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Article Excerpt
1. Introduction
CUSUM control charts find wide industrial use due to the fact that on-line measurement and distributed computing systems are now extensively used in Statistical Process Control (SPC) applications (Woodall and Montgomery, 1999). CUSUM charts are able to detect process shifts in both the mean and variance, and also identify the change points of the shifts (Khoo, 2005; Wu et al., 2007). Usually, two symmetrical CUSUM charts are used together to detect two-sided mean shifts. In a high-sided CUSUM chart for the detection of increasing mean shifts, the statistic [C.sub.t] to be updated for the tth sample can be written as
[C.sub.0] = [C.sub.t] = max(0, [C.sub.t - 1] + ([x.sub.t] - [[mu].sub.0]) - k), (1)
where [x.sub.t] is the tth sample value of a quality characteristic x following an independent and identical normal distribution, N ([mu], [[sigma].sup.2]), [[mu].sub.0] is the in-control mean of x and k is the reference parameter. The difference ([x.sub.t] - [[mu].sub.0]) is a sample value of the mean shift [delta][sigma]. In this article, it is assumed that the in-control mean [[mu].sub.0] and standard deviation [[sigma].sub.0] of x are known a priori; for example they can be estimated from field test records or historical data. Moreover, the standard deviation [sigma] is assumed to be constant, i.e., [sigma] [equivalent to] [[sigma].sub.0].
In some applications where the sample size (n) is larger than one, it may be safely assumed that x represents the sample mean of the n observations and [sigma] represents the standard deviation of this sample mean.
A conventional CUSUM chart often determines the reference parameter k based on a special mean shift [[delta].sub.s]. As a result, it performs optimally when the mean shift [delta] is equal to [[delta].sub.s]. However, it is very difficult, if not impossible, to predict the actual magnitude of [delta] for most applications (Reynolds and Stoumbos, 2004), and in turn there is no guarantee that a conventional CUSUM chart can always perform well during the operation. In order to make the CUSUM chart effective over a wide range of mean shifts, Lucas (1982) proposed the combined application of a CUSUM chart and an [bar.X] chart. Other researchers including Sparks (2000) and Zhao et al. (2005) have recommended the simultaneous use of two or three CUSUM charts.
Sparks (2000) proposed the concept of an adaptive CUSUM chart, the so-called ACUSUM chart, which adjusts the reference parameter k according to the on-line estimated value [^.[delta]] of the mean shift. It is expected that such an adaptive feature can make the ACUSUM chart more efficient in signaling a range of future expected but unknown mean shifts from a holistic viewpoint. Shu and Jiang (2006) simplified the design and implementation of the ACUSUM chart. Particularly, they developed a two-dimensional Markov procedure to evaluate the Average Run Length (ARL) of the ACUSUM chart and formulated a more applicable algorithm to compute the control limit based on [^.[delta]]. Both Sparks (2000) and Shu and Jiang (2006) used an EWMA procedure to estimate the mean shift [^.[delta]], and then set the reference parameter k equal to 0.5[^.[delta]].
Recently, Reynolds and Stoumbos (2004) and Jiao and Helo (2008) have found that an exponential w will influence the sensitivity of the CUSUM chart to mean shifts [delta] when the term ([x.sub.t] - [[mu].sub.0]) in Equation (1) is replaced by [([x.sub.t] - [[mu].sub.0]).sup.w]. In general, a larger w makes the CUSUM chart more effective for detecting larger [delta], and a smaller w makes it more sensitive to smaller [delta].
In this article, we propose an enhanced ACUSUM chart in which both k and w are adapted dynamically in accordance with the current estimated value [^.[delta]] of the mean shift. To differentiate from the version of Sparks (2000) and Shu and Jiang (2006) which is called the ACUSUM I chart, the one proposed in this article is referred to as the ACUSUM II chart. The roman numerals I and II also indicate the respective number of adaptable charting parameters. It is noted that the ACUSUM I chart can be regarded as a special case of the ACUSUM II chart, whereby w is equal to one. As revealed from the performance studies in the subsequent sections, the enhanced ACUSUM II chart with a new adaptive feature is able to increase the detection effectiveness by more than 20%, on average, compared with the ACUSUM I chart where only the reference parameter k is adapted.
The remainder of the article proceeds as follows. The design and implementation of the ACUSUM II chart is discussed in the next section. A performance comparison of the ACUSUM II chart with seven other single or combined CUSUM charts is presented subsequently. The discussions and conclusions are drawn in the last section.
2. ACUSUM II chart
In this section, the basic formulation of the ACUSUM II chart is first introduced. Then an optimization model is presented for the design of this chart. It is followed by the selection of the objective function for the optimization design. Finally, the implementation of the ACUSUM II chart is outlined.
2.1. Basic formulation
The quality characteristic x can be converted to z that follows a standard normal distribution when the process is in control:
z = x - [[mu].sub.0]/[sigma]. (2)
For the purpose of detecting increasing mean shifts, the statistic [C.sub.t] in the ACUSUM II chart will be updated by
[C.sub.0] = 0, [C.sub.t] = max(0, [C.sub.t - 1] + q - k), (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where the parameters k, w and q all depend on the current sample value [z.sub.t]. The statistic [C.sub.t] may increase or decrease depending on whether the sample value [z.sub.t] is larger or smaller than zero. However, [C.sub.t] is always decreased towards zero by the reference parameter k. When an increasing mean shift occurs, [C.sub.t] is likely to become increasingly larger. Sooner or later, a subsequent sample point will exceed the control limit H of the ACUSUM II chart, and thereby produce an out-of-control signal.
This article focuses on the high-sided ACUSUMI II chart. However, a symmetrical low-sided counterpart can be built straightforwardly:
[C.sub.0.sup.-] = 0, [C.sub.t.sup.-] = min(0, [C.sub.t - 1.sup.-] + q + k), (5)
where q is also determined by Equation (4).
The performance of a control chart is usually measured by the (ARL), which is the average number of samples required to signal an out-of-control case or produce a false alarm. The out-of-control AR[L.sub.1] is commonly used as an indicator of the power (or effectiveness) of the control chart, whereas the in-control AR[L.sub.0] for the false alarm rate. In this article, the out-of-control AR[L.sub.1] will be computed under the steady-state mode. It assumes that the process has reached its stationary distribution at the time when the process shift occurs. Since production processes often operate in an in-control condition for most or relatively long periods of time (Montgomery, 2005), the steady-state mode is therefore more realistic than the zero-state mode.
It seems desirable to adjust the parameters k and w (Equations (3) and (4) of an ACUSUM II chart continuously in accordance with the current estimated value [[^.[delta].sub.t] of the mean shift. However, studies on VSSI (Variable Sample Sizes and Sampling Intervals) CUSUM charts discover that merely using m (m = 2 or 3) sets of sample size and sampling interval may gain most of the benefits that can be reached by a VSSI CUSUM chart (Yu and Wu, 2004; Zhang and Wu, 2006), and are relatively easier to design and implement as well. For example, a VSI (Variable Sampling Intervals) CUSUM chart usually uses only two different sampling intervals, [h.sub.1] and [h.sub.2], alternatively according to the predicted process status. The smaller sampling interval [h.sub.1] is used when the sample point is close to the control limits; and the larger [h.sub.2] is adopted when the sample point is quite close to the target value. This suggests that, when implementing an ACUSUM II chart, one may use only m different sets of [k.sub.i] and [w.sub.i],(1 = 1, 2,..., m) and each set is best for detecting a particular discrete value of [[delta].sub.i] ([[delta].sub.min] < [[delta].sub.1] < ... < [[delta].sub.m] < [[delta].sub.max]). Here, [[delta].sub.min] and [[delta].sub.max] are the lower and upper bounds, respectively, representing the range of mean shifts of interest in the problem. Jointly, the m set of ([k.sub.i], [w.sub.i]) will optimize the overall performance of the ACUSUM II chart over the entire mean shift range. The ACUSUM II chart keeps on switching among the m different sets of ([k.sub.i], [w.sub.i]) depending on...
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