I have a system with $k$ tasks, where each task is a state in a Markov chain. I am working on allocating agents to each task. I would like to achieve a specific distribution of the agents among these tasks in the steady state. How do I optimize the Transition Probability Matrix (TPM) using OR-Tools?
Some info of my system: The system exists as a simulation of the process. Hence, each agent is simulated to make an actual choice about which state to go to from its current state based on the TPM. In one iteration, every agent makes a transition. I run this for a very large number of iterations to simulate the steady state.
Constraints: Standard constraints of a TPM
Optimization: Minimize difference between desired distribution vector and actual distribution vector
Edit: While I appreciate the analytical means of arriving at the values of the TPM, I am interested in optimizing the TPM based on the results of simulation. I suppose this shifts the focus from the conceptual approach behind finding the values in the TPM, to how I may use my own function/method to determine the suitability of each intermediate solution generated by the solver (and how the solver uses the suitability determined by the custom function) specifically in OR-Tools.