Two approaches to analytical modeling based on distribution fitting for manufacturing applications: continuous approximation and extreme value theory

Abstract

Statistics have been a handy toolset for practitioners in the industry for decades. Facing uncertainty is inevitable in a real-life environment, and knowing the probability distribution for any uncertain phenomenon makes understanding it easier. Therefore, distribution fitting has been applied extensively in industrial settings as an independent tool or part of a more sophisticated approach. In this dissertation, we utilized two analytical modeling approaches that apply distribution fitting as a crucial step: continuous approximation modeling and extreme value theory. In chapters three and four, we applied continuous approximation modeling in combination with a semi-Markov process to analyze the performance of a single machine with stochastic sequence-dependent setup times. More specifically, in chapter three, we studied sequential batch processing on a single machine with stochastic processing and setup times. In chapter four, we considered a reactive rescheduling approach with stochastic setup times. Both problems can be formulated as an asymmetric traveling salesperson problem. Using continuous approximation modeling, we could estimate the total time for the optimum sequence of jobs without finding the optimum solution. To do so, we had several runs of Monte-Carlo simulations for problems with different numbers of jobs. Then, we showed that a normal distribution can approximate the optimum total time, even for small problems. Then, we fitted a curve over the relation between the number of jobs and μ and σ of the normal distribution related to the optimum total time. Moreover, we applied semi- Markov process modeling to provide probability distributions for the system performance measures. We validated the respective models using Monte-Carlo simulations and provided illustrative use cases to demonstrate how the models can be employed. In chapter five, we applied extreme value theory to estimate the deepest valley on an additively manufactured surface (Sv) using linear measurement data. We applied the block maxima method from the extreme value theory, whereby the underlying distribution of the depth of individual valleys is modeled with a Gumbel distribution. The observed experimental results demonstrate that the methods can produce estimates that significantly outperform more straightforward benchmarks (e.g., Rv) and our proposed method achieve a relatively accurate estimation. The proposed methodology in chapter five can contribute to enabling a cheaper and more efficient way to quantify and estimate the surface roughness, consequently facilitating the investigation of its impacts on mechanical performance (especially fatigue) and the quality control of additively manufactured parts.