Transcript Chapter 1, Heizer/Render, 5th edition

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
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Outline
 PROCESS CAPABILITY
Process Capability Ratio (Cp)
 Process Capability Index (Cpk

 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-chartsandR -charts
 LCL and UCL
 p-charts and C-charts
C
pˆ
and Cpk
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Learning Objectives - Continued
When you complete this chapter, you should be
able to Identify or Define:
 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 signal 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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Types of
Statistical Quality Control
Statistical
Quality Control
Acceptance
Sampling
Process
Control
Variables
Charts
Attributes
Charts
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Natural and Assignable Variation
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Quality Characteristics
Variables
Attributes
 Characteristics that you
measure, e.g., weight, length
 May be in whole or in
fractional numbers
 Continuous random variables
 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.
Upper control limit
Frequency
Lower 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
Attributes
Charts
P
Chart
X
Chart
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C
Chart
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Statistical Process Control Steps
Start
Produce Good
Provide Service
Take Sample
No
Can we
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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Control Chart for Samples of 9 Boxes
Variation due to
assignable causes
17=UCL
Variation due to
natural causes
16=Mean
15=LCL
1 2
3 4
5
6
7
8 9 10 11 12
Sample Number
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Variation due to
assignable causes
Out of control
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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
Mean for
sample i
n
 Ri
i 
n
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Range for
sample i
R  i 1
n
# Samples
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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
12
0.266
1.716
0.284
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
LCLR  D3R
Range for Sample i
n
R 
 Ri
i 1
n
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# Samples
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Steps to Follow When Using
Control Charts
1. Collect 20 to 25 samples of n=4 or n=5 from a
stable process and compute the mean.
2. 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.
3. 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
4. Investigate points or patterns that indicate the
process is out of control. Assign causes for the
variations.
5. Collect additional samples and revalidate the
control limits.
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Mean and Range Charts
Complement Each Other
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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
UCLp  p  z
p (1  p )
n
LCLp  p  z
p (1  p )
n
k
n
 ni
i 1
k
z = 2 for 95.5% limits;
z = 3 for 99.7% limits
k
and p 
 xi
i 1
k
# Defective Items
in Sample i
 ni
i 1
p (1  p )
n
w heren  size of each sample
p 
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Size of sample i
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Percent Defective
P-Chart for Data Entry Example
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
UCLp=0.10
p  0.04
LCLp=0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sample Number
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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  3 c
Use 3 for 99.7%
limits
LCLc  c  3 c
# Defects in
Unit i
k
c 
 ci
i1
# Units Sampled
k
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Patterns to Look for in Control
Charts
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Deciding Which Control Chart to Use
 Using an X 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.


 Using the P-Chart:
We deal with fraction, proportion, or percent defectives
Observations are attributes that can be categorized in
two states
 Have several samples, each with many observations
 Assume a binomial distribution unless the number of
samples is very large – then assume a normal
distribution.


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Deciding Which Control Chart to Use
 Using a C-Chart:
Observations are attributes whose defects per unit of
output can be counted
 The number counted is often a small part of the
possible occurrences
 Assume a Poisson distribution
 Defects such as: number of blemishes on a desk,
number of typos in a page of text, flaws in a bolt of
cloth

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Process Capability Ratio, Cp
Upper Specification  Low erSpecification
Cp 
6σ
  standard dev iationof the process
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Process Capability Cpk
 Upper Specification Limit  x
C pk  minimum of 
, or
3

x  Low er Specification Limit 

3

w here x  process mean
  standard dev iation 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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6
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
3
4
Cut-Off
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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
2
AQL
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3
4
5
6
Indifference zone
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7
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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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 –
[ Some bad units
accepted; the “lot” is good
or bad]
(b) Statistical process
control – [Keep the
process in “control”]
(c) cpk >1 – [Design a
process that is in control]
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