Chapter Three Statistical Process Control (SPC)

CHAPTER TWO - THEORY OF CONTROL CHARTS13Methods And Philosophy of Statistical Process ControlProcess And Measurement System Capability Analysis2Control Charts for VariablesControl Charts for AttributesThis is another option for an Overview slide.

Control Charts for AttributesMany quality characteristics cannot be conveniently represented numerically. In such cases, each item inspected is classified as either conforming or nonconforming to the specifications on that quality characteristic. Quality characteristics of this type are called attributes.

Types of attribute control Chartsp chartsThis chart shows the fraction of nonconforming or defective product produced by a manufacturing process. It is also called the control chart for fraction nonconforming. np chartsThis chart shows the number of nonconforming. Almost the same as the p chart.c chartsThis shows the number of defects or nonconformities produced by a manufacturing process.u chartsThis chart shows the nonconformities per unit produced by a manufacturing process.

Dr.Gere K2P-ChartIn this chart, we plot the percent of defectives (per batch, per day, per machine, etc.). However, the control limits in this chart are not based on the distribution of rate events but rather on the binomial distribution (of proportions).

P-Construction for constant subgroup size

Select the quality characteristics.Determine the subgroup size and methodCollect the data.Calculate the trial central line and control limits.Establish the revised central line and control limits.Achieve the objective.

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Fraction Nonconforming Control Chart: Standard Given

Dr.Gere K4 And when P- unknown

Where

The number of defective chart (np -chart) is an alternative chart that may be substituted for the P-chart. The underlying distribution for the np-chart is the binomial. In the case of the np-chart, the sample size must be constant. Rather than calculating the proportion of defective items in a sample for plotting on a P-chart, the np-chart allows the actual number of defective units to be plotted directly. This eliminates the need for one calculation (p), thereby decreasing the probability of an error. The np- chart also is somewhat easier for production operators to understand.

Dr.Gere K5 But the possibility of confusion from having two types of charts might out weight the slight advantage of the np-chart and it would be better to use the p-chart even for constant subgroup size.

Dr.Gere K6Control limits for the np-chart are calculated as followsDr.Gere K7Frozen orange juice concentrate is packed in 6-oz cardboard cans. These cans are formed on a machine by spinning them from cardboard stock and attaching a metal bottom panel. By inspection of a can, we may determine whether, when filled, it could possibly leak either on the side seam or around the bottom joint. Such a nonconforming can has an improper seal on either the side seam or the bottom panel. Set up a control chart to improve the fraction of nonconforming cans produced by this machine.Example 30 samples of n = 50 cans each were selected at half-hour intervals over a three-shift period in which the machine was in continuous operation. The data are shown in the next TableDr.Gere K8

Dr.Gere K9SolutionCalculating fraction of defectives/ nonconforming where n = 50

the average of these individual sample fractions nonconforming is

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Dr.Gere K11calculate the upper and lower control limits as

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The sample fraction nonconforming from each preliminary sample is plotted on this chart. We note that two points those from samples 15 and 23, plot above the upper control limit, so the process is not in control.Dr.Gere K13Analysis of the data from sample 15 indicates that a new batch of cardboard stock was put into production during that half-hour period.The introduction of new batches of raw material sometimes causes irregular production performance, and it is reasonable to believe that this has occurred here. Furthermore, during the half-hour period in which sample 23 was obtained, a relatively inexperienced operator had been temporarily assigned to the machine, and this could account for the high fraction nonconforming obtained from that sample. Consequently, samples 15 and 23 are eliminated, and the new center line and revised control limits are calculated as

But the possibility of confusion from having two types of charts might out weight the slight advantage of the np-chart and it would be better to use the p-chart even for constant subgroup size.

Dr.Gere K14Control limits for the np-chart are calculated as followsDr.Gere K15Control charts for nonconformitiesControl charts can be developed for either the total number or the average number of nonconformities in a unit. These control charts are based on the Poisson distribution under the assumption that the number of opportunities for a nonconformity is large and the probability of occurrence of a nonconformity is small and constant, with the inspection unit being the same for each sample.

C-Chart: The vertical axis of C-Charts represents the number of defectives per unit or per day etc while the X-axis represents the number of samples. These charts are constructed on the assumption that quality attribute defects are very rare.

Dr.Gere K16C chart for number of nonconformities

Given a standard value of C andCi, is the number of nonconformities in the sample and ni is the sample size of the ith sampleU-Chart: In the U-Chart the vertical axis represents the rate of defectives (number defectives divided by number of samples) and the X-axis the number of samples. Unlike C-Chart, U-chart can be used for different sample sizes. U-Charts are constructed on the assumption that the numbers of defectives are rare, and hence control limits are based on the Poisson distribution.

Dr.Gere K17U chart for number of nonconformities

When no standard is given

U chart for number of nonconformities per unit

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20Advantages of Variable C ChartAdvantages of Attribute C ChartMore sensitive than attribute control charts.Therefore, variable control charts may alert us to quality problems before any actual "unacceptables" (as detected by the attribute chart) will occur. Montgomery (1985) calls the variable control charts leading indicators of trouble that will sound an alarm before the number of rejects (scrap) increases in the production process.

Allowing for quick summaries, that is, the engineer may simply classify products as acceptable or unacceptable, based on various quality criteria. Thus, attribute charts sometimes bypass the need for expensive, precise devices and time-consuming measurement procedures. More easily understood by managers unfamiliar with quality control procedures.

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