CHAPTER 6 Control Charts for Variables 6 1

CHAPTER 6 Control Charts for Variables

6 -1. INTRODUCTION Variable - a single quality characteristic that can be measured on a numerical scale. We monitor both the mean value of the characteristic and the variability associated with the characteristic.

VARIABLES DATA Examples Length, width, height Weight Temperature Volume

BOTH MEAN AND VARIABILITY MUST BE CONTROLLED

6 -2. CONTROL CHARTS FOR AND R Notation for variables control charts n - size of the sample (sometimes called a subgroup) chosen at a point in time m - number of samples selected = average of the observations in the ith sample (where i = 1, 2, . . . , m) = grand average or “average of the averages (this value is used as the center line of the control chart)

6 -2. CONTROL CHARTS FOR AND R Notation and values Ri = range of the values in the ith sample Ri = xmax - xmin = average range for all m samples is the true process mean is the true process standard deviation

6 -2. CONTROL CHARTS FOR AND R Statistical Basis of the Charts Assume the quality characteristic of interest is normally distributed with mean , and standard deviation, . If x 1, x 2, …, xn is a sample of size n, then the average of this sample is = (X 1 + X 2 + … + Xn)/n is normally distributed with mean, , and standard deviation,

6 -2. CONTROL CHARTS FOR AND R Statistical Basis of the Charts If n samples are taken and their average is computed, the probability is 1 - α that any sample mean will fall between: The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

CONTROL CHARTS FOR When the parameters are not known Compute Now UCL = LCL = R(bar)

CONTROL CHARTS FOR But, est = So, Let A 2 = Then, can be written as:

6 -2. CONTROL CHARTS FOR AND R Control Limits for the chart A 2 is found in Appendix VI for various values of n.

CONTROL CHART FOR R We need an estimate of R We will use relative range W = R/ , R = W Let d 3 be the standard deviation of W Std. Dev [R] = Std. Dev [W ] R = d 3 Use R to estimate R

CONTROL CHART FOR R est, R = d 3(R(bar)/d 2) UCL = R(bar) + 3 est, R = R(bar) + 3 d 3(R(bar)/d 2) CL = R(bar) LCL = R(bar) - 3 est, R = R(bar) - 3 d 3(R(bar)/d 2) Let D 3 = 1 – 3(d 3/d 2) And, D 4 = 1 + 3(d 3/d 2)

6 -2. CONTROL CHARTS FOR AND R Control Limits for the R chart D 3 and D 4 are found in Appendix VI for various values of n.

6 -2. CONTROL CHARTS FOR AND R Trial Control Limits The control limits obtained from equations (6 -4) and (6 -5) should be treated as trial control limits. If this process is in control for the m samples collected, then the system was in control in the past. If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

6 -2. CONTROL CHARTS FOR AND R Trial control limits and the out-of-control process Eqns 6. 4 and 6. 5 are trial control limits Determined from m initial samples Typically 20 -25 subgroups of size n between 3 and 5 Any out-of-control points should be examined for assignable causes If assignable causes are found, discard points from calculations and revise the trial control limits Continue examination until all points plot in control If no assignable cause is found, there are two options 1. 2. Eliminate point as if an assignable cause were found and revise limits Retain point and consider limits appropriate for control If there are many out-of-control points they should be examined for patterns that may identify underlying process problems

EXAMPLE 6. 1 THE HARD BAKE PROCESS

6 -2. CONTROL CHARTS FOR AND R Estimating Process Capability The x-bar and R charts give information about the performance or Process Capability of the process. Assumes a stable process. We can estimate the fraction of nonconforming items for any process where specification limits are involved. Assume the process is normally distributed, and x is normally distributed, the fraction nonconforming can be found by solving: P(x < LSL) + P(x > USL)

EXAMPLE 6 -1: ESTIMATING PROCESS CAPABILITY Determine est = R(bar)/d 2 =. 32521/2. 326=0. 1398 Where d 2 Specification limits are 1. 5 +. 5 microns. (1. 00, values are given in App. Table VI 2. 00) Assuming N(1. 5056, 0. 13982) Compute p = P(x < 1. 00) + P(x > 2. 00)

EXAMPLE 6 -1: ESTIMATING PROCESS CAPABILITY P = F([1. 00– 1. 5056]/0. 1398) +1 – F([2. 001. 5056]/. 1398) = F(-3. 6166) + 1 – F(3. 53648) =. 00035 These values from the APPENDIX 2 / 693 So, about 350 ppm will be out of specification

6 -2. CONTROL CHARTS FOR AND R Process-Capability Ratios (Cp) Used to express process capability. For processes with both upper and lower control limits, Use an estimate of if it is unknown. Cest, p = (2. 0 – 1. 0)/[6(. 1398)] = 1. 192

6 -2. CONTROL CHARTS FOR AND R If Cp > 1, then a low # of nonconforming items will be produced. If Cp = 1, (assume norm. dist) then we are producing about 0. 27% nonconforming. If Cp < 1, then a large number of nonconforming items are being produced.

6 -2. CONTROL CHARTS FOR AND R Process-Capability Ratios (Cp) The percentage of the specification band that the process uses up is denoted by **The Cp statistic assumes that the process mean is centered at the midpoint of the specification band – it measures potential capability.

EXAMPLE 6 -1: ESTIMATING BANDWIDTH USED Pest = (1/1. 192)100% = 83. 89% The process uses about 84% of the specification band.

PHASE II OPERATION OF CHARTS Use of control chart for monitoring future production, once a set of reliable limits are established, is called phase II of control chart usage (Figure 6. 4) A run chart showing individuals observations in each sample, called a tolerance chart or tier diagram (Figure 6. 5), may reveal patterns or unusual observations in the data

6 -2. CONTROL CHARTS FOR AND R Control Limits, Specification Limits, and Natural Tolerance Limits (See Fig 6. 6) Control limits are functions of the natural variability of the process (LCL, UCL) Natural tolerance limits represent the natural variability of the process (usually set at 3 -sigma from the mean) (LNTL, UNTL) Specification limits are determined by developers/designers. (LSL, USL)

6 -2. CONTROL CHARTS FOR AND R

6 -2. CONTROL CHARTS FOR AND R Control Limits, Specification Limits, and Natural Tolerance Limits There is no mathematical relationship between control limits and specification limits.

RATIONAL SUBGROUPS charts monitor between-sample variability (variability in the process over time) R charts measure within-sample variability (the instantaneous process variability at a given time) Standard deviation estimate of used to construct control limits is calculated from within-sample variability It is not correct to estimate using

6 -2. CONTROL CHARTS FOR AND R Guidelines for the Design of the Control Chart Specify sample size, control limit width, and frequency of sampling If the main purpose of the x-bar chart is to detect moderate to large process shifts, then small sample sizes are sufficient (n = 4, 5, or 6) If the main purpose of the x-bar chart is to detect small process shifts, larger sample sizes are needed (as much as 15 to 25).

6 -2. CONTROL CHARTS FOR AND R Guidelines for the Design of the Control Chart R chart is insensitive to shifts in process standard deviation when n is small The range method becomes less effective as the sample size increases May want to use S or S 2 chart for larger values of n > 10

6 -2. CONTROL CHARTS FOR AND R Guidelines for the Design of the Control Chart Allocating Sampling Effort Choose a larger sample size and sample less frequently? Or, choose a smaller sample size and sample more frequently? The method to use will depend on the situation. In general, small frequent samples are more desirable

CHANGING SAMPLE SIZE OF X-BAR AND R CHARTS 1 - variable sample size: size the center line changes continuously and the chart is difficult to interpret. (X-bar and s is more appropriate) 2 - Permanent or semi-permanent change: Introduction to Statistical Quality Control, 4 th Edition

6 -2. 3 CHARTS BASED ON STANDARD VALUES If the process mean and variance are known or can be specified, then control limits can be developed using these values: Constants are tabulated in Appendix VI

6 -2. 4 INTERPRETATION OF R CHARTS AND Patterns of the plotted points will provide useful diagnostic information on the process, and this information can be used to make process modifications that reduce variability. Cyclic Patterns ( systematic environmental changes ) Mixture ( tend to fall near or slightly outside the control limits – over control and adjustment too often) Shift in process level ( new workers, raw material, …) Trend ( continuous movement in one direction- wear out ) Stratification ( a tendency for the point to cluster artificially around the center line- e. g. Error in control limits computations )

6 -2. 6 THE OPERATING CHARACTERISTIC FUNCTION How well the and R charts can detect process shifts is described by operating characteristic (OC) curves. Consider a process whose mean has shifted from an incontrol value by k standard deviations. If the next sample after the shift plots in-control, then you will not detect the shift in the mean. The probability of this occurring is called the -risk.

6 -2. 6 THE OPERATING CHARACTERISTIC FUNCTION The probability of not detecting a shift in the process mean on the first sample is = P{LCL < Xbar < UCL = 1 = 0 + k } L= multiple of standard error in the control limits k = shift in process mean (# of standard deviations).

EXAMPLE We are using a 3 limit Xbar chart with sample size equal to 5 L =3 n=5 Determine the probability of detecting a shift to 1 = 0 + 2 on the first sample following the shift k =2

EXAMPLE, CONTINUED So, =F[3– 2 SQRT(5)]-F[-3– 2 SQRT(5)] = F(-1. 47) – F(-7. 37) =. 0708 P(not detecting on first sample) =. 0708 P(detecting on first sample) =1 -. 0708=. 9292

6 -2. 6 THE OPERATING CHARACTERISTIC FUNCTION The operating characteristic curves are plots of the value against k for various sample sizes If the shift is 1. 0σ and the sample size is n = 5, then β = 0. 75.

USE OF FIGURE 6 -13 Let k = 1. 5 When n = 5, =. 35 P(detection on 2 nd sample) =. 35(. 65) P(detection on 3 rd sample) =. 352(. 65) P(detection on rth sample) = r-1(1 - ) ARL = 1/(1 - ) 1/(1 -. 35) = 1/. 65 = 1. 54

OC CURVE FOR THE R CHART Use l = 1/ 0 ( the ratio of new to old process standard deviation ) in Fig. 6. 14 R chart is insensitive when n is small But, when n is large, W = R/s loses efficiency, and S chart is better

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6 -3. 1 CONSTRUCTION AND OPERATION OF AND S CHARTS S is an estimator of c 4 where c 4 is a constant depends on sample size n. The standard deviation of S is

6 -3. 1 CONSTRUCTION AND OPERATION OF AND S CHARTS If a standard is given the control limits for the S chart are: B 5, B 6, and c 4 are found in the Appendix VI for various values of n.

6 -3. 1 CONSTRUCTION AND OPERATION OF AND S CHARTS No Standard Given If is unknown, we can use an average sample standard deviation

6 -3. 1 CONSTRUCTION AND OPERATION OF AND S CHARTS Chart when Using S The upper and lower control limits for the where A 3 is found in the Appendix VI chart are given as

6 -3. 2 THE AND S CONTROL CHARTS WITH VARIABLE SAMPLE SIZE The and S charts can be adjusted to account for samples of various sizes. A “weighted” average is used in the calculations of the statistics. m = the number of samples selected. ni = size of the ith sample

6 -3. 2 THE AND S CONTROL CHARTS WITH VARIABLE SAMPLE SIZE The grand average can be estimated as: The average sample standard deviation is:

6 -3. 2 THE AND S CONTROL CHARTS WITH VARIABLE SAMPLE SIZE Control Limits

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS What if you could not get a sample size greater than 1 (n =1)? Examples include Automated inspection and measurement technology is used, and every unit manufactured is analyzed. The production rate is very slow, and it is inconvenient to allow samples sizes of n > 1 to accumulate before analysis The X and MR charts are useful for samples of sizes n = 1.

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS Moving Range Control Chart The moving range (MR) is defined as the absolute difference between two successive observations: MRi = |xi - xi-1| which will indicate possible shifts or changes in the process from one observation to the next.

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS X and Moving Range Charts The X chart is the plot of the individual observations. The control limits are where

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS X and Moving Range Charts The control limits on the moving range chart are:

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS Example Ten successive heats of a steel alloy are tested for hardness. The resulting data are Heat Hardness 1 52 6 52 2 51 7 50 3 54 8 51 4 55 9 58 5 50 10 51

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS Example

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS Interpretation of the Charts MR charts cannot be interpreted the same as R or charts. Since the MR chart plots data that are “correlated” with one another, then looking for patterns on the chart does not make sense. MR chart cannot really supply useful information about process variability.

6 -4. THE SHEWHART CONTROL CHART FOR INDIVIDUAL MEASUREMENTS The normality assumption is often taken for granted. When using the individuals chart, the normality assumption is very important to chart performance.

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