Transcript Process Control Charts

Operations
Management
Statistical Process Control
Supplement 6
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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



Process Capability
Acceptance Sampling


Operating Characteristic (OC) Curves
Average Outgoing Quality
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Learning Objectives
When you complete this chapter, you should
be able to :
 Identify or Define:













Natural and assignable causes of variation
Central limit theorem
Attribute and variable inspection
Process control
x charts and R charts
LCL and UCL
p-charts and C-charts
Cpk
Acceptance sampling
OC curve
AQL and LTPD
AOQ
Producer’s and consumer’s risk
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Learning Objectives - continued
When you complete this chapter, you should
be able to :
Describe or explain:

The role of statistical quality control
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Statistical Quality Control (SPC)
 Measures performance of a process
 Uses mathematics (i.e., statistics)
 Involves collecting, organizing, & interpreting
data
 Objective: provide statistical when assignable
causes of variation are present
 Used to


Control the process as products are produced
Inspect samples of finished products
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Figure S6.1
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Types of
Statistical Quality Control
Statistical
Quality Control
Process
Control
Variables
Charts
Acceptance
Sampling
Attributes
Charts
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Variables
S6-7
Attributes
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Quality Characteristics
Variables
 Characteristics that you
measure, e.g., weight,
length
 May be in whole or in
fractional numbers
 Continuous random
variables
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Attributes
 Characteristics for which
you focus on defects
 Classify products as either
‘good’ or ‘bad’, or count #
defects

e.g., radio works or not
 Categorical or discrete
random variables
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Statistical Process Control (SPC)
 Statistical technique used to ensure process is
making product to standard
 All process are subject to variability
Natural causes: Random variations
 Assignable causes: Correctable problems


Machine wear, unskilled workers, poor material
 Objective: Identify assignable causes
 Uses process control charts
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Process Control:
Three Types of Process Outputs
(a) In statistical control and
capable of producing within
control limits. A process with
only natural causes of
variation and capable of
producing within the
specified control limits.
Frequency
Lower control limit
Upper control limit
(b) In statistical control, but not
capable of producing within control
limits. A process in control (only
natural causes of variation are
present) but not capable of
producing within the specified
control limits; and
(c) Out of control. A process out of
control having assignable causes of
variation.
Size
Weight, length, speed, etc.
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The Relationship Between Population
and Sampling Distributions
Three population distributions
Distribution of sample means
Beta
Mean of sample means  x
x
Standard deviation of
 x 
the sample means
n
Normal
Uniform
 3 x  2  x  1 x
x
  x  2  x  3 x
(mean)
95.5% of all x fall within  2  x
99.7% of all x fall within  3 x
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Sampling Distribution of Means,
and Process Distribution
Sampling
distribution of the
means
Process
distribution of
the sample
xm
( mean )
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Process Control Charts
Plot of Sample Data Over Time
Sample Value
80
Sample
Value
UCL
60
40
Average
20
LCL
0
1
5
9
13
17
21
Time
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Control Chart Purposes
 Show changes in data pattern

e.g., trends

Make corrections before process is out of control
 Show causes of changes in data

Assignable causes


Data outside control limits or trend in data
Natural causes

Random variations around average
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Theoretical Basis
of Control Charts
Central Limit Theorem
As sample size
gets
large
enough,
sampling distribution
becomes almost
normal regardless of
population
distribution.
X
X
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Theoretical Basis
of Control Charts
Central Limit Theorem
Mean
Standard deviation
X 
x
x 
X 
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n
X
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Control Chart Types
Continuous
Numerical Data
Control
Charts
Categorical or Discrete
Numerical Data
Variables
Charts
R
Chart
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Attributes
Charts
P
Chart
X
Chart
S6-18
C
Chart
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Statistical Process Control Steps
Start
Produce Good
Provide Service
Take Sample
No
Assign.
Causes?
Yes
Inspect Sample
Stop Process
Create
Control Chart
Find Out Why
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X Chart
 Type of variables control chart

Interval or ratio scaled numerical data
 Shows sample means over time
 Monitors process average
 Example: Weigh samples of coffee & compute
means of samples; Plot
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X Chart
Control Limits
UCL x  x  A R
From
Table S6.1
LCL x  x  A R
n
x 
 xi
Sample
Mean at
Time i
i 
n
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Sample
Range at
Time i
n
R 
# Samples
S6-21
 Ri
i 1
n
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Factors for Computing Control
Chart Limits
Sample
Size, n
2
Mean
Upper
Lower
Factor, A2 Range, D4 Range, D3
1.880
3.268
0
3
1.023
2.574
0
4
0.729
2.282
0
5
0.577
2.115
0
6
0.483
2.004
0
7
0.419
1.924
0.076
8
0.373
1.864
0.136
9
0.337
1.816
0.184
10
0.308
1.777
0.223
0.184
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R Chart
 Type of variables control chart

Interval or ratio scaled numerical data
 Shows sample ranges over time

Difference between smallest & largest values in
inspection sample
 Monitors variability in process
 Example: Weigh samples of coffee &
compute ranges of samples; Plot
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R Chart
Control Limits
UCL R  D4 R
From Table S6.1
LCL R  D3R
Sample Range at
Time i
n
R 
 Ri
i 1
n
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# Samples
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Steps to Follow When Using
Control Charts
Collect 20 to 25 samples of n=4 or n=5 from a
stable process and compute the mean.
Compute the overall means, set approximate
control limits,and calculate the preliminary upper and
lower control limits.If the process is not currently
stable, use the desired mean instead of the overall
mean to calculate limits.
Graph the sample means and ranges on their
respective control charts and determine whether they
fall outside the acceptable limits.
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Steps to Follow When Using
Control Charts - continued
 Investigate points or patterns that indicate the
process is out of control. Assign causes for the
variations.
 Collect additional samples and revalidate the control
limits.
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Figure S6.5
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p Chart
 Type of attributes control chart

Nominally scaled categorical data
 e.g., good-bad
 Shows % of nonconforming items
 Example: Count # defective chairs & divide by
total chairs inspected; Plot

Chair is either defective or not defective
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p Chart
Control Limits
UCL p  p  z
p (  p )
n
LCL p  p  z
p (  p )
n
k
n
 ni
i 
k
k
and
p
z = 2 for 95.5% limits;
z = 3 for 99.7% limits
# Defective Items
in Sample i
 xi
i 1
k
 ni
Size of sample i
i 1
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c Chart
 Type of attributes control chart

Discrete quantitative data
 Shows number of nonconformities (defects)
in a unit


Unit may be chair, steel sheet, car etc.
Size of unit must be constant
 Example: Count # defects (scratches, chips
etc.) in each chair of a sample of 100 chairs;
Plot
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c Chart
Control Limits
UCLc  c  
c
LCLc  c  
c
# Defects in
Unit i
k
 ci
c  i1
k
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Use 3 for 99.7%
limits
# Units Sampled
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Figure S6.7
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Process Capability Cpk
 Upper Specification Limit  x
C pk  minimum of 
, or


x  Lower Specification Limit 



where x  process mean
  standard deviation of the process population
Assumes that the process is:
•under control
•normally distributed
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Meanings of Cpk Measures
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
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What Is
Acceptance Sampling?
 Form of quality testing used for incoming
materials or finished goods

e.g., purchased material & components
 Procedure



Take one or more 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
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What Is an
Acceptance Plan?
 Set of procedures for inspecting incoming
materials or finished goods
 Identifies



Type of sample
Sample size (n)
Criteria (c) used to reject or accept a lot
 Producer (supplier) & consumer (buyer)
must negotiate
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Operating Characteristics Curve
 Shows how well a sampling plan
discriminates between good & bad lots
(shipments)
 Shows the relationship between the
probability of accepting a lot & its quality
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OC Curve
100% Inspection
P(Accept Whole Shipment)
100%
Keep whole
shipment
0%
0
1
2
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Return whole
shipment
3
4
5
Cut-Off
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7 8 9 10
% Defective in Lot
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OC Curve with Less than 100%
Sampling
P(Accept Whole Shipment)
Probability is not 100%: Risk of
keeping bad shipment or
returning good one.
100%
Keep whole
shipment
Return whole
shipment
0%
0
1
2
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3
4
Cut-Off
5
6
7
8
9
10
% Defective in Lot
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AQL & LTPD
 Acceptable quality level (AQL)
Quality level of a good lot
 Producer (supplier) does not want lots with fewer
defects than AQL rejected

 Lot tolerance percent defective (LTPD)


Quality level of a bad lot
Consumer (buyer) does not want lots with more
defects than LTPD accepted
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Producer’s & Consumer’s Risk
 Producer's risk ()


Probability of rejecting a good lot
Probability of rejecting a lot when fraction
defective is AQL
 Consumer's risk (ß)


Probability of accepting a bad lot
Probability of accepting a lot when fraction
defective is LTPD
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An Operating Characteristic (OC)
Curve Showing Risks
100
95
 = 0.05 producer’s risk for AQL
75
Probability of
Acceptance
50
25
= 0.10
10
Consumer’s
risk for LTPD
0
0
1
Good
lots
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AQL
3
4
5
6
Indifference zone
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Percent
Defective
8
LTPD
Bad lots
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OC Curves for Different Sampling
Plans
P(Accept Whole Shipment)
n = 50, c = 1
100%
n = 100, c = 2
0%
0
1
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2
3 4 5 6 7 8
AQL
LTPD
% Defective in Lot
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10
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Average Outgoing Quality
AOQ 
Where:
( Pd )( Pa )( N  n )
N
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
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Developing a Sample Plan
 Negotiate between producer (supplier) and
consumer (buyer)
 Both parties attempt to minimize risk

Affects sample size & cut-off criterion
 Methods
MIL-STD-105D Tables
 Dodge-Romig Tables
 Statistical Formulas

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Statistical Process Control - Identify
and Reduce Process Variability
Lower
specification
limit
Upper
specification
limit
(a) Acceptance
sampling
(b) Statistical
process control
(c) cpk >1
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