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Statistical Process Control
PowerPoint presentation to accompany Heizer and Render Operations Management, Eleventh EditionPrinciples of Operations Management, Ninth
Edition
PowerPoint slides by Jeff Heyl
66
© 2014 Pearson Education, Inc.
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Learning Objectives1. Apply quality management tools for
problem solving
2. Identify the importance of data in quality management
6S–6S–22
Introduction
Statistical Quality Control
Statistical Process Control (SPC)
Acceptance Sampling (AS)
Statistical process control is a statistical technique that is widely used to ensure that the process meets standards.
Acceptance sampling is used to determine acceptance or rejection of material evaluated by a sample.
6S–6S–33
Introduction
FiringPreparing
the clay for throwing
Wedging Throwing Pinching pots
Painting
Pottery Making Process
6S–6S–44
Introduction
6S–6S–55
Statistical Process Control Chart (SPC)
Variability is inherent in every process.
Natural variation – can not be eliminated Assignable variation -- Deviation that can be
traced to a specific reason: machine vibration, tool wear, new worker.
Variation
Natural Variation
Assignable Variation
6S–6S–66
Statistical Process Control Chart (SPC)
The essence of SPC is the application of statistical techniques to prevent, detect, and eliminate defective products or services by identifying assignable variation.
6S–6S–77
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
UCL
LCL
Sample number
Mean
Out ofcontrol
Natural variationdue to chance
Abnormal variationdue to assignable sources
Abnormal variationdue to assignable sources
A control chart is a time-ordered plot obtained from an ongoing process
6S–6S–88
Statistical Process Control Chart (SPC)
Statistical Process Control Chart (SPC)
Control Charts
Control Charts for Variable Data
Control Charts for Attribute Data
-charts (for controlling central tendency)x
R-charts (for controlling variation)
p-charts (for controlling percent defective)
c-charts (for controlling number of defects)
Attribute Data (discrete): qualitative characteristic or condition, such as pass/fail, good/bad, go/no go.
Variable Data (continuous): quantifiable conditions along a scale, such as speed, length, density, etc.
6S–6S–99
1. Take random samples
2. Calculate the upper control limit (UCL) and the lower control limit (LCL)
3. Plot UCL, LCL and the measured values
4. If all the measured values fall within the LCL and the UCL, then the process is assumed to be in control and no actions should be taken except continuing to monitor.
5. If one or more data points fall outside the control limits, then the process is assumed to be out of control and corrective actions need to be taken.
Statistical Process Control Chart (SPC)
6S-10
x-Charts
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor from Table control chart factor from Table S6.1(page 241) S6.1(page 241)
xx ==average of the sample meansaverage of the sample means
6S–6S–1111
Range=18-13=5
Hour 1Hour 1
BoxBox Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1717
22 1313
33 1616
44 1818
55 1717
66 1616
77 1515
88 1717
99 1616
x-Charts
6S–6S–1212
Range=17-14=3
Hour 2Hour 2
BoxBox Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1414
22 1616
33 1515
44 1414
55 1717
66 1515
77 1515
88 1414
99 1717
RR ==(5+3)/2 = 4(5+3)/2 = 4
x-Charts
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor from Table control chart factor from Table S6.1 (page241) S6.1 (page241)
xx ==average of the sample meansaverage of the sample means
6S–6S–1313
Average=(17+13+…+16)/9=16.11
Hour 1Hour 1
BoxBox Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1717
22 1313
33 1616
44 1818
55 1717
66 1616
77 1515
88 1717
99 1616
x-Charts
6S–6S–1414
Average=(14+16+…+17)/9=15.22
Hour 2Hour 2
BoxBox Weight ofWeight ofNumberNumber Oat FlakesOat Flakes
11 1414
22 1616
33 1515
44 1414
55 1717
66 1515
77 1515
88 1414
99 1717
xx ==(16.11+15.22)/2 = 15.665(16.11+15.22)/2 = 15.665
x-Charts
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor from Table control chart factor from Table S6.1 (page241) S6.1 (page241)
xx ==average of the sample meansaverage of the sample means
6S–6S–1515
x-Charts Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.881.88 3.273.27 00
33 1.021.02 2.582.58 00
44 .73.73 2.282.28 00
55 .58.58 2.122.12 00
66 .48.48 2.002.00 00
77 .42.42 1.921.92 0.080.08
88 .37.37 1.861.86 0.140.14
99 .34.34 1.821.82 0.180.18
1010 .31.31 1.781.78 0.220.22
1111 .29.29 1.741.74 0.260.26
6S–6S–1616
x-Charts
Lower control limit Lower control limit (LCL)(LCL) = x - A = x - A22RR
Upper control limit Upper control limit (UCL)(UCL) = x + A = x + A22RR
wherewhere RR ==average range of the samplesaverage range of the samples
AA22 ==control chart factor from Table control chart factor from Table S6.1 (page241) S6.1 (page241)
xx ==average of the sample meansaverage of the sample means
6S–6S–1717
x-Charts Example S6.1: Eight samples of seven tubes were taken at random intervals. Construct the x-chart with 3- control limit. Is the current process under statistical control? Why or why not? Should any actions be taken?
Sample size = n = 7
A2 = ?
Sample number Mean Range
1 6.36 0.16 2 6.38 0.18 3 6.35 0.17 4 6.40 0.20 5 6.32 0.15 6 6.34 0.16 7 6.39 0.16 8 6.34 0.18
6S–6S–1818
x-Charts
6S–6S–1919
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.881.88 3.273.27 00
33 1.021.02 2.582.58 00
44 .73.73 2.282.28 00
55 .58.58 2.122.12 00
66 .48.48 2.002.00 00
77 .42.42 1.921.92 0.080.08
88 .37.37 1.861.86 0.140.14
99 .34.34 1.821.82 0.180.18
1010 .31.31 1.781.78 0.220.22
1111 .29.29 1.741.74 0.260.26
x-Charts
oz 29.6)17.0(42.036.62 RAxLCL
oz 36.68
34.6...38.636.6
x
oz 43.6)17.0(42.036.62 RAxUCL
oz 17.08
18.0...18.016.0
R
Example S6.1: Eight samples of seven tubes were taken at random intervals. Construct the x-chart with 3- control limit. Is the current process under statistical control? Why or why not? Should any actions be taken?
A2 = 0.42
Sample number Mean Range
1 6.36 0.16 2 6.38 0.18 3 6.35 0.17 4 6.40 0.20 5 6.32 0.15 6 6.34 0.16 7 6.39 0.16 8 6.34 0.18
6S–6S–2020
x-Charts Control Chart Control Chart for sample of for sample of 7 tubes7 tubes
6.43 = UCL6.43 = UCL
6.29 = LCL6.29 = LCL
6.36 = Mean6.36 = Mean
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
It is assumed that the central tendency of process is in control with 99.73% confidence. No actions need to be taken except to continuously monitor this process.
6S–6S–2121
Steps in Creating Charts
1. Take samples from the population and compute the appropriate sample statistic
2. Use the sample statistic to calculate control limits
3. Plot control limits and measured values
4. Determine the state of the process (in or out of control)
5. Investigate possible assignable causes and take actions
6S–6S–2222
R-Charts
Lower control limit Lower control limit (LCL)(LCL) = D = D33RR
Upper control limit Upper control limit (UCL)(UCL) = D = D44RR
wherewhere
RR ==average range of the samplesaverage range of the samples
DD33 and D and D44==control chart factors from control chart factors from Table S6.1 (Page 241) Table S6.1 (Page 241)
6S–6S–2323
R-Charts
6S–6S–2424
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.881.88 3.273.27 00
33 1.021.02 2.582.58 00
44 .73.73 2.282.28 00
55 .58.58 2.122.12 00
66 .48.48 2.002.00 00
77 .42.42 1.921.92 0.080.08
88 .37.37 1.861.86 0.140.14
99 .34.34 1.821.82 0.180.18
1010 .31.31 1.781.78 0.220.22
1111 .29.29 1.741.74 0.260.26
R-Charts
Average range R Average range R = 5.3 = 5.3 poundspoundsSample size n Sample size n = 5= 5From From Table S6.1Table S6.1 D D44 = ? = ? DD33 = ? = ?
Example S6.2Example S6.2
6S–6S–2525
R-Charts
6S–6S–2626
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.881.88 3.273.27 00
33 1.021.02 2.582.58 00
44 .73.73 2.282.28 00
55 .58.58 2.122.12 00
66 .48.48 2.002.00 00
77 .42.42 1.921.92 0.080.08
88 .37.37 1.861.86 0.140.14
99 .34.34 1.821.82 0.180.18
1010 .31.31 1.781.78 0.220.22
1111 .29.29 1.741.74 0.260.26
R-Charts
UCLUCLRR = D= D44RR
= (2.12)(5.3)= (2.12)(5.3)= 11.2 = 11.2 poundspounds
LCLLCLRR = D= D33RR
= (0)(5.3)= (0)(5.3)= 0 = 0 poundspounds
Average range R Average range R = 5.3 = 5.3 poundspoundsSample size n Sample size n = 5= 5From From Table S6.1Table S6.1 D D44 = 2.12, = 2.12, DD33 = 0 = 0
UCL = 11.2UCL = 11.2
Mean = 5.3Mean = 5.3
LCL = 0LCL = 0
Example S6.2Example S6.2
6S–6S–2727
R-Charts
oz 17.08
18.0...18.016.0
R
n=7
Example S6.3: Refer to Example S6.1. Eight samples of seven tubes were taken at random intervals. Construct the R-chart with 3- control limits. Is the current process under statistical control? Why or why not? Should any actions be taken?
D3 =? D4 = ?
Sample number Mean Range
1 6.36 0.16 2 6.38 0.18 3 6.35 0.17 4 6.40 0.20 5 6.32 0.15 6 6.34 0.16 7 6.39 0.16 8 6.34 0.18
6S–6S–2828
R-Charts
6S–6S–2929
Sample Size Sample Size Mean Factor Mean Factor Upper Range Upper Range Lower Lower RangeRange
n n AA22 DD44 DD3322 1.881.88 3.273.27 00
33 1.021.02 2.582.58 00
44 .73.73 2.282.28 00
55 .58.58 2.122.12 00
66 .48.48 2.002.00 00
77 .42.42 1.921.92 0.080.08
88 .37.37 1.861.86 0.140.14
99 .34.34 1.821.82 0.180.18
1010 .31.31 1.781.78 0.220.22
1111 .29.29 1.741.74 0.260.26
R-Charts
oz 17.08
18.0...18.016.0
R
Sample number Mean Range
1 6.36 0.16 2 6.38 0.18 3 6.35 0.17 4 6.40 0.20 5 6.32 0.15 6 6.34 0.16 7 6.39 0.16 8 6.34 0.18
6S–6S–3030
Example S6.3: Refer to Example S6.1. Eight samples of seven tubes were taken at random intervals. Construct the R-chart with 3- control limits. Is the current process under statistical control? Why or why not? Should any actions be taken?
oz 01.0)17.0(08.03 RDLCL
oz 33.0)17.0(92.14 RDUCL
D3 =0.08, D4 = 1.92
R-ChartsControl Chart Control Chart for sample of for sample of 7 tubes7 tubes
0.33 = UCL0.33 = UCL
0.01 = LCL0.01 = LCL
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
0.17 = R0.17 = R
The variation of process is in control with 99.73% confidence.
6S–6S–3131
Mean and Range Charts
R-chartR-chart(R-chart detects (R-chart detects increase in increase in dispersion)dispersion)
UCLUCL
LCLLCL
(a) The central tendency of process is in control, but its variation is not in control.
x-chartx-chart(x-chart does not (x-chart does not detect dispersion)detect dispersion)
UCLUCL
LCLLCL
6S–6S–3232
Mean and Range Charts(b) The variation of process is in control, but its central tendency is not in control.
R-chartR-chart(R-chart does not (R-chart does not detect changes in detect changes in mean)mean)
UCLUCL
LCLLCL
x-chartx-chart(x-chart detects (x-chart detects shift in central shift in central tendency)tendency)
UCLUCL
LCLLCL
6S–6S–3333
R-Chart and X-ChartExample S6.4: Seven random samples of four resistors each are taken to establish the quality standards. Develop the R-chart and the x-chart both with 3- control limits for the production process. Is the entire process under statistical control? Why or why not?
Sample Number Sample Range Sample Mean
1 3 100.5 2 2 101.5 3 4 100.0 4 1 99.5 5 2 99.0 6 5 97.0 7 4 101.0
D3 = 0, and D4 = 2.28
n = 4
R = (3 + 2 + … + 4)/7 = 3.0
0)0.3(03 RDLCL
6.84)0.3(28.24 RDUCL
6S–6S–3434
R-Chart and X-ChartControl Chart Control Chart for sample of for sample of 4 resistors4 resistors
6.84 = UCL6.84 = UCL
0 = LCL0 = LCL
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
3.0 = R3.0 = R
The variation of process is in control with 99.73% confidence.
6S–6S–3535
R-Chart and X-Chart
X= (100.5 + 101.5 + … + 101.0)/7 99.79
Sample Number Sample Range Sample Mean
1 3 100.5 2 2 101.5 3 4 100.0 4 1 99.5 5 2 99.0 6 5 97.0 7 4 101.0
n = 4, A2 = 0.73
R = (3 + 2 + … + 4)/7 = 3.0
97.6)0.3(73.079.992 RAxLCL
101.98)0.3(73.079.992 RAxUCL
6S–6S–3636
R-Chart and X-ChartsControl ChartControl Chart
101.98 = UCL101.98 = UCL
97.6 = LCL97.6 = LCL
99.79 = Mean99.79 = Mean
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
The central tendency of process is not in control with 99.73% confidence.
In conclusion, with 99.7% confidence, the entire resistor production process is not in control since its central tendency is out of control although its variation is under control.
6S–6S–3737
EX 1 in classA part that connects two levels should have a distance between the two holes of 4”. It has been determined that x-bar chart and R-chart should be set up to determine if the process is in statistical control. The following ten samples of size four were collected. Calculate the control limits, plot the control charts, and determine if the process is in control
No. of Sample Mean Range
1 4.01 0.04
2 3.98 0.06
3 4.00 0.02
4 3.99 0.05
5 4.00 0.06
6 3.97 0.02
7 4.02 0.02
8 3.99 0.04
9 3.98 0.05
10 4.01 0.066S–6S–3838
R-Chart and X-Chart
6S–6S–3939
Example S6.5: Resistors for electronic circuits are manufactured at Omega Corporation in Denton, TX. The head of the firm’s Continuous Improvement Division is concerned about the product quality and sets up production line checks. She takes seven random samples of four resistors each to establish the quality standards. Develop the R-chart and the chart both with 3- control limits for the production process. Is the entire process under statistical control? Why or why not?
# of sample Readings of Resistance (ohms)
1 99 100 102 101
2 101 103 101 101
3 98 102 101 99
4 99 100 99 100
5 99 99 98 100
6 95 100 97 96
7 101 99 101 103
R-Chart and X-Chart# of Sample 1 2 3 4 5 6 7
Sample range 3 2 4 1 2 5 4
Sample mean 100.5 101.5 100.0 99.5 99.0 97.0 101.0
3.07
4...23
R n=4 D3 =0 D4 = 2.28
03 RDLCL
84.60.328.24 RDUCL
6.84 = UCL6.84 = UCL
0 = LCL0 = LCL
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
3.0 = R3.0 = Rvariation of process is in control with 99.73% confidence.
6S–6S–4040
R-Chart and X-Chart# of Sample 1 2 3 4 5 6 7
Sample range 3 2 4 1 2 5 4
Sample mean 100.5 101.5 100.0 99.5 99.0 97.0 101.0
3.07
4...23
R
n=4
102.0 = UCL102.0 = UCL
97.6 = LCL97.6 = LCL
Sample numberSample number
|| || || || || || || || || || || ||11 22 33 44 55 66 77 88 99 1010 1111 1212
99.8 = X99.8 = X
central tendency of process is not in control with 99.73% confidence.
Thus, entire process is not in control.
6S–6S–4141
A2 =0.73
X= (100.5 + … + 101.0)/7 99.8
97.6)0.3(73.08.992 RAxLCL
102.0)0.3(73.08.992 RAxUCL
EX 2 in classA quality analyst wants to construct a sample mean chart for controlling a packaging process. Each day last week, he randomly selected four packages and weighed each. The data from that activity appears below. Set up control charts to determine if the process is in statistical control
Day Package 1 Package 2 Package 3 Package 4
Monday 23 22 23 24
Tuesday 23 21 19 21
Wednesday 20 19 20 21
Thursday 18 19 20 19
Friday 18 20 22 20
6S–6S–4242
Statistical Process Control Chart (SPC)
Control Charts
Control Charts for Variable Data
Control Charts for Attribute Data
-charts (for controlling central tendency)x
R-charts (for controlling variation)
p-charts (for controlling percent defective)
c-charts (for controlling number of defects)
Attribute Data (discrete): qualitative characteristic or condition, such as pass/fail, good/bad, go/no go.
Variable Data (continuous): quantifiable conditions along a scale, such as speed, length, density, etc.
6S–6S–4343
Control Charts for Attribute Data Categorical variables
Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure
Percentage of defects (p-chart) Number of defects (c-chart)
6S–6S–4444
P-Charts
n
ppzpzpUCL p
)1(ˆ
wherewhere pp ==mean percent defective overall the mean percent defective overall the samplessampleszz ==number of standard deviations = 3number of standard deviations = 3nn ==sample sizesample size
n
ppzpzpLCL p
)1(ˆ
6S–6S–4545
P-Charts
SampleSample NumberNumber PercentPercent SampleSample NumberNumber PercentPercentNumberNumber of Errorsof Errors DefectiveDefective NumberNumber of Errorsof Errors DefectiveDefective
11 66 .06.06 1111 66 .06.0622 55 .05.05 1212 11 .01.0133 00 .00.00 1313 88 .08.0844 11 .01.01 1414 77 .07.0755 44 .04.04 1515 55 .05.0566 22 .02.02 1616 44 .04.0477 55 .05.05 1717 1111 .11.1188 33 .03.03 1818 33 .03.0399 33 .03.03 1919 00 .00.00
1010 22 .02.02 2020 44 .04.04
Total Total = 80= 80
Example S6.6: Data-entry clerks at ARCO key in thousands of insurance records each day. One hundred records entered by each clerk were carefully examined and the number of errors counted. Develop a p-chart with 3- control limits and determine if the process is in control.
6S–6S–4646
P-Charts
040.0)20)(100(
80
examined records ofnumber Total
errors ofnumber Totalp
099.0100
)04.01(04.0304.0
)1(
n
ppzpUCL
0019.0100
)04.01(04.0304.0
)1(
n
ppzpLCL
n = 100
Because we cannot have a negative percent defective
040.020
04.0...05.006.0
samples ofNumber
defectivefraction Total ,
por
6S–6S–4747
P-Charts.11 .11 –.10 .10 –.09 .09 –.08 .08 –.07 .07 –.06 .06 –.05 .05 –.04 .04 –.03 .03 –.02 .02 –.01 .01 –.00 .00 –
Sample numberSample number
Per
cen
t d
efec
tive
Per
cen
t d
efec
tive
| | | | | | | | | |
22 44 66 88 1010 1212 1414 1616 1818 2020
UCL= 0.10UCL= 0.10
LCL= 0.00LCL= 0.00
p p = 0.04= 0.04
Possible assignable
causes present
Possible good assignable causes present
The process is not in control with 99.73% confidence.
6S–6S–4848
C-Charts A c-chart is used when the quality cannot be
measured as a percentage. Number of car accidents per month at a particular
intersection Number of complaints the service center of a hotel
receives per week Number of scratches on a nameplate Number of dimples found on a metal sheet
6S–6S–4949
C-Charts
wherewhere cc ==mean number defective overall the samplesmean number defective overall the samples
UCLUCL = c + = c + 33 c c LCLLCL = c - = c - 33 c c
6S–6S–5050
C-Charts
c c = 54 / 9= 54 / 9 = 6 = 6 complaints complaints //weekweek
|1
|2
|3
|4
|5
|6
|7
|8
|9
Week NumberWeek Number
Nu
mb
er o
f d
efec
tN
um
ber
of
def
ect14 14 –
12 12 –
10 10 –
8 8 –
6 6 –
4 –
2 –
0 0 –
UCLUCL = 13.35= 13.35
LCLLCL = 0= 0
c c = 6= 6
Example S6.7: Over 9 weeks, Red Top Cab company received the following numbers of calls from irate passengers: 3, 0, 8, 9, 6, 7, 4, 9, 8, for a total of 54 complaints. Determine the 3- control limits of a c-Chart.
Because we cannot have the negative number of defective records
UCLUCL = c + = c + 33 c c= 6 + 3 6= 6 + 3 6= 13.35= 13.35
LCLLCL = c - = c - 33 c c= 6 - 3 6= 6 - 3 6= - 1.35 => 0= - 1.35 => 0
The process is in control with 99.73% confidence.
6S–6S–5151
1. Effective quality management is data driven
2. There are multiple tools to identify and prioritize process problems
3. There are multiple tools to identify the relationships between variables
Managing Quality Summary
6S–6S–5252
Recommended