Fermatean fuzzy soft rough graphs: A hybrid model for multi-criteria decision-making in supply chain risk analysis under complex uncertainty

Ahad Khaled Mohamed Alharbi; Kholood Mohammad Alsager; Ahad Khaled Mohamed Alharbi; Kholood Mohammad Alsager

doi:10.3934/math.20261169

AIMS Mathematics

2026, Volume 11, Issue 9: 29430-29469. doi: 10.3934/math.20261169

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Research article

Fermatean fuzzy soft rough graphs: A hybrid model for multi-criteria decision-making in supply chain risk analysis under complex uncertainty

Department of Mathematics, College of Science, Qassim University, Buraydah, Saudi Arabia; Email: 461215456@qu.edu.sa

Received: 09 June 2026 Revised: 04 August 2026 Accepted: 19 August 2026 Published: 14 September 2026
MSC : 03B52, 03E72

Complex supply chain decision-making requires mathematical models capable of handling parametric, granular, and relational uncertainty simultaneously. Existing graph-based models address only part of these uncertainty types. To the best of our knowledge, no graph-based framework simultaneously models all three within a unified mathematical structure, motivating the proposed FFSRG model. In this paper, we proposed the Fermatean fuzzy soft rough graph (FFSRG), integrating Fermatean fuzzy sets, soft sets, rough sets, and graph theory into a unified decision-making framework. Vertex and edge approximation operators were defined, fundamental algebraic properties were established, aggregation operators preserving the Fermatean constraint were developed, and a six-step decision-making algorithm was proposed. The framework was validated through a supply chain risk assessment involving ten suppliers. Comparative analysis with six graph-based models demonstrated the effectiveness of the proposed approach. The proposed FFSRG achieved the highest ranking accuracy (91.2%) and uncertainty-handling performance (0.89). The results showed that integrating rough approximations with Fermatean fuzzy structures improves decision consistency and discrimination among alternatives. The FFSRG framework provides an effective tool for multi-criteria decision-making under complex uncertainty, with potential applicability beyond supply chain risk analysis.
- Fermatean fuzzy sets,
- soft sets,
- rough sets,
- fuzzy graphs,
- multi-criteria decision-making
Citation: Ahad Khaled Mohamed Alharbi, Kholood Mohammad Alsager. Fermatean fuzzy soft rough graphs: A hybrid model for multi-criteria decision-making in supply chain risk analysis under complex uncertainty[J]. AIMS Mathematics, 2026, 11(9): 29430-29469. doi: 10.3934/math.20261169

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Abstract

Complex supply chain decision-making requires mathematical models capable of handling parametric, granular, and relational uncertainty simultaneously. Existing graph-based models address only part of these uncertainty types. To the best of our knowledge, no graph-based framework simultaneously models all three within a unified mathematical structure, motivating the proposed FFSRG model. In this paper, we proposed the Fermatean fuzzy soft rough graph (FFSRG), integrating Fermatean fuzzy sets, soft sets, rough sets, and graph theory into a unified decision-making framework. Vertex and edge approximation operators were defined, fundamental algebraic properties were established, aggregation operators preserving the Fermatean constraint were developed, and a six-step decision-making algorithm was proposed. The framework was validated through a supply chain risk assessment involving ten suppliers. Comparative analysis with six graph-based models demonstrated the effectiveness of the proposed approach. The proposed FFSRG achieved the highest ranking accuracy (91.2%) and uncertainty-handling performance (0.89). The results showed that integrating rough approximations with Fermatean fuzzy structures improves decision consistency and discrimination among alternatives. The FFSRG framework provides an effective tool for multi-criteria decision-making under complex uncertainty, with potential applicability beyond supply chain risk analysis.

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