th powers of the membership degree and the non-membership degree is less than 1. Considering that human opinions about a certain thing are not limited to simply "yes" or "no", but also include "abstention" and "rejection", Mahmood et al. ^[22] proposed the concept of spherical fuzzy sets. It allows the sum of the membership degree, non-membership degree, and hesitation degree to be greater than 1. Based on the above extended fuzzy sets, a great many relevant MADM and pattern recognition methods have also been developed. Sarfraz ^[23] proposed a new aggregation operator based on the interval-valued Pythagorean fuzzy set to address the priority ranking problem in MADM issues. Akram et al. ^[2] combined the Pythagorean fuzzy set and the rough set and proposed new AHP and TOPSIS decision-making methods. Asif et al. ^[24] proposed new operators, such as the Pythagorean fuzzy Hamacher interactive weighted averaging by using the Hamacher t-norm, and applied them to MADM problems. Liu et al. ^[25] proposed the q-rung orthopair fuzzy weighted averaging operator and the q-rung orthopair fuzzy weighted geometric operator to address the decision information. Pethaperumal et al. ^[26] proposed a new distance measure for q-rung orthopair MFSs, and they applied it to decision-making problems. Radovanovic et al. ^[27] introduced a hybrid multi-criteria decision-making (MCDM) model combining spherical fuzzy AHP and Grey MARCOS methods. The above extensions of fuzzy sets and intuitionistic fuzzy sets are all aimed at expressing the uncertainty and fuzziness in human information more accurately.

Then, since the projection measure can take into account not only the distance, but also the included angle between the evaluated objects, it is a very suitable method for dealing with decision problems. Xu et al. ^[28] introduce the concepts of intuitionistic fuzzy numbers in terms of modulus, angle of entrapment, projective measure, etc. Zeng et al. ^[29] used the projection measure of intuitionistic fuzzy numbers in the determination of expert weight information in decision problems. Zheng et al. ^[30] proposed the concept of bidirectional projection to compute the projection measure.

Although extended intuitionistic fuzzy sets have more flexibility than intuitionistic fuzzy sets in expressing uncertain information, using a fixed value to represent the membership degree and non-membership degree lacks accuracy. Atanasov recently proposed a new extension of intuitionistic fuzzy sets called circular intuitionistic fuzzy sets (CIFS) ^[31]. CIFS fits the fuzziness and uncertainty of information to a circle in geometry, which has membership degree and non-membership degrees, and also uncertainty by adding a parameter radius. To some extent, CIFS retains the fuzziness and uncertainty of the evaluation information while maintaining its accuracy. Distance measures and similarity measures are very important to study in practical decision-making applications, existing distance measures, and similarity measures for intuitionistic fuzzy sets cannot be directly used for CIFS. Atanasov himself proposed four distances ^[32] on CIFS, based on the classical Euclidean and Hamming distances, among others, and the distance is calculated by employing parameters such as the radius r of membership and non-membership degrees. However, the specific meaning and origin of the distance formula has not been explained. Kahraman et al. ^[33,34] transformed the CIFS into ordinary intuitionistic fuzzy set through the radius for calculation in decision-making problems. This will lead to the loss of some original decision-making information and cause deviations in the calculation of decision-making results. Xu et al. ^[35] defined a new circular intuitionistic fuzzy sets distance through the underlying relationships among the membership degree, non-membership degree, hesitation degree, and radius in CIFS. Subsequently, many extended fuzzy sets based on circular intuitionistic fuzzy sets have also been proposed. Khan et al. ^[36] proposed the circular Pythagorean fuzzy set, and then generalized it to the new disc Pythagorean fuzzy set. They also gave the corresponding addition, subtraction, multiplication, and division operation rules, as well as the distance formula. Ashraf et al. ^[37] introduced the circular spherical fuzzy set and disc spherical fuzzy set, and established the ELECTRE method using the circular spherical fuzzy set. At the same time, circular intuitionistic fuzzy sets also have many applications in scoring methods ^[38] and pattern recognition ^[39]. It demonstrates the huge development potential of circular fuzzy sets.

Most of the existing methods only consider the distance and ignore the angle between different circular intuitionistic fuzzy sets when calculating the differences between them. In this paper, a new bidirectional projection measures of circular intuitionistic fuzzy sets will be given. Considering the differences in distance and angle between different CIFSs, the projection similarity is defined. When calculating the circular parameter r in CIFS, the method of taking the maximum value of different expert evaluations is used. When the value given by one of the experts is significantly different from that of the other experts, it will influence the value of r to a great extent. Therefore, in this paper, the calculation of the r parameter is improved by using the least squares method to fit the value of r.

The rest of the paper is organized as follows. The Section 2 briefly introduces some concepts of circular intuitionistic fuzzy sets and puts forward a new method for calculating the parameter r. Section 3 describes in detail the proposed bidirectional projection measurement method. In Section 4, an MADM method is established based on the proposed bidirectional projection measurement. Section 5 provides a computational example and makes a comparison with other methods. Finally, conclusions are given in Section 7.

2.   Preliminaries of circular intuitionistic fuzzy sets

This part introduces the basic concepts of circular intuitionistic fuzzy sets and proposes a new method for calculating the parameter $r$.

Definition 1. ^[31] Let X be a given fixed universe. A CIFS $C$ in X is an object of the form:

where the functions

denote the membership degree and non-membership degree of $C$, respectively, and for any $x \in X$, there is

a circle around each element, and

holds. Further,

is called the hesitancy degree of element x in circular intuitionistic fuzzy set $C$.

CIFS can be regarded as an extension of IFS. In geometry, IFS can be represented by a point, while CIFS can be represented by a circle with the center at $\mu_C(x), v_C(x)$ and a radius of $r$. The ways in which CIFS and IFS express decision-maker information are different. Different experts may give different evaluations of the same decision problem. We need to synthesize the various opinions given by different experts. The following definition gives how to generate the corresponding circular intuitionistic fuzzy set when there is a set of intuitionistic fuzzy pairs.

Definition 2. ^[31] A CIFS $C$ in a fixed universe X, for $x_{i}\in X$, and a set of intuitionistic fuzzy pairs have the form $\left\{\langle m_{i, 1}, n_{i, 1}\rangle, \langle m_{i, 2}, n_{i, 2}\rangle, \ldots\right\}$. Then, we calculate the circular fuzzy set as follows:

where $k_i$ denotes the number of decision makers. Let

where the radius of the $C$ is obtained through the maximum Euclidean distance.

In Definition 2, the approach of calculating parameter $r$ for Eq (2.3) by means of the maximum Euclidean distance is vulnerable to the influence of a few extremely incorrect decision makers. For example, four decision makers provide evaluation data $\left\{\langle 0.4, 0.5\rangle, \langle 0.5, 0.4\rangle, \langle 0.4, 0.4\rangle, \langle 1, 0\rangle \right\}.$ Obviously, the fourth expert provided evaluation information that is significantly different from that of the other three experts, when calculating the parameter $r$, efforts should be made to minimize the influence of the fourth expert on the calculation. However, in practice, and thus, when calculating the parameter $r$ using the maximum Euclidean distance, the information provided by the fourth expert is utilized to obtain $r$. This value of r exhibits an excessive disparity compared to the information from the evaluations of the other three experts.

Therefore, this paper proposes a method for obtaining the parameter of CIFS using the ordinary least squares

When the objective function reaches its minimum value, it means that, at this moment, the sum of the distances between the parameter $r$ and all the original intuitionistic fuzzy pairs is the closest, and the parameter $r$ preserves the information of the original intuitionistic fuzzy pairs to the greatest extent. However, when calculating the parameter $r$ using the maximum value method, it may be significantly affected by extreme values, and the overall original values may be neglected.

The objective function is differentiated with respect to the variable $r_i$. When the derivative is equal to zero, the minimum value is obtained when

and we will redefine the calculation method of the parameter $r$ of the CIFS below.

Definition 3. A CIFS $C$ in a fixed universe X, for $x_{i}\in X$, and a set of intuitionistic fuzzy pairs have the form $\left\{\langle m_{i, 1}, n_{i, 1}\rangle, \langle m_{i, 2}, n_{i, 2}\rangle, \ldots\right\}$. Then we calculate the circular fuzzy set as follows:

where $k_i$ denotes the number of decision makers. Let

As illustrated in Figure 1, we can see that there are five possible shapes of the circle in the circular intuitionistic fuzzy set. Several circles have incomplete points because the values of membership degree and non-membership degree need to satisfy certain relationships in practical applications. We give the following Definition 4.

Definition 4. ^[31] Let $C$ be a circular intuitionistic fuzzy set, and

where $O_r$ represents a function with a radius of r and a center at $\left(\mu_C(x), v_C(x)\right)$,

The following presents some basic operations of circular fuzzy sets that will be used in this paper.

Definition 5. ^[35] Let

be two circular intuitionistic fuzzy sets. Then,

Definition 6. ^[40] Let

be a circular intuitionistic fuzzy set. We define $c = \left(u_{\mathrm{c}}, v_c; r\right)$ as a circular intuitionistic fuzzy value, and a score function $S_c$ and an accuracy function $H_c$ of the circular intuitionistic fuzzy value are defined as follows:

where $H_c(c_r) \in[0, 1].$

Considering that the larger the value of $r$ for an circular intuitionistic fuzzy value, the more negative influence it should have on the score of circular intuitionistic fuzzy value. Therefore, we subtract the parameter $r$ when defining the scoring function.

Definition 7. ^[40]

Let

and

be two CIFVs, and the ranking rules are defined as follows:

1) If $S_c\left(\mathrm{c}_1\right) \succ S_c\left(\mathrm{c}_2\right)$, then $\mathrm{c}_1 \succ \mathrm{c}_2$;

2) If $S_c\left(\mathrm{c}_1\right) = S_c\left(\mathrm{c}_2\right)$ and $H_c\left(\mathrm{c}_1\right) \succ H_c\left(\mathrm{c}_2\right)$, then $\mathrm{c}_1 \succ \mathrm{c}_2$;

3) If $S_c\left(\mathrm{c}_1\right) = S_c\left(\mathrm{c}_2\right)$ and $H_c\left(\mathrm{c}_1\right) = H_c\left(\mathrm{c}_2\right)$, then $\mathrm{c}_1 = \mathrm{c}_2$.

3.   Bidirectional projection measures of circular intuitionistic fuzzy set

For any variable $x$ in the universe $X$, the circular intuitionistic fuzzy set can be regarded as a vector. When using distance to measure the similarity between two different circular intuitionistic fuzzy sets, the influence of the angle between vectors is not taken into account, and when considering the similarity between different circular intuitionistic fuzzy sets in this paper, both the included angle and the distance are taken into account. Currently, there is relatively little research on the projection model of two circular intuitionistic fuzzy sets. Therefore, this paper proposes a bidirectional projection measures of circular intuitionistic fuzzy sets.

3.1.   Some existing projection models

Definition 8. ^[28] Let

and

be two IFNs;

and then

$|\alpha|$ is called the module of $\alpha.$ It is the same for $|\beta|$.

is called the inner product of $\alpha$ and $\beta$.

is called the projection of $\alpha$ on $\beta$.

Definition 9. ^[28] Let $X$ be a finite universe,

and $A$ and $B$ be two IFSs in $X$. Then,

is called the projection of $A$ on $B$, and $(\mu_{\alpha_i}, v_{\alpha_i})$ denotes the i-th intuitionistic fuzzy number in A, where

As shown in Figure 2, the greater the projection value, the closer the intuitionistic fuzzy sets $A$ and $B$ are to each other. However, extensive research has revealed that, in certain cases, the above-mentioned method is unreasonable in determining the proximity between IFS A and B. For example, let $A, B$, and $C$ be three intuitionistic fuzzy sets,

we calculated and obtained

and

In fact, $A$ is closer to $B$ than $C$, however, the value of $\operatorname{Proj}_BC$ is larger than $\operatorname{Proj} _BA$.

In another case as shown in Figure 3, the projection value

makes it impossible to distinguish which intuitionistic fuzzy set is closer to $B$.

Hence, in order to overcome the shortcoming of the above projection model, a bidirectional projection measure has been proposed for interval numbers.

Definition 10. ^[41] Let

and

be two interval vectors. Then,

is called the module of $a$. It is the same for $\|b\|$.

is called the inner product of $a$ and $b$.

is called the bidirectional projection between a and b.

The bidirectional projection measure includes both the distance and the angle between vectors $a$ and $b$, as well as the bidirectional projection magnitude. Determine the proximity of $a$ and $b$ by projecting $a$ and $b$ onto each other. We find that when a = b, the projection value

and for any two interval vectors a and b, the bidirectional projection measure

satisfies the symmetry of the similarity measure. However, it also has some drawbacks. For example, for the interval vectors

but the angle difference is large. Obviously,

but it is unreasonable.

3.2.   The bidirectional projection measure

In this paper, the bidirectional projection measure is extended to circular intuitionistic fuzzy sets. Since the membership and non-membership degrees of circular intuitionistic fuzzy sets are represented by a circle, we extend the modulus, projection values, etc., of circular intuitionistic fuzzy sets to interval values. Next, the bidirectional projection measure of circular intuitionistic fuzzy sets is presented as follows.

Definition 11. Let

be a given universe,

and

are two CIFSs on X. Then,

is called the module of $C_1$, and $\|C_1\|$ is an interval number. The same is true with $\|C_2\|$. $r_1\left(x_i\right)$ denotes $r$ in the CIFS $C_1$ corresponding to $x_i$, and

is called the cosine of the included angle between $C_1$ and $C_2$.

Definition 12. Let

be a given universe, and

and

are two CIFSs on X. Then,

is called the projection of CIFS $C_2$ on the CIFS $C_1$, and $\operatorname{Proj}_{C_1}C_2$ is an interval number.

Definition 13. Let

be a given universe, and

and

be two circular intuitionistic fuzzy sets on X. Then,

is called the bidirectional projection measure between $C_1$ and $C_2$, where

and

Elements in the circular intuitionistic fuzzy set are represented by a circle. Their projection values, referring to the projection of ordinary vectors, are interval-valued, as shown in Figure 4.

The bidirectional projection measure takes into account not only the distance between different circular intuitionistic fuzzy sets, but also the angle between them. The bidirectional projection measure satisfies the following theorems. It can be seen that when the modulus length of the circular intuitionistic fuzzy set is larger, it indicates that the relationship between membership and non-membership is more explicit, and we obtain more information. While the magnitude of the angle reflects the proportional relationship between the membership degree and the non-membership degree, these factors will undoubtedly affect the similarity between different circular fuzzy sets.

Theorem 1. Let $C_1$ and $C_2$ be two circular intuitionistic fuzzy sets on the universe

and $\operatorname{BProj}(C_1, C_2)$ be the bidirectional projection measure on the CIFS. $\operatorname{BProj}(C_1, C_2)$ satisfies the following three properties:

a) $0 \leq \operatorname{BProj}\left(C_1, C_2\right) \leq 1$.

b) $\operatorname{BProj}\left(C_1, C_2\right) = \operatorname{BProj}\left(C_2, C_1\right)$.

c) $\operatorname{BProj}\left(C_1, C_2\right) = 1$ if and only if $C_1 = C_2$.

Proof. a) The conclusion clearly holds as obtained through Eq (3.11).

b) Through Eq (3.11), Theorem 1 b) clearly holds as well.

c) When

substituting into Eq (3.11), we can obtain that

The converse also holds.

□

3.3.   Example and comparison analysis

In order to demonstrate the rationality of calculating the parameter $r$ of the circular fuzzy set using the least squares method and the rationality of the bidirectional projection measure based on the circular fuzzy set. First, an example is given to compare the integrity of different fuzzy sets in representing human fuzzy information. Another example is provided to compare the superiority of the bidirectional projection similarity method based on circular intuitionistic fuzzy sets over other measurement methods.

Example 1. Suppose a company wants to evaluate the work capabilities of two employees. Corresponding fuzzy membership functions are established. The greater the membership degree and the smaller the non-membership degree, the stronger the work capability is indicated. The universe

represents two employees, and four experts conduct evaluations on them and provide the corresponding membership degrees and non-membership degrees,

The evaluation information of the second employee is

respectively using the intuitionistic fuzzy set, the interval-valued intuitionistic fuzzy set, the circular fuzzy set with parameter $r$ calculated in ^[31], and the circular fuzzy set with parameter $r$ calculated by the least squares method to represent the evaluation information.

By observing Table 1, we can find that when using intuitionistic fuzzy sets to represent the decision-making information of the two employees, it is completely consistent.

However, it is obvious that the evaluation information provided by the four experts is completely inconsistent. The evaluation information interval for the first employee is relatively large, while the evaluation interval for the second employee is more concentrated. Although the IVIFS makes a distinction in terms of numerical values, we find that the evaluation intervals of the two employees differ significantly, and in fact the actual gap is not that large. The CIFS retains as much information as possible from all the original evaluations. The method of calculating the parameter $r$ in ^[31] is vulnerable to the influence of extreme evaluation information. However, choosing the least squares method can effectively avoid this problem.

Example 2. Utilize the existing distances and measures of circular fuzzy sets, as well as the bidirectional projection similarity proposed in this paper, to compare the distances between different circular fuzzy sets. The results are shown in Table 2.

(1) By comparing the circular fuzzy sets of the first group and the second group, we find that the membership degree in $\langle x, \frac{3}{10}\sqrt{2} + 0.1, 0.1; 0 \rangle$ is larger than that in $\langle x, 0.3, 0.4; 0.1 \rangle$, and the non-membership degree in $\langle x, \frac{3}{10}\sqrt{2} + 0.1, 0.1; 0 \rangle$ is smaller than that in $\langle x, 0.3, 0.4; 0.1 \rangle$, so $\langle x, \frac{3}{10}\sqrt{2} + 0.1, 0.1; 0 \rangle$ should be greater than $\langle x, 0.3, 0.4; 0.1 \rangle$. However, the distances calculated by using the Euclidean distance in ^[32] are equal. It is not reasonable. It can be seen that the bidirectional projection measure proposed in this paper can effectively distinguish them.

(2) By observing the circular fuzzy sets in the third group, it can be found that when $A$ and $B$ are completely different, the similarity calculated using the method in ^[41] is 1, indicating complete similarity, which does not conform to the actual situation. However, the similarity calculated by using the bidirectional projection similarity proposed in this paper is 0.9615, which is in line with the actual situation.

(3) By observing the circular intuitionistic fuzzy sets in the fourth and fifth groups, it can be found that only the parameter $r$ of $A$ is different. Through the results, it is found that the parameter $r$ will affect the distance and similarity between circular fuzzy sets, which demonstrates the effectiveness of CIFS compared with intuitionistic fuzzy sets and extended intuitionistic fuzzy sets.

While the aforementioned example does not encompass all possible cases of circular intuitionistic fuzzy sets, it is representative of a subset of such sets. It can be observed that some existing measures for distinguishing different circular intuitionistic fuzzy sets exhibit certain limitations. The bidirectional projection measure proposed in this paper fully considers the effects of angle and distance on the similarity of circular intuitionistic fuzzy sets, and it is found to be more reasonable than some of the existing methods by example.

4.   Application of bidirectional projection measure in group decision making

In this section, for a group multi-attribute decision-making (GMADM) problem, this paper proposes a GMADM method based on the bidirectional projection measure of circular intuitionistic fuzzy sets. The basic procedure of the proposed method is illustrated in the following Figure 5. The detailed basic steps of the proposed method are as follows:

Step 1. For MADM problem, first, the set of alternatives

is determined for the problem. The set of decision criteria are

and the weight vector of the criterion is

$DMs$ represent a group of experts in various fields, denoted by $DM1, DM2, \ldots, DMk$.

Step 2. First, the evaluation linguistic of each expert for each criterion of all alternatives was collected, and then the evaluation linguistic was transformed into corresponding intuitionistic fuzzy pairs according to Table 3.

Table 3 presents the transformation relationship between human language evaluation values and intuitionistic fuzzy numbers, and transforms the evaluation languages of experts into intuitionistic fuzzy numbers. Here, we obtain $k$ matrices

where

and $\tilde{d}_{i j k}$ represents the performance evaluation of the i-th alternative on the j-th criterion by the k-th expert.

Step 3. Use Eq (2.3) to aggregate intuitionistic fuzzy pairs of the same alternative with the same criterion in $k$ matrices

to get the corresponding intuitionistic fuzzy numbers $\langle\mu_{i j}, v_{i j}\rangle$. Then, the corresponding $r$ length is calculated using Eq (2.7), and, subsequently, a matrix

is obtained, where

is a circular intuitionistic fuzzy number representing the evaluation value of the j-th criterion for the i-th alternative.

Step 4. The weight information of each attribute value is also evaluated through the linguistic values provided by experts, such as "very important" and "unimportant", and so on. The linguistic weight information of different criteria is quantified into the corresponding intuitionistic fuzzy numbers based on Table 4. Then, the r-length is calculated using Eq (2.7). We obtain the corresponding weight information matrix

where

represents the weight of the j-th criterion.

Step 5. Construct a weighted decision matrix

where

is calculated by Eq (2.8).

Step 6. Identifying an ideal solution

we categorize the $n$ criteria into two types: one type is the cost criteria $N_1$, and the other type is the benefit criteria $N_2$.

Use the method in Definition 7 to compare the ordinal relations of circular intuitionistic fuzzy sets.

Step 7. Calculate $\operatorname{Bproj}\left(\mathrm{A}_i, \mathrm{Y}\right)$, and rank the alternatives based on the value.

5.   Example analysis

5.1.   Illustrative example

The supplier selection example from ^[33] was chosen for calculations to test the effectiveness of the new approach. The example presented is a fast-moving consumer goods supplier selection problem. The detailed steps of the proposed algorithm are illustrated as follows:

Step 1. First, the set of alternative suppliers for the fast-moving consumer goods (FMCG) is determined, comprising four candidates. There are four alternatives ($A_1, A_2, A_3$, and $A_4$). They respectively represent four suppliers. Based on the literature review and expert opinions, seven decision criteria ($\left\{C_1, C_2, \ldots, C_7\right\}$) were determined for the fast-moving consumer goods (FMCG) supplier selection problem. These seven criteria, are respectively, price, quality, performance, delivery, flexibility, relationship closeness, and reputation. A panel of experts was established to evaluate the problem, consisting of five experts, namely $D M_1$–$D M_5$.

Step 2. Based on the linguistic evaluation values provided in Table 3, the 5 experts evaluated the 7 criteria for the four suppliers, resulting in Table 5. Then, according to Table 3, the linguistic values are transformed into the corresponding intuitionistic fuzzy pairs, resulting in Table 6.

Step 3. According to Eqs (2.3) and (2.7), the 5 matrices $\tilde{D}_k$ are aggregated to obtain the circular intuitionistic fuzzy set decision matrix

as shown in Table 7.

Step 4. Based on the weight information of the criteria provided by the decision-makers, Table 8 is obtained. Based on the transformation relationship of linguistic values into intuitionistic fuzzy numbers provided in Table 4, an intuitionistic fuzzy set weight information matrix is obtained. Then, by applying Eq (2.7), the radius $r$ of the circular intuitionistic fuzzy set is calculated to derive the weight information matrix of the circular intuitionistic fuzzy set (Table 9).

Step 5. Obtain the weighted decision matrix (Table 10) using Eq (2.8).

Step 6. Determining the ideal solution $Y$ (Table 11).

Step 7. Calculate the bidirectional projection similarity of each alternative to the ideal solution (Table 12).

The ranking of the alternative options is as follows:

5.2.   Comparative analysis

In ^[33], the calculated results are:

and we found that the optimal selection results are consistent with those of this paper, but the comparison results between the first and fourth options differ. Observing the weighted decision matrix in Table 10, we can see that $A_1$ has 4 criteria superior to $A_4$, while the other criteria where $A_4$ is better have only a small margin. It can be inferred that $A_1$ should be superior to $A_4$. At the same time, we use the intuitionistic fuzzy set projection method in ^[28] to compute the results for comparison (Table 13).

The optimal choice for all three methods is $A_3$, indicating the effectiveness of the method proposed in this paper. Furthermore, through the previous analysis, it can be concluded that $A_1$ should be superior to $A_2$ and $A_4$, which also demonstrates the rationality and accuracy of the method proposed in this paper.

6.   Managerial implications

This paper pioneers a method of using the geometric projection approach in CIFS to calculate the similarity, providing research ideas for the subsequent calculation of the similarity of related extended circular fuzzy sets. Additionally, relevant decision-making methods are presented, offering a new model for GMADM problems.

7.   Conclusions

Circular intuitionistic fuzzy sets have a stronger capability to express the uncertainty in expert evaluations compared to intuitionistic fuzzy sets. This paper proposes a novel method for calculating the parameter r of CIFS, and applies it to the definition of bidirectional projection measure for CIFS. We introduces a new method for calculating the similarity between circular intuitionistic fuzzy sets and demonstrates its rationality and correctness through examples. The new bidirectional projection measure method not only considers the distance between CIFS, but also takes into account the angles between them. The new bidirectional projection measure method fills the gap in the research of projection measures within the field of circular intuitionistic fuzzy sets. There are still some shortcomings in the research presented in this paper. There has been no further detailed discussion on various relationships between different circles, such as tangency, intersection, and inclusion, and the relationship between these relationships and the similarity among them. This can serve as a future research direction. Moreover, when using the operation operators between circular intuitionistic fuzzy sets, there has been no further comparison of their rationality. This can also be considered as a future research direction.

In further research, this method can be further extended to other types of decision-making data, such as Pythagorean fuzzy sets, spherical fuzzy sets, and so on, consider the process of incorporating parameter r into other extended fuzzy sets. This method can be applied to more complex decision-making problems. And, we may consider whether it can be applied to more problems, such as pattern recognition, risk assessment, and other issues.

Use of Generative-AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The author declares no conflict of interest regarding the publication for the paper.