How can Markov decision processes help you solve complex Operations Research problems?

3 How to formulate MDPs?

Formulating an MDP requires defining the state space, which is the set of possible states that the system can be in; the action space, which is the set of possible actions that the decision maker can take in each state; the transition function, which is the probability of moving from one state to another given an action; the reward function, which is the immediate reward or cost of taking an action in a state; and the discount factor, which is a parameter that determines how much future rewards are valued compared to immediate rewards.

4 How to solve MDPs?

To solve an MDP, you need to find a policy, which is a rule that specifies what action to take in each state. There are two main approaches to finding a policy: value iteration and policy iteration. Both approaches use dynamic programming to iteratively update the value function, which is the expected total reward of following a policy from a state. Value iteration starts with an arbitrary value function and updates it until it converges to the optimal value function, from which the optimal policy can be derived. Policy iteration starts with an arbitrary policy and alternates between evaluating its value function and improving it by choosing the best action in each state.

5 How to implement MDPs?

To implement an MDP, you need to have a computational tool that can perform the necessary calculations and store the results. There are many software packages and libraries that can help you with this task, such as MATLAB, Python, R, or Julia. You can also use online platforms or simulators that provide interactive environments for learning and testing MDPs. Some examples are OpenAI Gym, RL-Glue, and PyMDP Toolbox.

6 How to improve MDPs?

MDPs are not perfect, and they have some limitations and assumptions that may not hold in reality. For example, MDPs require that the state space, action space, transition function, reward function, and discount factor are known and fixed, which may not be the case in dynamic or uncertain environments. MDPs also assume that the states are fully observable, which may not be true if some information is hidden or noisy. To overcome these challenges, you can use extensions or variations of MDPs, such as partially observable MDPs, stochastic games, reinforcement learning, or approximate dynamic programming.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?