Circular Intuitionistic Fuzzy ELECTRE III Model for Group Decision Analysis

1Introduction

Decision theory is a rapidly evolving field, particularly in the context of multi-criteria and multi-actor (or group) decision-making processes. This process entails ranking, selecting, or assigning a set of alternatives that are evaluated based on several conflicting criteria, typically performed by a group of individuals. The input data for this procedure can originate from various types of information, including quantitative and qualitative data. In many cases, this information may be subject to imprecision, ambiguity, or uncertainty due to measurement errors, imperfect knowledge, subjectivity, and other factors. Consequently, many decision-making models in the literature have been proposed to incorporate fuzzy set (FS) theory as a means of addressing this issue (Zadeh, 1965). In FS, each element of a set is characterized by a membership degree with a value in the unit interval [0,1], while the non-membership degree complements it. For instance, in the context of a stock selection problem, when evaluating the potential of a particular stock, an investor assigns a value of 0.89, signifying a favourable view on 89% of the stock’s potential, while holding a contrary opinion on 11%. The value 0.89 denotes the membership degree, whereas 0.11 represents the non-membership degree, with both initially defined as precise values. Subsequently, FS has expanded to include interval-valued FS (IVFS) (Zadeh, 1975), type-2 FS (T2FS) (Zadeh, 1965), and hesitant FS (HFS) (Torra and Narukawa, 2009), among others, to handle the issue of uncertain membership degrees instead of precise membership degrees. Some extensive reviews on the development of fuzzy set theory and its application in decision analysis can be referred to in the works of Dubois (2011), Abdullah (2013) and Kahraman et al. (2016).

In some other studies, researchers have argued that there are cases that go beyond the complementary nature of membership and non-membership degrees, involving indeterminate or incomplete information. Atanassov (1986) introduced intuitionistic fuzzy set (IFS) as an extension of FS, where membership and non-membership degrees can exist independently within the unit interval, and the sum of the two degrees can be less than one. Additionally, a hesitancy degree was introduced to complement the membership and non-membership degrees. Similarly, IFS has undergone several developments, including the proposal of interval-valued IFS (IVIFS) (Atanassov and Gargov, 1989), type-2 IFS (T2IFS) (Zhao and Xiao, 2012), hesitant IFS (HIFS) (Beg and Rashid, 2014), among others, to represent uncertain membership and non-membership degrees (i.e. IF values). A substantial body of literature has reported on the expansion of multi-criteria and multi-actor decision-making models using these sets (Yusoff et al., 2011; Taib et al., 2016; Ecer et al., 2022).

Recently, Atanassov (2020) introduced another extension of IFS, aiming to enhance the flexibility in interpreting and representing IF values. This extension is known as a Circular Intuitionistic Fuzzy Set (CIFS). In CIFS, each element of a set is characterized as a circle, with membership and non-membership degrees determining the centre and radius, denoted as r, which represents the region of uncertainty. The primary distinction between CIFS and IVIFS lies in their representation of imprecision. IVIFS is depicted as a rectangle (bounded by lower and upper bounds of membership and non-membership degrees) within the IFS interpretation triangle (IFIT). In contrast, CIFS expresses its uncertainty region with equidistant boundaries from the centre. Fig. 1 provides a geometrical interpretation of IVIFS and CIFS.

Fig. 1

Geometrical interpretation of (a) IVIFS and (b) CIFS.

The development of the CIFS theory is still in its early stages, and therefore, limited research has been conducted on it. In the initial paper, Atanassov (2020) introduced CIFS, along with some fundamental operations and relations, where the radius was confined to the range of r∈[0,1]. Subsequently, Atanassov expanded the range of the radius to [0,2] in order to encompass the entire IFIT and also defined distance measures for CIFS (Atanassov and Marinov, 2021). Since then, there have been several studies aimed at advancing CIFS both in theory and practical applications. These include extensions of present worth analysis within the CIFS framework (Boltürk and Kahraman, 2022), distance and divergence measures tailored for CIFS (Khan et al., 2022), distance measure based on the theory of CIFS, considering the information of four aspects: membership degree, non-membership degree, radius and the assignment of hesitation degree (Xu and Wen, 2023), circular q-rung orthopair fuzzy set and some algebraic properties (Yusoff et al., 2023), generalized CIFS (Pratama et al., 2023), circular pythagorean fuzzy sets (Bozyiğit et al., 2023) and its applications in decision-making models (Çakir and Taş, 2023), particularly in Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) (Kahraman and Alkan, 2021; Alkan and Kahraman, 2022; Chen, 2023), as well as Multicriteria Optimization and Compromise Solution (VIKOR) (Kahraman and Otay, 2021; Otay and Kahraman, 2021; Büyükselçuk and Sarı, 2023).

One of the intriguing developments in the context of CIFS in group decision-making problem was introduced by Kahraman and Alkan (2021). Specifically, they proposed the inclusion of psychological behaviours within TOPSIS model under the CIFS environment, known as CIF-TOPSIS. These characteristics involve optimistic and pessimistic attitudes, which are represented as IF values derived from CIF values. The optimistic value is defined as the sum of the membership degree and the subtraction of the non-membership degree, both adjusted by the radius, r. Conversely, the pessimistic value is defined in the opposite manner. In general, the optimistic value reflects the group’s inclination toward higher membership degrees (validity), while the pessimistic value indicates a tendency toward higher non-membership degrees (non-validity). However, these operations have shown certain limitations. Specifically, they may result in optimistic and pessimistic values falling outside the circular area. Ideally, all values should remain within the circular region to accurately represent the data. Furthermore, Kahraman and Alkan (2021) restricted the radius, r to the range [0,1] in CIF-TOPSIS, which covers only a certain region within the IFIT. This limitation may lead to incomplete representation and is not inclusive of the entire dataset.

In addition to the models mentioned above, ELECTRE (ELimination Et Choix Traduisant la REalité – elimination and choice expressing reality) is another well-established approach for handling multi-criteria and multi-actor decision-making problems. It was developed in the late 1960 by Roy (1968). ELECTRE is a comprehensive outranking relation model with several variants designed for different decision problems, including ranking, selection, and assignment of alternatives. These variants include ELECTRE I (Roy, 1968), ELECTRE II (Roy and Bertier, 1973), ELECTRE III (Roy, 1978), ELECTRE IV (Hugonnard and Roy, 1982), and more. Numerous studies have explored the use of ELECTRE models based on fuzzy concepts and their development, including their application in FS (Mohamadghasemi et al., 2020), IFS (Rouyendegh, 2017; Qu et al., 2018). Notably, ELECTRE III, which relies on fuzzy outranking relations, has been widely utilized for its effectiveness in the ranking process. Numerous developments related to FS, IFS, and IVFS have been documented in the literature (Joshi, 2016; Hashemi et al., 2016; Peng et al., 2019; Ramya et al., 2023; Forestal and Pi, 2022). To the best of our knowledge, there has been no study proposing the integration of CIFS into ELECTRE models, particularly ELECTRE III.

Given the aforementioned limitations and research gaps, the research contribution of this paper is to achieve the following main objectives.

The remainder of this paper is structured as follows: Section 2 provides fundamental definitions and insights into IFS, CIFS, and ELECTRE III to clarify the proposed approach. In Section 3, we present a method for converting the group decision matrix and group weighting vector, based on CIF values, into optimistic and pessimistic forms of IF values. Section 4 introduces the proposed CIF-ELECTRE III model. We illustrate the application of this model with a numerical example in the context of stock selection in Section 5. Section 6 covers sensitivity analyses and a comparative discussion. In the conclusion (Section 7), we offer recommendations for future research.

2Preliminaries

This section recalls some basic definitions and related concepts of IFS, CIFS, and ELECTRE III model prior to the development of the proposed CIF-ELECTRE III.

2.2Circular Intuitionistic Fuzzy Set

Definition 3

Definition 3(Atanassov, 2020).

Let X be a finite non-empty set. A Circular Intuitionistic Fuzzy Set Ar (denoted CIFS Ar) in X can be defined as:

(4)

Ar={⟨x,μA(x),νA(x);r⟩|dx∈X},

where 0⩽μA(x)+νA(x)⩽1 for every x∈X, functions μA:X→[0,1] and νA:X→[0,1] are membership and non-membership degrees, respectively, and also r∈[0,1] is a radius of the circle around each element x∈X.

Similarly, the hesitancy function πA can be defined as πA(x)=1−(μA(x)+νA(x)) for every x∈X. Under the IFS interpretation triangle (IFIT), each element of CIFS is represented by a circle with centre at ⟨μA(x),νA(x)⟩ and radius, r>0. Note that, when r=0, CIFS is reduced to IFS (i.e. a single point). Alternatively, CIFS can also be defined as the following.

Let L∗={(m,n)|m,n∈[0,1]andm+n⩽1}, be an expression of L-fuzzy set, then Ar can be written in the form:

(5)

Ar={⟨x,Or(μA(x),νA(x))⟩|x∈X},

where:

Or(μA(x),νA(x))={(m,n)|m,n∈[0,1]and(μA(x)−m)2+(νA(x)−n)2⩽r}∩L∗={(m,n)|m,n∈[0,1]and(μA(x)−m)2+(νA(x)−n)2⩽randm+n⩽1}.

Remark 1.

Note that Definition 3 with radius r∈[0,1] is not exhaustive as the IFS triangle is only partly covered.

Recently, Atanassov and Marinov (2021) expanded the radius to r∈[0,2] with consideration that if the centre is at ⟨0,1⟩ or ⟨1,0⟩, the smallest radius distance needed to completely cover the IFS triangle is 2. The geometrical interpretation of CIFS for radii r=1 and r=2 are depicted in Fig. 2 (a) and (b), respectively.

Fig. 2

Geometrical interpretation of CIFS for (a) r=1 and (b) r=2.

Two different ways for constructing of CIFSs have been proposed recently (Atanassov, 2020): i) based on the geometrical interpretations of the standard IFSs (Atanassova, 2010), and ii) based on hesitant IFS (Torra, 2010; Chen et al., 2016). In the following, the algorithm for building the CIFS from IFSs is presented.

Definition 4

Definition 4(Atanassov, 2020).

Let gj be an element formed by a set of kj intuitionistic fuzzy pairs {(mj1,nj1),(mj2,nj2),…,(mjkj,njkj)}, where j=1,2,…,n. The membership function μ(gj) and non-membership function ν(gj) of CIFS can be obtained as follows:

(6)

⟨μ(gj),ν(gj)⟩=⟨Σh=1kjmjhkj,Σh=1kjnjhkj⟩,

and the radius, rj is the maximum of the Euclidean distance such that:

(7)

rj=max1⩽h⩽kj|(μ(gj)−mjh)2+(ν(gj)−njh)2|,

where kj is the number of input sources for gj. Then, for universe F={g1,g2,…,gn}, the CIFS can be represented in the following form:

(8)

Ar={⟨gj,μ(gj),ν(gj);r(gj)⟩|gj∈F}={⟨gj,Or(μ(gj),ν(gj))⟩|gj∈F}.

The above expressions, under the group decision setting, can simply be assumed as the aggregated values of a group of actors h=1,2,…,k with respect to each criterion j=1,2,…,n. The radius r can be considered as the bounded uncertain region between the collective value of group (centre of a circle) and the individual actors’ values. Kahraman and Alkan (2021) then proposed the conversion of CIF values to IF values by defining the concept of optimistic and pessimistic values, i.e. the attitudinal characters of the group of actors. By taking the effect of radius r on both membership and non-membership degrees, the IF values for optimistic and pessimistic can be generated.

Definition 5

Definition 5(Kahraman and Alkan, 2021).

Let A=⟨μj,νj;rj⟩ be a CIF value. Then the conversion of CIF value to IF values can be obtained as follows:

(9)

Ao=⟨μjo,νjo⟩=⟨μj+rj,νj−rj⟩,

(10)

Ap=⟨μjp,νjp⟩=⟨μj−rj,νj+rj⟩,

where Ao and Ap are termed optimistic IF value and pessimistic IF value, respectively.

Remark 2.

Note that Definition 5 has a limitation as these optimistic and pessimistic IF values are not within the circular area (or radius r) in the IFS triangle.

For example, the distance between the optimistic value and the centre of CIF value is obtained as:

(μjo−μj)2+(νjo−νj)2=(μj+rj−μj)2+(νj−rj−νj)2=(rj)2+(−rj)2=rj2≠rj.

Analogously, this also applies to the pessimistic value, where (μjp−μj)2+(νjp−νj)2=(−rj)2+(rj)2=rj2≠rj.

3Group Decision Matrix and Group Weighting Vector Based on CIFS

This section is devoted to the conversion of group decision matrix and group weighting vector to optimistic and pessimistic forms. In the previous work, Kahraman and Alkan (2021) proposed the formulation in Eqs. (9) and (10) to transform the CIF value to optimistic and pessimistic IF values, respectively. However, as stated in Remark 2, these values are not within the circular area of radius r in the IFS interpretation triangle. Hence, this formulation needs to be redefined. In the following, the definitions of group decision matrix and group weighting vector based on CIFS and their conversion to optimistic and pessimistic IF values are presented.

Upon checking the above definition, dijo and dijp clearly satisfy the condition of standard IFS, as can be demonstrated below:

Based on the Definiton 7, the concept of optimistic and pessimistic for weighting vector can be defined accordingly.

Likewise, the condition for standard IFS is fulfilled for wjo and wjp such that: 0⩽μjo+νjo⩽1 and 0⩽μjp+νjp⩽1. The difference between the proposed conversion IF values and the one used in Kahraman and Alkan (2021) is depicted in Fig. 3.

Fig. 3

Comparison of optimistic-pessimistic IF values for the proposed approach and Kahraman and Alkan’s approach.

Moreover, it can easily be shown that the distance between the optimistic IF value, dijo=⟨μijo,νijo⟩, such that μijo,νijo∈[0,1], and the centre of the CIF value dij=⟨μij,νij⟩ is exactly rij, which satisfies the boundary condition:

Analogously, it can be shown for dijp=⟨μijp,νijp⟩, such that μijp,νijp∈[0,1], its distance from the centre of the CIF value dij=⟨μij,νij⟩ is also rij. The same are true for wjo,wjp∈Orj(μj,νj), which results in optimistic and pessimistic in the range of Orj(μj,νj). However, there are two cases in optimistic and pessimistic IF values that the above condition is not satisfied. Specifically, either one of the degrees is not within [0,1] or both degrees are not elements of [0,1]. Let dijo=⟨μijo,νijo⟩ be an optimistic IF value (or dijp=⟨μijp,νijp⟩ be an pessimistic IF value), then:

Case (1): there is a possibility that φ⩽μijo⩽1, but νijo<0 occurs in the case of optimistic (or μijp<0, but φ⩽νijp⩽1 in the case of pessimistic), where φ=rij22.

For example, dij=⟨0.4,0.3;0.5⟩, then dijo=⟨μijo,νijo⟩=⟨0.75,−0.05⟩. This clearly contradicts the condition of IFS which is νijo≠[0,1].

Case (2): there is possibility that μijo>1 and νijo<0 occur in case of optimistic (or in case of pessimistic: μijp<0 and νijp>1).

For example, dij=⟨0.7,0.3;0.5⟩, then dijo=⟨μijo,νijo⟩=⟨1.05,−0.05⟩. This also contradicts the condition of IFS, which is μijo,νijo∉[0,1]. Fig. 4 demonstrates these two cases.

Fig. 4

Special cases for optimisitic decision element, when (a) νijo<0 and (b) μijo>1, νijo<0.

This leads to the following theorems for the elements of Do and Dp, as well as the Wo and Wp.