We prove part the first part of the proposition (weak duality) by induction. It is well-known that, by the value iteration algorithm's convergence, Q Consider a state s S and a feasible action a A (s). We use an induction proof. B (w), which follows by the convergence of value iteration.A.2 Proof of Theorem 1 Proof. Now we state the following lemma.

This article introduces an advanced analytical approach for predicting backorders in inventory management. Backorder refers to an order that cannot be immediately fulfilled due to stock depletion. Multiple classification techniques, including Balanced Bagging Classifiers, Fuzzy Logic, Variational Autoencoder - Generative Adversarial Networks, and Multi-layer Perceptron classifiers, are assessed in this work using performance evaluation metrics such as ROC-AUC and PR-AUC. Moreover, this work incorporates a profit function and misclassification costs, considering the financial implications and costs associated with inventory management and backorder handling. The study suggests that a combination of modeling approaches, including ensemble techniques and VAE, can effectively address imbalanced datasets in inventory management, emphasizing interpretability and reducing false positives and false negatives. This research contributes to the advancement of predictive analytics and offers valuable insights for future investigations in backorder forecasting and inventory control optimization for decision-making.

Supply chain management relies on accurate backorder prediction for optimizing inventory control, reducing costs, and enhancing customer satisfaction. However, traditional machine-learning models struggle with large-scale datasets and complex relationships, hindering real-world data collection. This research introduces a novel methodological framework for supply chain backorder prediction, addressing the challenge of handling large datasets. Our proposed model, QAmplifyNet, employs quantum-inspired techniques within a quantum-classical neural network to predict backorders effectively on short and imbalanced datasets. Experimental evaluations on a benchmark dataset demonstrate QAmplifyNet's superiority over classical models, quantum ensembles, quantum neural networks, and deep reinforcement learning. Its proficiency in handling short, imbalanced datasets makes it an ideal solution for supply chain management. To enhance model interpretability, we use Explainable Artificial Intelligence techniques. Practical implications include improved inventory control, reduced backorders, and enhanced operational efficiency. QAmplifyNet seamlessly integrates into real-world supply chain management systems, enabling proactive decision-making and efficient resource allocation. Future work involves exploring additional quantum-inspired techniques, expanding the dataset, and investigating other supply chain applications. This research unlocks the potential of quantum computing in supply chain optimization and paves the way for further exploration of quantum-inspired machine learning models in supply chain management. Our framework and QAmplifyNet model offer a breakthrough approach to supply chain backorder prediction, providing superior performance and opening new avenues for leveraging quantum-inspired techniques in supply chain management.

AI can analyze large amounts of data in order to identify patterns and hidden correlations, which would otherwise take humans considerably longer to comprehend and decipher. It can also be fed with a wide range of variables, which gives the engine more flexibility in its analysis. AI is revolutionising the overall healthcare landscape Automating healthcare is then, perhaps, a result of this changing healthcare landscape and the growing use and rapid adoption of technology. The healthcare sector is experiencing a dramatic shift in how goods and services are delivered. There are several driving factors in this transition, including a shift away from treating shorter episodes of illness towards a greater focus on long-term wellness and prevention. In addition consumers and patients alike show the deviation from and changing expectations.