Analytical assessment of the processing productivity of complex workpieces based on Markov models

<jats:p>Introduction. In multi product manufacturing, loading, unloading, and auxiliary movements of equipment components contribute significantly to the overall processing cycle time. It is hypothesized that transitioning from the processing of individual simple workpieces to complex workpieces – combining several parts in a single unit – can reduce unproductive time expenditures. However, existing analytical approaches do not provide a quantitative assessment of the productivity gain while accounting for the probabilistic nature of transitions between technological states. Purpose of the work is to develop analytical expressions for evaluating the productivity of a technological system when processing simple and complex workpieces on the basis of Markov models. The research objective is to establish a correspondence between the time parameters of conventional time and motion study (technical rating) and the transition intensities in a Markov chain, and to derive a formula for the relative gain. The technological process is represented as a directed graph, with transitions between states specified by intensities. The research methods include mathematical modeling based on the fundamental principles of Markov process theory, numerical simulation in the Maple environment, and the principles of group parts processing theory. Results and discussion. The developed mathematical models enable quantitative estimation of the fractions of time that the technological system spends in the states of loading, processing, auxiliary movements, and unloading. The analytical relationships obtained directly relate productivity to the transition intensities between states. It is shown that the key states determining processing efficiency are loading, unloading, and auxiliary movements. Numerical simulations performed in the Maple environment confirmed the validity of the derived formulas. The results obtained demonstrate the potential of the proposed approach for the analytical assessment of the processing productivity of complex workpieces. The derived expressions are valid for a wide range of surface configurations and can be used to calculate the gain achieved when combining an arbitrary number of parts.</jats:p>