Operations Management. Supplement 6 – Statistical Process Control. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e . Statistical Process Control (SPC). Variability is inherent in every process
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Supplement 6 –
Statistical Process Control
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 7e
Operations Management, 9e
Each of these represents one sample of five boxes of cereal
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Frequency
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Weight
SamplesTo measure the process, we take samples and analyze the sample statistics following these steps
(a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight
Figure S6.1
The solid line represents the distribution
Frequency
Weight
SamplesTo measure the process, we take samples and analyze the sample statistics following these steps
(b) After enough samples are taken from a stable process, they form a pattern called a distribution
Figure S6.1
Variation
Shape
Frequency
Weight
Weight
Weight
SamplesTo measure the process, we take samples and analyze the sample statistics following these steps
(c) There are many types of distributions, including the normal (bellshaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape
Figure S6.1
Frequency
Time
Weight
SamplesTo measure the process, we take samples and analyze the sample statistics following these steps
(d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable
Figure S6.1
Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes
(a) In statistical control and capable of producing within control limits
Frequency
Upper control limit
Lower control limit
(b) In statistical control but not capable of producing within control limits
(c) Out of control
Size
(weight, length, speed, etc.)
Process ControlFigure S6.2
Variables
Attributes
The mean of the sampling distribution (x) will be the same as the population mean m
x = m
s
n
The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n
sx =
Central Limit TheoremRegardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve
Three population distributions
Mean of sample means = x
Beta
Standard deviation of the sample means
s
n
Normal
= sx =
Uniform
      
3sx 2sx 1sx x +1sx +2sx +3sx
95.45% fall within ± 2sx
99.73% of all x
fall within ± 3sx
Population and Sampling DistributionsDistribution of sample means
Figure S6.3
For variables that have continuous dimensions
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x  zsx
where x = mean of the sample means or a target value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size
Setting Chart LimitsSample Weight of Number Oat Flakes
1 17
2 13
3 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
s = 1
Hour Mean Hour Mean
1 16.1 7 15.2
2 16.8 8 16.4
3 15.5 9 16.3
4 16.5 10 14.8
5 16.5 11 14.2
6 16.4 12 17.3
n = 9
UCLx = x + zsx= 16 + 3(1/3) = 17 ozs
LCLx = x  zsx = 16  3(1/3) = 15 ozs
Setting Control LimitsFor 99.73% control limits, z = 3
Variation due to assignable causes
Out of control
17 = UCL
Variation due to natural causes
16 = Mean
15 = LCL
Variation due to assignable causes
           
1 2 3 4 5 6 7 8 9 10 11 12
Out of control
Sample number
Setting Control LimitsControl Chart for sample of 9 boxes
For xCharts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x  A2R
where R = average range of the samples
A2 = control chart factor found in Table S6.1
x = mean of the sample means
Setting Chart LimitsSample Size Mean Factor Upper Range Lower Range
n A2D4D3
2 1.880 3.268 0
3 1.023 2.574 0
4 .729 2.282 0
5 .577 2.115 0
6 .483 2.004 0
7 .419 1.924 0.076
8 .373 1.864 0.136
9 .337 1.816 0.184
10 .308 1.777 0.223
12 .266 1.716 0.284
Control Chart FactorsTable S6.1
Average range R = .25
Sample size n = 5
UCLx = x + A2R
= 12 + (.577)(.25)
= 12 + .144
= 12.144 ounces
From Table S6.1
Setting Control LimitsAverage range R = .25
Sample size n = 5
UCLx = x + A2R
= 12 + (.577)(.25)
= 12 + .144
= 12.144 ounces
UCL = 12.144
Mean = 12
LCLx = x  A2R
= 12  .144
= 11.857 ounces
LCL = 11.857
Setting Control LimitsUpper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from Table S6.1
Setting Chart LimitsFor RCharts
Sample size n = 5
From Table S6.1 D4= 2.115, D3 = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
UCL = 11.2
Mean = 5.3
LCLR = D3R
= (0)(5.3)
= 0 pounds
LCL = 0
Setting Control LimitsThese sampling distributions result in the charts below
(Sampling mean is shifting upward but range is consistent)
UCL
(xchart detects shift in central tendency)
xchart
LCL
UCL
(Rchart does not detect change in mean)
Rchart
LCL
Mean and Range ChartsFigure S6.5
These sampling distributions result in the charts below
(Sampling mean is constant but dispersion is increasing)
UCL
(xchart does not detect the increase in dispersion)
xchart
LCL
UCL
(Rchart detects increase in dispersion)
Rchart
LCL
Mean and Range ChartsFigure S6.5
Take samples from the population and compute the appropriate sample statistic
Use the sample statistic to calculate control limits and draw the control chart
Plot sample results on the control chart and determine the state of the process (in or out of control)
Investigate possible assignable causes and take any indicated actions
Continue sampling from the process and reset the control limits when necessary
n
sp =
UCLp = p + zsp
^
^
LCLp = p  zsp
^
where p = mean fraction defective in the sample
z = number of standard deviations
sp = standard deviation of the sampling distribution
n = sample size
^
Control Limits for pChartsPopulation will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
1 6 .06 11 6 .06
2 5 .05 12 1 .01
3 0 .00 13 8 .08
4 1 .01 14 7 .07
5 4 .04 15 5 .05
6 2 .02 16 4 .04
7 5 .05 17 11 .11
8 3 .03 18 3 .03
9 3 .03 19 0 .00
10 2 .02 20 4 .04
Total = 80
(.04)(1  .04)
100
80
(100)(20)
sp = = .02
p = = .04
^
pChart for Data Entry.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
UCLp= 0.10
UCLp = p + zsp= .04 + 3(.02) = .10
^
Fraction defective
p = 0.04
LCLp = p  zsp = .04  3(.02) = 0
^
LCLp= 0.00
         
2 4 6 8 10 12 14 16 18 20
Sample number
pChart for Data Entry.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
UCLp= 0.10
UCLp = p + zsp= .04 + 3(.02) = .10
^
Fraction defective
p = 0.04
LCLp = p  zsp = .04  3(.02) = 0
^
LCLp= 0.00
         
2 4 6 8 10 12 14 16 18 20
Sample number
pChart for Data EntryPossible assignable causes present
LCLc = c 3 c
where c = mean number defective in the sample
Control Limits for cChartsPopulation will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
c = 54 complaints/9 days = 6 complaints/day
UCLc = c + 3 c
= 6 + 3 6
= 13.35
UCLc= 13.35
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
Number defective
LCLc = c  3 c
= 6  3 6
= 0
c = 6
LCLc= 0

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Day
cChart for Cab CompanyThree major management decisions:
Variables Data
Attribute Data
Attribute Data
Target
Lower control limit
Patterns in Control ChartsNormal behavior. Process is “in control.”
Figure S6.7
Target
Lower control limit
Patterns in Control ChartsOne plot out above (or below). Investigate for cause. Process is “out of control.”
Figure S6.7
Target
Lower control limit
Patterns in Control ChartsTrends in either direction, 5 plots. Investigate for cause of progressive change.
Figure S6.7
Target
Lower control limit
Patterns in Control ChartsTwo plots very near lower (or upper) control. Investigate for cause.
Figure S6.7
Target
Lower control limit
Patterns in Control ChartsRun of 5 above (or below) central line. Investigate for cause.
Figure S6.7