IJPAM: Volume 26, No. 4 (2006)


Zvi Retchkiman Konigsberg
Instituto Politécnico Nacional
Centro de Investigación en Computación - CIC
Mineria 17-2, Col. Escandon
Mexico D.F 11800, MEXICO
e-mail: mzvi@cic.ipn.mx

Abstract.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 is utilized as a tool for trajectory planning via an 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 properties framework it is shown that the DPPN theoretic notions of equilibrium and stability are those of the place-transitions Petri net. In the trajectory-dynamic properties framework, the utility function used for trajectory planning in the DPPN 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. This work makes firm steps toward the modelling and analysis of problems, in the field of decision process systems, using DPPN.

Received: September 29, 2005

AMS Subject Classification: 91B06, 93D05, 39A05, 39A11, 62C99

Key Words and Phrases: Decision Process Petri Nets, stability, Lyapunov methods, optimization

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 26
Issue: 4