Solution of structural and mathematical optimization problems using a new hybrid swarm intelligence optimization algorithm

Advances in Engineering Software 127 (2019) 106–123

Contents lists available at ScienceDirect

Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft

Solution of structural and mathematical optimization problems using a new hybrid swarm intelligence optimization algorithm

T

Ali Mortazavia, Vedat Toğanb, , Mahsa Moloodpoorc ⁎

a

Civil Engineering Department, Usak University, Usak/Turkey Civil Engineering Department, Karadeniz Technical University, 61080 Trabzon/Turkey c Mechanical Engineering Department, Ege University, Izmir/Turkey b

ARTICLE INFO

ABSTRACT

Keywords: Integrated particle swarm optimization (iPSO) Teaching and learning based optimization (TLBO) Hybrid optimization method

In this investigation a new optimization algorithm named as interactive search algorithm (ISA) is presented. This method is developed through modifying and hybridizing the affirmative features of recently developed integrated particle swarm optimization (iPSO) algorithm with the pairwise knowledge sharing mechanism of the teaching and learning based optimization (TLBO) method. Proposed ISA provides two different navigation schemes as Tracking and Interacting. Each agent based on its tendency factor can pick one of these two schemes for searching the domain. Additionally, ISA utilizes an improved fly-back technique to handle problem constraints. The proposed method is tested on a set of mathematical and structural optimization benchmark problems with discrete and continuous variables. Numerical results indicate that the new algorithm is competitive with other well-stablished metaheuristic algorithms.

1. Introduction Optimization techniques have widely been applied in the different fields of science and engineering to attain an optimal state for the desired systems. This target is usually met through the maximization or minimization of proper objective function(s) considering some specific constraints. In the structural optimization problem, mostly the weight of the system is designated as the objective function. The main goal is to minimize this objective function, such that all constraints stay feasible and no variable bounds are violated. For example, the displacement and stress limitations are two important constraints for this class of problems. To solve an optimization problem choosing an efficient method plays crucial role on the accuracy and computational time of the solution process. Generally, optimization methods can be categorized into two main groups: gradient based and non-gradient based techniques [1,2]. The gradient based methods require continuous objective functions and their gradients to compute the proper search direction and/or appropriate step size. These methods have rapid convergence rate and low computational cost. However, finding a continuous objective function for several optimization problems can be so difficult or even impossible. Also, due to their sensitivity to the starting point especially in the constrained problems with more complex search boundaries [3], they can get trapped into local minima on the more complex search spaces ⁎

[4]. These shortcomings cause to limit their usage in the more complicated optimization problems [5]. However, the non-gradient based techniques numerically examine the search space for the optimal solution via progressive-stochastic evaluation of the search space. Hence, they do not require any gradient of the objective function. Especially along with the developments in computer technology these methods gain more attention among the researchers. Associated to the non-gradient methods, metaheuristic algorithms provide a mathematical model which generally inspired from a natural phenomenon like physical principles or social laws. For example, the genetic algorithm (GA) [6–8], the particle swarm optimizer (PSO) and its variants [9–13], the ant colony optimization (ACO) and its enhanced variants [14–16], the teaching and learning based optimization (TLBO) and its improved versions [17–20], the water wave optimization (WWA) [21], virus optimization algorithm (VOA) [22] and the bat inspired algorithm (BIA) [23] are the metaheuristic methods which have widely been implemented in the structural optimization problems [24]. Although the aforementioned methods utilize different strategies to find the optimal solution, global and local search capabilities are commonly two important specifications of a metaheuristic optimizer. Establishing a proper balance between these two search strategies leads to obtain the optimal solution with lower computational cost and higher accuracy [25,26]. Meeting this aim for metaheuristic algorithms with higher number of adjustable parameters is more difficult, since

Corresponding author. E-mail addresses: [email protected] (A. Mortazavi), [email protected] (V. Toğan).

https://doi.org/10.1016/j.advengsoft.2018.11.004 Received 23 March 2018; Received in revised form 25 October 2018; Accepted 6 November 2018 0965-9978/ © 2018 Elsevier Ltd. All rights reserved.

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

the teaching and learning based optimizer (TLBO) [20] is combined with iPSO method in the current study. Based on the proposed philosophy (i.e. the peer-learning phase) pairs of individuals can directly communicate with each other to update their locations. So, this approach provides more democratic movement for the agents and allows them not to be mainly affected by two dominant agents (i.e. best and weighted particles). Subsequently, even in the further iterations, the proper level of diversity can be formed in the colony to increases the algorithm's search capacity. The proposed approach is named as interactive search algorithm (ISA). In ISA, each agent according to its tendency factor can reciprocally interact with other agents or move toward predefined standard dominant positions (e.g. the best agent). In contrast with TLBO algorithm, ISA is the single-phase algorithm which incorporates the global search scheme of iPSO with the interactive local search capability of TLBO to put forward an efficient search strategy. Also, to handle the constraints, instead of the conventional penalty function, ISA utilizes improved fly-back (IFB) method [9]. The new method is tested on several benchmark optimization problems (structural and mathematical) including continuous and discrete variables. Numerical results demonstrate that proposed method is competitive with other well-established metaheuristic techniques.

Table 1 The pseudo code for ISA. [Initialize population] While termination criteria is not met do Compute weighted agent for the swarm, using Eq. (8) For each agent do If τ ≥ τ0, in which τ0 = 0.3 Apply tracking paradigm to update current agent location using Eq. (7) Elseif τ < τ0, in which τ0 = 0.3 Apply interacting paradigm to update current agent location using Eq. (7) End End [Check constraints using IFB]: For each agent do If any component(s) of the current agent violate the numeric constraints Change violated component(s) with corresponding ones available in the weighted agent End If The current particle violates the characteristic constraints Reset the agent to its previous best position saved in the XP matrix Else

) and old f (X old [Compare the objective function value of the updated f (Xupdated i ) i agents] ) < f (Xiold) If f (Xupdated i Replace the old agent with updated agent and update the XP matrix Else reset the updated agent to its old position stored in XP End End End [Compare global best agent (XG) with weighted agent (XW)] If f(XW) < f(XG) Replace XG with XW End End

2. Formulation of the optimization problem Generally, an optimization problem is defined as minimizing (or maximizing) the value of an objective function meeting the predefined condition(s) and expressed mathematically as:

minimize f (X), X = [x1, x2, …, xD] D

(1)

D

where x ∈ ℜ , and ℜ is the D-dimensional search space defined by the side constraints:

they require effortful tuning of ad-hoc parameters depending on the essence of the problem. This condition becomes even more severe when the penalty approach is applied to handle the problem constraints as it can also add extra adjustable terms into the algorithm [27]. To overcome this problem, the self-adaptive penalty functions or parameterfree constraint handling methods can be utilized in the optimization process [28]. Integrated particle swarm optimizer (iPSO) originated by Mortazavi and Toğan [29] is an improved and modified version of the conventional particle swarm optimizer (PSO) to handle both continuous and discrete optimization problems. This method was tested on several structural problems [30,31] and the outcomes indicated that, despite of its competitive performance, the rapid drop of diversity level in the successive iterations have degraded its search capability. Weighted particle is the specific particle iPSO utilizes to navigate the colony. This particle is the weighted average of whole colony, and it enables the particles to share their experience during navigation process. However, the effect of the better particles on forming the weighted particle is much higher than the others due to their higher weight coefficients. Based on this fact, in the further iterations, the weighted particle lies so close to the best particle and practically both of them nearly spot the same location of the search space. Thus, the effect of the weighted particle is highly reduced or even vanished in the final iterations, so the population highly loses its diversity and any premature convergence is expected. On the other hand, due to this proximity if the best particle is trapped in a local minimum all colony agents are conducted toward the local minimum not only by the best particle but also by weighted particle. To mitigate this problem and guarantee a satisfactory level of diversity for the population, the philosophy of the peer-learning phase of

x l, i

xi

x u, i ,

i = 1, 2, …, D

(2)

where xl, i and xu, i are the lower and upper bounds for design variables in the design vector (X). To accept any x as a solution, it must be in the feasible region (x ∈ Ω⊆ℜD) defined by a set of nc additional equality or inequality constraints

gj (X)

0,

j = 1, 2, …, q

and hj (X) = 0 j = q + 1, …, nc

(3)

where q and (nc-q) are the number of inequality and equality constraints, respectively. 3. Interactive search algorithm (ISA) and its basis methods This section describes the proposed interactive search algorithm (ISA). Since this method hybridizes the affirmative features of TLBO and iPSO methods, initially both TLBO and iPSO algorithms are concisely described and consequently ISA is explained. 3.1. Teaching learning based optimization (TLBO) TLBO method is inspired from knowledge exchange process between teacher and students during the educational period. The mathematical model of this simulation has been established by Rao et al. [20]. This method is divided into two main phases as teaching phase and learning phase. In the teaching phase, initially all candidate solutions (i.e. agents) are evaluated, and the best agent is selected as teacher. In this phase, both teacher (Xteacher) and the mean value of all agents (Xmean) are utilized to guide all other members in the class. In the teaching phase, the new position of ith agent (Xnew, i) is obtained as below:

107

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Fig. 1. Flowchart of ISA.

108

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 2 Details of the considered numerical benchmark functions. ID

Name

BF1

Shifted Rotated High Conditioned Elliptic Function

BF2

Plot

Formulation*

F2 (z ) = z = (x

Schwefel's Problem 2.13

f_bias i 1 2 1 zi

D i=1

(106) D

−450

+ f _bias ,

o)*M , x = [x1, x2 , ...,xD ]

D i=1

F4 (x) =

−460

Bi (x ))2 + f _bias ,

(A i

x = [x1, x2 , ...,xD ] Ai =

D j=1

Bi (x ) =

(aij sin D j =1

j

+ bij cos j ) ,

(aij sin xj + bij cos xj ) ,

for i = 1, ...,D

BF3

BF4

⁎

Shifted Expanded Griewank's plus Rosenbrock's Function

Rotated Version of Hybrid Composition with Noise in Fitness

−130

if G (x ) = H (x ) =

xi2 xi D D i = 1 4000 i = 1 cos( i ) + D 1 2 xi + 1) 2 + (xi i = 1 (100(x i

1 1)2)

then F5 (x) = G (H (z1, z2)) + G (H (z2, z3)) + ... G (H (zD 1, zD )) + G (H (zD , z1)) + f _bias13

f16 (x) =

D i=1

(

k max k= 0

[ak cos(2 bk (xi + 0.5))])

D

k max k= 0

[ak cos(2 bk ·0.5)]

a = 0.5 , b = 3 , kmax = 20 (f16 f _bias16) be G(x), then F (x) = G (x)*(1 + 0.2|N (0, 1)|) + f _bias

Details about all given functions are available in CEC2005 database [34]

109

let

120

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 3 Results of sensitivity analysis carried out for the BF1 benchmark function.

Table 4 Results of sensitivity analysis carried out for the BF2 benchmark function.

No.

τ0

w

Objective function value error Best Mean

Worst

No.

τ0

w

Objective function value error Best Mean

Worst

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

5.23 5.20 5.18 5.19 5.22 5.21 5.19 5.11 5.16 5.18 5.20 5.12 5.09 5.09 5.13 5.22 5.18 5.12 5.15 5.17 5.24 5.21 5.14 5.17 5.20 5.27 5.25 5.21 5.23 5.26 5.28 5.26 5.23 5.27 5.29

5.63 5.56 5.54 5.59 5.62 5.60 5.55 5.52 5.54 5.56 5.57 5.54 5.51 5.53 5.55 5.61 5.58 5.55 5.57 5.63 5.62 5.61 5.58 5.60 5.65 5.64 5.62 5.60 5.63 5.67 5.68 5.66 5.63 5.65 5.69

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

4.95 4.92 4.89 4.91 4.93 4.93 4.90 4.86 4.88 4.91 4.87 4.81 4.81 4.80 4.85 4.89 4.87 4.85 4.90 4.92 4.91 4.89 4.88 4.92 4.94 4.92 4.91 4.90 4.93 4.95 4.94 4.93 4.91 4.95 4.97

5.08 5.03 5.02 5.04 5.05 5.05 5.02 5.00 5.01 5.03 5.02 5.00 4.97 4.96 5.01 5.03 5.01 5.00 5.02 5.05 5.04 5.03 5.01 5.02 5.06 5.05 5.04 5.02 5.06 5.07 5.07 5.06 5.04 5.05 5.08

0.25

0.30

0.35

0.40

0.50

0.60

Xnew, i = Xi + r (X teacher

TF Xmean)

5.41 5.40 5.33 5.39 5.42 5.37 5.30 5.23 5.25 5.31 5.26 5.21 5.20 5.23 5.28 5.29 5.22 5.24 5.28 5.30 5.31 5.28 5.27 5.32 5.35 5.34 5.33 5.31 5.35 5.37 5.38 5.35 5.32 5.36 5.39

(4)

Xj)

if f (X i) < f (Xj)

Xnew, i = X i + r (Xj

Xi)

otherwise

0.30

0.35

0.40

0.50

0.60

(PSO) via applying both the concept of weighted particle and IFB mechanism. The main role of the weighted particle XW is to prevent algorithm getting stuck into local minima when the current particle lies very close to its previous best position or to the global best particle. Since XW (see Eq. (8)) applies the experience of all particles considering their objective values, it can improve the swarm movements through the search space. iPSO is mathematically formulated as follows:

in which r is a random number in the interval [0,1], xi is the current location of the ith agent, TF= round[1 + rand(0, 1)], is the teaching factor. At this level, if xnew, i produces a better result, it is replaced with current agent xi and, otherwise, xi is kept as the same. In the learning phase of TLBO, all agents can interact with each other based on the fact that students can increase their knowledge via making dialogues. In this phase, the ith agent (Xi) based on its objective function value, f(Xi), can move toward or away from the randomly selected jth agent (xj) where i≠j. This phase can be mathematically stated as:

Xnew, i = X i + r (Xi

0.25

4.98 4.95 4.93 4.94 4.96 4.97 4.94 4.91 4.92 4.93 4.95 4.93 4.90 4.89 4.92 4.96 4.97 4.90 4.93 4.95 4.99 4.98 4.94 4.96 5.00 5.00 4.99 4.96 5.98 5.01 5.01 5.00 4.99 5.02 5.03

t + 1X

i

= tX i +

t+1

i

where t+1 t+1

(5)

( tXW

i

=

i

= wi ×

4i

2i

t

( t XG

i

tX ) i

+( tXP) j

where 1i = C1 × rand1i

If Xnew, i offers a better result, it replaces xi otherwise xi remains the same (see [17,20] ).

4i

1i

if rand 0i +

+

2i 3i

2i

+

( tXW

3i ) (

tXP j

tX ) i

+

if rand 0i >

t XP) j

= C2 × rand2i

3i

= C3 × rand3i

= C4 × rand4i

(6)

3.2. Integrated particle swarm optimizer (iPSO)

t+1

where t and t + 1 represent current and next steps, respectively. νi is the updated velocity, wi is the inertia factor of current velocity, and tνi is the current velocity of the ith agent. C1 = − (ϕ1i + ϕ2i), C2 = 2, C3 = 1, and C4 = 2 are acceleration factors, and randki (k = 0,1,…,4) is the random number selected from the [0, 1] interval. XP is the matrix storing the previous best positions of all agents, tXPj is the randomly

The standard particle swarm optimizer has been introduced by Kennedy and Eberhart [32]. This method is based on a metaphor of the social collaboration of animals in a colony such as fish or birds for finding food resources or avoiding enemies. Mortazavi and Toğan [9] improved the performance of conventional particle swarm optimizer

110

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 5 Results of sensitivity analysis carried out for the BF3 benchmark function.

Table 6 Results of sensitivity analysis carried out for the BF4 benchmark function.

No.

τ0

w

Objective function value error Best Mean

Worst

No.

τ0

w

Objective function value error Best Mean

Worst

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

0.70 0.64 0.61 0.63 0.66 0.63 0.60 0.55 0.59 0.61 0.59 0.51 0.48 0.48 0.55 0.61 0.54 0.48 0.55 0.60 0.66 0.63 0.59 0.62 0.64 0.69 0.65 0.62 0.66 0.70 0.73 0.69 0.65 0.67 0.71

1.04 0.99 0.88 1.00 1.03 1.02 1.00 0.90 0.95 0.97 0.99 0.97 0.97 0.96 0.98 0.99 0.98 0.89 0.97 1.00 1.01 1.00 0.97 0.98 0.99 1.03 1.02 0.99 1.00 1.01 1.05 1.03 1.01 1.02 1.04

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

2.18 2.16 2.15 2.17 2.19 2.15 2.11 2.09 2.12 2.13 2.10 2.06 2.01 2.02 2.02 2.12 2.09 2.06 2.08 2.10 2.15 2.13 2.10 2.12 2.14 2.17 2.15 2.11 2.14 2.16 2.19 2.16 2.14 2.17 2.18

2.36 2.34 2.33 2.35 2.37 2.35 2.33 2.31 2.32 2.34 2.32 2.31 2.28 2.28 2.30 2.33 2.32 2.29 2.30 2.31 2.34 2.33 2.31 2.32 2.35 2.36 2.35 2.33 2.34 2.37 2.37 2.36 2.34 2.35 2.38

0.25

0.30

0.35

0.40

0.50

0.60

0.89 0.87 0.85 0.86 0.88 0.85 0.82 0.79 0.83 0.86 0.79 0.76 0.73 0.72 0.75 0.81 0.78 0.69 0.76 0.80 0.83 0.80 0.78 0.79 0.82 0.84 0.82 0.81 0.83 0.85 0.86 0.85 0.82 0.81 0.87

selected agent among current XP, the vector tXG is the global best particle up to current step, and t + 1Xi and tXi respectively represent the updated position and current position of the ith agent [9]. Finally, w is randomly selected from [0.5, 0.55] and α = 0.4 [9].

t + 1X

0.25

0.30

0.35

0.40

0.50

0.60

2.28 2.24 2.20 2.21 2.25 2.27 2.22 2.17 2.19 2.24 2.25 2.20 2.16 2.18 2.23 2.26 2.25 2.23 2.27 2.28 2.29 2.28 2.24 2.25 2.26 2.30 2.27 2.26 2.28 2.29 2.31 2.29 2.28 2.30 2.32

= tX i + t + 1 i where if tracking paradigm 0 t+1

3.3. Interactive search algorithm (ISA) In order to create an effective search strategy, the best of TLBO and iPSO algorithms are modified and combined. This new hybrid algorithm offers two different paradigms to update the location of each agent inside the search domain. In the first paradigm, weighted agent xW , global best agent xG and prior best position captured by an agent x Pj are used for updating the current agent xi. This paradigm is denoted as tracking paradigm. The other paradigm is called interacting paradigm. It guides the agents via providing pairwise interaction between the swarm members to share their knowledge. In each iteration, an agent can pick one of these two paradigms based on its tendency factor (τ). From sensitivity analysis (see Section 4), most proper values for the tendency factor and weighted inertia (w) are determined as τ = 0.3 and w = 0.4. Consequently, ISA is mathematically formulated as following:

i

i

= wi ×

t

i

+(

1i

(

2i

( t XG

(

3i

( tXW

( tXPj

tX

tXP))+ j tXP)) j

i))+

if

<

t+1

i

= (rTi

(X i

Xj))

for f (X i) < f (Xj)

i

(rTi

(X j

X i))

for otherwise

t+1

=

0

interacting paradigm (7)

where τ is a random number uniformly selected from [0,1], ri is the vector of random numbers selected form interval of [0, 1], ϕki = Ck.ri (k = 1,2,3) are the vectors of coefficients, C1 = − (ϕ1i + ϕ2i), C2 = 2, and C3 = 1 are acceleration factors [9]. 1 ≤ i, j ≤ np while i ≠ j and np is the number of agents. It should be noted that sizes of ϕ1i, ϕ2i, ϕ3i and ri are equal to the problem dimension (D). Also, ⊗ denotes the Hadamard product (Element-wise multiplication) and τ0 is the tendency factor value set as τ0 = 0.3 from sensitivity analysis. In order to clarify the proposed algorithm, ISA's main procedure is summarized in Table 1 as pseudo code form and Fig. 1 as flowchart form. Proposed ISA handles

111

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

the problem constraints utilizing the improved fly-back (IFB) approach [29,31], which consists of three main steps. First, optimization constraints are divided into numeric constraints (that do not entail function evaluations) and characteristic constraints (evaluation of their constraint violation entails a new analysis). Second, if the agent violates any numeric constraint(s) the corresponding violated components are replaced with those available in the weighted agent. Third, the updated agent is evaluated. If it finds a better solution than the old agent, the latter is replaced with updated agent, if not it is reset to its prior best position. In Eq. (7), the weighted agent is defined as:

Table 7 Statistical comparison of optimization results obtained for functions BF1-BF4. Test function BF1

BF2

BF3

BF4

Best Mean Worst Std. Best Mean Worst Std. Best Mean Worst Std. Best Mean Worst Std.

this study PSO

TLBO

iPSO

ISA

7.39 7.62 8.14 2.17 5.19 5.38 5.49 0.89 1.19 1.27 1.39 0.55 2.44 2.57 2.70 1.01

5.46 5.92 7.01 0.75 5.18 5.31 5.45 0.25 1.21 1.29 1.43 0.42 2.34 2.42 2.51 0.48

6.81 7.21 7.63 1.02 5.04 5.18 5.42 0.20 0.57 0.75 1.17 0.41 2.23 2.27 2.31 0.12

5.09 5.20 5.51 0.11 4.80 4.89 4.96 0.09 0.48 0.69 0.89 0.33 2.01 2.16 2.28 0.17

N c¯ w X iP i=1 i

XW =

where w w N c¯iw = (c^i / i = 1 c^i ) in which w c^i

1

= 1

max k

max (f (XkP)) M

M (f (X kP))

f (XiP) +

k

min

1 k

M

(f (XkP)) +

Fig. 2. Best run convergence histories for (a) BF1, (b) BF2, (c) BF3 and (d) BF4.

112

,

i = 1, 2, …, N .

(8)

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 8 Results of sensitivity analysis carried out for the 25-bar truss problem. τ0

w

Structural weight (lb) Best Mean

Worst

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

546.84 546.55 546.04 546.98 547.25 546.66 546.12 545.89 546.10 546.79 545.84 545.14 545.09 545.90 545.62 546.21 545.92 545.37 546.19 546.84 546.87 546.33 545.82 546.51 547.04 547.11 546.74 546.20 547.39 547.76 547.80 547.25 546.85 547.79 548.03

548.94 548.16 547.64 548.22 548.81 548.75 547.85 546.51 547.11 547.90 548.25 545.99 545.90 545.88 547.63 548.37 547.08 546.28 547.33 547.82 548.63 547.55 547.37 547.68 548.32 548.77 548.16 547.81 548.29 548.73 548.84 548.36 548.07 548.65 548.96

0.25

0.30

25-bar truss-

0.35

0.40

Fig. 3. The diversity variations for TLBO, iPSO and ISA methods. 0.50

0.60

547.71 547.02 546.87 547.13 547.86 547.20 546.91 546.11 546.84 547.51 546.89 546.07 545.39 545.79 546.32 547.19 546.76 546.01 546.64 547.10 547.65 547.27 546.74 547.17 547.39 547.92 547.43 547.21 547.66 547.87 548.11 547.67 547.36 547.81 548.17

constraints, ii) Characteristic constraints. The former category controls the sizing variables’ bounds while the latter checks the violations of the stress and displacement limitations after the structural analysis [29–31,33]. 4. Comparison of ISA with its parental methods The ISA algorithm is tested in four mathematical optimization problems and four structural optimization problems. All benchmark problems considered in this study are commonly solved by optimization experts to evaluate performance of newly developed algorithms. The constrained optimization cases exhibit distinct kind of objective functions involving different number of design variables and a range of types and number of constraints. For any new algorithm developed, benchmark problems offer a good measure for demonstrating its ability over other algorithms proposed in advance. Therefore, these problems are commonly resolved again and again by the researchers who are developer of the algorithms. The developed algorithm is coded in MATLAB® using its graphic user interface (GUI) and it is linked to CSI SAP2000® via its open application programming interface (OAPI) module to analyze the structures. The built-in FEM based analyzer module performs analyzes based on the direct stiffness method. Computations for all test cases are

Fig. 4. Schematic of the spatial 25-bar tower.

Also, N indicates the population size and ε is a small positive w number to prevent division by zero if the denominator of c^i becomes equal to zero and it is set as ε = 1E − 6 in the current work. In the structural optimization problems, upper and lower limits for sizing variables are defined at the start of the algorithm while the members’ stresses and nodal displacements are declared via the structural analyses. In this class of optimization problems IFB separates problem constraints domain into two different categories: i) Numeric

113

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 9 Statistical comparison of optimization results for the 25-bar truss problem. Design Variables (in2)

BB-BC [37]

ABC [27]

mTLBO [18]

HPSSO [38]

CBO [39]

FPA [40]

This study PSO

TLBO

iPSO

ISA

A1 A2 A3 A4 A5 A6 A7 A8 Best (lb) Mean (lb) Worst (lb) Standard deviation (lb) NSAs

0.0100 2.0920 2.9640 0.0100 0.0100 0.6890 1.6010 2.6860 545.380 545.780 – 0.491 10,820

0.0110 1.9790 3.0030 0.0100 0.0100 0.6900 1.6790 2.6520 545.190 – – – 500,000

0.0100 1.9878 2.9914 0.0102 0.0100 0.6828 1.6775 2.6640 545.175 545.483 – 0.306 12,199

0.0100 1.9907 2.9881 0.0100 0.0100 0.6824 1.6764 2.6656 545.1640 – – – 13,326

0.0100 2.1297 2.8865 0.0100 0.0100 0.6792 1.6077 2.6927 544.31 545.256 – 0.294 9090

0.0100 1.8308 3.1834 0.0100 0.0100 0.7017 1.7266 2.5713 545.159 545.730 – 0.59 8149

0.010 2.052 3.001 0.0100 0.0100 0.6840 1.6160 2.6730 545.21 546.98 547.82 2.908 14,420

0.010 2.0740 2.9610 0.0100 0.0100 0.691 1.6170 2.6740 545.12 545.62 546.15 0.26 11,350

0.010 1.9100 2.798 0.0100 0.0100 0.708 1.8360 2.6400 545.09 545.70 545.98 0.25 5980

0.010 1.9100 2.7980 0.0100 0.0100 0.7080 1.8360 2.6400 545.09 545.39 545.90 0.22 5840

In order to determine the proper values of tendency factor (τ) and inertia coefficient (w), sensitivity analysis is performed for all benchmark functions. The corresponding results are presented in Table 3 through Table 7. The values reported in Tables 3–7 show the error on objective function value in logarithmic scale to make easy comparison. The error value is defined as (f(x)-f(x*)) where f(x) designates recorded optimum value during the optimization process and f(x*) indicates the global optimum value for each test function [34]. For these trials, the problem dimension for all functions is set as D = 30, ISA is run for 50 times on each function, the population size is selected as 30, and the number of objective function evaluations (OFEs) is set equal to 300,000. The results of sensitivity analysis indicate that τ0 and w must be selected in the range 0.30–0.40. Based on the reported values (Table 3 through 6), the best setting of internal parameters is τ0 = 0.3 and w = 0.4. For more clarity, Table 7 compares the best solution (Best), mean solution (Mean), worst solution (Worst) and standard deviation (Std.) obtained for each optimization algorithm over 50 independent runs. Fig. 2 shows convergence histories relative to best optimization run of each algorithm. It can be seen that ISA outperforms PSO, TLBO and iPSO under both aspects of convergence rate and achieved optimal solution. This demonstrates that the proposed ISA algorithm successfully integrates the best features of TLBO and iPSO.

Fig. 5. Spatial 25-bar truss problem: comparison of best run convergence curves of ISA and its parent algorithms.

4.1.1. Diversity analysis An important aspect of metaheuristic optimization, which relies on exploration and exploitation properties, is its ability to preserve an admissible amount of diversity among the population during the optimization process. Too low diversity would decrease the number of possible solutions but speeds up the convergence. However, too much diversity level puts forward several solutions, and consequently negatively affects the convergence behavior [35]. This problem becomes more severe for problems with complex search spaces. Thus, compromising between the diversity and convergence rate directly affects the efficiency and performance of the metaheuristic method. To compare the diversity of iPSO, TLBO and ISA, the diversity diagrams are generated and compared for the 3D sphere function problem, stated as:

carried out on a computer with 12GB DDR3 RAM, Intel Core i7-2.4 GHz CPU and 64 bit Windows 10 operating system. For each test case, 50 and 30 independent runs, respectively, are performed for the numerical and structural test problems due to stochastic nature of the proposed algorithm. All sensitivity analyses are performed for fixed size but randomly generated initial populations. 4.1. Numerical benchmark test functions In this section, four mathematical benchmark functions are examined. Selected test functions consist of different characteristics (e.g. quadratic, nonlinear) and have a number of equality and inequality constraints. The benchmark functions taken from CEC 2005 [34] are abbreviated as BF1, BF2, BF3 and BF4, respectively and listed in Table 2.

f (x ) = x12 + x 22 + x32

114

x

[ 10, 10]

(9)

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

The diversity is calculated based on the following formulation [36]:

Diversity (t ) =

1 NL

N

D

i=1

j=1

(x i j

x¯ j )

2

(10)

where t indicates the current step, N indicates the population size, L is the search space's longest diagonal length, D is the problem dimension x i j is jth component of the ith agent and x¯ j is the mean value of all jth components of all agents. The achieved diversity curves are plotted in Fig. 3. As can be seen from this figure, the diversity level of ISA is enclosed between diversity levels of TLBO as upper limit and iPSO as lower limit. This indicates that ISA preserves the best features of iPSO meanwhile improves the performance of the iPSO method via adding an admissible level of diversity to this method 4.2. Structural optimization problems The ISA algorithm is tested also on four classical structural optimization problems. Statistical results are obtained for 30 independent runs and compared with those available in the technical literature 4.2.1. Design of a spatial 25-bar tower The first test problem regards the spatial 25-bar truss shown in Fig. 4. The modulus of elasticity and density of the material are 10.0 Msi and 0.1 lb/in3, respectively. The allowable displacements for all nodes in all coordinate directions are restricted to ± 0.35 in. Truss members are collected into eight groups because of the symmetry about X and Y-axes follows: (1) A1, (2) A2-A5, (3) A6-A9, (4) A10-A11, (5) A12A13, (6) A14-A17, (7) A18-A21, and (8) A22-A25. The cross-sectional areas of the grouped members can range 0.01in2 ≤ Ai ≤ 3.4in2, where i = 1,..,8. The grouped cross-sectional areas of members are assumed as continuous sizing variables for this case. Allowable stresses (tension, compression) are: for the members grouped in A1, A4, and A5 (35, 35.092), A2 (35, 11.59), A3 (35, 17.305), A6 (35, 6.759), A7 (35, 6.959), and A8 (35, 11.082). The truss will be designed to resist on two loading conditions (in kips): for Case 1, on nodes 1 (PX = 1, PY = 10, PZ = −5), 3 (PX = 0.5, PY = 0, PZ = 0), and 6 (PX = 0.5, PY = 0, PZ = 0); for Case 2, on nodes 1 (PX = 0, PY = 20, PZ = −5), and 2 (PX = 0, PY = −20, PZ = −5). Table 8 shows the results of the sensitivity analysis carried out for determining the best setting of internal parameters τ0 and w. Using the proper values determined from the sensitivity analysis for the weighted inertia (w = 0.4), and the tendency factor (τ0 = 0.3), the solutions obtained in the present work are tabulated in Table 9 in comparison with those available in the related technical literature. The required number of structural analyses (NSAs) are also compared in this table. The computational cost of ISA (5840 structural analyses) is obtained by multiplying the population size (10) by the required iterations (584). As can be observed from the presented results, ISA finds only 0.143% heavier design than the design obtained by CBO weighting 544.31 lb [39]. However, ISA needs 3250 less structural analyses than CBO [39] to find the optimum design. The standard deviation of the present solution is lower than for the other cited methods. Furthermore, ISA is the fastest algorithm overall. The convergence rates of ISA and its parent algorithms are compared in Fig. 5. It is observed from this figure that PSO method has steady linear approach toward the optimum point. Although iPSO outperforms TLBO for general convergence rate, iPSO stagnates around a local minimum between 2800 and 3900 structural analyses. The main reason of this stagnation is that the weighted agent and global best agent both point the same location of

Fig. 6. Schematic of the spatial 72-bar truss. Table 10 Results of sensitivity analysis carried out for the 72-bar truss problem. τ0

w

Structural weight (lb) Best Mean

Worst

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

384.42 383.25 380.50 381.01 381.99 383.83 382.74 379.64 380.56 381.08 383.11 378.37 378.37 378.37 380.91 383.69 382.51 378.37 380.42 381.65 384.20 383.46 381.35 382.16 383.34 384.97 383.93 381.87 383.22 384.09 385.39 384.71 382.27 383.86 384.81

389.61 388.45 387.14 388.21 388.97 387.21 385.78 384.23 385.64 386.97 385.46 383.10 382.28 382.38 383.32 386.81 385.73 381.01 383.89 384.67 387.18 386.53 383.91 384.86 385.77 388.37 387.84 386.04 387.40 388.68 389.79 388.54 387.43 388.76 389.88

0.25

0.30

72-bar truss

0.35

0.40

0.50

0.60

385.31 384.27 382.84 383.63 384.57 384.38 383.29 381.20 382.69 383.15 383.79 381.87 380.10 379.08 381.48 384.26 382.97 378.65 381.63 382.41 385.72 384.33 382.05 383.34 383.99 386.23 385.47 383.62 384.79 385.52 387.54 386.81 384.98 385.11 386.43

115

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 11 Statistical comparison of optimization results for the 72-bar truss problem. Design Variables (in2)

BB-BC [37]

mTLBO [18]

CSP [41]

CBO [39]

ECBO [42]

FPA [40]

This study PSO

iPSO

TLBO

ISA

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 Best (lb) Mean Worst Standard deviation (lb) NSAs

1.8577 0.5059 0.1000 0.1000 1.2476 0.5269 0.1000 0.1012 0.5209 0.5172 0.1004 0.1005 0.1565 0.5507 0.3922 0.5922 379.850 382.080 – 1.912 19,679

1.8807 0.5142 0.1000 0.1000 1.2711 0.5151 0.1000 0.1000 0.5317 0.5134 0.1000 0.1000 0.1565 0.5429 0.4081 0.5733 379.632 379.759 – 0.149 21,542

1.9446 0.5026 0.1000 0.1000 1.2676 0.5099 0.1000 0.1000 0.5067 0.5165 0.1075 0.1000 0.1562 0.5402 0.4223 0.5794 379.970 381.560 – 1.803 10,500

1.9028 0.5180 0.1001 0.1003 1.2787 0.5074 0.1003 0.1003 0.5240 0.5150 0.1002 0.1015 0.1564 0.5494 0.4029 0.5504 379.694 379.896 – 0.079 15,600

1.8519 0.5141 0.1000 0.1000 1.2819 0.5091 0.1000 0.1000 0.5312 0.5173 0.1000 0.1000 0.1560 0.5572 0.4259 0.5271 379.77 380.39 – 0.801 18,000

1.8758 0.5160 0.1000 0.1000 1.2993 0.5246 0.1001 0.1000 0.4971 0.5089 0.1000 0.1000 0.1575 0.5329 0.4089 0.5731 379.095 379.534 – 0.272 9092

1.7430 0.5181 0.1000 0.1000 1.3079 0.5190 0.1000 0.1000 0.5140 0.5460 0.1000 0.1090 0.1610 0.5089 0.4970 0.5620 381.91 383.01 386.27 2.081 19,850

1.8577 0.5059 0.1000 0.1000 1.2476 0.5269 0.1000 0.1012 0.5209 0.5172 0.1004 0.1005 0.1565 0.5507 0.3922 0.5922 379.85 380.08 381.01 0.813 19,679

1.8601 0.5209 0.1000 0.1000 1.2710 0.5090 0.1000 0.1000 0.4850 0.5009 0.1000 0.1000 0.1680 0.5840 0.4330 0.5201 380.62 381.49 381.99 0.778 13,742

1.8329 0.5127 0.1000 0.1000 1.2635 0.5228 0.1000 0.1000 0.5273 0.5131 0.1000 0.1000 0.1568 0.5336 0.4110 0.5849 378.37 378.65 378.81 0.382 10,480

Fig. 7. Spatial 72-bar truss problem: comparison of best run convergence curves of ISA and its parent algorithms.

search domain, thus, all agents are highly conducted toward the spotted local minimum. During this period (i.e. from 2800 to 3900 structural analyses) TLBO algorithm, owing to its interactive behavior, becomes superior to iPSO algorithm. As expected, ISA method, via combining both methods, merges their strength points and finds a better design more quickly. 4.2.2. Design of a spatial 72-bar truss The second structural optimization problem solved in this study is the weight minimization of the spatial 72-bar shown in Fig. 6. The

Fig. 8. Schematic of the planar 200-bar truss.

116

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

can be seen that the ISA obtained the best design and was the 2nd fastest optimizer overall. Similar to PSO, TLBO displays an almost linear approach to the optimal point. iPSO is faster than PSO and TLBO but stagnation occurs whenever weighted and global best agents come very close. This occurs in the range 4000 to 5400 and 8000 to 9400 structural analyses. The interactive paradigm enables ISA to reduce cost function monotonically.

Table 12 Results of sensitivity analysis carried out for the 200-bar truss problem. τ0

w

Structural weight (lb) Best Mean

Worst

0.20

0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20 0.60 0.50 0.40 0.30 0.20

25,469.21 25,463.14 25,459.79 25,462.47 25,468.29 25,466.68 25,460.08 25,456.33 25,461.71 25,465.31 25,462.81 25,457.42 25,450.86 25,450.86 25,459.22 25,464.09 25,461.41 25,457.92 25,459.11 25,462.52 25,475.35 25,469.28 25,465.87 25,479.49 25,484.37 25,489.54 25,482.62 25,478.26 25,483.19 25,489.91 25,492.49 25,486.77 25,480.89 25,490.76 25,496.45

25,579.10 25,570.52 25,552.58 25,566.98 25,572.49 25,565.32 25,551.19 25,532.57 25,545.14 25,554.97 25,552.61 25,535.16 25,510.00 25,529.01 25,541.65 25,559.68 25,550.76 25,539.47 25,548.28 25,558.39 25,566.37 25,560.21 25,549.93 25,559.73 25,568.91 25,579.95 25,568.12 25,557.58 25,567.53 25,576.49 25,587.87 25,578.24 25,569.64 25,576.32 25,588.92

0.25

200-bar truss

0.30

0.35

0.40

0.50

0.60

25,496.22 25,492.36 25,487.44 25,493.59 25,498.91 25,493.79 25,489.67 25,484.10 25,490.92 25,495.65 25,490.05 25,487.79 25,475.02 25,479.02 25,488.56 25,496.46 25,491.61 25,486.54 25,494.63 25,501.39 25,502.05 25,598.12 25,592.32 25,500.69 25,507.30 25,509.57 25,501.49 25,496.22 25,506.76 25,511.25 25,513.52 25,508.94 25,502.41 25,509.09 25,519.23

4.2.3. Design of a planar 200-bar truss The third design example solved in this study regards the planar 200-bar truss schematized in Fig. 8. The members of the truss are linked together within 29 independent groups. The material's modulus of elasticity and density are 30 Msi and 0.283 lb/in3, respectively. There is no displacement constraint while both tensile and compressive members should be designed under the stress limitation of ± 10.0 ksi. The minimum cross-sectional area of members should be limited to 0.1 in2. The structure is subjected to three independent load cases: (i) (1, 0, 0) kips acting on nodes 1, 6, 15, 20, 29, 34, 43, 48, 57, 62 and 71, (ii) (0, −10, 0) kips acting on nodes 1–6, 8, 10, 12, 14–20, 22, 24, 26, 28–34, 36, 38, 40, 42–48, 50, 52, 54, 56–62, 64, 66, 68, 70–75, and (iii) previous two load conditions acting together. Table 12 presents the results of sensitivity analysis to find the best setting of w and τ0. Based on these results, values of internal parameters for ISA are set as w = 0.40 and τ0 = 0.30 for this test case. It is remarkable that for both w = 0.40 and w = 0.3 ISA finds results: hence, the value of inertia parameter can be selected from 0.3 ≤ w ≤ 0.40 range. However, based on the observed data it can be concluded that w = 0.40 can be the proper value for this parameter. Population size and number of required iterations for this test problem are 25 and 422, respectively. Table 13 compares the optimization results obtained by ISA and other metaheuristic algorithms. Remarkably, ISA designed the lightest structure and was the most robust optimizer. Standard deviation is only 14.66 lb and the structural weight 25,450.86 lb found by ISA in its best run is very close to the mean weight found over all independent runs. ISA completed the search process within only 10,550 structural analyses. The HPSACO algorithm [45] actually designed a lighter structure than ISA but this design violates stress constraints by almost 10%. Convergence curves of the best optimization run for ISA and its parent algorithms are shown in Fig. 9. It appears that PSO and TLBO show nearly the same convergence behavior, however the TLBO finds a better solution. On the other hand, ISA converges to the global optimum more quickly as it combines the best features properties of both TLBO and iPSO.

material density and modulus of elasticity are equal to those of the first example. Maximum allowable displacement for all nodes in all coordinate directions is restricted to ± 0.25 in. Element stresses must be less than ± 25 ksi. The truss will be designed to resist on two loading conditions: PX = 5 kips, PY = 5 kips, and PZ = −5 kips acting on node 17 for Condition 1 and PX = 0 kips, PY = 0 kips, and PZ = −5 kips acting on nodes 17, 18, 19, and 20 for Condition 2. Since the structure is symmetric about X and Y-axes, elements are grouped into 16 independent sizing variables: (1) A1–A4, (2) A5–A12, (3) A13–A16, (4) A17–A18, (5) A19–A22, (6) A23–A30, (7) A31–A34, (8) A35–A36, (9) A37–A40, (10) A41–A48, (11) A49–A52, (12) A53–A54, (13) A55–A58, (14) A59–A66 (15) A67–A70, and (16) A71–A72. The 16 continuous sizing variables thus defined can vary between 0.1 and 3 in2. Population size and number of required iterations for this test problem are 20 and 524, respectively. Results of sensitivity analysis on the effect of internal parameters for this design example are presented in Table 10. Thirty independent optimization runs were carried out for that purpose. The obtained value τ= 0.35 is different from its counterpart for the previous test problem because of the stochastic nature of the ISA metaheuristic search engine. Table 11 compares the optimization results obtained by ISA with those quoted in literature. The convergence curves relative to the best optimization runs of ISA and its parent algorithms are presented in Fig. 7. It

4.2.4. Design of a spatial 582-bar tower with a newly defined extended element grouping The last structural optimization example is the weight minimization of the spatial 582-bar tower schematized in Fig. 10. This test problem was defined and well established for the first time by Hasançebi et al. [23]. They categorized the size variables into 32 independent groups. This problem variant is solved also in this study and the corresponding results are presented in Table 16. However, in the current study to increase the problem complexity we refine the grouping into 58 independent groups while the symmetry of the tower is still preserved. The loads acting on the tower are: I The vertical load as −6.75 kips on each node II The horizontal load as 1.12 kips on each node in x- direction

117

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 13 Statistical comparison of optimization results for the 200-bar truss problem. Group (in2)

Member

SA [43]

IHS [44]

HPSACO [45]

TLBO [19]

This study PSO iPSO

TLBO

ISA

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16

1–4 5, 8, 11, 14, 17 19–24 18, 25, 56, 63, 94, 101, 132, 139, 170, 177 26, 29, 32, 35, 38 6, 7, 9, 10, 12, 13, 15, 16, 27, 28, 30, 31, 33, 34, 36, 37 39–42 43, 46, 49, 52, 55 57–62 64, 67, 70, 73, 76 44, 45, 47, 48, 50, 51, 53, 54, 65, 66, 68, 69, 71, 72, 74, 75 77–80 81, 84, 87, 90, 93 95–100 102, 105, 108, 111, 114 82, 83, 85, 86, 88, 89, 91, 92, 103, 104, 106, 107, 109, 110, 112, 113 115–118 119, 122, 125, 128, 131 133–138 140, 143, 146, 149, 152 120, 121, 123, 124, 126, 127, 129, 130, 141, 142, 144, 145, 147, 148, 150, 151 153–156 157, 160, 163, 166, 169 171–176 178, 181, 184,187, 190 158, 159, 161, 162, 164, 165, 167, 168, 179, 180, 182, 183, 185, 186, 188, 189 191–194 195, 197, 198, 200 196, 199

0.1468 0.9400 0.10 0.10 1.9400 0.2962 0.10 3.1042 0.10 4.1042 0.4034 0.1912 5.4284 0.10 6.4284 0.5734

0.1540 0.9410 0.10 0.10 1.9420 0.3010 0.10 3.1080 0.10 4.1060 0.4090 0.1910 5.4280 0.10 6.4270 0.5810

0.1033 0.9184 0.1202 0.1009 1.8664 0.2826 0.1 2.9683 0.1 3.9456 0.3742 0.4501 4.96029 1.0738 5.9785 0.78629

0.1460 0.9410 0.10 0.1010 1.9410 0.2960 0.10 3.1210 0.10 4.1730 0.4010 0.1810 5.4230 0.10 6.4220 0.5710

0.75901 0.90320 1.1 0.9952 2.13500 0.41930 1.00410 2.80520 1.03440 3.78420 0.52690 0.43020 5.26830 0.96850 6.04730 0.78250

0.146 0.941 0.1 0.101 1.941 0.296 0.1 3.121 0.1 4.173 0.401 0.181 5.423 0.1 6.422 0.571

0.1500 0.9460 0.1010 0.1 1.9450 0.2960 0.1 3.16101 0.1020 4.1990 0.4010 0.1810 5.4310 0.1 6.4280 0.5710

0.1469 0.9447 0.10 0.10 1.9405 0.2958 0.10 3.1040 0.10 4.104 0.4035 0.1916 5.4279 0.10 6.4279 0.5736

0.1327 7.9717 0.10 8.9717 0.7049

0.1510 7.9730 0.10 8.9740 0.7190

0.73743 7.3809 0.6674 8.3 1.19672

0.1560 7.9580 0.10 8.9580 0.7200

0.5920 8.18580 1.03620 9.20620 1.47740

0.156 7.958 0.1 8.958 0.72

0.1560 7.9610 0.1 8.9590 0.7220

0.1338 7.97326 0.10 8.9723 0.7054

0.4196 10.8636 0.10 11.8606 1.0339

0.4220 10.8920 0.10 11.8870 1.0400

1 10.8262 0.1 11.6976 1.388

0.4780 10.8970 0.10 11.8970 1.0800

1.83360 10.6110 0.98510 12.5090 1.97550

0.478 10.897 0.1 11.897 1.08

0.4910 10.909 0.1010 11.9850 1.0840

0.42076 10.8669 0.10 11.8673 1.0349

6.6818 10.8113 13.8404 25,445.6 – – – 9650

6.6460 10.8040 13.8700 25,491.9 – – – 19,670

4.9523 8.8 14.6645 25,156 – – –

6.4620 10.7990 13.9220 25,488.15 – – – 28,059

4.51490 9.8 14.5310 28,537.8 29,102.3 29,540.7 1523.13 25,980

6.462 10.799 13.922 25,488.2 25,556.3 25,499.5 56.32 12,630

6.4640 10.8020 13.9360 25,542.5 25,889.2 25,601.3 183.07 28,080

6.68437 10.8073 13.8448 25,450.86 25,475.02 25,480.88 14.66 10,550

A17 A18 A19 A20 A21 A22 A23 A24 A25 A26

A27 A28 A29 Best (lb) Mean Worst Standard deviation (lb) NSAs

III The horizontal load as 1.12 kips on each node in y- direction As listed in Table 14, sizing variables (cross-sectional areas of bars) are selected from a discrete set of 140 W-shape profiles. These sections are selected from steel structural profiles of AISC-ASD. The upper and lower bounds of cross-sectional areas are 6.16 in2 (39.74 cm2) and 215.0 in2 (1387.09 cm2), respectively. All nodal displacements should be limited to 3.15 in (8 cm), and since the stress and stability constraints are checked according to AISC-ASD89 stress limits should be restricted as follows: + i

= 0.6Fy i i < 0 i

0 (11)

and i are tensile and compressive stresses, respectively. where Also, σi− is a function of the slenderness ratio as follows: + i

1 i

=

2 i

2Cc2

Fy /

5 3

12 23

+ 2E 2 i

3 i

3 i 8Cc

for

8Cc3 i

for

i

< Cc

Cc

where Cc is the slenderness ratio defined as:

Fig. 9. Planar 200-bar truss problem: comparison of best run convergence curves of ISA and its parent algorithms.

118

(12)

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Fig. 10. Schematic of the spatial 582-bar tower.

119

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 14 Discrete cross-sections available for the sizing variables of the 582-bar tower problem. W-shape profile list taken from AISC code W27 × 178 W27 × 161 W27 × 146 W27 × 114 W27 × 102 W27 × 94 W27 × 84 W24 × 162 W24 × 146 W24 × 131 W24 × 117 W24 × 104 W24 × 94 W24 × 84 W24 × 76 W24 × 68 W24 × 62 W24 × 55 W21 × 147 W21 × 132

W21 × 122 W21 × 111 W21 × 101 W21 × 93 W21 × 83 W21 × 73 W21 × 68 W21 × 62 W21 × 57 W21 × 50 W21 × 44 W18 × 119 W18 × 106 W18 × 97 W18 × 86 W18 × 76 W18 × 71 W18 × 65 W18 × 60 W18 × 55

W18 × 50 W18 × 46 W18 × 40 W18 × 35 W16 × 100 W16 × 89 W16 × 77 W16 × 67 W16 × 57 W16 × 50 W16 × 45 W16 × 40 W16 × 36 W16 × 31 W16 × 26 W14 × 730 W14 × 665 W14 × 605 W14 × 550 W14 × 500

W14 × 455 W14 × 426 W14 × 398 W14 × 370 W14 × 342 W14 × 311 W14 × 283 W14 × 257 W14 × 233 W14 × 211 W14 × 193 W14 × 176 W14 × 159 W14 × 145 W14 × 132 W14 × 120 W14 × 109 W14 × 99 W14 × 90 W14 × 82

stress must not exceed the value of (

Table 15 Results of sensitivity analysis carried out for the 582-bar tower problem. τ0

w

Structural volume (m3) Best Mean

Worst

0.25

0.45 0.40 0.35 0.45 0.40 0.35 0.45 0.40 0.35 0.45 0.40 0.35 0.45 0.40 0.35 0.45 0.40 0.35

21.74 21.14 20.81 20.69 19.97 20.17 21.58 20.87 21.69 22.10 21.26 22.34 22.59 21.88 22.78 22.97 22.41 23.07

23.61 23.19 23.37 23.42 22.27 22.51 23.54 22.62 22.79 23.61 22.94 23.12 23.81 23.15 23.47 23.99 23.80 24.01

0.30 582-bar truss

0.35 0.40 0.50 0.60

Cc =

2 2E Fy

22.86 22.05 22.71 22.43 21.28 22.19 22.90 21.75 22.77 23.26 22.09 23.11 23.37 22.64 23.67 23.49 22.87 23.88

=

k i li ri

(13)

300 for tension members 200 for compression members

W12 × 136 W12 × 120 W12 × 106 W12 × 96 W12 × 87 W12 × 79 W12 × 72 W12 × 65 W12 × 58 W12 × 53 W12 × 50 W12 × 45 W12 × 40 W12 × 35 W12 × 30 W12 × 26 W12 × 22 W10 × 112 W10 × 100 W10 × 88

12 2E ) 23 i2

W10 × 77 W10 × 68 W10 × 60 W10 × 54 W10 × 49 W10 × 45 W10 × 39 W10 × 33 W10 × 30 W10 × 26 W10 × 22 W8 × 67 W8 × 58 W8 × 48 W8 × 40 W8 × 35 W8 × 31 W8 × 28 W8 × 24 W8 × 21

(AISC-ASD, 1989).

Sensitivity analysis on the effect of ISA internal parameters is carried out also for this test problem. However, based on the observation and conclusion attained from the former sensitivity analyses for the solved examples, 18 independent runs for the best setting of w and τ0 are considered (i.e. w = 0.4 and τ0 = 0.3). It should be noted that, for this case, as most complex example test problem, acceptation of w = 0.4 and τ0 = 0.3 gives considerably more accurate solution which proves that neglecting some small oscillations such an assumption for these parameters is admissible. Table 15 shows the corresponding results of the sensitivity analysis in terms of the best, worst and mean weights, and standard deviation. For the conventional structural scheme including 32 groups of elements, population size and number of required iterations are 30 and 194, respectively. For the structural scheme with 58 groups of elements (extended grouping), population size and number of iterations are 60 and 242, respectively. Tables 16 and 17 compare the optimization results of ISA and its basis algorithms. It can be seen from Table 16 that, in the classical 32variable problem, the present algorithm was overall the most efficient optimizer in terms of optimized volume and computational cost. IMBA [48] found the practically the same optimized volume as ISA (20.0688 vs. 20.074 m3), with a smaller standard deviation (0.0201 vs. 0.355 m3), but required about three times more structural analyses than ISA (15,300 vs. 5820). Once again, ISA obtained the best design, showed the smallest standard deviation on optimized volume, and required the lowest number of structural analyses. This makes ISA suited for average scale structural optimization problems including discrete variables. The variations of structural volume recorded for the best optimization runs of all tested algorithms are shown in Fig. 11. Again, ISA has the fastest convergence rate followed by iPSO. Up to 15,060 analyses, TLBO and PSO have almost the same convergence rate but the former algorithm then becomes considerably faster. Interestingly, the convergence history of ISA until 30,180 structural analyses is almost the same as iPSO, but then becomes similar to TLBO. This indicates

According to this code, maximum allowable slenderness ratio should be taken as 200 and 300 for compressive and tensile elements, respectively. In addition, the slenderness constraint should be specified as follows: i

W14 × 74 W14 × 68 W14 × 61 W14 × 53 W14 × 48 W14 × 43 W14 × 38 W14 × 34 W14 × 30 W14 × 26 W14 × 22 W12 × 336 W12 × 305 W12 × 279 W12 × 252 W12 × 230 W12 × 210 W12 × 190 W12 × 170 W12 × 152

(14)

where λi , li and ri are the slenderness ratio, length and radius of gyration of the ith member, respectively. For compression elements, if constraint on the slenderness ratio is not to be satisfied, the allowable

120

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 16 Statistical comparison of optimization results for the 582-bar tower problem with 32 groups. Design variables (cm2)

Optimal cross-sectional areas (cm2) CBO [46] PSO [47]

IMBA [48]

TLBO [49]

iPSO [29]

This study (ISA)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Volume (m3) Standard Deviation (m3) NSAs

W8 × 21(39.74) W12 × 79(149.68) W8 × 28(53.16) W10 × 60(39.74) W8 × 24(45.68) W8 × 21(39.74) W10 × 68(129.03) W8 × 24(45.68) W8 × 21(39.74) W14 × 48(90.96) W12 × 26(49.35) W21 × 62(118.06) W18 × 76(143.87) W12 × 53(100.64) W14 × 61(115.48) W8 × 40(75.48) W10 × 54(101.93) W12 × 26(49.35) W8 × 21(39.74) W14 × 43(81.29) W8 × 24(45.68) W8 × 21(39.74) W10 × 22(41.87) W8 × 24(45.68) W8 × 21(39.74) W8 × 21(39.74) W8 × 24(45.68) W8 × 21(39.74) W8 × 21(39.74) W6 × 25(47.35) W10 × 33(62.64) W8 × 28(53.16) 21.8376 1.67 6400

W8 × 21(39.74) W24 × 76(144.51) W8 × 21(39.74) W12 × 65(48.51) W8 × 21(39.74) W8 × 21(39.74) W10 × 54(101.93) W8 × 21(39.74) W8 × 21(39.74) W12 × 50(37.34) W8 × 21(39.74) W10 × 68(129.03) W24 × 76(144.51) W14 × 53(100.64) W12 × 79(58.93) W8 × 21(39.74) W12 × 65(48.51) W8 × 21(39.74) W8 × 21(39.74) W12 × 45(85.16) W8 × 21(39.74) W8 × 21(39.74) W16 × 26(49.54) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) 20.0688 2.01e-2 15,300

W8 × 21(39.74) W24 × 84(159.35) W8 × 21(39.74) W24 × 62(117.42) W8 × 21(39.74) W8 × 21(39.74) W16 × 57(108.39) W8 × 21(39.74) W8 × 21(39.74) W12 × 53(100.64) W8 × 21(39.74) W10 × 77(145.81) W21 × 83(156.77) W21 × 57(107.74) W18 × 76(143.87) W8 × 21(39.74) W10 × 22(41.87) W18 × 55(104.51) W8 × 21(39.74) W8 × 21(39.74) W14 × 30(57.09) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W10 × 22(41.87) W8 × 21(39.74) W8 × 21(39.74) W8 × 31(58.84) W8 × 21(39.74) W12 × 22(41.81) 20.304 – 15,550

W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W21 × 73(138.71) W12 × 53(100.64) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W18 × 76(143.87) W24 × 62(117.42) W10 × 49(92.90) W10 × 49(92.90) W12 × 79(58.93) W21 × 62(118.06) W14 × 43(81.29) W16 × 26(49.54) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 24(45.68) W8 × 24(45.68) W8 × 21(39.74) W16 × 67(127.10) W8 × 31(58.84) W8 × 24(45.68) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 21(39.74) W8 × 28(53.16) 20.9464 0.993 2360

W8 × 21 (39.74) W18 × 76 (56.64) W8 × 21 (39.74) W12 × 65 (48.51) W8 × 21 (39.74) W8 × 21 (39.74) W14 × 48 (35.81) W8 × 21 (39.74) W8 × 21 (39.74) W12 × 50 (37.34) W8 × 21 (39.74) W16 × 77 (57.40) W12 × 79 (58.93) W8 × 48 (35.81) W12 × 79 (58.93) W8 × 21 (39.74) W16 × 67 (50.04) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 48 (35.81) W8 × 21 (39.74) W8 × 21 (39.74) W12 × 26 (19.43) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) W8 × 21 (39.74) 20.074 0.355 5820

W8 × 21(39.74) W12 × 79(149.68) W8 × 24(45.68) W10 × 60(39.74) W8 × 24(45.68) W8 × 21(39.74) W8 × 48(90.97) W8 × 24(45.68) W8 × 21(39.74) W10 × 45(85.81) W8 × 24(45.68) W10 × 68(129.03) W14 × 74(140.65) W8 × 48(90.97) W18 × 76(143.87) W8 × 31(55.90) W8 × 21(39.74) W16 × 67(127.10) W8 × 24(45.68) W8 × 21(39.74) W8 × 40(75.48) W8 × 24(45.68) W8 × 21(39.74) W10 × 22(41.87) W8 × 24(45.68) W8 × 21(39.74) W8 × 21(39.74) W8 × 24(45.68) W8 × 21(39.74) W8 × 21(39.74) W8 × 24(45.68) W8 × 24(45.68) 22.3958 – 17,500

that, in the early steps, the tracking paradigm is the dominant search strategy for ISA algorithm while afterwards the interaction paradigm dominates the navigation of the colony. In summary, the distinctive features of iPSO and TLBO are combined together in the ISA algorithm.

peer-learning phase of TLBO algorithm is applied to emphasizing the role of agents’ communication on the navigation process by sharing their data through the pairwise interactions. Also, to provide more democratic movements, rather than a random scalar coefficient, the vector of random coefficients corresponding to each design variable is utilized for updating the location of each agent. The performance of proposed ISA method is tested on several mathematical functions and truss structure problems with continuous and discrete variables. The effects of internal parameters of ISA are studied by performing sensitivity analyses for all examples. The diversity analyses are also performed and compared for ISA and its parent algorithms. The optimization results reveal that the proposed ISA significantly outperforms than its parent algorithms (iPSO and TLBO) and is comparable with state-of-the-art metaheuristic methods in terms of computational cost and search ability. ISA is currently being tested in topology and layout optimization problems of skeletal structures with discrete sizing/layout variables.

5. Conclusion In this investigation, a new optimization algorithm called interactive search algorithm (ISA) is developed to solve constrained and non-constrained problems with discrete and continuous variables. This method, indeed, is a modified and hybridized version of iPSO and TLBO methods. Although extensive experiments proved the distinguished search ability of iPSO on different structural optimization problems, the high dependency of this method on the weighted particle to navigate the colony negatively affects its performance. The main reason of this limitation is the inability of iPSO to provide an adequate level of diversity to establish the steady balance between the exploration and exploitation phases. In the current study, to mitigate this drawback, the

121

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

Table 17 Statistical comparison of optimization results for the 582-bar tower problem with 58 groups. Design Variables

Optimal cross-sectional areas (in2) PSO

TLBO

iPSO

ISA

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Volume (m3) Standard Deviation (m3) NSAs

W12 × 22 (6.48) W27 × 102 (30) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 89 (26.2) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 57 (16.8) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 57 (16.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W14 × 90 (26.5) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W27 × 102 (30) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 88 (25.9) W10 × 22 (6.49) W8 × 21 (6.16) W24 × 62 (18.2) W8 × 21 (6.16) W8 × 21 (6.16) W24 × 68 (20.1) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 21.096 5.36 49,500

W12 × 22 (6.48) W10 × 88 (25.9) W8 × 24 (7.08) W12 × 22 (6.48) W16 × 89 (26.2) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W12 × 22 (6.48) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W10 × 68 (20) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W18 × 76 (22.3) W10 × 22 (6.49) W8 × 21 (6.16) W16 × 57 (16.8) W10 × 22 (6.49) W8 × 21 (6.16) W16 × 57 (16.8) W8 × 21 (6.16) W8 × 21 (6.16) W24 × 68 (20.1) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 20.5 0.882 20,000

W8 × 21 (6.16) W24 × 84 (24.7) W8 × 24 (7.08) W8 × 21 (6.16) W14 × 74 (21.8) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W14 × 61 (17.9) W18 × 76 (22.3) W21 × 83 (24.3) W12 × 40 (11.8) W8 × 21 (6.16) W18 × 76 (22.3) W8 × 24 (7.08) W8 × 21 (6.16) W24 × 62 (18.2) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 54 (15.8) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 40 (11.8) W8 × 21 (6.16) W8 × 21 (6.16) W10 × 39 (11.5) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 19.99 0.901 18,500

W8 × 21 (6.16) W24 × 84 (24.7) W8 × 24 (7.08) W8 × 21 (6.16) W14 × 74 (21.8) W8 × 24 (7.08) W8 × 21 (6.16) W18 × 71 (20.8) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 50 (14.7) W8 × 24 (7.08) W8 × 21 (6.16) W12 × 45 (13.2) W8 × 24 (7.08) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W10 × 68 (20) W18 × 76 (22.3) W12 × 65 (19.1) W24 × 68 (20.1) W12 × 40 (11.8) W8 × 21 (6.16) W14 × 61 (17.9) W10 × 22 (6.49) W8 × 21 (6.16) W21 × 57 (16.7) W10 × 22 (6.49) W8 × 21 (6.16) W10 × 54 (15.8) W8 × 21 (6.16) W8 × 21 (6.16) W16 × 36 (10.6) W8 × 21 (6.16) W8 × 21 (6.16) W12 × 30 (8.79) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) W8 × 21 (6.16) 19.97 0.501 14,520

122

Advances in Engineering Software 127 (2019) 106–123

A. Mortazavi et al.

[17] Dede T, Toğan V. A teaching learning based optimization for truss structures with frequency constraints. Struct Eng Mech 2015;53:833–45. [18] Camp CV, Farshchin M. Design of space trusses using modified teaching–learning based optimization. Eng Struct 2014;62–63:87–97. [19] Degertekin SO, Hayalioglu MS. Sizing truss structures using teaching-learningbased optimization. Comput Struct 2013;119:177–88. [20] Rao RV, Savsani VJ, Vakharia DP. Teaching–learning-based optimization: A novel method for constrained mechanical design optimization problems. Comput Aided Design 2011;43:303–15. [21] Zheng Y-J. Water wave optimization: A new nature-inspired metaheuristic. Comput Oper Res 2015;55:1–11. [22] Liang Y-C, Cuevas Juarez JR. A novel metaheuristic for continuous optimization problems: Virus optimization algorithm. Eng Optim 2016;48:73–93. [23] Hasançebi O, Teke T, Pekcan O. A bat-inspired algorithm for structural optimization. Comput Struct 2013;128:77–90. [24] Kaveh A, Ghazaan MI. Meta-heuristic algorithms for optimal design of real-size structures. Springer International Publishing; 2018. [25] Nickabadi A, Ebadzadeh MM, Safabakhsh R. A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput 2011;11:3658–70. [26] Peer ES, Bergh Fvd, Engelbrecht AP. Using neighbourhoods with the guaranteed convergence PSO. Swarm Intelligence Symposium, 2003. SIS '03. Proceedings of the 2003 IEEE. 2003. p. 235–42. [27] Sonmez M. Artificial Bee Colony algorithm for optimization of truss structures. Appl Soft Comput 2011;11:2406–18. [28] Kaveh A, Dadras A, Bakhshpoori T. Improved thermal exchange optimization algorithm for optimal design of skeletal structures. Smart Struct Syst 2018;21:263–78. [29] Mortazavi A, Toğan V. Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer. Struct Multidiscipl Optim 2016;54:715–36. [30] Mortazavi A, Toğan V, Nuhoğlu A. An integrated particle swarm optimizer for optimization of truss structures with discrete variables. Struct Eng Mech 2017;61:359–70. [31] Mortazavi A, Toğan V. Sizing and layout design of truss structures under dynamic and static constraints with an integrated particle swarm optimization algorithm. Appl Soft Comput 2017;51:239–52. [32] Kennedy J, Eberhart R. Particle swarm optimization. Neural Networks, 1995. Proceedings., IEEE International Conference on. 1944. 1995. p. 1942–8. [33] Mortazavi A, Toğan V. Weight minimization of truss structures with sizing and layout variables using integrated particle swarm optimizer. J Civil Eng Manage 2017. [34] P.N. Suganthan, N. Hansen, J.J. Liang, K. Deb, Y. Chen, A. Auger, S. Tiwari. Problem definitions and evaluation criteria for the CEC 2005 special session on realparameter optimization, Technical Report, Nanyang Technological University, Singapore, May 2005 AND KanGAL Report 2005005, IIT Kanpur, India, (2005). [35] Zhang B, Pan Q-k, Gao L, Zhang X-l, Sang H-y, Li J-q. An effective modified migrating birds optimization for hybrid flowshop scheduling problem with lot streaming. Appl Soft Comput 2017;52:14–27. [36] Tang K, Li Z, Luo L, Liu B. Multi-strategy adaptive particle swarm optimization for numerical optimization. Eng Appl Artif Intell 2015;37:9–19. [37] Camp C. Design of space trusses using big bang–big crunch optimization. J Struct Eng 2007;133:999–1008. [38] Kaveh A, Bakhshpoori T, Afshari E. An efficient hybrid Particle Swarm and Swallow Swarm Optimization algorithm. Comput Struct 2014;143:40–59. [39] Kaveh A, Mahdavi VR. Colliding bodies optimization method for optimum design of truss structures with continuous variables. Adv Eng Softw 2014;70:1–12. [40] Bekdaş G, Nigdeli SM, Yang X-S. Sizing optimization of truss structures using flower pollination algorithm. Appl Soft Comput 2015;37:322–31. [41] Kaveh A, Sheikholeslami R, Talatahari S, Keshvari-Ilkhichi M. Chaotic swarming of particles: A new method for size optimization of truss structures. Adv Eng Softw 2014;67:136–47. [42] Kaveh A, Ilchi Ghazaan M. Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Adv Eng Softw 2014;77:66–75. [43] Lamberti L. An efficient simulated annealing algorithm for design optimization of truss structures. Comput Struct 2008;86:1936–53. [44] Degertekin SO. Improved harmony search algorithms for sizing optimization of truss structures. Comput Struct 2012;92–93:229–41. [45] Kaveh A. Advances in metaheuristic algorithms for optimal design of structures. Springer International Publishing; 2017. [46] Kaveh A, Mahdavi VR. Colliding bodies optimization method for optimum discrete design of truss structures. Comput Struct 2014;139:43–53. [47] Hasançebi O, Çarbaş S, Doğan E, Erdal F, Saka MP. Performance evaluation of metaheuristic search techniques in the optimum design of real size pin jointed structures. Comput Struct 2009;87:284–302. [48] Sadollah A, Eskandar H, Bahreininejad A, Kim JH. Water cycle, mine blast and improved mine blast algorithms for discrete sizing optimization of truss structures. Comput Struct 2015;149:1–16. [49] Toğan V, Mortazavi A. Sizing optimization of skeletal structures using teachinglearning based optimization. Int J Optim Control 2017;7:130–41.

Fig. 11. Spatial 582-bar tower problem with extended grouping (58 sizing groups): comparison of best run convergence curves of ISA and its parent algorithms.

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.advengsoft.2018.11.004. References [1] Besset S, Jézéquel L. Optimization of structural dynamic behaviour based on effective modal parameters. Int J Numer Methods Eng 2007;70:523–42. [2] Lee KS, Geem ZW. A new structural optimization method based on the harmony search algorithm. Comput Struct 2004;82:781–98. [3] Arora JS. Introduction to optimum design (Fourth edition). Boston: Academic Press; 2017. [4] Noel MM. A new gradient based particle swarm optimization algorithm for accurate computation of global minimum. Appl Soft Comput 2012;12:353–9. [5] He S, Prempain E, Wu QH. An improved particle swarm optimizer for mechanical design optimization problems. Eng Optim 2004;36:585–605. [6] Li J-P. Truss topology optimization using an improved species-conserving genetic algorithm. Eng Optim 2015;47:107–28. [7] Toğan V, Daloğlu AT. An improved genetic algorithm with initial population strategy and self-adaptive member grouping. Comput Struct 2008;86:1204–18. [8] Toğan V, Daloğlu AT. Optimization of 3d trusses with adaptive approach in genetic algorithms. Eng. Struct 2006;28:1019–27. [9] Mortazavi A, Toğan V. Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer. Struct Multidisciplinary Optim 2016:1–22. [10] Lim WH, Mat Isa NA. An adaptive two-layer particle swarm optimization with elitist learning strategy. Inf Sci 2014;273:49–72. [11] Fan Q, Yan X. Self-adaptive particle swarm optimization with multiple velocity strategies and its application for p-Xylene oxidation reaction process optimization. Chemometrics Intell Lab Syst 2014;139:15–25. [12] Elsayed SM, Sarker RA, Mezura-Montes E. Self-adaptive mix of particle swarm methodologies for constrained optimization. Inf Sci 2014;277:216–33. [13] El-Maleh AH, Sheikh AT, Sait SM. Binary particle swarm optimization (BPSO) based state assignment for area minimization of sequential circuits. Appl Soft Comput 2013;13:4832–40. [14] Deng W, Chen R, He B, Liu Y, Yin L, Guo J. A novel two-stage hybrid swarm intelligence optimization algorithm and application. Soft. Comput 2012;16:1707–22. [15] Camp CV, Bichon BJ. Design of space trusses using ant colony optimization. J Struct Eng 2004;130:741–51. [16] Camp C, Bichon B, Stovall S. Design of steel frames using ant colony optimization. J Struct Eng 2005;131:369–79.

123