We consider the problem of adjusting a machine that starts production after an off-target setup operation. This setup adjustment problem was first studied by Grubbs (1954, 1983). A general formulation for setup adjustment problems is presented in this paper. The formulation uni.es some well-known process adjustment schemes, including Grubbs' harmonic and extended rules, adjustment methods based on stochastic approximation and recursive least squares, and a recent method on adaptive EWMA feedback controllers. The proposed formulation is Bayesian and based on a Kalman filter. The formulation allows us to show the equivalence of the setup process adjustment problem with a simple instance of what is called a Linear Quadratic Gaussian (LQG) controller in the control engineering literature. As an important byproduct, the LQG model allows us to solve more complicated setup adjustment problems with readily available techniques. Extensions to cases in which there are quadratic adjustment costs and in which the problem is multivariate are discussed. The multivariate setup adjustment solution is illustrated with a multihead filling machine example.
Key Words: Engineering Process Control, Kalman Filtering, Linear Quadratic Control, Setup Adjustment, Stochastic Approximation.
By ENRIQUE DEL CASTILLO, The Pennsylvania State University, University Park, PA 16802
RONG PAN, University of Texas at El Paso, El Paso, TX 79968
BIANCA M. COLOSIMO, Politecnico di Milano, Via Bonardi 9, 20133, Milan, Italy
PROCESS adjustment techniques based on the feedback principle have become an important resource in the toolkit used by quality engineers (see, for example, Box et al. (1994), Box and Luceño (1997), and Sachs et al. (1995)). A variety of techniques for process adjustment have been proposed and studied, partly due to interest in the integration of statistical process control (SPC) and engineering process control (EPC) methods. In this paper, we present a formulation that unifies several well-known process adjustment schemes, and we show how several extensions can be obtained using standard methods from control engineering.
To motivate the type of adjustment problems considered in this paper and to introduce some necessary notation, consider the setup adjustment problem first studied by Grubbs (1954). It is of interest to adjust a production process that manufactures discrete metal parts. Suppose, without loss of generality, that measurements Yt correspond to the deviations from target of some quality characteristic of the items as they are produced at discrete points in time t = 1, 2, .... Grubbs (1954) proposed a method for the adjustment of the machine in order to bring the process back to target if at start-up it was off-target by d units. The offset d is caused typically by systematic errors that exist during the setup operation or due to randomness in general. In some manufacturing processes, such as machining, an incorrect setup operation can result in severe consequences for the quality of the parts produced thereafter. This is analogous to Deming's "funnel experiment" when the funnel is initially off-target. If the adjustment cost is negligible, and if there is a significant cost associated with running the process off-target, it is evident that the process should be adjusted back to target.
This paper is organized as follows. First, a summary of the two adjustment rules proposed by Grubbs (1954) is presented. A Bayesian formulation of the setup adjustment problem based on a Kalman filter is developed next. It is then shown that the Kalman filter controller contains as particular cases several other adjustment techniques. We then present extensions of the basic model to more complex setup adjustment problems by noting the equivalence of the setup adjustment problem with linear quadratic control. These more complex cases include the multivariate case, which is numerically illustrated with application to a multiple head machine process.
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