A simulation-based learning automata framework for solving semi-Markov decision problems under long-run average reward

Many problems of sequential decision making under uncertainty, whose underlying probabilistic structure has a Markov chain, can be set up as Markov Decision Problems (MDPs). However, when their underlying transition mechanism cannot be characterized by the Markov chain alone, the problems may be set up as Semi-Markov Decision Problems (SMDPs). The framework of dynamic programming has been used extensively in the literature to solve such problems. An alternative framework that exists in the literature is that of the Learning Automata (LA). This framework can be combined with simulation to develop convergent LA algorithms for solving MDPs under long-run cost (or reward). A very attractive feature of this framework is that it avoids a major stumbling block of dynamic programming; that of having to compute the one-step transition probability matrices of the Markov chain for every possible action of the decision-making process. In this paper, we extend this framework to the more general SMDP. We also present numerical results on a case study from the domain of preventive maintenance in which the decision-making problem is modeled as a SMDP. An algorithm based on LA theory is employed, which may be implemented in a simulator as a solution method. It produces satisfactory results in all the numerical examples studied.