Evaluation of stochastic flow lines with provisioning of auxiliary material

Flow lines are often used to perform assembly operations in multi-stage processes. During these assembly operations, components that are relatively small, compared to the work pieces travelling down the flow line, are mounted to the work pieces at a given stage. Those components, or more generally, any kind of auxiliary material, are provisioned to the corresponding production stage in a repetitive but not necessarily deterministic manner using a certain delivery frequency, each time filling the local storage up to a predetermined order-up-to level. Just like random processing times, machine failures, and repairs, the randomness of the provisioning process can impact the long-term throughput of such a flow line. In this paper, we develop a fast and accurate analytical performance evaluation method to estimate the long-term throughput of a Markovian flow line of this type for the practically important case of limited buffer capacities between the production stages. We first give an exact characterization of a two-machine line of that type and show how to determine system state probabilities and aggregate performance measures. Furthermore, we show how to use this two-machine model as the building block of an approximate decomposition approach for longer flow lines. As opposed to previous decomposition approaches, even the state space of the two-machine lines can become so large that an exact solution of the Markov chains can become impractical. We hence show how to set up, train, and use an artificial neural network to replace the Markov chain solver embedded in the decomposition approach, which then leads to an accurate and extremely fast flow line evaluation tool. The proposed methodology is evaluated by a comparison with simulation results and used to characterize the structural patterns describing the behaviour of flow lines of this type. The method can be used to systematically consider the combined impact of the delivery frequency and the local order-up-to levels for the auxiliary material when designing a flow line of this type.