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SEMI M56-1103 © SEMI 2003 5 RELATED INFORMATION 1 EXTENSIONS OF THE METHODOLOGY NOTICE: This relate d informati on is not an official part of M56. It was derived fr om task force deli b erations during the developm ent o…

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SEMI M56-1103 © SEMI 2003 4
7.4 Calculate
α
and
β
as follows (Note 4). Note that
the symbols in the equations for
α
and
β
have the
following meanings:
f (x) = PDF of process characteristic x,
USL = upper specification limit,
LSL = lower specification limit, and
Φ(u) = Gaussian CDF (see Equation (1) in Section
7.3).
7.4.1 For a characteristic with only a USL, use the
following equations to calculate
α
and
β
:
∫
∞−
+−
Φ−=
USL
M
xxf
xUSL
d)(1
σ
δ
α
∫
∞
+−
Φ=
USL
M
xxf
xUSL
d)(
σ
δ
β
7.4.2 For a characteristic with only an LSL, use the
following equations to calculate
α
and
β
:
∫
∞
+−
Φ=
LSL
M
xxf
xLSL
d)(
σ
δ
α
∫
∞−
+−
Φ−=
LSL
M
xxf
xLSL
d)(1
σ
δ
β
7.4.3 For a characteristic with both upper and lower
specifications limits, use the following equations to
calculate
α
and
β
:
xxf
xLSL
xxf
xUSL
USL
LSL
M
USL
LSL
M
d)(
d)(1
∫
∫
+−
Φ
+
+−
Φ−=
σ
δ
σ
δ
α
xxf
xLSLxUSL
xxf
xLSLxUSL
USL
MM
LSL
MM
d)(
d)(
∫
∫
∞
∞−
+−
Φ−
+−
Φ
+
+−
Φ−
+−
Φ=
σ
δ
σ
δ
σ
δ
σ
δ
β
NOTE 5: Background information related to the calculation
of
α
and
β
is given in Related Information 3.
7.5 Use a binary decision model. If measurement-
based decisions are labeled as pass or fail and items are
inherently conforming or nonconforming, there are only
four outcomes:
• pass a conforming item,
• fail a conforming item (
α
error),
• pass a nonconforming item (
β
error), and
• fail a nonconforming item.
The probabilities associated with these outcomes are
1−
α
,
α
,
β
, and 1−
β
, respectively.
7.6 To define the cost model, assign costs to each of
the four decisions above on the basis of the business
model used for the manufacturing process as follows:
• c
pc
: cost of passing a conforming item,
• c
fc
: cost of failing a conforming item (
α
error),
• c
pn
: cost of passing a non-conforming item (
β
error), and
• c
fn
: cost of failing a non-conforming item.
7.6.1 Assign zero incremental cost to the two correct
outcomes (c
pc
and c
fn
).
7.6.2 Assign the incremental costs for the error out-
comes, c
fc
and c
pn
, on the basis of the business model
used for the manufacturing process.
7.7 Calculate the cost due to misclassification resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
fc
α
+ c
pn
β
, (2)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).
NOTE 6: See Related Information 4 for an example of this
calculation.
7.8 If it is desired to include costs for correct as well as
incorrect classification, calculate the total cost resulting
from measurement variability based on the assigned
incremental costs given in Section 7.6 and the
frequency of occurrence of
α
and
β
errors as follows:
cost = c
pc
(1
−α
) +
c
fc
α
+
c
pn
β
+ c
fn
(1−
β
), (3)
using the appropriate equations for α and β as given in
Section 7.4, depending on the nature of the
specification (LSL only, USL only, or both).
SEMI M56-1103 © SEMI 2003 5
RELATED INFORMATION 1
EXTENSIONS OF THE METHODOLOGY
NOTICE: This related information is not an official part of M56. It was derived from task force deliberations
during the development of the document. This related information was approved for publication by full letter ballot
procedures on September 3, 2003.
R1-1 Introduction
R1-1.1 In general, when a single characteristic on an
item is measured once on a single gauge and 100%
sampling is employed, the model will take the form
described in this practice.
R1-1.2 If the situation is more complex, the nature of
the model will be different. Factors that can affect the
nature of the model include the following:
• number of items examined (lot acceptance
sampling vs. 100% sampling),
• number of times an item is inspected (single vs.
multiple),
• effect of the inspection process on the item
(destructive vs. non-destructive),
• number of item characteristics examined for a
single decision (one vs. many), and
• cost functions associated with the business
decisions (fixed vs. variable).
R1-1.3 In addition, it is possible to have more than one
set of cost functions for each type of item that is
inspected by the metrology system. Because much of
this is context specific, it would be impossible to cover
all possible models. Other extensions to the model
include relaxation of the assumption of constant
variance, the introduction of variability in the bias, and
the use of guard banding. The foundation of all these
extensions, however, is the use of
α
and
β
.
R1-2 Extension to Multiple Gauges
R1-2.1 To extend the model to multiple gauges, one
must make the additional assumption that all measuring
gauges are measuring the same characteristic.
R1-2.1.1 In addition, define a conforming item as one
that all gauges show the measured characteristic to be
in specification.
R1-2.1.2 A non-conforming item is taken to be one in
which at least one gauge shows the measured
characteristic to be outside of specification.
R1-2.2 It is then possible to define a set of
α
and
β
error rates for each gauge. Let
α
1
,
α
2
, …,
α
n
be the
α
values and
β
1
,
β
2
, …,
β
n
be the
β
values associated with
the n different gauges.
R1-2.3 The overall
α
value,
α
T
, is calculated from the
equation:
α
T
=
∏
−
−−
n
i
i
1
)1(
απ
where:
,d)(
∫
∞−
=
USL
xxf
π
and
f (x) = PDF of the characteristic being measured.
NOTE 1: This equation is based on the probability P that the
item is conforming but that one or more gauges give a
conforming result:
α
T
= P[Item is conforming, ≥1 gauges show nonconforming]
= P[Item is conforming] −
P[Item is conforming, All gauges show conforming]
R1-2.4 The overall
β
value,
β
T
, is calculated from the
equation:
β
T
=
∏
=
n
i
i
1
.
β
NOTE 2: This equation is based on the probability P that the
item is nonconforming but that all gauges give a conforming
result:
β
T
= P[Item is nonconforming, All gauges show pass]
R1-3 Extension to Multiple Inspections with
the Same Metrology System
R1-3.1 Multiple inspection with the same metrology
system is a special case of inspection with multiple
gauges. If the same measurement system is used to
measure the item characteristic repeatedly, one merely
lets
α
i
=
α
and
β
i
=
β
for all i, as the
α
and
β
error rates
will not change for the same gauge.
R1-4 Extension to Decisions Based on
Multiple Characteristics
R1-4.1 It is also possible to develop a model where a
decision is based on more than one characteristic. As
with the case of multiple gauges, several assumptions
must be made.
SEMI M56-1103 © SEMI 2003 6
R1-4.1.1 The characteristics being measured are
independent or an independent combination of their
values is used.
R1-4.1.2 All tests are performed before a decision to
reject or pass is made.
R1-4.1.3 F(x) has been deconvolved from F·G.
R1-4.1.4 A conforming item is defined as one in which
all measured characteristics are shown to be in
specification.
R1-4.1.5 A non-conforming item is taken to be one in
which at least one characteristic is outside of
specification.
R1-4.2 It is then possible to define a set of
α
and
β
errors for each gauge. Let
α
1
,
α
2
, …,
α
n
be the
α
values and
β
1
,
β
2
, …,
β
n
be the
β
values associated with
the n different characteristics.
R1-4.3 The overall
α
value,
α
T
, is calculated from the
equation:
α
T
=
∏∏
==
−−
n
i
n
i
ii
11
)1(
απ
where:
,d)(
∫
∞−
=
USL
ii
xxf
π
and
f
i
(x) = PDF of the characteristic i.
NOTE 3: This equation is based on the probability P that all
characteristics conform, but that at least one test failed:
α
T
= P[All characteristics conform, At least one test failed]
= P[All characteristics conform] −
P[All characteristics conform, All tests passed]
R1-4.4 The overall
β
value,
β
T
, is calculated from the
equation:
β
T
=
∏∏
==
−−
n
i
n
i
ii
p
11
)1(
α
where:
p
i
= proportion of observations within specification
for characteristic i.
NOTE 4: This equation is based on the probability P that one
or more characteristics are nonconforming but that all tests
passed:
β
T
= P[≥1 characteristic nonconforming, All tests passed]
= P[All pass] −
P[All characteristics conform, All tests passed]