T.-Y. Chen is a professor at Chang Gung University, Taiwan, affiliated with the Department of Industrial and Business Management and the Graduate Institute of Management. She holds a BS in transportation engineering, an MS in civil engineering, and a PhD in traffic and transportation from National Chiao Tung University. Her research focuses on multi-criteria decision analysis, fuzzy decision-making, and intelligent management support, earning her recognition as an Honourary Member of the Phi Tau Phi Scholastic Honour Society.
This paper presents a multiple-criteria choice model, the circular intuitionistic fuzzy (C-IF) ELECTRE, designed to resolve C-IF ambiguities through built-in circular functions. Joint generalized scoring functions establish contrast relationships between C-IF evaluation values, facilitating concordance and discordance analyses for option ranking. The efficacy of C-IF ELECTRE I and II—leveraging tools such as the prioritization Boolean matrix, average outflows and inflows, and overall net flow—is validated through a multi-expert supplier evaluation, with outcomes benchmarked against alternative methods. A comparative analysis explores the impact of parameter variations, underscoring how integrating C-IF sets with ELECTRE enhances decision-making in complex, multifaceted environments.
multiple-criteria choice modelcircular intuitionistic fuzzy (C-IF)ELECTREjoint generalized scoring functioncomparative analysisThe author sincerely thanks the National Science and Technology Council, Taiwan (NSTC 111-2410-H-182-012-MY3), and Chang Gung Memorial Hospital, Linkou, Taiwan (BMRP 574), for their essential financial support.Introduction
Multiple-criteria analysis is crucial for addressing decision-making challenges in practical scenarios (Chen, 2024; Liu, 2024). In the ELECTRE (i.e. ÉLimination Et Choix Traduisant la REalité in French) framework, evaluative criteria serve as standards to assess and prioritize options based on their performance (Akram et al., 2023a, 2023b). However, uncertainty can disrupt the ELECTRE process, leading to inaccurate assessments, inconsistent decisions, and suboptimal choices, undermining decision-makers’ confidence (Ramya et al., 2023; Wu et al., 2023). Insufficient or unreliable data may introduce biases, distort rankings, and increase subjective judgments, compromising reliability (Liu et al., 2023; Yüksel and Dinçer, 2023). This heightened risk complicates the appraisal of options and limits the efficacy of ELECTRE-based analysis (Zhang et al., 2023; Zhou et al., 2022).
To meet the demands of uncertain environments, researchers have expanded the ELECTRE methodology to various fuzzy scenarios. Beyond conventional fuzzy models, recent efforts have focused on higher-order fuzzy frameworks. Akram et al. (2023b) introduced the fuzzy ELECTRE IV method using triangular fuzzy numbers for imprecise data. Wu et al. (2023) developed a triadic decision model combining ELECTRE III with attribute ratio evaluation in a spherical fuzzy setting for customer selection. Wang and Chen (2021) used T-spherical fuzzy score functions and Minkowski distance-dependent indices to design T-spherical fuzzy ELECTRE I and II methods. Yüksel and Dinçer (2023) evaluated sustainability in circular industrialization using quantum spherical fuzzy ELECTRE. Zhang et al. (2023) created an ELECTRE II method with cosine similarity measures to assess financial logistics firms’ efficiency using double hierarchy hesitant fuzzy information. Ramya et al. (2023) proposed a disposal technique selection framework for e-waste using ELECTRE III with wiggly hesitant Pythagorean fuzzy sets. Zhou et al. (2022) developed a Fermatean fuzzy ELECTRE method using Jensen-Shannon divergence and cross-entropy to handle uncertain decision-maker and criteria weights. Akram et al. (2023a) investigated a 2-tuple linguistic Fermatean fuzzy ELECTRE II for group decision-making with linguistic variables. Pinar and Boran (2022) introduced a q-rung picture fuzzy ELECTRE model, integrating the technique for order preference by similarity to ideal solutions (TOPSIS) for group decision analysis. These advancements are a result of continuous work to improve the applicability and efficacy of ELECTRE-based strategies for tackling challenging and ambiguous decision-making situations.
Applying fuzzy extensions of the ELECTRE-based outranking methodology to multi-criteria analysis enhances decision-making, particularly in uncertain environments. These approaches incorporate fuzzy logic to handle uncertainty, offering deeper insights for decision-makers. However, advancements specifically tailored to the emerging circular intuitionistic fuzzy (C-IF) sets remain limited. Intuitionistic fuzzy (IF) sets, introduced by Atanassov (1986), include degrees of hesitancy alongside membership and non-membership, addressing vagueness more effectively than conventional fuzzy sets (Chen, 2024; Liu, 2024). Their integration with various tools enables more precise handling of uncertain information (Liu et al., 2023). Atanassov later extended IF sets into C-IF sets, which represent circles with radii encompassing membership and non-membership components (Atanassov, 2020; Çakır and Taş, 2023). C-IF sets have demonstrated versatility across realistic applications (Alinejad et al., 2024; Ci, 2024; Kong, 2024) and hold promise for future advancements in ELECTRE-based methods.
C-IF sets offer greater flexibility by integrating membership, hesitancy, and non-membership components (Ci, 2024; Jameel et al., 2024). Represented by circular shapes, they capture multiple qualities or attributes, with the circular function enhancing their expressiveness in a triangular distribution space (Kong, 2024; Pratama et al., 2024). When the circle’s radius is set to zero, C-IF sets revert to standard IF sets. C-IF sets excel in handling uncertain, imprecise information, making them suitable for diverse decision-making tasks (Alinejad et al., 2024; Chen, 2023a). Applications include decision assistance and support (Chen, 2023b, 2024), supplier evaluation and selection (Çakır and Taş, 2023; Chen, 2024), sustainable renewable energy management (Jameel et al., 2024), biomass resource strategies (Alinejad et al., 2024), food supply chain monitoring (Alsattar et al., 2024), pattern recognition and medical diagnosis (Khan et al., 2022), and public health risk assessments (Kong, 2024). These examples underscore the adaptability and utility of C-IF sets for addressing uncertainty in practical applications across various domains.
The versatility of C-IF sets makes them valuable for decision-making; however, effective analysis requires reliable methods to interpret and manipulate uncertain information (Pinar and Boran, 2022; Zhang et al., 2023). Chen (2023b) formulated two generalized C-IF distance metrics – three-term and four-term Minkowski-like measures – designed to handle imperfect information. The four-term approach accounts for radius, membership, non-membership, and hesitancy, offering a complete representation of C-IF numbers. In contrast, the three-term model excludes hesitancy. This study adopts the four-term model for its comprehensive nature, providing a solid foundation for developing a C-IF ELECTRE approach to better address research objectives.
C-IF sets are highly effective for managing complex and ambiguous information. However, there remains a significant gap in developing ELECTRE-based methods tailored for C-IF decision environments. While existing ELECTRE approaches have shown success in handling uncertainty across various fuzzy frameworks – including spherical fuzzy (Wu et al., 2023), T-spherical fuzzy (Wang and Chen, 2021), quantum spherical fuzzy (Yüksel and Dinçer, 2023), double hierarchy hesitant fuzzy (Zhang et al., 2023), normal wiggly Pythagorean hesitant fuzzy (Ramya et al., 2023), Fermatean fuzzy (Zhou et al., 2022), 2-tuple linguistic Fermatean fuzzy (Akram et al., 2023a), and q-rung picture fuzzy (Pinar and Boran, 2022) – their application within C-IF contexts remains limited. This study addresses this gap with the following motivations:
Uncertainty Challenges: Current ELECTRE-based approaches struggle with complex uncertainties, highlighting the need for more robust and enhanced methods.
Limited C-IF Applications: Despite progress in C-IF methods, there has been little integration of ELECTRE in C-IF multiple-criteria analysis, highlighting a key research gap.
Distance Measurement Importance: Accurate measurement of C-IF distances is crucial for distinguishing complex information. This study adopts the four-term Minkowski-like distance model for its completeness in developing the C-IF ELECTRE approach.
This research aims to develop the C-IF ELECTRE, a multiple-criteria decision model for discrete decisions involving conflicting or incomparable criteria. It integrates circular intuitionistic fuzziness with a joint generalized scoring function, based on Hezam et al.’s (2023) natural exponential function, to address uncertainty. The concepts of aggressive and cautious IF estimates (inspired by Chen, 2023a) are employed to provide upper and lower estimations within the C-IF context, with several theorems outlining their properties and relationships. The C-IF ELECTRE approach involves establishing concordance and discordance sets to determine when one alternative is superior, equal, or inferior to another. C-IF ELECTRE I uses consistency and inconsistency indicators to build a dominance graph for partial-priority rankings. C-IF ELECTRE II introduces consistency-dependent outflow, inconsistency-dependent inflow, and net flow values for complete-priority rankings. Applied to supplier evaluations, the method aligns with comparative results and highlights the impact of parameter settings, such as distance and divergence measures, on ranking outcomes.
This study offers key advancements in ELECTRE-based decision-making:
Joint Generalized Scoring Function: Introduces an inclination parameter to capture the decision-maker’s aggressive, neutral, or cautious tendencies, integrating these attitudes into assessments.
C-IF ELECTRE Framework: Develops a practical model tailored for C-IF contexts with step-by-step algorithms for defining problems, calculating scores, measuring consistency/inconsistency, and generating rankings.
C-IF ELECTRE I and II Techniques: Combines C-IF sets with ELECTRE to manage complex multi-criteria decisions, validated through multi-expert supplier assessments, producing reliable, consistent rankings aligned with comparative methods.
This article is structured as follows: Section 1 highlights research motivations and the need for innovations in ELECTRE-based methodologies for C-IF contexts. Section 2 covers fundamental mathematical notations for IF and C-IF configurations. Section 3 develops a joint generalized scoring function to address C-IF uncertainty. Section 4 introduces the C-IF ELECTRE model for complex decision-making in uncertain contexts. Section 5 demonstrates the application of C-IF ELECTRE I and II techniques through a multi-expert supplier evaluation case. Section 6 analyses the impact of inclination parameter settings on results, highlighting the approach’s advantages. Section 7 provides conclusions and suggests directions for future research.
Foundational Concepts Related to C-IF Sets(Atanassov, <xref ref-type="bibr" rid="j_infor575a_ref_005">1986</xref>)<italic>.</italic>
Assume ℵ is a nonempty set of elements. Let uI(χ) and vI(χ) be functions mapping ℵ to [0,1], representing the degree to which an element χ∈ℵ belongs to or does not belong to an IF set I, respectively, subject to 0⩽uI(χ)+vI(χ)⩽1. The IF set I within ℵ is defined as:
I={⟨χ,uI(χ),vI(χ)⟩|χ∈ℵ}.
Let i(χ)=(uI(χ),vI(χ)) represent an IF number within the IF set I. The degree of hesitancy is given by hI(χ)=1−uI(χ)−vI(χ). The scoring mechanism for i(χ) uses a natural exponential function with Euler’s number e as:
M(i(χ))=12[uI(χ)−vI(χ)+hI(χ)·(e(uI(χ)−vI(χ))e(uI(χ)−vI(χ))+1−12)+1].
Let L∗ represent an L-fuzzy set, defined as L∗={⟨ℓ,ℓ′⟩|ℓ,ℓ′∈[0,1]andℓ+ℓ′⩽1}. The membership degree uC(χ):ℵ⟶[0,1] and non-membership degree vC(χ):ℵ⟶[0,1] capture the extent to which χ∈ℵ belongs to or does not belong to a C-IF set C. These degrees satisfy 0⩽uC(χ)+vC(χ)⩽1. The degree of hesitancy is calculated as: hC(χ)=1−uC(χ)−vC(χ). The built-in circular function Or has a radius rC(χ):ℵ⟶[0,2] with the centre at (uC(χ),vC(χ)). A C-IF number is represented as c(χ)=(uC(χ),vC(χ);rC(χ)). The C-IF set C and its circular function Or within the domain ℵ are structured as follows: C={⟨χ,uC(χ),vC(χ);rC(χ)⟩|χ∈ℵ}={⟨χ,Or(uC(χ),vC(χ))⟩|χ∈ℵ},Or(uC(χ),vC(χ))={⟨ℓ,ℓ′⟩|ℓ,ℓ′∈[0,1]and(uC(χ)−ℓ)2+(vC(χ)−ℓ′)2⩽rC(χ)}∩L∗.
A standard fuzzy set describes membership, while an IF set adds flexibility by allowing for hesitation between membership and non-membership. Building on this, the C-IF set introduces a circular structure, capturing more complex, ambiguous characteristics. Unlike IF sets, C-IF sets represent uncertainty more precisely using their circular form. Figure 1 compares IF and C-IF sets within a triangular space defined by the vertices (0,0), (1,1), and (0,1), illustrating their overlap and differences. In this space, the C-IF number c(χ) is represented by the centre (uC(χ),vC(χ)) and radius rC(χ) of the circular function Or, while the IF number i(χ) corresponds to the point (uI(χ),vI(χ)). Figure 1 shows five ways to depict Or, demonstrating how it constrains Or(uC(χ),vC(χ)) within the L-fuzzy set L∗. When the radius rC(χ)=0 for all elements in the domain ℵ, the C-IF set C reduces to an IF set I, and the C-IF number c(χ) becomes equivalent to the IF number i(χ).
The visualization of the relationship between standard IF and C-IF constructs.
Chen (2023b) introduced two generalized C-IF distance metrics to overcome the limitations of traditional measures and improve adaptability. These metrics use triadic and quadripartite representations of C-IF Minkowski-like distances. The triadic model includes radius, membership, and non-membership, while the quadripartite version adds hesitancy, providing a more comprehensive assessment of C-IF dimensionality.
Consider two C-IF numbers, c(χ)=(uC(χ),vC(χ);rC(χ)) and c(χ′)=(uC(χ′),vC(χ′);rC(χ′)). A positive integer ξ∈Z+ serves as the metric parameter. The C-IF Minkowski-like distance between them is defined using three-term and four-term strategies as follows: DM0ξ(c(χ),c(χ′))=12(12|rC(χ)−rC(χ′)|+12(|uC(χ)−uC(χ′)|ξ+|vC(χ)−vC(χ′)|ξ)ξ),DMξ(c(χ),c(χ′))=12(12|rC(χ)−rC(χ′)|+12(|uC(χ)−uC(χ′)|ξ+|vC(χ)−vC(χ′)|ξ+|hC(χ)−hC(χ′)|ξ)ξ).
These distance measures quantify the dissimilarity between C-IF numbers, varying by approach (three-term or four-term) and the chosen metric parameter ξ. The triadic model omits hesitancy, while the quaternary model offers a more exhaustive representation. This research adopts the four-term Minkowski-like distance as the foundation for the C-IF ELECTRE methodology due to its comprehensive coverage of all relevant dimensions, ensuring alignment with the study’s objectives.
Joint Generalized Scoring Function
This section aims to develop a joint generalized scoring function that addresses the C-IF uncertainty. Utilizing a natural exponential-based scoring mechanism (Hezam et al., 2023), it effectively handles IF scenarios. Building on Chen’s (2023a) concepts of aggressive and conservative estimates, this section introduces practical notions of aggressive and cautious IF estimates. It also explores their fundamental properties, highlighting their role in representing upper and lower bounds within C-IF information.
For a C-IF number c(χ)=(uC(χ),vC(χ);rC(χ)), its aggressive IF estimate icα(χ) and cautious IF estimate icβ(χ) are defined as follows: icα(χ)=(uIα(χ),vIα(χ))=(min{1,uC(χ)+rC(χ)2},max{0,vC(χ)−rC(χ)2}),icβ(χ)=(uIβ(χ),vIβ(χ))=(max{0,uC(χ)−rC(χ)2},min{1,vC(χ)+rC(χ)2}).
For a C-IF numberc(χ), the aggressive IF estimateicα(χ)=(uIα(χ),vIα(χ))and cautious IF estimateicβ(χ)=(uIβ(χ),vIβ(χ))follow a quasi-ordering relationship: icα(χ)≽Qicβ(χ). When defined asicα(χ)=(uC(χ)+rC(χ)/2,vC(χ)−rC(χ)/2)andicβ(χ)=(uC(χ)−rC(χ)/2,vC(χ)+rC(χ)/2), both estimates exhibit equal hesitancy:hIα(χ)=hIβ(χ).
The conditions for icα(χ)≽Qicβ(χ) are uIα(χ)⩾uIβ(χ) and vIα(χ)⩽vIβ(χ) for each χ∈ℵ. From Eqs. (7) and (8), it follows that min{1,uC(χ)+rC(χ)/2}⩾max{0,uC(χ)−rC(χ)/2}, leading to uIα(χ)⩾uIβ(χ). Since max{0,vC(χ)−rC(χ)/2}⩽min{1,vC(χ)+rC(χ)/2}, it follows that vIα(χ)⩽vIβ(χ). It is established that icα(χ)≽Qicβ(χ) if and only if uIα(χ)⩾uIβ(χ) and vIα(χ)⩽vIβ(χ). Given icα(χ)=(uC(χ)+rC(χ)/2,vC(χ)−rC(χ)/2) and icβ(χ)=(uC(χ)−rC(χ)/2,vC(χ)+rC(χ)/2), the degrees of hesitancy are: hIα(χ)=1−uC(χ)−rC(χ)/2−vC(χ)+rC(χ)/2=1−uC(χ)−vC(χ) and hIβ(χ)=1−uC(χ)−vC(χ). Thus, icα(χ) and icβ(χ) have the same degrees of hesitancy. □
The scoring mechanismsM(icα(χ))andM(icβ(χ))exhibit the fundamental properties: (1)0⩽M(icα(χ))⩽1and0⩽M(icβ(χ))⩽1; (2)M(icα(χ))=1andM(icβ(χ))=1ificα(χ)=(1,0)andicβ(χ)=(1,0), respectively; and(3)M(icα(χ))=0andM(icβ(χ))=0ificα(χ)=(0,1)andicβ(χ)=(0,1), respectively.
Applying Eq. (2), the scoring mechanisms M(icα(χ)) and M(icβ(χ)) are established:
M(icα(χ))=12[(uIα(χ)−vIα(χ))+hIα(χ)·(e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12)+1],M(icβ(χ))=12[(uIβ(χ)−vIβ(χ))+hIβ(χ)·(e(uIβ(χ)−vIβ(χ))e(uIβ(χ)−vIβ(χ))+1−12)+1].
Given the property in (1), the aggressive IF estimate icα(χ) satisfies uIα(χ)+vIα(χ)+hIα(χ)=1, where uIα(χ), vIα(χ), and hIα(χ) are constrained to the interval [0, 1]. The scoring mechanism uses the natural exponential function. Since 0⩽uIα(χ)−vIα(χ)⩽1, we have 0<e(uIα(χ)−vIα(χ))⩽e1≈2.71828. This leads to the following outcomes:
0<e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1⩽ee+1,−12<e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12⩽ee+1−12=e−12e+2=0.2311.
Aa a result, it is known that M(icα(χ))⩽1, because:
uIα(χ)−vIα(χ)+hIα(χ)·(e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12)⩽uIα(χ)+vIα(χ)+hIα(χ)⩽1,12[(uIα(χ)−vIα(χ))+hIα(χ)·(e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12)+1]⩽12(1+1)=1.
On the other hand, it is inferred that M(icα(χ))⩾0, because:
uIα(χ)−vIα(χ)+hIα(χ)·(e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12)+uIα(χ)+vIα(χ)+hIα(χ)=2uIα(χ)+hIα(χ)·(e(uIα(χ)−vIα(χ))e(uIα(χ)−vIα(χ))+1−12+1)>2uIα(χ)+hIα(χ)(−12+1)⩾0.
It follows that 0⩽M(icα(χ))⩽1 and 0⩽M(icβ(χ))⩽1. According to property (2), M(icα(χ))=1 and M(icβ(χ))=1 if icα(χ)=(1,0) and icβ(χ)=(1,0), respectively. In these cases, both aggressive and cautious IF estimates yield a score of 1. For property (3), M(icα(χ))=0 and M(icβ(χ))=0 if icα(χ)=(0,1) and icβ(χ)=(0,1). Here, both IF numbers yielding (0,1) lead to a score of 0. □
For the aggressive and cautious IF estimatesicα(χ)andicβ(χ), the scoring mechanisms satisfy the inequalityM(icα(χ))⩾M(icβ(χ))for eachχ∈ℵ.
Consider the scoring mechanism M(i(χ)) for an IF number i(χ)=(uI(χ),vI(χ)) in Eq. (2). Expanding the formula of M(i(χ)), the following representation is yielded:
M(i(χ))=12+12uI(χ)−12vI(χ)−14hI(χ)+12hI(χ)·(e(uI(χ)−vI(χ))e(uI(χ)−vI(χ))+1).
To find the first partial derivative of M(i(χ)) in terms of uI(χ), we treat vI(χ) and hI(χ) as constants. The derivatives of the first four terms with respect to uI(χ) are 0, 0.5, 0, and 0, respectively. For the fifth term, which has uI(χ) in the exponent, we apply the chain rule. To simplify this process, we introduce a new function:
H(i(χ))=e(uI(χ)−vI(χ))e(uI(χ)−vI(χ))+1.
Applying the quotient rule, the following outcome can be generated:
dH(i(χ))duI(χ)=(e(uI(χ)−vI(χ))+1)·e(uI(χ)−vI(χ))−e(uI(χ)−vI(χ))·e(uI(χ)−vI(χ))(e(uI(χ)−vI(χ))+1)2=e(uI(χ)−vI(χ))(e(uI(χ)−vI(χ))+1)2.
The first partial derivative of M(i(χ)) regarding uI(χ) can be obtained:
∂M(i(χ))∂uI(χ)=12+12hI(χ)e(uI(χ)−vI(χ))(e(uI(χ)−vI(χ))+1)2⩾0.
Since the partial derivative ∂M(i(χ))/∂uI(χ) is non-negative, the scoring mechanism M(i(χ)) is non-decreasing with respect to uI(χ) when vI(χ) and hI(χ) are held constant. Similarly, to find the first partial derivative of M(i(χ)) in terms of vI(χ), we differentiate the expression while treating uI(χ) and hI(χ) as constants. The result of differentiating the function H(i(χ)) with respect to vI(χ) is as follows:
dH(i(χ))dvI(χ)=(e(uI(χ)−vI(χ))+1)·(−e(uI(χ)−vI(χ)))−e(uI(χ)−vI(χ))·(−e(uI(χ)−vI(χ)))(e(uI(χ)−vI(χ))+1)2=−e(uI(χ)−vI(χ))(e(uI(χ)−vI(χ))+1)2.
The first partial derivative of M(i(χ)) concerning vI(χ) is generated:
∂M(i(χ))∂vI(χ)=−12−12hI(χ)e(uI(χ)−vI(χ))(e(uI(χ)−vI(χ))+1)2⩽0.
Thus, one can conclude that the scoring mechanism M(i(χ)) is non-increasing in relation to vI(χ) when u(χ) and hI(χ) are fixed. As established in Theorem 1, the aggressive and cautious IF estimates have a quasi-ordering relationship: icα(χ)≽Qicβ(χ), leading to uIα(χ)⩾uIβ(χ) and vIα(χ)⩽vIβ(χ). Given that M(i(χ)) is non-decreasing in uI(χ) and non-increasing in vI(χ), it can be deduced that M(icα(χ))⩾M(icβ(χ)). □
Consider c(χ)=(uC(χ),vC(χ);rC(χ)) as a C-IF number within the C-IF set C. Let φ be an inclination parameter ranging from 0 to 1. The joint generalized scoring function for c(χ), denoted as Sφ(c(χ)), is formulated as follows:
Sφ(c(χ))=φ·M(icα(χ))+(1−φ)·M(icβ(χ)).
The following characteristics are fulfilled by the joint generalized scoring functionSφ(c(χ))(φ∈[0,1])of a C-IF numberc(χ): (1)0⩽Sφ(c(χ))⩽1; (2)Sφ(c(χ))=M(icα(χ))whenφ=1; and(3)Sφ(c(χ))=M(icβ(χ))whenφ=0.
From Definition 6, the inclination parameter φ ranges from 0 to 1. As per Theorem 2, the scoring mechanisms for icα(χ) and icβ(χ) are also bounded between 0 and 1. Given properties 0⩽φ⩽1, 0⩽M(icα(χ))⩽1, and 0⩽M(icβ(χ))⩽1, we conclude that 0⩽Sφ(c(χ))⩽1. For properties (2) and (3), the equations Sφ(c(χ))=M(icα(χ)) and Sφ(c(χ))=M(icβ(χ)) follow from the cases where φ=1 and φ=0, respectively. □
When the inclination parameter φ exceeds 0.5, it indicates that the decision-maker places greater importance on the aggressive IF estimate, reflecting an aggressive stance. Conversely, if φ is less than 0.5, the decision-maker favours the cautious IF estimate, demonstrating a cautious attitude. When φ equals 0.5, both aggressive and cautious perspectives are given equal weight, reflecting a neutral disposition.
The inclination parameter φ in the joint generalized scoring function Sφ(c(χ)) represents the decision-maker’s psychological predisposition toward aggressive, neutral, or cautious attitudes. Assigning a specific value to φ reflects the decision-maker’s personal tendencies or preferences. Within Sφ(c(χ)), φ indicates a neutral tendency or preference for either aggressive or cautious outcomes.
Proposed C-IF ELECTRE Approach
This section introduces the C-IF ELECTRE decision-making model, designed to address complex discrete choice situations with multiple, incommensurable, and contradictory criteria. Proposed within the C-IF uncertain context, this model offers a novel approach for tackling these intricate decision-making challenges.
Suggested Methodology
From a mathematical perspective, a multiple-criteria choice problem involves a set of m choice options, signified as O={O1,O2,…,Om}, and a set of n evaluative criteria, denoted as E={E1,E2,…,En}. The decision-maker can distinguish between two subsets of evaluative criteria: EB, which includes beneficial criteria to be maximized, and EN, which includes non-beneficial criteria to be minimized. These subsets are mutually exclusive (EB∩EN=∅) and together comprise the entire set of criteria (EB∪EN=E).
The significance of a criterion Ej∈E is represented by its C-IF weight wj=(ωj,ϖj;rj), where ωj∈[0,1] and ϖj∈[0,1] indicate the degree of membership and non-membership of Ej in the fuzzy concept of “importance.” The radius rj∈[0,2] reflects the extent of uncertainty within the circular structure. The degree of hesitancy is given by ℏj=1−ωj−ϖj. A C-IF set W represents the weight characteristic, defined as follows: W={⟨Ej,ωj,ϖj;rj⟩Ej∈E}={⟨Ej,Or(ωj,ϖj)⟩|Ej∈E},Or(ωj,ϖj)={⟨ℓ,ℓ′⟩|ℓ,ℓ′∈[0,1],(ωj−ℓ)2+(ϖj−ℓ′)2⩽rj,andℓ+ℓ′⩽1}.
The C-IF evaluation value of a choice option Ok (where k=1,2,…,m) assessed by an evaluative criterion Ej (where j=1,2,…,n) is represented as ckj=(ukj,vkj;rkj). The hesitation is calculated using the formula hkj=1−ukj−vkj. The fuzzy characteristic associated with each choice option Ok∈O is represented on this wise: Ck={⟨Ej,ukj,vkj;rkj⟩Ej∈E}={⟨Ej,Or(ukj,vkj)⟩|Ej∈E},Or(ukj,vkj)={⟨ℓ,ℓ′⟩|ℓ,ℓ′∈[0,1],(ukj−ℓ)2+(vkj−ℓ′)2⩽rkj,andℓ+ℓ′⩽1}.
Based on Definition 5, the aggressive IF estimate ikjα=(ukjα,vkjα) and cautious IF estimate ikjβ=(ukjβ,vkjβ) related to the C-IF evaluation value ckj are calculated using the formulas: ikjα=(min{1,ukj+rkj/2},max{0,vkj−rkj/2}) and ikjβ=(max{0,ukj−rkj/2},min{1,vkj+rkj/2}). The degrees of hesitancy are given by: hkjα=1−ukjα−vkjα and hkjβ=1−ukjβ−vkjβ. As per Definition 2, the natural exponential function-based scoring mechanisms for ikjα and ikjβ are defined as: M(ikjα)=(1/2)·{(ukjα−vkjα)+hkjα·[e(ukjα−vkjα)/(e(ukjα−vkjα)+1)−(1/2)]+1} and M(ikjβ)=(1/2)·{(ukjβ−vkjβ)+hkjα·[e(ukjβ−vkjβ)/(e(ukjβ−vkjβ)+1)−(1/2)]+1}. Utilizing these results and setting the inclination parameter φ, the joint generalized scoring function is generated as: Sφ(ckj)=φ·M(ikjα)+(1−φ)·M(ikjβ). This scoring function will be used to establish the sets of concordance and discordance for all pairs of choice options.
The contrast relationship between the C-IF evaluation values ckj=(ukj,vkj;rkj) and clj=(ulj,vlj;rlj) for choice options Ok and Ol assessed by Ej can be identified using the joint generalized scoring function-based relations: “≻S” (more advantageous than), “∼S” (indifferent), and “≺S” (more disadvantaged than), as follows:
For Ej∈EB: (a) If Sφ(ckj)>Sφ(clj), then ckj≻Sclj; (b) If Sφ(ckj)=Sφ(clj), then ckj∼Sclj; and (c) If Sφ(ckj)<Sφ(clj), then ckj≺Sclj.
For Ej∈EN: (a) If Sφ(ckj)<Sφ(clj), then ckj≻Sclj; (b) If Sφ(ckj)=Sφ(clj), then ckj∼Sclj; and (c) If Sφ(ckj)>Sφ(clj), then ckj≺Sclj.
The concordance set Cφ(Ok/Ol) and the discordance set Dφ(Ok/Ol) for the case where choice option Ok outperforms Ol (Ok,Ol∈O and k≠l) can be defined using the joint generalized scoring function-based relations in Definition 7, as seen below: Cφ(Ok/Ol)={Ej|ckj≽ScljforEj∈E}={Ej|(Sφ(ckj)⩾Sφ(clj)fo rEj∈EB),(Sφ(ckj)⩽Sφ(clj)forEj∈EN)},Dφ(Ok/Ol)={Ej|ckj≺ScljforEj∈E}={Ej|(Sφ(ckj)<Sφ(clj)forEj∈EB),(Sφ(ckj)>Sφ(clj)forEj∈EN)}.
The concordance setCφ(Ok/Ol)and the discordance setDφ(Ok/Ol)satisfy the following characteristics in a fixed setting of the inclination parameter φ: (1)Cφ(Ok/Ol)∩Dφ(Ok/Ol)=∅; (2)Cφ(Ok/Ol)∪Dφ(Ok/Ol)=E; (3)Cφ(Ok/Ol)=E∖Dφ(Ok/Ol); and(4)Cφ(Ok/Ol)=Dφ(Ol/Ok)whenSφ(ckj)≠Sφ(clj)forEj∈E.
Properties (1) to (3) are trivially valid. Property (1) states that Cφ(Ok/Ol) and Dφ(Ok/Ol) do not share any common criteria. Property (2) indicates that the union of Cφ(Ok/Ol) and Dφ(Ok/Ol) includes all criteria in the set E. Property (3) clarifies that Cφ(Ok/Ol) contains all criteria in E except those in Dφ(Ok/Ol). When Sφ(ckj)≠Sφ(clj), property (4) determines that Cφ(Ok/Ol)={Ej|(Sφ(ckj)>Sφ(clj)forEj∈EB),(Sφ(ckj)<Sφ(clj)forEj∈EN)}. Conversely, it is known that Dφ(Ol/Ok)={Ej|(Sφ(clj)<Sφ(ckj)forEj∈EB),(Sφ(clj)>Sφ(ckj)forEj∈EN)}. Therefore, it is concluded that Cφ(Ok/Ol)=Dφ(Ol/Ok). □
This study employs the joint generalized scoring function with W to define consistency indicators between choice pairs. Given the C-IF weight wj=(ωj,ϖj;rj), the aggressive and cautious IF estimates are: iWjα=(ωjα,ϖjα)=(min{1,ωj+rj/2},max{0,ϖj−rj/2}) and iWjβ=(ωjβ,ϖjβ)=(max{0,ωj−rj/2},min{1,ϖj+rj/2}). The hesitancy degrees are: ℏjα=1−ωjα−ϖjα and ℏjβ=1−ωjβ−ϖjβ. Using Definition 2, the scoring mechanisms for iWjα and iWjβ are based on the natural exponential function:
M(iWjα)=12[(ωjα−ϖjα)+ℏjα·(e(ωjα−ϖjα)e(ωjα−ϖjα)+1−12)+1],M(iWjβ)=12[(ωjβ−ϖjβ)+ℏjβ·(e(ωjβ−ϖjβ)e(ωjβ−ϖjβ)+1−12)+1].
Using the derived results and assuming the same inclination parameter φ, the joint generalized scoring function is defined as: Sφ(wj)=φ·M(iWjα)+(1−φ)·M(iWjβ). This function combines the scoring mechanisms M(iWjα) and M(iWjβ) for aggressive and cautious IF estimates, weighted by φ and (1−φ), to produce an overall score for wj.
When the choice option Ok outperforms Ol (Ok,Ol∈O and k≠l), the consistency indicator ICφ(Ok/Ol) is defined via the joint generalized scoring functions as:
ICφ(Ok/Ol)=∑Ej∈Cφ(Ok/Ol)Sφ(wj)·|Sφ(ckj)−Sφ(clj)|∑j′=1nSφ(wj′)·|Sφ(ckj′)−Sφ(clj′)|.
Assuming thatSφ(ckj)≠Sφ(clj)for at least one evaluative criterionEj∈E, without loss of generality, the consistency indicatorICφ(Ok/Ol)within a fixed setting of the inclination parameter φ satisfies the boundary condition:0⩽ICφ(Ok/Ol)⩽1.
Applying Theorem 4, we know 0⩽Sφ(ckj)⩽1 and 0⩽Sφ(clj)⩽1, implying 0⩽|Sφ(ckj)−Sφ(clj)|⩽1. Since the joint generalized scoring function is non-negative, ICφ(Ok/Ol)⩾0. Given that Cφ(Ok/Ol)⊆E by Theorem 5 (where Cφ(Ok/Ol)∪Dφ(Ok/Ol)=E), it follows: ∑Ej∈Cφ(Ok/Ol)Sφ(wj)·|Sφ(ckj)−Sφ(clj)|⩽∑Ej∈ESφ(wj)·|Sφ(ckj)−Sφ(clj)|=∑j′=1nSφ(wj′)·|Sφ(ckj′)−Sφ(clj′)|. Thus, ICφ(Ok/Ol)⩽1. In cases of zero denominator issues, at least one |Sφ(ckj)−Sφ(clj)| must be non-zero, as Sφ(ckj)≠Sφ(clj) for at least one criterion Ej∈E. Hence, 0⩽ICφ(Ok/Ol)⩽1. □
AssumingSφ(ckj)≠Sφ(clj)forEj∈E, the consistency indicatorICφ(Ok/Ol)exhibits the ensuing properties for a fixed inclination parameter φ: (1)ICφ(Ok/Ol)+ICφ(Ol/Ok)=1; (2)∑k=1,k≠lm∑l=1,l≠kmICφ(Ok/Ol)=m(m−1)/2; and(3)I‾Cφ=0.5, whereI‾Cφis the average consistency indicator across allICφ(Ok/Ol), withk≠l.
As per Definition 8, the concordance sets are derived as: Cφ(Ok/Ol)={Ej∣(Sφ(ckj)⩾Sφ(clj)forEj∈EB),(Sφ(ckj)⩽Sφ(clj)forEj∈EN)} and Cφ(Ol/Ok)={Ej∣(Sφ(clj)⩾Sφ(ckj)forEj∈EB),(Sφ(clj)⩽Sφ(ckj)forEj∈EN)}. Assuming Sφ(ckj)≠Sφ(clj) for Ej∈E, the simplified expressions become:Cφ(Ok/Ol)={Ej∣(Sφ(ckj)>Sφ(clj)forEj∈EB),(Sφ(ckj)<Sφ(clj)forEj∈EN)} and Cφ(Ol/Ok)={Ej∣(Sφ(clj)>Sφ(ckj)forEj∈EB),(Sφ(clj)<Sφ(ckj)forEj∈EN)}. Thus, Cφ(Ok/Ol)∪Cφ(Ol/Ok)=E. For property (1), the calculation follows:
ICφ(Ok/Ol)+ICφ(Ol/Ok)=∑Ej∈Cφ(Ok/Ol)Sφ(wj)|Sφ(ckj)−Sφ(clj)|+∑Ej∈Cφ(Ol/Ok)Sφ(wj)|Sφ(clj)−Sφ(ckj)|∑j′=1nSφ(wj′)·|Sφ(ckj′)−Sφ(clj′)|=∑Ej∈ESφ(wj)·|Sφ(ckj)−Sφ(clj)|∑Ej∈ESφ(wj)·|Sφ(ckj)−Sφ(clj)|=1.
The sum of consistency indicators for Ok outperforming Ol and vice versa equals 1, satisfying property (1). Property (2) asserts that the total sum of consistency indicators for all option pairs (excluding self-pairs) is m(m−1)/2. This follows from the relation ICφ(Ok/Ol)+ICφ(Ol/Ok)=1, leading to:
∑k=1,k≠lm∑l=1,l≠kmICφ(Ok/Ol)=∑k=1,k≠lm(ICφ(Ok/Ol)+ICφ(Ol/Ok))=∑k=1,k≠lm1=m(m−1)2.
This confirms property (2), representing the total pairwise comparisons. The average consistency indicator I‾Cφ is computed over all ICφ(Ok/Ol) values, as:
I‾Cφ=∑k=1,k≠lm∑l=1,l≠kmICφ(Ok/Ol)m(m−1)=m(m−1)2/m(m−1)=12.
Accordingly, property (3) is confirmed. □
This study introduces an inconsistency indicator for each choice pair using the joint generalized scoring function Sφ and the C-IF Minkowski-like distance DM0ξ or DMξ. In Definition 4, DM0ξ(ckj,clj) employs a three-term model (radius, membership, non-membership), excluding hesitancy. Conversely, the four-term distance DMξ(ckj,clj) incorporates all key elements: radius, membership, non-membership, and hesitancy. This study adopts the four-term Minkowski-like distance model to fully capture the uncertainty in C-IF information and leverage its comprehensive representation. By setting ξ=1 (Manhattan) or ξ=2 (Euclidean), the four-term distance is computed as:
DMξ(ckj,clj)=12{12|rkj−rlj|+12(|ukj−ulj|ξ+|vkj−vlj|ξ+|hkj−hlj|ξ)ξ}.
When choice option Ok outperforms Ol (where Ok,Ol∈O and k≠l), the inconsistency indicator IDφ(Ok/Ol) is formulated using the (four-term strategy-based) C-IF Minkowski-like distance and the joint generalized scoring function as follows:
IDφ(Ok/Ol)=∑Ej∈Dφ(Ok/Ol)DMξ(ckj,clj)·|Sφ(ckj)−Sφ(clj)|∑j′=1nDMξ(ckj′,clj′)·|Sφ(ckj′)−Sφ(clj′)|.
Assuming thatSφ(ckj)≠Sφ(clj)for at least one evaluative criterionEj∈E, without loss of generality, the inconsistency indicatorIDφ(Ok/Ol)within a fixed setting of the inclination parameter φ satisfies the boundary condition: 0⩽IDφ(Ok/Ol)⩽1.
Both the C-IF Minkowski-like distance and the joint generalized scoring function possess the non-negativity property. Next, this theorem’s proving procedure follows a resemble approach to that of Theorem 6. □
AssumingSφ(ckj)≠Sφ(clj)forEj∈E, the inconsistency indicatorIDφ(Ok/Ol)satisfies the following beneficial characteristics for a fixed inclination parameter φ: (1)IDφ(Ok/Ol)+IDφ(Ol/Ok)=1; (2)∑k=1,k≠lm∑l=1,l≠kmIDφ(Ok/Ol)=m(m−1)/2; and(3)I‾Dφ=0.5, whereI‾Dφrepresents the average inconsistency indicator calculated by considering allIDφ(Ok/Ol)values forOk,Ol∈O, andk≠l.
By Definition 8, the discordance sets are defined as follows: Dφ(Ok/Ol)={Ej∣(Sφ(ckj)<Sφ(clj)forEj∈EB),(Sφ(ckj)>Sφ(clj)forEj∈EN)} and Dφ(Ol/Ok)={Ej∣(Sφ(clj)<Sφ(ckj)forEj∈EB),(Sφ(clj)>Sφ(ckj)forEj∈EN)}. With the precondition Sφ(ckj)≠Sφ(clj) for Ej∈E, it implies that Dφ(Ok/Ol)∪Dφ(Ol/Ok)=E. The C-IF Minkowski-like distance exhibits symmetry: DMξ(ckj,clj)=DMξ(clj,ckj). Property (1) can be confirmed as follows: ΣEj∈Dφ(Ok/Ol)DMξ(ckj,clj)·|Sφ(ckj)−Sφ(clj)|+ΣEj∈Dφ(Ol/Ok)DMξ(clj,ckj)·|Sφ(clj)−Sφ(ckj)|=ΣEj∈EDMξ(ckj,clj)·|Sφ(ckj)−Sφ(clj)|. Consequently, IDφ(Ok/Ol)+IDφ(Ol/Ok)=ΣEj∈EDMξ(ckj,clj)·|Sφ(ckj)−Sφ(clj)|/ΣEj∈EDMξ(ckj,clj)·|Sφ(ckj)−Sφ(clj)|=1. Thus, property (1) is satisfied. The proofs for properties (2) and (3) follow a similar approach as in Theorem 7. In summary, property (1) states that the sum of the inconsistency indicators for Ok and Ol equals one. Property (2) indicates the total number of possible pairwise discordance comparisons is m(m−1)/2. Property (3) asserts that, on average, the inconsistency indicators reflect a balanced level of inconsistency between the choice options. □
To construct the C-IF ELECTRE I prioritization procedure, this study first compares the consistency indicator ICφ(Ok/Ol) with the average consistency indicator I‾Cφ for Ok,Ol∈O and k≠l. This comparison allows for the establishment of the consistency entry BCφ(Ok/Ol) and the creation of the consistency Boolean matrix BCφ. The process investigates and articulates consistencies between pairs of choice options as follows: BCφ(Ok/Ol)=1ifICφ(Ok/Ol)⩾I‾Cφ,0ifICφ(Ok/Ol)<I‾Cφ,BCφ=−BCφ(O1/O2)⋯BCφ(O1/Om)BCφ(O2/O1)−⋯BCφ(O2/Om)⋮⋮⋱⋮BCφ(Om/O1)BCφ(Om/O2)⋯−.
This study compares the inconsistency indicator IDφ(Ok/Ol) with the average inconsistency indicator I‾Dφ to generate the inconsistency entry BDφ(Ok/Ol) for Ok,Ol∈O and k≠l. The inconsistency Boolean matrix BDφ is constructed. The following procedure assesses and depicts the inconsistencies between pairs of choice options: BDφ(Ok/Ol)=1ifIDφ(Ok/Ol)⩽I‾Dφ,0ifIDφ(Ok/Ol)>I‾Dφ,BDφ=−BDφ(O1/O2)⋯BDφ(O1/Om)BDφ(O2/O1)⋯BDφ(O2/Om)⋮⋮⋱⋮BDφ(Om/O1)BDφ(Om/O2)⋯−.
Using the specified φ, this paper formulates the overall prioritization entry BOφ(Ok/Ol) for Ok,Ol∈O (k≠l) and constructs the overall prioritization Boolean matrix BOφ as: BOφ(Ok/Ol)=BCφ(Ok/Ol)·BDφ(Ok/Ol),BOφ=−BOφ(O1/O2)⋯BOφ(O1/Om)BOφ(O2/O1)−⋯BOφ(O2/Om)⋮⋮⋱⋮BOφ(Om/O1)BOφ(Om/O2)⋯−.
If BOφ(Ok/Ol)=1, it indicates that Ok is preferred to Ol based on both consistency and inconsistency indicators. If BOφ(Ok/Ol)=0, it means that either Ok is less preferred than Ol or they are considered unrelated. A dominance graph can be constructed to illustrate a partial priority ranking of the m options based on the results from the overall prioritization Boolean matrix BOφ, forming the basis for the C-IF ELECTRE I prioritization process.
This study presents the concepts of consistency-dependent average outflow, inconsistency-dependent average inflow, and overall net flow for ranking choice options in the C-IF ELECTRE II prioritization process. These concepts are inspired by the leaving and entering flows developed by Wang and Chen (2021). Specifically, the consistency-dependent average outflow ACφ(Ok), the inconsistency-dependent average inflow ADφ(Ok), and the overall net flow NOφ(Ok) for choice option Ok are defined as follows: ACφ(Ok)=∑l=1,l≠kmICφ(Ok/Ol)m−1,ADφ(Ok)=∑l=1,l≠kmIDφ(Ok/Ol)m−1,NOφ(Ok)=ACφ(Ok)−ADφ(Ok).
The average outflow, average inflow, and overall net flow satisfy these properties: (1) 0⩽ACφ(Ok)⩽1; (2) 0⩽ADφ(Ok)⩽1; and (3) −1⩽NOφ(Ok)⩽1. A higher value of NOφ(Ok) indicates a better option. The C-IF ELECTRE II process ranks options in O by descending net flows, offering a thorough assessment to guide decision-making effectively.
Suggested Algorithm
Figure 2 presents a systematic framework for applying the C-IF ELECTRE I and II approaches to multiple-criteria choice problems under C-IF uncertainty. The C-IF ELECTRE I method involves several stages: (1) Problem Definition: Identifies choice options and evaluative criteria; (2) Data Creation: Establishes C-IF weights for criteria and generates C-IF evaluation values for options; (3) C-IF Scoring: Computes aggressive and cautious IF estimates, and determines scoring mechanisms based on hesitancy degrees; (4) Consistency/Inconsistency Indexing: Identifies concordance and discordance sets, calculates Minkowski-like distances, and determines consistency/inconsistency indicators; and (5) C-IF ELECTRE I Prioritization: Produces overall prioritization entries and Boolean matrices, and creates a dominance graph for partial-priority ranking. The C-IF ELECTRE II approach follows similar steps but incorporates average outflows and inflows alongside net flows to establish complete-priority rankings via overall net flows.
The systematic procedure of the C-IF ELECTRE I and II approaches.
The implementation steps of the C-IF ELECTRE I approach are as follows:
Problem definition stage: See Steps I.1 and I.2
Step I.1. Define a multiple-criteria decision problem with evaluative criteria E={E1,E2,…,En} and choice options O={O1,O2,…,Om}.
Step I.2. Split criteria into beneficial EB and non-beneficial EN subsets.
Data creation stage: See Steps I.3 and I.4
Step I.3. Establish C-IF weights wj=(ωj,ϖj;rj) for each criterion Ej and construct the weight characteristic W using Eqs. (10) and (11).
Step I.4. Generate C-IF evaluation values ckj=(ukj,vkj;rkj) for each option Ok under criterion Ej and build the fuzzy characteristic Ck using Eqs. (12) and (13).
C-IF scoring stage: See Steps I.5–I.7
Step I.5. Compute the aggressive IF estimates ikjα=(ukjα,vkjα) and iWjα=(ωjα,ϖjα) using Eq. (7), as well as the cautious IF estimates ikjβ=(ukjβ,vkjβ) and iWjβ=(ωjβ,ϖjβ) by Eq. (8).
Step I.6. Derive hesitancy degrees hkjα, hkjβ, ℏjα, and ℏjβ, as well as scoring mechanisms M(ikjα), M(ikjβ), M(iWjα), and M(iWjβ) using the natural exponential function in Eq. (2).
Step I.7. Designate a value to the inclination parameter φ∈[0,1], and generate joint scoring functions Sφ(ckj) and Sφ(wj) using Eq. (9).
Consistency/inconsistency indexing stage: See Steps I.8–I.10
Step I.8. Employ Eqs. (14) and (15), respectively, to establish the concordance set Cφ(Ok/Ol) and the discordance set Dφ(Ok/Ol) (Ok,Ol∈O and k≠l).
Step I.9. Allocate a value for the metric parameter ξ∈Z+, and derive the C-IF Minkowski-like distance DMξ(ckj,clj) using the four-term strategy described in Eq. (6).
Step I.10. Determine the consistency indicator ICφ(Ok/Ol) and the inconsistency indicator IDφ(Ok/Ol) using Eqs. (16) and (17), respectively.
C-IF ELECTRE I prioritization stage: See Steps I.11–I.13
Step I.11. Compare the consistency indicator ICφ(Ok/Ol) with the average consistency indicator I‾Cφ=0.5 (from Theorem 7) to obtain the consistency entry BCφ(Ok/Ol) and the consistency Boolean matrix BCφ, as provided in Eqs. (18) and (19), respectively.
Step I.12. Contrast the inconsistency indicator IDφ(Ok/Ol) with the average inconsistency indicator I‾Dφ=0.5 (from Theorem 9) to acquire the inconsistency entry BDφ(Ok/Ol) and the inconsistency Boolean matrix BDφ, as presented in Eqs. (20) and (21), respectively.
Step I.13. Produce the overall prioritization entry BOφ(Ok/Ol) and the overall prioritization Boolean matrix BOφ using Eqs. (22) and (23), respectively. Construct a dominance graph to indicate the partial-priority ranking of options in the set O.
The implementation steps of the C-IF ELECTRE II approach include:
Steps II.1 to II.10. Same as Steps I.1 to I.10.
C-IF ELECTRE II prioritization stage: See Steps II.11 and II.12
Step II.11. Calculate the consistency-dependent average outflow ACφ(Ok) the inconsistency-dependent average inflow ADφ(Ok) using Eqs. (24) and (25), respectively.
Step II.12. Obtain the overall net flow NOφ(Ok) using Eq. (26), and then rank the options in descending order based on their overall net flow values.
Application to a Problem with Supplier Evaluation
This section delves into the practical quandary of multi-expert supplier appraisal, as posed by Otay and Kahraman (2022), through the application of the proposed C-IF ELECTRE framework. Moreover, the operationalization of the advocated C-IF ELECTRE I and II methodologies is elucidated, highlighting their prowess in both efficacy and efficiency.
The C-IF ELECTRE approach is applied to evaluate and select suppliers in an engineering company, focusing on a specific component from various supplied options. Initially, potential suppliers are listed as choice options. Environmental considerations, particularly pollution control and ISO standards, are prioritized in the assessment. Suppliers failing to meet these criteria are eliminated from further analysis. The remaining options—Supplier Options #1 to #3—are evaluated across three key dimensions: cost, service, and technology/quality. These dimensions are assessed through nine evaluative criteria, including payment terms, on-time delivery, and quality management systems. Figure 3 illustrates the hierarchical structure of the multi-expert supplier evaluation, detailing the relationships among dimensions, evaluative criteria, and supplier options.
The hierarchical structure of the multi-expert supplier evaluation issue.
Step I.1 was conducted based on the problem setting from Otay and Kahraman (2022) and the hierarchical structure shown in Fig. 3. Three choice options were evaluated: Supplier Options #1, #2, and #3, represented as O={O1,O2,O3}. These suppliers were assessed across three dimensions: (1) the cost dimension, consisting of price (E1), terms of payments (E2), and handling and transportation (E3); and (2) the service dimension, encompassing flexibility (E4), on-time delivery (E5), and past performance (E6); and (3) the technology and quality dimension, including quality management systems (E7), technological capability (E8), and R&D studies (E9). The set of evaluative criteria was defined as E={E1,E2,…,E9}.
Step I.2 divides the set E into two subsets: EB (beneficial criteria) and EN (non-beneficial criteria). Beneficial criteria reflect attributes where higher values are preferred, while non-beneficial criteria favour lower values. For instance, price (E1) and terms of payments (E2) are non-beneficial, as lower values are preferred. In contrast, quality management systems (E7) and technological capability (E8) are beneficial, with higher values indicating desirable traits. Although the nine criteria E1,E2,…,E9 include both types, they are standardized as beneficial following Otay and Kahraman (2022). The sets of beneficial and non-beneficial criteria were defined as EB={E1,E2,…,E9} and EN=∅.
The data creation stage focuses on establishing the C-IF weight wj and C-IF evaluation value ckj as outlined in Steps I.3 and I.4. This study follows a structured nine-point linguistic rating scale, as proposed by Chen (2024), involving the following steps: (1) Select a Linguistic Scale: Use a nine-point scale to streamline evaluations; (2) Collect Evaluations: Gather semantic evaluations from decision-makers regarding choice options and criterion importance; (3) Convert to IF Numbers: Transform the semantic evaluations into corresponding IF numbers; (4) Calculate C-IF Evaluation Values: For each decision-maker’s assessment, define an IF number. Then, average these IF numbers to find the centre and determine the radius based on maximum deviation; and (5) Establish C-IF Weights: Similarly, calculate C-IF weights by averaging the importance weights to find the center and radius. These steps synthesize insights from multiple decision-makers into a cohesive C-IF framework for evaluation values and weights.
In the context of multi-expert supplier evaluation, the data from Otay and Kahraman (2022) were used, consolidating evaluations from three procurement specialists to derive collective C-IF weights. In Step I.3, the C-IF weight wj is presented as a C-IF number (ωj,ϖj;rj) in the second column of Table 1. Using Eqs. (10) and (11), the weight characteristic was established as: W={⟨Ej,ωj,ϖj;rj⟩|Ej∈{E1,E2,…,E9}}={⟨Ej,Or(ωj,ϖj)⟩|Ej∈{E1,E2,…,E9}}, where Or(ωj,ϖj)={⟨ℓ,ℓ′⟩∣ℓ,ℓ′∈[0,1],[(ωj−ℓ)2+(ϖj−ℓ′)2]0.5⩽rj,andℓ+ℓ′⩽1} for j∈{1,2,…,9}.
Information regarding the C-IF weight wj and the C-IF evaluation value ckj.
Ej
(ωj,ϖj;rj)
(u1j,v1j;r1j)
(u2j,v2j;r2j)
(u3j,v3j;r3j)
E1
(0.573,0.360;0.158)
(0.346,0.528;0.271)
(0.624,0.254;0.163)
(0.618,0.277;0.100)
E2
(0.327,0.589;0.112)
(0.439,0.439;0.142)
(0.650,0.250;0.000)
(0.675,0.207;0.190)
E3
(0.456,0.469;0.158)
(0.346,0.528;0.271)
(0.569,0.327;0.112)
(0.629,0.267;0.114)
E4
(0.454,0.480;0.174)
(0.401,0.480;0.198)
(0.698,0.194;0.074)
(0.615,0.267;0.178)
E5
(0.571,0.366;0.174)
(0.578,0.316;0.098)
(0.718,0.175;0.102)
(0.618,0.277;0.100)
E6
(0.342,0.574;0.161)
(0.618,0.277;0.100)
(0.737,0.140;0.145)
(0.776,0.096;0.199)
E7
(0.324,0.592;0.074)
(0.578,0.316;0.098)
(0.615,0.267;0.178)
(0.729,0.166;0.115)
E8
(0.577,0.360;0.074)
(0.574,0.293;0.341)
(0.598,0.296;0.072)
(0.569,0.327;0.112)
E9
(0.458,0.473;0.000)
(0.528,0.346;0.271)
(0.569,0.327;0.112)
(0.598,0.296;0.072)
∗: Refer to Otay and Kahraman (2022).
Specifics on the aggressive IF estimates iWjα and ikjα, alongside the cautious IF estimates iWjβ and ikjβ.
Outcomes associated with the aggressive IF estimates iWjα=(ωjα,ϖjα) and ikjα=(ukjα,vkjα)
Ej
(ωjα,ϖjα)
(u1jα,v1jα)
(u2jα,v2jα)
(u3jα,v3jα)
E1
(0.685,0.248)
(0.538,0.336)
(0.739,0.139)
(0.689,0.206)
E2
(0.406,0.510)
(0.539,0.339)
(0.650,0.250)
(0.809,0.073)
E3
(0.568,0.357)
(0.538,0.336)
(0.648,0.248)
(0.710,0.186)
E4
(0.577,0.357)
(0.541,0.340)
(0.750,0.142)
(0.741,0.141)
E5
(0.694,0.243)
(0.647,0.247)
(0.790,0.103)
(0.689,0.206)
E6
(0.456,0.460)
(0.689,0.206)
(0.840,0.037)
(0.917,0.000)
E7
(0.376,0.540)
(0.647,0.247)
(0.741,0.141)
(0.810,0.085)
E8
(0.629,0.308)
(0.815,0.052)
(0.649,0.245)
(0.648,0.248)
E9
(0.458,0.473)
(0.720,0.154)
(0.648,0.248)
(0.649,0.245)
Outcomes associated with the cautious IF estimates iWjβ=(ωjβ,ϖjβ) and ikjβ=(ukjβ,vkjβ)
(ωjβ,ϖjβ)
(u1jβ,v1jβ)
(u2jβ,v2jβ)
(u3jβ,v3jβ)
E1
(0.461,0.472)
(0.154,0.720)
(0.509,0.369)
(0.547,0.348)
E2
(0.248,0.668)
(0.339,0.539)
(0.650,0.250)
(0.541,0.341)
E3
(0.344,0.581)
(0.154,0.720)
(0.490,0.406)
(0.548,0.348)
E4
(0.331,0.603)
(0.261,0.620)
(0.646,0.246)
(0.489,0.393)
E5
(0.448,0.489)
(0.509,0.385)
(0.646,0.247)
(0.547,0.348)
E6
(0.228,0.688)
(0.547,0.348)
(0.634,0.243)
(0.635,0.237)
E7
(0.272,0.644)
(0.509,0.385)
(0.489,0.393)
(0.648,0.247)
E8
(0.525,0.412)
(0.333,0.534)
(0.547,0.347)
(0.490,0.406)
E9
(0.458,0.473)
(0.336,0.538)
(0.490,0.406)
(0.547,0.347)
In Step I.4, the C-IF evaluation value ckj, expressed as a C-IF number (ukj,vkj;rkj), was derived from the aggregated evaluation data in Otay and Kahraman’s (2022) study and is shown in the third to fifth columns of Table 1. The C-IF characteristic for each Ok was established using Eqs. (12) and (13) as follows: Ck={⟨Ej,ukj,vkj;rkj⟩|Ej∈{E1,E2,…,E9}}={⟨Ej,Or(ukj,vkj)⟩|Ej∈{E1,E2,…,E9}}, where Or(ukj,vkj)={⟨ℓ,ℓ′⟩∣ℓ,ℓ′∈[0,1],[(ukj−ℓ)2+(vkj−ℓ′)2]0.5⩽rkj,andℓ+ℓ′⩽1} for k∈{1,2,3} and j∈{1,2,…,9}.
Table 2 presents the aggressive and cautious IF estimates for each criterion and supplier option. In Step I.5, the aggressive IF estimates iWjα=(ωjα,ϖjα) and ikjα=(ukjα,vkjα) are generated using Eq. (7). For example, with w1=(0.573,0.360;0.158) and c21=(0.624,0.254;0.163), Eq. (7) yields: iW1α=(min{1,0.573+0.158/2},max{0,0.360−0.158/2})=(0.685,0.248) and i21α=(min{1,0.624+0.163/2},max{0,0.254−0.163/2})=(0.739,0.139). These results are shown in the upper portion of Table 2. Similarly, the cautious IF estimates iWjβ=(ωjβ,ϖjβ) and ikjβ=(ukjβ,vkjβ) were generated using Eq. (8) and presented in the bottom half of Table 2.
In Step I.6, the degrees of hesitancy ℏjα, hkjα, ℏjβ, and hkjβ for the estimates iWjα, ikjα, iWjβ, and ikjβ were computed. For instance, for iW1α, hesitancy was calculated as: ℏ1α=1−ω1α−ϖ1α=1−0.685−0.248=0.067. The scoring mechanisms M(iWjα), M(ikjα), M(iWjβ), and M(ikjβ) were then derived using the natural exponential function approach from Eq. (2). For example, for i21α=(0.739,0.139) with h21α=0.122:
M(i21α)=12[(0.739−0.139)+0.122·(e(0.739−0.139)e(0.739−0.139)+1−12)+1]=0.809.
The results, including the scoring mechanisms M(iWjα) and M(ikjα) for aggressive IF estimates (left part), and M(iWjβ) and M(ikjβ) for cautious IF estimates (right part), are illustrated in Table 3.
Specifics on the scoring mechanisms M(iWjα), M(ikjα), M(iWjβ), and M(ikjβ).
Outcomes related to aggressive IF estimates
Outcomes related to cautious IF estimates
Ej
M(iWjα)
M(i1jα)
M(i2jα)
M(i3jα)
M(iWjβ)
M(i1jβ)
M(i2jβ)
M(i3jβ)
E1
0.722
0.604
0.809
0.748
0.494
0.208
0.572
0.602
E2
0.447
0.603
0.705
0.878
0.286
0.397
0.705
0.603
E3
0.607
0.604
0.705
0.769
0.379
0.208
0.543
0.603
E4
0.612
0.603
0.812
0.809
0.362
0.315
0.705
0.549
E5
0.729
0.705
0.852
0.748
0.479
0.564
0.705
0.602
E6
0.498
0.748
0.913
0.967
0.265
0.602
0.701
0.705
E7
0.416
0.705
0.809
0.872
0.310
0.564
0.549
0.706
E8
0.663
0.894
0.707
0.705
0.557
0.396
0.603
0.543
E9
0.492
0.792
0.705
0.707
0.492
0.396
0.543
0.603
In Step I.7, the inclination parameter φ is assigned a value within [0,1] to reflect the decision-maker’s preference between aggressive and cautious IF estimates. Here, setting φ=0.5 indicates equal importance to both. Next, the joint generalized scoring functions S0.5(wj) and S0.5(ckj) were computed for each wj and ckj using Eq. (9). For instance, the calculation for S0.5(c21) is: S0.5(c21)=0.5×0.809+(1−0.5)×0.572=0.691.
In the example with φ=0.5, the expressions of Eqs. (14) and (15) were simplified since all criteria were beneficial. The concordance and discordance sets were defined as:
C0.5(Ok/Ol)={Ej|S0.5(ckj)⩾S0.5(clj)forEj∈E},D0.5(Ok/Ol)={Ej|S0.5(ckj)<S0.5(clj)forEj∈E}.
As outlined in Step I.8, Table 4 presents the results for C0.5(Ok/Ol) and D0.5(Ok/Ol) when option Ok outperforms Ol (k≠l), along with relevant explanations.
Specifics on the concordance and discordance sets for φ=0.5 (C0.5(Ok/Ol) and D0.5(Ok/Ol)).
Ok
Ol
C0.5(Ok/Ol)
D0.5(Ok/Ol)
Explanation
O1
O2
∅
{E1,E2,…,E9}
S0.5(c1j)<S0.5(c2j) for j∈{1,2,…,9}
O3
{E8}
{E1,E2,…,E7,E9}
S0.5(c18)⩾S0.5(c38) and S0.5(c1j)<S0.5(c3j) for j∈{1,2,…,7,9}
O2
O1
{E1,E2,…,E9}
∅
S0.5(c2j)⩾S0.5(c1j) for j∈{1,2,…,9}
O3
{E1,E4,E5,E8}
{E2,E3,E6,E7,E9}
S0.5(c2j)⩾S0.5(c3j) for j∈{1,4,5,8} and S0.5(c2j)<S0.5(c3j) for j∈{2,3,6,7,9}
O3
O1
{E1,E2,…,E7,E9}
{E8}
S0.5(c3j)⩾S0.5(c1j) for j∈{1,2,…,7,9} and S0.5(c38)<S0.5(c18)
O2
{E2,E3,E6,E7,E9}
{E1,E4,E5,E8}
S0.5(c3j)⩾S0.5(c2j) for j∈{2,3,6,7,9} and S0.5(c3j)<S0.5(c2j) for j∈{1,4,5,8}
In Step I.9, the metric parameter ξ was set to 1 and 2, reflecting the common use of the Manhattan (ξ=1) and Euclidean (ξ=2) metrics in practice. The C-IF distances DM1(ckj,clj) and DM2(ckj,clj) were computed using the four-term technique from Eq. (6):
DM1(ckj,clj)=12[12|rkj−rlj|+12(|ukj−ulj|+|vkj−vlj|+|hkj−hlj|)],DM2(ckj,clj)=12(12|rkj−rlj|+12(|ukj−ulj|2+|vkj−vlj|2+|hkj−hlj|2)).
These formulas account for differences in parameters r, u, v, and h between two C-IF evaluation values (ckj and clj) to calculate the distance measures DM1 and DM2. Given c13=(0.346,0.528;0.271) and c23=(0.569,0.327;0.112) with hesitancy degrees of 0.126 and 0.104, the distances were: DM1(c13,c23)=DM1(c23,c13)=(1/2)·[(1/2)·|0.271−0.112|+(1/2)·(|0.346−0.569|+|0.528−0.327|+|0.126−0.104|)]=0.168, and DM2(c13,c23)=DM2(c23,c13)=(1/2)·{(1/2)·|0.271−0.112|+[(1/2)·(|0.346−0.569|2+|0.528−0.327|2+|0.126−0.104|2)]0.5}=0.163. These Manhattan- and Euclidean-like distances, detailed in Table 5, quantify the dissimilarity between the C-IF evaluation values, reflecting differences across their parameters.
Specifics on the C-IF Manhattan- and Euclidean-like distances DM1(ckj,clj) and DM2(ckj,clj).
Outcome of the C-IF Manhattan-like distances
Outcome of the C-IF Euclidean-like distances
DM1(c1j,c2j),
DM1(c1j,c3j),
DM1(c2j,c3j),
DM2(c1j,c2j),
DM2(c1j,c3j),
DM2(c2j,c3j),
Ej
DM1(c2j,c1j)
DM1(c3j,c1j)
DM1(c3j,c2j)
DM2(c2j,c1j)
DM2(c3j,c1j)
DM2(c3j,c2j)
E1
0.177
0.196
0.034
0.176
0.192
0.033
E2
0.156
0.135
0.089
0.151
0.134
0.086
E3
0.168
0.197
0.031
0.163
0.192
0.031
E4
0.192
0.114
0.078
0.190
0.114
0.076
E5
0.072
0.021
0.052
0.072
0.020
0.051
E6
0.084
0.126
0.041
0.080
0.120
0.040
E7
0.053
0.082
0.079
0.050
0.081
0.076
E8
0.109
0.098
0.030
0.108
0.097
0.029
E9
0.077
0.105
0.030
0.074
0.102
0.029
Following Step I.10, the consistency indicator IC0.5(Ok/Ol) and inconsistency indicator ID0.5(Ok/Ol) (for φ=0.5) were derived using Eqs. (16) and (17). In the case where O3 outperforms O2, Table 4 shows the concordance set C0.5(O3/O2)={E2,E3,E6,E7,E9} and the discordance set D0.5(O3/O2)={E1,E4,E5,E8}. The consistency indicator IC0.5(O3/O2) was then computed as:
IC0.5(O3/O2)=∑Ej∈{E2,E3,E6,E7,E9}S0.5(wj)·|S0.5(c3j)−S0.5(c2j)|∑j′=19S0.5(wj′)·|S0.5(c3j′)−S0.5(c2j′)|=(0.367·|0.741−0.705|+0.493·|0.686−0.624|+0.382·|0.836−0.807|+0.363·|0.789−0.679|+0.492·|0.655−0.624|)/(0.608·|0.675−0.691|+0.367·|0.741−0.705|+0.493·|0.686−0.624|+0.487·|0.679−0.759|+0.604·|0.675−0.779|+0.382·|0.836−0.807|+0.363·|0.789−0.679|+0.610·|0.624−0.655|+0.492·|0.655−0.624|)=0.458.
By employing Eq. (17) and utilizing the C-IF Manhattan-like distance DM1(ckj,clj), the inconsistency indicator ID0.5(O3/O2) can be produced in this fashion:
ID0.5(O3/O2)=∑Ej∈{E1,E4,E5,E8}DM1(c3j,c2j)·|S0.5(c3j)−S0.5(c2j)|∑j′=19DM1(c3j′,c2j′)·|S0.5(c3j′)−S0.5(c2j′)|=(0.034·|0.675−0.691|+0.078·|0.679−0.759|+0.052·|0.675−0.779|+0.030·|0.624−0.655|)/(0.034·|0.675−0.691|+0.089·|0.741−0.705|+0.031·|0.686−0.624|+0.078·|0.679−0.759|+0.052·|0.675−0.779|+0.041·|0.836−0.807|+0.079·|0.789−0.679|+0.030·|0.624−0.655|+0.030·|0.655−0.624|)=0.451.
Using the C-IF Euclidean-like distance DM2(ckj,clj), the inconsistency indicator ID0.5(O3/O2) equals 0.454. Additional results are provided in Table 6’s third, fifth, and seventh columns.
Specifics on the consistency/inconsistency indicators and entries for φ=0.5.
Use of the DM1 measure
Use of the DM2 measure
Ok
Ol
IC0.5(Ok/Ol)
BC0.5(Ok/Ol)
ID0.5(Ok/Ol)
BD0.5(Ok/Ol)
ID0.5(Ok/Ol)
BD0.5(Ok/Ol)
O1
O2
0.000
0 (<I‾C0.5)
1.000
0 (>I‾C0.5)
1.000
0 (>I‾C0.5)
O3
0.019
0 (<I‾C0.5)
0.990
0 (>I‾C0.5)
0.990
0 (>I‾C0.5)
O2
O1
1.000
1 (⩾I‾C0.5)
0.000
1 (⩽I‾C0.5)
0.000
1 (⩽I‾C0.5)
O3
0.542
1 (⩾I‾C0.5)
0.549
0 (>I‾C0.5)
0.546
0 (>I‾C0.5)
O3
O1
0.981
1 (⩾I‾C0.5)
0.010
1 (⩽I‾C0.5)
0.010
1 (⩽I‾C0.5)
O2
0.458
0 (<I‾C0.5)
0.451
1 (⩽I‾C0.5)
0.454
1 (⩽I‾C0.5)
Following Step I.11, the average consistency indicator I‾C0.5 is 0.5 (Theorem 7). Using Eq. (18), the consistency entry BC0.5(Ok/Ol) was generated by comparing IC0.5(Ok/Ol) with I‾C0.5. The fourth column of Table 6 presents these comparisons. The consistency Boolean matrix BC0.5 was then constructed using Eq. (19):
BC0.5=−001−110−.
In Step I.12, the average inconsistency indicator I‾D0.5 was yielded to be 0.5 (Theorem 9). By comparing ID0.5(Ok/Ol) with I‾D0.5, the inconsistency entry BD0.5(Ok/Ol) was derived using Eq. (20). Results based on the C-IF Manhattan-like distance DM1 and Euclidean-like distance DM2 are shown in the sixth and eighth columns of Table 6, respectively. Table 6 summarizes the consistency and inconsistency indicators IC0.5(Ok/Ol) and ID0.5(Ok/Ol), along with their entries BC0.5(Ok/Ol) and BD0.5(Ok/Ol) for φ=0.5. As noted from Eq. (21), both distance measures yield the same inconsistency Boolean matrix BD0.5, as shown:
BD0.5=−001−010−.
Following Step I.13, the overall prioritization entry BO0.5(Ok/Ol) was calculated using BO0.5(Ok/Ol)=BC0.5(Ok/Ol)·BD0.5(Ok/Ol) from Eq. (22). These results were then used to build the overall prioritization Boolean matrix BO0.5 via Eq. (23), as shown:
BO0.5=−001−010−.
Continuing the procedure in Step I.13, a dominance graph was created to illustrate the partial-prioritization ranking of the three supplier options, as shown in Fig. 4. The overall outranking relationships using the C-IF ELECTRE I techniques were depicted utilizing orange arrows: O2≽O0.5O1 and O3≽O0.5O1. Consistency-based relationships are shown with green arrows: O2≽C0.5O1, O2≽C0.5O3, and O3≽C0.5O1. Inconsistency-based relationships are indicated by yellow arrows: O2≽D0.5O1 and O3≽D0.5O1. The ranking from the C-IF analytic hierarchy process (AHP) and VIKOR (i.e. VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian) methodology in Otay and Kahraman (2022) was O2≽O3≽O1. The outranking relationships from the C-IF ELECTRE I approach with φ=0.5 and ξ=1,2 align closely with the findings of Otay and Kahraman.
The dominance graph for the multi-expert supplier evaluation issue.
In the C-IF ELECTRE II approach, Steps II.1–II.10 mirror Steps I.1–I.10. In Step II.11, Eq. (24) calculates the consistency-dependent average outflow AC0.5(Ok). For example, for O1:AC0.5(O1)=∑l=1,l≠13IC0.5(O1/Ol)/(3−1)=[IC0.5(O1/O2)+IC0.5(O1/O3)]/2=(0.000+0.019)/2=0.010. Moreover, AC0.5(O2)=0.771 and AC0.5(O3)=0.720. In Step II.11, Eq. (25) computes the inconsistency-dependent average inflow AD0.5(Ok). Using the C-IF Manhattan-like distance measure DM1:AD0.5(O1)=∑l=1,l≠13ID0.5(O1/Ol)/(3−1)=[ID0.5(O1/O2)+ID0.5(O1/O3)]/2=(1.000+0.990)/2=0.995. Additionally, AD0.5(O2)=0.275 and AD0.5(O3)=0.231. Using the C-IF Euclidean-like distance measure DM2:AD0.5(O1)=∑l=1,l≠13ID0.5(O1/Ol)/(3−1)=[ID0.5(O1/O2)+ID0.5(O1/O3)]/2=(1.000+0.990)/2=0.995, AD0.5(O2)=0.273, and AD0.5(O3)=0.232.
Finally, Eq. (26) calculates the overall net flow NO0.5(Ok). Using the C-IF Manhattan-like distance measure DM1, the calculations yield: NO0.5(O1)=AC0.5(O1)−AD0.5(O1)=0.010−0.995=−0.985, NO0.5(O2)=0.496, and NO0.5(O3)=0.489. Using the C-IF Euclidean-like distance DM2:NO0.5(O1)=AC0.5(O1)−AD0.5(O1)=0.010−0.995=−0.985, NO0.5(O2)=0.498, and NO0.5(O3)=0.488. Arranging these in descending order, the complete prioritization ranking is O2≽N0.5O3≽N0.5O1 for both measures DM1 and DM2. This aligns with the preference order from the C-IF AHP and VIKOR methodology in Otay and Kahraman (2022), confirming that the outranking relationships from the current C-IF ELECTRE II approach with φ=0.5 and ξ=1,2 are consistent with their findings.
The proposed C-IF ELECTRE I and II techniques have demonstrated feasibility and efficiency in addressing multiple-criteria supplier assessments. By integrating C-IF theory with the established ELECTRE framework, these methodologies provide a comprehensive and reliable decision-making toolset. They effectively consider both beneficial and non-beneficial criteria while accounting for the decision-maker’s attitudes toward aggression and caution. The rankings of supplier options generated from these techniques are coherent and dependable, aligning with the findings from the C-IF AHP and VIKOR approach by Otay and Kahraman (2022). This indicates that the C-IF ELECTRE I and II methods significantly enhance decision-making in complex supplier evaluation scenarios.
The C-IF ELECTRE approach provides valuable insights into how decision-makers evaluate suppliers across multiple criteria. Key dimensions—such as cost, service, technology, and quality—greatly influence supplier rankings, particularly factors like price, on-time delivery, and technological capability. By incorporating both aggressive and cautious estimates, this method addresses uncertainty and adapts to varying market conditions and expert judgments, improving the reliability of recommendations.
Key advantages of the C-IF ELECTRE methodology include: (1) Refined Uncertainty Representation: Circular structures capture uncertainty, offering a balanced view of fluctuating factors like supplier reliability; (2) Flexibility in Risk Management: The inclination parameter allows customization based on the decision-maker’s risk tolerance, accommodating aggressive or cautious evaluations; and (3) Improved Sensitivity to Complex Criteria: This approach effectively manages complex, interdependent criteria, making it ideal for multidimensional supplier evaluations where factors are interconnected.
Despite its strengths, the C-IF ELECTRE approach has limitations: (1) Computational Complexity: The integration of C-IF sets and the need for precise calibration of parameters (φ and ξ) increases computational intricacy, potentially burdening decision-makers without advanced tools or expertise; and (2) Limited Sensitivity to Extremes: The approach may not adequately respond to suppliers excelling or underperforming in specific criteria, risking the omission of those with exceptional performance in niche areas but lacking broader competencies.
Comprehensive Comparative Analysis
This section examines the effects of different inclination parameter settings on the C-IF ELECTRE approach’s results. It also incorporates the Chebyshev distance metric (ξ→∞) within the C-IF Minkowski-like distance measure, alongside the previously used Manhattan and Euclidean metrics. Moreover, it evaluates divergence functions proposed by Khan et al. (2022) to compare the results generated by the C-IF ELECTRE techniques.
The C-IF Chebyshev-like distance between ckj and clj is derived by setting the metric parameter ξ in Definition 4 to infinity. This is calculated using the four-term strategy as:
DM∞(ckj,clj)=12{12|rkj−rlj|+max{|ukj−ulj|,|vkj−vlj|,|hkj−hlj|}}.
Divergence measures in fuzzy contexts help identify differences between fuzzy sets. For C-IF sets, Khan et al. (2022) developed various divergence functions to address higher-order uncertainties and evaluated their performance. They established five divergence functions based on chi-square and Canberra distances, as well as the exponential function. This study will adopt these five divergence measures to calculate inconsistency indicators. Let DEϵ(ckj,clj) represent the divergence measure between ckj and clj, where ϵ is an identifier parameter from the set {1,2,…,5}. The formulas for the five divergence measures based on chi-square distances (DE1 and DE2), Canberra distances (DE3andDE4), and the exponential function (DE5) are presented below: DE1(ckj,clj)=(ukj−ulj)21+ukj+ulj+(vkj−vlj)21+vkj+vlj+(rkj−rlj)21+rkj+rlj,DE2(ckj,clj)=[(ukj−ulj)21+ukj+ulj+(vkj−vlj)21+vkj+vlj]·e(rkj−rlj)21+rkj+rlj,DE3(ckj,clj)=|ukj−ulj|1+ukj+ulj+|vkj−vlj|1+vkj+vlj+|rkj−rlj|1+rkj+rlj,DE4(ckj,clj)=[|ukj−ulj|1+ukj+ulj+|vkj−vlj|1+vkj+vlj]·e|rkj−rlj|1+rkj+rlj,DE5(ckj,clj)=12·(e2−1)[(ukj−ulj)·(e4·ukj1+ukj+ulj−e4·ulj1+ukj+ulj)+(vkj−vlj)·DE5(ckj,clj)=(e4·vkj1+vkj+vlj−e4·vlj1+vkj+vlj)+(rkj−rlj)·(e4·rkj1+rkj+rlj−e4·rlj1+rkj+rlj)].
In the initial comparative analysis, the inclination parameter φ was systematically varied from 0 to 1 in increments of 0.1. Using these eleven configurations, the study applied the C-IF ELECTRE to the same supplier evaluation issue. Figure 5 illustrates the results in two parts: Fig. 5(a) shows the distribution of consistency indicators ICφ(Ok/Ol) among option pairs. The distribution remains stable for φ values between 0 and 0.4, but shifts significantly when φ exceeds 0.5. As φ increases, ICφ(O2/O1), ICφ(O2/O3), and ICφ(O3/O1) decrease, while ICφ(O1/O2), ICφ(O3/O2), and ICφ(O1/O3) increase. Figure 5(b) displays the consistency-dependent average outflows ACφ(Ok) for each option. It reveals that the average outflow of O1 is consistently the lowest across all φ values. As φ increases from 0 to 0.6, O2 has the highest average outflow, followed by O3. From φ=0.7 to 1, O3 surpasses O2 in average outflow, indicating a shift in advantage. These findings suggest that O2 is favoured at lower φ values, while O3 gains an advantage at higher φ values.
Comparison outcomes of consistency indicators and consistency-dependent average outflows.
In the second comparative analysis, the study examined the combined effects of different inclination parameter values (φ=0,0.1,…,1) and various C-IF distance and divergence measures on inconsistency indicators. It evaluated the C-IF Minkowski-like distance DMξ with metric parameters ξ=1,2,∞, alongside five divergence measures DEϵ based on chi-square, Canberra, and exponential functions. The study tested these eight measures: Manhattan-like DM1, Euclidean-like DM2, Chebyshev-like DM∞, chi-square DE1 and DE2, Canberra DE3andDE4, and exponential DE5. It compared outcomes for various parameter combinations, as shown in Fig. 6: (1) Similar trends: Inconsistency indicators calculated with DM1, DM2, and DM∞ showed high consistency. Likewise, results from divergence measures DE1, DE3, and DE5 aligned closely with those from the Minkowski-like distances; (2) Distinct patterns: Line charts in Fig. 6(e) and (g) revealed unique trends, suggesting these cases differ from others; and (3) Stable results: Indicators based on DE2 and DE4 remained relatively consistent across all φ values, contrasting with other measures that captured parameter variations more effectively. These findings highlight the C-IF ELECTRE framework’s robustness in accommodating diverse distance and divergence measures.
Comparison outcomes of inconsistency indicators for distinct C-IF distance/divergence measures.
The third comparative analysis considered key parameters: inclination (φ=0,0.1,…,1), metric (ξ=1,2,∞), and identifier (ε=1,2,…,5). Figure 7 contrasts outcomes of inconsistency-dependent average inflows across various C-IF distance/divergence measures, including Manhattan, Euclidean, and Chebyshev distances (DM1, DM2, and DM∞) from Chen (2023b) and five divergence measures (DE1–DE5) from Khan et al. (2022). Key findings include: (1) Consistent trends: Radar charts in Fig. 7(a)–(d), (f), and (h) show similar patterns across distance measures DM1, DM2, and DM∞ and divergence measures DE1, DE3, and DE5; (2) Distinct cases: Fig. 7(e) (DE2) and Fig. 7(g) (DE4) reveal unique patterns, indicating these scenarios differ from others; and (3) Disadvantage patterns: Across all φ values, option O1 shows the highest average inflow (disadvantage). For φ from 0 to 0.4, O2 has the lowest inflow, followed by O3. When φ exceeds 0.5, O3 becomes the least disadvantaged, followed by O2. These results confirm that the C-IF ELECTRE techniques produce consistent and reliable findings, regardless of the measurement method used.
Comparison outcomes of inconsistency-dependent average inflows.
The fourth comparative study considered the parameters φ=0,0.1,…,1, ξ=1,2,∞, and ε=1,2,…,5. Figure 8 illustrates the overall net flow NOφ(Ok) for each option, reflecting both the advantages from consistency indicators and disadvantages from inconsistency indicators. When using C-IF Manhattan, Euclidean, and Chebyshev distances (DM1, DM2, and DM∞), or divergence measures based on Canberra distances (DE3,DE4) and the exponential function (DE5), the rankings were O2≽NφO3≽NφO1 for φ values from 0 to 0.5, and shifted to O3≽NφO2≽NφO1 for φ values from 0.6 to 1. With the chi-square-based divergence measure DE1, the rankings remained O2≽NφO3≽NφO1 for φ between 0 and 0.4, but shifted to O3≽NφO2≽NφO1 for φ between 0.5 and 1. Using DE2, the shift occurred later, with rankings changing from O2≽NφO3≽NφO1 (φ=0 to 0.6) to O3≽NφO2≽NφO1 (φ=0.7 to 1). These findings reveal how rankings vary with φ values under different distance and divergence measures, offering insights into the options’ performance across varying conditions.
Comparison outcomes of overall net flows.
Conclusions
The proposed C-IF ELECTRE I and II approaches effectively handle uncertainties, hesitancies, and imprecise data through C-IF theory. They integrate a scoring mechanism, consistency and inconsistency indexing, and prioritization steps. Applied to supplier evaluation tasks, these methods demonstrate practical utility and reliability. The key contributions include: introducing the joint generalized scoring function, capturing decision-maker preferences, developing the C-IF ELECTRE framework, and validating its benefits through multi-expert supplier assessments.
The C-IF ELECTRE methodology, while effective for handling uncertainty in supplier evaluations or other decision-making issues, requires careful parameter tuning, may be complex for non-experts, and could face limitations in other fuzzy contexts or decision-making scenarios, requiring further validation and adaptation for broader applicability. Future research could broaden the application of the C-IF ELECTRE approach and strengthen its validity through comparisons with other methods and model quality metrics. While this study focused on supplier evaluation, future work should apply the approach to diverse fields—such as healthcare, project management, and environmental sustainability—to assess its versatility. Comparative analyses with other decision-making models, including traditional and fuzzy ELECTRE methods, would highlight the unique strengths of the C-IF ELECTRE. Additionally, evaluating its stability and reliability through metrics like accuracy and consistency would provide valuable quantitative insights, further validating its effectiveness and practical value.
Acknowledgements
The author thanks the editor and anonymous referees for their insightful feedback, which greatly enhanced the paper’s quality.
ReferencesAkram, M., Bibi, R., Deveci, M. (2023a). An outranking approach with 2-tuple linguistic Fermatean fuzzy sets for multi-attribute group decision-making. Engineering Applications of Artificial Intelligence, 121, 105992.Akram, M., Zahid, K., Deveci, M. (2023b). Multi-criteria group decision-making for optimal management of water supply with fuzzy ELECTRE-based outranking method. Applied Soft Computing, 143, 110403.Alinejad, S., Alimohammadlou, M., Abbasi, A., Mirghaderi, S.-H. (2024). Smart-Circular strategies for managing biomass resource challenges: a novel approach using circular intuitionistic fuzzy methods. Energy Conversion and Management, 314, 118690.Alsattar, H.A., Mourad, N., Zaidan, A.A., Deveci, M., Qahtan, S., Jayaraman, V., Khalid, Z. (2024). Developing IoT sustainable real-time monitoring devices for food supply chain systems based on climate change using circular intuitionistic fuzzy set. IEEE Internet of Things Journal, 11(16), 26680–26689.Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96.Atanassov, K.T. (2020). Circular intuitionistic fuzzy sets. Journal of Intelligent & Fuzzy Systems, 39(5), 5981–5986.Çakır, E., Taş, M.A. (2023). Circular intuitionistic fuzzy decision making and its application. Expert Systems with Applications, 225120076.Chen, T.-Y. (2023a). A circular intuitionistic fuzzy evaluation method based on distances from the average solution to support multiple criteria intelligent decisions involving uncertainty. Engineering Applications of Artificial Intelligence, 117, 105499.Chen, T.-Y. (2023b). Evolved distance measures for circular intuitionistic fuzzy sets and their exploitation in the technique for order preference by similarity to ideal solutions. Artificial Intelligence Review, 56(7), 7347–7401.Chen, T.-Y. (2024). A circular intuitionistic fuzzy assignment model with a parameterized scoring rule for multiple criteria assessment methodology. Advanced Engineering Informatics, 61, 102479.Ci, Q. (2024). Fuzzy aggregation for multi-sensor indoor localization: integrating heterogeneous data sources. IEEE Access, 12, 131993–132015.Hezam, I.M., Rani, P., Mishra, A.R., Alshamrani, A.M. (2023). A combined intuitionistic fuzzy closeness coefficient and a double normalization-based WISP method to solve the gerontechnology selection problem for aging persons and people with disability. AIMS Mathematics, 8(6), 13680–13705.Jameel, T., Riaz, M., Aslam, M., Pamucar, D. (2024). Sustainable renewable energy systems with entropy based step-wise weight assessment ratio analysis and combined compromise solution. Renewable Energy, 235, 121310.Khan, M.J., Kumam, W., Alreshidi, N.A. (2022). Divergence measures for circular intuitionistic fuzzy sets and their applications. Engineering Applications of Artificial Intelligence, 116, 105455.Kong, X. (2024). Complex circular intuitionistic fuzzy Heronian mean aggregation for dynamic air quality monitoring and public health risk prediction. IEEE Access, in press. https://doi.org/10.1109/ACCESS.2024.3458172.Liu, H. (2024). Enhanced CoCoSo method for intuitionistic fuzzy MAGDM and application to financial risk evaluation of high-tech enterprises. Informatica, 48(5), 1–14.Liu, W., Du, Y., Chang, J. (2023). A new intuitionistic fuzzy best worst method for deriving weight vector of criteria and its application. Artificial Intelligence Review, 56, 11071–11093.Otay, İ., Kahraman, C. (2022). A novel circular intuitionistic fuzzy AHP&VIKOR methodology: an application to a multi-expert supplier evaluation problem. Pamukkale University Journal of Engineering Sciences, 28(1), 194–207.Pinar, A., Boran, F.E. (2022). A novel distance measure on q-rung picture fuzzy sets and its application to decision making and classification problems. Artificial Intelligence Review, 55(2), 1317–1350.Pratama, D., Yusoff, B., Abdullah, L. (2024). Some operations on circular intuitionistic fuzzy sets. AIP Conference Proceedings, 2905(1), 030029.Ramya, L., Narayanamoorthy, S., Manirathinam, T., Kalaiselvan, S., Kang, D. (2023). An extension of the hesitant Pythagorean fuzzy ELECTRE III: techniques for disposing of e-waste without any harm. Applied Nanoscience, 13(3), 1939–1957.Wang, J.-C., Chen, T.-Y. (2021). A T-spherical fuzzy ELECTRE approach for multiple criteria assessment problem from a comparative perspective of score functions. Journal of Intelligent & Fuzzy Systems, 41(2), 3751–3770.Wu, M., Song, J., Fan, J. (2023). ITARA and ELECTRE III three-way decision model in the spherical fuzzy environment and its application in customer selection. Journal of Intelligent & Fuzzy Systems, 44(6), 10067–10084.Yüksel, S., Dinçer, H. (2023). Sustainability analysis of digital transformation and circular industrialization with quantum spherical fuzzy modeling and golden cuts. Applied Soft Computing, 138, 110192.Zhang, R., Xu, Z., Gou, X. (2023). ELECTRE II method based on the cosine similarity to evaluate the performance of financial logistics enterprises under double hierarchy hesitant fuzzy linguistic environment. Fuzzy Optimization and Decision Making, 22(1), 23–49.Zhou, L.P., Wan, S.P., Dong, J.Y. (2022). A Fermatean fuzzy ELECTRE method for multi-criteria group decision-making. Informatica, 33(1), 181–224.