Heizer supp 06

Operations
Management
Supplement 6 –
Statistical Process
Control
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 7e
Operations Management, 9e
© 2008 Prentice Hall, Inc. S6 – 1

Outline
 Statistical Process Control (SPC)
 Control Charts for Variables
 The Central Limit Theorem
 Setting Mean Chart Limits (x-Charts)
 Setting Range Chart Limits (R-Charts)
 Using Mean and Range Charts
 Control Charts for Attributes
 Managerial Issues and Control Charts


Outline – Continued

 Process Capability
 Process Capability Ratio (Cp)
 Process Capability Index (Cpk )
 Acceptance Sampling
 Operating Characteristic Curve
 Average Outgoing Quality


Learning Objectives
When you complete this supplement
you should be able to:

1. Explain the use of a control chart
2. Explain the role of the central limit
theorem in SPC
3. Build x-charts and R-charts
4. List the five steps involved in
building control charts


Learning Objectives
When you complete this supplement
you should be able to:

5. Build p-charts and c-charts
6. Explain process capability and
compute Cp and Cpk
7. Explain acceptance sampling
8. Compute the AOQ


Statistical Process Control
(SPC)
 Variability is inherent
in every process
 Natural or common
causes
 Special or assignable causes
 Provides a statistical signal when
assignable causes are present
 Detect and eliminate assignable
causes of variation

Natural Variations
 Also called common causes
 Affect virtually all production processes
 Expected amount of variation
 Output measures follow a probability
distribution
 For any distribution there is a measure
of central tendency and dispersion
 If the distribution of outputs falls within
acceptable limits, the process is said to
be “in control”

Assignable Variations
 Also called special causes of variation
 Generally this is some change in the process
 Variations that can be traced to a specific
reason
 The objective is to discover when
assignable causes are present
 Eliminate the bad causes
 Incorporate the good causes


Samples
To measure the process, we take samples
and analyze the sample statistics following
these steps
Each of these
represents one
(a) Samples of the sample of five
product, say five boxes of cereal
boxes of cereal
taken off the filling
Frequency
# #

machine line, vary # # #
from each other in # # # #
weight # # # # # # #

# # # # # # # # # #

Figure S6.1 Weight

Samples
these steps
The solid line
represents the
(b) After enough distribution
samples are
taken from a
stable process,
they form a Frequency
pattern called a
distribution

Figure S6.1 Weight

Samples
these steps
(c) There are many types of distributions, including
the normal (bell-shaped) distribution, but
distributions do differ in terms of central
tendency (mean), standard deviation or
variance, and shape Figure S6.1
Frequency

Central tendency Variation Shape

Weight Weight Weight

Samples
these steps
(d) If only natural
causes of
variation are Frequency
present, the Prediction
output of a
process forms a
distribution that Tim
e
is stable over Weight
time and is Figure S6.1
predictable

Samples
these steps
?
? ??
?
(e) If assignable ?
? ?
?
causes are ?
? ?
?
?? ?
??
present, the Frequency
?

process output is Prediction
not stable over
time and is not
predicable Tim
e
Weight
Figure S6.1


Control Charts
Constructed from historical data, the
purpose of control charts is to help
distinguish between natural variations
and variations due to assignable
causes


Process Control
(a) In statistical
control and capable
of producing within
Frequency control limits

Lower control limit Upper control limit
(b) In statistical
control but not
capable of producing
within control limits

(c) Out of control

Size
(weight, length, speed, etc.) Figure S6.2


Types of Data
Variables Attributes
 Characteristics that  Defect-related
can take any real characteristics
value  Classify products
 May be in whole or as either good or
in fractional bad or count
numbers defects
 Continuous random  Categorical or
variables discrete random
variables

Central Limit Theorem
Regardless of the distribution of the
population, the distribution of sample means
drawn from the population will tend to follow
a normal curve
1. The mean of the sampling
distribution (x) will be the same x=µ
as the population mean µ
2. The standard deviation of the
sampling distribution (σ x) will
σ
equal the population standard σx =
n
deviation (σ ) divided by the
square root of the sample size, n

Population and Sampling
Distributions
Three population Distribution of
distributions sample means
Mean of sample means = x
Beta
Standard
deviation of σ
the sample = σx =
Normal n
means

Uniform

| | | | | | |

-3σ x -2σ x -1σ x x +1σ x +2σ x +3σ x
95.45% fall within ± 2σ x
99.73% of all x
fall within ± 3σ x Figure S6.3


Sampling Distribution
Sampling
distribution
of means

Process
distribution
of means

x=µ
(mean)
Figure S6.4


Control Charts for Variables
 For variables that have
continuous dimensions
 Weight, speed, length,
strength, etc.
 x-charts are to control
the central tendency of the process
 R-charts are to control the dispersion of
the process
 These two charts must be used together


Setting Chart Limits
For x-Charts when we know σ
Upper control limit (UCL) = x + zσ x
Lower control limit (LCL) = x - zσ x
where x = mean of the sample means
or a target value set for the process
z = number of normal standard
deviations
σx = standard deviation of the
sample means
= σ/ n
© 2008 Prentice Hall, Inc.
σ = population standard S6 – 21

Setting Control Limits
Hour 1 Hour Mean Hour Mean
Sample Weight of 1 16.1 7 15.2
Number Oat Flakes 2 16.8 8 16.4
1 17 3 15.5 9 16.3
2 13 4 16.5 10 14.8
3 16 5 16.5 11 14.2
4 18 6 16.4 12 17.3
n=9 5 17
6 16 For 99.73% control limits, z = 3
7 15
8 17 UCLx = x + zσ x = 16 + 3(1/3) = 17 ozs
9 16
Mean 16.1 LCLx = x - zσ x = 16 - 3(1/3) = 15 ozs
σ= 1

Control Chart
for sample of Variation due
Out of to assignable
9 boxes control causes

17 = UCL

Variation due to
16 = Mean natural causes

15 = LCL

Variation due
| | | | | | | | | | | |
to assignable
1 2 3 4 5 6 7 8 9 10 11 12 Out of causes
Sample number control


For x-Charts when we don’t know σ

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

Control Chart Factors
Sample Size Mean Factor Upper Range Lower
Range
n A2 D4 D3
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
Table S6.1

Process average x = 12 ounces
Average range R = .25
Sample size n = 5


Sample size n = 5

UCLx = x + A2R
= 12 + (.577)(.25)
= 12 + .144
= 12.144 ounces
From
Table S6.1


Sample size n = 5

UCLx = x + A2R UCL = 12.144
= 12 + (.577)(.25)
= 12 + .144 Mean = 12
= 12.144 ounces

LCLx = x - A2R LCL = 11.857

= 12 - .144
= 11.857 ounces

R – Chart

 Type of variables control chart
 Shows sample ranges over time
 Difference between smallest and
largest values in sample
 Monitors process variability
 Independent from process mean


For R-Charts

Upper 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


Average range R = 5.3 pounds
Sample size n = 5
From Table S6.1 D4 = 2.115, D3 = 0

UCLR = D4R UCL = 11.2
= (2.115)(5.3)
= 11.2 pounds Mean = 5.3

LCLR = D3 R LCL = 0

= (0)(5.3)
= 0 pounds

Mean and Range Charts
(a)
These (Sampling mean is
sampling shifting upward but
distributions range is consistent)
result in the
charts below

UCL
(x-chart detects
x-chart shift in central
tendency)
LCL
UCL
(R-chart does not
R-chart detect change in
mean)
LCL
Figure S6.5

Mean and Range Charts
(b)
These
(Sampling mean
sampling
is constant but
distributions
dispersion is
result in the
increasing)
charts below

UCL
(x-chart does not
x-chart detect the increase
in dispersion)
LCL
UCL
(R-chart detects
R-chart increase in
dispersion)
LCL
Figure S6.5

Steps In Creating Control
Charts
1. Take samples from the population and
compute the appropriate sample statistic
2. Use the sample statistic to calculate control
limits and draw the control chart
3. Plot sample results on the control chart and
determine the state of the process (in or out of
control)
4. Investigate possible assignable causes and
take any indicated actions
5. Continue sampling from the process and reset
the control limits when necessary

Manual and Automated
Control Charts


Control Charts for Attributes
 For variables that are categorical
 Good/bad, yes/no,
acceptable/unacceptable
 Measurement is typically counting
defectives
 Charts may measure
 Percent defective (p-chart)
 Number of defects (c-chart)

Control Limits for p-Charts
Population will be a binomial distribution,
but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics

UCLp = p + zσ p
^
p(1 - p)
σp =
^
n
LCLp = p - zσ p
^

p = mean fraction defective in the sample
z = number of standard deviations
σp = standard deviation of the sampling distribution
^

n = sample size

p-Chart for Data Entry
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
80 (.04)(1 - .04)
p= = .04 σp = = .02
(100)(20) ^
100

UCLp = p + zσ p = .04 + 3(.02) = .10
^

LCLp = p - zσ p = .04 - 3(.02) = 0
^

.11 –
.10 – UCLp = 0.10
.09 –
Fraction defective

.08 –
.07 –
.06 –
.05 –
.04 – p = 0.04
.03 –
.02 –
.01 – LCLp = 0.00
.00 – | | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
Sample number

UCLp = p + zσ p = .04 + 3(.02) = .10
^
Possible
LCLp = p - zσ p = .04 - 3(.02) = 0
^ assignable
causes present
.11 –
.10 – UCLp = 0.10
.09 –
Fraction defective

.08 –
.07 –
.06 –
.05 –
.04 – p = 0.04
.03 –
.02 –
.01 – LCLp = 0.00
.00 – | | | | | | | | | |
2 4 6 8 10 12 14 16 18 20
Sample number

Control Limits for c-Charts
Population will be a Poisson distribution,
but applying the Central Limit Theorem
allows us to assume a normal distribution
for the sample statistics

UCLc = c + 3 c LCLc = c - 3 c

c = mean number defective in the sample


c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day

UCLc = c + 3 c 14 –
UCLc = 13.35
14

Number defective
=6+3 6 12 –
= 13.35 10 –
8 –
6 – c= 6
LCLc = c - 3 c 4 –
=6-3 6 2 – LCLc = 0
0 – | | | | | | | | |
=0 1 2 3 4 5 6 7 8 9
Day


Managerial Issues and
Control Charts
Three major management decisions:

 Select points in the processes that
need SPC
 Determine the appropriate charting
technique
 Set clear policies and procedures


Which Control Chart to Use
Variables Data
 Using an x-chart and R-chart:
 Observations are variables
 Collect 20 - 25 samples of n = 4, or n =
5, or more, each from a stable process
and compute the mean for the x-chart
and range for the R-chart
 Track samples of n observations each


Attribute Data
 Using the p-chart:
 Observations are attributes that can
be categorized in two states
 We deal with fraction, proportion, or
percent defectives
 Have several samples, each with
many observations


Attribute Data
 Using a c-Chart:
 Observations are attributes whose
defects per unit of output can be
counted
 The number counted is a small part of
the possible occurrences
 Defects such as number of blemishes
on a desk, number of typos in a page
of text, flaws in a bolt of cloth


Patterns in Control Charts

Upper control limit

Target

Lower control limit
Normal behavior.
Process is “in control.”
Figure S6.7



Upper control limit

Target

Lower control limit
One plot out above (or
below). Investigate for
Figure S6.7 cause. Process is “out
of control.”


Upper control limit

Target

Lower control limit
Trends in either
direction, 5 plots.
Figure S6.7 Investigate for cause of
progressive change.


Upper control limit

Target

Lower control limit
Two plots very near
lower (or upper)
Figure S6.7 control. Investigate for
cause.


Upper control limit

Target

Lower control limit
Run of 5 above (or
below) central line.
Figure S6.7 Investigate for cause.



Upper control limit

Target

Lower control limit
Erratic behavior.
Investigate.
Figure S6.7


Process Capability
 The natural variation of a process
should be small enough to produce
products that meet the standards
required
 A process in statistical control does not
necessarily meet the design
specifications
 Process capability is a measure of the
relationship between the natural
variation of the process and the design
specifications

Process Capability Ratio
Upper Specification - Lower Specification
Cp =
6σ
 A capable process must have a Cp of at
least 1.0
 Does not look at how well the process
is centered in the specification range
 Often a target value of Cp = 1.33 is used
to allow for off-center processes
 Six Sigma quality requires a Cp = 2.0

Insurance claims process

Process mean x = 210.0 minutes
Process standard deviation σ = .516 minutes
Design specification = 210 ± 3 minutes

Cp =
6σ




Cp =
6σ
213 - 207
= = 1.938
6(.516)




Cp =
6σ
213 - 207
= = 1.938 Process is
6(.516) capable


Process Capability Index

Upper Lower
Cpk = minimum of Specification - x , x - Specification
Limit Limit
3σ 3σ

 A capable process must have a Cpk of at
least 1.0
 A capable process is not necessarily in the
center of the specification, but it falls within
the specification limit at both extremes

New Cutting Machine
New process mean x = .250 inches
Process standard deviation σ = .0005 inches
Upper Specification Limit = .251 inches
Lower Specification Limit = .249 inches


New Cutting Machine
(.251) - .250
Cpk = minimum of ,
(3).0005


New Cutting Machine
(.251) - .250 .250 - (.249)
Cpk = minimum of ,
(3).0005 (3).0005

Both calculations result in
New machine is
.001
Cpk = = 0.67 NOT capable
.0015


Interpreting Cpk

Cpk = negative number

Cpk = zero

Cpk = between 0 and 1

Cpk = 1

Cpk > 1
Figure S6.8

Acceptance Sampling
 Form of quality testing used for
incoming materials or finished goods
 Take samples at random from a lot
(shipment) of items
 Inspect each of the items in the sample
 Decide whether to reject the whole lot
based on the inspection results
 Only screens lots; does not drive
quality improvement efforts

Acceptance Sampling
 Form of quality testing used for
incoming materials or finished goods
 Take samples at random from a lot
Rejected lots can be:
(shipment) of items
 Inspect each of theReturnedthethe
 items in to sample
supplier
 Decide whether to reject the whole lot
 Culled for
based on the inspection results
defectives
 Only screens lots; does not drive
(100% inspection)
quality improvement efforts

Operating Characteristic
Curve
 Shows how well a sampling plan
discriminates between good and
bad lots (shipments)
 Shows the relationship between
the probability of accepting a lot
and its quality level


The “Perfect” OC Curve
Keep whole
shipment
P(Accept Whole Shipment)

100 –

75 –
Return whole
50 – shipment

25 –
Cut-Off
0 –
| | | | | | | | | | |

0 10 20 30 40 50 60 70 80 90 100
% Defective in Lot


An OC Curve
Figure S6.9
100 –
95 – α = 0.05 producer’s risk for AQL

75 –

Probability
of 50 –
Acceptance

25 –

10 –
β = 0.10 0 |– | | | | | | | | Percent
0 1 2 3 4 5 6 7 8 defective
AQL LTPD
Consumer’s
risk for LTPD Good Indifference
Bad lots
lots zone


AQL and LTPD
 Acceptable Quality Level (AQL)
 Poorest level of quality we are
willing to accept
 Lot Tolerance Percent Defective
(LTPD)
 Quality level we consider bad
 Consumer (buyer) does not want to
accept lots with more defects than
LTPD

Producer’s and Consumer’s
Risks
 Producer's risk (α )
 Probability of rejecting a good lot
 Probability of rejecting a lot when the
fraction defective is at or above the
AQL
 Consumer's risk (β )
 Probability of accepting a bad lot
 Probability of accepting a lot when
fraction defective is below the LTPD

OC Curves for Different
Sampling Plans

n = 50, c = 1

n = 100, c = 2


Average Outgoing Quality

(Pd)(Pa)(N - n)
AOQ =
N

where
Pd = true percent defective of the lot
Pa = probability of accepting the lot
N = number of items in the lot
n = number of items in the sample


Average Outgoing Quality
1. If a sampling plan replaces all defectives
2. If we know the incoming percent
defective for the lot

We can compute the average outgoing
quality (AOQ) in percent defective

The maximum AOQ is the highest percent
defective or the lowest average quality
and is called the average outgoing quality
level (AOQL)

Automated Inspection

 Modern
technologies
allow virtually
100%
inspection at
minimal costs
 Not suitable
for all
situations

SPC and Process Variability
Lower Upper
specification specification
limit limit
(a) Acceptance
sampling (Some
bad units accepted)

(b) Statistical process
control (Keep the
process in control)

(c) Cpk >1 (Design
a process that
is in control)
Process mean, µ Figure S6.10

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