Unit IV

Quality Control
Methods
 Introduction
• Quality characteristics of manufactured products have received much
attention from design engineers and production personnel as well as from
those concerned with financial management.

• In a manufacturing context, no matter how carefully machines are

calibrated, environmental factors are controlled, materials and other inputs
are monitored, and workers are trained, diameter will vary from bolt to
bolt, some plastic sheets will be stronger than others, some circuit boards
will be defective whereas others are not, and so on.

• We might think of such natural random variation as uncontrollable

background noise.
 Introduction
• There are, however, other sources of variation that may have a fatal
impact on the quality of items produced by some process.

• Such variation may be attributable to contaminated material,

incorrect machine settings, unusual tool wear, and the like.

• These sources of variation have been termed assignable causes.

• “Control charts” provide a mechanism for recognizing situations
where assignable causes may be adversely affecting product quality.

• Once a chart indicates an out-of-control situation, an investigation

can be launched to identify causes and take corrective action.
 Introduction
• A basic element of control charting is that samples have
been selected from the process of interest at a sequence of
time points. Depending on the aspect of the process under
investigation, some statistic, such as the sample mean or
sample proportion of defective items, is chosen.

• The value of this statistic is then calculated for each sample

in turn.

• A traditional control chart then results from plotting these

calculated values over time.
 Introduction
• The chart has a center line and two control limits (See the figure below).

The basis for the choice of a center line is sometimes a target value or

design specification.
 Introduction
• If the points on the chart all lie between the two control limits, the
process is deemed to be in-control. That is, the process is believed
to be operating in a stable fashion reflecting only natural random
variation.
• An out-of-control signal occurs whenever a plotted point falls
outside the limits. This is assumed to be attributable to some
assignable cause, and a search for such causes commences.

• The limits are designed so that an in-control process generates very

few false alarms, whereas a process not in control quickly gives rise
to a point outside the limits.
 𝑿-Chart
𝑿-Chart or simply X-Chart is a control chart which is used to monitor the mean of a
process. With this chart, we can easily identify trends or patterns which helps in
maintaining product quality.

Its components are:

1. Centerline (CL): The overall process mean (𝑿)

2. Upper Control Limit (UCL): The maximum acceptable process mean

3. Lower Control Limit (LCL): The minimum acceptable process mean

The control limits are calculated as follows:

𝑈𝐶𝐿 = X + A2 𝑅 and 𝐿𝐶𝐿 = X − A2 𝑅

Where,

X: Average of sample means, A2 : Factor based on sample size n and is obtained using
control chart table, 𝑅: Average range of the samples
 Example

Example: The following table gives data on

moisture content for specimens of a certain type

of fabric. Calculate the control limits, construct

the 𝑿-control chart, and determine if the process

is in-control or out-of-control.
Example
 Example
Sol:
• We start by calculating the sample means (𝑋𝑖 ) 𝑖 = 1,2, … , 8. By calculations
𝑋1 = 12.72,…, 𝑋8 = 13.18, and the overall mean 𝑋 = 12.92.
• Then we calculate the ranges (𝑅𝑖) and the overall range (𝑅). Here, 𝑅1 = 1.2
,…,𝑅8 = 1.3.
The average of 𝑅𝑖 gives 𝑅 = 1.425.
• For our example, 𝑛 = 5, by looking at the control chart tables, 𝐴2 = 0.577.
Then 𝑈𝐶𝐿 = X + A2 𝑅 = 12.92 + 0.577 ∗ 1.425 = 13.74 and
𝐿𝐶𝐿 = X − A2𝑅 = 12.92 − 0.577 ∗ 1.425 = 12.10
X-chart pictured in the next slide shows that all points lie in the limits. Thus the
process is in-control.
 14
Example

13.5

13

Sample Means
12.5 LCL
CL
UCL

12

11.5

11
1 2 3 4 5 6 7 8
 R-Chart
Range chart or simply R-Chart is a control chart which is used to
monitor the process variability by tracking the range of sample
data. With this chart, we can identify the trends in the dispersion
of measurements.

The components of R-Chart are:

Lower Control Limit (LCL): Minimum allowable process range

Centerline (CL): The average range (R)

Upper Control Limit (UCL): Maximum allowable process range

 R-Chart
The control limits are calculated as follows:
𝐿𝐶𝐿 = 𝐷3 𝑅
𝑈𝐶𝐿 = 𝐷4 𝑅
Where,
𝑅: Average range of the samples
𝐷3 , 𝐷4 : Control chart factors based on the sample size which can be
obtained using control chart tables.
Example: By using the previous example’s data, calculate limits for R-
Chart, construct the R-chart and check if the process is out-of-control
or in-control.
 R-Chart
Sol:

The average of ranges (R)= 1.425 , and by using the control chart
table, for n=5, 𝐷3 = 0 and 𝐷4 = 2.114. Then,
𝐿𝐶𝐿 = 𝐷3 𝑅 = 0 ∗ 1.425 = 0
𝑈𝐶𝐿 = 𝐷4 𝑅 = 2.114 ∗ 1.425 = 3.01245
𝐶𝐿 = 𝑅 = 1.425

By using the R-chart in the next slide, we can conclude that the
process is in control since no point is out of the limits.
 R-Chart
3.5

2.5

2 Ranges (Ri)
LCL
CL
1.5
UCL

0.5

0
1 2 3 4 5 6 7 8
 p-Chart
The p-chart or proportion control chart is used to detect changes in the
proportion of defective items in a process.
The components of the p-chart are calculated as follows:
𝑝𝑖
• Centerline (CL): The average proportion of defects 𝑝 = 𝑘
, where
k is the total number of samples and 𝑝𝑖 is the proportion defective for
𝑑𝑖
each sample 𝑝𝑖 = 𝑛

𝑝 1−𝑝 𝑝 1−𝑝
• 𝐿𝐶𝐿 = 𝑝 − 3 and 𝑈𝐶𝐿 = 𝑝 + 3 ,
𝑛 𝑛

Where, n is the sample size.

• If 𝐿𝐶𝐿 is negative, it is set to zero.
 p-Chart
Example: A sample of 100 cups from a particular
dinnerware pattern was selected on each of 25
successive days, and each sample was examined for
defects. The resulting numbers of unacceptable cups
are summarized in the table in next slide. Calculate
limits for p-chart, construct the chart and check if
the process is out-of-control or in-control.
 p-Chart
Day (i) 1 2 3 4 5 6 7 8
Number of 7 4 3 6 4 9 6 7
unacceptable
cups

Day (i) 9 10 11 12 13 14 15 16
Number of 5 3 7 8 4 6 2 9
unacceptable
cups

Day (i) 17 18 19 20 21 22 23 24 25
Number of 7 6 7 11 6 7 4 8 6
unacceptable
cups
 p-chart
Sol:
1.52
𝑝= = 0.0608
25
𝑝 1− 𝑝 0.0608 1 − 0.0608
𝐿𝐶𝐿 = 𝑝 − 3 = 0.0608 − 3
𝑛 100
= −0.0109
𝑝 1− 𝑝 0.0608(1 − 0.0608)
𝑈𝐶𝐿 = 𝑝 + 3 = 0.0608 + 3
𝑛 100
= 0.1325
Since the 𝐿𝐶𝐿 is negative, it is therefore set to zero.
The p-chart is pictured in the next slide and indicates that the process is
in-control as no point is beyond the control limits.
 p-chart
0.14

0.12

0.1

0.08
pi
LCL
Mean p
0.06
UCL

0.04

0.02

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
 c-chart
The count control chart or simply c-chart is used to
identify changes that affect defect counts. Unlike for p-
chart which deals with the proportion of defects, c-chart
deals with the number of defects in a production process.
Its components are calculated as follows:
Centerline (CL): The average number of defects (𝑐 )

𝐿𝐶𝐿 = 𝑐 − 3 𝑐 and 𝑈𝐶𝐿 = 𝑐 + 3 𝑐.

If 𝐿𝐶𝐿 is negative, it is set to zero
 c-chart
Example:
A company manufactures metal panels that are baked after first being
coated with a slurry of powdered ceramic. Flaws sometimes appear in
the finish of these panels, and the company wishes to establish a control
chart for the number of flaws. The number of flaws in each of the 24
panels sampled at regular time intervals are as follows:
7, 10, 9, 12, 13, 6, 13, 7, 5, 11, 8, 10, 13, 9, 21, 10, 6, 8, 3, 12, 7, 11, 14,
10.

Use c-chart to investigate if the number of flaws in process is in-control

or out-of-control.
 c-chart
Sol:
235
c = CL = = 9.79
24

𝐿𝐶𝐿 = 𝑐 − 3 𝑐 = 9.79 − 3 9.79 = 0.40

𝑈𝐶𝐿 = 𝑐 + 3 𝑐 = 9.79 + 3 9.79 = 19.18

By looking at the c-chart pictured in the next slide, we observe that the
point corresponding to 15th panel lies above the upper control limit.
Thus the process is out-of-control.
After getting this unusual result, an investigation should be done to
know its real cause. Probably, the slurry used on that panel is of
unusually low viscosity (which is an assignable cause).
 c-chart
25

20

15
Number of Flaws
LCL
CL

10 UCL

0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24