Chapter 1: Statistical Process Control

Statistical Process Control

The application of statistical techniques to determine whether a process is delivering what a customer wants. Control charts are primarily used to detect defects.

c. Indicators of out of control conditions

• A trend in the observations (the process is drifting). • A sudden or step change in the observations. • A run of five or more observations on the same side of the mean (If we flip a coin and get "heads" five times in a row, we become suspicious of the coin or of the coin flipping process.) • Several observations near the control limits (Normally only 1 in 20 observations are more than 2 standard deviations from the mean.) • One or more observations outside of the control limits

Variation of outputs

a) Performance measurements -Variables — service or product characteristics measured on a continuous scale ex; weight length ⇒ Advantage: if defective, we know by how much — the direction and magnitude of corrections are indicated. ⇒ Disadvantage: measurements typically involve special equipment, employee skills, exacting procedures, and time and effort.

e) Assignable causes

• Any cause of variation that can be identified and eliminated. • Change in the mean, spread, or shape of a process distribution is a symptom that an assignable cause of variation has developed. • After a process is in statistical control, SPC is used to detect significant change, indicating the need for corrective action.

Control charts

a. A control chart has a nominal value, or center line, which can be the process's historic average or a target that managers would like the process to achieve. b. A sample characteristic measured above the upper control limit (UCL) or below the lower control limit (LCL) indicates that an assignable cause probably exists. • Steps for using a control chart: ⇒ Take a random sample, measure the quality characteristic, and calculate a variable or attribute performance measure. ⇒ Plot the statistic; if it falls outside the control limits, look for assignable causes. ⇒ Eliminate the cause if it degrades performance. Incorporate the cause if it improves performance. Recalculate the control chart. ⇒ Periodically repeat the procedure.

Examples of process change that can be detected by SPC

a. A decrease in the average number of complaints per day at a hotel b. A sudden increase in the proportion of defective gear boxes c. An increase in the time to process a mortgage application d. A decline in the number of scrapped units at a milling machine e. An increase in the number of scrapped units at a milling machine

Control charts for variables used to monitor the mean and the variability of the process distribution.

a. R-Charts (also known as range charts) • Monitor process variability • First remove assignable causes of variation. • While process is in statistical control, collect data to estimate the average range of output that occurs. • To establish the upper and lower control limits for the Rchart, we use Table 3.1, which provides two factors; D3 and D4. These factors establish the UCLR and LCLR at three standard deviations above and below .

Control charts for attributes

a. p-chart — used for controlling the proportion defective generated by the process • Sampling for a p-chart involves a yes or no decision, based on the binomial distribution • Take a random sample of n units. • Count the number of defectives. • Proportion defective = number of defectives ÷ sample size • Plot sample proportion defective on a chart. If it is outside the range between the upper and lower control limits, search for an assignable cause. If a cause is found, do not use these data to determine the control limits. • Two things to note: ⇒ The lower control limit cannot be negative ⇒ When the number of defects is less than the LCL, then the system is out of control in a good way. We want to find the assignable cause. Find what was unique about this event that caused things to work out so well.

a) Performance measurements

• Attributes — a characteristic counted in discrete units, (yes-no, integer number) that could be quickly counted. ex; erroneous forms ⇒ Used to determine conformance to complex specifications, or when measuring variables is too costly. ⇒ Advantages: ° Quickly reveals when quality has changed, provides an integer number of how many are defective. ° Requires less effort, and fewer resources than measuring variables. ⇒ Disadvantages: ° Doesn't show by how much they were defective, the direction and magnitude of corrections are not indicated. ° Requires more observations, since each observation provides little information.

b) Sampling

• Complete inspection ⇒ Used when ° Costs of failure are high relative to costs of inspection. ° Inspection is automated • Sampling plans ⇒ Used when ° Inspection costs are high ° Inspection destroys the product ⇒ Sampling plans include ° Sample size, n random observations ° Time between successive samples ° Decision rules that determine when action should be taken

d) Common causes

• Random, or unavoidable sources of variation within a process. • Characteristics of distributions ⇒ Location—measured by the mean of the distribution ⇒ Spread—measured by the range or standard deviation ⇒ Shape—whether the observations are symmetrical or skewed

b. c-chart — used for controlling the number of defects when more than one defect can be present in a service or product

• Take a random sample of one. • Inspect the quality attribute . • Count the number of defects. • Plot the number of defectives on a chart. If it is outside the range between the upper and lower control limits, search for the assignable cause. If a cause is found do not use these data to determine the control limits. ⇒ The Poisson distribution mean and standard deviation are both described using the same number, . The mean equals , and the standard deviation equals . ⇒ We set upper and lower control limits in a manner similar to pcharts.

c) Sampling distributions

• The distribution of sample means can be approximated by the normal distribution (reference the central limit theorem described in statistics texts). • sample range • standard deviation

d) Two types of error are possible with the use of control charts

• Type I • Type II

b) x Charts

• Used to see whether the process is generating an output, on average, consistent with a target value set by management or with past performance. • The process average is plotted on the - chart after the process variability is in control. • The upper and lower control limits can be established in two ways. • If the standard deviation of the process distribution is known, we could place UCL and LCL at "z" standard deviations away from the mean, depending on the desired confidence level. Or we could use Table 3.1 to find A2, which when multiplied by the previously determined , places UCL and LCL three standard deviations above and below the mean.

Using x- and R-charts to monitor a process

⇒ Construct the R-chart. ⇒ Compute the range for each sample. ⇒ Plot the ranges on the R-chart. If process is not in statistical control, find the assignable causes and repeat the construction of the R-chart. ⇒ Construct -chart. ⇒ Compute the mean for each sample. ⇒ Plot the sample means on the -chart. If all sample means are within the control limits, the process is in statistical control in terms of the process average. If the process is not in statistical control, find the assignable cause and repeat the construction of the -chart.