Applied Soft Computing 2 (2002) 140–151 Multistage decision-making using simulated annealing applied to a fuzzy automaton Jiri Pospichal ∗ , Vladimir Kvasnicka Department of Mathematics, Slovak Technical University, 81237 Bratislava, Slovak Republic Received 22 February 2002; received in revised form 21 September 2002; accepted 15 October 2002 Abstract A new optimization method that stochastically builds up a solution step-by-step in combination with simulated annealing is used for multistage decision-making of finite-state automaton. The quality of the new algorithm for larger scale problems was tested by two tasks: (1) maximizing the probability of goal satisfaction with fuzzy goals subject to fuzzy constraints and (2) minimizing the length of decision sequence leading to a specified termination state. The new method required a number of evaluations of solutions, which was smaller by orders of magnitude in comparison with a “classical” genetic algorithm. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy automaton; Multistage decision-making; Simulated annealing; Optimization; Genetic algorithm 1. Introduction A multistage decision-making is similar to a com- plex problem solving, in which a suitable sequence of decisions is to be found. The task can be inter- preted as a series of interactions between a decision maker and an outside world, at each stage of which some decisions are available and their immediate ef- fect can be easily computed. In the end, some goals should be reached or one or more variables optimized. The immediate effect of the decision steps is in a way related to the level of satisfaction of these goals or to the final value of the optimized variables, in a similar way as local optimization is related to global optimization. ∗ Corresponding author. Tel.: +421-2-5249-5177; fax: +421-2-5249-3198. E-mail addresses: pospich@cvt.stuba.sk (J. Pospichal), kvasnic@cvt.stuba.sk (V. Kvasnicka). URL: http://math.chtf.stuba.sk/staff/pospichal/pospich.htm Decision-making and control are typically treated as two fields with distinct methods for solving prob- lems, and yet they are closely related. In both cases, a sequence of decisions is required. However, in process control, the complexity of the problem to be solved involves dynamic environment; thus, the decision se- quence can not be prepared in advance. In this paper, the gap between decision-making and control in the field of fuzzy decisions and fuzzy control is not that wide, since the problems presumed to be suitable for the proposed algorithm would be more complex than typical multistage decision problems, but less complex than process control tasks that require a controller for dynamic environment. Such a controller represents a typical example of a connection between the fuzzy paradigm and optimization, since the controller may be guided by fuzzy rules, which in turn may be opti- mized by evolutionary methods [6]. However, this is not the case in the presented paper. Our aim is much simpler: the goal is to find a se- quence of decision steps (which in our examples will 1568-4946/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S1568-4946(02)00058-3