Optimal off-line measurement schedule to support process and quality management
Systematic methods and tools for managing the complexity
Process Operation, Monitoring & Analysis (T4-2P)
Keywords: quality control, scheduling, Ornstein-Uhlenbeck process, stochastic dynamics, papermaking
In industrial processes, such as in papermaking, the product quality is measured from end product samples in laboratory. This information is typically used for three purposes: firstly to validate on-line sensors, secondly to manage quality parameters not measurable on line – e.g. strength of paper – and thirdly to decide whether to accept or reject the product batch. Each of the purposes set requirements on how much uncertainty may be tolerated in the quality estimate. It is a common practice to measure all quality parameters at regular time intervals. However, this is costly and may limit possibilities to measure those quality parameters that would be most important for overall uncertainty management and thus for decision support.
Quality parameters are typically statistically dependent and thus measuring a subset of them provides information about the others. A measurement schedule describes which of the quality parameters are measured at which time instants. We seek for optimal measurement schedule over a time horizon such that it minimizes the costs of measurement while maintaining the uncertainty within the constraints set by the application. Our quality information dynamics consists of continuous degradation towards the joint a priori probability density of quality, and occasional updates when new measurements are made. We describe the degradation with probability density function dynamics, Fokker-Planck equation, and the updating with probabilistic description of measurement and by applying Bayesian combination of earlier degraded information and of fresh measurement information. With this information dynamics we are able to assess the quality uncertainty at any time with any measurement schedule.
We show that if a priori information is multivariate Gaussian, the natural degradation between the measurements is according to Ornstein-Uhlenbeck (OU) process. Assuming that measurement uncertainties are normally distributed and that a subset X(2)is measured while X(1) is estimated, we derive the recursion for covariance matrix of measurement information about X=[X(1) X(2)] between two time instances. The recursion is expressed in terms of covariance matrices for the quality information and a priori information, the measurement uncertainty covariance matrix and the diffusion matrix for the OU process.
Our presentation provides the general theory for measurement information dynamics, and then discusses how the recursion can be used to find the lowest cost measurement schedule satisfying the constraints on uncertainty in quality information. We show examples of optimal schedules and how they depend on the time horizon of optimization. Optimal schedules are determined with a genetic algorithm. The results are discussed from the point of view of process and quality management in papermaking industry.
See the full pdf manuscript of the abstract.
Presented Wednesday 19, 13:30 to 15:00, in session Process Operation, Monitoring & Analysis (T4-2P).
