Intelligent Control and Automation, 2010, 1, 1-14
doi:10.4236/ica.201.11001 Published Online August 2010 (
Copyright © 2010 SciRes. ICA
Parametric Tolerance Analysis of Mechanical Assembly by
Developing Direct Constraint Model in CAD and Cost
Competent Tolerance Synthesis
Govindarajalu Jayaprakash1, Karuppan Sivakumar2, Manoharan Thilak3
1Department of Mechanical Engineering, Shivani College of Engineering, Tiruchirapalli, India
2Department of Mechanical Engineering, Bannari Amman Institute of Technology, Sathiyamangalam, India
3Department of Mechanical Engineering, T. R. P. Engineering College, Tiruchirapalli, India
Received February 17, 2010; revised March 21, 2010; accepted June 1, 2010
The objective of tolerance analysis is to check the extent and nature of variation of an analyzed dimension or
geometric feature of interest for a given GD & T scheme. The parametric approach to tolerance analysis is
based on parametric constraint solving. The accuracy of simulation results is dependent on the user-defined
modeling scheme. Once an accurate CAD model is developed, it is integrated with tolerance synthesis model.
In order to make it cost competent, it is necessary to obtain the cost-tolerance relationships. The neural net-
work recently has been reported to be an effective statistical tool for determining relationship between input
factors and output responses. This study deals development of direct constraint model in CAD, which is in-
tegrated to an optimal tolerance design problem. A back-propagation (BP) network is applied to fit the
cost-tolerance relationship. An optimization method based on Differential Evolution (DE) is then used to
locate the combination of controllable factors (tolerances) to optimize the output response (manufacturing
cost plus quality loss) using the equations stemming from the trained network. A tolerance synthesis problem
for a motor assembly is used to investigate the effectiveness and efficiency of the proposed methodology.
Keywords: Tolerance Analysis, Tolerance Synthesis, CAD Integration, Optimization, Neural Network
1. Introduction
Tolerance is the allowable range of variation from design
intent in a dimension. As one of many design variables,
the role of dimensional tolerances is to restrict the
amount of size variation in a manufacturing feature while
ensuring functionality. Although the ideal amount of
feature variation is zero, it is neither feasible nor eco-
nomical to meet the ideal due to a variety of process fac-
tors including machine tool accuracy, material property
variation, process effects, etc. Determining the allowable
amount of dimensional variation at design stage impacts
the manufacturing costs incurred during the actual proc-
essing of the part. Tolerance analysis involves modeling
of the relations among variation, tolerance and cost. Tol-
erance analysis is conducted using variation propagation
models that compute how part, subassembly, and process
variations propagate to final product variation, which is
related to product quality. Variations are typically trans-
lated to tolerances using statistic principles, and analyti-
cal models are then used to estimate cost as a function of
tolerance. The variation of the analyzed dimension arise
form the accumulation of dimensional and/or geometric
variations in the tolerance chain. The analysis include: 1)
the contributor, i.e., The dimensions or features that
causes variations in the analyzed dimension, 2) the sensi-
tivities with respect to each contributor, 3) the percent
contribution to variation from each contributor, and 4)
worst case variations, statistical distribution, and accep-
tance rates. Analysis approaches can be classified as 1)
one-dimensional (1D), two-dimensional (2D), and
three-dimensional (3D), according to dimensionality; 2)
worst case (i.e., 100% acceptance rate) and statistical
(i.e., less than 100% acceptance rate), according to
analysis objective; 3) dimensional and dimensional +
geometric, according to the type of variation included; or
4) part level and assembly level, according to the analy-
sis level. Popular analysis methods are manual 1D chart,
linearized 2D/3D analysis, and the Monte Carlo simula-
Copyright © 2010 SciRes. ICA
At the present time, three different and disconnected
communities i.e., designers/draftsmen, engineering ana-
lysts, and design researchers are using vastly different
tools and techniques for tolerance analysis. Cultural and
educational differences between these communities have
isolated them from one another and made them unaware
of the others’ techniques. The draftsmen community uses
a manual procedure called tolerance charts; but can do
only worst-case analysis, and is conducted in only one
direction at a time. Meanwhile, the engineering analysis
community uses computer-aided tolerancing software
CATs. These CAT packages can do both worst-case and
statistical analyses. Many researchers [1-4] provided
good surveys of GD&T modeling for CATs; Some re-
searcher [5] discussed the simplification of feature based
models for tolerance analysis, and used the linear pro-
gramming approach to tolerance analysis involving
geo-metric tolerances [5]; Guilford et al. [2] introduced a
CAT system using the variation modeling and feasibility
space approaches.
2. Background
2.1. 1D Tolerance Charts
Tolerance charting is a manual bookkeeping procedure
for 1D stack calculations. The analyst typically works
with engineering drawings [6-7]. Since the method is
limited to worst-case analysis, the analyst positions parts
in assemblies to yield each of the worst-cases minimum
or maximum value of the analyzed dimension, i.e., sepa-
rate charts have to be constructed for each worst-case.
Since no algebraic expression for the analyzed dimension
in terms of the contributors is generated by this method,
no statistical analysis can be performed. Also, contribu-
tors not in the direction of analysis are ignored, which
may yield significant errors in most cases. The limita-
tions associated with this method, as practiced today, are
as follows. 1) It is done manually, and since each type of
tolerance is handled differently, the user must remember
all the rules correctly while constructing the charts to
obtain correct results, making the process tedious and
prone to errors. 2) It deals with one direction at a time,
ignoring possible contributions from other directions,
which often leads to inaccurate results. 3) The charting
procedure is only capable of the worst-case analysis only;
no statistical analysis is available. 4). This method has
not been widely integrated with existing CAD systems.
2.2. Parametric Tolerance Analysis
Most CAT packages take advantage of the same para-
metric/variational approach used in CAD systems and
apply the Monte Carlo simulation to tolerance analysis
[8-10]. This section will give a brief description of pa-
rametric approach to tolerance analysis. In the parametric
approach, the analyzed dimension is expressed as an al-
gebraic function an equation, or a set of equations that
relates the analyzed dimension to those on which it de-
pends i.e., contributors. The function is either linearized
or directly used for the Monte Carlo simulation in the
nonlinear analysis. Results commonly available are the
lists of contributors, sensitivities, and percentage contri-
butions, and the tolerance accumulation for worst-case
and statistical cases.
2.3. Cost Competent Tolerancing
Aspects such as design for quality, quality improvement
and cost reduction, asymmetric quality losses, charts for
optimum quality and cost, minimum cost approach, cost
of assemblies, development of cost tolerance models
[11-15] have been explored in the quality area of toler-
ance synthesis. Experiments (DOE) approach was used
in robust tolerance design, the cases of ‘nominal the best’,
‘smaller the better’, ‘larger the better’, and asymmetric
loss function, were investigated [16] and allocation of
tolerances of products with asymmetric quality loss was
presented [14]. The combined effect of manufacturing
cost and quality loss was also investigated under the re-
straints of process capability limits, design functionality
restriction and product quality requirements by using
tolerance chart optimization for quality and cost [12].
Relationships between the product cost and tolerances
have also been investigated. An analytical method was
proposed for determining tolerances for mechanical parts
with objectives of minimizing manufacturing costs [17].
Minimizing the cost of assembly was investigated
mathematically in which it was observed that widening
the tolerance of more expensive part and a tightening of
tolerances on cheaper parts could result in major reduc-
tion in cost of the assembly [18]. Exhaustive search,
zero–one, SQP and Univariate methods were evaluated
for performing a combined minimum cost tolerance al-
location and process selection [19]. The production cost
tolerance and hybrid tolerance models based on empiri-
cal cost tolerance data of manufacturing processes like
punching, turning, milling, grinding and casting were
introduced [20]. The robust design by tolerance alloca-
tion considering quality and manufacturing cost and op-
timizing tolerance allocation based on manufacturing
cost were also investigated [21,22]. It involved develop-
ment of relationship between part tolerances and assem-
bly tolerances to provide a quantitative measure of prod-
uct quality using the quality loss function concept intro-
duced by Taguchi. Numerical optimization was used to
balance manufacturing cost and product quality. The
possibility of using statistics and probability methods for
Copyright © 2010 SciRes. ICA
allocation of tolerances has also been explored with a
view to developing tools for tolerance synthesis.
Relationship between function and dimensional varia-
tion on assembly was considered as logical basis for se-
lection of tolerances [23]. A probabilistic model for tol-
erance synthesis was developed [24] in which the reli-
ability indices were associated with either assembly con-
dition or component dimension. Strategies to compute
small changes or gradient in tolerance values were also
developed using probability theory [25]. Tolerance cost
models based on the distribution function zone [26], tol-
erance optimization problem using system of experi-
mental design [27] and using Monte–Carlo simulations
[28] approaches, were also investigated in application of
statistical methods for tolerance synthesis.
2.4. Neural Network-Based Cost-Tolerance
Neural networks have received a lot of attention in many
research and application areas. One of the major benefits
of neural networks is the adaptive ability of their gener-
alization of data from the real world. Exploiting this ad-
vantage, many researchers apply neural networks for
nonlinear regression analysis and have reported positive
experimental results in their applications [29]. Recently,
neural networks have received a great deal of attention in
manufacturing areas. Zhang and Huang [30] presented an
extensive review of neural network applications in
manufacturing. Neural networks are defined by Rumel-
hart and McClelland [31] as `massively parallel inter-
connected networks of simple (usually adaptive) ele-
ments and their hierarchical organizations which are in-
tended to interact with objects of the real world in the
same way as biological nervous systems do. The ap-
proach towards constructing the cost} tolerance rela-
tionships is based on a supervised back-propagation (BP)
neural network. Among several well-known supervised
neural networks, the BP model is the most extensively
used and can provide good solutions for much industrial
application [32].
A BP network is a feed-forward network with one or
more layers of nodes between the input and output nodes.
An imperative item of the BP network is the iterative
method that propagates the error terms required to adopt
weights back from nodes in the output layer to nodes in
lower layers. The training of a BP network involves three
stages: the feed forward of the input training pattern, the
calculation and BP of the associated error, and the ad-
justment of the weights. After the network reaches a sat-
isfactory level of performance, it will learn the relation-
ships between input and output patterns and its weights
can be used to recognize new input patterns.
Figure 1 depicts a BP network with one hidden layer.
The hidden nodes of the hidden layer perform an impor-
Figure 1. Architecture of a three-layer BP network.
tant role in creating internal representation. The follow-
ing nomenclatures are used for describing the BP learn-
ing rule.
netpi = net input to processing unit i in pattern p (a pat-
tern corresponding to a vector of factors),
wij = connection weight between processing unit I and
processing unit j,
api = activation value of processing unit i in pattern p,
pi = the effect of a change on the output of unit I in
pattern p,
gpi = target value of processing unit i,
= learning rate.
The net inputs and the activation values of the middle
processing nodes are calculated as follows:
netw a
a1exp net
The net input is the weighed sum of activation values
of the connected input units plus a bias value. Initially,
the connection weights are assigned randomly and are
varied continuously. The activation values are in turn
used to calculate the net inputs and the activation values
of the output processing units using the same Equations
(1) and (2).
Once the activation values of the output units are cal-
culated, we compare the target value with activation
value of each output unit. The discrepancy is propagated
 
pipipi ipi
δgafnet (3)
For the hidden processing units in which the target
values are unknown, instead of Equation (3), the follow-
ing equation is used to calculate the discrepancy. It takes
the form
iipipk ki
δfnetδw (4)
From the results of Equations (3) and (4), the weights
between processing units are adjusted using
wεδ a
Copyright © 2010 SciRes. ICA
2.5. Differential Evolution (DE)
Differential Evolution is an improved version of Genetic
Algorithm for faster optimization [33]. Unlike simple
GA that uses binary coding for representing problem
parameters, Differential Evolution (DE) uses real coding
of floating point numbers. Among the DE’s advantages
are its simple structure, ease of use, speed and robust-
The simple adaptive scheme used by DE ensures that the
mutation increments are automatically scaled to the cor-
rect magnitude. Similarly DE uses a non-uniform cross-
over in that the parameter values of the child vector are
inherited in unequal proportions from the parent vectors.
For reproduction, DE uses a tournament selection where
the child vector competes against one of its parents. The
overall structure of the DE algorithm resembles that of
most other population based searches. The parallel ver-
sion of DE maintains two arrays, each of which holds a
population of NP, D-dimensional, real valued vectors.
The primary array holds the current vector population,
while the secondary array accumulates vectors that are
selected for the next generation. In each generation, NP
competitions are held to determine the composition of
the next generation. Every pair of vectors (Xa, Xb) de-
fines a vector differential: Xa – Xb. When Xa and Xb are
chosen randomly, their weighted differential is used to
perturb another randomly chosen vector Xc. This process
can be mathematically written as Xc = Xc + F (Xa – Xb).
The scaling factor F is a user supplied constant in the
range (0 < F < 1.2). The optimal value of F for most of
the functions lies in the range of 0.4 to 1.0 [33]. Then in
every generation, each primary array vector, Xi is tar-
geted for crossover with a vector like Xc to produce a
trial vector Xt. Thus the trial vector is the child of two
parents, a noisy random vector and the target vector
against which it must compete. The non-uniform cross-
over is used with a crossover constant CR, in the range 0
< CR < 1. CR actually represents the probability that the
child vector inherits the parameter values from the noisy
random vector. When CR = 1, for example, every trial
vector parameter is certain to come from Xc. If, on the
other hand, CR = 0, all but one trial vector parameter
comes from the target vector. To ensure that Xt differs
from Xi by at least one parameter, the final trial vector
parameter always comes from the noisy random vector,
even when CR = 0. Then the cost of the trial vector is
compared with that of the target vector, and the vector
that has the lowest cost of the two would survive for the
next generation. In all, just three factors control evolu-
tion under DE, the population size, NP; the weight ap-
plied to the random differential, F; and the crossover
constant, CR.
The general convention used is DE/x/y/z. DE stands
for Differential Evolution, x represents a string denoting
the vector to be perturbed, y is the number of difference
vectors considered for perturbation of x, and z stands for
the type of crossover being used (exp:exponential; bin:
binomial). Thus, the working algorithm outlined above is
the strategy of DE, i.e..DE/rand/1/bin. Hence the pertur-
bation can be either in the best vector of the previous
generation or in any randomly chosen vector. Similarly
for perturbation either single or two vector differences
can be used. For perturbation with a single vector differ-
ence, out of the three distinct randomly chosen vectors,
the weighted vector differential of any two vectors is
added to the third one. Similarly for perturbation with
two vector differences, five distinct vectors, other than the
target vector are chosen randomly from the current popu-
lation. Out of these, the weighted vector difference of each
pair of any four vectors is added to the fifth one for per-
turbation. In binomial crossover, the crossover is per-
formed on each of the D variables whenever a randomly
picked number between 0 and 1 is within the CR value.
2.5.1. Pseudo Code for DE
The pseudo code of DE used in the present study is given
· Choose a seed for the random number generator.
· Initialize the values of D, NP, CR, F and MAXGEN
(maximum generation).
· Initialize all the vectors of the population randomly.
The variables are normalized within the bounds. Hence
generate a random number between 0 and 1 for all the
design variables for initialization.
for i = 1 to NP
{ for j = 1 to D
Xi, j = Lower bound + random number *( upper bound -
lower bound)}
· All the vectors generated should satisfy the constraints.
Penalty function approach, i.e., penalizing the vector by
giving it a large value, is followed only for those vectors,
which do not satisfy the constraints.
· Evaluate the objective function of each vector. Here is
the value of the objective function to be minimized cal-
culated by a separate function defunct. objective ()
for i = 1 to NP
Ci = defunct. objective ()
· Find out the vector with the minimum objective value
i.e. the best vector so far.
Cmin = C1 and best = 1
for i = 2 to NP
{ if (Ci < Cmin)
then Cmin = Ci and best = i }
· Perform mutation, crossover, selection and evaluation
of the objective function for a
specified number of generations.
While (gen < MAXGEN)
{ for i = 1 to NP
· For each vector Xi (target vector), select three distinct
Copyright © 2010 SciRes. ICA
vectors Xa, Xb and Xc (select five, if two vector differ-
ences are to be used) randomly from the current popula-
tion (primary array) other than the vector Xi
{ r1 = random number * NP
r2 = random number * NP
r3 = random number * NP
} while
(r1 = i) OR (r2 = i) OR (r3 = i) OR (r1 = r2) OR (r2 = r3)
OR (r1 = r3)
· Perform crossover for each target vector Xi with its
noisy vector Xn,i and create a trial vector, Xt, i. Per-
forming mutation creates the noisy vector.
· If CR = 0 inherit all the parameters from the target
vector Xi, except one which should be from Xn, i.
· for binomial crossover
{ p = random number
for n = 1 to D
{if ( p < CR )
Xn, i = Xa, i + F (Xb, i - Xc, i)
Xt, i = Xn, i
} else Xt, i = Xi, j
· Again, the NP noisy random vectors that are generated
should satisfy the constraint and the penalty function
approach is followed as mentioned above.
· Perform selection for each target vector, Xi by com-
paring its objective value with that of the trial vector, Xt, i;
whichever has the minimum objective will survive for the
next generation.
Ct, i = defunct. Objective ()
if (Ct, i < Ci )
new Xi = Xt, i
else new Xi = Xi}
/* for i = 1 to NP */
· Print the results (after the stopping criteria is met).
The stopping criterion is maximum number of genera-
3. Parametric Approach Using Direct CAD
In the parametric approach, the analyzed dimension is
expressed as an algebraic function an equation, or a set
of equations that relates the analyzed dimension to those
on which it depends, i.e., contributors. The function is
either linearized or directly used for the Monte Carlo
simulation in the nonlinear analysis. Results commonly
available are the lists of contributors, sensitivities, and
percentage contributions, and the tolerance accumulation
for worst-case and statistical cases.
3.1. Linearized Tolerance Analysis
In this type of analysis, partial derivatives are calculated
for each contributor; the derivatives give the sensitivity
for each contributor from which worst case and variance
can be determined. In general, the dimension to be ana-
lyzed, A, can be expressed as a function of independent
variables (contributors), di, i.e.,
12 n
Afd ,d,.......,d (6)
To perform a linearized tolerance analysis, this func-
tion f, usually called the design function, is linearized
about the variables nominal values i
d, using the Tay-
lor’s series expansion, as follows:
1n i
i1 i
A fd,....,dd
After linearization, both worst-case and statistical
analyses can be performed. For worst-case analysis, the
mean and worst-case variance of A are computed from
equation below
12 n
12 n
ff f
Add .....d
dd d
 
 
 
12 n
12 n
ff f
Add .....d
dd d
 
 
For statistical analysis, the mean of A is computed us-
ing the same equation as (8), but statistical variance of A
is obtained from this equation
22 2
22 2
Ad1d2 dn
12 n
ff f
dd d
 
 
 
 
 
  (10)
The percentage contribution of di is computed as
 (11)
The acceptance rate can also be computed if the cor-
responding design limits of A are supplied. In the earlier
Sfd,....d d
 is the sensitivity of A
to the contributor di, and
ddd  represents the
contributors’ perturbation ranges (tolerances) about their
respective nominal values. Tolerance sensitivity is an
essential aspect of tolerance analysis for mechanical as-
semblies in 2D and 3D space while the sensitivity is
nonzero constants (usually 1.0) for 1D analysis. The
contributors’ tolerances are usually assumed to corre-
spond to n sigma standard deviations (typically n=6, i.e.,
) in statistical analysis. Si and Ci are useful
measures for redesign.
3.2. Direct Constraint Model in CAD
In parametric CAD systems, constraint equations based
Copyright © 2010 SciRes. ICA
on geometric and dimensional relations are used to
model a design. By perturbing the variables in these
equations, some kind of sensitivity and tolerance analysis
can be performed [32]. The design process using such a
system is as follows. 1) First, create the nominal topol-
ogy to obtain a model exhibiting the desired geometric
elements and connectivity between the elements, but
without the dimensions. 2) Next, describe the required
properties between the model entities in terms of geo-
metric constraints, which define the desired mathemati-
cal relationships between the numerical variables of the
model entities. 3) Third, the modeling system applies a
general solution procedure to the constraints, resulting in
an evaluated model where the declared constraints are
satisfied. 4) Create variants of the model by changing the
values of the constrained variables. After each change, a
new instance of the model is created by re-executing the
constraint solution procedure.
As can be seen from the earlier process, if the user
specifies the dimension of interest, the system solution
procedure can also obtain that value for a specific in-
stance of the model. If one variable is perturbed at a time,
this variable’s sensitivity can be studied by comparing
this perturbation’s effect on the dimension of interest.
With the sensitivities of each variable and their perturba-
tion ranges tolerances, both linearized and non-linearized
analyses can be performed. Therefore, tolerance analysis
functionality is just an extension or by-product of para-
metric solid modeling.
4. An Application
In this study, the Pro/E wildfire version 3.0 parametric
modeling software package is used to develop the direct
constraint model. Linear tolerance analysis (statistical
analysis) is carried out for the problem, in which partial
derivatives are calculated for each contributor and the
derivatives give the sensitivity for each contributor from
which variance can be determined. The response variable
in this study is Total cost which is sum of manufacturing
cost and quality losses and it is expressed as
ijijjij Mik
j1 k1
Where m is the total number of components from q
assembly dimensions in a finished product, Kj the cost
coefficient of the jth resultant dimension for quadratic
loss function, Uij the jth resultant dimension from the ith
experimental results, ij the jth resultant variance of sta-
tistical data from the ith experimental results, Tj the de-
sign nominal value for the jth assembly dimension, tik the
tolerance established in the ith experiment for the kth
component, and CM(tik) the manufacturing cost for the
tolerance tik.
This application is related to motor assembly [13]
which consists of an x-base, crank, shaft and motor base.
Figures 2-7 are graphic representation of the motor as-
sembly with dimensioning and tolerancing schemes. Ta-
ble 1 provides some relevant information for these fig-
ures. The ordering number in the first row of Table 1 is
also given in Figures 3-7 for the purpose of easy asso-
ciation. The objective is to determine an appropriate tol-
erance allocation so that there is sufficient clearance be-
tween the crank and x-base.
Figure 2. A motor assembly drawing (Jeang, 1999).
Figure 3. X-base.
Figure 4. X-base.
Copyright © 2010 SciRes. ICA
Table 1. Dimensioning and tolerancing schemes for motor assembly.
and size no. 1 2 3 4 5 6 7 8 9 10
Component X-base motor basemotor base motor shaftmotor shaftmotor shaftmotor motor crank crank
feature flatness profile flatness size perpendicular-
ity profile size position size
Illustration Surface on
Surface on
motor base
Surface on
the bottom of
motor base
Size of the
shaft (with
target value
ity of shaft
of shaft
size of
Hole posi-
tion of mo-
Hole size
of crank
Hole perpen-
dicularity of
– – – – –
Influence on
clearance? yes no yes yes yes no no no no no
Figure 5. Motor base.
Figure 6. Shaft.
Figure 7. Crank.
A parametric model of motor assembly is created in
order to develop constraint equations based on geometric
and dimensional relations (Figure 8). The direct con-
straint model in CAD is created as follows. 1) First, a
nominal topology to obtain a model exhibiting the de-
sired geometric elements and connectivity between the
elements is created, but without the dimensions. 2) Next,
the required properties between the model entities is de-
scribed in terms of geometric constraints, which define
the desired mathematical relationships between the nu-
merical variables of the model entities. 3) Thirdly, a
general solution procedure is applied to the constraints,
which results in an evaluated model where the declared
constraints are satisfied. 4) Finally, more variants of the
model are created by changing the values of the con-
strained variables. After each change, a new instance of
the model is created by re-executing the constraint solu-
tion procedure.
Copyright © 2010 SciRes. ICA
Once the constraint equation is developed (Equation
13), sensitivity and tolerance analysis can be performed
by perturbing the variables in that equation. Then, the
user has to specify the dimension of interest; the system
solution procedure will obtain that value for the specific
instance of the model. In this application the dimension
of interest is clearance between the crank and x-base
which is 0.89 cm (Figure 9). The variables are X1 (motor
shaft size), X2 (motor shaft perpendicularity), X3 (x-base)
and X4 (motor base flatness). The number of levels for
each variable is three. Table 3 shows the variables and
levels of experiment with 27 runs. Each variable’s sensi-
tivity can be studied by comparing this perturbation’s
effect on the dimension of interest by perturbing that
variable alone. After determining the sensitivities of each
variable and their perturbation ranges tolerances, both
linearized analyses can be performed. Thus the tolerance
analysis functionality is found to be an extension or by-
product of parametric solid modeling.
 
 
XSin 0.5771
XSin0.5771 +2
XSin 0.99369XSin 0.99369
Then neural network model of cost-tolerance function
is developed as follows. The 2/3rd of experimental results
drawn randomly from Table 3 are used to train the neu-
ral network. Before applying the neural network for
modeling, the architecture of the network has been de-
cided; i.e. the number of hidden layers and the number of
neurons in each layer. As there are 4 inputs and 1 output,
the number of neurons in the input and output layer has
to be set to 4 and 1 respectively. Also, the back propaga-
tion architecture with one hidden layer is enough for
majority of the applications. Hence only one hidden layer
has been adopted. A procedure was employed to opti-
mize the number of neurons in the hidden layer. Accord-
ingly, an experimental approach was adopted, which
involves testing the trained neural networks against the
remaining 1/3rd of experimental results. Experimental
and predicted outputs for different number of neurons
have been compared. The regression statistics for differ-
ent architecture are determined and listed in Table 4 and
the same have been plotted against the number of neu-
rons as shown in Figure 10. It is observed that the re-
gression statistics were minimized with 7 neurons.
Hence, 4-7-1 is the most suitable network for the task
under consideration. The training function used in this
research is Gradient descent with momentum back-
propagation. The transfer function used in this research is
Figure 8. Motor assembly (Pro/E Wildfire 3.0 model).
Copyright © 2010 SciRes. ICA
Figure 9. Motor assembly–Value of Clearance.
Figure 10. Error versus the number of neurons in a hidden
Table 2. Tolerance costs for each factor at various levels.
Lower level Middle level Upper level
x1 $18.07 $13.63 $12.82
x2 $35.18 $24.68 $21.90
x3 $279.61 $170.39 $108.57
x4 $29.87 $19.62 $17.98
tan-sigmoid and gradient. Descent w/momentum weight/
bias learning function has been used. Figure 11 shows
the schematic diagram of the neural network. The learning
rate = 0.7, momentum = 0.65 and training epochs = 2000.
The weights (and biases) are randomly initialized be-
tween –0.5 and 0.5.Once the neural network gets trained,
it can provide the result for any arbitrary value of input
data set. Table 5 shows the experimental result and the
model prediction. It is observed that the prediction based
on an ANN model is quite close to the experimental ob-
Figure 11. Schematic diagram of the neural network.
Copyright © 2010 SciRes. ICA
Table 3. Experiment design.
X-Base Flatness
Motor base
Flatness (x2)
Motor shaft size
Motor shaft perpen-
dicularity (x4)
Total cost in
$ TC(X)
1 0.15 0.075 0.1 0.08 228.9
2 0.15 0.075 0.1 0.04 239.5
3 0.15 0.075 0.05 0.08 361.3
4 0.15 0.075 0.05 0.04 373.0
5 0.15 0.1 0.075 0.08 266.1
6 0.15 0.1 0.075 0.04 277.6
7 0.15 0.05 0.075 0.08 277.2
8 0.15 0.05 0.075 0.04 289.0
9 0.15 0.1 0.1 0.06 228.4
10 0.15 0.1 0.05 0.06 361.1
11 0.15 0.05 0.1 0.06 240.8
12 0.15 0.05 0.05 0.06 372.5
13 0.2 0.075 0.075 0.08 274.6
14 0.2 0.075 0.075 0.04 286.3
15 0.1 0.075 0.075 0.08 267.0
16 0.1 0.075 0.075 0.04 278.7
17 0.2 0.075 0.1 0.06 236.3
18 0.2 0.075 0.05 0.06 369.9
19 0.1 0.075 0.1 0.06 228.6
20 0.1 0.075 0.05 0.06 362.0
21 0.2 0.1 0.075 0.06 274.6
22 0.2 0.05 0.075 0.06 290.9
23 0.1 0.1 0.075 0.06 266.7
24 0.1 0.05 0.075 0.06 278.3
25 0.15 0.075 0.075 0.06 271.2
26 0.15 0.075 0.075 0.06 268.4
27 0.15 0.075 0.075 0.06 269.9
The neural network model for the above problem is
developed as per the approach discussed previously.
Based on those discussions, the BP network of 4-7-1
architecture produces the best performance (refer Table
4) and the same is adopted to generate the neural net-
work based cost-tolerance function under this case study.
At this point, the relationship between input factors
X = (x1, x2, x3, x4) = (x-base flatness, motor base flat-
ness, motor shaft size, motor shaft perpendicularity), and
output response F(X) (total cost defined by Equation 12)
can be revealed from the constructed neural network.
The solution of the motor assembly case can be found by
solving the following mathematical models:
12 34
MaximizeF XFx,x,x,x
subject to 0.1x0.2,
A clearance of 0.89 cm has to be maintained between
Copyright © 2010 SciRes. ICA
Table 4. Regression statistics for each network architecture.
Mean Prediction
error %
R square
4-4-1 4.498492501 0.943112 12.06117
4-5-1 2.035675506 0.987509 6.426893
4-6-1 4.100965211 0.902752 16.94439
4-7-1 1.735294596 0.989658 5.608395
4-8-1 3.495912167 0.96686 10.32931
4-9-1 5.935647182 0.725488 33.33576
4-10-1 4.749441739 0.850217 22.10918
Table 5. Comparison of experimental results with the ANN
model prediction.
Total cost in $
TC(X) Experimen-
tal results
Predicted values by
Prediction error
239.517 233.5125993 2.506878729
373.02 370.1791799 0.761573146
228.449 229.8694101 0.621762439
361.1 353.4342438 2.12289012
266.993 269.1789878 0.818743499
236.277 234.1075089 0.918198157
274.648 270.9355255 1.351720937
266.679 251.1227928 5.833307915
271.192 269.3409074 0.682576422
Maximum prediction error for each output
in this row in % 5.833307915
Minimum prediction error for each output in
this row in % 0.621762439
Mean prediction error for each output in this
row in % 1.735294596
Table 6. The DE specific data.
Variable type Real variable
Population size 100
Cross over type binomial
No of difference vector 1
Vector to be perturbed random
Total no of generation 100
No of variables 4
motor base and crank. The functional constraint is the
constraint equation developed using the parametric
model (Equation 13). The Problem (14) is solved by the
proposed Differential evolution discussed in the previous
section. The outline of the proposed optimization strat-
egy is shown in Figure 12.
The optimization strategy is explained as follows. Ini-
tially the cost-tolerance function is established by the
neural network model. Once the neural network based
cost–tolerance function is established, and then optimi-
zation of the problem (Equation 14) is carried out using
Differential evolution (DE). The DE optimization pro-
gram determines the set of tolerance with minimum cost.
Those tolerance values are assigned to the Direct con-
straint model in CAD and the clearance value is deter-
mined. If the clearance value is less than the desired one,
Set initial population of tolerance
Clearance is less than
the desired one
Find the set of tolerance with minimum cost
Improve population using DE
Evaluate the Neural network based cost-tolerance
Determine the clearance value from the CAD
Figure 12. The outline of the proposed optimization strategy.
Copyright © 2010 SciRes. ICA
the process terminates, otherwise the process is repeated
from the beginning.
The least cost is found to be $ 232.1907; the solution
converges in the 49th generation (Figure 13). The value
of cost obtained by this method is lesser than that of the
value obtained by response surface methodology [13]
which was $238.5191 and the constraint equation pro-
vided, ensures that the values of variables satisfies the
functional constraint. The values of the variables are as
follows. x-base flatness, x1 = 0.1, motor base flatness, x2
= 0.081125, motor shaft size, x3 = 0.1, and the motor
shaft perpendicularity, x4 = 0.07799. It is fond that the
proposed hybrid methodology with BP and DE can solve
tolerance synthesis problem effectively.
5. Discussion
Many products are now routinely designed with the aid
of computer modeling. With an input consisting of de-
signable engineering parameters and parameters repre-
senting manufacturing process conditions, computer
simulation generates an output, which is the product’s
quality characteristic. Then a standard statistical analysis
is performed based on this output. The finite element
analysis of mechanical components and the design of
electronic circuits are two important application areas
where computer modeling is widely used. With the cur-
rent development of computer aided tolerancing software
(CATS) it becomes possible to put tolerance design via
computer modeling into practical use. Major CATs use
the abstracted feature-parameter model for the Monte
Carlo simulation based tolerance analysis. The major
problems with these CAT packages are as follows: 1)
Cumbersome work is needed in model creation. First,
parametric CAD uses a combination of 2D constraint
solving with 3D sweep and loft operations; this con-
straint model is not suitable for tolerance analysis. Sec-
ond, STEP standards for tolerances exist but vendors do
not provide translators, so importing a CAD model with
GD & T is only partially achieved, i.e., the GD & T and
assembly constraint information is lost after importing.
Therefore, the user must recreate the tolerance specs in
CATs manually using CAD entities. 2) There is a lack of
an underlying mathematical model for geometric varia-
tions. First, the quality of the analysis depends on the
expertise of the person creating the CATS model; this is
a problem. The results should depend on the GD & T
scheme only and not by trial and error or any “tricks”.
Second, the Monte Carlo simulation does not produce a
closed form solution; the solution changes with the
number of simulations performed; one can never be sure
of the worst-case results. Third, since the dependent di-
mensions cannot be expressed explicitly by one equation
in terms of all contributors, the contributors and sensi-
Figure 13. Solution history.
tivities are determined numerically by trial and error. In
this study, a parametric analysis method based on direct
constraint model is proposed which addresses the above
problems in using CATs packages. First, the parametric
tolerance analysis borrows its concept from the mature
parametric CAD; there by it can be easily integrated with
CAD system. Second, a constraint equation is developed
after developing the direct constraint model, which en-
sures that dependent dimension can be expressed explic-
itly in terms of contributors. The constraint equation is
then used as a functional constraint in the optimization
method. Due to the above reasons, parametric tolerance
analysis based on direct constraint model in CAD is
more suitable for tolerance analysis of simple problems
than that of CATs package.
6. Conclusions
In this research, the parametric tolerance analysis of
given application problem is performed by developing a
direct constraint model in CAD. This method is found to
be more suitable for tolerance analysis of simple prob-
lems than that of CATs package. And the proposed op-
timization strategy is found to provide better formulation
of cost–tolerance relationships for empirical data. BP
network architecture of configuration 4-7-1 generates a
suitable model for cost-tolerance relationship of R2 value
0.99993, there by eliminating errors due to curve fitting
in case of regression fitting. And it also generates more
robust outcomes of tolerance synthesis. The proposed
optimization strategy obtains an optimal solution better
than that of Response surface methodology (RSM)
(Jeang, 1999). The CAD model developed ensures that
the values of variables satisfy the functional constraint.
This study proposes a parametric tolerance analysis of
mechanical assembly by developing direct constraint
model in cad and cost competent tolerance synthesis
Copyright © 2010 SciRes. ICA
based on BP learning and DE based optimization algo-
rithm. The constraint equation developed ensures that the
proposed values of controllable factors (tolerances) sat-
isfy the assembly constraint, even before the start of
manufacturing process. There by reducing scrap and re-
work cost.
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