First Semester Dr. Abed Schokry SQC Chapter 9: Cumulative Sum and Exponential Weighted Moving Average Control Charts

Transcription

1 Department of Industrial Engineering First Semester Dr. Abed Schokry SQC Chapter 9: Cumulative Sum and Exponential Weighted Moving Average Control Charts Learning Outcomes After completing this chapter you should be able to Discuss the difference between Capability and Stability, Set up and use control charts for monitoring the process mean, Design a cusum control chart for mean to obtain specific ARL performance, Incorporate initial response feature into the cusum control chart, Set up and use EWMA control charts for monitoring the process mean, Design an EWMA control chart for mean to obtain specific ARL performance, Explain why the EWMA control chart is robust to the assumption of normality, Understand the advantage of cusum and EWMA control charts relative to Shewhart control charts 1

2 The Difference Between Capability and Stability? Once again, a process is capable if individual products consistently meet specifications. A process is stable if only common variation is present in the process. Average Run Length The Average Run Length (ARL) is the average number of points that most be plotted before a point indicates an out-of-control condition. For the Shewhart chart this can be calculated as ARL=1/p where p is the probability that any one point exceeds the control limit. 2

3 Average Run Length ARL should be long when the process is in-control and short when the process is out-of-control. For a 3 Shewhart chart and an underlying normally distributed process we have ARL=1/(.0027)= when the process is in-control For many practical applications this chart has proven very effective and robust. It should be used more. Average Run Length (ARL) The Average Run Length is the number of points that, on average, will be plotted on a control chart before an out of control condition is indicated If the process is in control: ARL=1/ α If the process is out of control: ARL=1/(1- β) α - the probability of a Type I error, β - the probability of a Type II error. 3

4 Introduction Chapters 4 through 6 focused on Shewhart control charts. Major disadvantage of Shewhart control charts is that it only uses the information about the process contained in the last plotted point. Two effective alternatives to the Shewhart control charts are the cumulative sum (CUSUM) control chart and the exponentially weighted moving average (EWMA) control chart. Especially useful when small shifts are desired to be detected. Process Control Charts Other Control Charts Moving Average Chart. The moving average chart is an interesting chart that is used for monitoring variables and measurement on a continuous scale. The chart uses past information to predict what the next process outcome will be. Using this chart, we can adjust a process in anticipation of its going out of control. Cusum Chart. The cumulative sum, or cusum, chart is used to identify slight but sustained shifts in a universe where there is no independence between observations. 4

5 The Cumulative-Sum Control Chart Basic Principles: The Cusum Control Chart for Monitoring the Process Mean The cusum chart incorporates all information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a target value. If 0 is the target for the process mean, x j is the average of the jth sample, then the cumulative sum control chart is formed by plotting the quantity i Ci (x j 0) j1 CUSUM Control Chart Incorporates all the information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a target value, 0 CUSUM can be used to monitor process mean defectives defects variance CUSUM can have sample size n 1 We concentrate on sample size n = 1 5

6 CUSUM charts CUSUM is short for cumulative sums. As measurements are taken, the difference between each measurement and the bench mark value is calculated, and this is cumulatively summed up. If the processes are in control, measurements do not deviate significantly from the bench mark, so measurements greater than the bench mark and those less than the bench mark averaged each other out, and the CUSUM value should vary narrowly around the bench mark level. If the processes are out of control, measurements will more likely to be on one side of the bench mark, so the CUSUM value will progressively depart from that of the bench mark. CUSUM charts CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process xn are assigned weights wn, and summed as follows: So=0 Sn+1=max(0, Sn+xn-wn) When the value of S exceeds a certain threshold value, a change in value has been found. The above formula only detects changes in the positive direction. When negative changes need to be found as well, the min operation should be used instead of the max operation, and this time a change has been found when the value of S is below the (negative) value of the threshold value. 6

7 Basic Principle of CUSUM Plot C i CUSUM sample statistic Example: Say target 0 = 10 i i C x j 1 j 0 Obs i x i (x i 0 ) i C x = = = = 0.55 If the process remains in-control, C i remains near 0 i j 1 j 0 Tabular CUSUM Control Chart x i ~ N( 0, ) - quality characteristic CUSUM works by compiling the statistics: C i+ = accumulated deviations above negative) 0 (resets to 0 if it would go C i = accumulated deviations below negative) 0 (resets to 0 if it would go The Tabular CUSUM Record following values in table: C max 0, x K C i i 0 i1 C max 0, K x C where starting values are C i 0 i i1 0 C0 0 7

8 Tabular CUSUM Control Chart Cont'd Let 1 = out-of-control value then 1 0 K 2 K is reference value chosen halfway between target 0 and out-ofcontrol value With shift expressed in std dev units, i.e., C i K 2 and accumulate deviations from μ 0 that are greater than K C i C i C i and are reset to zero upon becoming negative How to Determine if Process Out-of-Control? H - decision interval C C If i or i exceed the decision interval (H), the process is considered out-of-control Rule of thumb value for H Choose H to be five times the process standard deviation, H = 5 Counters N + and N record the number of consecutive periods the CUSUM C i and C i rose above zero, respectively. The counters can be used to indicate when the shift most likely occurred 8

9 Notes about CUSUM control charts Do not apply zone rules Do not apply run rules Successive values of C and C are not independent i i Choosing Proper Type of Control Chart: Individuals Charts Use (x & MR), MA, EWMA, or CUSUM charts when: 1. Repeated measures do not make sense 2. Inconvenient / impossible to obtain more than one measurement per sample 3. Automated testing allows you to measure every unit (EWMA chart may be best) 4. Data becomes available very slowly and waiting for a larger sample is impractical. 9

10 Improving Cusum Performance for Large Shifts: The Combined Shewhart-Cusum Scheme More on Cusums Cusums are often used to determine if a process has shifted off a specified target because it is easy to calculate the required adjustment One-sided cusums are often useful Cusums can also be used to monitor variability Cusums are available for other sample statistics (ranges, standard deviations, counts, proportions) Rational subgroups and cusums 10

11 The Cusum V-Mask 11

12 Drawbacks of Cusum V-Mask Only for two-sided schemes Headstart cannot be implemented (Headstart: Fast Initial Response, FIR) Range of arms V-mask unclear Interpretation parameters (angle,...) not well determined Fast Initial Response Cusum When an out-of-control signal is received, the Cusum user should search for assignable cause. If a cause was NOT found, the sums might be left as is. If the sum stays above the threshold value, a cause does exist but it was not detected. Start another search. The Sum should be reset, when an assignable cause is detected and removed. The sum could be reset to 0 Or to head start values (h/2, -h/2) if multiple causes may exist, this is referred as Fast Initial Response 12

13 The Fast Initial Response (FIR) Cusum K = 3, H = 12, headstart = H/2 = 6 H = 12 implies that the cusum signals at sample 3 Without the headstart, it would not signal until sample 6 13

14 Advantages and Disadvantages of the CUSUM The CUSUM chart is very effective for small shifts and when the subgroup size n=1. The CUSUM is relatively slow to respond to large shifts. Also, special patterns are hard to see and analyze. 14

15 EWMA charts Exponentially Weighted Moving Average is a statistic that averages the data in a way that gives less and less weight to data as they are further removed in time. EWMA(t)=λY(t)+(1 λ)ewma(t 1), t=1,2,,n. EWMA(0) is the mean of historical data (target) Y(t) is the observation at time t n is the number of observations to be monitored including EWMA(0) 0<λ 1 is a constant that determines the depth of memory of the EWMA. The equation is due to Roberts (1959). The EWMA chart is sensitive to small shifts in the process mean, but does not match the ability of Shewhart-style charts to detect larger shifts EWMA control procedure can be made sensitive to a small or gradual drift in the process, whereas the Shewhart control procedure can only react when the last data point is outside a control limit. Exponentially-weighted Moving Average (EWMA) Chart The idea of moving averages of successive (adjacent) samples can be generalized. In principle, in order to detect a trend we need to weight successive samples to form a moving average; however, instead of a simple arithmetic moving average, we could compute a geometric moving average. It is also called Geometric Moving Average chart. EWMA Y (1 ) EWMA ( t1) t t UCL EWMA ks LCL EWMA ks

16 Exponentially-weighted Moving Average (EWMA) Chart EWMA Charts are generally used for detecting small shifts in the process mean. They will detect shifts of.5 sigma to 2 sigma much faster. They are, however, slower in detecting large shifts in the process mean. In addition, typical run tests cannot be used because of the inherent dependence of data points. EWMA Charts may also be preferred when the subgroups are of size n=1. UCL EWMA1 ks 2 EWMA ( t1) Yt (1 ) EWMAt LCL EWMA1 ks 2 where λ is the weighting factor. The factor k is chosen generally to be 2 or 3. EWMA charts: Ideas Generalization of the run rules for the Shewhart control charts - signal occurs when m successive observations exceed k1*sigma control limit - signal occurs when (m-1) out of m successive observations exceed k2* sigma control limit 16

17 EWMA charts: Ideas Economical design of control charts how to determine the sample size, the interval between samples, and the control limits that will yield approximately maximum average net income Different approaches 17

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19 Robustness of EWMA to Non-normal Process Data 19

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21 Extensions of the EWMA Fast initial response feature Monitoring variability Monitoring count data The EWMA as a predictor of process level 9.3 The Moving Average Control Chart 21

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