ISO 16337:2021

INTERNATIONAL ISO
STANDARD 16337
First edition
2021-04
Application of statistical and related
methods to new technology and
product development process —
Robust tolerance design (RTD)
Application des méthodes statistiques et des méthodes liées aux
nouvelles technologies et de développement de produit — Plans
d'expériences robustes
Reference number
ISO 16337:2021(E)
©
ISO 2021

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ISO 16337:2021(E)

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© ISO 2021
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ISO 16337:2021(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Robust tolerance design . 2
4.1 General . 2
4.2 RTD experimentation . 4
4.2.1 Data generation . . 4
4.2.2 Experimental design for data collection . 4
4.2.3 Analysis of variance . 7
4.3 Tolerance determination .10
4.3.1 Estimating total variance if tolerance is changed .10
4.3.2 Deciding tolerance .11
5 RTD case study (1) — Stabilizing a circuit by using theoretical formula .12
5.1 Experimentation .12
5.1.1 Objective . .12
5.1.2 Experimental design for data collection and analysis of variance .13
5.2 Tolerance determination .16
6 RTD case study (2) — Stabilizing the piston by using a simulation experiment .18
6.1 Experimentation .18
6.1.1 Objective . .18
6.1.2 Experimental design for data collection and analysis of variance .18
6.2 Tolerance determination .21
Bibliography .26
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ISO 16337:2021(E)

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
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This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 8, Application of statistical and related methodology for new technology and product
development.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/ members .html.
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ISO 16337:2021(E)

Introduction
The designer of a product typically decides the specifications of the product and passes them on to the
manufacturing section for use in manufacturing the product. The specifications include the designed
nominal values and tolerances for the parts and/or elements of the product. The optimum nominal
values of the design parameters are determined by robust parameter design (RPD), and the optimum
tolerances are determined by robust tolerance design (RTD).
RPD, as described in ISO 16336, is applied to the product prior to RTD. In RPD, the major noise factors are
used to evaluate robustness as measured by the signal-to-noise ratio, which represents the variability
of product output. It is a measure for comparing robustness between levels of control factors. RPD
identifies the combination of the values of the design parameters as an optimum RPD condition for
minimizing the variability, that is, maximizing the robustness.
RTD, as described in this document, is a method for selecting the degree of errors of the parts or
elements of the product from the viewpoint of variability under the optimum RPD condition, that is,
the combination of optimum nominal values of the design parameters. If a manufactured product has
errors from the designed nominal values, the product output will deviate from the designed value. The
error in a design parameter should be smaller than the designed error limit to keep the product output
within the designed variability. This is why the design parameters need a tolerance.
The design of a product can be finalized by setting the optimum error limits of the design parameters
by using RTD. The expected variance in output of a product manufactured with errored parts or
elements can be estimated using RTD. After RPD is used to identify a set of optimum values for the
design parameters, RTD is used to check whether the estimated variance is smaller than the target
variance under the optimum RPD condition.
RPD can be used to set the optimum nominal values of the design parameters without increasing
manufacturing cost while RTD is closely related to the manufacturing cost. Smaller tolerances, meaning
higher-grade parts or elements, result in higher costs, while larger tolerances, meaning lower-grade
parts or elements, result in lower costs. To finalize the product design, the cost of manufacturing the
product is considered. The loss function in the Taguchi methods is used to transform the benefits of an
improvement in quality into a monetary amount, the same as a cost.
The cost of the improvement and the benefits of the improvement in quality should be balanced in
deciding the tolerances. RPD and RTD together provide a cost-effective way of optimizing product design.
If RPD cannot achieve the product variability smaller than the target variability, the tolerances of the
design parameters are reduced to improve the variability, but smaller tolerances result in higher costs.
On the other hand, if RPD can achieve the product variability much smaller than the target variability,
the tolerances of the design parameters are increased to reduce manufacturing cost, so larger tolerances
result in lower costs.
Products manufactured with optimum nominal values and tolerances of design parameters are robust
to noise situations under usage conditions after shipment. Robust products minimize users’ quality
losses due to defects, failures, and quality problems.
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INTERNATIONAL STANDARD ISO 16337:2021(E)
Application of statistical and related methods to new
technology and product development process — Robust
tolerance design (RTD)
1 Scope
This document specifies guidelines for applying the robust tolerance design (RTD) provided by the
Taguchi methods to a product in order to finalize the design of the product.
NOTE 1 RTD is applied to the target product to set the optimum tolerances of the design parameters around
the nominal values. RTD identifies the effects of errors in the controllable design parameters on product output
and estimates the total variance of the product output if the tolerances are changed. Hence, RTD achieves the
target variance of the output from the viewpoints of robustness, performance, and cost.
NOTE 2 The tolerance expresses a maximum allowable error in the value of a design parameter in the
manufacturing process. In a perfect world, the parts or elements of every product have the designed nominal
values of the design parameters. However, actual manufacturing does not reproduce the exact designed nominal
values of the design parameters for all products. The actual products have errors in the values of their parts or
elements. These errors are supposed to be within the designed tolerances.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 16336, Applications of statistical and related methods to new technology and product development
process — Robust parameter design (RPD)
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 16336 apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// www .electropedia .org/
3.1
tolerance
difference between the upper specification limits and lower specification limits
3.2
robust tolerance design
RTD
method of setting optimum tolerances from the viewpoints of robustness, performance, and cost
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ISO 16337:2021(E)

4 Robust tolerance design
4.1 General
A company’s product design section normally gives the specifications of a product, that is, the nominal
values and tolerances of the design parameters, to the manufacturing section. The manufacturing
section uses the designed specifications in manufacturing the product. When specifications specify the
limits of a design parameter as m±Δ , the parameter value x in the manufacturing process should
satisfy the following restriction:
mx−≤ΔΔ≤+m , (1)
where m and Δ denote a nominal value and its permissible difference, respectively. Only the symmetric
(±Δ ) case is discussed in this document. In the symmetric case, the tolerance is 2Δ, and the permissible
difference Δ is half the tolerance.
If the absolute error of a design parameter is larger than the specified permissible difference Δ, the
variability in the product output cannot meet the designed performance and specifications.
RTD is used by the design section to set the optimum tolerance for each design parameter to achieve
the designed performance, which is evaluated based on the total variance of the product output. The
permissible difference of a design parameter is the maximum allowable error around the nominal value
in the manufacturing process, and it is closely related to the cost of manufacturing.
The optimum nominal values of the design parameters can be identified by robust parameter design
[1]
(RPD) through robustness measure, signal-to-noise ratio . The selection of a robust product by setting
the nominal values as the optimum values using RPD prior to RTD is highly recommended. RPD can
optimize the target product by choosing the optimum combination of design parameter nominal values
[2]
from the viewpoint of the variability of the product output without increasing the cost .
If RPD cannot achieve a target variability, RTD is used to identify possible tolerances for achieving the
target variability even at a higher cost. Smaller tolerances result in smaller variability, but this requires
upgrading the parts or elements of the product, which leads to higher manufacturing cost. RTD is used
to investigate the balance between product quality and improvement cost.
Even if RPD achieves the target variance, RTD is used, in some cases, to identify larger tolerances than
those considered in RPD. Larger tolerances mean larger variability, but if the increased variability
satisfies the target variability, the larger tolerances are applicable as they lead to reduced cost of
manufacturing the designed product.
The purpose of RTD is to achieve the target variability by setting optimum tolerances from the
viewpoints of robustness, performance, and cost. For this purpose, RTD estimates the total variance of
the output of the designed product if the tolerance of a design parameter is changed. The total variance
can be estimated based on the results of analysis of variance (ANOVA).
Assume that a value x of design parameter F has a linear effect on output y of the product, as shown in
Figure 1 a). If the present permissible difference of x in F is Δ = Δ, the error distribution of F affects
P
output y with a magnitude of βΔ. If the permissible difference Δ of F is reduced to new permissible
difference Δ = λΔ [λ<1 in Figure 1 a)], the effect of changing Δ in F on the output is reduced to λβΔ,
N
and the variance in y due to changing Δ in F is reduced from the present variance V to new variance
FP
2
V = λ V . As a result, the total output variance is reduced from V to V [Figure 1 b)].
FN FP TP TN
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ISO 16337:2021(E)

a) Linear dependence on x in F b) Change in total variance
Figure 1 — Effect of changing Δ in design parameter F on total variance
The new total variance V can be estimated as
TN
2
VV=+VV=+λ V , (2)
TN FN eFPe
Δ
N
where λ= is assumed.
Δ
P
If the tolerance of a design parameter is reduced, that is, λ<1, the magnitude of error of the design
parameter becomes smaller, and the total output variance is reduced. A smaller tolerance means that
an upgraded part or element is used, so the cost of producing the new design can be higher than that of
the present design.
If the tolerance of a design parameter is enlarged, that is, λ>1, the magnitude of error of the design
parameter becomes larger, and the total output variance is enlarged. A larger tolerance means that a
down-graded part or element is used, so the cost of producing the new design can be smaller than that
of the present design.
RTD comprises two steps, as follows.
1) RTD experimentation: Collect data on the designed product, and analyse the data to determine the
dependence of the product output on the design parameters.
2) Tolerance determination: Estimate the total variance if a tolerance is changed, and compare the
effects in quality with the cost of the change to identify the optimum tolerance.
RTD experiments collect the output data of the designed product in which there are errors in the product
design parameters, and estimate the total variance and its dependence on the design parameters. The
experimental design plan is used to collect the data under the combination of design parameter errors.
The ANOVA results show the effects of errors in the design parameters on the product output. The
product output has a target variance from the viewpoints of robustness and performance.
In RTD experiments, the design parameters are taken as noise factors. A noise factor is an experimental
factor which is taken into experiment for the purpose of estimating its variability. The variance in the
linear effect of errors in the design parameters is estimated.
In RPD, on the other hand, the design parameters are taken as control factors. A control factor is an
experimental factor which is taken into experiment for the purpose of selecting the optimum level of the
factor. Designers can fix the nominal values of design parameters to the optimum RPD values. However,
in actual manufacturing, the parts or elements of the product invariably have errors, so the designer
cannot specify the error of a design parameter. The designer can set only the permissible difference Δ
as an error limit.
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ISO 16337:2021(E)

Design parameter errors cause variability in product output. If the error of a design parameter has a
linear effect on the product output, the output variance can be changed by resetting the tolerance of the
design parameter. RTD experimentation is used to determine the contributions of the effects of errors
in design parameters to the product output.
In the tolerance determination step of RTD, the change in the output variance due to resetting a
tolerance is estimated, and the designer selects optimum tolerance for achieving the target output
variance. The optimum tolerance can be determined by balancing the effect in quality due to a tolerance
[3]
change against the cost of the tolerance change .
4.2 RTD experimentation
4.2.1 Data generation
RTD experimentation is used to determine the design parameters’ linear effects for the designed
product. The relationship between the output by the product and the errors in the design parameters is
investigated. The output data can be generated in three ways:
1) by using a theoretical formula,
2) by experimentation with an actual product;
3) by simulation experimentation.
When the theoretical relationship between the product output and the design parameters is known,
the output data can be directly calculated for various combination of the design parameter values.
RTD offers multi-factor design as an experimental design for generating the output data in various
combinations of the level of experimental factors, as shown in case study (1) in Clause 5. ANOVA is used
for analysing the dependence of the product output on the factors.
Mathematical analysis can be applied in this case. Mathematical analysis consists of using variance
estimates for a system by, for example, propagating an input variance through the system via Taylor
[4]
series expansions of moment generating functions .
If an actual product can be constructed, it can be used for experimentation, and the data output can be
collected using the actual experiment. However, in many cases, it is difficult to set the intended levels
of the errors of design parameters in an actual product because the noise levels cannot be controlled
within the error distribution of the design parameters. Simulation experimentation can be used in such
cases. This is why simulation experiments are often used in RTD. A simulation program can provide the
product output data, as shown in case study (2) in Clause 6.
4.2.2 Experimental design for data collection
RTD experimentation is used for collecting output data for the designed product under the combinations
of design parameter errors. There are many design parameters, and a multi-factor experimental design
is used to generate various such combinations. The purpose of RTD experimentation is to determine
the main effects of experimental factors. An orthogonal array plan is recommended as a multi-factor
experimental plan for collecting the data as it is an efficient way to collect data for an RTD experiment.
An orthogonal array plan can reduce the number of experimental runs compared with a full-factorial
plan for the same number of factors and can assign the maximum number of factors in a plan for the
same number of experimental runs. The main effects of factors can be estimated under the condition of
a balanced combination of the other factors’ levels. The choice of the orthogonal array depends on the
[3]
numbers of factors and their levels .
An example orthogonal array (L ) is shown in Table 1. Seven experimental factors with three levels
18
(B-H) and one factor with two levels (A) can be assigned to the columns in the array. Rows represent
the experimental run. The number in each cell represents the level of the factor assigned to the column.
The experimental run of low No. 1 should be performed under the combination of factor’s levels
A1B1C1D1E1F1G1H1.
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ISO 16337:2021(E)

For RTD, the design parameters are assigned to the columns as noise factors. For the purpose of
estimating the linear and non-linear effects of a factor, each factor has at least three levels. However,
if the proportional property is obvious for a factor, a two-level setting is sufficient. Two-level factor
is assigned to the first column. The last column in Table 1 shows the output data y calculated for the
i
combination of factors’ levels shown in the cells in the same low
Table 1 — Example of orthogonal array L and output data
18
Column 1 2 3 4 5 6 7 8 Data
output

A B C D E F G H
No.
1 1 1 1 1 1 1 1 1 y
1
2 1 1 2 2 2 2 2 2 y
2
3 1 1 3 3 3 3 3 3 y
3
4 1 2 1 1 2 2 3 3 y
4
5 1 2 2 2 3 3 1 1 y
5
6 1 2 3 3 1 1 2 2 y
6
7 1 3 1 2 1 3 2 3 y
7
8 1 3 2 3 2 1 3 1 y
8
9 1 3 3 1 3 2 1 2 y
9
10 2 1 1 3 3 2 2 1 y
10
11 2 1 2 1 1 3 3 2 y
11
12 2 1 3 2 2 1 1 3 y
12
13 2 2 1 2 3 1 3 2 y
13
14 2 2 2 3 1 2 1 3 y
14
15 2 2 3 1 2 3 2 1 y
15
16 2 3 1 3 2 3 1 2 y
16
17 2 3 2 1 3 1 2 3 y
17
18 2 3 3 2 1 2 3 1 y
18
Table 2 shows an example of level setting of factors for RTD. The upper and lower permissible differences
are assumed to be the same for simplicity. The levels of the factors are set around nominal value m with
level width d. Nominal value m is set to an optimum value by RPD from the viewpoint of robustness.
Level width d is set from the actual standard deviation of the design parameter if it is known.
Table 2 — Example of level settings of factors for RTD
Factor 1 2 3
A m − d m + d —
A A A A
B m − d m m + d
B B B B B
C m − d m m + d
C C C C C
D m − d m m + d
D D D D D
E m − d m m + d
E E E E E
F m − d m m + d
F F F F F
G m − d m m + d
G G G G G
H m − d m m + d
H H H H H
When the actual standard deviation σ of the error in the design parameter is not exactly known, the
x
Δ Δ
assumption σ = or σ = can be applied.
x x
2 3
When the actual standard deviation σ of the error in the design parameter is known, the level width d
x
and the levels of the factors are set as follows.
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ISO 16337:2021(E)

For a two-level factor, d=σ :
x
X1: First level      xm=−σ , (3)
1 x
X2: Second level      xm=+σ . (4)
2 x
3
For a three-level factor, d= σ :
x
2
3
X1: First level      xm=−dm=− σ , (5)
1 x
2
X2: Second level      xm= , (6)
2
3
X3: Third level      xm=+dm=+ σ . (7)
3 x
2
2
Setting the level of the factors in this way makes the estimated variance σ of output y caused by the
y
22
linear effect of the error in the factor βσ , where β represents the linear coefficient of the relationship
x
yx=β between output y and input x.
If yi==11,,nj,, ,r represents the output from j-th run in r repeated runs on i-th level x in n
()
ij i
level factor, the linear coefficient β and the sum of squares of linear effect S are calculated as
β
n r
()xx−−()yy
∑∑
iij
i=1 j=1
β= , (8)
n
2
rx()−x
∑
i
i=1
2
n r
 
 
()xx−−()yy
∑∑
iij
n
 
i=1 j=1 
2 2
S = =−rx()x ⋅β . (9)
∑
β i
n
i==1
2
rx()−x
∑
i
i=1
For a two-level factor A with levels xx=−d and xx=+d , the sum of squares of linear effect S is
1 2 β
22
calculated as Sr=⋅2d β . If the linear effect of factor A is significant, S approximately represents
β β
2 2
2rσ , where 2r denotes the number of data items and σ denotes the variance of each. If level width
y y
22 22 2 2
d is set to σ , Sr==22drβσ βσ≅2r . Then variance σ in output y caused by the linear effect
x β xy y
22 2
of the error in the factor becomes σβ= σ .
yx
For a three-level factor B with levels xx=−dx, =x , and xx=+d , the sum of squares of linear effect
12 3
22
S is calculated as Sr=⋅2d β . If the linear effect of the factor is significant, S approximately
β β β
2 3
represents 3rσ , where 3r denotes the number of data items. If the level width d is set to σ ,
y x
2
3
22 22 22 2 2
Sr==22drβσ⋅⋅βσ=≅33rrβσ . Then variance σ in output y caused by the linear effect of
β xx y y
2
22 2
the error in the noise factor becomes σβ= σ .
yx
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ISO 16337:2021(E)

4.2.3 Analysis of variance
ANOVA is used to identify the linear effects of the factors and the ratios of their contributions to the
total variance.
The ANOVA calculations for orthogonal array L are as follows.
18
Total sum of squares:
18
2
()y
∑
i
18 18
i=1
2 2
Sy=−()yy=− . (10)
∑∑
T i i
18
i==1 i 1
The total sum of squares is decomposed into sum of squares S of the linear effect of each factor and
•
sum of squares S of the error as follows:
e
SS=+SS++SS++SS++SS+ . (11)
TA BCll DEll FGll Hl e
For calculating the factor effects, the sum of the data for each factor level is calculated:
Yy=+yy++yy++yy++yy+
A1 12 34 56 789
Yy=+y +++yyy ++y yyyy++
A2 10 11 12 13 14 15 16 17 18
Yy=+yy+ +++yy y
B1 1 23101112
Yy=+yy++yy++y (12)
B2 45 61314 115
Yy=++yy +++yyy
B3 789 16 17 18

Yy=+yy++yy++y
H3 3 47121417
Table 3 summarizes the calculated sums of data.
Table 3 — Sums of data for each factor level
Sum of data
Factor
Level 1 Level 2 Level 3
A Y Y -
A1 A2
B Y Y Y
B1 B2 B3
C Y Y Y
C1 C2 C3
D Y Y Y
D1 D2 D3
E Y Y Y
E1 E2 E3
F Y Y Y
F1 F2 F3
G Y Y Y
G1 G2 G3
H Y Y Y
H1 H2 H3
For a two-level fact
...