Establishment and Solution of a Multi-Stage Decision Model Based on Hypothesis Testing and Dynamic Programming Algorithm

This paper introduces a novel multi-stage decision-making model that integrates hypothesis testing and dynamic programming algorithms to address complex decision-making scenarios. Initially, we develop a sampling inspection scheme that controls for both Type I and Type II errors using a simple random sampling method without replacement, ensuring the randomness and representativeness of the sample while minimizing selection bias. Through the application of hypothesis testing theory, a hypothesis testing model concerning the defect rate is established, and formulas for the approximate distribution of the sample defect rate and the minimum sample size required under two different scenarios are derived. Subsequently, a multi-stage dynamic programming decision model is constructed. This involves defining the state transition functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T_{k}\left(n_{k}, s_{k}\right)$</tex> and stage-specific objective functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$V_{k, n}$</tex>, followed by obtaining six optimal decision strategies under various conditions through backward recursion. The results demonstrate the model's potent capability for multi-stage decision-making and its high interpretability, offering significant advantages in practical applications.