Pythagorean fuzzy N-bipolar soft sets-based multi-criteria decision-making framework for sustainability evaluation and risk assessment in manufacturing industries

Maha M Saeed; Sagvan Y Musa; Baravan A Asaad; Zanyar A Ameen

doi:10.1038/s41598-025-15126-1

Pythagorean fuzzy N-bipolar soft sets-based multi-criteria decision-making framework for sustainability evaluation and risk assessment in manufacturing industries

Sci Rep. 2025 Aug 13;15(1):29648. doi: 10.1038/s41598-025-15126-1.

Authors

Maha M Saeed¹, Sagvan Y Musa^{2

3}, Baravan A Asaad^{4

5}, Zanyar A Ameen⁶

Affiliations

¹ Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O.Box 80203, Jeddah, 21589, Saudi Arabia.
² Department of Mathematics, College of Education, University of Zakho, Zakho, 42002, Iraq.
³ Department of Computer Science, College of Science, Knowledge University, Kirkuk Road, Erbil, 44001, Iraq.
⁴ Department of Mathematics, College of Science, University of Zakho, Zakho, 42002, Iraq.
⁵ Department of Computer Science, College of Science, Cihan University-Duhok, Duhok, 42001, Duhok, Iraq.
⁶ Department of Mathematics, College of Science, University of Duhok, Duhok, 42001, Duhok, Iraq. zanyar.ameen@uod.ac.

Abstract

Sustainability evaluation in manufacturing industries is increasingly vital for promoting responsible growth and long-term competitiveness amid environmental, social, and economic challenges. Effective decision-making (DM) under uncertainty is crucial for managing multiple, often conflicting sustainability objectives. In this paper, we propose a novel hybrid model, termed Pythagorean fuzzy N-bipolar soft sets (PFNBSSs), which integrates Pythagorean fuzzy sets (PFSs), N-soft sets (NSSs), and bipolar soft sets (BSSs) within a unified multi-criteria decision-making (MCDM) framework. For theoretical purposes, we define basic operations and algebraic properties of PFNBSSs, supported by illustrative examples. To demonstrate practical applicability, the PFNBSS model is applied to assess sustainability practices in manufacturing industries through two numerical examples: one focusing on positive and negative sustainability indicators, and another emphasizing comparative sustainability risk assessment across diverse manufacturing sectors. Detailed interpretations of computational results and their relevance in practical DM are provided. This is followed by a comparative analysis confirming the superior discrimination power and expressive capability of the PFNBSS model over existing alternatives. The paper concludes with a critical evaluation of the model and suggestions for future research.

Keywords: MCDM; N-soft sets; Pythagorean fuzzy N-bipolar soft sets; Pythagorean fuzzy sets; Sustainable Manufacturing Evaluation.