Dealing with imperfect information is a common task in everyday life, as uncertainty and ambiguity inevitably permeate any real-world system, creating challenges in the decision-making process [1,2]. Whether it is selecting the best alternative in business, diagnosing medical conditions, or making strategic decisions in engineering or finance, decision-makers are often confronted with complex situations where data is incomplete, imprecise, or ambiguous [3]. In such cases, conventional methods that rely on precise numerical values may fall short. To solve this problem, various effective theories have been established, like fuzzy sets [4,5], rough sets [6,7], Z-numbers [8,9], and Evidence theory [10,11]. These theories have a wide range of applications in various domains, such as supplier selection [12], decision-making [3], risk analysis [13], and medical diagnosis [14].

Among them, fuzzy sets, introduced by Zadeh [15], have become a foundational concept in modeling uncertainty. The notion of fuzzy sets allows for the representation of vague concepts by assigning a membership degree (MD) to elements in a set, providing a flexible way to handle ambiguous information. However, the classical fuzzy sets still face limitations when dealing with more complex uncertainty. Intuitionistic fuzzy sets (IFSs), developed by Atanassov [16], extended fuzzy sets by introducing a nonmembership degree (ND) along with the MD, providing a more comprehensive model of uncertainty. However, IFSs impose a strict constraint where the sum of the MD and ND must be less than or equal to one. This restriction can be limiting when dealing with high levels of ambiguity, as it fails to fully account for situations where both degrees exhibit substantial uncertainty. Building on IFSs, Pythagorean fuzzy sets (PFSs), introduced by Yager [17], offer a more flexible framework. PFSs allow the sum of the squares of the MD and ND to be less than or equal to one, relaxing the constraint imposed by IFSs. This relaxation provides more flexibility in modeling uncertainty and has attracted research focused on the application of PFSs [18,19]. Recently, Senapati and Yager [20] introduced Fermatean fuzzy sets (FFSs), which offer a further generalization by relaxing the constraints of IFSs and PFSs. FFSs use a more generalized relationship between the MD and ND, where the sum of the cubes of the MD and ND must be less than or equal to one. This generalized structure allows FFS to capture a broader range of uncertainty, making them highly suitable for decision-making scenarios that involve significant ambiguity and conflicting information. So far, FFS has attracted a lot of attention [21–23]. For example, Ref. [21] introduced several Fermatean fuzzy weighted average, weighted geometric, weighted power average, and weighted power geometric operators and applied them for the multiattribute decision-making (MADM) problem. Kakati et al. [22] suggested Fermatean fuzzy Archimedean Heronian mean and geometric Heronian mean operators, and developed a new MADM method for sustainable urban transport.

In practical applications, especially in fields involving pattern classification, clustering, decision-making, and artificial intelligence, measuring the discrepancy between two sets is crucial. Interestingly, distance/divergence/similarity operators for IFSs and PFSs have received great attention, and many operators have been developed. For example, some researchers have extended several widely used operators, including Hamming distance [24,25], Euclidean distance [24,26], Hausdorff distance [27,28], Jensen-Shannon divergence [29–31], Hellinger distance [32,33], cosine similarity [34,35], Jaccard similarity [36,37] and Dice similarity [38], to the environments of IFSs and PFSs, and have validated their effectiveness in scenarios such as pattern recognition, multiattribute decision-making (MADM) and medical diagnosis. Furthermore, more innovative distance and similarity operators for IFSs and PFSs can be found in the literature [39–41]. Several operators of distance and similarity have been developed for FFSs to enhance their practical utility. For example, Senapati and Yager [20] first introduced a Fermatean fuzzy operator based on Euclidean distance. Later, Onyeke and Ejegwa [42] pointed out that Senapati and Yager’s work contradicts the axiomatic of distance, and developed an enhanced distance operator for FFSs. Sahoo [43] introduced several similarity operators for FFSs and applied them to group decision-making. Kirisci [44] combined cosine similarity and Euclidean distance operators for FFSs, applying these concepts within the TOPSIS framework. Subsequently, lIU [45] highlighted deficiencies in Kirisci’s approach and introduced an improved cosine similarity operator. Ref. [46] designed the Hellinger distance and the triangular divergence for FFSs, respectively. Liu [14] proposed a new triangular divergence operator for FFSs to overcome the limitations of the previous work [46]. Ref. [47] suggested some t-conorms-based distance operators for FFSs. Liu [14] presented some similarity operators between FFSs based on Tanimoto and Sørensen coefficients.

Furthermore, many MADM models have been developed to achieve successful outcomes in solving real-life problems that contain different alternatives and multiple criteria in the decision-making process [48–50]. Fuzzy logic is frequently used in the literature to overcome the uncertainties that occur in the decision-making process in MADM models [51–54]. Among them, the alternative ranking order method accounting for the two-step normalization (AROMAN) model is a novel ranking technique [55], which uses a two-step normalization mechanism to ensure that selections can be compared objectively and fairly. Currently, AROMAN has been extended to some fuzzy frameworks to solve various decision-making problems. Ref. [55] introduced the AROMAN model, which provides a new solution to the delivery decision process of cargo bicycles. Ref. [56] combined the AROMAN model and FUCOM weighting method in an interval-type 2 fuzzy environment and demonstrated its potential in practical decision-making problems. Ref. [57] integrated the MEREC weighting technique and the extended AROMAN model to assess Turkey’s sustainable competitiveness level.

Despite some progress, there remain several significant research gaps:

Some distance/divergence operators for FFSs largely ignore the hesitation degree (HD). These operators may generally fail to accurately represent the differences between FFSs, especially in real-world decision-making scenarios where HD plays a key role.

The research objective of this paper is to fill these gaps by introducing a unified divergence operator that accounts for all three components (MD, ND, and HD) of FFSs, providing a more reliable and intuitive operator for comparing the sets. Further, by integrating the new operator with the AROMAN model, we seek to enhance the effectiveness of decision-making systems, thereby contributing to both theoretical and practical advancements in the field.

The main contributions of this paper are summarized as:

We propose a unified parametric divergence operator for FFSs, which considers the MD, NMD, and HD.

The paper is organized as follows: Section 2 reviews the basic concepts and the existing distance/divergence operators. Section 3 proposes a unified parametric divergence operator for FFSs and offers some numerical comparisons between existing operators and the proposed operator. Applications to pattern recognition, hierarchical clustering, and MADM are provided in Section 4. The conclusion is given in Section 5.

In this section, we briefly review some foundational concepts about IFSs, PFSs, and FFSs and introduce the existing distance/divergence operators.

Determining the discrepancy between FFSs remains a challenging yet critical aspect of decision-making processes. To address this, we introduce a unified parametric divergence operator for FFSs in this section. Unlike traditional divergence measures that only focus on a single aspect of membership information, our operator simultaneously incorporates the three core components of FFSs, namely the membership degree (MD), non-membership degree (NMD), and hesitation degree (HD). This design ensures that all sources of uncertainty are taken into account when comparing two fuzzy information sets. Moreover, the parameter δ provides a flexible mechanism for adjusting the sensitivity of the divergence: different values of δ emphasize different balance points between ℱ and 𝒢, allowing the operator to generalize several well-known divergences. We then explore the fundamental properties of the divergence operator, examine its connections to established operators, and analyze its efficacy. Overall, the proposed divergence operator proves more effective in differentiating FFSs and circumvents counterintuitive outcomes that may arise from using partial or incomplete measures.

Definition 10: Let ℱ={⟨oi,ℱΦ(oi),ℱΛ(oi)⟩|oi∈O} and 𝒢={⟨oi,𝒢Φ(oi),𝒢Λ(oi)⟩|oi∈O} be two FFSs in O, the parametric divergence between FFSs can be defined as: (25)𝒟δ(ℱ,𝒢)=1mδ(δ−1)∑i=1m(∑τ((ℱτ3(oi))δ(𝒢τ3(oi))1−δ−δℱτ3(oi)+(δ−1)𝒢τ3(oi)))where τ∈{Φ,Λ,Ξ} and δ∈R∖{0,1}.

This formulation highlights two important characteristics. First, the operator explicitly aggregates the divergence across all three components Φ (membership), Λ (non-membership), and Ξ (hesitation). Although Ξ is mathematically determined by Φ and Λ, its explicit inclusion does not create redundancy. On the contrary, it guarantees that the divergence takes into account the complete information of an FFS, including the degree of hesitation, which is critical in many decision-making scenarios. Moreover, including Ξ does not break the essential properties of the operator: the divergence remains non-negative, equals zero only when ℱ=𝒢, and is symmetric between the two sets. Thus, the operator offers a more comprehensive representation of uncertainty. Second, the parametric form with exponent δ enables a continuum of divergence behaviors. When δ approaches particular values, the operator reduces to well-known measures such as the Jensen–Shannon divergence, Hellinger distance, or χ2 divergence. Thus, our proposal serves as a unifying generalization that bridges multiple classical operators.

The parametric divergence operator for FFSs can also be written in an expanded form as: (26)𝒟δ(ℱ,𝒢)=1mδ(δ−1)∑i=1m(((ℱΦ3(oi))δ(𝒢Φ3(oi))1−δ−δℱΦ3(oi)+(δ−1)𝒢Φ3(oi))+((ℱΛ3(oi))δ(𝒢Λ3(oi))1−δ−δℱΛ3(oi)+(δ−1)𝒢Λ3(oi))+((ℱΞ3(oi))δ(𝒢Ξ3(oi))1−δ−δℱΞ3(oi)+(δ−1)𝒢Ξ3(oi)))

This equivalent expression shows how each component contributes to the overall divergence: the cubic powers ensure the operator respects the Fermatean structure, while the subtraction terms −δℱτ3(oi) and (δ−1)𝒢τ3(oi) correct for scaling and maintain non-negativity. Importantly, the explicit presence of Ξ increases the sensitivity of the measure when hesitation changes.

Example 1: To illustrate this effect, we compare divergence values computed with and without including Ξ, using (26) with δ=0.5:

Case 1: ℱ={⟨0.60,0.20⟩}⇒ℱΞ=0.9189, 𝒢={⟨0.75,0.30⟩}⇒𝒢Ξ=0.8199.

𝒟with Ξ(0.5)=0.1179, 𝒟without Ξ(0.5)=0.0795, an increase of 48.30%.

Case 2: ℱ={⟨0.50,0.30⟩}⇒ℱΞ=0.9465, 𝒢={⟨0.60,0.20⟩}⇒𝒢Ξ=0.9189.

𝒟with Ξ(0.5)=0.0391, 𝒟without Ξ(0.5)=0.0360, an increase of 8.61%.

These results confirm that including hesitation does not introduce redundancy but instead ensures that the divergence is responsive to uncertainty. When hesitation varies strongly, the divergence appropriately grows larger; when hesitation is nearly constant, the divergence remains close to the two-dimensional variant.

Example 2: Consider the following example where δ=0.6. We have two FFSs ℱ and 𝒢, given by: ℱ={⟨0.37,0.45⟩,⟨0.28,0.19⟩},𝒢={⟨0.42,0.38⟩,⟨0.32,0.73⟩}

The calculations give 𝒟(0.6)(ℱ,𝒢)=0.3204 and 𝒟(0.6)(𝒢,ℱ)=0.3885.

These results, where 𝒟(0.6)(ℱ,𝒢)≠𝒟(0.6)(𝒢,ℱ), illustrate the asymmetry of 𝒟(0.6). The asymmetrical nature highlights an important limitation: the measure depends on the order of the input sets. In many decision-making applications, however, symmetry is desirable, since the divergence between ℱ and 𝒢 should intuitively equal that between 𝒢 and ℱ. To address this, we further extend the operator to its symmetric form, as shown below.

Definition 11: Let ℱ={⟨oi,ℱΦ(oi),ℱΛ(oi)⟩|oi∈O} and 𝒢={⟨oi,𝒢Φ(oi),𝒢Λ(oi)⟩|oi∈O} be two FFSs in O, the symmetric parametric divergence between FFSs can be defined as: (27)𝒟Lδ(ℱ,𝒢)=12[𝒟δ(ℱ,ℱ+𝒢2)+𝒟δ(𝒢,ℱ+𝒢2)]

Here, the operator computes the divergence of each set with respect to their average and then averages the results, ensuring symmetry. This construction not only eliminates the order-dependence problem but also aligns with the structure of Jensen–Shannon divergence, thereby reinforcing the interpretability and robustness of the proposed measure in practical scenarios.

The following properties are derived from the 𝒟L(δ)(ℱ,𝒢).

Property 1 (Non-negativity): 𝒟L(δ)(ℱ,𝒢)⩾0.

Proof: For each i and τ, we consider (ℱτ3(oi))δ(𝒢τ3(oi))1−δ−δℱτ3(oi)+(δ−1)𝒢τ3(oi). We need to show that this expression is non-negative: (ℱτ3(oi))δ(𝒢τ3(oi))1−δ≥δℱτ3(oi)−(δ−1)𝒢τ3(oi)

We set x=ℱτ3(oi) and y=𝒢τ3(oi), where 0≤x,y≤1: f(x,y)=xδy1−δ−δx+(δ−1)y

Case 1: 0<δ<1

In this case, the function f(x,y) is convex where 0≤x,y≤1.

By Jensen’s inequality and the properties of convex functions, we have xδy1−δ≥δx+(1−δ)ywhich implies that xδy1−δ−δx+(δ−1)y≥0

Thus, the expression is non-negative for 0<δ<1.

Case 2: δ>1

In this case, the roles of x and y are reversed: f(x,y)=xδy1−δ−δx+(δ−1)y

As xδ grows faster than linear, we again leverage the fact that the convexity of power functions ensures: xδy1−δ≥δx+(1−δ)y

This ensures the term is non-negative, similarly to the case for 0<δ<1.

Summing over all i and all τ, we conclude: 𝒟δ(ℱ,𝒢)=1mδ(δ−1)∑i=1m∑τ((ℱτ3(oi))δ(𝒢τ3(oi))1−δ−δℱτ3(oi)+(δ−1)𝒢τ3(oi))≥0

This establishes that 𝒟δ(ℱ,𝒢) is non-negative.

As a result, 𝒟Lδ(ℱ,𝒢)=12[𝒟δ(ℱ,ℱ+𝒢2)+𝒟δ(𝒢,ℱ+𝒢2)]≥0.□

Property 2 (Non-degeneracy): 𝒟L(δ)(ℱ,𝒢)=0 if and only if ℱ=𝒢.

Proof: We assume two FFSs ℱ=𝒢: 𝒟L(δ)(ℱ||𝒢)=𝒟L(δ)(𝒢||𝒢)=𝒟L(δ)(ℱ||ℱ)=1mδ(δ−1)∑i=1m(∑τ((ℱτ3(oi))δ(ℱτ3(oi))1−δ−δℱτ3(oi)+(δ−1)ℱτ3(oi)))=0.

If 𝒟L(δ)(ℱ||𝒢)=0, then 12[𝒟(δ)(ℱ||ℱ+𝒢2)+𝒟(δ)(𝒢||ℱ+𝒢2)]=0implies 𝒟(δ)(ℱ||ℱ+𝒢2)=𝒟(δ)(𝒢||ℱ+𝒢2)=0.

Thus, we have ℱ=ℱ+𝒢2,𝒢=ℱ+𝒢2.

Solving these, we conclude ℱ=𝒢.

This completes the proof of non-degeneracy, demonstrating that 𝒟L(δ)(ℱ||𝒢)=0 if and only if ℱ=𝒢. ◻

Property 3 (Symmetry): 𝒟L(δ)(ℱ,𝒢)=𝒟L(δ)(𝒢,ℱ).

Proof: We consider two FFSs ℱ and 𝒢: 𝒟L(δ)(ℱ||𝒢)=12[𝒟(δ)(ℱ||ℱ+𝒢2)+𝒟(δ)(𝒢||ℱ+𝒢2)]and 𝒟L(δ)(𝒢||ℱ)=12[𝒟(δ)(𝒢||ℱ+𝒢2)+𝒟(δ)(ℱ||ℱ+𝒢2)].

By comparing these expressions, it is clear that: 𝒟L(δ)(ℱ||𝒢)=𝒟L(δ)(𝒢||ℱ).

This equality confirms that 𝒟L(δ) is symmetric. Thus, the symmetry Property is proven. ◻

Property 4: When δ=2, 𝒟L(δ) simplifies into the χ2 divergence (𝒟χ2): (28)𝒟L(2)(ℱ,𝒢)=𝒟χ2(ℱ,𝒢)where (29)𝒟χ2(ℱ,𝒢)=14m∑i=1m(∑τ(ℱτ3(oi)−𝒢τ3(oi))2ℱτ3(oi)+𝒢τ3(oi))

Proof: For two FFSs ℱ and 𝒢 in O, when δ=2, we have: 𝒟Lδ(ℱ,𝒢)=14m∑i=1m{∑τ[(ℱτ3(oi))2(ℱτ3(oi)+𝒢τ3(oi)2)−1−2ℱτ3(oi)+ℱτ3(oi)+𝒢τ3(oi)2]+∑τ[(𝒢τ3(oi))2)(ℱτ3(oi)+𝒢τ3(oi)2)−1−2𝒢τ3(oi)+(ℱτ3(oi)+𝒢τ3(oi)2)]}=14m∑i=1m[∑τ(2(ℱτ3(oi))2ℱτ3(oi)+𝒢τ3(oi)−2ℱτ3(oi)+ℱτ3(oi)+𝒢τ3(oi)2)+∑τ(2(𝒢τ3(oi))2ℱτ3(oi)+𝒢τ3(oi)−2𝒢τ3(oi)+ℱτ3(oi)+𝒢τ3(oi)2)]=14m∑i=1m[∑τ(2((ℱτ3(oi))2+(𝒢τ3(oi))2)−(ℱτ3(oi)+𝒢τ3(oi))2ℱτ3(oi)+𝒢τ3(oi))]=14m∑i=1m(∑τ(ℱτ3(oi)−𝒢τ3(oi))2ℱτ3(oi)+𝒢τ3(oi))=𝒟χ2(ℱ,𝒢)◻

Property 5: When δ→1, 𝒟L(δ) converges to the Jensen-Shannon divergence (𝒟JS): (30)𝒟L(1)(ℱ,𝒢)=𝒟JS(ℱ,𝒢)where (31)𝒟JS(ℱ,𝒢)=12m∑i=1m(∑τℱτ3(oi)ln⁡2ℱτ3(oi)ℱτ3(oi)+𝒢τ3(oi)+𝒢τ3(oi)ln⁡2𝒢τ3(oi)ℱτ3(oi)+𝒢τ3(oi))

Proof: For two FFSs ℱ and 𝒢 in O, when δ→1, we have:

limδ→1𝒟L(δ)(ℱ,𝒢)=limδ→1{∂∂δ[∑i=1m(∑τ(ℱτ3(oi))δ(ℱτ3(oi)+𝒢τ3(oi)2)1−δ−δℱτ3(oi)+(δ−1)(ℱτ3(oi)+𝒢τ3(oi)2)+∑τ(𝒢τ3(oi))δ(ℱτ3(oi)+𝒢τ3(oi)2)1−δ−δ𝒢τ3(oi)+(δ−1)(ℱτ3(oi)+𝒢τ3(oi)2))]∂∂δ2mδ(δ−1)}=limδ→1{[∑i=1m(∑τ(ℱτ3(oi))δ(ℱτ3(oi)+𝒢τ3(oi)2)1−δln⁡2ℱτ3(oi)ℱτ3(oi)+𝒢τ3(oi)−ℱτ3(oi)+(ℱτ3(oi)+𝒢τ3(oi)2)+∑τ(𝒢τ3(oi))δ(ℱτ3(oi)+𝒢τ3(oi)2)1−δln⁡2𝒢τ3(oi)ℱτ3(oi)+𝒢τ3(oi)−𝒢τ3(oi)+(ℱτ3(oi)+𝒢τ3(oi)2))]2m(2δ−1)}=12m∑i=1m[∑τ(ℱτ3(oi)ln⁡2ℱτ3(oi)ℱτ3(oi)+𝒢τ3(oi)+𝒢τ3(oi)ln⁡2𝒢τ3(oi)ℱτ3(oi)+𝒢τ3(oi))]=𝒟JS(ℱ,𝒢)

◻

Property 6: When δ=12, 𝒟L(δ) simplifies into the Hellinger distance (𝒟Hel): (32)𝒟L(12)(ℱ,𝒢)=𝒟Hel(ℱ,𝒢)where (33)𝒟Hel(ℱ,𝒢)=1m∑i=1m{∑τ[(ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2+(𝒢τ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2]}

Proof: For two FFSs ℱ and 𝒢 in O, when δ=12, we have:

𝒟Lδ(ℱ,𝒢)=−2m∑i=1m{∑τ[(ℱτ3(oi))12(ℱτ3(oi)+𝒢τ3(oi)2)12−12ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2]+∑τ[(𝒢τ3(oi))12(ℱτ3(oi)+𝒢τ3(oi)2)12−12𝒢τ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2]}=−2m∑i=1m{∑τ−12(ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2−12∑τ(ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2}=1m∑i=1m{∑τ[(ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2+(ℱτ3(oi)−ℱτ3(oi)+𝒢τ3(oi)2)2]}

◻

Property 7: When δ→0, 𝒟L(δ) converges to the arithmetic-geometric divergence (𝒟AG): 𝒟L(12)(ℱ,𝒢)=𝒟AG(ℱ,𝒢)where (34)𝒟AG(ℱ,𝒢)=1m∑i=1m(∑τ(ℱτ3(oi)+𝒢τ3(oi))2ln⁡ℱτ3(oi)+𝒢τ3(oi)2ℱτ3(oi)∗𝒢τ3(oi))