imperialist competitive algorithm for engineering ... - Semantic Scholar

ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 11, NO. 6 (2010) PAGES 675-697

IMPERIALIST COMPETITIVE ALGORITHM FOR ENGINEERING DESIGN PROBLEMS A. Kaveh*a and S. Talataharib a Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran b Department of Civil Engineering, University of Tabriz, Tabriz, Iran

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ABSTRACT

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Imperialist Competitive Algorithm (ICA) is one of the recent meta-heuristic algorithms proposed to solve optimization problems. The Imperialist Competitive Algorithm is based on a socio-politically inspired optimization strategy. This paper presents four different variants of this algorithm. These methods are applied to some engineering design problems and a comparison is made among the results of these algorithms and other meta-heuristics. The results show the efficiency and capabilities of the ICA in finding the optimum design.

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Keywords: Imperialist competitive algorithm; meta-heuristic algorithms; engineering design problems

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1. INTRODUCTION

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In the recent decades, many mathematical algorithms have been developed to solve various engineering optimization problems. However due to some disadvantages of these methods to handle engineering problems [1], meta-heuristic algorithms, such as Genetic Algorithms (GAs) proposed by Holland [2] and Goldberg [3], Particle Swarm Optimization (PSO) developed by Eberhart and Kennedy [4], Ant Colony Optimization (ACO) formulated by Dorigo et al. [5], Harmony Search (HS) suggested by Geem et al. [6], Big Bang–Big Crunch algorithm (BB–BC) proposed by Erol and Eksin [7] and improved by the authors [8], and Charged System Search (CSS) developed by Kaveh and Talatahari [9-11] are widely utilized for solving optimization engineering design problems. These methods are mostly powerful tools for obtaining the solution of optimization problems. Recently a new meta-heuristic algorithm, so called Imperialist Competitive Algorithm (ICA) is proposed by Atashpaz et al. [12,13] and applied to structural optimum design by the Kaveh and Talatahari [14]. ICA is a socio-politically motivated optimization algorithm which is similar to many other evolutionary algorithms, and starts with a random initial population or empires. Each individual agent of an empire is called country and the

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E-mail address of the corresponding author: [email protected] (A. Kaveh)

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countries are categorized into two types; colony and imperialist state that collectively form empires. Imperialistic competitions among these empires form the basis of the ICA. During this competition, weak empires collapse and powerful ones take possession of their colonies. Imperialistic competitions converge to a state in which there exists only one empire and its colonies are in the same position and have the same cost as the imperialist [13]. This article utilizes ICA for the solution of some engineering problems. In order to fulfill this aim, four different ICA-based methods are developed. Some changes on the original ICA are performed to improve the exploration and exploitation abilities of the algorithm. There are some constraints in the engineering design problems which must be handled. Several techniques are available to handle the constraints for meta-heuristic algorithms. The feasible-based constrained approach is one of the powerful and reliable approaches used by many researchers [15]. In this paper, a modified feasible-based mechanism handles the constraints [16]. Simulation results and comparisons based on various well-known constrained engineering design problems demonstrate the efficiency of the present algorithm and its variants.

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2. ENGINEERING OPTIMIZATION PROBLEMS

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2.1 Statement of the optimization design problem Many engineering design problems can be formulated as constrained optimization problems:

find {x} to minimize

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subject to :

g j ({x})  0

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f cost ({x})

j  1,2,...., n g

hk ({x})  0

k  1,2,...., nh

xi,min  xi  xi ,max

i  1,2,...., d

where { x}  [ x1 , x 2 ,..., x d ]T denotes the decision vector;

(1)

f cost is the cost function

(objective function); xi,min and xi,max are the minimum and the maximum permissible values for the ith variable, respectively; ng is the number of inequality constraints and nh is the number of equality constraints. In common practice, an equality constraint hk ({x})  0 can be replaced by an inequality constraint | hk ({x}) |   0 , where  is a small tolerant amount. Thus all constraints can be transformed into inequality constraints [17]. 2.2 Constraint handling approach The aim of a constraint optimization is to search for feasible solutions with better objective values. {x} is a feasible solution if it satisfies all the constraints. In this paper, a modified feasible-based mechanism is utilized to handle the constraints which can be described as follows [16]:

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Level 1: Any feasible solution is preferred to any infeasible solution. Level 2: Infeasible solutions containing slight violence of the constraints (from 0.01 in the first iteration to 0.001 in the last iteration) are considered as feasible solutions. Level 3: Between two feasible solutions, the one having better objective function value is preferred. Level 4: Between two infeasible solutions, the one having smaller sum of constraint violation is preferred. This sum is calculated as ng



nh





Viol   max 0, g j ({x})   max 0, | h j ({x}) |  j 1



(2)

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j 1

By using feasible-based rule in the first and fourth levels, the search tends to the feasible region rather than infeasible region, and in the third level the search tends to the feasible region with good solutions. For most of the engineering optimization problems, the global minimum is located on or close to the boundary of a feasible design space. Applying the level 2, the particles will approach to the boundaries and can fly to the global minimum with a higher probability.

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3. IMPERIALIST COMPETITIVE ALGORITHM IMPLEMENTATION As an alternative to the conventional mathematical approaches, the meta-heuristic optimization techniques have widely been utilized and improved to obtain engineering optimum design solutions. Many of these methods are created by the simulation of the natural processes. As an example, GA is a particular evolutionary algorithm, which aims at simulating natural genetic variation and natural selection. The inspiration source of PSO, formulated by Edward and Kennedy in 1995 [4], is the social behavior of animals such as bird flocking or fish schooling. Ant Colony Ooptimization simulates the foraging behavior of real ants. Harmony Search imitates the musical performance process which takes place when a musician searches for a better state of harmony. As another example, Big Bang–Big Crunch algorithm relies on one of the theories of the evolution of the universe. Also, Charged System Search utilizes the governing laws of physics and mechanics, [9]. This paper presents a new optimization algorithm which contrary to the above mentioned methods, is not based on phenomena from the nature.

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3.1 General aspects [13] ICA simulates the social political process of imperialism and imperialistic competition. This algorithm is population based process in which each individual of the population is called a country. Some of the best countries (in optimization terminology, countries with lower cost) are selected to be the imperialist states and the remaining countries form the colonies of these imperialists. All the colonies of initial countries are divided among the imperialists based on their power. The power of each country, the counterpart of fitness value, is inversely proportional to its cost. The imperialist states together with their colonies form

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some empires. After forming initial empires, the colonies in each empire start moving toward their relevant imperialist country. This movement is a simple model of assimilation policy which was pursued by some of the imperialist states. The total power of an empire depends on both the power of the imperialist country and the power of its colonies. This fact is modeled by defining the total power of an empire as the power of the imperialist country plus a percentage of mean power of its colonies. Then the imperialistic competition begins among all the empires. Any empire that is not able to succeed in this competition and cannot increase its power (or at least prevent losing its power) will be eliminated from the competition. The imperialistic competition will gradually result in an increase in the power of the powerful empires and a decrease in the power of weaker ones. Weak empires will loose their power and ultimately they will collapse. The movement of colonies toward their relevant imperialist states along with competition among empires and also the collapse mechanism will hopefully cause all the countries to converge to a state in which there exist just one empire in the world and all the other countries are colonies of that empire. In this ideal new world, colonies will have the same position and power as the imperialist.

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3.2 Original ICA Each agent or country is formed of an array of variable values and the related cost of a country is found by evaluation of the cost function fcost of the corresponding variables using Eq. (1). Total number of initial countries is set to Ncountry and the number of the most powerful countries to form the empires is taken as Nimp. The remaining Ncol of the initial countries will be the colonies each of which belongs to an empire. In this paper, 10 percent of countries belong to empires and the remaining are used as colonies. To form the initial empires, the colonies are divided among imperialists based on their power. That is, the number of colonies of an empire should be directly proportionate to its power. In order to proportionally divide the colonies among imperialists, the normalized cost of an imperialist is defined as ( imp , n ) ( imp ,i ) Cn  f cos  max ( f cos ) (3) t t

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i

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( imp , n ) where f cos is the cost of the nth imperialist and Cn is its normalized cost. The initial t colonies are divided among empires based on their power or normalized cost, and for the nth empire it will be as follows

     Cn  (4) NCn  Round  N  N col  imp   Ci   i 1  where NCn is the initial number of colonies associated to the nth empire which are selected randomly among the colonies. These colonies along with the nth imperialist form the nth empire.

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In ICA, the assimilation policy, pursued by some of former imperialist states, is modeled by moving all the colonies toward the imperialist. This movement is shown in Figure 1(a) in which a colony moves toward the imperialist by a random value that is uniformly distributed between 0 and β×d:

{x}new  {x}old  U (0,   d )  {V1}

(5)

where β is a parameter greater than 1 and d is the distance between colony and imperialist. β>1 causes the colonies to get closer to the imperialist state from both sides. β >>1 gradually results in a divergence of colonies from the imperialist state, while a very close value to 1 for β reduces the search ability of the algorithm. {V1} is a vector which its start point is the previous location of the colony and its direction is toward the imperialist locations. The length of this vector is set to unity.

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(a)

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(b) Figure 1. Movement of colonies to its new location in the original ICA: (a) toward their relevant imperialist, (b) in a randomly deviated direction

To increase the searching around the imperialist, a random amount of deviation is added to the direction of movement. Figure 1(b) shows the new direction which is obtained by deviating the previous location of the country as great as θ. In this figure, θ is a random

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number with uniform distribution as

  U ( , )

(6)

where  is a parameter that adjusts the deviation from the original direction. In most of implementations, a value of about 2 for β [12] and about 0.1 (Rad) for  results in good convergence of countries to the global minimum. If the new position of a colony is better than that of the corresponding imperialist (considering the cost function), the imperialist and the colony change their positions and the new location with lower cost becomes the imperialist. Then the other colonies move toward this new position. Imperialistic competition is another strategy utilized in the ICA methodology. All empires try to take the possession of colonies of other empires and control them. The imperialistic competition gradually reduces the power of weaker empires and increases the power of more powerful ones. The imperialistic competition is modeled by just picking some (usually one) of the weakest colonies of the weakest empires and making a competition among all empires to possess these (this) colonies. Based on their total power, in this competition, each of empires will have a likelihood of taking possession of the mentioned colonies. Total power of an empire is mainly affected by the power of imperialist country. But the power of the colonies of an empire has an effect, albeit negligible, on the total power of that empire. This fact is modeled by defining the total cost as

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NCn

( col ,i )  f cos t

( imp ,n ) TCn  f cos   t

i 1

NCn

h c

(7)

where TCn is the total cost of the nth empire and  is a positive number which is considered to be less than 1. A small value for  causes the total power of the empire to be determined by just the imperialist and increasing it will add to the role of the colonies in determining the total power of the corresponding empire. The value of 0.1 for  is found to be a good value in most of the implementations [12]. Similar to Eq. (3), the normalized total cost is defined as

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NTCn  TCn  max (TCi )

(8)

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where NTCn is the normalized total cost of the nth empire. Having the normalized total cost, the possession probability of each empire is evaluated by

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Pn 

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NTCn

(9)

N imp

 NTCi

i 1

When an empire loses all its colonies, it is assumed to be collapsed. In this model implementation where the powerless empires collapse in the imperialistic competition, the corresponding colonies will be divided among the other empires. Moving colonies toward imperialists are continued and imperialistic competition and implementations are performed during the search process. When the number of iterations reaches a pre-defined value, the search process is stopped. The pseudo-code of the ICA algorithm is presented in Figure 2.

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Step 1: Initialization. Define the optimization problem; Select some random points as new position of colonies; Initialize the empires.

Step 2: Colonies Movement. Move the colonies toward their relevant imperialist.

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Step 3: Imperialist Updating. If the new colony has lower cost than that of imperialist, exchange the positions of that colony and the imperialist.

Step 4: Imperialistic Competition. Pick the weakest colony from the weakest empire and give it to

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the empire that has the most likelihood to possess it.

Step 5: Implementation. Eliminate the powerless empires.

Step 6: Terminating Criterion Control. Repeat Steps 2-5 until a terminating criterion is satisfied.

Figure 2. The pseudo-code of the ICA algorithm.

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3.3 Different Variants of ICA-based methods Recently, the authors presented a variant of the ICA for the structural optimum design [14]. Here, we present three other variants of the ICA method by simplifying the previous method of [14]. These variants are designed considering different movement approaches for countries as follow:

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3.3.1 ICA-1 method According to Figure 3(a), when a colony is deviated by θ from its previous direction of movement, it can be located on any point in the circle shown in the figure. In the first version of the ICA, we define a random vector, {V2 } , to determine the location of the colony. This vector is designed in a manner that it is perpendicular to the line crossing the previous location of colony and the imperialistic (i.e. {V1} ) and its length is equal to unity. Therefore, Eqs. (5) and (6) are modified and replaced by the following equation

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{x}new  {x}old  U (0,   d )  {V1}  tan( )  d  {V2 }, {V2 }  {rand } | {V1}  {V2 }  0, || {V2 } || 1

(10)

3.3.2 ICA-2 method As the second version, we utilize the different random values for the different components of the solution vector. Since these random values are not necessarily the same, the colony is deviated automatically without requiring the definition of θ, as shown in Figure 3(b). Thus we will have

{x}new  {x}old    d  {rand }  {V1}

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(11)

where {rand} is a random vector and the sign "  " denotes an element-by-element multiplication. It should be noted that in Eq. (5), U (0,   d ) creates one random number.

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3.3.3 ICA-3 method The third ICA-based algorithm utilizes the combination of the properties of the previous two versions. For ICA-3, not only different random values are used similar to ICA-2, but also the definition of θ is taken as that of ICA-1 in order to increase the exploration of the algorithm as follows: {x}new  {x}old    d  {rand }  {V1}  tan( )  d  {V2 }, (12) {V2 }  {rand } | {V1}  {V2 }  0, || {V2 } || 1

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Figure 3(c) describes the performance of this version. As a result, this algorithm will have the most exploration ability among the others. 3.3.4 ICA-4 method [14] As the last method, Eq. (12) is modified as shown in Figure 3(d). Here, considering the two first term of this equation, a point out of the colony-imperialistic contacting line can be obtained as indicated in the figure. Then, it is possible to obtain the orthogonal colonyimperialistic contacting line instead of the random vector defined by Eq. (12). Thus the final location of colony will be as

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{x}new  {x}old    d  {rand }  {V1}  U (1,1)  tan( )  d  {V2 }, {V1}  {V2 }  0, || {V2 } || 1

(13)

Comparing to ICA-3, in this method {V2 } is not a random vector, since this vector must be crossed the point obtained from the two first terms and also be perpendicular to {V1} . Therefore, we use a random value by using U (1,1) for the third term of the Eq. (13) which changes its value in addition to its direction by using the negative values.

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(a)

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(b)

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(c)

(d) Figure 3. Performance of variants ICA-based algorithms: (a) ICA-1, (b) ICA-2, (c) ICA-3, (d) ICA-4

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4. SIMULATION AND ANALYSIS Some well-studied numerical examples are taken from the optimization literature to examine the efficiency of the proposed approaches. The examples contain some uni-modal and multimodal mathematical functions presented in section 4.1 and some engineering design problems investigated in section 4.2. These examples have been previously solved using a variety of other techniques, which is useful to show the validity and effectiveness of the proposed algorithms. For each example, 30 independent runs are carried out using each ICA-based algorithm and compared to other methods.

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4.1 Mathematical examples In this section a number of benchmark functions are optimized using ICA-based algorithms and the results are compared to each other to recognize the best version of the ICA method among the others. The description of these test problems is provided in Table 1. The number of variables (d) is set to 10. Figure 4 shows the final results of the ICA-based algorithms in the 30 independent runs. For all functions, ICA-2 has the worst performance. This shows that utilizing different random values without considering deviation angle is not sufficient and it is necessary to add some other factors in order to increase the exploration ability. For the first and fifth functions, the ICA-1 has better performance than others while for the reminding ICA-4 is the best algorithm. For fourth function both ICA-1 and IAC-4 in all 30 independent runs can find the global minimum. Table 2 presents the statistics data for these algorithms. Small values indicate the stability of the algorithm. Considering the results of Table 2 and Figure 4, one can be conclude that ICA-1 and ICA-4 are better algorithms which utilize both different random numbers and deviation angle.

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Table 1: Specifications of the benchmark problems Function name

F2 F3

Function

{x}  [100,100]d

f ({x})   xi2

{x}  [1.28,1.28]d

f ({x})   ixi4  rand [0,1)

0.0

{x}  [1,1]d

d   f ({x})   exp  0.5 xi2  i 1  

–1.0

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F1

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F4

{x}  [ 600,600]d

F5

{x}  [30,30]d

F6

Global minimum

Interval

{x}  [5.12,5.12]d

d

0.0

i 1

d

i 1

x 1 d 2 d  xi   cos( i ) 4000 i 1 i 1 i

0.0

100( xi 1  xi2 ) 2  ( xi  1) 2

0.0

f ({x})  1 

f ({x}) 

d 1 i 1

d



f ({x})   xi2  10 cos(2xi )  10 i 1



0.0

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Table 2: Statistics data for the ICA-based algorithm Function name F1

F2

F3

F4

ICA-1

ICA-2

ICA-3

ICA-4

minimm

4.05e–13

0.1169

2.11e–7

1.06e–12

mean

5.89e–11

0.2903

2.10e–5

4.57e–11

median

8.76e–12

0.2710

1.40e–5

2.07e–11

Std

1.11e–10

0.1286

1.68e–5

minimum

0.0052

0.0049

0.0044

mean

0.0250

0.0507

0.0222

median

0.0238

0.0471

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Std

0.0106

0.0224

minimm

–1.00

–0.9984

mean

–1.00

–0.9953

median

–1.00

Std

0.00

minimum

1.08e–12

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0.0179 0.0173

0.0068

–1.00

–1.00

–1.00

–1.00

–0.9954

–1.00

–1.00

0.0017

2.50e–7

0.00

0.1414

2.84e–6

6.44e–13

1.98e–10

0.5756

6.57e–5

5.28e–11

4.65e–11

0.5327

4.43e–5

1.86e–11

3.54e–10

0.2562

5.65e–5

7.07e–11

0.0431

1228

0.1597

0.4145

mean

4.468

5293

27.6

6.110

median

3.627

5210

19.05

7.555

Std

4.676

2766

23.34

4.753

minimum

9.025

16.48

6.544

4.975

mean

19.82

35.11

14.10

9.725

median

20.26

25.44

14.49

9.997

Std

5.751

5.527

3.583

2.832

median

minimum

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r A Std

F6

0.0064

0.0093

mean

F5

S f 0.0223

4.49e–11

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(a) F1

(b) F2 -1

0

10

10

ICA-1 ICA-2 ICA-3 ICA-4

-5

ICA-1 ICA-2 ICA-3 ICA-4

Final result

Final result

10

-10

-2

10

10

-15

10

-3

1

5

10

15 # of run

20

25

10

30

1

5

10

(c) F3 5

10

ICA-2 ICA-3

-0.993 -0.994

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0

10 Final result

Final result

-0.995 -0.996 -0.997

-5

10

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-0.998

-10

10

-0.999 -1

-15

5

10

15 # of run

20

25

10

30

(e) F5

iv

6

10

4

Final result

10

h c

2

10

0

10

-2

1

5

r A 10

15 # of run

25

30

20

25

30

ICA-1 ICA-2 ICA-3 ICA-4

1

5

10

15 # of run

20

25

ICA-1 ICA-2 ICA-3 ICA-4

30

(f) F6

40

ICA-1 ICA-2 ICA-3 ICA-4

35 30

Final result

1

20

(d) F4

-0.992

10

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15 # of run

25 20 15 10 5 0

1

5

10

15 # of run

20

25

30

Figure 4. Comparing the final results of mathematical results obtained by the ICA-based algorithms 4.2 Engineering design problems Here, a set of four engineering design optimization problems was chosen to evaluate the performance of our proposed algorithm. The number of objective function evaluations per run is set to 20,000. This value for a GA-based algorithm of Coello [18], and for a PSO-based method of He and Wang [17] is equal to 900,000 and 200,000 respectively. For the evolution strategies presented by Montes and Coello [19], it is 25,000. Thus, it can be pointed out that ICA-based algorithm has a better convergence speed compared to other meta-heuristics. The detailed descriptions of the test problems are presented in the following subsections.

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4.2.1 A tension/compression spring design problem This problem consists of minimizing the weight of a tension/compression spring subject to constraints on shear stress, surge frequency and minimum deflection as shown in Figure 5. The design variables are the mean coil diameter D (=x1); the wire diameter d (=x2) and the number of active coils N (=x3). The detailed information of the problem is stated in Table 3. Table 3: Specifications of the tension/compression spring problem Cost function

f cost ({x})  ( x3  2) x2 x12

Constraint functions

g1 ({x})  1 

g 2 ({x}) 

x23 x3 71785 x14

4 x22  x1 x2 12566( x2 x13  x14 )

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

g 4 ({x}) 

iv

Variable regions

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h c

1

5108 x12

140.45 x1

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g 3 ({x})  1 

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0

x22 x3

1  0

0

x1  x2 1  0 1.5

0.05  x1  2

0.25  x2  1.3 2  x3  15

This problem has been solved by Belegundu [20] using eight different mathematical optimization techniques (only the best results are shown). Arora [21] also solved this problem using a numerical optimization technique called a constraint correction at the constant cost. Coello [18] as well as Coello and Montes [22] solved this problem using GAbased method. Additionally, He and Wang [17] utilized a co-evolutionary particle swarm optimization (CPSO). Recently, Montes and Coello [19] used various evolution strategies to solve this problem. Table 4 presents the best solution and statistical information of this problem obtained using the ICA-based algorithms and compares the results with solutions reported by other researchers. The best feasible solutions obtained by ICA-1 and ICA-4 are better than those previously reported. In addition, as shown in the table, the average searching quality of ICA4 is superior to those of other methods. Moreover, the standard deviation of the results by ICA-1 and ICA-4 in 30 independent runs for this problem among the others is the smallest.

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Figure 5. Tension/compression spring

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Table 4: Optimum results for the tension/compression string design Methods

x1 (d)

x2 (D)

x3 (N)

Best result

Belegundu [20]

0.050000

0.315900

14.250000

0.0128334

Arora [21]

0.053396

0.399180

9.185400

Worst result

Std. Dev.

N/A

N/A

N/A

N/A

N/A

Coello [18]

0.051480

0.351661

11.632201

0.012769

0.012822

3.9390e–5

Coello & Montes [22]

0.051989

0.363965

10.890522

0.0126810

0.0127420

0.012973

5.9000e–5

He & Wang [17]

0.051728

0.357644

11.244543

0.0126747

0.012730

0.012924

5.1985e–5

Montes & Coello [19]

0.051643

0.355360

11.397926

0.012698

0.013461

0.16485

9.6600e–4

ICA-1

0.0516910

0.3572141

11.243772

0.0126406

0.0126856

0.0127655

3.6806e–5

ICA-2

0.0512504

0.3466918

11.887441

0.0126462

0.0128562

0.0131925

1.9118e–4

ICA-3

0.052137

0.368067

10.643529

0.0126503

0.0127568

0.0130979

1.191e–4

ICA-4

0.051774

0.359229

11.127122

0.0126407

0.0126702

0.0126925

2.1815e–5

iv

h c

r A

S f

o e 0.0127303

0.0127048

Mean of results

N/A

4.2.2 A welded beam design problem The welded beam structure, shown in Figure 6, is a practical design problem that has been often used as a benchmark problem for testing different optimization methods. The objective is to find the minimum fabricating cost of the welded beam subject to constraints on shear stress ً( ) , bending stress ( ) , buckling load ( Pc ) , end deflection ( ) , and side constraint. There are four design variables, namely h( x1 ) , l (  x2 ) , t ( x3 ) and b( x4 ) . The mathematical formulation of the problem is presented in Table 5.

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Figure 6. Welded beam structure

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Table 5: Specifications of the welded beam problem Cost function

689

f cost ({x})  1.10471x12 x2  0.04811x3 x4 (14.0  x2 )

S f

Constraint functions

g1 ({x})   ({x})   max  0

g 2 ({x})   ({x})   max  0 g 3 ({x})  x1  x4  0

o e

g 4 ({x})  0.10471x12  0.04811x3 x4 (14.0  x2 )  5.0  0 g 5 ({x})  0.125  x1  0 g 6 ({x})   ({x})   max  0

v i h

g 7 ({x})  P  Pc ({x})  0 x2  ( " ) 2 2R P MR , '  J 2 x1x2

 ({x})  ( ' ) 2  2 ' "

c r

A

 '

M  P( L 

x2 ),R  2

x22  x1  x3    4  2 

2

  x 2  x  x  2   J  2 2 x1 x2  2   1 3     12  2    

 ({x}) 

6 PL x4 x32

,  ({x}) 

4 PL3 Ex33 x4

x32 x46 x 36  Pc ({x})  1 3 2  L 2 L  P  6000lb , L  14in , 4.013E

E 4G

   

E  30 106 psi , G  12106 psi

Variable regions

0.1  x1, 4  2 0.1  x2,3  10

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Deb [23], Coello [18] and Coello and Montes [22] solved this problem using GA-based methods. Radgsdell and Phillips [24] compared optimal results of different optimization methods that were mainly based on mathematical optimization algorithms containing APPROX (Griffith and Stewart’s successive linear approximation), DAVID (Davidon– Fletcher–Powell with a penalty function), SIMPLEX (Simplex method with a penalty function), and RANDOM (Richardson’s random method) algorithms. Also, He and Wang [17] using CPSO, and Montes and Coello [19] using evolution strategies solved this problem. The comparison of results is shown in Table 6. The ICA-4 result, which was obtained after 20,000 searches, was better than those reported by He and Wang [17] who got the best results between others. The reported number of function evolution for by He and Wang is 200,000. The statistical simulation results are also shown in Table 6. From Table 6, it can be seen that the worst solutions found by ICAbased methods are better than the best solution found by Ragsdell and Phillips [24] and the best solution found by Deb [23]. In addition, the standard deviation of the results by all ICAbased methods in 30 independent runs is very small.

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Table 6: Optimum results for the welded beam design Methods

x1 (h)

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x2 (l)

x3 (t)

x4 (b)

Best result

Mean of results

Worst result

Std. Dev.

0.2444

2.3815

N/A

N/A

N/A

Regsdell & Phillips [24] APPROX

0.2444

6.2189

8.2915

DAVID

0.2434

6.2552

8.2915

0.2444

2.3841

N/A

N/A

N/A

iv

SIMPLEX

0.2792

5.6256

7.7512

0.2796

2.5307

N/A

N/A

N/A

RANDOM

0.4575

4.7313

5.0853

0.6600

4.1185

N/A

N/A

N/A

Deb [23]

0.248900

6.173000

8.178900

0.253300

2.433116

N/A

N/A

N/A

Coello [18]

0.208800

3.420500

8.997500

0.210000

1.748309

1.771973

1.785835

0.011220

Coello & Montes [22]

0.205986

3.471328

9.020224

0.206480

1.728226

1.792654

1.993408

0.074713

He & Wang [17]

0.202369

3.544214

9.048210

0.205723

1.728024

1.748831

1.782143

0.012926

Montes & Coello [19]

0.199742

3.612060

9.037500

0.206082

1.737300

1.813290

1.994651

0.070500

ICA-1

0.205241

3.494641

9.038958

0.205747

1.727903

1.764821

1.815100

0.029470

ICA-2

0.209046

3.322341

9.251953

0.204679

1.760374

1.856047

1.973904

0.061941

ICA-3

0.207822

3.437025

9.007861

0.207055

1.732465

1.816112

1.900021

0.045870

ICA-4

0.205703

3.47106

9.036654

0.205731

1.724906

1.742214

1.793435

0.017831

h c

r A

4.2.3 A pressure vessel design problem A cylindrical vessel is capped at both ends by hemispherical heads as shown in Figure 7. The

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objective is to minimize the total cost, including the cost of material, forming and welding. Table 7 presents the details of the algorithm in which x1 is the thickness of the shell (Ts), x2 is the thickness of the head (Th), x3 is the inner radius (R) and x4 is the length of cylindrical section of the vessel, not including the head (L). Ts and Th are integer multiples of 0.0625 inch, the available thickness of rolled steel plates, and R and L are continuous.

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Figure 7. Schematic of pressure vessel

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Table 7: Specifications of the pressure vessel problem Cost function

f cost ({x})  0.6224 x1 x3 x4  1.7781x2 x32  3.1661x12 x4  19.84 x12 x3

v i h

g1 ({x})   x1  0.0193x3  0

Constraint functions

g 2 ({x})   x2  0.00954 x3  0

Variable regions

A

c r

4 g 3 ({x})  x32 x4  x33  1,296,000  0 3 g 4 ({x})  x 4  240  0 0  x1, 2  99

10  x3, 4  200

The approaches applied to this problem include genetic adaptive search (Deb and Gene [25]), a GA-based co-evolution model (Coello [18]), an augmented Lagrangian multiplier approach (Kannan and Kramer [26]), a branch and bound technique (Sandgren [27]), a feasibility-based tournament selection scheme (Coello and Montes [22]), a co-evolutionary particle swarm optimization (He and Wang [17]), an evolution strategy (Montes and Coello [19]). The best solutions and their statistical simulation results obtained by the above mentioned approaches and by the ICA methods are listed in Table 8.

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Table 8: Optimum results for the pressure vessel Methods

Sandgren [27] Kannan & Kramer [26] Deb & Gene [25] Coello [18] Coello & Montes [22] He & Wang [17] Montes & Coello [19] ICA-1 ICA-2 ICA-3 ICA-4

x1 (Ts)

x2 (Th)

x3 (R)

x1 (L)

Best result

Mean of results

Worst result

Std. Dev.

1.125000

0.625000

47.70000

117.7010

8129.103

N/A

N/A

N/A

1.125000

0.625000

58.29100

43.6900

7198.042

N/A

N/A

N/A

0.937500

0.500000

48.32900

112.6790

6410.381

N/A

N/A

N/A

0.812500

0.437500

40.32390

200.0000

6288.744

6293.843

6308.149

7.4133

0.812500

0.437500

42.09739

176.6540

6059.946

6177.253

6469.322

130.9297

0.812500

0.437500

42.09126

176.7465

6061.077

6147.133

6363.804

86.4545

0.812500

0.437500

42.09808

176.6405

6059.745

6850.004

7332.879

426.0000

0.812500 0.812500 0.812500 0.812500

0.437500 0.437500 0.437500 0.437500

42.0759 41.8061 42.0939 42.0983

176.9162 180.2938 176.8357 176.6379

6062.468 6095.652 6063.617 6059.728

6179.125 6261.452 6197.265 6100.212

6280.668 6376.825 6379.918 6208.985

67.8235 73.8965 90.6012 40.4317

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From Table 8, it can be seen that the best solution found by ICA-4 is better than the best solutions found by other techniques. Also, the average searching quality of ICA-based algorithm is better than those of other methods, and even the worst solutions found by these methods are better than the best solutions found by Deb and Gene [25], Kannan and Kramer [26] and Sandgren [27].

c r

A

Figure 8. Speed reducer

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4.2.4 Speed reducer design problem The design of the speed reducer [28] shown in Figure 8, is considered with the face width x1, module of teeth x2, number of teeth on pinion x3, length of the first shaft between bearings x4, length of the second shaft between bearings x5, diameter of the first shaft x6, and diameter of the first shaft x7 (all variables continuous except x3 that is integer). The weight of the speed reducer is to be minimized subject to constraints on bending stress of the gear teeth, surface stress, transverse deflections of the shafts and stresses in the shaft. The problem is defined in Table 9. Table 9: Specifications of the speed reducer problem

D I

f cost ({x})  0.7854 x1 x22 (3.3333x32  14.9334 x3  43.0934)

Cost function

 1.508 x1 ( x62  x72 )  7.4777( x63  x73 )  0.7854( x4 x62  x5 x72 ) g1 ({x}) 

Constraint functions

g 2 ({x}) 

g 4 ({ x }) 

v i h

c r

A

g 6 ({x}) 

1.0

110 x63 1.0 85 x73

x1 x22 x32 1.93x43

x2 x3 x64

1  0

1  0

1 . 93 x 53

x 2 x 3 x 74

1  0

2

 745 x4     16.9  10 6  1  0  x x   2 3  2

 745 x5     157.5  10 6  1  0  x x   2 3 

g 7 ({x})  g 8 ({x})  g 9 ({x}) 

g10 ({x})  g11 ({x}) 

Variable regions

1  0

S f

397.5

o e g 3 ({x}) 

g 5 ({x}) 

27

x1 x22 x3

x 2 x3 1  0 40 5 x2 x1 x1 12 x2

1  0 1  0

1.5 x6  1.9 x4 1.1x7  1.9 x5

1  0 1  0

2.6  x1  3.6;0.7  x2  0.8 17  x3  28;7.3  x4  8.3 7.8  x5  8.3;2.9  x6  3.9;5.0  x7  5.5

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This example is solved by two constrained particle swarm optimizer algorithms previously (COPSO [29], SiC-PSO [30]). A total of 30,000 and 24,000 objective function evaluations per run was considered for this COPSO and SiC-PSO algorithms, respectively. However, when 20,000 function evaluations are used for ICA-based algorithm similar to the previous examples, in all runs the optimum results are obtained. So, we set 5,000 function evaluations as the maximum number for the ICA-based algorithm. As shown in Table 10, even with this small number of function evaluations, ICA-4 could find the best result almost in the all runs. Table 10 summarizes the statistical information of the PSO- and the ICAbased algorithms. In addition, Figure 9 shows the all final results in 30 runs obtained by ICA-based methods for these engineering problems to simplify the comparison.

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Table 10: Optimum results for the speed reducer design Methods

Best result

Mean of results

Std. Dev.

Hernandez et al. [29]

2,996.372448

2,996.4085

0.0286

Cagnina et al. [30]

2,996.348165

2,996.3482

ICA-1

2997.22

3,003.1012

ICA-2

3,009

ICA-3

3,004

ICA-4

2,996.348165

Final result

0.013

h c

0.0129 0.0128 0.0127

1

5

r A 10

15 # of run

20

25

Final result

6300 6250

2,996.1022

0.0000 ICA-1 ICA-2 ICA-3 ICA-4

1.75 1.7

1

5

10

15 # of run

20

25

30

(b) Welded beam structure 3070

ICA-1 ICA-2 ICA-3 ICa-4

3060 3050 3040 3030 3020 3010

6100 6050

7.1682

1.8

ICA-1 ICA-2 ICA-3 ICA-4

6150

3,016.2011

1.85

30

6200

15.351

1.9

Final result

6350

3,041.2100

1.95

(a) Tension/compression spring 6400

4.7300

2

Final result

ICA-1 ICA-2 ICA-3 ICA-4

0.0131

0.0126

o e

iv

0.0132

S f

0.0000

3000 1

5

10

15 # of run

20

25

(c) Pressure vessel

30

2990

1

5

10

15 # of run

20

25

30

(d) Speed reducer

Figure 9. Comparing the final results of the engineering examples obtained by the ICA-based algorithms

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5. CONCLUDING REMARKS Imperialist competitive algorithm (ICA) is a novel meta-heuristic search method that uses imperialism and imperialistic competition process as a source of inspiration. This algorithm starts with some random initial countries. Some of the best countries are selected to be the imperialist states and all the other countries form the colonies of these imperialists. The colonies are divided among the imperialists based on their power. Then, the colonies start moving toward their relevant imperialist state. This movement is a simple model of assimilation policy that is pursued by some imperialist states. In addition, the imperialistic competition strategy gradually reduces the power of weaker empires and increases the power of more powerful ones. This is modeled by picking some of the weakest colonies of the weakest empires and making a competition among all empires to possess this colony. This process is continued and when an empire loses all its colonies, it will be collapsed. Here, four different rules are presented to describe the movement process of colonies. According to the first rule, after moving the colony toward its imperialist, a random vector orthogonal to the colony-imperialistic contacting line is utilized to deviate the movement direction of the colony. This deviation was forecasted to increase the search ability of the algorithm. For the second variant, the different random values are utilized for the components of the solution vector without requiring the use of θ. The results of the ICA-2 approach for the mathematical and engineering numerical examples indicate inferiority of this version compared to the others. This proves the fact that to have a better algorithm, the use of a deviation angle in the movement process is necessary. Combining variants 1 and 2 forms the variant ICA-3. Due to using random values in the direct movement step (similar to that defined in ICA-2) and in the deviation step (similar to that described in ICA-1), the exploration ability is quite high for this variant. However, the balance between exploration and exploitation of the method is not regarded sufficiently suitable and as a result it is the second weakest algorithm. In the last variant as described in Ref. [14], the direct movement step is considered as defined for ICA-2, however for the deviation step, the orthogonal colony-imperialistic contacting line is utilized. This variant has directed the search process toward the global optimum in all the numerical examples studied in the article. Having a good search ability in addition to a powerful exploitation makes this variant superior compared to the other ICA-based algorithms. The comparison of the results with other meta-heuristic methods for engineering examples shows that ICA-1 and ICA-4 methods have a good convergence rate and can reach to better solutions. Comparing ICA-1 and ICA-4 methods, it can be observed that ICA-4 [14] has a better performance while the complexity of the ICA-1 is less and its results are not considerably worse than ICA-4. To sum up, these two presented variants of the new algorithm are powerful optimization methods which can easily be utilized for engineering problems to find the optimum designs.

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Acknowledgement: The first author as the fellow of the Iranian Academy of Sciences is grateful for the support.

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