Quality Analysis in Acyclic Production Networks

Abstract

The production network under examination consists of a number of workstations. Each workstation is a parallel configuration of machines performing the same kind of tasks on a given part. Parts move from one workstation to another and at each workstation a part is assigned randomly to a machine. We assume that the production network is acyclic, that is, a part does not return to a workstation where it previously received service. Furthermore, we assume that the quality of the end product is additive, that is, the sum of the quality contributions of the machines along the production path. The contribution of each machine is modeled by a separate random variable.Our main result is the construction of estimators that allow pairwise and multiple comparison of the means and variances of machines in the same workstation. These comparisons then may lead to the identification of unreliable machines. We also discuss the asymptotic distributions of the estimators that allow the use of standard statistical tests and decision making.

Estimations of means and variances in a Markov linear model

“…In contrast to classic linear approaches as linear regression and ANOVA, our approach allows us to model different variances for each category and the error term to be arbitrary. It weakens the assumptions of independent predictors of a previous work [8], in the following way: the value of X (k+1) conditioned on X (k) is independent of X (k− 1) , for all reasonable choices of k.…”

“…The proposed model applies in a very general setting. However, it is motivated by real-world applications in the quality control of parallel production networks, see also the previous work [8], and we think it might be useful to have this particular use case in mind.…”

“…In Section 2, we define the model using Markov chains and directed acyclic graphs. In Section 3 we summarize the results on the case of independent covariates from [8]. Section 4 contains the main results.…”

“…Proof. We give the proof from [8] since it illustrates the idea behind the construction of the estimators. Using twice the law of large numbers and the continuous mapping theorem, we obtain…”