Transcript Chapter4a Process Capability

Chapter 5a Process Capability
This chapter introduces the topic of process
capability studies. The theory behind
process capability and the calculation of Cp
and Cpk is presented
Specification Limit &
Process Limit
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Look at indv. values and avg. values of x’s
Indv x’s values n = 84 - considered as population
Avg’s
n = 21 - sample taken
X and X x = same (in this case)
Normally distributed individual x’s and
 x avg. values
having same mean, only the spread is different  >
σ
Relationship σ x 
= popu. Std. dev. of avg’s
n
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If n = 5 = 0.45
 =  x popn std dev of indiv. x’s
SPREAD OF AVGS IS HALF OF SPREAD FOR INDV.
VALUES
Relationship between
population and sample values
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Assume Normal Dist.
‘Estimate’ popu. std. dev.
ˆ
s
c4

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4(n  1)
c4  4n  3 ; n = 84
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 σˆ

4.16 (c4 = 0.99699)
0.99699
= 4.17
x


n

= 2.09
4.17
4
Central Limit Theorem
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‘If the population from which samples are
taken is NOT normal, the distribution of
SAMPLE AVERAGES will tend toward
normality provided that sample size, n, is at
least 4.’
Tendency gets better as n
Standardized normal for distribution of
averages
Z = x μ
σ
n
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Central Limit Theorem
is one reason why
control chart works
No need to worry about
distribution of x’s is not
normal, i.e. indv.
values.
Averages distribution
will tend to ND
Control Limits &
Specifications
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Control limits - limits for
avg’s, and established
as a func. of avg’s
Specification limits allowable variation in
size as per design
documents e.g. drawing
 for individual values
estimated by design
engineers
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Control limits, Process spread, Dist of
averages, & distribution of individual values
are interdependent. – determined by the
process
C. Charts CANNOT determine process meets
spec.
Process Capability &
Tolerance
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When spec. established without knowing
whether process capable of meeting it or
not serious situations can result
Process capable or not – actually looking
at process spread, which is called process
capability (6)
Let’s define specification limit as tolerance
(T) : T = USL -LSL
3 types of situation can result
the value of 6 < USL-LSL
the value of 6 = USL - LSL
the value of 6 > USL - LSL
Case I and Case II situations
Case 3 situation
Process Capability
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2.
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5.
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1.
2.
3.
4.
5.
Procedure (s – method)
Take subgroup size 4 for 20 subgroups
Calculate sample s.d., s, for each subgroup
Calculate avg. sample s.d. s = s/g
Calculate est. population s.d. ˆ o  s c 4
Calculate Process Capability = 6ˆ
R - method
Same as 1. above
Calculate R for each subgroup
Calculate avg. Range, R= R/g
Calculate σˆ o  R d2
Calculate Calculate 6 ˆ o
Process Capability (6) And
Tolerance
Cp - Capability Index
T = U-L
Cp = 1 
Case II 6 = T
Cp > 1 
Case I 6 < T
Cp < 1 
Case III 6 > T
Usually Cp = 1.33 (de facto
std.)
 Measure of process
performance
 Shortfall of Cp - measure
not in terms of nominal or
target value >>> must use
Cpk
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Formulas
Cp = (T)/6
Cpk = Z(min)
3
Z (USL) =
USL  x
σ
x  LSL

Example
Determine Cp and Cpk for a
process with average 6.45,
 = 0.030, having USL =
6.50 , LSL = 6.30 -- T = 0.2
L
6.30
U
T
6.45
x =
6.50
Solution
Cp= T/6= 0.2/6(0.03)=1.11
Cpk = Z(min)/3
Z(U) = (USL -x)/ 
=
6.50-6.45)/0.03 = 1.67
Z(L) = (x –LSL)/  = 6.456.30)/0.03 = 5.00
Cpk = 1.67/3 = 0.56
Process NOT capable since
not centered. Cp > 1 doesn’t
mean capable. Have to
check Cpk
Comments On Cp, Cpk
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Cp does not change when process center
(avg.) changes
Cp = Cpk when process is centred
Cpk  Cp always this situation
Cpk = 1.00 de facto standard
Cpk < 1.00  process producing rejects
Cp < 1.00  process not capable
Cpk = 0  process center is at one of spec.
limit (U or L)
Cpk < 0  i.e. – ve value, avg outside of limits
Exercise
1.
2.
Find Cp, Cpk
x = 129.7 (Length of
radiator hose)
 = 2.35
Spec. 130.0  3.0
What is the %
defective?
Find Cp, Cpk
Spec.U = 58 mm
L = 42 mm
 = 2 mm
When = 50
When = 54
3. Find Cp, Cpk
U = 56
L = 44
=2
When = 50
When = 56