ELECTRE III is a well-established outranking relation model used to address the ranking of alternatives in multi-criteria and multi-actor decision-making problems. It has been extensively studied across various scientific fields. Due to the complexity of decision-making under uncertainty, some higher-order fuzzy sets have been proposed to effectively model this issue. Circular Intuitionistic Fuzzy Set (CIFS) is one such set recently introduced to handle uncertain IF values. In CIFS, each element of the set is characterized by a circular area with a radius,

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,

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 (

Recently, Atanassov (

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 (

One of the intriguing developments in the context of CIFS in group decision-making problem was introduced by Kahraman and Alkan (

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 (

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

To explore the CIFS theory and integrate it with the ELECTRE III model for group decision analysis, extending the model’s capabillities.

To redefine the operations used to generate optimistic and pessimistic IF values from the group decision matrix and group weighting vector. Additionally, suggest conditional rules to ensure that every element of the set remains within the circular area.

To demonstrate the applicabillity of the proposed model through a case study of stock-picking process.

To conduct a comprehensive comparative analysis with existing models and then justify sensitivity analyses to explore the impact of

The remainder of this paper is structured as follows: Section

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.

Let

Several metric methods in terms of score and accuracy functions have been presented to compare two or more IF values (see, for example, Garg,

For IFS

Moreover, let

If

If

if

if

Let

Similarly, the hesitancy function

Let

Note that Definition

Recently, Atanassov and Marinov (

Geometrical interpretation of CIFS for (a)

Two different ways for constructing of CIFSs have been proposed recently (Atanassov,

Let

The above expressions, under the group decision setting, can simply be assumed as the aggregated values of a group of actors

Let

Note that Definition

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

ELECTRE III is a preference model for ranking purposes. In a general case, it is used for multi-criteria decision-making problem (single actor). However, it also can be directly extended to the case of multi-actor (or group) problem. The following are the important notations related to the basic data and the construction of ELECTRE III model.

Let

Moreover, under the group decision setting, let

The main feature of ELECTRE III compared to the other variants of ELECTRE family is a type of preference so-called pseudo-criterion. Pseudo-criterion is a multilevel threshold approach, as can be defined in the following.

A pseudo-criterion is a function

The main process in ELECTRE III is the pairwise comparison of alternatives with respect to each criterion. These pairwise comparisons are presented in concordance and discordance indices. The concordance index

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 (

Let

Upon checking the above definition,

Based on the Definiton

Let

Likewise, the condition for standard IFS is fulfilled for

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,

Analogously, it can be shown for

Case (1): there is a possibility that

For example,

Case (2): there is possibility that

For example,

Special cases for optimisitic decision element, when (a)

This leads to the following theorems for the elements of

It is clear that, for

For

For

For

The proof of this theorem is straightforward based on proofs of Theorem

The final outranking procedure in the CIF-ELECTRE III model is significantly influenced by the aformentioned definitions and theorems. For both the optimistic and pessimistic scenarios, separate scores underlie the ranking process. After identifying the optimistic and pessimistic scores, the composite ratio is defined. This ratio parameterizes the attitudinal character of the group of actors in order to determine the final score before the ranking process.

Let

At the end, this composite ratio,

Framework of CIF-ELECTRE III.

In this section, a novel CIF-ELECTRE III is presented. The framework of the proposed model is demonstrated in Fig.

As a case study of the stock-picking process, this section shows how the suggested strategy was put into practice. Specifically, an investor intends to invest in one of the pre-screened stocks in Bursa Malaysia (i.e. under the technology sector). To finalize the decision, a set of criteria based on the fundamental analysis is considered for the further appraisal of stocks, i.e. using the forward-looking scenario analysis. The details of the decision-making process are demonstrated as follows.

Three financial analysts

Initially, using the predetermined linguistic scale such in Table

Linguistic scale for (a) rating of alternatives and (b) weighting of criteria.

Linguistic terms | IFVs for alternatives | |

m | n | |

VVG/VVH | 0.9 | 0.1 |

VG/VH | 0.7775 | 0.0625 |

G/H | 0.6775 | 0.1625 |

MG/MH | 0.5775 | 0.2625 |

F/M | 0.4775 | 0.3635 |

MB/ML | 0.3775 | 0.4625 |

B/L | 0.2188 | 0.5438 |

VB/VL | 0.0688 | 0.6938 |

VVB/VVL | 0.1 | 0.9 |

(a) |

Linguistic terms | IFVs for alternatives | |

m | n | |

VI | 0.9 | 0.1 |

I | 0.5813 | 0.1058 |

M | 0.3313 | 0.3563 |

U | 0.1813 | 0.5063 |

VU | 0.1 | 0.9 |

(b) |

Linguistic decision matrix for each analyst.

Criteria | Decision makers | |||||

VG | MG | VG | F | B | ||

VG | G | VG | MB | B | ||

G | MG | VG | F | B | ||

VG | VG | MG | MB | B | ||

G | VG | G | F | F | ||

MG | VG | MG | MG | B | ||

G | F | VG | F | F | ||

F | G | VVG | MG | MG | ||

G | MG | VG | F | MG | ||

F | G | MG | G | F | ||

F | VG | VG | F | MG | ||

F | G | G | F | G |

At this stage, the linguistic data provided by analysts are converted to their corresponding IF values to compute the group decision matrix

Group decision matrix

Optimistic decision matrix,

Optimistic decision matrix, |
Pessimistic decision matrix, |
|||||||

From the group decision matrix (Table

Indifferent, preference and veto thresholds.

L | L | M | M | |

MH | ML | MH | H | |

VH | H | VH | VH |

Deterministic format of optimistic-pessimistic decision matrix and thresholds.

0.782 | 0.715 | 0.648 | 0.115 | 0.515 | 0.315 | 0.115 | 0.115 | |

0.515 | 0.715 | 0.515 | 0.715 | 0.248 | 0.715 | 0.115 | 0.448 | |

0.715 | 0.515 | 0.715 | 0.715 | 0.715 | 0.248 | 0.715 | 0.315 | |

0.191 | 0.347 | 0.315 | 0.515 | −0.085 | −0.085 | 0.048 | −0.018 | |

−0.325 | 0.120 | 0.382 | 0.515 | −0.325 | −0.476 | 0.115 | 0.115 | |

−0.325 | −0.325 | 0.115 | 0.115 | −0.325 | −0.325 | 0.115 | 0.115 | |

0.315 | −0.085 | 0.315 | 0.515 | 0.315 | −0.085 | 0.315 | 0.515 | |

0.715 | 0.515 | 0.715 | 0.715 | 0.715 | 0.515 | 0.715 | 0.715 |

Linguistic preferences for criteria weights provided by analysts.

VI | I | I | U | |

VI | I | M | M | |

I | I | I | U |

Optimistic weight and pessimistic weight for criteria.

(0.794,0.102;0.213) | (0.581,0.106;0) | (0.498,0.189;0.236) | (0.231,0.456;0.141) | |

(0.98,0) | (0.581,0.106) | (0.665,0.02) | (0.33,0.355) | |

(0.643,0.252) | (0.581,0.106) | (0.331,0.356) | (0.131,0.556) | |

0.99 | 0.738 | 0.821 | 0.487 | |

0.696 | 0.738 | 0.488 | 0.287 |

After that, determine the optimistic concordance index

Optimistic and pessimistic concordance indices.

1 | 0.567 | 0.581 | 0.886 | 0.886 | 1 | 0.566 | 0.159 | 1 | 1 | ||

0.431 | 1 | 0.491 | 1 | 1 | 0.709 | 1 | 0.464 | 1 | 1 | ||

0.557 | 0.693 | 1 | 0.873 | 1 | 0.604 | 0.66 | 1 | 1 | 1 | ||

0.161 | 0.282 | 0.126 | 1 | 0.869 | 0.345 | 0.236 | 0.059 | 1 | 0.952 | ||

0.226 | 0.372 | 0.126 | 0.431 | 1 | 0.351 | 0.280 | 0.103 | 0.388 | 1 |

Optimistic discordance indices for each criterion.

Criteria 1 | Criteria 2 | Criteria 3 | Criteria 4 | |||||||||||||||||

1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0.425 | 0.425 | 0 | 0 | |

0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | |

0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | |

0.713 | 0.046 | 0.546 | 1 | 0 | 0.713 | 0.046 | 0.546 | 1 | 0 | 0.046 | 0 | 0.213 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | |

1 | 1 | 1 | 0.479 | 1 | 1 | 1 | 1 | 0.479 | 1 | 0 | 0 | 0.046 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |

Pessimistic discordance indices for each criterion.

Criteria 1 | Criteria 2 | Criteria 3 | Criteria 4 | ||||||||||||||||

1 | 0 | 0 | 0 | 0 | 1 | 0.808 | 0.059 | 0 | 0 | 1 | 0 | 0.713 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

0 | 1 | 0.379 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0.713 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

0 | 0 | 1 | 0 | 0 | 0.253 | 0.919 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

0.713 | 0.046 | 1 | 1 | 0 | 0.808 | 1 | 0.697 | 1 | 0 | 0 | 0 | 0.879 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |

1 | 0.646 | 1 | 0 | 1 | 1 | 1 | 1 | 0.794 | 1 | 0 | 0 | 0.713 | 0 | 1 | 0 | 0 | 0 | 0 | 1 |

The credibility index for (a) optimistic and (b) pessimistic.

Based on the result, the first ranking is stock,

Final ranking and score.

Alternatives | Score | Rank |

1.43 | 3 | |

2.09 | 2 | |

2.22 | 1 | |

−2.14 | 4 | |

−3.60 | 5 |

To check the robustness and stability of the proposed model, some sensitivity analyses, as well as a comparison analysis are carried out. These analyses are conducted to analyse various scenarios that might amend the result of the proposed model. First, a sensitivity analysis with respect to

The score alternatives with different

Degree of optimisitic attitude |
|||||||||||

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |

0.95 | 1.04 | 1.14 | 1.24 | 1.34 | 1.43 | 1.53 | 1.63 | 1.72 | 1.82 | 1.92 | |

2.59 | 2.49 | 2.39 | 2.29 | 2.19 | 2.09 | 1.99 | 1.89 | 1.79 | 1.69 | 1.59 | |

2.42 | 2.38 | 2.34 | 2.40 | 2.26 | 2.22 | 2.18 | 2.14 | 2.10 | 2.06 | 2.02 | |

−2.13 | −2.13 | −2.13 | −2.13 | −2.13 | −2.14 | −2.14 | −2.14 | −2.14 | −2.14 | −2.14 | |

−3.82 | −3.78 | −3.73 | −3.69 | −3.65 | −3.60 | −3.56 | −3.52 | −3.47 | −3.43 | −3.39 |

Result of CIF-ELECTRE III.

In Table

Sensitivity analysis of CIF-ELECTRE III based on (a) criteria weights and (b) thresholds.

Comparison of ELECTRE III under different sets and CIF-TOPSIS.

Model | Information | Ranking |

Proposed model | ||

IF-ELECTRE III | – | |

IVIF-ELECTRE III | – | |

CIF-TOPSIS | ||

Furthermore, we compare the ranking results of our proposed model with ELECTRE III under two different sets: IF-ELECTRE III and IVIF-ELECTRE III (Hashemi

In Table

Moreover, for a more in-depth analysis of the effect of the final decision concerning the optimistic and pessimistic attitudes, we apply the tranquillity measure proposed by Yager (

Tranquillity measures on CIF-TOPSIS and CIF-ELECTRE III for

CIF-TOPSIS | 0.2649 | 0.2714 | 0.2778 | 0.2843 | 0.2907 | 0.2937 | 0.2946 | 0.2913 | 0.2879 | 0.2846 | 0.2812 |

CIF-ELECTRE III | 0.3676 | 0.3589 | 0.3498 | 0.3417 | 0.3437 | 0.3457 | 0.3478 | 0.3500 | 0.3522 | 0.3463 | 0.3363 |

As shown in Table

In this paper, we propose an extension of the ELECTRE III model within the context of the CIFS environment for group decision analysis. We introduce several extensions to the group decision matrix (referred to as the CIF decision matrix) and the group weighting vector (referred to as the CIF weighting vector). These extensions specifically address CIFS conditions, focusing on optimistic and pessimistic attitudes. We construct these attitudinal attributes based on a set of conditional rules, ensuring that every element remains confined within a circular area defined by a radius

However, our model has certain limitations and room for future development. Firstly, the model currently represents the attitudinal character of the entire group and does not account for individual actor attitudes. Secondly, it focuses exclusively on homogeneous group decision-making scenarios. Thirdly, in this model, CIF data is simplified to IF data for computational convenience. Future work could involve addressing individual actor attitudinal characteristics separately, accommodating heterogeneous group decision-making, and developing an algorithm that does not require converting CIF information to IF values. Lastly, it is essential to note that this study primarily deals with a simplified case involving a small group of just three experts. Discrepancies in final rankings may arise in large-scale group decision-making (LS-GDM) and more complex decision-making scenarios. Therefore, further application and analysis of our proposed method under LS-GDM are warranted.

The authors express their gratitude to the anonymous reviewers for their valuable comments and remarks that have improved the quality of the paper.