Optimization of Mechanical Design Problems Using Improved Differential Evolution Algorithm

AMAE Int. J. on Production and Industrial Engineering, Vol. 02, No. 01, June 2011

Optimization of Mechanical Design Problems Using
Improved Differential Evolution Algorithm
Millie Pant, Radha Thangaraj and V. P. Singh
Department of Paper Technology,
Indian Institute of Technology Roorkee, India.
millifpt@iitr.ernet.in, t.radha@ieee.org, singhfpt@iitr.ernet.in
type of problem (for example Quadratic Programming
Problems, Geometric Programming Problems etc). Keeping in
view the limitations of traditional techniques researchers have
proposed the use of stochastic optimization methods and
intelligent algorithms for solving NLP which may be
constrained or unconstrained. Some examples are: Genetic
Algorithms [1] – [3], Ant Colony Optimization [4], Chaos
Optimization Algorithm [5], Particle Swarm Optimization [6],
Differential Evolution [7] etcetera. Based on the research
efforts in literature, constraint handling methods have been
categorized in a number of classes [8] - [10]:
 Reject infeasible solutions
 Penalty function methods
 Convert the constrained problem to an unconstrained
problem
 Preserving feasibility methods
 Pareto ranking methods
 Repair methods
In the present research paper we have concentrated our
work to DE, which is comparatively a newer addition to the
class of population based search techniques. DE is a
stochastic, population based search strategy developed by
Storn and Price [7] in 1995. It is a novel evolutionary approach
capable of handling no-differentiable, non-linear and
multimodal objective functions. DE has been designed as a
stochastic parallel direct search method, which utilizes
concepts borrowed from the broad class of EAs. The method
typically requires few, easily chosen control parameters. This
paper presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The structure of the paper is as follows: in section
II, we have briefly explained the Differential Evolution
Algorithm, in section III; we have defined and explained the
proposed ICDE algorithm. Section IV deals with experimental
settings and test problems, Section V gives the numerical
results and discussion and finally the paper conclude with
section VI.

Abstract— Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
Index Terms—Differential Evolution, optimization,
Mechanical design problems, constraint optimization.

I. INTRODUCTION
Many real-world optimization problems are solved subject
to sets of constraints. The search space in COPs consists of
two kinds of solutions: feasible and infeasible. Feasible points
satisfy all the constraints, while infeasible points violate at
least one of them. Therefore the final solution of an
optimization problem must satisfy all constraints. A
constrained optimization problem may be distinguished as a
Linear Programming Problem (LPP) and Nonlinear
Programming Problem (NLP). In this paper we have
considered NLP problems where either the objective function
or the constraints or both are nonlinear in nature. The general
NLP is given by nonlinear objective function f, which is to be
minimized/maximized with respect to the design variables

x  ( x1 , x2 ,....., xn ) and the nonlinear inequality and
equality constraints. This can be formulated by,

Minimize / Maximize f (x )

Subject to : g j ( x )  0, j  1,......, p
hk ( x )  0, k  1,......, q

(1)
II. DIFFERENTIAL E VOLUTION ALGORITHM
(2)

DE shares a common terminology of selection, crossover
and mutation operators with GA however it is the application
of these operators that make DE different from GA. Whereas,
in GA crossover plays a significant role, it is the mutation
operator which effects the working of DE [11].
The working of DE may be described as follows:
Mutation: For a D-dimensional search space, for each target

xi min  xi  xi max (i  1,......,n) .
where p and q are the number of inequality and equality
constraints respectively. There are many traditional methods
in the literature for solving NLP. However, most of the
traditional methods require certain auxiliary properties (like
convexity, continuity etc.) of the problem and also most of
the traditional techniques are suitable for only a particular
© 2011 AMAE

DOI: 01.IJPIE.02.01.205

vector X i , g at the generation g, its associated
16

mutant vector is generated via certain mutation strategy. The
most frequently used mutation strategies implemented in the
DE codes are listed below.
DE/rand/1(S1): Vi , g  X r1 , g  F * ( X r2 , g  X r3 , g )
DE/rand/2 (S2):

The proposed ICDE algorithm uses the mean zero standard
deviation one normal distribution for initializing the population and uses the following three selection criteria: After calculating the trial vector (i) If the trial vector and the target
vector are feasible then select the best one (ii) If both the
particles are infeasible then select the one having smaller
constraint violation (iii) If one is feasible and the other one is
infeasible then select the feasible one. Also at the end of
every iteration, the particles are sorted by using the three
criteria: (a) Sort feasible solutions in front of infeasible solutions (b) Sort feasible solutions according to their fitness
function values (c) Sort infeasible solutions according to
their constraint violations. The computational steps of ICDE
algorithm is given below:
Step 1 Initialize the population using
normal distribution with mean zero and
standard deviation one.
Step 2 For all particles
Evaluate the objective function
Calculate the constraint violation
End for
Step 3 While stopping criterion is not
satisfied
Do
Step 3.1 Mutation
For all particles
G e n e r a t e a m u t a t e d v e c t o r V i,g
corresponding to the target vector Xi,g
via one of the equations (3) to (7)
End for
Step 3.2 Crossover //Generate trial
vector Ui,g
For all particles
Select jrand {1,…,D}
For j = 1 to D
If (rand(0,1) d” CR or j= jrand)
Then Ui,g = Vi,g
Else Ui,g = Xi,g
End if
End for
End for
Step 3.3 Selection
For all particles
Set Xi,g+1 according to the three selection
criteria
End for
Step 3.4 Sort the particles using the
three sorting rules
Step 3.5 Go to next generation
Step 4 End while

(3)

Vi, g  X r1 , g  F * ( X r2 , g  X r3 , g )  F * ( X r4 , g  X r5 , g )
(4)
(5)

DE/bes t/1 (S3): Vi, g  X best, g  F * ( X r1 , g  X r2 , g )
DE/best/2 (S4):

Vi , g  X best, g  F * ( X r1 , g  X r2 , g )  F * ( X r3 , g  X r4 , g )
(6)
DE/rand-to-best/1 (S5):
Vi , g  X r1 , g  F * ( X best , g  X r2 , g )  F * ( X r3 , g  X r4 , g )

(7)
where r1 , r2 , r3 , r4 , r5 {1,2,...., NP} are randomly chosen
integers, must be different from each other and also different
from the running index i. F (>0) is a scaling factor which
controls the amplification of the difference vector. X best , g is
the best individual vector with the best fitness value in the
population at generation g.
Crossover: In order to increase the diversity of the perturbed
parameter vectors, crossover is introduced [12]. The parent
vector is mixed with the mutated vector to produce a trial
vector u ji , g 1

(rand j  CR) or ( j  jrand )
v
u ji , g 1   ji , g 1 if
x ji , g if (rand j  CR) and ( j  jrand ) (8)

where j = 1, 2,……, D; rand j  [0,1] ; CR is the crossover
constant

takes

values

in

the

range

[0,

1]

and j rand  (1,2,....., D ) is the randomly chosen index.
Selection: Selection is the step to choose the vector between
the target vector and the trial vector with the aim of creating
an individual for the next generation. The simple flow of DE
algorithm is given in Fig 1.

Fig 1 Flow of DE algorithm

IV. EXPERIMENTAL SETTINGS AND TEST PROBLEMS

III. PROPOSED ICDE ALGORITHM

In order to make a fair comparison of all versions of DE
algorithms, we fixed the same seed for random number
generation so that the initial population is same for both the
algorithms. The population size is taken as 50. The crossover
constant CR is set as 0.95 and the scaling factor F is set as

The proposed algorithm ICDE is a simple algorithm for
solving constraint optimization problems, it is easy to
implement. It differs from unconstrained optimization
algorithm only in the place of initialization, selection of
particles to the next generation and sorting the final results.
© 2011 AMAE

DOI: 01.IJPIE.02.01.205

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0.7. For each algorithm, the stopping criteria is to terminate
the search process when one of the following conditions is
satisfied: (1) the maximum number of generations is reached
(assumed 10000 generations), (2) | fmax - fmin | < 10-4 where f is
the value of objective function. A total of 30 runs for each
experimental setting were conducted. If the run satisfies the
second stopping condition then that run is called successful
run. Also we implemented the new idea to five versions of DE
algorithm. To check the efficiency of the proposed ICDE
algorithm we have tested it on four optimization problems
arising commonly in the field of Mechanical engineering. All
the problems considered here are highly non linear in nature
and are subject to various constraints. The mathematical
models of these problems may be given as:

Fig 2 Heat Exchanger Network Design Problem

C. Gas Transmission Compressor Design (GTCD) [15]
The mathematical model is,
f ( x )  8.61105 x11 / 2 x2 x3 2 / 3 x4 1 / 2  3.69 10 4 x3

A. Weight Minimization of a Speed Reducer (WMSR) [13]
The mathematical model of this problem is,
Min

 7.72 108 x11 x2 0.219  765.43 10 6 x11
Subject to
x4 x 2 2  x2 2  1

f ( x)  0.7854 x1 x2 2 (3.3333 x3 2  14.9334 x3  43.0934)

20  x1  50 , 1  x 2  10 , 20  x3  50 , 0.1  x 4  60

 1.508x1 ( x6 2  x7 2 )  7.477( x6 3  x7 3 )  0.7854( x4 x6 2  x5 x7 2 )

D. Optimal Design of Industrial refrigeration System (ODIRS)
[16]

Subject to
x1 x2 2 x3  27 , x1 x 2 2 x3 2  397.5 , x2 x6 4 x3 x4 3  1.93 ,
A1 B11  1100

f ( x)  63098.88 x 2 x 4 x12  5441.5 x 2 2 x12  115055.5 x 21.664 x 6

Where A1  [(745 x4 x2 1 x3 1 ) 2  16.96116 ]0.5 , B1  0.1x6 3

 6172.27 x2 2 x6  63098.88x1 x3 x11  5441.5x12 x11

A2 B2 1  850

W h e r e

 115055.5x11.664 x5  6172.27 x1 2 x5  140.53x1 x11
A2  [(745 x5 x 2 1 x3 1 ) 2  15.7510 6 ]0.5

,

 281.29 x3 x11  70.26 x12  281.29 x1 x3  281.29 x3 2

B2  0.1x7 3

 14437 x81.8812 x12 0.3424 x10 x14 1 x12 x7 x9 1

x2 x 3  40 , x1 x2 1  5 , x1 x2 1  12 , 1.5 x6  x 4  1.9 ,

 20470.2 x7 2.893 x110.316 x12
Subject to

1.5 x7  x5  1.9 .
2.6  x1  3.6 ,

0.7  x 2  0.8 ,

1.524 x7 1  1 , 1.524 x8 1  1 , 0.07789 x1  2 x7 1 x9  1 ,

17  x3  28 ,

7.3  x4  8.3 , 7.3  x5  8.3 , 2.9  x6  3.9 , 5  x7  5.5

7.05305 x9 1 x12 x10 x8 1 x 2 1 x14 1  1 , 0.0833x131 x14  1 ,

B. Heat Exchanger Network Design (HEND) [14]

47.136 x2 0.333 x10 1 x12  1.333x8 x13 2.1195
 62.08x132.1195 x12 1 x8 0.2 x10 1  1 ,

Minimize f ( x)  x1  x 2  x3
Subject to

0.04771x10 x81.8812 x12 0.3424  1 ,

1  0.0025( x4  x6 )  0 , 1  0.0025( x 5  x 7  x 4 )  0 ,
1  0.01( x8  x 5 )  0

0.0488 x9 x71.893 x110.316  1 ,

,

0.0193x2 x4 1  1 ,

 x1x6  833.33252 x4  100 x1  83333.333  0 ,

0.0298 x1 x5 1  1 ,

 x2 x7  1250 x5  x2 x4  1250 x4  0 ,

0.056 x 2 x6 1  1 ,

2 x9 1  1 , 2 x10 1  1 , x12 x111  1 ,

 x3x8  1250000  x3x5  2500x5  0 ,

0.0099 x1 x3 1  1 ,

0.001  xi  5 , i  1,...,14
V. EXPERIMENTAL RESULTS AND DISCUSSION

100  x1  10000 , 1000  xi  10000 (i  2,3) ,

Tables I – IV gives the numerical results given four real
life constrained optimization problems. These problems occur
frequently in the field of mechanical designs. The comparison
criteria for the algorithms is done in terms of best, average
and worst fitness function values, NFE, std, SR and time. For
© 2011 AMAE

DOI: 01.IJPIE.02.01. 205

18

all the algorithms, NFE represents the number of function
evaluations, which helps in determining the convergence of
the algorithm. Lesser value of NFE implies faster convergence.
std represents the standard deviation which tells the stability
of the algorithms. Smaller std implies that the algorithm is
more stable. SR represents the success rate, which signifies
the efficiency of an algorithm. It tells us how many times the
algorithm was able to converge successfully within 1% of the
true global optima. For all the problems we compared our
results with those available in literature. Form Tables I to IV,
the

© 2011 AMAE

DOI: 01.IJPIE.02.01.205

results obtained by the different DE versions and the ones
available in literature are given. From the numerical results it
is quite visible that the versions of DE gave better results
than the ones available in literature. In terms of best, average
and worst fitness function values all the algorithms gave
more or less similar results. However in terms of NFE, SR and
time taken, the algorithms showed different behavior.

19

The first problem is involves the design of a speed reducer
for small aircraft engine. It has a nonlinear objective function
and it consists of eleven inequality constraints and seven
unknown variables. For this problem all the DE versions gave
same results in terms of best, worst and average fitness
function values. If we compare the NFC and convergence
time then DE/rand-to-best/1 is better than all other versions.
The second problem addresses the design of a heat exchanger
network as shown in Fig 2. It has three equality constraints,
three inequality constraints and eight decision variables. For
this problem also all the algorithms gave same result in
comparison best fitness function value. In comparison of
average fitness value DE/best/1 gave slightly worse value
than other algorithms. But in terms of convergence time it is
better than all other versions. The third problem is a gas
transmission compressor design problem. For this problem
DE/best/1 gave better result in terms of standard deviation,
NFE and convergence time. DE/rand/1 gave better result
than other algorithms in terms of best fitness function value.

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VI CONCLUSION
In this paper, we proposed an Improved DE algorithm
called ICDE for solving constrained optimization problems.
ICDE differs from the basic DE in the initialization, selection
and sorting phases. These modifications are embedded on
various versions of DE and their efficiency is validated on a
set of four real life engineering design problems, occurring
frequently in the field of mechanical engineering. We would
like to add that in the present article though we have
considered only four problems, the preliminary numerical
results obtained, show that the proposed modifications are
beneficial for solving constrained optimization problems
effectively. Moreover, this is a general technique/ modification
and can be applied to any population based search technique
like Genetic Algorithms, Particle Swarm Optimization etc.
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Computers and Industrial Engineering, Vol. 29, pp. 531 – 535,
1995.

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DOI: 01.IJPIE.02.01.205

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