SMDP semi-Markov decision problem

decision-making processes in situations where the system dynamics are not fully Markovian. In this framework, decision-makers aim to maximize some objective function by choosing actions based on the current state and elapsed time since the last transition. SMDPs find applications in various fields such as operations research, artificial intelligence, and control theory.

In a Markov Decision Problem (MDP), decisions are made based on the current state of the system, and the future state is independent of the past given the current state. However, in certain scenarios, this assumption does not hold, and the future state may depend not only on the current state but also on the time elapsed since the last transition. SMDPs provide a solution to such problems by incorporating additional information about the time dimension.

To understand SMDPs, let's start with the basic elements of the framework. An SMDP consists of a set of states, actions, rewards, and transition probabilities. The states represent the possible configurations of the system, while actions represent the available choices that can be made. The decision-maker receives a reward based on the state and action taken. Transition probabilities determine the likelihood of moving from one state to another after taking a specific action.

However, in SMDPs, the time spent in each state is not restricted to a single step. Instead, it follows a so-called "holding time distribution." This distribution captures the elapsed time since the last transition. The holding time can be continuous or discrete, depending on the problem at hand. By considering the holding time, SMDPs enable decision-makers to model scenarios where the duration of a particular state affects the decision-making process.

To formalize SMDPs, we introduce a concept called the "sojourn time." The sojourn time refers to the duration spent in a specific state before making the next transition. It follows the holding time distribution and affects the transition probabilities. The transition probabilities in an SMDP are defined as the probability of moving from one state to another within a specific time interval, given the current state and the sojourn time.

The objective of solving an SMDP is to find a policy that maximizes the expected total reward over an infinite time horizon. A policy in an SMDP maps each state and elapsed time to an action. The value of a state in an SMDP is defined as the expected total reward when following a particular policy. By computing the value function, we can determine the optimal policy that maximizes the expected total reward.

Solving SMDPs involves several techniques. One common approach is to use dynamic programming, which involves breaking down the problem into smaller subproblems and solving them iteratively. Value iteration and policy iteration are two widely used dynamic programming algorithms for solving SMDPs.

In value iteration, the algorithm iteratively updates the value function for each state by considering the expected rewards and transition probabilities. The process continues until the values converge to their optimal values. Once the values are known, the optimal policy can be derived by choosing the action that maximizes the expected total reward at each state.

Policy iteration, on the other hand, alternates between two steps: policy evaluation and policy improvement. In policy evaluation, the algorithm computes the value function for a given policy. In policy improvement, the algorithm updates the policy by choosing the action that maximizes the expected total reward based on the current value function. The process continues until the policy converges to the optimal policy.

Other solution methods for SMDPs include linear programming, reinforcement learning, and simulation-based approaches. Linear programming can be used to solve SMDPs with a finite number of states and actions by formulating the problem as a linear program. Reinforcement learning algorithms, such as Q-learning and Monte Carlo methods, can be adapted to solve SMDPs by estimating the value function through interactions with the environment. Simulation-based approaches involve generating sample paths to estimate the expected total reward and derive an optimal policy.

In conclusion, a Semi-Markov Decision Problem (SMDP) extends the framework of Markov Decision Problems (MDPs) by incorporating additional information about the elapsed time since the last transition. SMDPs allow decision-makers to model scenarios where the duration of a particular state affects the decision-making process. By considering the holding time distribution and sojourn times, SMDPs provide a more accurate representation of real-world decision-making problems. Various solution methods, including dynamic programming, linear programming, reinforcement learning, and simulation-based approaches, can be used to solve SMDPs and derive optimal policies.