Pre-control and Some Simple Alternatives

Transcription

1 Pre-control and Some Simple Alternatives Stefan H. Steiner Dept. of Statistics and Actuarial Sciences University of Waterloo Waterloo, N2L 3G1 Canada Pre-control, also called Stoplight control, is a quality monitoring scheme similar to a control chart. However, the goal of Pre-control is the prevention of nonconforming units. The appeal of Pre-control is due mainly to its ease of implementation, lack of assumptions, and reported successes. However, it has come under much criticism since many of its decision rules appear ad hoc. In addition, there is some confusion regarding different versions of Pre-control suggested in the literature. This article compares the various Pre-control schemes and contrasts them with more traditional control charts such as acceptance control charts and X charts. Based on this analysis the conditions under which Pre-control is appropriate are laid out. Also, alternatives that retain much of the simplicity of Pre-control but have improved operating characteristics are suggested. Key Words: Acceptance Control Charts; Grouped Data; Modified Precontrol; Operating Characteristic; Process Capability; Stoplight Control

2 2 1. Introduction Pre-control (PC), sometimes called Stoplight control, was developed to monitor the proportion of nonconforming units produced in a manufacturing process. Implementation is typically very straight-forward. All test units are classified into one of three groups: green, yellow, or red, where the colors loosely correspond to good, questionable, and poor quality products (see Figure 1). Decisions to stop and adjust the process or to continue operation are made based on the number of green, yellow, and red units observed in a small sample. The goal of PC is to detect when the proportion of nonconforming units produced becomes too large. Thus, PC monitors the process to ensure that process capability ( C pk ) remains large. PC was initial proposed in 1954 (see Shainin and Shainin, 1989 and Ermer and Roepke, 1991 for more details) as an easier alternative to Shewhart charts. However, since that time, at least three different versions of PC have been suggested in the literature. In this article the three versions are called: Classical PC Two-stage PC Modified PC Classical PC refers to the original formulation of Pre-control as described by Shainin and Shainin (1989) and Traver (1985). Two-stage PC is a modification discussed by Salvia (1988) that improves the method s operating characteristics by taking an additional sample if the initial sample yields ambiguous results. Modified PC was suggested by Gurska and Heaphy (1991), and represents a departure from the philosophy of Classical and Two-stage PC in that it compromises between the design philosophy of Shewhart type charts and the simplicity of application of PC.

3 3 PC schemes are defined by two attributes; their group classification method and their decision criteria. A third criteria, the qualification procedure, is the same for all three PC versions, and is this not discussed in this article. The most important difference between the three versions of PC is the group classification method. Classical PC and Two-stage PC base the classification of units on engineering tolerance or specification limits. A unit is classified as green if its quality dimension of interest falls into the central half of the tolerance range. A yellow unit has a quality dimension that falls into the remaining tolerance range, and a red unit falls outside the tolerance range. Assuming, without loss of generality, that the upper and lower specification limits (USL and LSL) are 1 and 1 respectively, group classification is based on the endpoints: t = [ 1,.5,.5, 1]. The classification scheme is illustrated in Figure 1. The colored circle can be used for the ease of the operators as a dial indictor overlay (Shainin and Shainin, 1989). Green Yellow Yellow Red Red Yellow Green Yellow Red LSL Measurement USL Figure 1: PC Classification Criteria Let the quality dimension of interest be Y. Then, given a probability density function for observations f( y), the group probabilities for Classical and Two-stage PC are given by (1.1)..5 ( ) = P g = f y P green P yellow y=.5.5 ( ) = P y = f y P red ( )dy 1 ( )dy + f( y)dy (1.1) y= 1 ( ) = P r = f y y=.5 1 ( )dy + f( y)dy y= y=1

4 4 Using Modified PC, on the other hand, the group classification scheme is the same as that used with Classical PC and Two-stage PC with one very important difference. Modified PC uses control limits, as defined in Shewhart charts, rather than tolerance limits to define the boundaries between green, yellow and red units. As shown later, this change is fundamental and makes Modified PC much more like a Shewhart chart than like Classical PC. Group probabilities for the Modified PC procedure are given in (1.2). Note that to avoid confusion, through this article the current process mean and standard deviation are denoted µ and σ, whereas estimates of the in-control mean and standard deviation used to set the control limits for Modified PC and Shewhart type charts are denoted µ c and σ c. P m P m µ c +1.5σ c ( green) = f( y)dy µ c 1.5σ c µ c 1.5σ c µ c +3σ c ( yellow) = f( y)dy + f( y)dy (1.2) P m µ c 3σ c µ c +1.5σ c µ c 3σ c ( red) = f( y)dy + f( y)dy µ c +3σ c The group probabilities given by Equations (1.1) and (1.2) are the same when µ c = 0 and σ c = 1/3. This makes sense because in that case control limits and tolerance limits are the same. It has been suggested that Classical PC and Two-stage PC are only applicable if the current process spread (six process standard deviations) covers less than 88% of the tolerance range (Traver, 1985). With specification limits at ±1, as defined previously, this condition corresponds to the constraint σ < The second important distinction between the three PC versions is their decision criteria. With Classical PC the decision to continue operation or to adjust the process is based on only one or two sample units. The decision rules are given below: Classical PC Decision Procedure Sample two consecutive parts A and B If part A is green continue operation (no need to measure B).

5 5 If part A is yellow measure part B. If part B is green continue operation, otherwise stop and adjust process. If part A is red stop and adjust process (no need to measure B). The Two-stage PC and Modified PC decision criteria are more complicated since they are based on a two-stage procedure. If the initial two observations do not provide clear evidence regarding the state of the process, additional observations (up to three more) are taken. The decision criteria for Two-stage and Modified PC is given below: Two-stage and Modified PC Decision Procedure Sample two consecutive parts. If either part is red stop process and adjust. If both parts are green continue operation. If either or both of the parts are yellow continue to sample, up to an additional three units. In the combined sample, if three greens are obtained continue operation; and if three yellows or a single red are observed stop the process. The advantage of this more complicated decision procedure is that more information regarding the state of the process is obtained and thus decision errors are less likely. The disadvantage is that, on average, larger sample sizes are needed, and thus implementation is more demanding. Table 1 summarizes the comparison of the three versions of PC. Table 1: Comparison of Pre-control Versions Pre-control Version Group Classification Decision Criteria Classic tolerance limits two observations Two-stage tolerance limits five observations Modified control limits five observations Based on these comparisons it can be concluded that the purpose of the various versions of PC are quite different. By design, Classical PC, and Two-stage PC tolerate

6 6 some deviation in the target mean of a process so long as the proportion nonconforming does not become too large. Thus, process stability is not required, and Classical and Twostage PC can be quickly implemented since there is no need to estimate the current process mean and standard deviation and show stability. Thus Classical and Two-stage PC are very similar to Acceptance Control Charts. By contrast, the goal of Modified PC is to monitor the process for deviations from stability. As a result, mean drifts are not tolerated, and Modified PC is very similar to Shewhart type charts such as X charts. 2. Comparison with Traditional Control Charting Methods In this section the various PC methods are compared with appropriate traditional control charting techniques. In addition, the performance of PC under some special situations is explored. Classical and Two-stage PC are compared with Acceptance Control Charts (ACCs), and Modified PC is compared with an X chart. Previous comparisons of Classical and Two-stage PC with more traditional control charts were made with X & R charts (Mackertich, 1990, Ermer and Roepke, 1991), however this is the wrong comparison. For both ACC and Classical and Two-stage PC the goal is not statistical control, but rather producing parts within specification. An ACC is designed to monitor a process when the process variability is much smaller than the specification (tolerance) spread. Under that assumption moderate drifts in the mean (from the target value) are tolerable since they do not yield a significant increase in the proportion of nonconforming units. Like Classical and Two-stage PC, ACCs are based on specification limits. However, ACC limits are derived based on a distributional assumption and require a known and constant process standard deviation. ACC limits are usually set assuming a normal process, although if justified other assumptions could be made. For more information on the design of ACCs see Ryan (1989). Modified PC, on the other hand, bases the grouping criteria on the control limits, and thus Modified PC has the same goal as a Shewhart X chart, namely statistical control of a process.

7 7 PC and traditional control charting techniques differ in a number of ways. The first obvious difference is that PC uses only information from the classified (or grouped) observations, whereas traditional control charts ( X charts and ACCs) use variables data. Grouping data results in decision rules that are easy to implement, but clearly, the grouping also discards information. The loss of information can be quantified by calculating the expected statistical (Fisher) information available in the PC grouped observations. Calculating the Fisher information available in grouped data is discussed in Kulldorff (1961) and Steiner, Geyer and Wesolowsky (1995). Figure 2 shows the expected information about the mean and standard deviation for various parameter values. The plots are scaled so that variables data (infinite number of groups) would provide an expected information content of unity for all values of µ and σ. The expected information graph about µ is symmetric about µ = 0, thus negative mean values are not shown. Figure 2 shows that the group classification scheme used in PC is not very efficient when µ is close to zero and/or σ is small. The small amount of information available when µ equals zero may not be a major concern if at that mean value it is very easy to conclude that the process should continue operation. This would be the case, for example, if the process standard deviation were very small compared with the specification range. This issue is explored further in Section Expected Information Expected Information µ σ Figure 2: Expected Information about µ (left, assume σ =0.2933) and σ (right, assume µ =0), t = [ 1,.5,.5, 1]

8 8 PC and traditional control charts also differ in their decision rules. All control charts require some decision rules to determine whether the process should be allowed to continue operation or if it should be stopped. It is difficult to evaluate the effectiveness of the decision rules in isolation from the grouping criteria. However, the overall effectiveness of various control charts can be compared through their Operating Characteristic (OC) curve. An OC curve shows the probability a control chart signals (or fails to signal) for different parameter values. OC curves for ACCs and X charts can be derived from results given in Ryan (1989). The probability Classical and Two-stage PC schemes continue operating, called the probability of acceptance of the process, can be found by calculating the probability of each combination of green and yellow units that leads to acceptance. For Classical and Two-stage PC the results are given by equations (2.1) and (2.2), see Salvia (1988). P accept P accept ( Classical) = Pg + PyPg (2.1) ( Two stage) = Pg + Py ( ) 2 2PgPy(1 Pg 3 3Pg 2 Py) Py 2 1 Pg 3 ( ) (2.2) where Pg and Py are given by Equations (1.1). Figure 3 shows the OC curves for Classical PC, Two-stage PC, as derived from (2.1) and (2.2), and an ACC with sample size n = 2. Figure 3 shows that an ACC with sample size n = 2 is superior to Classical PC since it has significantly better power to detect mean shifts. Similarly, the Two-stage PC scheme is superior to an ACC with n = 2. Note that different σ values simply shift the OC curves horizontally without changing the ranking. Also, a process whose quality characteristic is approximately normally distributed is assumed. A different underlying distribution for the quality characteristic may dramatically change the probabilities of acceptance, but generally all the considered procedure are similarly effected.

9 Probability of Acceptance Classical Precontrol ACC n = stage Precontrol Mean Shift in Sigma Units Figure 3: OC curve Classical and Two-stage PC versus ACC with n=2 N(µ, σ =.2933) process Note however, that on average the Two-stage PC scheme requires a larger sample size. Two-stage PC uses a partially sequential procedure and requires sample sizes of between one and five units. In the appendix, the average sampling number of the Twostage PC procedure, denoted En ( ) is derived. En ( ) depends on the group probabilities, given by (1.1), that are in turn dependent on the process parameters µ and σ. Figure 4 shows plots of En ( ) versus the process mean assuming normality. For example, En;µ ( = 0,σ =.29333) = 2.4, and En;µ ( = 2σ,σ =.29333) = 3.5, En ( ) En ( ) µ µ ( ) for Two-stage PC as function of µ Figure 4: Average sample size En σ = on left, σ = 0.2 on right, N( µ, σ ) process

10 10 These results suggest a comparison of Two-stage PC with ACCs having larger sample sizes. OC curves of Two-stage PC and ACCs with sample sizes of 3, 4 and 5 are shown in Figure 5. Clearly, of the compared curves the PC procedure has least power to detect mean shifts. In addition ACCs are likely easier to implement since they require fixed sample sizes. On the other hand, the power of Two-stage PC is fairly competitive with an ACC with n = 3, and ACCs have the disadvantage of requiring an estimate of σ, an assumption that σ does not change, and precise or variables measurements rather than grouped data. In addition, Two-stage PC requires smaller sample size on average when the process is centered at µ = 0. In conclusion, Two-stage PC appears a reasonable alternative to ACCs when little process knowledge is available or if estimating the process mean and standard deviation are expensive stage Precontrol Probability Accept ACC n=4 ACC n=5 ACC n= Mean Shift in Sigma Units Figure 5: Comparison of Two-stage PC and ACC with n=3, n=4, n=5 N(µ, σ =.29333) process Classical and Two-stage PC schemes are designed to prevent defectives, but they do not adapt to changes in process variability. Figures 6 and 7 explore the relation between the probability Classical and Two-stage PC signal and the probability of a defect. Figure 6

11 11 shows a contour plot of the probability of a defect for various combinations of µ and σ, assuming USL = 1 and LSL = 1. The probability of a defect, or a nonconforming unit, is given by P r in Equations (1.1). Figure 7 shows contour plots of probability that Classical and Two-stage PC signal (one minus the probability of acceptance as given by (2.1) and (2.2). Ideally Figures 6 and 7 would look similar, however, this is not the case. The PC methods are too conservative when σ is small. When σ is small compared with the specification limits a significant drift in the process mean can be tolerated before the proportion defective becomes large. However, using PC, large mean values together with small σ values are likely to lead to a signal since then many yellow units would be observed. As a result, PC is not applicable when σ is very small compared with the specification limits Standard Deviation e Mean Figure 6: Contour plot of the probability of a defect

12 12 Standard Deviation e Standard Deviation e Mean Mean Figure 7: Contour plot of P(signal) for Classical & Two-stage PC 0.9 ACCs do not suffer from this shortcoming since they adjust their control limits for different σ values. Table 2 gives specific values shown on the contour plots and equivalent values from an ACC with n = 5 for a direct comparison. Most enlightening are the two cases: µ = 0, σ =.2, and µ =.6, σ =.1. In both cases the probability of a defect is fairly small and approximately the same. However, when µ = 0, σ =.2 the probability the PC scheme signals is very small, whereas when µ =.6, σ =.1 the probability of a signal using PC is very large. Clearly, when σ is small, PC rejects some processes that yield very few defects. This is appropriate if the goal is also to keep the mean centered on the target value. However, the goal of PC is defect detection and prevention. Thus, PC is too conservative when σ is small, say when 6σ covers less than 40% of the tolerance range. As shown in Table 2 an ACC with n = 5 does not suffer from this problem to the same degree. Table 2: Probability of defect and signals P(signal) P(signal) µ σ P(defect) Classical PC Two-stage PC P(signal) ACC n = ~ e e 18 ~ e e e e e not applicable e e e

13 13 Modified PC, unlike Classical and Two-stage PC, classifies units based on control limits. As a result, Modified PC is the same in philosophy as an X control chart, since its goal is to monitor the stability of a process. With Modified PC and X charts ideally any drift in the process mean would be detected. The operating characteristics of Modified PC can be determined using Equation (2.2) and (1.2) with µ = 0, σ = 1/3. Modified PC is compared with traditional X control charts in Table 3. The average sample size of Modified PC can be determined through the result in the appendix setting µ = 0 and σ = 1/3. A plot of the result would be similar to those shown in Figure 4. Table 3: Modified PC compared with X control charts P signal P signal P signal P signal µ Modified X chart X chart X chart PC n = 3 n = 4 n = ±1σ ±2σ Table 3 shows that Modified PC has a very high false alarm rate that is an order of magnitude larger than that for X charts. When the process is stable ( µ = 0) a Modified PC chart signals a problem on average almost 2.4% of the time. This false alarm rate is too high, since searching for assignable causes is usually time consuming and expensive, and too many false alarms greatly reduces the credibility of the control chart and may lead to it being ignored. In summary, Classical and Two-stage PC are fairly competitive with ACCs, except under certain circumstances. When the process variation is very small compared with the tolerance range ( 6σ <.4T, where T is the tolerance range), Classical and Two-stage PC lead to undesirable results such as rejecting a process that is producing virtually all parts within specification. On the other hand, when the process variation is relatively large compared to the tolerance range (6σ >.88T) Classical and Two-stage PC yield

14 14 excessively large false alarm rates. However, if 6σ lies between 40% and 88% of the tolerance range (with specification limits of 1 and 1, > σ >.13333) Classical or Two-stage PC yield good results. Modified PC, on the other hand, is designed to have the same goal as a Shewhart chart. Thus, Modified PC is similar to Two-stage PC with 6σ equal to 100% of the tolerance range. As a result, Modified PC is not a good method since it yields too many false alarms. 3. Alternatives to Traditional Pre-control As discussed in Section 2, PC has a number of important advantages over traditional control charts, mainly in terms of simplicity of implementation. However, its design choices, in particular the grouping criteria and the decision rules, appear quite ad hoc. We may ask if the performance of PC could be improved while retaining its simplicity? In this section, three simple variations of PC called Ten Unit PC, Mean Shift PC, and Simplified PC are considered. Each variation is very similar to Two-stage PC, and requires only minor modifications in the design. The goal is not to develop the optimal approach under particular assumptions, but rather to consider simple changes that retain the flavor of PC. If optimal charts are desired, and more process information is available, the grouped data ACCs and Shewhart type charts developed by Steiner, Geyer, and Wesolowsky (1994, 1995) should be considered. One reason why Two-stage PC is fairly competitive with ACCs in terms of power is due to its partially sequential decision procedure. As a result, one possible improvement approach is to make the decision process more sequential. A totally sequential procedure is theoretically feasible, but would be difficult to implement since it would occasionally require large sample sizes and defining the acceptance/rejection criteria would be difficult. A version of PC that allows a maximum of ten units at each decision point is a compromise. The proposed decision rules for Ten Unit PC are given below. This sampling procedure should be repeated each time the process is to be monitored.

15 15 Ten Unit PC Decision Process Take sample units one at a time defining #G and #Y as the number of green and yellow units observed so far respectively. Stop the process if there are any red units, or if at any time #Y #G 2 together with #Y 3, or if at any time #Y 5. Continue operation of the process and stop sampling if at anytime #G #Y 2. Otherwise, continue to sample. By enumerating all the possible paths to acceptance and rejection of Ten Unit PC it is possible to derive expressions for the probability of a signal and the average sample size. Table 4 shows a comparison of the operating characteristics of the proposed Ten Unit PC scheme and Two-stage PC for various combinations of process mean and standard deviation. Table 4 shows that Ten Unit PC has lower false alarm rates, more power to detect two-sigma shifts in the mean, and requires approximately the same average number of units as Two-stage PC. Table 4: Comparison of Ten Unit and Two-stage PC P(signal) En ( ) P(signal) µ σ Two-stage PC Two-stage PC Ten Unit PC En ( ) Ten Unit PC ±1σ ±2σ / ±1σ 1/ ±2σ 1/ A second improvement idea stems from the realization that using PC, units are classified into one of five groups (see Figure 1) but only three groups (green, yellow and red) are used during the decision process. As a result, important information pertaining to

16 16 the process mean is ignored. The proposed Mean Shift PC utilizes this information. Mean Shift PC is a very simple adaptation of Two-stage PC since it does not requiring any changes in the data collection process. The Mean Shift PC method classifies observations into one of four groups: Green, Yellow+, Yellow, and Red. Red units above or below the specification limits are not distinguished because they are very rare and automatically lead to a signal in any case. The Mean Shift PC decision rules, given below, are similar to those used in Two-stage PC. Mean Shift PC Decision Process Sample two consecutive parts. If either unit is red stop the process. If both units are green continue operation. If either or both units are yellow, sample an additional three units. If in the combined sample three greens are obtained continue the process. If three Yellow+ or three Yellow or a single Red are observed then stop the process. Table 5: Comparison of Two-stage PC and Mean Shift PC P µ σ signal Twostage PC Shift P signal Mean PC 0 1/ ±1σ 1/ ±2σ 1/ ±1σ ±2σ Table 5 compares the power of the Mean Shift PC and Two-stage PC. The table shows that Mean Shift PC has a much smaller false alarm rate than Two-stage PC, and virtually identical power to detect two sigma mean shifts. This advantage in terms of operating characteristics will typically outweigh the slightly more complicated decision

17 17 process. Thus, Mean Shift PC is a good alternative to Two-stage PC, though the process standard deviation should also be monitored in some way. The third variation, called Simplified PC uses only green and yellow units. The group probabilities and decision process for Simplified PC are given below:.5 ( ) = P g = f y P green P yellow y=.5.5 ( ) = P y = f y ( )dy ( )dy + f( y)dy (3.1) y= y=.5 Simplified PC Decision Process Sample five consecutive parts If three or more units are yellow, then stop the process Otherwise continue operation. The loss of efficiency using Simplified PC rather than Two-stage PC is very slight when the process variation is in the range where PC is appropriate. This only small loss of efficiency arises because the probability of a red unit is usually very small, and thus the red classification provides very little information. In fact, as designed, Simplified PC has a slightly smaller false alarm rate, though not quite as much power to detect mean shifts. Table 6 shows a comparison. Compared with Two-stage PC, Simplified PC has the advantage of requiring less effort in the group classification, and the decision rules are more straight forward. In addition, although Simplified PC requires a larger sample size on average often the required sample sizes are the same and implementation is usually easier if the sample size is fixed.

18 18 Table 6: Comparison of Two-stage PC and Simplified PC P µ σ signal Twostage PC Simplified P signal PC 0 1/ ±1σ 1/ ±2σ 1/ ±1σ ±2σ Simplified PC is also appealing since it can fairly easily be adapted when the process variation is much smaller than the tolerance range. As shown in Section 2, PC is not appropriate if the process standard deviation is less than around 115th of the tolerance range. When σ is small, observing yellow units with the traditional grouping criterion as defined by (1.1), is not necessarily evidence of a problem. See Figures 6 and 7. In this circumstance the Simplified PC group limits [.5,.5] can be replaced with the group limits [ c, c], where c is closer to the tolerance limits (.5 < c < 1). By adjusting the group limits it is possible to only detecting process mean shifts that yield an excessive proportion of non-conforming units. The decision process remains unchanged. The best value for c depends on the process standard deviation and the acceptable and rejectable process levels (APL and RPL). A good choice for c is the midpoint between the APL mean and the RPL mean. APL and RPL mean values can be estimated from acceptable and unacceptable process capability values. Process capability is defined as C pk = min( ( USL µ ) 3σ, ( µ LSL) 3σ), where LSL and USL is the lower and upper specification limits respectively. For example, assume the process standard deviation is σ =.1, and that an acceptable process capability value is C pk = 4/3 (an average of 6 defects out of 100,000 units) whereas C pk = 2/3 (an average of 94 defects out of 10,000 units) is unacceptable. Then looking only at the upper specification limit (the problem is symmetric on the lower limit) the corresponding APL and RPL mean values are µ =.6 and µ =.8 respectively. In this case choosing c =.7 is reasonable. Figure 8 illustrates the operating

19 19 characteristics obtained in this example for various parameter values. For example, the probability of a signal when µ =.6 is.031 and the probability of a signal when µ =.8 is.969. If this false alarm rate is too large adjusting the decision barrier c slightly higher may be appropriate. An OC curve for Two-stage PC would show a much great chance of rejecting the process at the acceptable mean value µ = P signal µ Figure 8: OC curve of the adapted Simplified PC procedure 4. Conclusions This article compared and contrasts previously recommended PC approaches and suggests simple variations that improve performance. Classical and Two-stage PC are compared with acceptance control charts rather than X charts since this is a more appropriate comparison. It is concluded that Classical and Two-stage PC are good methods when the process standard deviation σ lies in the range T 15 σ 11T 75, where T is the tolerance range. Two-stage PC is preferred over Classical PC unless the additional sampling required is very onerous. Modified PC, on the other hand, has the same goal as an X chart, but is shown to have an excessively large false alarm rate, and is thus not recommended. Three alternatives to PC are suggested that retain the great advantage of very simple implementation. The Ten Unit approach is recommended when improved operating

20 20 characteristics are desired and taking some additional observations is not difficult. The Mean Shift approach yields better results than Two-stage PC when the goal is the detection of process mean shifts, and Simplified PC utilizes a simplified classification and decision procedure while providing similar operating characteristics. Simplified PC can also easily be adapted to handle situations when the process standard deviation is smaller than T 15. Appendix In this appendix an expression for the average sampling number En ( ) of the Twostage PC scheme, discussed in Section 2, is derived. En ( ) is determined by calculating the probability of each possible path to either acceptance or rejection of the process. Below in (A.1), p i is the probability that the Two-stage PC reaches a decision in i units. For example, Two-stage PC would stop after two units if we observe either two green units (process acceptable), or if the first observation is yellow or green and the second observation is red (process rejectable). p 1 = p r, p 2 = p 2 g + p r ( p y + p g ), p 3 = p r p 2 3 y + 2 p g p y p r + p y (A.1) p 4 = 3p y 2 p g p r + 2 p g 2 p y p r + 2 p g 3 p y + 3p y 3 p g, p 5 = 5p y 2 p g 2 p r + 5p y 3 p g 2 + 5p g 3 p y 2 5 En ( )= ip i References i=1 Ermer, D.S., Roepke, J.R. (1991), An Analytical Analysis of Pre-control, ASQC Quality Congress Transactions - Milwaukee, Gurska, G.F., Heaphy M.S. (1991), STOP LIGHT CONTROL - Revisited, ASQC Statistics Division Newsletter, Fall. Kulldorff, G. (1961), Contributions to the Theory of Estimation from Grouped and Partially Grouped Samples, John Wiley and Sons, New York. Mackertich, N. A. (1990), Pre-control vs. Control Charting: A Critical Comparison, Quality Engineering, 2,

21 21 Ryan, T.P. (1989), Statistical Methods for Quality Improvement, John Wiley & Sons, New York. Shainin, D., Shainin, P. (1989), Pre-control Versus X & R Charting: Continuous or Immediate Quality Improvement?, Quality Engineering, 1, Salvia, A.A. (1988), Stoplight Control, Quality Progress, September, Steiner, S.H., Geyer, P.L., Wesolowsky, G.O. (1994), Control Charts based on Grouped Data, International Journal of Production Research., 32, Steiner, S.H., Geyer, P.L., Wesolowsky G.O. (1995), "Shewhart Control Charts to Detect Mean Shifts Based on Grouped Data," submitted to Quality and Reliability Engineering International. Traver, R.W. (1985), Pre-control: A Good Alternative to X -R Charts, Quality Progress, September,