Minimum-Cost Tolerance Allocation 14-9
14.7 2-D Example: One-way Clutch Assembly
The application of tolerance allocation to a 2-D assembly will be demonstrated on the one-way clutch
assembly shown in Fig. 14-6. The clutch consists of four different parts: a hub, a ring, four rollers, and four
springs. Only a quarter section is shown because of symmetry. During operation, the springs push the
rollers into the wedge-shaped space between the ring and the hub. If the hub is turned counterclockwise,
the rollers bind, causing the ring to turn with the hub. When the hub is turned clockwise, the rollers slip,
so torque is not transmitted to the ring. A common application for the clutch is a lawn mower starter.
(Reference 5)
c
c
Vector Loop
Ring
Hub
Roller
Spring
φ
φ
b
a
2
e
2
Figure 14-6 Clutch assembly with vector
loop
The contact angle φ between the roller and the ring is critical to the performance of the clutch. Variable
b, is the location of contact between the roller and the hub. Both the angle φ and length b are dependent
assembly variables. The magnitude of φ and b will vary from one assembly to the next due to the variations
of the component dimensions a, c, and e. Dimension a is the width of the hub; c and e/2 are the radii of the
roller and ring, respectively. A complex assembly function determines how much each dimension contrib-

consecutive vectors, but it vanishes identically.
h
x
= 0 = b + c sin(φ) - e sin(φ) (14.4)
h
y
= 0 = a + c + c cos(φ) - e cos(φ) (14.5)
Eqs. (14.4) and (14.5) may be solved for φ explicitly:






−
+
=
−
ce
ca
1
cosφ
(14.6)
The sensitivity matrix [S] can be calculated from Eq. (14.6) by differentiation or by finite difference:
[ S] =






c
b
a
b
eca
φφφ
The tolerance sensitivities for δφ are in the top row of [S]. Assembly variations accumulate or stackup
statistically by root-sum-squares:
(
)
∑
=
2
)(
jij
xS δδφ
= .01159 radians = .664 degrees
where δφ is the predicted 3σ variation, δx
j
is the set of 3σ component variations.
By worst case:
∑
=
jij
xS δδφ
= .01691 radians = .9688 degrees
where δφ is the predicted extreme variation.
14.8 Allocation by Scaling, Weight Factors
Once you have RSS and worst case expressions for the predicted variation δφ, you may begin applying
various allocation algorithms to search for a better set of design tolerances. As we try various combina-

tionality factor P is applied to δa and δe, while δφ is set to the maximum tolerance of ±.017453 radians
(±1° ).
( ) ( ) ( )( ) ( ) ( )
0008.6272.20004.5483.10004.6469.2017453.
017453.
131211
PP
ePScSaPS
xS
jij
++=
++=
∑
=
δδδ
δδφ
Solving for P:
P = 1.0429
δa = (1.0429)(.004)=.00417 in.
δe = (1.0429)(.0008)=.00083 in.
14.8.2 Proportional Scaling by Root-Sum-Squares
(
)
(
)
( ) ( ) ( )
( ) ( )( ) ( )( )( ) ( ) ( )( )
222
2
13

tolerances and scale them to match the 1.0 degree limit. The original tolerances had a ratio of 5:1. The final
ratio will be the product of 1:2 and 5:1, or 2.5:1. The sensitivities do not affect the ratio.
14-12 Chapter Fourteen
(
)
(
)
( )( ) ( ) ( )( )
( ) ( )( )( ) ( )( )( ) ( ) ( )( )( )
222
2
13
2
12
2
11
2
0008.30/206272.20004.5483.10004.30/106469.2017453.
30/2030/10017453.
PP
ePScSaPS
xS
jij
+−+−=
++=
∑
=
δδδ
δδφ
Solving for P:





=
i
i
k/k
k/
i
ii
i
T
SBk
SBk
T
(
)
( ) ( )
2/2
1
2/1
2
11
1
2
1
++
+




+
=
11
1
11
11
1
11
1 i
i
k/k
k/
i
ii
i
ASM
T
SBk
SBk
S
TST
( )
( ) ( )
∑
++
+



SBk
SBk
S
TST
Part Dimension Process Nominal Sensitivity B k Minimum Maximum
(inch) Tolerance Tolerance
Hub a Mill 2.1768 -2.6469 .1018696 .45008 .0025 .006
Roller c Lap .9000 -10.548 .000528 1.130204 .00025 .00045
Ring e Grind 4.0000 2.62721 .0149227 .79093 .0005 .0012
The cost data for computing process cost is shown in Table 14-8:
Table 14-8 Process tolerance cost data for the clutch assembly
Minimum-Cost Tolerance Allocation 14-13
14.9.1 Minimum Cost Tolerances by Worst Case
To perform tolerance allocation using a Worst Case Stackup Model, let T
1
= δa, and T
i
= δe, then S
1
= S
11
,
k
1
= k
a
, and B
1
= B
a

e
k
aa
ee
a
SBk
SBk
ScSaS δδδ
( )
( )( )( )
( )( )( )
( ) ( )
790931450081
7909311
627221018696045008
64692014922779093
62722000454831064692017453
./.
)./(
da . . . a d. . 





++=
The only unknown is δa, which may be found by iteration. δe may then be found once δa is known.

δa = .0038 in. δe = .0012 in. C = $4.30
The relationship between the resulting three pairs of tolerances is very clear when they are plotted as
shown in Fig. 14-7. Tol e and Tol a are plotted as points in 2-D tolerance space. The feasible region is
bounded by a box formed by the upper and lower process limits, which is cut off by the Worst Case limit
curve. The original tolerances of (.004, .0008) lie within the feasible region, nearly touching the WC Limit.
Extending a line through the original tolerances to the WC Limit yields the proportional scaling results
found in section 14.2 (.00417, .00083), which is not much improvement over the original tolerances. The
minimum cost tolerances (OptWC) were a significant change, but moved outside the feasible region. The
feasible point of lowest cost (Mod WC) resulted at the intersection of the upper limit for Tol e and the WC
Limit (.0038, .0012).
Tol a
Tol e
0
0.001
0.002
0.003
0.004
0.005
0 0.002 0.004
Original
Opt WC
Mod WC
WC Limit
Opt WC
Original
Mod WC
WC Limit
Feasible Region
Figure 14-7 Tolerance allocation
results for a Worst Case Model

da .
a d. .






+
+=
Solving for δa by iteration and δe as before:
δa = .00409 in.
( )( )( )
( )( )( )
( )
( ) ( )
790932450082
7909321
00409
62722101869645008
64692014922779093
./.
)./(
. de 

Mod RSS
RSS Limit
Opt RSS
Mod RSS
RSS Limit
Original
Feasible Region
Figure 14-8 Tolerance allocation
results for the RSS Model
( ) ( ) ( )
(
)
( ) ( )
2/22
2/2
13
112
13
2
12
2
11
++
+







design. The optimization preferred to drive Tol e to a much larger value. One way to enlarge the feasible
region is to select an alternate process for dimension e. Instead of grinding, suppose we consider turning.
The process limits change to (.002< δe <.008), with B
e
= .118048 k
e
= 45747. Table 14-9 shows the revised
data.
Table 14-9 Revised process tolerance cost data for the clutch assembly
Part Dimension Process Nominal Sensitivity B k Minimum Maximum
(inch) Tolerance Tolerance
Hub a Mill 2.1768 -2.6469 .1018696 .45008 .0025 .006
Roller c Lap .9000 -10.548 .000528 1.130204 .00025 .00045
Ring e Turn 4.0000 2.62721 .118048 .45747 .002 .008
Milling and turning are processes with nearly the same precision. Thus, B
e
and B
a
are nearly equal as
are k
e
and k
a
. The resulting RSS allocated tolerances and cost are:
δa =.00434 in. δe = .00474 in. C = $2.54
The new optimization results are shown in Fig. 14-9. The feasible region is clearly much larger and the
minimum cost point (Mod Proc) is on the RSS Limit curve on the region boundary. The new optimum point
has also changed from the previous result (Opt RSS) because of the change in B
e
and k

The modified optimization results are shown in Fig. 14-10. The feasible region is the smallest yet due
to the tight Worst Case (WC) Limit. The minimum cost point (Mod Proc) is on the WC Limit curve on the
region boundary.
Tol a
Tol e
0
0.001
0.002
0.003
0.004
0.005
0 0.002 0.004
Original
Opt WC
Mod WC
Mod Proc
WC Limit
Opt WC
Original
Mod WC
Mod Proc
WC Limit
Feasible Region
Figure 14-10 Tolerance allocation
results for the modified WC Model
Cost reductions can be achieved by comparing cost functions for alternate processes. If cost-versus-
tolerance data are available for a full range of processes, process selection can even be automated. A very
systematic and efficient search technique, which automates this task, has been published. (Reference 4)
It compares several methods for including process selection in tolerance allocation and gives a detailed
description of the one found to be most efficient.

eight, the difference between WC and RSS would have been much more significant.
It should be noted that tolerances specified at the process limit may not be desirable. If the process
is not well controlled, it may be difficult to hold it at the limit. In such cases, the designer may want to back
off from the limits to allow for process uncertainties.
14.12 References
1. Chase, K. W. and A. R. Parkinson. 1991. A Survey of Research in the Application of Tolerance Analysis to the
Design of Mechanical Assemblies: Research in Engineering Design. 3(1):23-37.
2. Chase, K. W., J. Gao and S. P. Magleby. 1995. General 2-D Tolerance Analysis of Mechanical Assemblies with
Small Kinematic Adjustments. Journal of Design and Manufacturing. 5(4): 263-274.
3. Chase, K.W. and W.H. Greenwood. 1988. Design Issues in Mechanical Tolerance Analysis. Manufacturing
Review. March, 50-59.
4. Chase, K. W., W. H. Greenwood, B. G. Loosli and L. F. Hauglund. 1989. Least Cost Tolerance Allocation for
Mechanical Assemblies with Automated Process Selection. Manufacturing Review. December, 49-59.
5. Fortini, E.T. 1967. Dimensioning for Interchangeable Manufacture. New York, New York: Industrial Press.
6. Greenwood, W.H. and K.W. Chase. 1987. A New Tolerance Analysis Method for Designers and Manufacturers.
Journal of Engineering for Industry, Transactions of ASME. 109(2):112-116.
7. Hansen, Bertrand L. 1963. Quality Control: Theory and Applications. Paramus, New Jersey: Prentice-Hall.
8. Jamieson, Archibald. 1982. Introduction to Quality Control. Paramus, New Jersey: Reston Publishing.
9. Pennington, Ralph H. 1970. Introductory Computer Methods and Numerical Analysis. 2nd ed. Old Tappan,
New Jersey: MacMillan.
10. Speckhart, F.H. 1972. Calculation of Tolerance Based on a Minimum Cost Approach. Journal of Engineering for
Industry, Transactions of ASME. 94(2):447-453.
11. Spotts, M.F. 1973.Allocation of Tolerances to Minimize Cost of Assembly. Journal of Engineering for Industry,
Transactions of the ASME. 95(3):762-764.
RSS Cost Allocation Results
0 0.002 0.004 0.006
Original
Opt RSS
Mod RSS
Mod Proc

various processes and the relative cost of tightening tolerances. Relative costs were used to eliminate the
effects of inflation. The resulting chart, Table 14A-1, appears in References 7 and 8. Least squares curve
fits were performed at Brigham Young University and are presented here for the first time. The Reciprocal
Power equation, C = A + B/T
k
, presented in Chapter 14, was used as the empirical function. Fig. 14A-1
shows a typical plot of the original data and the fitted data. The curve fit procedure was a standard
nonlinear method described in Reference 9, which uses weighted logarithms of the data to convert to a
linear regression problem. Results are tabulated in Table 14A-2 and plotted in Figs. 14A-2 and 14A-3.
Turn
0
0.5
1
1.5
2
2.5
0 0.002 0.004 0.006 0.008 0.01 0.012
Tolerance
Cost
Size 4: Data
Size 4: Fitted
Size 5: Data
Size 5: Fitted
Size 6: Data
Size 6: Fitted
Figure 14A-1 Plot of cost-versus-tolerance for fitted and raw data for the turning process
Minimum-Cost Tolerance Allocation 14-19
Table 14A-1 Relative cost of obtaining various tolerance levels
14-20 Chapter Fourteen
Table 14A-2 Cost-tolerance functions for metal removal processes

0.600-0.999 0.04682158 0.565492 0.0006 0.0015
1.000-1.499 0.04204992 0.6021191 0.0008 0.002
1.500-2.799 0.04809684 0.6021191 0.001 0.0025
2.800-4.499 0.06929088 0.565492 0.0012 0.003
4.500-7.799 0.09203907 0.5409254 0.0015 0.004
Turn / bore / shape
0.000-0.599 0.07201641 0.46822793 0.0008 0.003
0.600-0.999 0.085969502 0.45747142 0.001 0.004
1.000-1.499 0.101233386 0.44723008 0.0012 0.005
1.500-2.799 0.11800302 0.4389869 0.0015 0.006
2.800-4.499 0.11804756 0.45747142 0.002 0.008
4.500-7.799 0.12576137 0.46536684 0.0025 0.01
7.800-13.599 0.15997103 0.4389869 0.003 0.012
13.600-20.999 0.15300611 0.46822793 0.004 0.015
Mill
0.000-0.599 0.0862308 0.4259173 0.0012 0.003
0.600-0.999 0.10878812 0.4044547 0.0015 0.004
1.000-1.499 0.09544417 0.4431399 0.002 0.005
1.500-2.799 0.10186958 0.4500798 0.0025 0.006
2.800-4.499 0.14399071 0.4044547 0.003 0.008
4.500-7.799 0.12976209 0.4431399 0.004 0.01
7.800-13.599 0.13916564 0.4500798 0.005 0.012
13.600-20.999 0.17114563 0.4259173 0.006 0.015
Drill
0.000-0.599 0.00301435 1.0955124 0.003 0.005
0.600-0.999 0.00085791 1.3801824 0.004 0.006
1.000-1.499 0.00318631 1.1906627 0.005 0.008
1.500-2.799 0.00644133 1.0955124 0.006 0.01
2.800-4.499
0.00223316

8
0 0.0005 0.001 0.0015 0.002 0.0025
0
0.5
1
1.5
2
0 0.004 0.008 0.012
0
2
4
6
0 0.001 0.002 0.003 0.004
0
1
2
3
4
0 0.001 0.002 0.003 0.004
14-22 Chapter Fourteen
B k
Lap / Hone
0
0.002
0.004
0.006
0 2 4 6 8 10 12 14 16 18
Grind / Diamond turn
0
0.05

0
0.5
1
1.5
0 2 4 6 8 10 12 14 16 18
Minimum-Cost Tolerance Allocation 14-23
B k
Turn / bore / shape
Mill
Drill
Figure 14A-3 continued Plot of coefficients versus size for cost-tolerance functions
0
0.5
1
1.5
0 2 4 6 8 10 12 14 16 18
0
0.5
1
1.5
0 2 4 6 8 10 12 14 16 18
0
0.5
1
1.5
0 1 2 3 4
0
0.005
0.01
0 1 2 3 4

Dallas, Texas
Mr. Stoddard is a senior software developer of CE/TOL SixSigma Tolerance Optimization System, a
tolerance analysis application developed by Raytheon Systems Company. He received his master’s
degree in mechanical engineering from Brigham Young University. As a graduate student, Mr. Stoddard
Chapter
15
15-2 Chapter Fifteen
worked with Dr. Kenneth Chase, founder of ADCATS, on research related to the automation of the
tolerance modeling process. In his thesis, “Characterizing Kinematic Variation in Assemblies from
Geometric Constraints,” he developed an approach to automatic kinematic joint recognition.
Marvin Law
Raytheon Systems Company
Dallas, Texas
Mr. Law is a senior software developer at Raytheon Systems Company. He is involved in researching,
designing, and implementing the CE/TOL SixSigma Tolerance Optimization System. Marvin received
his master’s degree in mechanical engineering from Brigham Young University in 1996. At BYU, Mr. Law
was a research assistant for Dr. Kenneth Chase, founder of the Association for the Development for
Computer-Aided Tolerancing Systems (ADCATS). For his graduate thesis, “Multivariate Statistical
Analysis of Assembly Tolerance Specifications,” he developed methods for mathematically characteriz-
ing and performing simultaneous statistical analysis of multiple design requirements.
15.1 Background Information
The steady increase of computing capability over the past several years has made powerful engineering
analysis tools, such as Computer-Aided Design (CAD) and Finite Element Analysis, available to every
engineer. Computer-Aided Tolerancing (CAT) systems that use the CAD geometry to derive mathematical
tolerance models are now becoming available. These CAT systems hold great promise in automating
tolerancing tasks that used to be performed by hand or with computer spreadsheets, outside of the CAD
environment.
This chapter will introduce an automated tolerance analysis process and discuss the different com-
ponent technologies available that can be used to automate the steps in the tolerancing process.
15.1.1 Benefits of Automation

Automation of the characterization process requires the definition of a finite set of design measure-
ments. This set must be general enough to mathematically characterize all the classes of design require-
ments that may exist. Typical types of design measurements include:
• Gap - Measurable distance between two features along a specified direction
• Angle - Measurable angle between two specified surfaces about a specified axis
• Position - Measurable deviation from a specified location within a specified plane
This set is general enough that most design requirements can be described with one or more of these
design measurements.
With an automation tool the process by which a design measurement is defined is also important.
This process must be intuitive and easy to use. In cases where a tolerance analysis tool is integrated with
a CAD system, the process can be simplified by mapping the definition of the design measurement to
physical features within the geometry. This gives associativity and context to the definition of the critical
design measurement.
15.2.2 Characterizing the Model Function
Figure 15-1 Tolerancing process
Input
Variables
Manufacturing
Process
Capability
Model
Function
Design
Measurement
Design
Requirements
Analysis
Allocation
Tolerance Model
15-4 Chapter Fifteen

kinematic model can then be solved to find the resulting position of the assembled components.
If the assembly is overconstrained so that parts cannot adjust their relative positions to account for
variation, deformation of the components will occur. This is typically the case when sheetmetal parts are
used. Sheetmetal parts are brought together by fixtures and rigidly fastened together. Once the fixtures are
removed, the resulting assembly deforms to minimize its internal stress state. These deformation adjust-
ments can be described by overlaying a finite element model of the components. This finite element model
can then be solved to find the stresses and strains that will result from variation in the component parts
and predict how the assembly will deform.
A comprehensive model function will include the effects of all these sources of variation and their
corresponding methods of propagation.
15.2.2.2 Model Form
The model function must be captured in mathematical form for computer automation. It must be deter-
mined whether an exact or an approximation model will be used. Explicit equations (y = f(x
1
x
n
)) rather
than implicit equations (y = f(y, x
1
x
n
)) are desired to perform tolerance analysis because analytical
rather than brute force methods can be used. Exact models, however, can often only be expressed in
implicit form for complex assembly models that include all sources of variation.
An alternative to an exact mathematical model is an approximation model. This approximation model
can be of any order, but typically a first- or second-order approximation is used. The approximation model
is defined by finding sensitivities of critical features to each input variable of interest. These sensitivities
can be reasoned geometrically or calculated numerically. Once the sensitivity model is produced, it can be
used as the basis for analytical algorithms of tolerance analysis and optimization.
One useful mathematical model of the assembly is the CAD model. A CAD model has a full mathemati-

The final step in the building of a tolerance model is the characterization of the input variables. The model
function is the means of transforming how a change in the inputs will change the outputs. The input
variables to the model function are assumed to vary based on variation in the different manufacturing and
assembly processes. Tolerance ranges are also supplied for each variable as a limit of acceptable variation.
The discussion of the analysis process in the next section will show that the type of analysis performed
drives the type and form of the input data. Worst case analysis only requires tolerance limits while
statistical analysis requires a defined distribution on the variation of each variable.
Input variable data can come from several sources. The variable definitions, along with some or all of
the tolerance data, can be extracted from a CAD system. The statistical distribution information must come
from manufacturing data, as will be discussed in section 15.5.
A complete tolerance model is therefore composed of quantitative design measurements, a compre-
hensive model function and characterized input variables. This comprehensive tolerance model becomes
the basis from which tolerance analysis algorithms can be performed.
15.3 Automating Tolerance Analysis
While many tolerance analysis algorithms are simple enough to be applied without automation, there are
great benefits in automating tolerance analysis calculations. Automating the analysis calculations can
reduce effort and errors. Also, with automation, more advanced analysis methods can be implemented to
provide greater accuracy than simple analysis methods.
The Worst Case and RSS methods discussed in Chapter 9, and the DRSS and SRSS methods dis-
cussed in Chapter 11 are all simple enough to be used without automation. For example, the RSS method
is frequently used to solve simple 1-D tolerance stacks by hand. Very little data is required to use these
four methods. The formulas for each of these methods only require tolerances, derivatives and, in some
cases, Cpk values as inputs. Of course, these four methods are also easily automated by programming a
computer spreadsheet or programming software code.
There are two advanced tolerance analysis methods that are not easily applied without some form of
automation: the Method of System Moments and Monte Carlo Simulation. While both these methods are
more complicated to implement and require more input data, both offer better accuracy and more capability
than Worst Case, RSS, DRSS, or SRSS. Commercial CAT systems are generally based on one of these two
methods. The next two sections will describe these advanced methods in detail.
15.3.1 Method of System Moments

x
x
y
1
2
2
2
)(µµ (15.2)
∑








∂
∂
=
=
n
i
i
i
x
x
y
1
3

y
µµµµ








∂
∂
∑ ∑








∂
∂
+
∑






Series approximation. A second-order approximation improves the accuracy of the approximation for non-
linear functions. The trade-off with the second-order formulation is that the four MSM equations become
much more complex. The four second-order MSM equations can be found in Cox. (Reference 3)
The RSS, DRSS, and SRSS are first-order MSM methods derived from Eq. (15.2), the variance equa-
tion. Taking the square root of Eq. (15.2) yields the RSS formula, a formula for the standard deviation of the
model function. (See Chapter 9 for another derivation of the RSS formula.) Unlike the RSS, DRSS, and
SRSS methods, however, MSM allows the input variable to be characterized by any statistical distribu-
tion, including nonnormal distributions. Note that the four MSM equations include the first four statisti-
cal moments of the input variables. These four moments are calculated from the probability distributions
of the input variables.
In summary, MSM is an advanced tolerance analysis method similar to RSS, but more general. MSM
adds the capability of nonnormal input variables and a nonnormal estimate of the model function. Also, if
a second-order approximation is used, MSM can provide a more accurate approximation for nonlinear
model functions. The computation time for MSM is very small. In addition, once sensitivities are calcu-
lated, only the four MSM equations need to be re-evaluated whenever the distribution characteristics of
the input variables change. This quality makes MSM very attractive for rapid design iteration.
15-8 Chapter Fifteen
15.3.2 Monte Carlo Simulation
Monte Carlo Simulation (MCS) is another advanced tolerance analysis method. MCS is a statistical
technique based on random number generation. For the MCS method, each input variable is characterized
by a statistical distribution. A random value is selected from each input variable distribution and then
plugged into the model function. The resulting function value is then stored. To simulate manufacturing,
the process of randomly selecting the input values and then storing the resultant function value is
repeated many times. The stored function values can be plotted in a histogram, used to calculate the
standard deviation of the model function or used to calculate other metrics. The sample size, the number
of times the simulation is run, determines the accuracy of the analysis. The larger the sample size, the more
accurate the analysis. A typical sample size is 5000 assemblies. Obviously, this type of method must be
automated.
In contrast to MSM, MCS does not use an approximation of the model function. No derivatives are
required for MCS. This can be useful if the model function happens to be discontinuous. However, since

are fit with a distribution. For MCS, a distribution is fit to the histogram of the simulations. Distribution
fitting is automated by using tabular data or numerical methods for known distribution types. The distri-
bution types that are most commonly automated are the normal distribution, Lambda distribution, and the
Pearson and Johnson families of distributions. (References 8 and 9)
Automating the Tolerancing Process 15-9
In addition to fitting a distribution to the output of the MSM and MCS methods, the distribution
types of the input variables must also be defined. Ideally, for the input variables, the designer can define
specific distributions based on actual manufacturing data. If this data is not available, however, a distribu-
tion can be assumed from the tolerance value. For example, frequently it is assumed that variables are
normally distributed, the mean is equal to the nominal, and the standard deviation is equal to one-third the
tolerance value.
15.4 Automating Tolerance Optimization
One of the biggest benefits of automating the tolerance analysis algorithm is the opportunity to combine
the automated analysis method with a tolerance optimization method. Tolerance optimization is the pro-
cess of finding the optimal set of tolerances to meet certain design objectives. These design objectives
might be assembly cost, assembly quality, and/or part quality. Tolerance optimization and allocation
methods are presented in Chapter 11 and Chapter 14.
The analysis methods based on derivatives such as the Method of System Moments (MSM) have an
advantage over Monte Carlo Simulation (MCS) with respect to optimization. These derivatives provide
valuable information to optimization methods so that an optimal solution may be found quickly and
efficiently. The MCS method has been successfully used with optimization methods, but in order to have
reasonable computation time, sample sizes are usually set at 500 assemblies. Accuracy is sacrificed at
sample sizes this small.
15.5 Automating Communication Between Design and Manufacturing
Automating the creation, analysis, and optimization of the tolerance model is the first part of the tolerance
automation process. Automating the communication between design and manufacturing is the second
part.
One of the main purposes of automating the tolerancing process is to reduce problems in the transi-
tion of a product from design to manufacturing. A major cause of transition problems is a lack of commu-
nication. Designers often don’t understand manufacturing processes and capabilities. Manufacturing