Download Statistical Quality Control

Statistical Quality Control
Simple applications of Statistics in
TQM
Introduction
• The production processes are not perfect!
• Which means that the output of these processes will not be
perfect – correct and deterministic.
• Successive runs of the same production process will produce
non-identical parts.
• Alternately, seemingly similar runs of the production process
will vary, by some degree, and impart the variation into the
some product characteristics.
• Because of these variations in the products, we need
probabilistic models and robust statistical techniques to
analyze quality of such products.
2
Introduction
• No matter how carefully a production process is controlled,
these quality measurements will vary from item to item, and
there will be a probability distribution associated with the
population of such measurements.
• If all important sources of variations are under control in a
production process, then the slight variations among the
quality measurements usually cause no serious problems.
• Such a process should produce the same distribution of quality
measurements no matter when it is sampled, thus this is a
“stable system.”
3
Introduction
• Objective of quality control is to develop a scheme for
sampling a process, making a quality measurement of interest
on sample items, and then making a decision as to whether or
not the process is in the stable state, or “in control.”
• If the sample data suggests that the process is “out of control,”
a cause is for the abnormality is sought.
• A common method for making these decisions involves the
use of control charts.
• These are very important and widely used techniques in
industry, and everyone in the industry, even if not directly
related to quality control, should be aware of these.
4
SPC
Statistical process control
• Methodology for monitoring a process to identify special
causes of variation and signaling the need to take corrective
action.
• When special causes are present, the system said to be
statistically out of control.
• If the variations are due to common causes alone, the process
is said to be in statistical control.
• SPC relies heavily on control charts.
5
Quality control measurements
• Attributes – A performance characteristics that is either
present or absent in the product or service under consideration.
• Examples: Order is either complete or incomplete; an invoice
can have one, two, or more errors.
• Attributes data are discrete and tell whether the characteristics
conforms to specifications.
• Attributes measurements typically represented as proportions
or rates. e.g. rate of errors per opportunity.
• Typically measured by “Go-No Go” gauges.
6
Quality control measurements
• Variable – Continuous data that is concerned with degree of
conformance to specifications.
• Generally expressed with statistical measures such as averages
and standard deviations.
• Sophisticated instruments (caliper) used.
• In statistical sense, attributes inspection less efficient than
variable inspection.
• Attribute data requires larger sample than variable inspection
to obtain same amount of statistical information.
• Most quality characteristics in service industry are attributes.
7
The x chart
• A typical quality control plan requires sampling one or more
items from a production process periodically, and making
the appropriate quality measurements.
• Usually more than one items are measured each time to
increase accuracy and measure variability.
• The x chart helps the quality control person decide whether
the center (or average, or the location of central tendency) of
the measurement has shifted.
8
The x chart
• Suppose that n observations are to be made at each time in
which the process is checked.
• Let Xi be the ith observation (i = 1,…,n) at the specified time
point.
• Let X j be the average of n observations at time j. If
E  X i    and Var  X i    2
for a process in control, then
normally distributed with
Xj
should be approximately
EX j    and VarX j    2 / n.
9
The x chart
• Now since X j has approximately a normal distribution, we
can find the interval that will have a probability of 1- α of
containing X j .
• This interval is   z   
 /2
 n
• If the mean and variation of the population (the µ and σ) is
known then,
  


z
,
 /2
we can set the lower control limit at
 n
and the upper control limit at   z / 2   ,
 n
10
The x chart
• If a control chart is being constructed for a new process, the µ
and σ (the population parameters) will not be known and
hence must be estimated from the sample data.
• For establishing control limits, it is generally recommended
that at least 30 time points be sampled before the control
limits are calculated.
• For each of the k samples, we compute the sample mean, the
sample variance, and the range ( X j , S j , R j ) .
• Note that the range of a sample is the difference between the
largest and smallest value in the sample.
11
The
x chart
It has been found that :
 n 1  2
2
S `2j  
 S j is a better measure than S j .
 n 
To form an unbiased estimator of  from sample means X j
1 k
we can calculate : X   X j .
k j 1
In the similar fashion, we calculate average of S `2j quantities 1 k
S `  S `2j .
k j 1
However, this is not unbiased estimator of population std. deviation.
12
The
x chart
It can be made unbiased by dividing by a constant c2 from ref. tables.
Thus,
S`
is an unbiased estimator for  .
c2
So, the control limit of   z / 2

n
becomes : X  z / 2
S`
c2 n
For, 99.74% level of significan ce we have z / 2  3.
Now, we can write the control limit as X  A1S `,
where A1 
3
c2 n
, value for which is obtained from ref. tables.
13
The x chart
• Computation of the mean standard deviation can be avoided by
using the range data instead of calculating the adjusted
standard deviation.
1 k
R   Rj.
k j 1
Though R has a relationsh ip with σ , it is not an unbiased estimator.
R
However, the ratio
is! Here, d 2 is obtained from std. ref. tables.
d2
14
The x chart
Thus, the new control limit becomes :
X  z / 2
R
d2 n
where A2 
or X  A2 R ,
3
d2 n
, indicating 99.74% level of significan ce.
15
The r-Chart
• Deciding whether the center of the distribution of quality
measurements has shifted up or down may not be enough.
• It is frequently of interest to decide if the variability of the
process measurements has significantly increased or
decreased.
• A process that suddenly starts turning out highly variable
products could cause severe problems in the operations.
• Since we already have established the fact that there is a
relationship between the ranges (Rj) and σ, it would be natural
to base our control chart for variability on the Rj’s.
• A control chart for variability could be based on the adjusted
std dev also (Sj), but use of ranges provides nearly as much
accuracy for much less computation.
16
The r-Chart
• Using the same argument as control chart for central tendency
(x-bar), one could argue that almost all the Rj’s should be
within three standard deviations of the mean of Rj. Now
E R j   d 2 ;
Var R j   d 32 2 ;
where d 2 and d 3 are found in std ref. tables.
The confidence interval, then, is :
d 2  z / 2 d 3   d 2  z / 2 d 3 
17
The r-Chart
• As in the previous chart, if σ is not specified, it must be
estimated from the data. The best estimator of σ based on the
range is R / d 2, and the estimator of the control then becomes:

d3 
R
d 2  z / 2 d3 ; or R 1  z / 2 
d2
d2 

Once again, at 99.74% level of significan ce, z / 2  3.
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The r-Chart


d3 
d3 
Letting, 1  3   D3 ; and 1  3   D4 , the control limit becomes
d2 
d2 


R D , R D .
3
4
Values of D3 and D4 are found in statistica l tables.
19
Control charts for variable data
• Two basic types: x-bar-chart and R-chart – one used to
monitor centering of the process, the other for variation.
• Construction and establishing statistical control:
k
R
k
R
i 1
k
i
;
x
x
i 1
i
k
UCLR  D4 R;
UCLx  x  A2 R
LCLR  D3 R;
LCLx  x  A2 R
20
Interpreting patterns in control charts
•
1.
2.
3.
4.
•
General rules to determine whether a process is in control:
No points outside the control limits.
The number of points above and below the center line are
about the same.
Points seem to fall randomly above and below the center
line.
Most points are near center line, and only a few are close to
control limits.
Basic assumption: Central Limit Theorem.
21
Interpreting patterns in control charts
• One point outside control limits:
measurement or calculation error, power surge, a broken tool,
incomplete operation.
• Sudden shift in process average:
new operator or inspector, new machine setting.
• Cycles:
operator rotation or fatigue at the end of shift, different gauges,
seasonal effects such as temperature and humidity.
• Trends:
x-bar-chart – learning effect, dirt or chip buildup, tool wear, aging
of equipment; R-chart (increasing trend) – gradual decline in
material quality; R-chart (decreasing trend) – improved skills,
better materials.
22
Interpreting patterns in control charts
• Hugging the center line:
sample taken over various machines canceling out the variation
within the sample.
• Hugging the control limits:
sample taken over various machines not canceling out the
variation within the sample.
• Instability:
difficult to identify causes. Typically, over-adjustment of
machine.
• Always, R-chart analysis before the x-bar-chart analysis.
23
Interpreting patterns in control charts
Downward trend in R-chart…
24
Interpreting patterns in control charts
….causes smaller variation in x-bar-chart.
25
Process capability
• Particularly process characteristics study: Performed over a
period of time under actual operating conditions to capture the
variations in material and operators.
• Here we differentiate between the control limits and specified
tolerance limits (also called specification limits).
• Control limits are obtained from the data itself. Probability of
finding similar data within the calculated control limits can be
found.
• Specification limits are specified by product/process designers.
• Process capability study checks the probability of getting
product dimension within the specified limits.
26
Process capability
• Note that we are interested in finding individual product
dimensions within limits (and not the sample means).
• We need to use the estimate for population variance and
not variance of sample mean.
• From the previous analyses, we have:
R
 S`
Population std.dev. :   or 
c2
d2


Std. dev. of sample mean :
n
27
Process capability
• Let the specified tolerance limits be (a, b).
• To determine that the process of capable of producing parts
within these limits, we calculate the probability of finding
parts outside these limits. Hence, calculate –
bx
x a
z1   and z2  


• We calculate the probability area to the right of z1 and left
of z2 from the standard normal table.
• Addition of these two probabilities gives us the probability
that a part will not meet the specifications.
28
Process capability
From the numerical example, let a  16.0, b  16.75.
x  16.37; S ` 0.31

0.31
For n  5, c2  0.8407.   
 0.37.
0.8407
16.75  16.37
z1 
 1.027
0.37
16.37  16.0
z2 
 1.0
0.37
Prob. to the right of z1 is 0.5 - 0.348  0.152
Prob. to the left of z 2 is 0.5 - 0.341  0.159
Total prob. that product does not meet the specificat ion  31.1%
29
Process capability
From the numerical example, let a  16.0, b  16.75.
x  16.37; R  0.85
 0.85
For n  5, d 2  2.326.   
 0.365.
2.326
16.75  16.37
z1 
 1.041
0.365
16.37  16.0
z2 
 1.013
0.365
Prob. to the right of z1 is 0.5 - 0.3508  0.149
Prob. to the left of z 2 is 0.5 - 0.3438  0.156
Total prob. that product does not meet the specificat ion  30.5%
30
Process capability
• PCI – Process Capability Index
UTL  LTL
.

6
For the numerical example :
b  a 16.75  16.0
Cp   
 0.337.
6
6 * 0.37
Cp 
US  16.37  3 * 0.37  17.48; LS  16.37  3 * 0.37  15.26
Alternatel y,
16.75  16.0
Cp 
 0.337.
17.48  15.26
31
The p-Chart
• The control charts we looked at so far were applicable to
quality measurements that possessed continuous probability
distribution.
• This sampling is commonly referred to as “sampling by
variables.”
• In many cases we merely want to assess whether or not a
certain item is defective.
• We can observe a number of defective items from a particular
sample of series of samples. This is referred as “sampling by
attributes.”
32
The p-Chart
• Suppose a series of k independent samples, each of size n, is
selected for a particular process.
• Let p denote the proportion of defective items in the
population (total production for a certain time period) for a
process in control.
• Let Xi denote the number of defective items in the ith sample.
• Then Xi is a binomial variable, assuming random sampling
from a large lot, with mean E(Xi) = np, and Var(Xi) = np(1-p).
• We typically consider the fraction of defective items in a
sample (Xi/n) rather than the observed number of defectives.
33
The p-Chart
All sample fractions should lie within th ree standard deviations
of their mean, or in the interval
p (1  p )
p3
n
since, E  X i / n   p and Var  X i / n   p (1  p ) / n.
Since p is unknown, we must estimate the control limits,
using the data from k samples.
The unbiased estimator of p :
k
total number of defective observed
p

total sample size
X
i 1
nk
i
.
34
The p-Chart
• Thus the estimated control limits are given by:
p(1  p)
p 3
.
n
35
The c-Chart
• In many quality control problems the particular items being
subjected to inspection may have more than one defect.
• We may wish to count # of defects instead of merely
classifying at item as to whether or not it is defective.
• If Ci denotes the # of defects observed in the ith inspected
item, we can safely assume that Ci has a Poisson distribution.
• Let this Poisson distribution have a mean of λ for a process in
control.
• For us to be 99.74% sure, almost all the Ci’s should fall within
three standard deviations of the mean if the process is in
control.
36
The c-Chart
Since E Ci    and Var Ci    , the control limits are :
  3 .
If k items are inspected, then the unbiased estimator of  is :
1 k
C   Ci .
k i 1
The estimated control limits then become :
C 3 C .
37
Acceptance sampling by attributes
• Inspections in which a unit of product is classified simply as
defective or non-defective is called inspection by attributes.
• Lots of items, either raw materials or finished products, are
sold by a producer to consumer with some guarantee for
quality.
• We determine quality, here, by the proportion p of defective
items in the lot.
• To check this characteristics of the lot, the consumer will
sample some of the items, test them, and observe the number
of defectives.
38
Acceptance sampling by attributes
• Generally, some defectives are allowed because defective-free
lots may be too expensive for the consumer to purchase.
• But if the number of defectives is too large, the consumer will
reject the lot and return it to the producer.
• Sampling is generally used because the cost of inspecting the
entire lot may be too high or the inspection process may be
destructive.
• Before sampling inspection takes place for a lot, the consumer
must have in mind a proportion p0 of defectives that will be
acceptable.
• Thus, if the true proportion of defectives p is no greater than
p0, the consumer wants to accept the lot, and reject otherwise.
39
Acceptance sampling by attributes
• This maximum proportion of defectives satisfactory to the
consumer is called the Acceptable Quality Level (AQL).
• So, we have a hypothesis-testing problem.
• We are testing null hypothesis of the true defectives
proportions being less than or equal to the acceptable
proportion; against an alternative hypothesis that the true
proportion is actually greater than the acceptable one.
H o : p  p0 against H a : p  p0 .
40
Acceptance sampling by attributes
• A random sample of n items is selected from the lot, and the
number of defectives Y is observed.
• The decision concerning rejection or non-rejection of Ho (and
consequently rejecting or accepting the lot) will be based on
the observed value of Y.
• The probability of Type I error (α) in this problem is the
probability that the lot is rejected by the consumer when, in
fact, the proportion of defectives is satisfactory to the
consumer.
• This is referred to as the producer’s risk.
41
Acceptance sampling by attributes
• The value of α calculated at p = p0 is the upper limit to the
proportion of good lots rejected by the sampling plan being
considered.
• The probability of Type II error β calculated for some
proportion of defective p1, where p1 > p0, represents the
probability that an unsatisfactory lot will be accepted by the
consumer.
• This is referred to as the consumer’s risk.
• The value of β calculated at p1 = p is the upper limit to the
proportion of bad lots accepted by the sampling plan being
considered, for all p  p1 .
42
Acceptance sampling by attributes
• The null hypothesis will be rejected if the observed value of Y
is larger than some constant a.
• Since the lot is accepted if Y is less than or equal to a, the
constant a is called the acceptance number.
• In this context, the significance level α is given by:
  PReject H 0 when p  p0 
 PY  a, p  p0 
 1    PNot rejecting H 0 when p  p0 
 PAccepting the lot when p  p0   P( A)
43
Acceptance sampling by attributes
• Since p0 may not be known precisely, it is of interest to see
how P(A) behaves as a function of true p, for a given n and a.
• A plot of these probabilities is called an operating
characteristic curve (OC curve).
44
Acceptance sampling by attributes
Example of OC curve plotting:
• Suppose that one sampling plan calls for (n = 10, a = 1) while
another calls for (n = 25, a = 3). Plot the OC curve for both of
them.
• If the plant using these plans can operate with 30% defective
raw materials, considering the price, but cannot operate
efficiently if the proportion gets close to 40%, which plan do
you recommend?
Solution: We need to calculate the P(A) for various values of p.
• For each plan, P(A) is the binomial probability of finding at
the most a number of defects in n trials.
45
OC curve plotting
• For the first sampling plan, we have:
p
0
0.1
0.2
0.3
0.4
0.6
P(A)
1
0.736
0.376
0.15
0.046
0.002
0.4
• For the second plan:
p
0
0.1
0.2
0.3
P(A)
1
0.764
0.234
0.033 0.002
46
OC curve plotting
47
OC curve plotting
• Note that OC curve for first plan (n = 10, a = 1) drops slowly
and does not get appreciably small until p is in the
neighborhood of 0.4.
• For p around 0.3 this plan still has fairly high probability of
accepting the lot.
• The other curve for the second plan (n = 25, a = 3) drops
much more rapidly and falls to a small P(A) at p = 0.3.
• Hence, the second plan would be better for inspection if the
plant can afford to sample n = 25 items out of each lot before
making a decision.
48
Deming’s kp rule
• Objective is to minimize the average total cost of inspection of
incoming materials and final products for processes that are
stable.
• It calls for 0% or 100% inspection.
• It can be shown that, if the process is stable, the distribution of
the nonconforming items in the sample is independent of the
distribution of nonconforming items in the remainder of the
lot.
• Items are submitted in lots, and a random sample is drawn
from that lot to make a decision regarding the entire lot.
49
Deming’s kp rule
Assumptions
• Inspection process is completely reliable.
• All items are inspected prior to moving forward to the next
customer in the process.
• The vendor provides the buyer with an extra supply of items in
order to replace any nonconforming items that are found. The
cost is assumed to included in the contract.
50
Deming’s kp rule
Notations
• p: average proportion of nonconforming items in the lots
• k1: cost of initial inspection of an item
• k2: cost of repair or reassembly due to the usage of a
nonconforming item
• k: average cost to find a conforming item from the additional
supply to replace a detected nonconforming item = k1/(1-p)
• xi =1 if item i is nonconforming; 0 otherwise.
51
Deming’s kp rule
• The unit cost of initial inspection and replacement, if an
item is nonconforming, is given by:
k1  kxi if item i inspected
C1  
if item i not inspected
0
• Unit cost to repair and replace a nonconforming item:
(k2  k ) xi if item i not inspected
C2  
if item i inspected
0
• These costs components are mutually exclusive (if one is
present, other is zero.)
52
Deming’s kp rule
• If the item is inspected, the total cost is (k1+kxi). If the item is
not inspected, the total cost is (k2+k)xi.
• It can be extended to the whole lot, where the proportion of
nonconforming items is p.
• So, the average total cost per item if items are inspected is
(k1+kp). And the average total cost per item if items are not
inspected is (k2+k)p.
• The break-even point for p is obtained by equating the average
total costs.
• We get: p = k1/k2.
53
Deming’s kp rule
• If k1/k2 is greater than p, conduct no inspection. This situation
occurs if the proportion of nonconforming items is low, cost of
inspection is high, and cost of repairing a nonconformity is
low.
• If k1/k2 is less than p, conduct 100% inspection. In this
situation, it is quite expensive for a nonconforming item to be
allowed into production.
• If k1/k2 is equal to p, either no or 100% inspection may be
conducted. Usually, if the estimated value of p is not very
reliable, 100% inspection is conducted.
54
Critique of the kp rule
• Estimating proportion p requires sampling. Contradiction: for
sampling, initially some inspection would have to be carried
out.
• Proportion p may not be stationary over a long period of time.
Once again, sampling may be needed to verify.
• Estimating the costs is not easy. Linearity of costs may not
hold.
• What 100% inspection policy, if it involves destructive
testing?
55
Acceptance sampling by variables
• At times the characteristics under study involves a
measurement on each sampled item.
• In these cases, one could base the decision to accept or reject
on the measurements themselves, rather than simply on the
number of defective items.
• Defective items would mean those items having measurements
that do not meet the standards.
• When looking at actual measurements obtained from variables
method, the experimenter may gain some insights into degree
of nonconformance and may be able to quickly suggest
methods of improvements.
56
Acceptance sampling by variables
• For variables, arriving at the correct sample size and correct
acceptance factor is slightly tedious and depends on the nonstandard statistical tables.
• US Military standards (MIL-STD-414) is one such standard
for calculating the same.
57