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Nonlinear Profile Monitoring for Oven-Temperature Data

Problem: The monitoring of a key input, temperature, in a manufacturing process produces large amounts of data. It is difficult to determine an appropriate control-chart methodology that allows the chart user to determine when there are problems with this step of the manufacturing process. Current approaches for process monitoring involve the output data gathered after the process has been completed. It would be preferable to establish process monitoring on the process inputs. However, this is challenging when there is a large amount of process input data. Current phase I monitoring of the process inputs involve the use of individual control charts on some selected data from the temperature profiles that represent some features determined based on expert judgment. This approach does not use all the data nor does it take into account the potential correlation that exists among the selected data. Approach: We propose the use of a nonlinear model for modeling the profiles, thereby reducing the profiles to a smaller set of parameter estimates. For this nonlinear model data reduction approach, the parameter estimates and residual variability can then be used in the appropriate monitoring procedure. We show that a control chart based on the classical covariance-matrix estimate fails to detect large significant process changes, but the successive differences covariance matrix performs better. The statistic based on the successive differences is modified to account for the correlation between the profiles. We illustrate both the phase I and phase II analysis for these data. Results: The proposed data reduction approach and monitoring procedure makes use of all the available data and detects important process shifts where the interpretation of the nonlinear model parameters facilitates the root-cause investigation. This parametric approach can be easily automated using existing statistical software and results in a smaller number of control charts, which is a manageable way to determine the current state of the process. We highlight some issues that are raised by this particular dataset that have not been adequately addressed in the profile-monitoring literature.