Statistical Process Control Methods from the Viewpoint of Industrial Application

Statistical process control is a major part of industrial statistics and consists not only of control charting, but also of capability analysis, of design of experiments and other statistical techniques. This paper reviews and comments some available techniques for the univariate as well as for the multivariate case from the viewpoint of industrial application. 1 Distribution of a Quality Characteristic Many statistical process control activities assume that the quality characteristic under study exhibits a stable performance or in other words that the process is stable and capable. The stability of a process can be defined as the stability of the underlying probability distribution over time and very often this can be described as the stability of the distribution parameters over time. Only if the stability assumption is met by the process, the calculation of capability indices is meaningful and may be used in practice for evaluating process performance. Mathematically, we can, of course, calculate always capability indices, but for an unstable process these indices have no real significance, because there are not identified assignable causes in the process. Thus, a correct identification of the type of probability distribution is not sufficient without the assurance that it is stable over time. In the case that the process is not stable, the probability distribution of the quality characteristic may vary from time-point to time-point. In such a case one could use a mixture of probability distributions as model, which, therefore, can be looked upon as an indication for an unstable process. Each quality characteristic has a “proper nature” defined by physical and technical conditions. For instance a meaningful flatness-characteristic is defined as a parameter naturally bounded by null. In such a case the normal distribution cannot describe reality sufficiently well and a different model has to be selected. One of many possibilities would be to select the folded normal distribution [1] and [8]. Of course, any model should always be justified by experiments, i.e. by samples. If a sample leads to reject the hypothesis of the assumed model, then the situation must be analyzed again, with respect to various features as the resolution of the measurement gauge, mixture appearance, etc. Table 1 gives some recommendations for the selection of an appropriate model for some frequently relevant quality characteristics (see also [1]). A decision for a specific model to be used to describe the random variations exhibited by a quality characteristic, should be based on any available information about the physical and 50 Constantin Anghel technical conditions on the one hand and on a sufficiently large sample on the other hand, collected under “ideal” conditions (one operator, one supplier, optimal resolution of the gauge). Subsequent samples of the quality characteristic should be utilized for verifying the hypothesis that the initial distribution has not changed. In case of a rejection of the null hypothesis, an explanation has to be found by an additional analysis. Table 1: Recommended Models for Various Quality Characteristics Characteristic Model Characteristic Model geometric dimensions: straightness folded normal parallelism folded normal evenness folded normal rectangularity folded normal round form folded normal inclination (angularity) folded normal cylinder form folded normal position eccentric Rayleigh line form folded normal coaxiality (concentricity) eccentric Rayleigh surface form folded normal symmetry folded normal circularity flatness folded normal a) form folded normal b) position eccentric Rayleigh life length Weibull, Hjorth resistance normal voltage normal capacitance normal pressure normal viscosity normal These models should be looked upon as a first and preliminary attempt which, in any case, must be justified or discarded by a subsequent analysis of the situation at hand. 2 Data Collection and the Resolution of the Gauge Any data collection is the result of a measurement process and, therefore, the gauge has a high importance beginning with its resolution and its capability. A resolution below 2% of the tolerance is considered as consistent with industrial practice. Above this level the measurement values are assigned to classes by the measurement procedure and the random character of the sample will partly be lost. If we take samples of size two, then the expectation of range is 1.128 times the process standard deviation. This means that the expected distance between any two randomly selected measurements is about 1.1sσ, where σ denotes the process standard deviation. Whenever the difference between two successive measurements is less than the resolution of the gauge, the difference is set to zero. Thus, a larger resolution yields a larger number of zeros, and consequently results in a too small value of the average range (process variability) leading to action limits for an X̄-chart as well as for an R-chart, which are too tight (Figure 1). The consequences are too many false alarms, and often the wrong decision that the process is not stable [13]. Statistical Process Control Methods 51 1 Example Consider a quality characteristic with tolerance interval [L,U ] = [28.7, 31.3], which is evaluated by a gauge with resolution r = 0.01 (≈ 4% of the length of the tolerance interval). 50 samples of size n = 2 are taken the results of which are represented in Table 1 and Figures 1. Table 2: Observed values: 50 samples of size n = 2. observed value x frequency 29.98 1 29.99 3 30.00 28 30.01 41 30.02 23 30.03 2 30.04 2 Figure 1: X-chart and R-chart for the 50 samples of size n = 2. The uncertainty about the measurement result determines the gauge capability, which often is specified by four times the standard deviation 4σ. Clearly, the gauge capability affects the determination of the actual value of the process capability given by Cp. If the ratio of the measurement uncertainty (4σ) and the tolerance U − L, with L lower tolerance limit and U upper tolerance limit, is 25%, then a measured process capability (with measurement variability added to process variability) of 1.33 means for the actual index (without measurement variability) a value of 1.55 and a measured value of 1.67 corresponds to a real value of 2.2 (Figure 2). 52 Constantin Anghel Figure 2: The effect of measurement variability (uncertainty) on the determination of process capability Cp. Conversely, for a given actual process capability (without measurement variability) of 1.33, the capability with added measurement variability ( T = 25%) reaches only 1.2 and an actual value of 1.67 is decreased to 1.41 (Figure 3). Figure 3: The effect of measurement variability (uncertainty) on the verification of a given process capability Cp. Thus, information about gauge capability is very important to avoid false decisions with respect to process capability and with respect to releasing false alarms. Statistical Process Control Methods 53 3 Capability Indices As mentioned above, it makes sense to define capability indices for a stable process. Of course, it is possible to use the definitions and to calculate these indices, even if the distribution of the relevant quality characteristics change in time. However, only a stable process assures the predictions based on the stated capability index and, therefore, using the knowledge of what happened today for predicting what we expect tomorrow makes sense only if the situation does not change. From the viewpoint of application, it is desirable to define capability indices being independent of the special type of distribution allowing the comparison of two or more processes with different underlying distributions. The most widely used capability indices for the normal distribution are the following: