Accuracy of Semi-Analytical and Numerical Approaches in the Evaluation of Serial Bernoulli Production Lines

Abstract

The manufacturing industry has a great impact on the economic growth of countries. It is, therefore, crucial to master the skills of the production system by mathematical tools that enable the evaluation of the production systems’ performance measures. Four mathematical approaches toward the modeling of steady-state behavior of serial Bernoulli production lines were considered in this study, namely, the analytical approach, the finite state method, the aggregation procedure, and numerical modeling. The accuracy of the performance measures determined using the semi-analytical methods and the numerical approach was validated using numerous theoretical examples and the results obtained using the analytical model. All of the considered methods demonstrated relevant reliability, regardless of the different theoretical backgrounds.

Towards Digital Twinning of Fabrication Lines in Shipyards

“…Indeed, much work has been conducted in the field of production system engineering during the last three decades to bring Markov chains into play, especially regarding the modeling of complex and large-scale systems, such as serial lines, splitting lines, assembly systems, job shops, flexible manufacturing cells, re-entrant lines, and others, including quality checks, reworking stations, customer demands, lean design, improvability, bottleneck identification, and different machine reliability formulations [33]. Typically, these problems are tackled using semi-analytical approaches based on the Markovian framework, such as the decomposition technique [34], aggregation procedure [35], or finite-state method [36], as the analytical solution has proven to be highly sensitive to the scale of the state space [37]. Regardless of the method, the underlying goal of such mathematical models is the evaluation of the overall equipment efficiency [38,39] and key performance indicators, such as the production rate, throughput, work-in-process, probability of starvation, probability of blockage, and residence time [33].…”

Transient Response of Homogenous and Nonhomogenous Bernoulli Production Lines

“…First, the analytical model will be presented using the homogenous and nonhomogenous formulation of the generalized transition matrix [12], direct solution of balance equations, and modal superposition approach. Second, the finite-state method [32,33] will be further extended to the case of transient evaluation of homogenous and nonhomogenous systems to cope with well-known CPU (Central Processing Unit) and memory storage issues of large-scale and transition-rich stochastic systems. It is expected that this new approach will enable the integration of a more realistic Markovian framework within continuously developing Digital Twinning platforms, yielding more reliable predictive analytics as well as maintenance scheduling conditioned to lesser production losses.…”

Statistical Evaluation of Semi-Analytical, Analytical, and Numerical Models of the Serial Production Lines

“…The aggregation method, finite state method, and numerical approach were never statistically compared by the analytical approach. Therefore, the data from the supplement of the paper [16] and the software STATISTICA will be used in this work to fill the gap. To get an overview of the interaction between the input data and the output data a design of experiment approach with the software Design Expert is used on an illustrative example.…”

“…Such comparison will be done for 12 lines in 4 cases with 3, 4, 5 and 6 machines. The data generated in [16] will be used. Longer production lines with more than 6 machines in a line are not considered because the CPU demand for the analytical approach is too high to get results in a reasonable time.…”

“…The 12 production lines for each case are getting out of 200 randomly generated production lines. To sample out the 12 lines, the performance measures by the analytical approach for each case will be the base for the application of the following criteria's, [16]…”