Transcript TQM - σχολή μηχανικών μεταλλείων

Chapter 9A. Process Capability & Statistical Quality
Control
Outline:
 Basic Statistics
 Process Variation
 Process Capability
 Process Control Procedures


Variable data
 X-bar chart and R-chart
Attribute data
 p-chart
 Acceptance Sampling

Operating Characteristic Curve
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Focus
 This technical note on statistical quality control (SQC) covers the
quantitative aspects of quality management
 SQC is a number of different techniques designed to evaluate
quality from a conformance view

How are we doing in meeting specifications?
 SQC can be applied to both manufacturing and service processes
 SQC techniques usually involve periodic sampling of the process
and analysis of data
Sample size
 Number of samples

 SQC techniques are looking for variance
 Most processes produce variance in output

we need to monitor the variance (and the mean also) and correct
processes when they get out of range
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Basic Statistics
Normal Distributions have a
mean (μ) and a standard
deviation (σ)
For a sample of N observations:
3
μ
+3
99.7%
N
Mean
X   xi N
i 1
where:
xi = Observed value
N = Total number of observed values
 x
N
Standard Deviation
 
i 1
i
X

2
N
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Statistics and Probability
 SQC relies on central limit theorem and normal dist.
 We establish the Upper Control Limits (UCL) and the Lower
Control Limits (LCL) with plus or minus 3 standard deviations.
Based on this we can expect 99.7% of our sample observations to
fall within these limits.
 Acceptance sampling relies on Binomial and Hyper geometric
probability concepts
3 2
+2 +3
a/2
a/2
99.7%
a  Prob. of
Type I error
LCL
UCL
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Basic Stats
 Using SQC,


samples of a process output are taken, and
sample statistics are calculated
 The purpose of sampling is to find when the
process has changed in some nonrandom way

The reason for the change can then be quickly
determined and corrected
 This allows us to detect changes in the actual
distribution process
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Variation
 Random (common) variation is inherent in the
production process.
 Assignable variation is caused by factors that can
be clearly identified and possibly managed
 Using a saw to cut 2.1 meter long boards as a
sample process

Discuss random vs. assignable variation
 Generally, when variation is reduced, quality
improves.

It is impossible to have zero variability. T or F ?
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Taguchi’s View of Variation
Traditional view is that quality within the LS and US is good and
that the cost of quality outside this range is constant, where Taguchi
views costs as increasing as variability increases, so seek to achieve
zero defects and that will truly minimize quality costs.
High
High
Incremental
Cost of
Variability
Incremental
Cost of
Variability
Zero
Zero
Lower Target
Spec
Spec
Upper
Spec
Traditional View
Lower
Spec
Exhibits
TN8.1 &
TN8.2
Target
Spec
Upper
Spec
Taguchi’s View
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Process Capability
 Tolerance (specification, design) Limits


Bearing diameter 1.250 +- 0.005 inches
LTL = 1.245 inches UTL = 1.255 inches
 Process Limits



The actual distribution from the process
Run the process to make 100 bearings, compute the mean and
std. dev. (and plot/graph the complete results)
Suppose, mean = 1.250, std. dev = 0.002
 How do they relate to one another?
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Tolerance Limits vs. Process Capability
Specification Width
Actual Process Width
Specification Width
Actual Process Width
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Process Capability Example
 Design Specs: Bearing diameter 1.250 +- 0.005 inches

LTL = 1.245 inches UTL = 1.255 inches
 The actual distribution from the process  mean = 1.250, s = 0.002

+- 3s limits  1.250 +- 3(0.002)  [1.244, 1.256]

Anew process,  std. dev. = 0.00083
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Process Capability Index, Cpk
 Capability Index shows how well parts being produced fit
into design limit specifications
 X  LTL
UTL - X 

C pk = min 
or

3
3


 Compute the Cpk for the bearing example.

Old process, mean = 1.250, s = 0.002

What is the probability of producing defective bearings?

New process, mean = 1.250, s= 0.00083, re-compute the Cpk

When the computed (sample) mean = design (target) mean,
what does that imply?
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The Cereal Box Example
 Recall the cereal example. Consumer Reports has just published
an article that shows that we frequently have less than 15 ounces
of cereal in a box.
 Let’s assume that the government says that we must be within ±
5 percent of the weight advertised on the box.
 Upper Tolerance Limit = 16 + 0.05(16) = 16.8 ounces
 Lower Tolerance Limit = 16 – 0.05(16) = 15.2 ounces
 We go out and buy 1,000 boxes of cereal and find that they
weight an average of 15.875 ounces with a standard deviation of
0.529 ounces.
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Cereal Box Process Capability
 Specification or Tolerance Limits
Upper Spec = 16.8 oz,
 Lower Spec = 15.2 oz

 Observed Weight
 X  LTL UTL  X 
C pk  Min
;

3 
 3

Mean = 15.875 oz, Std Dev = 0.529 oz

What does a Cpk of 0.4253 mean?
 Many companies look for a Cpk of 1.3 or better… 6-Sigma
company wants 2.0!
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Types of Statistical Sampling
1. Sampling to accept or reject the immediate lot of
product at hand (Acceptance Sampling).
 Attribute (Binary; Yes/No; Go/No-go information)



Defectives refers to the acceptability of product across a
range of characteristics.
Defects refers to the number of defects per unit which may
be higher than the number of defectives.
p-chart application
2. Sampling to determine if the process is within
acceptable limits (Statistical Process Control)
 Variable (Continuous)


Usually measured by the mean and the standard deviation.
X-bar and R chart applications
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Evidence for Investigation…
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Control Limits
 If we establish control limits at
+/- 3 standard deviations, then we
would expect 99.7% of our
observations to fall within these
limits
UCL
LCL
UCL
LCL
UCL
LCL
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x
Attribute Measurements (p-Chart)
 Item is “good” or “bad”
 Collect data, compute average fraction bad (defective)
and std. dev. using:
T ot alNumber of Defect ives
p=
T ot alNumber of Observat ions
sp =
 The, UCL, LCL using:
p (1- p)
n
UCL = p + Z sp
LCL = p - Z sp
 Excel time!
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Variable Measurements (x-Bar and R Charts)
 A variable of the item is measured (e.g., weight,
length, salt content in a bag of chips)

Note that the item (sample) is not declared good or bad
 Since the actual the standard deviation of the process
is not known (and it may indeed fluctuate also) we
use the sample data to compute the UCL & LCL
x Chart Cont rolLim it s
R Chart ControlLimits
UCL = x + A 2 R
UCL = D 4 R
LCL = x - A 2 R
LCL = D3 R
For 3-sigma limits, factors A2 , D3 , and D4 and are given in Exhibit
9A.6, p. 341
 Excel time!

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Acceptance Sampling vs. SPC
 Sampling to accept or reject the immediate lot of
product at hand (Acceptance Sampling).


Determine quality level
Ensure quality is within predetermined (agreed) level
 Sampling to determine if the process is within
acceptable limits - Statistical Process Control (SPC)
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3
Acceptance Sampling
Advantages
Economy
Less handling damage
Fewer inspectors
Upgrading of the inspection
job
 Applicability to destructive
testing
 Entire lot rejection
(motivation for
improvement)





 Disadvantages
 Risks of accepting “bad” lots
and rejecting “good” lots
 Added planning and
documentation
 Sample provides less
information than 100-percent
inspection
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A Single Sampling Plan
 A Single Sampling Plan simply requires two
parameters to be determined:
1. n
the sample size (how many units to sample
from a lot)
2. c
the maximum number of defective items that
can be found in the sample before the lot is
rejected.
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RISK
 RISKS for the producer and consumer in sampling
plans:
 Acceptable Quality Level (AQL)

Max. acceptable percentage of defectives defined by
producer.
 a (Producer’s risk)

The probability of rejecting a good lot.
 Lot Tolerance Percent Defective (LTPD)

Percentage of defectives that defines consumer’s rejection
point.
  (Consumer’s risk)

The probability of accepting a bad lot.
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Operating Characteristic Curve
Probability of acceptance
The OCC brings the concepts of producer’s risk, consumer’s risk,
sample size, and maximum defects allowed together
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
a = .05 (producer’s risk)
n = 99
c=4
B =.10
(consumer’s risk)
1
2
3
AQL
4
5
6
7
8
The shape or slope
of the curve is
dependent on a
particular
combination of the
four parameters
9 10 11 12
LTPD
Percent defective
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9
Example: Acceptance Sampling
Zypercom, a manufacturer of video interfaces, purchases
printed wiring boards from an outside vender, Procard.
Procard has set an acceptable quality level of 1% and
accepts a 5% risk of rejecting lots at or below this level.
Zypercom considers lots with 3% defectives to be
unacceptable and will assume a 10% risk of accepting a
defective lot.
Develop a sampling plan for Zypercom and determine a
rule to be followed by the receiving inspection personnel.
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10
Developing A Single Sampling Plan
 Determine:
a?
?
AQL?
 LTPD?

 Divide LTPD by AQL  0.03/0.01 = 3
 Then find the value for “c” by selecting the value in the TN8.10
“n(AQL)”column that is equal to or just greater than the ratio above (3).
 Thus, c = 6
 From the row with c=6, get  nAQL = 3.286 and divide it by AQL
 3.286/0.01 = 328.6, round up to 329,  n = 329
Sampling Plan:
Take a random
sample of 329 units
from a lot.
Reject the lot if more
than 6 units are
defective.
c
0
1
2
3
4
LTPD/AQL
n AQL
44.890
10.946
6.509
4.890
4.057
0.052
0.355
0.818
1.366
1.970
c
5
6
7
8
9
LTPD/AQL
3.549
3.206
2.957
2.768
2.618
n AQL
2.613
3.286
3.981
4.695
5.426
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