Hospitals must allocate beds and staff effectively under volatile arrival patterns, where scheduling errors can cause preventable harm. This study introduces a fairness-aware, forecast-informed reinforcement learning framework for hospital scheduling, explicitly integrating fairness constraints, short-term demand forecasts, and SHAP-based explainability. The state fuses patient and system context with short-horizon demand forecasts, mean arrivals ($\widehat{\lambda }$), and volatility (${\widehat{\sigma }}^2$). The reward jointly optimizes efficiency, equity, and safety by penalizing waiting, diversions/transfers, ICU misuse, overtime, and cross-ward disparity. Using a benchmark-aligned synthetic cohort (60k visits over one year), the approach is compared against First-Come-First-Served (FCFS) and ablations without forecast features. The learned policy halves the mean waiting time (from 215.3 to 102.5 min), reduces diversions/transfers (from 27.6% to 7.8%), improves ICU match accuracy (from 93.4% to 95.1%), raises the fairness index by 45%, and cuts staff overtime by 56%. Adding forecast signals yields further gains over forecast-naive DQN (9% shorter waits; 28% fewer diversions/transfers), with robustness under demand surges and triage-mix shifts. By unifying equity constraints, anticipatory context, and explanation, the method turns reactive queues into proactive, auditable control and is extensible to perioperative flow, disaster triage, and outpatient capacity management.

Linear systems often involve coefficients that are uncertain or imprecise due to inherent variability and vagueness in the data. In scenarios where only approximate or vague knowledge of the system parameters is available, traditional fuzzy logic is commonly employed. However, conventional fuzzy logic may be inadequate when defining a membership degree with a single, precise value proves difficult. In such cases, Single-Valued Trapezoidal Neutrosophic Numbers (SVTrNNs) offer a more suitable framework, as they account for indeterminacy, alongside truth and falsity. The solution of Single-Valued Trapezoidal Neutrosophic Linear Equations (SVTrNLEs) was explored in this study using an embedding approach. The approach reformulates the SVTrNLEs into an equivalent crisp linear system, enabling the application of conventional solution methods. The solution was then obtained using either the matrix inversion method or the gradient descent optimization algorithm implemented in PyTorch. The robustness and adaptability of gradient-based optimization techniques were thoroughly assessed. The learning process minimizes the residual error iteratively, with convergence behaviour and numerical stability analyzed across various parameter configurations. The results demonstrate rapid convergence, proximity to exact solutions, and significant robustness to parameter variability, highlighting the efficacy of gradient descent for solving uncertain linear systems. These findings provide a foundation for the extension of gradient-based methods to more complex systems and broader applications. Furthermore, the existence and uniqueness of the neutrosophic solution to an n × n linear system were rigorously analyzed, with numerical examples provided to assess the reliability and efficiency of the proposed methods.