In this paper, we focus on the performance of adjustment rules for a machine that produces items in batches and that can experience errors at each setup operation performed before machining a batch. The adjustment rule is applied to compensate for the setup offset in order to bring back the process to target. In particular, we deal with the case in which no prior information about the distribution of the offset or about the within-batch variability is available. Under such conditions, adjustment rules that can be applied are Grubbs' rules, the exponentially-weighted moving average (EWMA) controller and the Markov chain Monte Carlo (MCMC) adjustment rule, based on a Bayesian sequential estimation of unknown parameters that uses MCMC simulation. The performance metric of the different adjustment rules is the sum of the quadratic off-target costs over the set of batches machined. Given the number of batches and the batch size, different production scenarios (characterized by different values of the lot-to-lot and the within-lot variability and of the mean offset over the set of batches) are considered. The MCMC adjustment rule is shown to have better performance in almost all of the cases examined. Furthermore, a closer study of the cases in which the MCMC policy is not the best adjustment rule motivates a modified version of this rule which outperforms alternative adjustment policies in all the scenarios considered. executed given the intrinsic complexity of these procedures (e.g. the fixturing of workpieces can only be automated for very simple part geometry). Even in cases when automatic setup operations are available, batches of the same part type are often affected by different initial offsets because the conditions of the machine, materials and operators vary from time to time. In this case, procedures designed to automatically compensate for the initial offset or setup error can significantly improve the outgoing quality of the parts processed. Assuming a quadratic off-target cost function, Grubbs 1 presented two adjustment rules aimed at adjusting for a potential initial offset present at setup of a single batch (what we refer to as Grubbs' 'harmonic' rule) and over a set of batches (what we call Grubbs' 'extended' rule), respectively 2 . The second rule, in particular, is optimal for a quadratic off-target cost function when parameters characterizing both within-batch and betweenbatch variability are known in advance. Del Castillo et al. 3 showed how Grubbs' extended rule has a Bayesian interpretation based on a Kalman filter when prior knowledge of parameters characterizing both batch-to-batch and within-batch distributions is required. Hence, Grubbs' extended rule can only be applied after a set of batches has already been machined in order to have accurate estimates of the required parameters. To start adjusting when no estimates are available, e.g. when a new product has to be processed or a new process is installed, simpler adjusting rules such as the harmonic rule and a discrete integral controller or...

No abstract

A bounded adjustment strategy is an important link between statistical process control and engineering process control (or closedloop feedback adjustment). The optimal bounded adjustment strategy for the case of a single variable has been reported in the literature and recently a number of publications have enhanced this relationship (but still for a single variable). The optimal bounded adjustment strategy for a multivariate processes (of arbitrary dimension) is derived in this article. This uses optimization and exploits a symmetry relationship to obtain a closed-form solution for the optimal strategy. Furthermore, a numerical method is developed to analyze the adjustment strategy for an arbitrary number of dimensions with only a one-dimensional integral. This provides the link between statistical and engineering process control in the important multivariate case. Both infinite-and finite-horizon solutions are presented along with a numerical illustration.

A new sequential Monte Carlo (SMC) adjustment method is presented for solving the machine setup adjustment problem when process parameters are unknown. In setup adjustment problems, the mean of the distribution of the quality characteristic of parts can change from lot to lot due to an improper setup operation. It is shown how a first SMC approach has performance equivalent to a recently proposed Markov chain Monte Carlo method but at a small fraction of the computational cost, allowing for on-line control. A second, modified SMC rule that avoids unnecessary adjustments that can inflate the variance is also presented. A simulation approach is presented that allows tuning of the modified SMC rule to provide robust adjustment with respect to the unknown process parameters. Applications in short-run manufacturing processes are discussed.

No abstract