Modeling, analysis and optimization of decision process systems

This paper introduces a new modeling paradigm for developing decision process representation called Decision Process Petri Nets (DPPN). It extends the place-transitions Petri net theoretic approach by including the Markov decision process. Place-transitions Petri nets (PN) are used for process representation taking advantage of the formal semantic and the graphical display. Markov decision processes are utilized as a tool for trajectory planning via a utility function. The main point of the DPPN is its ability to represent the mark-dynamic and trajectory-dynamic properties of a decision process. Within the mark-dynamic framework the theoretic notions of equilibrium and stability are those of the place-transitions Petri net. In the trajectory-dynamic framework, the utility function used for trajectory planning is optimized, via a Lyapunov like function, obtaining as a result new characterizations for final decision points (optimum point) and stability. Moreover, it is shown that the DPPN mark-dynamic and Lyapunov trajectory-dynamic properties of equilibrium, stability and final decision points (optimum point) converge under certain restrictions. An algorithm for optimum trajectory planning that makes use of the graphical representation of the place-transitions Petri net and the utility function is proposed. The work presented here makes firm steps toward the modelling and analysis of decision problems in several fields as: management, ecological systems, defense and homeland security issues and terrorism.