______________________________________________________________________________________ 30. Detecting Drifts Versus Shifts in the Process Mean In studying the performance of control charts, most of our attention is directed towards describing what will happen on the chart following a sustained shift in the process parameter. This is done largely for convenience, and because such performance studies must start somewhere, and a sustained shift is certainly a likely scenario. However, a drifting process parameter is also a likely possibility. Aerne, Champ, and Rigdon (1991) have studies several control charting schemes when the process mean drifts according to a linear trend. Their study encompasses the Shewhart control chart, the Shewhart chart with supplementary runs rules, the EWMA control chart, and the Cusum. They design the charts so that the in-control ARL is 465. Some of the previous studies of control charts with drifting means did not do this, and different charts have different values of ARL0, thereby making it difficult to draw conclusions about chart performance. See Aerne, Champ, and Rigdon (1991) for references and further details. They report that, in general, Cusum and EWMA charts perform better in detecting trends than does the Shewhart control chart. For small to moderate trends, both of these charts are significantly better than the Shewhart chart with and without runs rules. There is not much difference in performance between the Cusum and the EWMA. 31. Run Sum and Zone Control Charts The run sum control chart was introduced by Roberts (1966), and has been studied further by Reynolds (1971) and Champ and Rigdon (1997). For a run chart for the sample mean, the procedure divides the possible values of x into regions on either side of the center line of the control chart. If P0 is the center line and V 0 is the process standard deviation, then the regions above the center line, say, are defined as [PV00AnAii / , P 010 V / ni ), 0,1,2,..., a for 0 AAA012 AAaa 1 f, where the constants Ai are determined by the user. A similar set of regions is defined below the center line. A score is assigned to each region, say si for the ith region above the center line and s-i for the ith region below the center line. The score si is nonnegative, while the score s-i is nonpositive. The run sum chart operates by observing the region in which the subgroup averages fall and accumulating the scores for those regions. The cumulative score begins at zero. The charting procedure continues until either the cumulative score reaches or exceeds either a positive upper limit or a negative lower limit in which case an out-of-control signal is generated, or until the subgroup average falls on the other side of the center line in which case the scoring starts over with the cumulative score starting according to the current value of x . Jaehn (1987) discusses a special case of the run sum control chart, usually called the zone control chart. In the zone control chart, there are only three regions on either side of the center line corresponding to one-, two-, and three-sigma intervals (as in the western Electric rules), and the zone scores are often taken as 1,2,4, and 8 (this is the value 55 ______________________________________________________________________________________ assigned to a point outside the three-sigma limits and it is also the total score that triggers an alarm). Davis, Homer, and Woodall (1990) studied he performance of the zone control chart and recommended the zone scores 0, 2, 4, and 8 (or equivalently, 0, 1, 2, and 4). Champ and Rigdon (1997) use a Markov chain approach to study the average run length properties of several versions of the run sum control chart. They observe that the run sum control chart can be designed so that it has the same in-control ARL as a Shewhart with supplementary runs rules and better ARL performance than the Shewhart chart with runs rules in detecting small or moderate sized shifts. Their results are consistent with those of Davis, Homer, and Woodall (1990). Jin and Davis (1991) give a FORTRAN computer program for finding the ARLs of the zone control chart. Champ and Rigdon (1997) also compare the zone control chart to the Cusum and EWMA control charts. They observe that by using a sufficient number of regions, the zone control chart can be made competitive with the Cusum and EWMA, so it could be a viable alternative to these charts. 32. More About Adaptive Control Charts Section 9-5 of the text discusses adaptive control charts; that is, control charts on which either the sample size or the sampling interval, or both, are changed periodically depending on the current value of the sample statistic. Some authors refer to these schemes as variable sample size (VSS) or variable sampling interval (VSI) control charts. A procedure that changes both parameters would be called a VSS/SI control chart. The successful application of these types of charts requires some flexibility on the part of the organization using them, in that occasionally larger than usual samples will be taken, or a sample will be taken sooner than routinely scheduled. However, the adaptive schemes offer real advantages in improving control chart performance. The textbook illustrates a two-state or two-zone system; that is, the control chart has an inner zone in which the smaller sample size (or longest time between samples) is used, and an outer zone in which the larger sample size (or shortest time between samples) is used. The book presents an example involving an x chart demonstrating that an improvement of at least 50% in ATS performance is possible if the sampling interval can be adapted using a two-state system. An obvious question concerns the number of states: are two states optimal, or can even better results be obtained by designing a system with more that two states? Several authors have examined this question. Runger and Montgomery (1993) have shown that for the VSI control chart two states are optimal if one is considering the initial-state or zero-state performance of the control chart (that is, the process is out of control when the control chart is started up). However, if one considers steady-state performance (the process shifts after the control chart has been in operation for a long time), then a VSI control chart with more than two states will be optimal. These authors show that a well-designed two-state VSI control chart will perform nearly as well as the optimal chart, so that in practical use, there is little to be gained in operational performance by using more than two states. Zimmer, Montgomery and Runger (1998) 56 ______________________________________________________________________________________ consider the VSS control chart and show that two states are not optimal, although the performance improvements when using more that two states are modest, and mostly occur when the interest is in detecting small process shifts. Zimmer, Montgomery and Runger (2000) summarize the performance of numerous adaptive control chart schemes, and offer some practical guidelines for their use. They observe that, in general, performance improves more quickly from adapting the sample size than from adapting the sampling interval. Tagaras (1998) also gives a nice literature review of the major work in the field up through about 1997. Baxley ((1995) gives an interesting account of using VSI control charts in nylon manufacturing. Park and Reynolds (1994) have presented an economic model of the VSS control chart, and Prabhu, Montgomery, and Runger (1997) have investigated economic-statistical design of VSS/SI control charting schemes. 33. Multivariate Cusum Control Charts In Chapter 10 the multivariate EWMA (or MEWMA) control chart is presented as a relatively straightforward extension of the univariate EWMA. It was noted that several authors have developed multivariate extensions of the Cusum. Crosier (1988) proposed two multivariate Cusum procedures. The one with the best ARL performance is based on the statistic 1 1/2 Ciii ^`()()SX116c SX ii where ­ 0, if Cki d Si ® ¯()(1/),SXii1 kC i if C i ! k with S0 = 0, and k > 0. An out of control signal is generated when 11/2 YHiii ()SSc6 ! where k and H the reference value and decision interval for the procedure, respectively. Two different forms of the multivariate cusum were proposed by Pignatiello and Runger (1990). Their best-performing control chart is based on the following vectors of cumulative sums: i DXij ¦ jil i 1 and 11/2 MCiiii max{0,(DDc6 ) kl } where kll!0,ii 11 1 if MC i !0 and l i 1 otherwise . An out of control signal is generated if MCi > H. 57 ______________________________________________________________________________________ Both of these multivariate Cusums have better ARL performance that the Hotelling T2 or the chi-square control chart. However, the MEWMA has very similar ARL performance to both of these multivariate Cusums and it much easier to implement in practice, so it should be preferred. 34. Guidelines for Planning Experiments Coleman and Montgomery (1993) present a discussion of methodology and some guide sheets useful in the pre-experimental planning phases of designing and conducting an industrial experiment. The guide sheets are particularly appropriate for complex, high- payoff or high-consequence experiments involving (possibly) many factors or other issues that need careful consideration and (possibly) many responses. They are most likely to be useful in the earliest stages of experimentation with a process or system. Coleman and Montgomery suggest that the guide sheets work most effectively when they are filled out by a team of experimenters, including engineers and scientists with specialized process knowledge, operators and technicians, managers and (if available) individuals with specialized training and experience in designing experiments.