Introduction to Statistical Quality Control, 4th Edition,Chapter 8,Cumulative Sum and Exponentially Weighted Moving Average Control Charts,Introduction to Statistical Quality Control, 4th Edition,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.,Introduction to Statistical Quality Control, 4th Edition,8-1. The Cumulative-Sum Control Chart,8-1.1 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, is the average of the jth sample, then the cumulative sum control chart is formed by plotting the quantity,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Let xi be the ith observation on the process If the process is in control then Assume is known or can be estimated. Accumulate derivations from the target 0 above the target with one statistic, C+ Accumulate derivations from the target 0 below the target with another statistic, C C+ and C- are one-sided upper and lower cusums, respectively.,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,The statistics are computed as follows: The Tabular Cusumstarting values areK is the reference value (or allowance or slack value)If either statistic exceed a decision interval H, the process is considered to be out of control. Often taken as a H = 5,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Selecting the reference value, K K is often chosen halfway between the target 0 and the out-of-control value of the mean 1 that we are interested in detecting quickly. Shift is expressed in standard deviation units as 1= 0+, then K is,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Example 8-1 0 = 10, n = 1, = 1 Interested in detecting a shift of 1.0 = 1.0(1.0) = 1.0 Out-of-control value of the process mean: 1= 10 + 1 = 11 K = and H = 5 = 5 (recommended, discussed in the next section) The equations for the statistics are then:,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Example 8-1,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Example 8-1 The cusum control chart indicates the process is out of control. The next step is to search for an assignable cause, take corrective action required, and reinitialize the cusum at zero. If an adjustment has to be made to the process, may be helpful to estimate the process mean following the shift.,Introduction to Statistical Quality Control, 4th Edition,8-1.2 The Tabular or Algorithmic Cusum for Monitoring the Process Mean,Example 8-1 If an adjustment has to be made to the process, may be helpful to estimate the process mean following the shift. The estimate can be computed fromN+, N- are counters, indicating the number of consecutive periods that the cusums C+ or C- have been nonzero.,Introduction to Statistical Quality Control, 4th Edition,8-1.4 The Standardized Cusums,It may be of interest to standardize the variable xi. The standardized cusums are then,Introduction to Statistical Quality Control, 4th Edition,8-1.5 Rational Subgroups,Shewhart chart performance is improved with rational subgrouping Cusum is not necessarily improved with rational subgrouping Only if there is significant economy of scale or some other reason for taking larger samples should rational subgrouping be considered with the cusum,Introduction to Statistical Quality Control, 4th Edition,8-1.6 Improving Cusum Responsiveness for Large Shifts,Cusum control chart is not as effective in detecting large shifts in the process mean as the Shewhart chart. An alternative is to use a combined cusum-Shewhart procedure for on-line control. The combined cusum-Shewhart procedure can improve cusum responsiveness to large shifts.,Introduction to Statistical Quality Control, 4th Edition,8-1.7 The Fast Initial Response or Headstart Feature,These procedures were introduced to increase sensitivity of the cusum control chart upon start-up. The fast initial response (FIR) or headstart sets the starting values equal to some nonzero value, typically H/2. Setting the starting values to H/2 is called a 50 percent headstart.,