Emergency hospital selection in Peshawar using fractional Diophantine neural network for risky decision-making

Abstract In emergency medical scenarios, rapid and accurate hospital selection is crucial, especially in densely populated urban areas like Peshawar, Pakistan. This study proposes a novel hybrid decision-making model called the Fractional Diophantine Fuzzy Neural Network (FDFNN), which integrates fractional Diophantine fuzzy information with neural network structures and Hamacher aggregation operators. The key innovation lies in the model’s ability to effectively handle uncertainty, incomplete expert input, and unknown weight vectors—challenges often encountered in real-world multi-criteria decision-making (MCDM) problems. We applied the proposed model to evaluate six hospitals in Peshawar, considering five critical attributes for emergency care. Three Experts information was gathered and used to construct three decision matrices ( $${\mathcal{E}}_{1},{\mathcal{E}}_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> and $${\mathcal{E}}_{3})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , each representing different expert perspectives. The proposed model processed these matrices, aggregating the expert evaluations using the Hamacher operators and accounting for uncertainty in the data. After processing the input data, the model ranked the hospitals and identified the most suitable option for emergency situations based on the selected criteria. Based on expert opinions and the results of the proposed model, Rehman Medical Institute ( $${\mathbb{Q}}_{4}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> ), was identified as the best choice. To validate the effectiveness of the proposed model, we conducted a comparative analysis using established MCDM techniques, including TOPSIS, WASPAS, WS, WP, MOORA, and GRA. The results showed that the proposed model outperformed traditional methods in terms of decision accuracy and reliability. Additionally, the study explored the sensitivity of the model to changes in parameters ( $$\theta$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> and $$f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> ), providing further insights into the robustness of the decision outcomes. Furthermore, we also discuss the advantages and limitations of the proposed method, including its applicability in real-time decision-making scenarios and its potential for large-scale implementation.