Least-Cost Planning Sequence Estimation in Labeled Petri Nets

This paper develops a recursive algorithm for estimating the least-cost planning sequence in a manufacturing system that is modeled by a labeled Petri net. We consider a setting where we are given a sequence of labels that represents a sequence of tasks that need to be executed during a manufacturing process, and we assume that each label (task) can potentially be accomplished by a number of different transitions which represent alternative ways of accomplishing a specific task. The processes via which individual tasks can be accomplished and the interactions among these processes in the given manufacturing system are captured by the structure of the labeled Petri net. Moreover, each transition in this net is associated with a nonnegative cost that captures its execution cost (e.g., in terms of the amount of workload or power required to execute the transition). Given the sequence of labels (i.e., the sequence of tasks that has to be accomplished), we need to identify the transition firing sequence(s) (i.e., the sequence(s) of activities) that has (have) the least total cost and accomplishes (accomplish) the desired sequence of tasks while, of course, obeying the constraints imposed by the manufacturing system (i.e., the dynamics and structure of the Petri net). We develop a recursive algorithm that finds the least-cost transition firing sequence(s) with complexity that is polynomial in the length of the given sequence of labels (tasks). An example of two parallel working machines is also provided to illustrate how the algorithm can be used to estimate least-cost planning sequences.