Document Type : Research Paper
Authors
^{1} Departement of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
^{2} Faculty of Mechanical Engineering, Semnan University, Semnan, 1911135131, Iran
Abstract
Keywords
Optimization of Hybrid Composite Laminate Based on the Frequency using Imperialist Competitive Algorithm
H. Hemmatian^{a}^{*}, A. Fereidoon^{b}, H. Shirdel^{b}
^{a}Departement of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
^{b}Faculty of Mechanical Engineering, Semnan University, Semnan, 1911135131, Iran
paper INFO 

ABSTRACT 
Paper history: Received 11 July 2014 Received in revised form 21 August 2014 Accepted 11 September 2014 
Imperialist competitive algorithm (ICA) is a new sociopolitically motivated global search strategy. The ICA is applied to hybrid composite laminates to obtain minimum weight and cost. The approach which is chosen for conducting the multiobjective optimization was the weighted sum method (WSM). The hybrid composite Laminates are made of glass/epoxy and carbon/epoxy to combine the lightness and economical attributes of the first with highstiffness property of the second in order to make tradeoff between the cost and weight as the objective functions and natural flexural frequency as a constraint. The results were evaluated for different weighting factors (a) including optimum stacking sequences, and number of plies made of either glass or carbon fibers using the ICA, and were compared with those using the genetic algorithm (GA) and ant colony system (ACS). The comparisons confirmed the advantages of hybridization and revealed that the ICA outperformed the GA and ACS in terms of function’s value and constraint accuracy. 



Keywords: Composite laminate Hybridization Stacking sequence Imperialist competitive algorithm Frequency 


© 2014 Published by Semnan University Press. All rights reserved. 
Laminated composite materials find a wide range of applications in structural design, particularly in the field of automotive, aerospace and marine engineering. This is primarily due to the high specific strength and stiffness values with minimum weight that these type of materials offer.
The design of a structural component using composites involves both material and structural design. Laminated composites are usually designed due to the designer’s needs by choosing the thickness, orientation and number of lamina. The thickness and orientation of the lamina are usually limited to some set values due to manufacturing limitations. Searching for the optimum solution in laminated composite structures is a discrete optimizing problem [1, 2].
In all applications, it is ideal to have the stiffest, lightest, and the most economical structures [3]. These three requirements normally act against each other and may come in compromise with the help of hybridization of composite laminates in which the highstiffness material, generally more expensive and heavier, is used in the outer layers to provide enough rigidity and stiffness [4, 5]. The material used in the inner layers should bear lesser cost, be lighter, and have lowstiffness. Deflection, stress, and natural frequencies are some supplementary aspects which have been investigated in hybrid laminates in a multiobjective optimization process [68]. Javidrad and Nouri [9] applied a modified simulating annealing to minimize a cost function composed of the difference between the effective stiffness properties and weight of the considered laminate from the number of layers and the fiber angle of each layer.
Dynamic properties of composites are analysed for investigating structural issues [1012] and optimal designs of composites with frequency objective or constraint which are carried out extensively [13, 14].
Singleobjective maximization of the fundamental frequency for laminated plates was given by using continuous design variables [1517]. The same design for crossply laminates was studied by Duffy and Adali [18] and for anisotropic laminates by Adali [19]. Apalak et al. [20] carried out the layer optimization for maximum fundamental natural frequency of composite laminates by a genetic algorithm.
Fukunaga et al. optimized the composite structures under natural frequency constraints, where only the thickness of each lamina is taken as the design variable [21]. Also, the optimization algorithm of laminate frequency issue has been investigated by Narita [22]; Narita and Hodgkinson [23]. Minimum cost design of laminated plates undergoing free vibrations was conducted by Adali and Duffy [24]. Farshi et al. presented a method based on the Ritz approximations for minimum thickness. multilayer rectangular composite laminates is presented which is based on a layerwise optimizing procedure under natural frequency limitations [25].
Adali and Verijenco [26] optimized stacking sequence design of symmetric hybrid laminates undergoing free vibrations for fundamental frequency and frequency separation.
Regarding multiobjective optimization, Tahani et al. optimized the fundamental frequency and cost using the genetic algorithm (GA) [27], and Kolahan et al. also solved the same problem with the help of simulated annealing (SA) [28]. Reliability based optimization of composite laminates for frequency constraint was investigated by Hao et al. [29].
Hemmatian et al. [5] and Grosset et al. [30] optimized the number and angle of glass/epoxy and carbon/epoxy layers in order to get minimal cost and weight subject to the first natural frequency by using elitist ant system and GA, respectively. Abachizadeh and Tahani applied ACS on hybrid laminate composite for minimum cost and weight and natural frequency as a constraint [31].
Recently a new metaheuristic algorithm, so called imperialist competitive algorithm (ICA) is proposed by Atashpaz et al. [32, 33]. ICA is a sociopolitically 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 countries are categorized into two types; colony and imperialist state that collectively form empires.
ICA is applied to structural optimum design by the Kaveh and Talatahari [34, 35]. Abdi et al. used ICA to find the optimal design of laminated composite structures due to the various failure criteria [36]. Also ICA has been developed for optimum design of composite plates based on weight and cost by Mozafari et al. [37]. Composite plates under thermal buckling loads are optimized using ICA [38]. Thermal buckling loads of laminated composite plates are maximized for a given total thickness. Esmaeilzadeh modified the empire movement toward the superior empire for balancing the exploration and exploitation abilities of the ICA [39].
In this study, weight and cost of symmetric balanced hybrid laminates were optimized considering the first natural flexural frequency as the design constraint. Results were compared with those obtained using GA [30] and ACS [31].
Consider a simply supported symmetric hybrid laminated plate with the length a, width b, and total thickness h in the x, y, and z directions, respectively, as is shown in Fig. 1. Each layer has the thickness t and is idealized as a homogeneous orthotropic material. The total thickness of the laminate is equal to h=N×t which N is the total number of layers.
Figure 1. Geometry of composite laminate. 
The hybrid laminate is made of N_{i} inner and N_{o} outer layers so that N=N_{i}+N_{o}. The governing equation of motion within the classical laminated plate theory for the described symmetric laminate is given as follows [40]:
(1)
where w is the deflection in the z direction, h is the total thickness, and r is the mass density averaged in the thickness direction which is given by:
(2)
where r^{(k)}denotes the mass density of material in the k^{th} layer. The bending stiffnesses in Eq. (1) are defined as:
(3)
where is the transformed reduced stiffness of the k^{th} layer [5]. A general form of solution for w in the natural vibration mode (m, n) is presented as:
(4)
where is the natural flexural frequency of the vibration mode (m, n) and .Substituting Eq. (4) into Eq. (1) yields:
(5)
Different mode shapes are obtained by inserting different values of m and n where for the fundamental flexural frequency, both are put equal to one. Finally, the fundamental frequency (first frequency) is given as [5]:
(6)
Imperialist competitive algorithm (ICA) is a new sociopolitically motivated global search strategy that has recently been introduced for dealing with different optimizing tasks [32]. This evolutionary optimizing strategy has shown great performance in both convergence rate and better global optima [4144].
Like other evolutionary ones, the proposed algorithm starts with an initial population (countries in the world). Some of the best countries in the population are selected to be the imperialists and the rest form the colonies of these imperialists. All the colonies of initial population are divided among the mentioned imperialists based on their power. The power of an empire which is the counterpart of the fitness value in GA, is inversely proportional to its cost. After dividing all colonies among imperialists, these colonies start moving toward their relevant imperialist country. The total power of an empire depends on both the power of the imperialist country and the 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 can’t/can not increase its power (or at least prevent decreasing its power) will be eliminated from the competition. Weak empires will lose their power and ultimately they will collapse. The movement of colonies toward their relevant imperialists along with competition among empires and also the collapse mechanism will hopefully cause all the countries to converge to a state in which there exists just one empire in the world and all the other countries are colonies of that empire. In this ideal new world, colonies have the same position and power as the imperialist [32].
3.1. Creation of Initial Empires
An array of variable values to be optimized is generated. In the GA, this array is called “chromosome”, but in ICA the term “country” is used for this array. In an N_{var}dimentional optimisation problem, a country is a 1×N_{var} array. This array is defined as following
(7)
where p_{i} are the variables to be optimized. The variable values in the country are represented as floating point numbers. The cost of a country is found by evaluation of the cost function at variables:
(8)
To start the optimisation algorithm, initial countries of size N_{Country}is produced. We select N_{imp} of the most powerful countries to form the empires. The remaining N_{col} of the initial countries will be the colonies each of which belongs to an empire. To form the initial empires, the colonies are divided among imperialists based on their power. That is, the initial number of colonies of an empire should be directly proportionate to its power. To proportionally divide the colonies among imperialists, the normalized cost of an imperialist is defined by
(9)
where c_{n} is the cost of the n^{th} imperialist and C_{n} is its normalized cost. Having the normalized cost of all imperialists, the normalized power of each imperialist is defined by
(10)
The initial colonies are divided among empires based on their power. Then the initial number of colonies of the n^{th} empire will be
(11)
where N.C.n is the initial number of colonies of the nth empire and Ncol is the total number of initial colonies. To divide the colonies, N.C.n of the colonies are randomly chosen and given to the nth imperialist. These colonies along with the nth imperialist form the nth empire. The bigger empires have greater number of colonies while weaker ones have lesser.
3.2. Moving the Colonies of an Empire toward the Imperialist
Movement of colonies toward their relevant imperialist is shown in Fig. 2. The colony moves toward the imperialist by movement vector where x and θ are random variables with uniformed (or any proper) distribution:
(12)
(13)
where β is a number greater than 1 and d is the distance between the colony and the imperialist state. γ is a parameter that adjusts the deviation from the original direction.
Figure 2. Movement of colonies toward their relevant imperialist in a randomly deviated direction [32] 
3.3. Revolution; A Sudden Change in SocioPolitical Characteristics of a Country
Revolution is a fundamental change in power or organizational structures that takes place in a relatively short period of time that the colony randomly changes its position in the sociopolitical axis. The revolution rate in the algorithm indicates the percentage of colonies in each colony which will randomly change their position.
3.4. Exchanging Positions of the Imperialist and a Colony
While moving toward the imperialist, a colony might reach to a position with lower cost than the imperialist. In this case, the imperialist and the colony change their positions. Then the algorithm will continue by the imperialist in the new position and the colonies will be assimilated by the imperialist in its new position.
3.5. Total Power of an Empire
Total power of an empire is mainly affected by the power of imperialist country. However the power of the colonies of an empire has an effect, albeit negligible, on the total power of that empire. This fact is modelled by defining the total cost of an empire:
(14)
Where T.C.n is the total cost of the nth empire and ξ is a positive small number.
3.6. Total Imperialistic Competition
All empires try to take the possession of colonies of other empires and control them. The imperialistic competition gradually brings about a decrease in the power of weaker empires and an increase in the power of more powerful ones. To start the competition, first a colony of the weakest empires is chosen and then the possession probability of each empire is found. The possession probability P_{P} is proportionate to the total power of the empire. The normalized total cost of an empire is simply obtained by
(15)
Where, T.C._{n} and N.T.C_{n} are the total cost and the normalized total cost of empire, respectively. Having the normalized total cost, the possession probability of each empire is given by
(16)
To divide the mentioned colonies among empires vector P is formed as following:
(17)
Then the vector R with the same size as P whose elements are uniformly distributed random numbers is created,
(18)
Then vector D is formed by subtracting R from P
(19)
Referring to vector D the mentioned colony (colonies) is handed to an empire whose relevant index in D is maximized.
3.7. The Powerless Empires
Powerless empires will collapse in the imperialistic competition and their colonies will be divided among other empires. An empire collapses when it loses all of its colonies.
3.8. Convergence
After a while all the empires except the most powerful one will collapse and all the colonies will be under the control of this unique empire. In this ideal new world all the colonies will have the same positions and same costs and they will be controlled by an imperialist with the same position and cost as themselves. In such a condition, the algorithm will be stopped. The Fig. 3 shows the flowchart of the proposed algorithm.
The design problem here is the selection of the optimal stacking sequence of hybrid composite laminate to obtain the simultaneous minimization of the weight and cost of a rectangular plate with the length a=0.9144 m, width b=0.762 m, and layer thickness t=0.127 mm subjected to a constraint on the first natural flexural frequency having lower bound of 25 Hz. This frequency is calculated based on the formulations presented in section 2 (Eq. (6)). The concept of hybridization is studied by using a twomaterial composite in which the lightness and highstiffness carbon/epoxy layers and inexpensive lowstiffness glass/epoxy layers are considered. In this way, beside providing suitable structural rigidity, cost reduction which is always a significant and worthy goal can be achieved. The stiffnesstoweight ratio of carbon/epoxy is about four times higher than that of glass/epoxy with E/r=0.087 against E/r=0.022. However, it is also more expensive, with a cost per kilogram that is 8 times higher than that of glass/epoxy. If the higher priority is based on the weight, then, carbon/epoxy will be preferred. But if the cost is the paramount, the optimum laminate will contain glass/epoxy plies. The design of this simple rectangular plate leads us to study the tradeoff between these two objective functions. The problem was investigated here, as mentioned in Section 1, is taken from Grosset et al. [30] and Abachizadeh and Tahani [31] and the results are compared with those obtained. The fiber orientation may take any value from a set of 19 angles ranging from to in steps of . The laminate was considered symmetric and balanced. Being symmetric is a practical assumption which is a great advantage in problem simplification as only half of the laminate is needed for optimization. In addition, the requirement that the laminate is balanced can be easily enforced by using pairs of ±q layers at symmetric state.
Figure 3. Flowchart of the proposed algorithm [26] 
This assumption was taken in order to minimize shearextension and bendingtwisting effects. Although plies and layers may not need to come in pairs, they are treated like other angles due to programming necessities but with half the normal thickness to simulate a single ply. The numbers of pair layers vary from 6 to 11 for half of the Laminate.
Table 1 shows the numbers designated to each material and related angles. As present in Table 1, numbers 0 to 19 and 20 to 38 indicate the glass/epoxy and the carbon/epoxy materials, respectively, with corresponding angles.
Multiobjective optimization is an important research topic for researchers. This is due to the multiobjective nature of real world problems. It is difficult to compare the results of one multiobjective method to another, because there is not a unique optimum in multiobjective optimization as in single objective optimization. Hence, the best solution in multiobjective terms may be decided by the decision makers. Recently, multiobjective metaheuristic and evolutionary procedures have become very popular for multiobjective optimization.
The increasing acceptance of metaheuristic algorithms is due to their ability to: (1) find multiple solutions in a single run, (2) work without derivatives, (3) converge speedily to Paretooptimal solutions with a high degree of accuracy, (4) handle both continuous function and combinatorial optimization problems with ease, (5) be less susceptible to the shape or continuity of the Pareto tradeoff curve. These issues are a real concern for the techniques of mathematical programming [45].
Table 1. The numbers for glass/epoxy and carbon/epoxy materials with related angles. 

Angles 
Glass/epoxy 
Carbon/epoxy 
0 
1 
20 
+5/5 
2 
21 
+10/10 
3 
22 
+15/15 
4 
23 
+20/20 
5 
24 
+25/25 
6 
25 
+30/30 
7 
26 
+35/35 
8 
27 
+40/40 
9 
28 
+45/45 
10 
29 
+50/50 
11 
30 
+55/55 
12 
31 
+60/60 
13 
32 
+65/65 
14 
33 
+70/70 
15 
34 
+75/75 
16 
35 
+80/80 
17 
36 
+85/85 
18 
37 
90 
19 
38 
In this paper, the Pareto set is generated by optimizing a convex combination of the two objectives, weight (W) and cost (C) for a series of values for the multiplier α as:
(20)
(21)
(22)
where a, b, t, N_{Ca}, N_{Gl}, r_{Ca}, and r_{Gl} are the length, width, thickness, the number of carbon/epoxy layer, the number of glass/epoxy layer, the density of carbon/epoxy and glass/epoxy materials, respectively.
The properties of glass/epoxy and carbon/epoxy laminates are presented in Table 2 [46]. A code with a system of 100 countries and 3 imperialists was developed in MATLAB 2011 based on the ICA. The performance of ICA is satisfactory using the revolution rate of 0.4 and ξ=0.02. Each stacking sequence is treated as one country. The objective function (F) of any country is calculated based on laminate stacking sequence (angle, material and situation of each layer) and ICA process finally lead to best stacking sequence in 500 iterations.
According to the literature [30, 31], by assigning a = 0, 0.7, 0.8, 0.87, 0.93, 0.96, and 1, the combined objective function (F) is minimized by using a singleobjective optimizer based on ICA. Running the program for 10 times in order to ensure the convergence, the best results were obtained and compared with GA and ACS. The optimal stacking sequences and optimum values of the multiobjective function for a = 0 are given in Table 3. Figure 4 shows the value of multiobjective function (F) with respect to the number of iterations for a = 0. Pair 11 with 42 layers was obtained as an optimal number of layers for multiobjective function with a = 0.
By observing Fig. 4, the convergence was achieved in less than 110 iterations. The optimal stacking sequence and optimum values of multiobjective function for a = 0.7, 0.8, 0.87, 0.93 and 0.96 are given in Table 4.
Table 2. Mechanical properties of glass/epoxy and carbon/epoxy. 

Glass /epoxy 
Carbon /epoxy 
Parameters 
43.4 
137.9 
Longitudinal modulus (GPa) 
8.89 
8.96 
Transverse modulus (GPa) 
4.55 
7.1 
Inplane shear modulus (GPa) 
0.27 
0.3 
Poisson ratio 
1970 
1587 
Material density (kg/m^{3}) 
0.127 
0.127 
Layer thickness (mm) 
1 
8 
Cost factor 
Figure 4. Multiobjective function value (F) with respect to the number of iterations for a = 0 
Figure 5 shows the value of multiobjective function with respect to the number of iterations for a = 0.7, 0.8, 0.87, 0.93, 0.96. Pairs 8 and 9 with 32 layers were obtained as an optimal number of layers for multiobjective function with a = 0.7. Pairs 6, 7 and 8 with 24 layers were obtained as an optimal number of layers for F with a = 0.8 and a = 0.87. Pairs 6, 7 and 8 with 22 layers were obtained as an optimal number of layers for F with a = 0.93. Pairs 6, 7, 8 and 9 with 22 layers were obtained for a = 0.96.
The optimal stacking sequence and optimum value of F for a = 1 is given in Table 5. The values of multiobjective function versus the number of iterations for a = 1 are shown in Fig. 5. Pairs 6, 7, 8, 9 and 10 with 22 layers were obtained as an optimal number of layers for optimum multiobjective function with a = 1. As can be seen, the convergence was achieved in less than 50 iterations.
The number of layers and materials are two important factors for multiobjective optimization. Also, the angles of composite layers play a role in determining the first natural frequency. When the value of weighting factor (a) is set to zero, the problem is reduced to singleobjective optimizing problem for cost minimization. The ICA can be used for a laminate with the layers which is completely made of glass/epoxy plies as is shown in Table 3. For a = 1, the only active objective is the weight and consequently a laminate is completely made of carbon/epoxy layers as is shown in Table 5. The carbon/epoxy is stiffer than glass/epoxy, and can fulfil the requirement for the minimum value of the first natural frequency with lesser number of plies.
All applied methods including GA, ACS, and ICA achieve optimum designs in which the layers are made of carbon/epoxy in the outer layers and those are made of glass/epoxy in the inner ones. This creates a sandwichtype composite where the structural function is assured by the stiff carbon layers, placed on the outside, where their contribution to the flexural properties of the laminate is maximal, while inner layers are merely used to increase the distance of the outer plies from the neutral plane and to reduce the total cost.
The Contribution of layers with angles ranging from to is to maximize the first natural frequency of the plate. The appearance of or plies is due to the reduction in weight and cost. Although these plies may not contribute much to the frequency, it is advantageous to use them. Unlike other angles, they do not come in pairs which save unnecessary additional weight and cost. In addition, the plies always come into view in the inner layers where they are the least damaging for the performance of the plate.
Table 6 shows the comparisons of the ICA, GA, and ACS for each given a. It should be noted that with the intention of illustrating the material of each layer in the final stacking sequence notation, the glass/epoxy layers are shown by plain numbers, while the carbon/epoxy layers are represented by underlined numbers.
Table 3. Stacking sequences and optimum values for multiobjective function with a = 0.
F 
Frequency (Hz) 
Weight (Kg) 
Cost 
Stacking sequences 
Best 
7.3217 
25.0842 
7.3217 
7.3217 
10 4 15 16 11 9 15 4 18 11 1 
1 
25.2398 
4 14 7 4 7 9 15 5 1 11 11 
2 

25.2871 
14 10 15 18 14 4 5 3 3 4 1 
3 

25.1706 
9 2 7 15 15 12 5 3 1 11 7 
4 

25.0668 
9 2 7 15 15 12 5 3 1 11 7 
5 

25.3650 
14 4 10 9 13 2 19 2 16 5 14 
6 

25.0762 
9 3 10 12 5 8 10 16 2 1 10 
7 

25.2852 
10 14 19 10 13 5 3 4 17 13 4 
8 

25.0613 
4 3 4 4 10 17 13 6 1 3 8 
9 

25.4732 
10 10 4 2 5 3 9 17 17 9 1 
10 
Table 4. Stacking sequences and optimum values for multiobjective function for a = 0.7, 0.8, 0.87, 0.93 and 0.96
F 
Frequency (Hz) 
Weight (Kg) 
Cost 
Stacking sequences 
Best 
a 
6.6225 
25.0256 
5.4429 
9.3750 
34 9 5 7 12 12 9 9 
1 
0.7 
25.3566 
28 9 9 14 4 16 10 6 
2 

25.1308 
33 15 3 4 14 8 13 4 
3 

25.2970 
28 9 10 15 7 5 3 5 
4 

25.1993 
33 10 14 15 17 5 1 9 1 
5 

25.0591 
28 4 10 2 3 1 7 14 1 
6 

25.1328 
28 4 3 19 10 14 1 5 15 
7 

25.0520 
28 9 17 4 14 17 1 4 1 
8 

6.1364 
25.0982 
3.7771 
15.5736 
33 28 33 4 11 14 
1 
0.8 
25.0745 
28 28 34 17 18 7 
2 

25.1695 
28 33 28 14 3 17 
3 

25.0133 
28 34 33 18 17 4 
4 

25.0089 
28 29 23 10 2 5 
5 

25.0659 
29 28 34 9 13 17 
6 

25.0624 
28 34 23 3 5 8 
7 

25.0183 
28 34 38 38 7 14 12 
8 

25.0481 
28 28 38 38 9 16 17 
9 

25.1675 
34 28 34 15 1 1 1 1 
10 

5.3107 
25.1975 
3.7771 
15.5736 
28 33 34 14 14 10 
1 
0.87 
25.1254 
28 29 34 15 7 5 
2 

25.2269 
28 28 34 15 14 15 
3 

25.1167 
34 28 28 16 5 8 
4 

25.0360 
29 28 29 5 14 14 
5 

25.0177 
34 28 28 1 11 2 19 
6 

25.0350 
28 38 34 38 15 2 3 
7 

25.0035 
33 34 38 38 3 3 14 
8 

25.1040 
34 34 38 38 14 5 16 
9 

25.0397 
34 38 34 38 10 19 1 15 
10 

4.5336 
25.0825 
3.1573 
22.8181 
34 29 23 28 23 19 
1 
0.93 
25.0074 
28 28 34 31 24 19 
2 

25.0132 
34 28 29 23 1 35 
3 

25.0705 
28 34 24 28 24 1 
4 

25.1259 
34 28 34 24 27 1 
5 

25.2532 
34 28 34 34 23 1 
6 

25.0803 
34 34 29 28 1 21 
7 

25.0338 
34 28 29 38 26 20 1 
8 

25.0434 
28 28 24 38 34 38 1 
9 

25.0536 
28 33 23 38 38 38 38 1 
10 

3.9438 
25.1622 
3.1573 
22.8181 
28 33 29 28 23 1 
1 
0.96 
25.0898 
28 34 33 34 1 22 
2 

25.0006 
28 33 28 21 35 19 
3 

25.0767 
34 29 23 33 23 19 
4 

25.0668 
34 33 23 33 27 1 
5 

25.1325 
34 34 28 34 1 33 
6 

25.0232 
28 34 33 38 38 1 35 
7 

25.0291 
28 38 34 38 28 31 19 
8 

25.0867 
33 34 38 38 28 38 38 19 
9 

25.0119 
34 33 38 38 38 38 38 1 38 
10 
a = 0.7 
a = 0.8 
a = 0.87 
a = 0.93 
a = 0.96 
a = 1 
Figure 5. Multiobjective function value (F) with respect to the number of iterations for a = 0.7, 0.8, 0.87, 0.93, 0.96 and 1 
Table 5. Stacking sequences and optimum values for F with a = 1.
F 
Frequency (Hz) 
Weight (Kg) 
Cost 
Stacking sequences 
Best 
3.0895 
25.2313 
3.0895 
24.7164 
28 23 23 29 29 38 
1 
25.1387 
23 33 38 29 34 33 
2 

25.1849 
34 28 38 23 32 22 
3 

25.0577 
23 33 38 34 26 28 
4 

25.1193 
38 33 29 28 27 29 
5 

25.4151 
28 33 29 38 34 22 
6 

25.4765 
34 28 28 38 38 38 28 
7 

25.0230 
38 28 38 28 26 21 38 
8 

25.0450 
29 38 29 38 22 24 38 
9 

25.1800 
38 38 28 28 29 28 20 
10 

25.1183 
38 29 28 38 38 38 38 32 
11 

25.1277 
28 38 38 38 38 38 24 33 
12 

25.1021 
38 28 23 38 38 38 38 23 
13 

25.0101 
38 33 38 38 38 38 38 38 25 
14 

25.0079 
29 38 38 38 38 38 38 38 35 
15 

25.0854 
38 34 38 38 38 38 38 38 22 
16 

25.0217 
38 38 28 38 38 38 38 38 38 38 
17 

25.1227 
34 38 38 38 38 38 38 38 20 38 
18 

25.1359 
28 38 38 38 38 38 38 38 38 38 
19 

25.0346 
38 38 34 38 38 38 38 38 38 38 
20 
Table 6. Comparisons of optimization results for the ICA against GA [30] and ACS [31]
F 
Frequency (Hz) 
Weight (Kg) 
Cost 
Stacking sequences 
Pairs number of half of Laminate 
Method 
a 
7.32 
25.47 
7.32 
7.32 

11 
ICA 
0 
7.32 
25.82 
7.32 
7.32 

11 
GA 

7.32 
25.07 
7.32 
7.32 
11 
ACS 

6.62 
25.29 
5.44 
9.37 

8 
ICA 
0.7 
6.62 
25.19 
5.44 
9.37 

9 

6.62 
25.10 
5.44 
9.37 

8 
GA 

6.62 
25.09 
5.44 
9.37 

8 
ACS 

6.13 
25.16 
3.77 
15.57 

6 
ICA 
0.8 
6.13 
25.04 
3.77 
15.57 

7 

6.13 
25.16 
3.77 
15.57 

8 

6.19 
25.88 
4.61 
12.52 

7 
GA 

6.19 
25.42 
4.61 
12.52 

7 
ACS 

5.31 
25.22 
3.77 
15.57 

6 
ICA 
0.87 
5.31 
25.03 
3.77 
15.57 

7 

5.31 
25.03 
3.77 
15.57 

8 

5.47 
25.08 
4.19 
14.02 

7 
GA 

5.47 
25.11 
4.19 
14.02 

7 
ACS 

4.53 
25.25 
3.15 
22.81 

6 
ICA 
0.93 
4.53 
25.03 
3.15 
22.81 

7 

4.53 
25.05 
3.15 
22.81 

8 

4.67 
25.38 
3.71 
17.47 

7 
GA 

4.60 
25.02 
3.09 
24.72 

6 
ACS 

3.94 
25.16 
3.15 
22.81 

6 
ICA 
0 .96 
3.94 
25.02 
3.15 
22.81 

7 

3.94 
25.08 
3.15 
22.81 

8 

3.94 
25.01 
3.15 
22.81 

9 

4.27 
26.07 
3.64 
19.37 

6 
GA 

3.95 
27.07 
3.09 
24.72 

6 
ACS 

3.08 
25.41 
3.08 
24.71 

6 
ICA 
1 
3.08 
25.47 
3.08 
24.71 

7 

3.08 
25.12 
3.08 
24.71 

8 

3.08 
25.08 
3.08 
24.71 

9 

3.08 
25.12 
3.08 
24.71 

10 

3.09 
25.14 
3.09 
24.72 

6 
GA 

3.09 
25.10 
3.09 
24.72 

6 
ACS 
The quantities in the GA were converted to the SI system. It can be seen that the ICA has outperformed the GA and ASC. Furthermore, in terms of the constraint accuracy, the ICA is superior over GA and ACS algorithms. However, the main advantage of the GA over both ACS and ICA is the orientation of angles which makes them easier to construct laminated composites.
The problem of obtaining minimum cost and weight using ICA in hybrid laminate composites was investigated. The symmetric balanced laminate made of glass/epoxy or/and carbon/epoxy layers was chosen with certain geometrical specifications. The material and number of layers were the design of variables as well as the fiber orientations. The optimizing process was constrained by the first natural frequency of the plate to be not less than a predefined value (25 Hz). The results were evaluated for different weighting factors (a) based on WSM and compared to those obtained by the GA and ACS. This comparison confirms the superiority of the ICA in terms of the function’s value and constraint accuracy over GA and ACO. When cost was a primary consideration the plate was made from glass/epoxy, and when weight was a primary consideration, it was made of carbon/epoxy. The less stiff glass/epoxy layers, when they have to be used, always appear in the inner layers. This creates a sandwich type composite where the structural function is assured by the stiff carbon layers, placed on the outside, where their contribution to the flexural properties of the laminate is maximal/maximized, while inner layers are merely used to increase the distance of the outer plies from the medial plane. It certainly cannot be stated that metaheuristic algorithms have advantages over each other, since it depends on the structure of the problems and algorithm parameters. The ICA has reached to the same sequences and in some cases has reached to the better sequences of mentioned algorithms and it has offered minimum values for multiobjective function. This algorithm is so useful and as a new method is competitive with other heuristic algorithms.
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