## 3 Study Methodology
Figure 4: Methodology of the study. These variables are not only clinically relevant but also easily identifiable at the time of triage, ensuring their operational feasibility. In our scenario, a manual formula is particularly valuable, where decisions must be made quickly and accurately. Although ML models are effective, they require advanced technical infrastructure and continuous training and validation processes, which can hinder their real-time implementation, particularly in resource-limited hospitals. On the other hand, a manual formula can be seamlessly integrated into triage workflows, activating automatically upon the patient's arrival. For example, the formula developed in this study allows for the immediate identification of patients at risk of prolonged LOS, facilitating the activation of FT pathways. Moreover, the simplicity of the manual model reduces dependence on advanced technologies, ensuring its accessibility across a wide range of hospital environments. The transparency and traceability of manual formulas are also essential for ensuring that clinical decisions are fair and equitable. In critical settings, resource prioritisation processes (e.g., bed allocation or fast-track access) must be based on comprehensible and defensible criteria. ML systems, due to their lack of clear traceability, may produce biased outcomes that are difficult to detect and rectify. For this reason, the manual formula developed in this study is based on variables selected through expert consensus, integrating statistical evidence with ethical and clinical considerations. This ensures that decisions are consistent with the fundamental values of the healthcare system. Although ML models are often associated with high levels of accuracy, this study demonstrated that a well-designed manual formula can achieve comparable results. Using the train-test technique, the results of the manual formula showed an appropriate balance between sensitivity and specificity, similar to the performance of advanced automated algorithms. This highlights the validity of manual formulas as effective predictive tools, particularly when simplicity and traceability are prioritised. The development of a manual predictive formula allows for the integration of clinical expertise with statistical analysis, providing a robust, transparent, and real-time applicable tool. While ML systems have a role in large-scale research and prediction, in critical settings such as EDs--where trust, traceability, and clinical relevance are paramount--a manual formula designed by experts emerges as a more suitable alternative. Thus, the use of consensus formulas ensures understanding and acceptance by healthcare professionals while guaranteeing that clinical decisions are ethically defensible and aligned with principles of fairness in patient care. However, it is essential to ensure that the developed formula is capable of producing accurate and reliable results. To this end, this study compared the outcomes generated by the manual predictive formula with those obtained through commonly used machine learning systems, such as Random Forest, Support Vector Machines, and Multi-Layer Perceptron. This comparison evaluated the performance of the manual formula in terms of precision, sensitivity, and specificity, determining whether it could achieve results comparable to automated models. Similar to the approach outlined by [PERSON] et al. (2016), who demonstrated the efficacy of ensemble learning and Grad-CAM visualisation for pneumon-coniosis detection in chest X-rays, this study emphasises the importance of traceability and clinical relevance. Unlike complex machine learning models, the manual formula ensures transparency and interpretability, which are critical in healthcare settings, particularly in environments such as emergency departments where decisions must be made quickly and justifiably.