Multi-Period Newsvendor Problem

The Multi-Period Newsvendor Problem is a classic inventory management challenge that involves determining optimal order quantities over multiple periods to maximize profit. This problem can be modeled as a Markov Decision Process (MDP), where states represent current inventory levels, actions denote order quantities, and rewards are associated with sales profits. On the MATLAB platform, three primary algorithms can be employed to solve this MDP model: value iteration, policy iteration, and reinforcement learning algorithms.

Value iteration algorithm approximates the optimal value function through iterative updates of state-value functions. In each iteration, the algorithm updates each state's value using the Bellman optimality equation based on current value function estimates. This process continues until the value function converges to a stable solution. The advantage of value iteration lies in its simplicity and intuitiveness, though it may face computational efficiency challenges with large state spaces. In MATLAB implementation, this typically involves nested loops for state transitions and reward calculations using matrix operations.

Policy iteration algorithm alternates between two steps: policy evaluation and policy improvement. During policy evaluation, the algorithm computes the value function under the current policy, while policy improvement updates the policy based on the current value function. Policy iteration generally converges faster than value iteration but requires more computational effort per iteration. MATLAB implementations often utilize linear algebra solvers for policy evaluation and argmax operations for policy updates.

Reinforcement learning algorithms like Q-learning or SARSA are suitable when the model is unknown or difficult to precisely model. These algorithms learn optimal policies through interaction with the environment without requiring prior knowledge of state transition probabilities. In MATLAB, the Reinforcement Learning Toolbox provides built-in functions for implementing these algorithms, including environment setup, agent configuration, and training loop management.

Each of these three methods has distinct advantages and limitations, making them suitable for different scenarios. In practical applications, the choice of algorithm should be based on the specific characteristics of the problem, such as state space size, model availability, and computational resources, when solving the Multi-Period Newsvendor Problem.