Extraction of exact symbolic stationary probability formulas for Markov chains in finite space with application to production lines. part II: unveiling accurate formulas for very short serial production lines without buffers (three- and four-stations) - Boulas, Konstantinos S.; Dounias, Georgios D.; Papadopoulos, Chrissoleon T. (Frontiers in Manufacturing Technology, 2025)
IntroductionOver the past seven decades, a significant volume of research has been dedicated to manufacturing systems due to their importance in the worldwide economy. Much of this research has focused on using Markov stochastic modeling to formulate manufacturing systems problems. During the research effort, numerous numerical methods have been developed for solving such systems; however, relatively few formulas have been proposed. That is because even small systems are characterized by the well-known state explosion problem.MethodsIn short serial production lines, the underlying Markov chain is depicted as a graph of the transition diagram, which is constructed by implementing an algorithm. The steady-state probabilities are extracted in the symbolic form of two polynomial ratios. That is accomplished by employing a recently introduced method that assigns probabilities in symbolic form on the graph anti-arborescences. Finally, the performance metrics of the short production line can be obtained in exact closed-form expressions via its known definition from extant literature using straightforward algebraic operations.ResultsThe closed-form formulae for the performance metrics of short serial production lines (e.g., throughput, maximum utilization, work in process, blocking probability for the second station, probability of the third station being idle, etc.) with two, three, and four stages, absent buffers, are presented herein for the first time in the extant literature. The proposed algorithm results shed light on the well-known phenomenon of production lines known as the “bowl phenomenon”. Comprehending the formula structure enables the formulation of a straightforward model for throughput estimation for fully balanced short serial production lines using genetic programming for lines up to thirteen stages.DiscussionThe enormous size of the exact formulas highlights the need for more computational support for production lines larger than four stages without buffers. A comprehensive understanding of the underlying principles governing exact formulas will facilitate the implementation of innovative mathematical approaches to problem-solving. This understanding will also enable artificial intelligence to derive precise mathematical relationships with reduced complexity, thereby fostering intuition in production lines.