A novel dominance based pythagorean fuzzy rough set model

Iftikhar Ul Haq; Tanzeela Shaheen; Wajid Ali; Adeel Arshad; Muhammad Kumail

doi:10.55976/dma.320251437101-121

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A novel dominance based pythagorean fuzzy rough set model

Iftikhar Ul Haq ¹ , Tanzeela Shaheen ² , Wajid Ali ³ ^* , Adeel Arshad ⁴ , Muhammad Kumail ⁵

1. Department of Mathematics, Air University, PAF Complex E-9 Islamabad, 44230, Pakistan
2. Department of Mathematics, Air University, PAF Complex E-9 Islamabad, 44230, Pakistan
3. Department of Mathematics, Air University, PAF Complex E-9 Islamabad, 44230, Pakistan
4. Department of Mathematics, Air University, PAF Complex E-9 Islamabad, 44230, Pakistan
5. Department of Mathematics, Air University, PAF Complex E-9 Islamabad, 44230, Pakistan

Correspondence: wajidali00258@gmail.com

DOI: https://doi.org/10.55976/dma.320251437101-121

Received

28 July 2025
Revised

09 September 2025
Accepted

20 October 2025
Published

08 November 2025

Pythagorean fuzzy sets Fuzzy ordered information system Rough sets Dominance relation

Abstract

The dominance-based rough sets extend conventional rough sets by substituting the equivalence relation with a dominance relation. Nonetheless, the current dominance relations remain overly limited in practical utility, as they consistently require a precise decrease or increase for every attribute to be considered. Indeed, for numerous practical scenarios, it is sufficient to utilize the ascending or descending arrangement of fractional characteristics instead of considering all attributes or focusing only on the overall assessment of objects. This research established two novel dominance relations derived from the observed phenomenon. Subsequently, we formulated a comprehensive rough set models based on these relations, enabling us to define overall assessments and specific criteria for individual attributes. First, we established a broader form of dominance relations and created a rough set model rooted in this generalized dominance concept. We achieved this by employing the Pythagorean fuzzy additive operator to combine the individualized attribute values of every object in Pythagorean fuzzy environments, resulting in an overall evaluation. Next, we introduced a different form of dominance relationship indicated as the "generalized -dominance relation," along with the corresponding "generalized -dominance rough set model." This is accomplished by integrating a parameter , which is within the range of (0, 1), into the general dominance relationship. The inclusion of this parameter allows us to manage the number of attributes that fulfill dominance relationships, leading to the derivation of decision rules encompassing both "at least" and "at most" conditions. The objective is to develop new dominance relations and rough set models in Pythagorean fuzzy settings, including a generalised -dominance relation for flexible attribute evaluation. As a result, the models generate effective decision rules under "at least" and "at most" conditions, with numerical examples validating their applicability. The proposed models support decision making in complicated and unpredictable situations by producing exact rules that are helpful in planning, risk analysis, and supply chain management. They can be used by managers to more clearly and transparently assess options, establish priorities, and justify group decisions.

<h5>Introduction</h5>
Pawlak <a href="#reference_list_0">[1]</a> introduced the notion of rough sets in 1982 to provide a mathematical framework for addressing imprecision, ambiguity, and uncertainty in data analysis <a href="#reference_list_1">[2-7]</a>. This concept has been widely applied in various domains such as machine learning, pattern identification, decision analysis, processes regulation, knowledge discovery in databases, and expert systems <a href="#reference_list_7">[8-13]</a>, yielding successful outcomes in practical problem-solving scenarios.
Currently, investigations into rough sets have brought forth numerous noteworthy topics for consideration. These include challenges related to attribute reduction <a href="#reference_list_13">[14-16]</a>, models involving approximation operators <a href="#reference_list_16">[17-19]</a>, the development of axiomatic systems <a href="#reference_list_19">[20-22]</a>, expansions and variations of rough sets <a href="#reference_list_22">[23-26]</a>, and more. Within these areas of study, the exploration of rough set generalizataions stands out as a vital and significant aspect of rough set theory. These generalizations have demonstrated their worth as effective tools for tackling a range of practical issues.
Pawlak's rough sets were created to handle datasets in which each object is limited to having a distinct, discrete value for each attribute. Additionally, in these datasets, every attribute can establish either an equivalence relation within the domain of discourse or a partition of discourse. However, this equivalence or partition approach proves to be constraining in many applications because it cannot detect inconsistencies that arise from the assessment of criteria involving attributes with sequential domains, such as product excellence, market position, and debt ratio <a href="#reference_list_23">[24]</a>. Over the last 35 years, various generalized rough set approaches have been developed to address this issue. One particularly significant approach is the dominance−based rough set approach (DRSA) <a href="#reference_list_26">[27-29,38]</a>, which replaces the equivalence relation with a dominance relation. This substitution enables the resolution of ranking issues related to criteria. Consequently, DRSA offers a practical solution for addressing the common inconsistencies found in MADM problems involving exemplary decisions.
The DRSA and Pawlak's rough set model <a href="#reference_list_0">[1]</a> have established a strong basis for managing uncertainty and ranking issues in multicriteria decision making (MCDM). Building on this basis, a wide range of studies have expanded fuzzy and rough-set frameworks to include interval-valued, neutrosophic, hypersoft, Q-rung, and Pythagorean fuzzy environments. These studies have also integrated aggregation operators and decision methods such as TOPSIS for real-world selection problems <a href="#reference_list_30">[31]</a>. As an illustration of how distance-based MCDM tools supplement rough-set ranking techniques <a href="#reference_list_31">[32]</a>, TOPSIS and its fuzzy variants have been widely used for selection problems in engineering and medicine. Examples of these include medical-clinic selection for diagnosis and the selection of automotive alternatives using intuitionistic fuzzy TOPSIS <a href="#reference_list_32">[33, 34]</a>. In parallel, the mathematical primitives for expressing richer and precise information and for establishing suitable approximations and aggregation procedures are provided by the algebraic and operational foundations of generalized set structures, such as interval-valued neutrosophic hypersoft sets <a href="#reference_list_34">[35, 36]</a>, interval-valued fuzzy soft matrices, and neutrosophic hypersoft matrices.
When combining attribute-level information in MCDM and medical diagnosis applications, recent developments in aggregation and interaction operators <a href="#reference_list_29">[30, 37]</a> (such as interaction geometric, Einstein average, and extensions for q-rung orthopair contexts) demonstrate how aggregation behavior can be tuned and how interaction effects can be captured. The usefulness of combining specialized uncertainty models with ranking techniques is demonstrated by application-driven studies, such as the selection of UAVs for precision agriculture using interval-valued q-rung TOPSIS <a href="#reference_list_38">[39-42]</a> and group decision-making under interval-valued probabilistic-linguistic T-spherical fuzzy information for cloud-storage selection <a href="#reference_list_42">[43]</a>.
Moreover, research from other practical fields, such as innovative Möbius transformation-based image encryption algorithms, demonstrates the wide range of applications for sophisticated mathematical and fuzzy/neutrosophic tools beyond classical MCDM, and highlights areas where methods can be cross-fertilized <a href="#reference_list_43">[44]</a>. Our approach addresses the need to handle rich interval, orthopair, and neutrosophic information, support both global (overall) and selective (per-attribute) comparisons, and enable flexible aggregation and decision rules for "at least" and "at most" cases in MADM problems. These studies collectively drive our development of generalized dominance and generalized δ-dominance relations within Pythagorean fuzzy ordered information systems.
Current procedures in the domain of DRSA primarily concentrate on extending methods and creating algorithms to reduce attributes <a href="#reference_list_44">[45-47, 30, 48-51]</a>. The concept of DRSA was originally introduced by Greco et al.<a href="#reference_list_26">[27]</a>, who established that condition attributes serve as criteria and are prioritized based on preference. Consequently, the acquired knowledge consists of a collection of dominance classes, which represent sets of objects determined by a dominance relationship. As an illustration, Greco et al. <a href="#reference_list_45">[46]</a> introduced the dominance-based rough fuzzy set approach by integrating DRSA within the framework of fuzzy logic. In this particular approach, the approximation of the vague objective is achieved by employing a dominance relationship instead of a fuzzy relationship. In reference <a href="#reference_list_46">[47]</a>, the traditional concept of dominance was extended to establish a one-on-one comparison-based dominance relationship. This extension was used to examine the ordinal characteristics of the preferred levels in pairs of objects. Błaszczynski et al. <a href="#reference_list_44">[45, 52]</a> introduced the VC-DRSA, which incorporates the concept of variable precision rough sets from reference <a href="#reference_list_52">[53]</a> into DRSA and it explores various versions of VC-DRSA. These variations yield broader, lower approximations compared to those computed using DRSA. In other words, they define the lower approximation as specific objects that exhibit a robust, though not necessarily causal, relationship with the sets being approximated. This is achieved by adding a parameter that controls the coherence of the objects incorporated within the lower approximations. Hu <a href="#reference_list_53">[54]</a> introduced an algorithm for determining the reductions in the VC-DRSA. This approach applies to interval-valued intuitionistic fuzzy information systems, where objects are characterized by imprecise evaluations. The concept of dominance-based rough approximations and their implementations in this context has been further developed in these references <a href="#reference_list_54">[55, 48-50, 56]</a>.
The dominance relationship in the aforementioned study on dominance rough sets are all established based on the notion of the classical dominance relationship. This conventional dominance relation represents a strong preference among elements in a set, often referred to as "outranking," and signifies a preference for one object over another concerning each criterion. However, this conventional dominance relation is somewhat limiting because it requires that objects satisfy ranking criteria for all information system, rather than just partial ones. Consequently, when constructing dominance categories using conventional dominance relations, even if one attribute of an object is less superior compared to another, it will result in the exclusion of the former from the dominance category of the latter. This characteristic can be particularly significant, especially when dealing with MADM <a href="#reference_list_57">[58-60]</a> problems involving large sets of elements.
Novelty and contribution of the study
The study introduces two new dominance relations—generalized dominance and generalized -dominance within Pythagorean fuzzy systems, enabling both overall and selective attribute evaluations. It extends DRSA by deriving decision rules for “at least” and “at most” cases, supported by numerical examples, thus broadening rough set applications in MADM.
As widely recognized, in numerous group decision scenarios, our primary focus is often on identifying entities with the most favorable composite evaluations while disregarding their individual attribute values. In other words, even if an entity exhibits subpar values in certain individual attributes, it should still be considered among the optimal choices as long as its composite evaluation surpasses that of its counterparts. Conversely, in many real-world situations, it is imperative not only to identify entities with superior composite evaluations but also to ensure that they outperform others in specific individual attributes. For instance, in the context of a computer audit risk assessment, when seeking entities that outperform audit objects  in terms of overall evaluation, our task entails calculating the overall evaluations of all entities and selecting those that outperform . In specific cases, it is important to identify objects that not only have an overall evaluation that surpasses  but also outperform  in individual attributes. To manage the extent to which an object must outperform others across various criteria, we introduced a parameter  (where 0 <  ≤ 1). Taking these two factors into consideration, we established two new dominance relations within Pythagorean fuzzy information systems. These relations account for both the holistic evaluations of objects and the ranking of objects according to specific criteria, rather than evaluating each criterion individually. The Pythagorean operation is employed to combine all attribute values into a single Pythagorean value, which in turn forms the basis for overall evaluations of objects. Initially, this process leads to the creation of a new dominance relationship, known as the generalized dominance relation, by ranking objects based on their overall evaluations. This relationship helps identify the group of objects with superior comprehensive assessments. Next, the generalized -dominance relation is established within the framework of dominance Pythagorean fuzzy ordered information systems, and it is determined by incorporating the parameter  into the generalized dominance relation. Currently, two distinct dominance relations have been developed to align with decision makers' preferences regarding "overall evaluations" and "individual attribute values." This generalized dominance relation is then utilized in the RSA to introduce the concept of a generalized DRSA within the framework of a dominance-based Pythagorean fuzzy ordered information system.
Research gap and motivation of the study
Classical DRSA relies on strict dominance relations, which require all criteria to be satisfied, limiting its usefulness in real-world problems where partial dominance is acceptable. VC-DRSA relaxes this restriction but still bases dominance on conventional definitions, not addressing cases where composite evaluations should outweigh individual weaknesses. There is a lack of models in the existing literature that integrate overall evaluations with selective attribute dominance in fuzzy environments, and insufficient exploration of Pythagorean fuzzy information systems for generalized dominance-based rough sets.
In many decision-making contexts, the focus is on selecting entities with superior composite evaluations, even if they are weaker in some individual attributes. Conversely, in some applications, decision-makers require assurance that entities also outperform in specific individual attributes. To reconcile these perspectives, flexible dominance relations that can model both overall performance and selective attribute strength are needed. This motivated the development of generalized dominance and δ-dominance relations, offering decision-makers more nuanced tools for handling imprecision and uncertainty in MADM problems.
Furthermore, we introduced the generalized DRSA by integrating the generalized -dominance relationship, which enables us to generate decision rules for both "at least" and "at most" scenarios. We supported these concepts with numerical examples for better understanding. It is important to note that these two novel approaches introduced in this article differ from VC-DRSA. Our models primarily focus on establishing a broader dominance relationship, which can result in more inclusive dominant categories compared to those produced by DRSA. This distinction arises because our novel dominance relations solely emphasize the order of the comprehensive assessment value, irrespective of individual attribute orders, or impose preferences on the order of only certain attributes. Additionally, the  Approxs in our approach still employ the “inclusion” relation, akin to DRSA. In contrast, VC-DRSA is similar to DRSA in terms of dominance relations, but it introduces the consistency level parameter  ∈ (0, 1] to relax the conditions for inclusion in  Approxs. This parameter governs the consistency of approximations. Consequently, the  Approxs of VC-DRSA expand upon those of DRSA, as they can encompass objects that, in line with the traditional interpretation, would not be included in the  Approxs. VC-DRSA can identify reliances that DRSA might overlook and provides a more robust foundation for generating rules.
The organization of this article can be outlined as follows: In Section 2, we presented a concise overview of the initial considerations of our study, which encompass the explanation of Pythagorean fuzzy set notations, fundamental operations associated with Pythagorean fuzzy sets, and a method for determining the dominance classes of objects based on their attribute values. Section 3 is dedicated to the development of a generalized dominance rough set model by defining a broader dominance relation in a pythagorean fuzzy environment. In section 4, we proposed the concept of generalized -dominance rough sets by integrating a generalized -dominance relation, which allows us to derive "at least" and "at most" decision rules. Finally, we concluded the research in section 5 with a summary of our findings and a glimpse into potential avenues for future research. Figure 1 is added to show the whole pipline of the proposed research work.
<img src="https://file.luminescience.cn/690eb0468f0c91762570310" alt="" width="500" height="416" />
Figure 1. Flowchart of whole research work
 
The full text of this article can be viewed by clicking <a href="https://ojs.luminescience.cn/DMA/article/view/437/404">here</a>
<h5>Limitations</h5>
Without thorough validation on substantial or diverse real-world datasets, this study is primarily restricted to theoretical development and an instructive example. The usefulness of the proposed parameter δ has not been fully investigated, particularly in terms of its sensitivity and robustness in various circumstances.
<h5>Conclusions and future work</h5>
In this article, we established a generalized dominance relationship by evaluating each object with the condition attribute set, ultimately created a generalized dominancebased rough set model in a Pythagorean fuzzy context. We also explored various advantageous characteristics of this model, demonstrating its effectiveness in addressing numerous real-world issues, particularly those involving multicriteria group decisionmaking with extensive data sets. In addition, to accommodate specific needs for distinct attribute values in practical scenarios, we proposed an additional rough set model referred to as the generalized -dominance rough set model within the Pythagorean fuzzy framework. This model introduces constraints on selected distinct attributes based on the generalized dominancebased rough set model. The introduction of the parameter  results in the generation of decision rules. Finally, we provided an illustrative example to clarify the concept.
Practical and managerial implications: The proposed models support decision-making in complex and uncertain environments by generating precise rules that are useful in areas such as supply chain management, risk analysis, and planning. Managers can use these models to evaluate alternatives, set priorities, and justify group decisions with greater clarity and transparency.
This work extends classical and intuitionistic rough set approaches by using the Pythagorean fuzzy framework and introducing the novel -dominance model with attribute-specific constraints to enhance decision accuracy.
Future studies should investigate the integration of these two novel models with the VC-DRSA, which presents a significant and intriguing topic for further exploration.
<h5>Abbreviations</h5>
IS                    Information System
PFOIS             Pythagorean Fuzzy Ordered Information System
DPFOIS          Dominance Based Pythagorean Fuzzy Ordered Information System
RSA                 Rough Set Approach
DRSA              Dominance Based Rough Set Approach
MADM            Multi-Attribute Decision Making
VC-DRSA         Variable Consistency Dominance-Based Rough Set Approach
DPFDS             DominanceBased Pythagorean Fuzzy Decision System
Approxs          Lower/Upper Approximations
<h5>Declaration of interests</h5>
The authors state that they have not identified any financial or personal conflicts that might have influenced the research detailed in this publication. Furthermore, they assert that there are no conflicting interests related to the publication of this article.
<h5>Authors' contribution</h5>
All authors contributed equally to the conception, design, analysis, and preparation of this manuscript. All authors have read and approved the final version of the paper.
<h5>Data availability statement</h5>
No publicly available datasets were used or generated during this study. All data supporting the findings are included within the article.

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Haq, I. U., Shaheen, T., Ali, W., Arshad, A., & Kumail, M. (2025). A novel dominance based pythagorean fuzzy rough set model. Decision Making and Analysis, 3(1), 101–121. https://doi.org/10.55976/dma.320251437101-121

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