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Computer-Aided Design 43 (2011) 207–218

Contents lists available at ScienceDirect

Computer-Aided Design

journal homepage: www.elsevier.com/locate/cad

Simultaneous optimal selection of design and manufacturing tolerances with

alternative manufacturing process selection

K. Sivakumara, C. Balamuruganb,∗, S. Ramabalanc

aDepartment of Mechanical Engineering, Bannari Amman Institute of Technology, Sathyamangalam-638401, TamilNadu, India

bDepartment of Mechanical Engineering, M.A.M. College of Engineering, Tiruchirappalli-621105, TamilNadu, India

cEGS Pillay Engineering College, Nagppattinam, TamilNadu, India

a r t i c l e i n f o

Article history:

Received 17 February 2010

Accepted 16 October 2010

Keywords:

Tolerance design

Alternative manufacturing process

selection

Intelligent algorithms-Elitist

Non-dominated Sorting Genetic

Algorithm (NSGA-II)

Multi-objective Particle Swarm

Optimization (MOPSO)

a b s t r a c t

Tolerance specification is an important part of mechanical design. Design tolerances strongly influence

the functional performance and manufacturing cost of a mechanical product. Tighter tolerances normally

produce superior components, better performing mechanical systems and good assemblability with

assured exchangeability at the assembly line. However, unnecessarily tight tolerances lead to excessive

manufacturing costs for a given application. The balancing of performance and manufacturing cost

through identification of optimal design tolerances is a major concern in modern design. Traditionally,

design tolerances are specified based on the designer’s experience. Computer-aided (or software-based)

tolerance synthesis and alternative manufacturing process selection programs allow a designer to verify

the relations between all design tolerances to produce a consistent and feasible design. In this paper,

a general new methodology using intelligent algorithms viz., Elitist Non-dominated Sorting Genetic

Algorithm(NSGA-II)andMultiObjectiveParticleSwarmOptimization(MOPSO)forsimultaneousoptimal

selection of design and manufacturing tolerances with alternative manufacturing process selection is

presented. The problem has a multi-criterion character in which 3 objective functions, 3 constraints and 5

variables are considered. The average fitness factor method and normalized weighted objective functions

method are separately used to select the best optimal solution from Pareto optimal fronts. Two multi-

objectiveperformancemeasuresnamelysolutionspreadmeasureandratioofnon-dominatedindividuals

are used to evaluate the strength of Pareto optimal fronts. Two more multi-objective performance

measures namely optimiser overhead and algorithm effort are used to find the computational effort of

NSGA-II and MOPSO algorithms. The Pareto optimal fronts and results obtained from various techniques

are compared and analysed.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The tolerance design problem becomes more complex in the

presence of alternative processes (or machines) for manufac-

turing of each dimension. This is because the manufacturing

cost–tolerance characteristics differ from process to process, and

from machine to machine. All costs incurred during a product’s

life cycle can be divided in two main categories: manufacturing

cost, which occurs before the product reaches the customer; and

quality loss, which occurs after the product is sold. A loose tol-

erance (low manufacturing cost) indicates that the variability of

product quality characteristics shall be great (high quality loss).

On the other hand, a tight tolerance (high manufacturing cost)

∗Corresponding author. Tel.: +91 4312 650550; fax: +91 4312 650377.

E-mail addresses: ksk71@rediffmail.com (K. Sivakumar), rlc_bal@yahoo.co.in

(C. Balamurugan).

indicates the variability of product quality characteristics shall be

small (little quality loss). Hence, there arises a need to adjust the

design tolerance between quality loss and manufacturing cost and

to reach economic balance during product tolerance design.

In earlier years, researchers focused their attention on obtain-

ing the best tolerance allocation in such a way that product design

not only meets the functional requirements but also minimum

manufacturing cost. In order to solve the tolerance allocation

problem, various numerical methods were employed to deal with

complicated computations associated with tolerance design mod-

els. Ye and Salustri [1] introduced a new concurrent engineering

method for tolerance allocation and constructed a nonlinear opti-

mization model to implement the method. The model minimized

quality loss and manufacturing cost simultaneously in a single

objective function by setting both process tolerances and design

tolerances. Singh et al. [2,3] explored the application of genetic

algorithms to obtain the optimal solution to a set of tolerance

design problems with simple dimension chain involving sets of

0010-4485/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cad.2010.10.001

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K. Sivakumar et al. / Computer-Aided Design 43 (2011) 207–218

alternative processes. Prabhaharan et al. [4] used GA for optimal

toleranceallocationtohelpdesignandmanufacturingengineersto

overcome the shortcomings in the conventional tolerance stack

analysis and allocation system. Prabhaharan et al. [5] intro-

duced a continuous ant colony algorithm, a kind of metaheuristic

approachasanoptimizationtoolforminimizingthecriticaldimen-

sion deviation and allocating the cost-based optimal tolerances.

Krishna and Rao [6] used a scatter search method to simultane-

ously allocate both design and manufacturing tolerances based on

minimum total manufacturing cost. Huang and Shiau [7] obtained

the optimized tolerance allocation of a sliding vane rotary com-

pressor’s components for the required reliability with the mini-

mum cost and quality loss. Huang and Zhong [8] established the

sequential linear optimization models based on the process

capabilities. This approach releases the working tolerances,

reduces manufacturing costs, and enhances the acceptance rate of

machined parts. Singh et al. [9] introduced GA to obtain an opti-

mal solution to the advanced tolerance synthesis problem by con-

sidering continuous cost function. This method works for both

single and multiple tolerance stack-ups that share one or more

individual tolerances. Siva Kumar et al. [10] used a hybrid algo-

rithm (Tabu search + Heuristic algorithm) for optimum tolerance

allocation in complex assemblies with alternative process selec-

tion.GonzalezandSanchez[11]developedamethodologytoallow

an automatic tolerance allocation capable of minimizing manufac-

turing costs based on a statistical approach. Huang and Zhong [12]

used linear programming methods to obtain process tolerances

concurrently from an assembly by using the information of pro-

cess planning. Nonlinear optimal models have been established to

minimize the total manufacturing cost. Forouraghi [13] developed

a methodology to allow an automatic tolerance allocation capa-

ble of minimizing manufacturing costs based on a particle swarm

optimizer. Siva Kumar and Stalin [14] used a Lagrange multiplier

method to simultaneously allocate both design and manufactur-

ing tolerances based on minimum total manufacturing cost. Wu

et al. [15] developed a methodology to allow an automatic toler-

anceallocationcapableofminimizingbothmanufacturingcostand

quality loss based on a Monte Carlo simulation. Muthu et al. [16]

used a metaheuristic method to balance the manufacturing cost

and quality loss to achieve near optimal design and process toler-

ances simultaneously for minimum combined manufacturing cost

and quality loss over the life of the product. Forouraghi [17,18]

introduced a new design methodology which brings together

concepts from diverse fields of robust design, statistical quality

assurance, genetic algorithms, and tolerance analysis in order to

concurrently address tolerance and robust design in a unified

manner without considering alternative manufacturing process

selection.

From the literature review, the previous work on tolerance

allocation can be summarised as follows: 1. Scatter search method,

sequential linear method, Lagrange multiplier method, Monte

Carlo simulation, Linear programming methods, etc. were used

to solve the tolerance allocation problem, 2. A lot of research

has been carried out on the manufacturing cost only without

considering the quality loss of the product, 3. In some research

works, total manufacturing cost and quality loss analysis were

consideredusingatraditionalmethod;However,theseapproaches

suffer from following drawbacks: 1. Closed form solution is limited

only to small portion of the spectrum of the real world tolerancing

problems. These approaches are applicable to the objective

functions involving only simple cost functions, such as ‘reciprocal

powerfunction’(reciprocal,andreciprocalsquarefunctionsarethe

special cases of this function) and ‘exponential function’, subjected

only to traditional tolerance stack-up constraints (based on the

worst case and the root sum square criteria), 2. These approaches

are not applicable to discontinuous and/or non-differentiable cost

functions, as it requires a continuous first derivative, 3. Cost

functions with preferred process limits cannot be handled using

thismethod,4.Assembliesinvolvinginterrelateddimensionchains

(more than one assembly response functions with some common

dimensions) are difficult to handle, 5. These methods can attract

local minima. In order to overcome this problem, the process

should be started from various initial guesses, 6. The method

proposed by Forouraghi [13,17,18] does not consider alternative

manufacturing process selection.

To overcome the drawbacks of conventional optimization

approaches, intelligent optimization techniques such as GA,

simulated annealing [4,19–22], NSGA-II and MOPSO can be

used. Intelligent algorithms have proven successful in handling

many real-world multiobjective concurrent engineering problems

[23–26]. The advantages of intelligent techniques are 1. They

are a population-based search techniques, so a global optimal

solution is possible, 2. They do not need any auxiliary information

like gradients, derivatives, etc. 3. They can solve complex and

multimodel problems for global optimality, 4. They are problem

independent, i.e. Suitable for solving all types of problems, 5. They

offer Pareto optimal fronts that offer more number of optimal

solutions for the user’s choice, 6. They use an average fitness factor

method to find the best optimal solution from the Pareto optimal

fronts.

But the major limitation of the tolerance design work by Singh

et al. [21,22] is that only one objective function (minimization of

manufacturing cost) was considered. But a real world tolerance

design problem have more than one objective function. Such

important objective functions are quality loss and stack-up

tolerance.

To overcome the limitations of Singh et al. [21,22] work,

in this paper two intelligent optimization techniques NSGA-II

and MOPSO are proposed to do optimal tolerance design for

mechanical assemblies by considering all important objective

functions (Minimization of manufacturing cost, quality loss and

tolerance stack up), yield design constraint and machining

tolerance constraints. The average fitness factor method and

normalized weighted objective function method are separately

used to select the best optimal solution from Pareto optimal fronts

(optimal solution trade-offs). Two multiobjective performance

measures namely solution spread measure and ratio of non-

dominated individuals are used to evaluate the strength of Pareto

optimal fronts. Two more multiobjective performance measures

namely optimizer overhead and algorithm effort are used to find

the computational effort of the NSGA-II and MOPSO algorithms.

The significant contributions of our paper to the tolerance field

are1.Itproposesanovelapproachtodoalternativemanufacturing

process selection and tolerance allocation simultaneously, 2. This

paper simultaneously considers the minimization of tolerance

stack-up, manufacturing cost and the quality loss as objective

functions, 3. It improves the recent optimization model proposed

by Singh et al. [21,22] by adding two new objective functions

(qualitylossandtolerancestack-up),and4.Theproposedapproach

using NSGA-II and MOPSO solves all drawbacks of the methods

reported in literature [1–16,19–22].

The rest of this study is organized as follows: The Section 2

deals the problem definition. Section 3 presents the problem

formulation. Section 4 presents the proposed NSGA-II and MOPSO

techniques to obtain the optimal solutions. Section 5 deals two

methods and four multi-objective performance metrics used for

evaluating the proposed algorithms. Section 6 illustrates two

numerical examples. The running procedures for NSGA-II and

MOPSO algorithms are given in Section 6.3. In Section 7, the results

obtained from various methods are presented and compared. The

conclusions are presented in Section 8.

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209

Fig. 1. Machined layers and manufacturing tolerances [27].

2. Problem definition

2.1. Stock removal allowances

Singh et al. [27] addressed the stock removal allowances

manufacturing tolerance selection is greatly affected by the

amount of stock removal allowance. The stock removal allowance

is the layer of material to be removed from the surface of a

work-piece to obtain the required accuracy and surface quality

through machining. It greatly influences the quality and the

production efficiency of the machined features. Excessive stock

removal allowance will increase the consumption of material,

machining time, tools and power, while machining the work-piece

by a more expensive downstream process, results in an increase

in the manufacturing cost. On the other hand, with an insufficient

stock removal allowance, the defective surface layer caused by

the preceding operation cannot be rectified. The concept of stock

removal allowance has been depicted in Fig. 1.

The amount of stock removal allowance is the difference

between the dimensions obtained in the preceding operation

and the current operation. Since the manufacturing dimensions

are not fixed, and each of them is associated with some

tolerance, the actual stock removals from work-piece surfaces

vary in a certain range. The stock removal based on the nominal

working dimensions in successive operations is considered its

nominal value. Variation in the stock removal is the sum of

manufacturing tolerances in the proceeding and the current

operation. An appropriate stock removal allowance is required

for each successful manufacturing operation. This yields a set of

necessary constraints during simultaneous selection of design and

manufacturing tolerances.

2.2. Selection of machining process

The selection of machining process including equipment accu-

racy, set-up mode, machining sequence and cutting parameters is

strongly affected by the tolerance of the part to be machined. So it

is important to do a simultaneous selection of the best machining

process while allocating the tolerance.

2.3. Manufacturing cost

Manufacturing cost usually increases as the tolerance of qual-

ity characteristics in relation to the ideal value is reduced, because

more refined and precise operations are needed while the accept-

able ranges of output are reduced. Conversely, large tolerances are

less costly to achieve as they require less precise manufacturing

processes; but they usually result in poor performance, premature

wear and part rejection.

Assembly A (Overrunning clutch assembly, Singh et al. [22])

Total assembly manufacturing cost is the summation of all

the costs associated with manufacturing operations of all the

individual dimensions. The following expressions (Eqs. (1) and

(2)) represent the formulation of the manufacturing cost under

consideration for optimization. A 5th order polynomial, one of

the non-traditional cost functions, has been used to represent the

cost–tolerance relationship mathematically.

Casm= CX1+ CX3

(1)

Casm=

3

−

i=1,i̸=2

3

−

j=1

(a0+ a1t + a2t2+ a3t3+ a4t4+ a5t5)|xij.

(2)

Assembly B (Knuckle joint with three arms, Singh et al. [21])

The total assembly manufacturing cost can be represented by

Eq. (3). Minimization of the total cost constitutes the objective

function of the optimization problem

Casm= CX1+ 2CX2a+ CX2b+ 2CX3a+ CX3b+ CX4+ CX5.

The following tolerance-cost function is used to find the manufac-

turing cost of a process tolerance.

(3)

2.4. Quality loss function

Variability in the production process is unavoidable due to

inconsistency in tool work piece, material and process parameters.

Haq et al. [19] suggest that given ideal target value, an evaluation

function associated with deviations from the target can be

developed. In this study it is referred to as the quality loss function.

This loss function is a quadratic expression for measuring the cost

of the average value versus the target value and the variability of

product characteristics in terms of monetary loss due to product

failure in the eyes of the consumers. The quality loss function (QL)

is:

QL =

A

T2

I −

i=1

σ2

i.

(4)

For a three sigma tolerance design,

ti

3.

Then, (4) is rewritten as

σi=

QL =

A

9T2

I −

i=1

t2

i

(5)

where, T is the tolerance stack-up limit of the dimensional chain,

A is the quality loss cost.

3. Problem statements

The improved optimization model opens up possibilities for

tolerance control in order to achieve selection of the best

manufacturing processes, economical assembly manufacturing

cost and the minimum quality loss of the product. Formulation

of the simple optimal tolerance synthesis involves framing of

the objective functions and constraints. The objective functions

considered are: minimum tolerance stack-up (Z1 for Assembly

A and Z1, Z2,Z3for Assembly B), minimum total manufacturing

cost of the assembly (Z2 for Assembly A and Z4 for Assembly

B) (summation of the manufacturing cost involved in the

manufacture of all the toleranced dimensions) and minimum

quality loss function (Z3for Assembly A and Z5for Assembly B).

Thefunctionalrequirementsoftheassembly(stack-upconditions),

process precision limits and process selection conditions form the

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K. Sivakumar et al. / Computer-Aided Design 43 (2011) 207–218

constraints of the problem. The variables are tolerances of all part

dimensions. The optimization model used by Singh et al. [22] is

improved by adding a new objective function, minimization of

quality loss function (Z3for Assembly A and Z5for Assembly B).

A combined objective function approach is used here. Combined

objective functions for Assemblies A and B are

3.1. Assembly A (overrunning clutch assembly)

Minimize:

Z1= ∆Y = ξ1t13|x13+ ξ2t2+ ξ3t33|x33

Z2 = Casm

=

i=1,i̸=2

A

9T2

i=1

(6)

3

−

3

−

j=1

(a0+ a1tij+ a2t2

ij+ a3t3

ij+ a4t4

ij+ a5t5

ij)|xij

(7)

Z3= QL =

I −

t2

ij.

(8)

3.2. Assembly B (Knuckle joint with three arms)

Minimize:

Z1= ∆Y1= t1j+ t2j

Z2= ∆Y2= 3t2j+ t3j

Z3= ∆Y3= 3t3j+ t4j+ t5j

Z4 = Casm

= CX1+ 2CX2a+ CX2b+ 2CX3a+ CX3b+ CX4+ CX5

A

9T2

i=1

Constraints:

−

Variables:

Process precision limits : tmin

i = 1 to n,j = 1 to mi

where

Y = Assembly response function (Assembly A).

Casm= Total assembly manufacturing cost.

ξ1,ξ2,ξ3 = Value of sensitivity for component dimensions X1,

X2,X3,

QL = Quality loss function.

∆Y1= Tolerance stack-up for dimension Y1(Assembly B).

∆Y2= Tolerance stack-up for dimension Y2(Assembly B).

∆Y3= Tolerance stack-up for dimension Y3(Assembly B).

Cij(tij)= Cost of producing dimension xiby the process j maintain-

ing tolerance tij.

mi = Number of the available alternative processes for

producing dimension xi.

n = Number of the component dimensions.

Tkasm= Permissible variation in the kth assembly dimension,

known as assembly tolerance.

tij= Tolerance on the dimension xiproduced by the jth process.

(9)

(10)

(11)

(12)

Z5= QL =

I −

t2

ij.

(13)

Stack-up conditions -

n

i=1

tij≤ Tkasm.

(14)

ij

≤ tij≤ tmax

ij

4. Proposed methods

In this section, the proposed intelligent optimization tech-

niques NSGA-II and MOPSO are described.

4.1. Elitist non-dominated sorting genetic algorithm (NSGA-II)

Kalyanmoy Deb proposed the NSGA-II algorithm [29]. Essen-

tially, NSGA-II differs from non-dominated sorting Genetic Algo-

rithm (NSGA) implementation in a number of ways. First, NSGA-II

uses an elite-preserving mechanism, thereby assuring the preser-

vation of previously found good solutions. Second, NSGA-II uses a

fast non-dominated sorting procedure. Third, NSGA-II does not re-

quire any tunable parameter thereby making the algorithm inde-

pendent of the user.

Initially, a random parent population Po is created. The

population is sorted based on the non-domination. A special book-

keeping procedure is used in order to reduce the computational

complexity to O(MN2). Each solution is assigned a fitness equal to

its non-dominated level (1 is the best level). Thus, minimization

of fitness is assumed. Binary tournament selection, recombination,

and mutation operators are used to create a child population Qoof

size N. Thereafter the algorithm below is used in every generation.

Rt= PtUQt

F = fast-non-dominated-sort (Rt)

Pt+1= φ and i = 1

Until |Pt+1| + |Fi| ≤ N

Pt+1= Pt+1UFi

crowding-distance-assignment (Fi)

i = i + 1

Sort (Fi∝n)

Pt+1= Pt+1UPt+1[1 : (N − |Pt+1|)]

Qt+1= make-new-pop (Pt+1)

t = t + 1

First, a combined population Rt

allows parent solutions to be compared with the child population,

thereby ensuring elitism. The population Rtis of size 2N. Then,

the population Rtis sorted according to non-domination and non-

dominated fronts F1,F2, and so on are found. The algorithm is

illustrated in the following:

The new parent population Pt+1is formed by adding solutions

from the first front F1and continuing to other fronts successively

until the size exceeds N. Individuals of each front are used to

calculate the crowding distance — the distance between the

neighboringsolutions.Thereafter,thesolutionsofthelastaccepted

front are sorted according to a crowded comparison criterion and a

totalofN pointsarepicked.Sincethediversityamongthesolutions

is important, the crowded comparison criterion uses a relation

αnas follows: solution i is better than solution j in relation αnif

(irank < jrank) or ((irank = jrank) and (idistance > jdistance)). That is,

between two solutions with differing non-domination ranks the

preference is the point with the lower rank. Otherwise, if both

the points belong to the same front, then the preference is the

point which is located in a region with a smaller number of points

(or with larger crowded distance). This way, solutions from less

dense regions in the search space are given importance in deciding

which solutions to choose from Rt. This constructs the population

Pt+1. This population of size N is now used for selection, crossover

and mutation to create a new population Qt+1of size N. A binary

tournament selection operator is used but the selection criterion

is now based on the crowded comparison operator αn. The above

procedure is continued for a specified number of generations. It

is clear from the above description that NSGA-II uses (i) a faster

non-dominated sorting approach, (ii) an elitist strategy, and no

niching parameter. It has been proved that the above procedure

has O(MN2) computational complexity. The complexity is order

of (M× Population size (N)2). Non-dominated or pareto-optimal

solutions are those solutions in the set which do not dominate

=

PtUQt is formed. This

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K. Sivakumar et al. / Computer-Aided Design 43 (2011) 207–218

211

Non-dominated

sorting

Crowding

distance

sorting

F1

F2

F3

Rejected

Pt+1

Rt

Qt

Pt

Fig. 2. An iteration procedure of the NSGA-II algorithm [29].

each other, i.e., neither of them is better than the other in all the

objective function evaluations. The solutions on each pareto-front

are pareto-optimal with respect to each other.

Pseudo Code for NSGA-II is explained in the Appendix A. Fig. 2

shows an iteration of the proposed NSGA-II procedure.

4.1.1. NSGA-II operators

We have conducted a study to select a particular choice of

the NSGA-II parameters. The values of the parameters of NSGA-

II technique used in this study give the best optimal results. The

following are the values of the parameters of NSGA-II technique

used in this study:

Variable type= Real variable, Population size= 100, Crossover

probability = 0.6, Real-parameter mutation probability = 0.01,

Real-parameter SBX parameter = 10, Real-parameter mutation

parameter = 100, Total number of generations = 100.

4.2. Multi objective particle swarm optimization (MOPSO) [30]

The proposed algorithm MOPSO extends the single-objective

PSO algorithm to handle multiobjective optimization problems.

It incorporates the mechanism of crowding distance computa-

tion into the algorithm of PSO specifically on the global best

selection and in the deletion method of an external archive of

non-dominated solutions. The crowding distance mechanism to-

gether with a mutation operator maintains the diversity of non-

dominated solutions in the external archive. Also MOPSO has a

constraint handling mechanism for solving constrained optimiza-

tion problems. Fig. 3 shows an iteration of the proposed MOPSO

procedure. Pseudo Code for MOPSO is explained in the Appendix B.

4.2.1. MOPSO operators

The following are the values of the parameters of MOPSO

technique that have been used to obtain the best optimal results:

Population size = 100, Mutation probability = 0.5, Total number

of generations = 100, Inertia weight = 0.4.

5. Performance measures and methods for multi-objective

optimization

In this section, two methods and four performance metrics are

recommended and applied to examine the strength and weak-

nesses of the proposed multi-objective intelligent optimization

algorithms. Two methods (normalized weighting objective func-

tions and average fitness factor) are used to select the best opti-

mal solution. Two multiobjective performance measures namely

solution spread measure and ratio of non-dominated individuals

are used to evaluate the Pareto optimal fronts. Two more multi-

objective performance measures namely optimizer overhead and

algorithm effort are used to find the computational effort of an

optimization algorithm. These methods and metrics are chosen

since they have been widely used for performance comparisons in

multi-objective optimization [31].

Fig. 3. An iteration procedure of the MOPSO algorithm [30].

5.1. Normalized weighting objective functions

Multiple objectives are combined into scalar objective via a

weight vector. Weights may be assigned through direct assign-

ment, eigenvector method, empty method, minimal information

method,randomlydeterminedoradaptivelydetermined.Iftheob-

jective functions are simply weighted and added to produce a sin-

gle fitness, the function with the largest range would dominate

evolution. A poor input value for the objective function with a

largerrangemakestheoverallvaluemuchworsethanapoorvalue

fortheobjectivefunctionwithasmallerrange.Toavoidthis,allob-

jective functions are normalized to have same range. Also normal-

izingparametersmakeallobjectivefunctionsasunitlessfunctions.

For our problem, the combined objective function(fc) is defined as

follows:

Assembly A (Overrunning clutch assembly)

Minimize fc= W1× Z1/N1+ W2× Z2/N2+ W3× Z3/N3.

Assembly B (Knuckle joint with three arms)

(15)

Minimize fc = W1× Z1/N1+ W2× Z2/N2+ W3× Z3/N3

+W4× Z4/N4+ W5× Z5/N5.

The values of N1 = 0.1, N2 = 100 and N3 = 10 for Assembly A

(Overrunning clutch assembly) and the values of N1= N2= N3=

N5= 1.0 and N4= 100 for Assembly B (Knuckle joint with three

arms) are normalizing parameters of objective functions Z1, Z2, Z3,

Z4and Z5.

W1, W2, W3, W4and W5are the weights given to objective

functions 1, 2, 3, 4 and 5 respectively. Here the normalized

weighting objective functions method is used only to select the

best optimal solution from Pareto optimal fronts obtained from

NSGA-II and MOPSO. So we can give any weight to all objective

functions. But the condition is that summation of all weight values

should be 1. It means the total weight should be 100%. W1= W2=

W3 = 0.333 for Assembly A (overrunning clutch assembly) and

W1 = W2 = W3 = W4 = W5 = 0.2 for Assembly B (knuckle

joint with three arms). It means we are giving equal weight to both

objective functions.

For Assembly A (overrunning clutch assembly), the original

values of first, second and third objective functions from NSGA-II

(16)