Optimal Design of Steel Structures Using Innovative Black Widow Algorithm Hybridized with Greedy Sensitivity-Based Particle Swarm Optimization Technique

1. Introduction

1.1. Motivation

Optimization, as an approach to adopt the best solution, is one of the enhanced most active areas in various engineering design problems [ 1 ]. Computing technologies are the basis for the rapid growth of meta-heuristic algorithms [ 2 - 4 ]. These algorithms perform strategic searches without the need to calculate the implicit gradients of optimization formulations [ 5 ]. They are generally classified into three categories, including methods based on 1. physical and chemical processes, 2. swarm-based, and 3. evolution-based and biological [ 6 ]. Swarm-based algorithms refer to how members of a group interact with each other or with their environment. Particle Swarm Optimization (PSO) [ 7 ] is one of the most popular algorithms in this category. Among the most recent ones in this category are the Mountain Gazelle Optimizer (MGO) [ 8 ], Wild Horse Optimizer (WHO) [ 9 ], Pelican Optimization Algorithm (POA) [ 10 ], Seagull Optimization Algorithm (SOA) [ 11 ], Coyote Optimization Algorithm (COA) [ 12 ], Squirrel Search Algorithm (SSA) [ 13 ], Alpine Skiing Optimization (ASO) [ 14 ] and African Vultures Optimization Algorithm (AVOA) [ 15 ], etc. The answer obtained by these algorithms usually converges reasonably close to the optimal solution. Hence, the optimal solutions obtained by such algorithms are also called quasi-optimal [ 16 ]. This has encouraged researchers to come up with more efficient algorithms for finding optimal global solutions.

1.2. Research gap

A detailed review of the algorithms used in solving civil engineering problems shows that mostly combined and improved versions of the algorithms are used in order to offer more generalised and acceptable global results. For example, Kaveh and Talatahari used an improved Ant Colony Optimizer (IACO) for frame optimization [ 17 ]. An adaptive hybrid evolutionary firefly algorithm has been developed for shape and size optimization of truss structures [ 18 ]. An enhanced colliding bodies optimization method [ 19 ] proposes a mechanism to avoid falling into local optimum. Similarly, An enhanced imperialist competitive algorithm is also proposed to solve the problem of local optimum [ 20 ]. Or in another study, Differential evolution is combined with the eagle strategy [ 21 ]. Differential evolution has also been used in combination with the symbiotic organisms search process As such, one may refer to studies conducted using PSO-WOA [ 22 ], Hybrid Ant Lion Optimizer (ALO) [ 23 ], and Hybrid Tabu search algorithm [ 24 ]. One of the main reasons researchers have developed many different optimization algorithms recently, could be the diversity on problems seeking global optimum solutions. For instance, an optimization algorithm may work well for functions minimization but struggles for highly constrained industrial or multi-objective problems. To solve such issues, improved, enhanced, and hybrid versions of algorithms are used and sometimes lead to the construction of new algorithms with a completely different approach, such as multi-start algorithms. However, it is impossible to name one unique algorithm to elaborate for all optimization problems. However, a challenging passion has always been existed among researchers to outperform one more generalised promissing technique to tackle yet more diversed vaeiety of problems.

1.3. Current research contribution

Swarm-based methods use the three principles of evaluation, comparison, and imitation that exist in collaborative behaviors in nature [ 25 ]. In this study, by making one of the most active crowd intelligence techniques known as PSO, attempts will be made to achieve a technique that can reduce the computational efforts needed to handle problems in hand by introducing a hybridised meta-heuristic algorithm. The standard PSO approach alone often ends up in local optima, due to a premature convergence and therefore a failure to achieve an accurate design solution. But if somehow the search space is split wisely into suboptimal spaces, the problem of getting stuck in the local optimum may be solved automatically. This idea will be the basis of the proposed design of the authors, and from there mostly combined approaches are used to solve engineering design problems. It can be used as an optimization aid for other methods at the beginning of the solution. The theory of the method is explained in detail below.

1.4. Paper organization

First, a literature review for engineering problems will be given. The next section introduces the mathematical models for the benchmark problems that will be solved later in this study. The black widow algorithm will then be described; This algorithm will be used to combine with the proposed technique. A description of the proposed PSO-based technique will then be given. The following section adds the proposed technique to the black widow algorithm to complete the model. In the following section the performance of the hybrid approach will be evaluated by solving some benchmark problems in the field of structural design and compoarisons with data available in the literature will be assessed. The conclusions and recommendations will follow in the final sections and challenges of the proposed technique will be highlighted.

2. Algorithms and engineering problems

Optimization has been widely used in various fields of civil engineering, such as reliability [ 26 , 27 ], dampers [ 28 - 32 ], different types of concrete strength estimation [ 33 , 34 ], earthquake engineering [ 35 , 36 ], etc. This part examines a practical application of optimization in steel structures. For the discrete optimization of steel frames in 1991, Balling used a SA algorithm [ 37 ]. The total weight minimization of a six-story, asymmetrical building was the objective function in that study. In a related effort, Balling's frame was optimized discretely by May and Balling using a filtered SA (FiSA) strategy [ 38 ]. Discrete optimization was carried using the linearized branch and bound (LBandB) strategy. The impact of various neighborhood sizes on the effectiveness of the LB was subjected to a sensitivity analysis. Moreover, a number of case studies were used to examine the impact of various FiSA hyperparameter settings. The wide-range (W) shape sections from AISC were used to create 11 groups of structural elements for columns and girders in both studies.

Pezeshk [ 39 ] proposed an automated method based on the non-linear optimum design of steel frame structures in 2000. The design process adhered to the specifications set forth by AISC-LRFD and the available W-section elements. Fuzzy discrete multicriteria optimization (FDMCO) of steel frames was studied by Sarma and Adeli in 2000 [ 40 ]. In order to achieve this, the objective function was established as total cost minimization given three concurrent design criteria as follows: minimizing cost, minimizing weight and minimizing the number of different section types. To achieve the best performance, four different combinations of the FDMCO's effective parameters were looked at.

A decade later, Kaveh et al. [ 41 ] suggested using a discrete design variable-based performance-based seismic design of steel frames with an ACO algorithm as a solver. Based on the lateral drift, four performance levels were taken into account in the structure's nonlinear analysis (i.e. operational, ready for occupancy right away, life-safety-related, and collapse-prevention). Four earthquakes with probabilities of 50%, 20%, 10%, and 2% over a 50-year period were used to calculate the seismic loadings.

Alberdi and Khandelwal [ 42 , 43 ] conducted a comparison study on the effectiveness of nine algorithms in 2015 for reducing the weight of steel frames. Regardless of the variable space and the initial trials, the effectiveness of the employed algorithms was evaluated in terms of convergence consistency. The best solvers in this case study were DDHS and TS, according to the results of the simulations. A modified dolphin echolocation optimization (MDEO) algorithm was used by Gholizadeh and Poorhoseini [ 44 ] to optimize steel frames. The step locations were chosen for this modified algorithm using one-dimensional Gauss chaotic maps. A comparison of the proposed algorithm's performance with the original dolphin echolocation (DEO) algorithm and some other algorithms that had previously been applied to the same examples was made. For the MDEO algorithm to perform at its peak, a sensitivity analysis of a critical parameter called power was carried out.

In 2019, an investigation was carried by Bybordiani and Kazemzadeh Azad [ 45 ] into the ideal configuration of a steel-braced frame with dynamic soil-structure interaction. A rigid base and a half-space were thought to be the foundations of typical steel frames. The unbounded soil domain was modeled using a typical massless foundation. Based on two sets of ground motions, the seismic time-history analysis was applied to the model. The optimization was handled by the BB-BC algorithm. Using five optimization algorithms PSO, CSS, TLBO, GWO, and improved GWO (IGWO)—Zakian [ 46 ] tackled steel moment-resisting frames while taking into account natural frequency constraints. Eigenvalue analysis was used to determine the structure's natural frequency to achieve this. According to the results, the top solvers were TLBO, IGWO, and PSO. Steel concentrically braced frame (SCBF) collapse-performance-aided optimization was taken into account by Hassanzadeh and Gholizadeh [ 47 ] using the CMO algorithm. Three main steps were suggested to achieve this goal: (1) size and topology optimization based on seismic performance-based analysis; (2) producing fragility curves for the optimal solutions using incremental dynamic analysis; and (3) comparing fixed and optimized braces configurations in terms of minimum weight and collapse capacity. Based on the three levels of risk, the performance-based analysis was carried out: 1. instant occupancy, and 2. life safety; and 3. collapse prevention. (Please send me his work. The cross-section and location of the braces in the frame were designated as the design variables. According to the findings, topology optimization produced a more significant response with a noticeably higher level of collapse safety.

Kaveh [ 48 ] used a variety of optimization algorithms in 2020 to solve steel frame optimization problems, including the ABC, BB-BC, cyclical parthenogenesis algorithm (CPA), CS, thermal exchange optimization (TEO), water evaporation optimization algorithm (WEOA), and TLBO algorithms. WEO, CS, and TEO were the best optimizers for more accurate solution ns, while TEO, TLBO, and WEO had faster convergence rates.

In recent years, many metaheuristic algorithms have been created to address the shortcomings of earlier ones as much as possible. The applications of those algorithms to real-world and benchmark engineering problems have since been the subject of numerous investigations. The complexity of structural engineering-related problems has been found to make them the most difficult of all. The engineering optimization research community has therefore given them a lot of attention. However, a comparative analysis that outlines the salient aspects of the available studies in this field is lacking. This study proposes a hybrid optimization technique, using a developed approach for determining the best designs for steel structural problems.

3. Optimization of steel structures

One of the main purposes of optimizing steel frames is to minimize their weight (W) according to some engineering design constraints. W may be defined as:

$\begin{array}{cc}W={\sum }_{\mathrm{i=1}}^N{\gamma }_i.x_i.l_i& \mathrm{(1)}\end{array}$

Where ${\gamma }_i$ : i^th element density

$x_i$ : i^th member variable

$l_i$ : i^th member length

N : Number of members

According to the American Institute of Steel Construction (AISC) code, any structural design must satisfy the following constraints:

3.1. Resistance limitations

3.1.1. Allowable stress design (ASD) method

Real structures are subjected to different loads that mostly affect the interaction of forces on the members, as a result of which the design must be according to Equations given in the AISC-ASD regulation. It states that:

$\begin{array}{cc}{C}_1=\frac{f_a}{F_a}+\frac{f_{\mathrm{bx}}}{F_{\mathrm{bx}}}+\frac{f_{\mathrm{by}}}{F_{\mathrm{by}}}\le \mathrm{1.0}& \mathrm{(2)}\end{array}$

$\begin{array}{cc}{C}_2=\frac{f_a}{\mathrm{0.6}{F}_y}+\frac{f_{\mathrm{bx}}}{F_{\mathrm{bx}}}+\frac{f_{\mathrm{by}}}{F_{\mathrm{by}}}\le \mathrm{1.0}& \mathrm{(3)}\end{array}$

$\begin{array}{cc}{C}_3=\frac{f_a}{F_a}+\frac{C_{\mathrm{mx}}{f}_{\mathrm{bx}}}{(1-\frac{f_a}{{\mathrm{F´}}_{\mathrm{ex}}})F_{\mathrm{bx}}}+\frac{C_{\mathrm{my}}{f}_{\mathrm{by}}}{(1-\frac{f_a}{{\mathrm{F´}}_{\mathrm{ey}}})F_{\mathrm{by}}}\le \mathrm{1.0}& \mathrm{(4)}\end{array}$

Where F_a : Axial allowable stresses

F_b : Flexural allowable stresses

f_a : Corresponding stresses due to the axial force

f_b : Corresponding stresses due to the bending moment

$\frac{1}{1-\frac{f_a}{{\mathrm{F´}}_e}}$ : Amplification of the inter-span moment (the P-δ effect).

C_m: impact of the gradient's moment

In compression and Euler stress computation, calculation of the effective length (K) is required. Thus,

$\begin{array}{cc}{F}_e=\frac{{\pi }^{2}E}{{\mathrm{KL}/r}^2}& \mathrm{(5)}\end{array}$

Where F_e : Euler stresses

K : Effective length factor that calculated by means of the following equations [ 49 ]:

for braced frames:

$\begin{array}{cc}K=\frac{3G_A{G}_B+\mathrm{1.4}\left(G_A{G}_B\right)+\mathrm{0.64}}{3G_A{G}_B+2\left(G_A{G}_B\right)+\mathrm{1.28}}& \mathrm{(6)}\end{array}$ and for unbraced frames:

$\begin{array}{cc}K=\sqrt{\frac{\mathrm{1.6}{G}_A{G}_B+4\left(G_A{G}_B\right)+\mathrm{7.5}}{G_A{G}_B+\mathrm{7.5}}}& \mathrm{(7)}\end{array}$

Where:

$\begin{array}{cc}G=\frac{\Sigma {\left(\frac{\mathrm{EI}}{L}\right)}_{\mathrm{column}}}{\Sigma {\left(\frac{\mathrm{EI}}{L}\right)}_{\mathrm{Beam}}}& \mathrm{(8)}\end{array}$

In the above equation, GA and GB refer to the coefficient of relative stiffness of the column at both ends.

3.1.2. Load and resistance factor design (LRFD) method

The strength constraints, C4 and C5, were determined as indicated in the AISC-LRFD against both axial and bending forces:

$\begin{array}{cccc}{C}_4=\frac{P_u}{2Q_c{P}_n}+(\frac{M_{\mathrm{ux}}}{Q_b{M}_{\mathrm{nx}}}+\frac{M_{\mathrm{uy}}}{Q_b{M}_{\mathrm{ny}}})\le 1& ;& \frac{P_u}{Q_c{P}_n}<\mathrm{0.2}& \mathrm{(9)}\end{array}$

$\begin{array}{cccc}{C}_5=\frac{P_u}{Q_c{P}_n}+\frac{8}{9}(\frac{M_{\mathrm{ux}}}{Q_b{M}_{\mathrm{nx}}}+\frac{M_{\mathrm{uy}}}{Q_b{M}_{\mathrm{ny}}})\le 1& ;& \frac{P_u}{Q_c{P}_n}\ge \mathrm{0.2}& \mathrm{(10)}\end{array}$

Where P_u: Required axial strength

P_n: Nominal axial strength

M_ux: Required flexural strength (major axis)

M_nx: Nominal flexural strength (major axis)

M_uy: Required flexural strength (minor axis)

M_ny: Nominal flexural strength (minor axis)

φ_c: Resistance factor for compressionφb: Resistance factor for bending

3.2. Displacement limitations

According to the AISC, the maximum lateral displacement is given by:

$\begin{array}{cc}{\nu }^{∆}=R-\frac{{∆}_T}{H}\le 0& \mathrm{(11)}\end{array}$

and inter-story displacements:

$\begin{array}{ccc}{\nu }_j^d=R_t-\frac{d_j}{h_j}\le 0& j=\mathrm{1,2,}\mathrm{...},\mathrm{ns}& \mathrm{(12)}\end{array}$

where R : Maximum drift index

∆_T : Maximum lateral displacement

H: Structure’s elevation

d_j : Drift of inter-story

h_j : elevation of the j^th story

ns : No. of stories

R_t : Permitted drift index (For stories)

4. Review of black widow optimization (BWO) algorithm

The BWO algorithm [ 50 ] is very similar to the Genetic Algorithm (GA) as an evolutionary computational method. The algorithm starts with the initial population of the black widow spider and goes through three other steps, including Procreate, Cannibalism, and Mutation. One of the main advantages of this algorithm is in the Cannibalism stage, where the significantly degraded black widow spiders are eliminated. This may avoid destructive scattered individuals in the process. Fig. 1 indicates the flowchart of the BWO algorithm. The following is a brief description of the steps of the BWO algorithm:

Fig. 1. Flowchart of the BWO Algorithm.

4.1. Initial population

In multivariate optimization problems, a black widow is a design vector containing a set of variables for the problem. Thus, for the design number ides, it is given as shown below:

$\begin{array}{cc}{\mathrm{design}}_{\mathrm{ides}}={\mathrm{Widow}}_{\mathrm{ides}}={\left[x_1,x_2,\mathrm{...},x_{N_{\mathrm{var}}}\right]}_{\mathrm{ides}}& \mathrm{(13)}\end{array}$

And the objective function is defined as follows:

$\begin{array}{cc}{\mathrm{Objective\; Function}}_{\mathrm{ides}}=f{\left(\mathrm{widow}\right)}_{\mathrm{ides}}=f{\left(x_1,x_2,\mathrm{...},x_{N_{\mathrm{var}}}\right)}_{\mathrm{ides}}& \mathrm{(14)}\end{array}$

The Fitness function for minimizing or maximizing a problem may be computed using different constraints handling approaches. In the present study, the following method, based on Adaptive Penalty Formulation was utilized [ 51 ]:

After calculating the objective function of all individuals of the population, also the amount of violation of design constraints and as a result, the amount of normalized fitness of each design of the population, the maximum and minimum of the fitness functions in the population are computed as the sequence followed from Eqn. 15 until it leads to Eqn. 21:

$\begin{array}{cc}\stackrel{\sim }{f}\left(x\right)=\frac{f\left(x\right)-f_{\min }}{f_{\max }-f_{\min }}& \mathrm{(15)}\end{array}$

The rate of normalized violation for each individual of the population is equal to:

$\begin{array}{cc}c\left(x\right)=\frac{1}{m}{\sum }_{\mathrm{j=1}}^m\frac{v_j\left(x\right)}{v_{\mathrm{max,j}}}& \mathrm{(16)}\end{array}$

v_j(x) is the maximum value of the j^th constraint and v_max,j is the maximum value of the j^th constraint violation in the population. The modified fitness value is then formulated as follows:

$\begin{array}{cc}n\left(x\right)=\{\begin{array}{cc}c\left(x\right)& \mathrm{if}{r}_f=0\\ \sqrt{\stackrel{\sim }{f}{\left(x\right)}^2+c{\left(x\right)}^2}& \mathrm{otherwise}\end{array}& \mathrm{(17)}\end{array}$

Where r_f is equal to the number of feasible solutions per total population.

To account for the violation of constraints in a design, however small they may be, a penalization term as given in Eqn. 18 is used:

$\begin{array}{cc}p\left(x\right)=(1-r_f)\times X\left(x\right)+r_f\times Y\left(x\right)& \mathrm{(18)}\end{array}$

Where X(x) and Y(x) maybe defined as in Eqns. 19 and 20, respectively.

$\begin{array}{cc}X\left(x\right)=\{\begin{array}{cc}0& \mathrm{if}{r}_f=0\\ c\left(x\right)& \mathrm{otherwise}\end{array}& \mathrm{(19)}\end{array}$

$\begin{array}{cc}Y\left(x\right)=\{\begin{array}{cc}0& \mathrm{for\; feasible\; solutions}\\ \stackrel{\sim }{f}\left(x\right)& \mathrm{for\; infeasible\; solutions}\end{array}& \mathrm{(20)}\end{array}$

Finally, the value of the fitness function for each member of the population is obtained from the following equation:

$\begin{array}{cc}F\left(x\right)=n\left(x\right)+p\left(x\right)& \mathrm{(21)}\end{array}$

To start the optimization process then, a N_pop × N_var matrix is formed. The parents are then randomly selected to perform the reproductive stage of the mating operation, after which the female will eat the male black widow.

4.2. Procreate

The act of fertilization between male and female and childbearing is called reproduction. The phenomenon of reproduction in living organisms is done in several ways. In some animals or insects, a male mates with several females to produce offspring. In the original BWO algorithm, males and females were paired each time they reproduced. In this article, an improved version of this operator is applied. Some males and females are mated in the new operator at each stage. This practice simulates polygamy among black widow spiders. As it produces more children in each step, this action makes the search space more and better explored at this stage. Generally, the search area studied in this operator has increased in an improved way. The primary operator of this method is described following. In this algorithm for multiplication, an 1 × N_var array is produced, known as alpha which contains random values between zero and one. They are used to generate children y₁ and y₂ from parents represented by x₁ and x₂ as shown in Eqn. 22:

$\begin{array}{cc}\{\begin{array}{c}{y}_1=\propto \times x_1+(1-\propto )\times x_2\\ y_2=\propto \times x_2+(1-\propto )\times x_1\end{array}& \mathrm{(22)}\end{array}$

4.3. Cannibalism

Three types of cannibalism have been observed in the behavior of this type of spider; The first is sexual cannibalism, in which a female black widow eats a male she has mated after mating. Another type of cannibalism is sibling eating, in which strong spiders eat their weak siblings, and the third type of cannibalism is often seen when a mother spider eats her baby. In this algorithm, the rate of cannibalism (CR) is determined based on the number of survivors, an accomplishment related to the values of the Fitness functions as a result of which the weak spiders are spotted and eaten.

4.4. Mutation

At this point, after randomly selecting Mutepop (based on mutation rate) spiders from the total population, each of the chosen widows randomly exchanges three elements in their array. Table 1 shows the pseudo-code of the BWO.

5. Greedy sensitivity-based PSO (GSPSO) technique

The GSPSO algorithm generally consists of two parts: In the first part, the potential region is identified using a method derived from the Greedy search approach. Then, using an approach inspired by the univariate technique [ 52 ], the variables of the best design change values stepwise with normalized perturbation values until the change of variables causes violation of constraints or somewhat greater than a step before. In the second part, the search is pursued using the PSO search engine in a more confined range for each variable according to the findings of the previous step. More details of these steps may be summarized below:

5.1. Greedy partitioning the search space

In a constrained minimization problem say, an effective way to evaluate a reasonable restrained search space containing the optimum design is to allocate a certain boundary to the problem initially. Then, using a greedy-wise search approach, the search area is divided into a number of sub-parts. To the number of sub-parts defined, designs will be generated whose design variables for each design contain the mean value in that sub-part, followed by the fitness function evaluation, accordingly. With regard to the fitness functions determined, the fittest design will be selected to carry sensitivity analysis upon its design variables, as a result of which a new confined boundary for each variable will be obtained. The search space for all problems attempted in this research is divided into three sub-parts.

5.2. Relaxing variables to desired values

According to the above description, to set the best values for the variables primarily at this stage, the first two directions of motion are created for each variable, one to reduce the arbitrary value and the other to increase it. In each direction, the objective function and the constraint violation are calculated, and according to Eq. 21, the fitness function and the direction in which the fitness value is improved will be determined. Preferably the value by which the constraint violation is zero will be selected.

5.3. Moving variables in the endorsed direction

After determining the optimal path for all variables, their values change to improve the performance of the target. Then the univariate method is activated. Thus, the first variable changes its value while the other variables are kept constant, and the fitness functions are calculated. Until the fitness function's value is improved, the changes are sustained; otherwise, the corresponding variable will relax to its last value. This procedure is repeated for all variables until an improved performance at this stage is found. Also, during this process, if the value of a variable reaches the neighborhood of either bound and shows a further tendency to change more, it is allowed to enter the neighboring region until no improvement occurs.

5.4. PSO algorithm takes action

At this stage, the PSO algorithm comes into play, where, according to Fig. 2, by centralizing the value found for each variable a predefined n-design randomly set of values are produced within a 20% margin to its either side. This is repeated for all variables in the first iteration. The margin P narrows itself through iterations based on P = 20.5 - 0.5 (current_iteration).

Fig. 2. Flowchart of the GSPSO Algorithm.

6. A hybrid black WIDOW and GSPSO algorithm (BGP)

The BGP algorithm is a hybrid optimization technique that combines the BWO evolutionary algorithm with the GSPSO swarm intelligence algorithm. The GSPSO operator has been used to boost the BWO. The output of this combination shows that the presence of congestion intelligence not only reduces the number of analyses but also, to some extent, improves the final response. For problems with continuous variables where many variables are involved or with large catalog lists to select values for variables from, the proposed algorithm may be effective because initially, using the greed search step, the search area is reduced and a proper range of changes for each variable is determined to some extent. Then, using BWO, by removing the Sub-optimal solutions, a set of most probable solutions enters GSPSO. There, a more accurate exploration of the search space for each variable yields a more accurate range of changes. The end of this operation is left to the powerful engine of the PSO algorithm to finally get the best solution that should perform the global optimum solution. The above explanations are summarized in the flowchart of the BGP algorithm, illustrated in Fig. 3.

Fig. 3. Flowchart of the BWO-GSPSO (BGP).

7. Examples - results and discussions

7.1. Constrained mathematical function

7.1.1. Keane’s bumpy function (KBF)

The mathematical function selected here to evaluate the proposed algorithm's applicability is one of the most challenging functions in the literature due to its highly bumpy manner. Keane’s global optimization problem is a constrained multimodal minimization problem defined as:

$\begin{array}{cc}\mathrm{Minimize}f\left(x\right)=-\left|\right\{{\sum }_{\mathrm{i=1}}^m{\cos }^4\left(x_i\right)-2{\prod }_i^m{\cos }^2\left(x_i\right)\}/{\left({\sum }_{\mathrm{i=1}}^{m}i{x}_i^2\right)}^{\mathrm{0.5}}|& \mathrm{(23)}\end{array}$

subject to:

$\begin{array}{cc}{g}_1\left(x\right)=\mathrm{0.75}-{\prod }_{\mathrm{i=1}}^m{x}_i<0,g_2\left(x\right)={\sum }_{\mathrm{i=1}}^m{x}_i-\mathrm{7.5}m<0;x_i<\mathrm{10}& \mathrm{(24)}\end{array}$

The number of variables for this problem is set to 50. Each method was run 30 times, and the maximum number of iterations was set to 500 as a termination criterion for consistency in all cases. The results are documented in Table 2. The "Best" indicates the best fitness value found throughout the runs, and “Mean" and "Median" are computed according to the following:

“$\mathrm{Mean}=\frac{\mathrm{SUM\; of\; the\; terms}}{\mathrm{number\; of\; terms}}$” and “$\mathrm{Median}=\mathrm{middle\; of\; the\; set\; of\; numbers}$”

They are related to 30 times running of the optimization procedure.

Fig. 4. Convergence history of the Bumpy function.

As illustrated in Fig. 4, the BGP algorithm performed the best optimal solution after 30 iterations, where a more accurate outcome was accomplished compared to the other three algorithms.

7.2. Design examples

Firstly, it is to be noted that in solving the engineering design problems through the BGP algorithm proposed here, discrete variables were used, the catalog list details of which are given distinctly for each problem attempted. The examples studied utilize two types of cross-sectional areas; W sections are used for frames and pipes for bridges. The sections are sorted from small to large to ease the optimization procedure. It should be noted that the maximum number of analyzes, for the first three attempted structural problems with 10, 15, 24 number of story frames, is set to 20,000.

7.2.1. One-Bay ten-story frame

Fig. 5 shows the shapes and loads on a one-span, ten-story frame structure. This frame is one of the standard design problems. The catalog list of the beam elements uses all the W sections specified in the AISC code without restriction, but the column catalog list is limited to groups W12 and W14 only. This frame is designed according to the AISC-LRFD code, and the permissible limit for drift floors is smaller than the floor height, divided by 300. The modulus of elasticity (E) is 200 GPa, and the yield stress (f_y) is 248.2 MPa.

According to Fig. 6 and Table 3, the proposed BGP algorithm resulted in more satisfactory results than the BWO algorithm by exhibiting a faster convergence and a more comprehensive optimum weight for the structure.The number of analyzes required by BGP is about 75% less than that of BWO, and the weight has been reduced by about 4%. In the following, some more benchmark structural problems are examined to more firmly evaluate the potential of the proposed algorithm.

Fig. 5. Visual view of the one-bay ten-story structure.

Fig. 6. Convergence history of the 1-bay 10-story frame.

7.2.2. Three-Bay fifteen-story frame

A 3-span, 15-story frame structure with 105 members, including the loading, is illustrated in Fig. 7. Yield stress and the modulus of elasticity equal 248.2 MPa and 200 GPa, respectively. There are 1 and 10 group elements as the total design variables for the beams and columns, respectively. The catalog list of group elements for both beams and columns is selected from all the profiles of the AISC W-section list. This frame is also designed according to the AISC-LRFD code.

Fig. 7. Visual view of three-bay fifteen-story structure.

As shown in Fig. 8 and the results recorded in Table 4, the optimum design of the frame is obtained after 4,400 analyses and a minimum weight of 400.26 KN using BGP. The optimum design found using BWO is 445.22 KN with 18500 frame analyses. The BGP algorithm reduced the total number of analyzes by 76.22% and the weight by 10.1%.

Fig. 8. Convergence history of the 3-bay 15-story frame.

7.2.3. Three-Bay twenty four-story frame

The third example is a 24-story 3-span frame problem with 168 elements, including 72 beams and 96 columns. This frame has been studied by many researchers [ 57 ]-[ 59 ]. The loading and the grouping of the elements are shown in Fig. 9. Thus, the total number of design variables is 20, four of which are for the beams and sixteen for the columns. f_y is equal to 230.3 MPa, and the modulus of elasticity is equal to 205 GPa. The frame is aimed for the optimum design according to the AISC-LRFD code. The maximum inter-story drift index is limited to 1⁄300, and the allowable displacement of the last floor is limited to 292.61 mm.

Fig. 9. Visual view of the three-bay twenty four-story structure.

According to Fig. 10 and Table 5, the weight of 897.09 KN is obtained for the mentioned frame after 7100 analyses using BGP. For BWO, weight of the structure has obtained 1089.94 KN after 15500 analyses. The BGP algorithm successfully reduced the number of analyses by 54.19% and the weight by 17.69%.

Fig. 10. Convergence history of the 3-bay 24-story frame.

7.2.4. Two-Bay two-story irregular steel space frame

The Two-Bay Two-Story Irregular Steel Space Frame (Fig. 11) has twenty-one members, two groups of beams, and three groups of columns arranged as variables for it. The drift ratio limits are defined as 4 cm for top-story and 1 cm for inter-story, where H=4 m is the elevation of each story. The maximum beams deflection is restricted to 13.9 mm. This Space Frame is designed according to the AISC-LRFD code. The number of analyzes for this three-dimensional structure is set at 10,000. The results of Table 6 and Fig. 12 show that BGP has also shown acceptable performance in solving this problem.

Fig. 11. 2-story 21-member irregular space frame.

Fig. 12. Convergence history of the 2-story 21-member irregular space frame.

7.2.5. Real footbridge

In this example, a real footbridge is subjected to weight optimization to evaluate the proposed approach. SAP2000v22 software has been used for design based on AISC/ASD 360-16 code in the optimization process. To connect MATLAB to the SAP2000v22, a toolbox developed for MATLAB under the name SM Toolbox [ 63 ] was used.

Fig. 13 shows the dimensions of the footbridge. The constructed footbridge weight is 112.78 KN pursuant to the available drawings.

Fig. 13. Visual view of the Footbridge.

Table 7 indicates the grouping of the footbridge members and their color:

Fig. 14 illustrates parts for each group of the footbridge.

Fig. 14. Show groups according to the colors listed.

Table 8 lists the pipe sections used for optimization. The f_y and elasticity’s modulus (E) is set to 2400 MPa and 200 GPa, respectively.

7.2.5.1. Loads applied to the footbridge

Dead & live loads and wind loads as equivalent nodal loads and the direction of their application and coefficients related to earthquake loads are shown in Table 9.

Table 10 and Fig. 15 show the optimal solutions found using GA, PSO, BWO and the combined BGP algorithms.

Fig. 15. Convergence history of the footbridge.

As one could also assess in the latest problems, the proposed BGP algorithm has a high performance in finding the global optimum solution with a lower number of analyses compared to other algorithms available in Table 10.

7.3. Discussions of optimization results

The convergence histories obtained by the BGP method for the first three design examples, including building frames (Figs. 6, 8, and 10), show that the proposed technique started its work from lower values at the beginning of the process. One of the goals of the GSPSO was to provide a desirable and processed initial population for the rapid advancement of optimization operations, and the results indicate the achievement of this initial goal. Providing a good initial population to achieve the global optimum is also very promising. For example, the convergence history of the 15-story frame exhibits a significant difference in convergence speed without causing premature convergence. Therefore, the number of analyzes has been greatly reduced. These results are not only compared to the original algorithm (BWO) but are generally less than what is reported in the references. The evaluation results of the method, both on design examples and mathematical examples, indicate the reliability of the proposed BGP method. BWO algorithm was chosen as an algorithm that was not used before to evaluate design examples in the literature. Any other meta-heuristic algorithm can be combined with the GSPSO technique. Therefore, according to the obtained results, it is strongly recommended as a proposed option for researchers who intend to increase the efficiency of a meta-heuristic. Especially in problems where the global optimum cannot be guaranteed, the present technique can provide the minimum optimal weight for the design of structures with standard sections with a very good difference.

8. Conclusions

This study introduces an intelligent search engine called Greedy Sensitivity-based PSO (GSPSO). This method partitions the search space in an organized way. The way this technique works is that by specifying the sensitivity of each variable in the created subsection, it starts changing the variables and searching the space of the solution space. This operation continues until an optimal subpart is determined for each variable, then this information is given to an optimization algorithm. In this literature, the well-known PSO method is used. The role of PSO in the second stage is to find a population of optimal designs according to the input information from the previous step. Next, when GSPSO introduces an optimal population, these data are given to the Black Widow Optimization (BWO) algorithm to perform the final optimization operation. The newly introduced BWO algorithm has a high ability to find the optimal solution by detecting the suboptimal areas of the search space and removing them. Due to the use of PSO and BWO algorithms, the name of the proposed method is BWO hybridized with Greedy Sensitivity-based PSO (BGP) technique.

To evaluate the performance of the BGP algorithm, weight minimization was performed on three two-dimensional steel frames and two three-dimensional structures. The results indicate that the proposed method has significantly reduced the number of analyzes in finding the optimal solution. Also, referring to the optimal solution obtained from BGP compared to other combined methods, it can be acknowledged that the proposed method has high accuracy in finding the optimal global solution. This way, the weight obtained in engineering design problems has improved significantly compared to other methods.

Funding

This research received no external funding.

Conflicts of interest

The authors declare no conflict of interest.

Appendix

Table A 1 deals with the abbreviations mentioned in this article.

References

1. Sinha GR. Modern Optimization Methods for Science, Engineering and Technology. IOP Publishing; 2019. http://doi.org/10.1088/978-0-7503-2404-5.Publisher Full Text
2. Ahmad S, Mehfuz S, Mebarek-Oudina F, Beg J. RSM analysis based cloud access security broker: a systematic literature review. Cluster Comput. 2022; 25:3733-63. http://doi.org/10.1007/s10586-022-03598-z. Publisher Full Text
3. Farhan M, Omar Z, Mebarek-Oudina F, Raza J, Shah Z, Choudhari R V, et al. Implementation of the One-Step One-Hybrid Block Method on the Nonlinear Equation of a Circular Sector Oscillator. Comput Math Model. 2020; 31:116-32. http://doi.org/10.1007/s10598-020-09480-0. Publisher Full Text
4. Nyo MT, Mebarek-Oudina F, Hlaing SS, Khan NA. Otsu’s thresholding technique for MRI image brain tumor segmentation. Multimed Tools Appl. 2022; 81:43837-49. http://doi.org/10.1007/s11042-022-13215-1. Publisher Full Text
5. Van TH, Tangaramvong S, Limkatanyu S, Xuan HN. Two-phase ESO and comprehensive learning PSO method for structural optimization with discrete steel sections. Adv Eng Softw. 2022; 167:103102. Publisher Full Text
6. Ghasemi MR, Ghasri M, Salarnia A. Soccer league optimization-based championship algorithm (SLOCA): A fast novel meta-heuristic technique for optimization problems. Adv Comput Des. 2022; 7:297-319. http://doi.org/10.12989/acd.2022.7.4.297. Publisher Full Text
7. Slowik A. Particle Swarm Optimization. Ind. Electron. Handb. - Five Vol. Set, vol. 4, 2011, p. 1942-8.http://doi.org/10.1007/978-3-319-46173-1_2.Publisher Full Text
8. Abdollahzadeh B, Gharehchopogh FS, Khodadadi N, Mirjalili S. Mountain Gazelle Optimizer: A new Nature-inspired Metaheuristic Algorithm for Global Optimization Problems. Adv Eng Softw. 2022; 174:103282. Publisher Full Text
9. Naruei I, Keynia F. Wild horse optimizer: A new meta-heuristic algorithm for solving engineering optimization problems. Eng Comput. 2022; 38:3025-56.
10. Trojovský P, Dehghani M. Pelican Optimization Algorithm: A Novel Nature-Inspired Algorithm for Engineering Applications. Sensors. 2022; 22:855. Publisher Full Text
11. Dhiman G, Kumar V. Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems. Knowledge-Based Syst. 2019; 165:169-96. http://doi.org/10.1016/j.knosys.2018.11.024. Publisher Full Text
12. Pierezan J, Dos Santos Coelho L. Coyote Optimization Algorithm: A New Metaheuristic for Global Optimization Problems. 2018 IEEE Congr. Evol. Comput. CEC 2018 - Proc., 2018, p. 1-8. http://doi.org/10.1109/CEC.2018.8477769.Publisher Full Text
13. Jain M, Singh V, Rani A. A novel nature-inspired algorithm for optimization: Squirrel search algorithm. Swarm Evol Comput. 2019; 44:148-75. http://doi.org/10.1016/j.swevo.2018.02.013. Publisher Full Text
14. Yuan Y, Ren J, Wang S, Wang Z, Mu X, Zhao W. Alpine skiing optimization: A new bio-inspired optimization algorithm. Adv Eng Softw. 2022; 170:103158. Publisher Full Text
15. Abdollahzadeh B, Gharehchopogh FS, Mirjalili S. African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems. Comput Ind Eng. 2021; 158:107408. Publisher Full Text
16. Iba K. Reactive Power Optimization by Genetic Algorithm. IEEE Trans Power Syst. 1994; 9:685-92. http://doi.org/10.1109/59.317674. Publisher Full Text
17. Kaveh A, Talatahari S. An improved ant colony optimization for the design of planar steel frames. Eng Struct. 2010; 32:864-73. http://doi.org/10.1016/j.engstruct.2009.12.012. Publisher Full Text
18. Lieu QX, Do DTT, Lee J. An adaptive hybrid evolutionary firefly algorithm for shape and size optimization of truss structures with frequency constraints. Comput Struct. 2018; 195:99-112. http://doi.org/10.1016/j.compstruc.2017.06.016. Publisher Full Text
19. Kaveh A, Ilchi Ghazaan M. Enhanced colliding bodies optimization for design problems with continuous and discrete variables. Adv Eng Softw. 2014; 77:66-75. http://doi.org/10.1016/j.advengsoft.2014.08.003. Publisher Full Text
20. Maheri MR, Talezadeh M. An Enhanced Imperialist Competitive Algorithm for optimum design of skeletal structures. Swarm Evol Comput. 2018; 40:24-36. http://doi.org/10.1016/j.swevo.2017.12.001. Publisher Full Text
21. Talatahari S, Gandomi AH, Yang XS, Deb S. Optimum design of frame structures using the Eagle Strategy with Differential Evolution. Eng Struct. 2015; 91:16-25. http://doi.org/10.1016/j.engstruct.2015.02.026. Publisher Full Text
22. Mashayekhi MR, Shirpour A, Sadeghi R. Finding Optimum Parameters of Passive Tuned Mass Damper by PSO, WOA, and Hybrid PSO-WOA (HPW) Algorithms. J Soft Comput Civ Eng. 2023; 7:72-92. http://doi.org/10.22115/scce.2023.352340.1489. Publisher Full Text
23. Vu Hong Son P, Soulisa FV. A Hybrid Ant Lion Optimizer (ALO) Algorithm for Construction Site Layout Optimization. J Soft Comput Civ Eng. 2023; 7:50-71. http://doi.org/10.22115/scce.2023.365303.1540. Publisher Full Text
24. Tayfur B, Yilmaz H, Daloğlu AT. Hybrid tabu search algorithm for weight optimization of planar steel frames. Eng Optim. 2021; 53:1369-83. http://doi.org/10.1080/0305215X.2020.1793977. Publisher Full Text
25. Gholizadeh S, Milany A. An improved fireworks algorithm for discrete sizing optimization of steel skeletal structures. Eng Optim. 2018; 50:1829-49. http://doi.org/10.1080/0305215X.2017.1417402. Publisher Full Text
26. Hosseinaei S, Ghasemi MR, Etedali S, Chan THT. Reliability-based optimal control design for seismic-excited structures: A hybrid IS-MTLBO pseudo-double loop method. Structures. 2022; 44:1204-18. http://doi.org/10.1016/j.istruc.2022.07.040. Publisher Full Text
27. Rezaeemanesh M, Ghasemi SH, Rezaeemanesh M. Dual target optimization of two dimensional truss using cost efficiency and structural reliability sufficiency. J Soft Comput Civ Eng. 2020; 4:98-111. http://doi.org/10.22115/SCCE.2020.244833.1252. Publisher Full Text
28. Kamgar R, Samea P, Khatibinia M. Optimizing parameters of tuned mass damper subjected to critical earthquake. Struct Des Tall Spec Build. 2018; 27:e1460. Publisher Full Text
29. Salimi M, Kamgar R, Heidarzadeh H. An evaluation of the advantages of friction TMD over conventional TMD. Innov Infrastruct Solut. 2021; 6:1-12. http://doi.org/10.1007/s41062-021-00473-5. Publisher Full Text
30. Khatibinia M, Gholami H, Kamgar R. Optimal design of tuned mass dampers subjected to continuous stationary critical excitation. Int J Dyn Control. 2018; 6:1094-104. http://doi.org/10.1007/s40435-017-0386-7. Publisher Full Text
31. Kamgar R, Gholami F, Zarif Sanayei HR, Heidarzadeh H. Modified Tuned Liquid Dampers for Seismic Protection of Buildings Considering Soil-Structure Interaction Effects. Iran J Sci Technol - Trans Civ Eng. 2020; 44:339-54. http://doi.org/10.1007/s40996-019-00302-x. Publisher Full Text
32. Dadkhah M, Kamgar R, Heidarzadeh H, Jakubczyk-Galczyńska A, Jankowski R. Improvement of performance level of steel moment-resisting frames using tuned mass damper system. Appl Sci. 2020; 10:3403. Publisher Full Text
33. Wadhawan S, Bassi A, Singh R, Patel M. Prediction of Compressive Strength for Fly Ash-Based Concrete: Critical Comparison of Machine Learning Algorithms. J Soft Comput Civ Eng. 2023; 7:68-110. http://doi.org/10.22115/scce.2023.353183.1493. Publisher Full Text
34. Ghasemi MR, Ghasri M, Salarnia AH. ANFIS-TLBO Hybrid Approach to Predict Compressive Strength of Rectangular FRP Columns. Int J Optim Civ Eng. 2022; 12:399-410.
35. Ghasemi SH, Bahrami H, Akbari M. Classification of Seismic Vulnerability Based on Machine Learning Techniques for RC Frames. J Soft Comput Civ Eng. 2020; 4:13-21. http://doi.org/10.22115/scce.2020.223322.1186. Publisher Full Text
36. Hosseinaei S, Ghasemi MR, Etedali S, Chan THT. Sensitivity and Reliability Analyses in to Actively Controlled Structures Under Earthquake. Int J Struct Stab Dyn. 2022; 22:2250124. Publisher Full Text
37. Balling RJ. Optimal Steel Frame Design by Simulated Annealing. J Struct Eng. 1991; 117:1780-95. http://doi.org/10.1061/(asce)0733-9445(1991)117:6(1780). Publisher Full Text
38. May SA, Balling RJ. A filtered simulated annealing strategy for discrete optimization of 3D steel frameworks. Struct Optim. 1992; 4:142-8. http://doi.org/10.1007/BF01742735. Publisher Full Text
39. Pezeshk S, Camp C V, Chen D. Design of Nonlinear Framed Structures Using Genetic Optimization. J Struct Eng. 2000; 126:382-8. http://doi.org/10.1061/(ASCE)0733-9445(2000)126:3(382). Publisher Full Text
40. Sarma KC, Adeli H. Fuzzy Discrete Multicriteria Cost Optimization of Steel Structures. J Struct Eng. 2000; 126:1339-47. http://doi.org/10.1061/(asce)0733-9445(2000)126:11(1339). Publisher Full Text
41. Kaveh A, Farahmand Azar B, Hadidi A, Rezazadeh Sorochi F, Talatahari S. Performance-based seismic design of steel frames using ant colony optimization. J Constr Steel Res. 2010; 66:566-74. http://doi.org/10.1016/j.jcsr.2009.11.006. Publisher Full Text
42. Alberdi R, Khandelwal K. Comparison of robustness of metaheuristic algorithms for steel frame optimization. Eng Struct. 2015; 102:40-60. http://doi.org/10.1016/j.engstruct.2015.08.012. Publisher Full Text
43. Alberdi R, Murren P, Khandelwal K. Connection topology optimization of steel moment frames using metaheuristic algorithms. Eng Struct. 2015; 100:276-92. http://doi.org/10.1016/j.engstruct.2015.06.014. Publisher Full Text
44. Gholizadeh S, Poorhoseini H. Optimum design of steel frame structures by a modified Dolphin echolocation algorithm. Struct Eng Mech. 2015; 55:535-54. http://doi.org/10.12989/sem.2015.55.3.535. Publisher Full Text
45. Bybordiani M, Kazemzadeh Azad S. Optimum design of steel braced frames considering dynamic soil-structure interaction. Struct Multidiscip Optim. 2019; 60:1123-37. http://doi.org/10.1007/s00158-019-02260-4. Publisher Full Text
46. Zakian P. Meta-heuristic design optimization of steel moment resisting frames subjected to natural frequency constraints. Adv Eng Softw. 2019; 135:102686. Publisher Full Text
47. Hassanzadeh A, Gholizadeh S. Collapse-performance-aided design optimization of steel concentrically braced frames. Eng Struct. 2019; 197:109411. Publisher Full Text
48. Kaveh A, Biabani Hamedani K, Milad Hosseini S, Bakhshpoori T. Optimal design of planar steel frame structures utilizing meta-heuristic optimization algorithms. Structures. 2020; 25:335-46. http://doi.org/10.1016/j.istruc.2020.03.032. Publisher Full Text
49. Dumonteil P. Simple Equations for Effective Length Factors. AISC Eng J. 1992; 29:111-5.
50. Hayyolalam V, Pourhaji Kazem AA. Black Widow Optimization Algorithm: A novel meta-heuristic approach for solving engineering optimization problems. Eng Appl Artif Intell. 2020; 87:103249. Publisher Full Text
51. Tessema B, Yen GG. An adaptive penalty formulation for constrained evolutionary optimization. IEEE Trans Syst Man, Cybern Part ASystems Humans. 2009; 39:565-78. http://doi.org/10.1109/TSMCA.2009.2013333. Publisher Full Text
52. Khajeh A, Ghasemi MR, Arab HG, Khajeh A. Hybrid Particle Swarm Optimization, Grid Search Method and Univariate Method To Optimally Design Steel Frame Structures. Int J Optim Civ Eng. 2017; 7:171-89.
53. Geethaikrishnan C, Mujumdar PM, Sudhakar K, Adimurthy V. a robust and efficient hybrid algorithm for global optimization. 2009 IEEE Int. Adv. Comput. Conf. IACC 2009, 2009, p. 486-91. http://doi.org/10.1109/IADCC.2009.4809059.Publisher Full Text
54. Gupta N, Khosravy M, Mahela OP, Patel N. Plant Biology-Inspired Genetic Algorithm: Superior Efficiency to Firefly Optimizer. Appl. Firefly Algorithm its Var. Case Stud. New Dev., Springer; 2020, p. 193-219. http://doi.org/10.1007/978-981-15-0306-1_9.Publisher Full Text
55. Erol OK, Eksin I. A new optimization method. Adv Eng Softw. 2006; 37:106-11.
56. Kaveh A, Talatahari S. Hybrid algorithm of harmony search, particle swarm and ant colony for structural design optimization. Stud Comput Intell. 2009; 239:159-98. http://doi.org/10.1007/978-3-642-03450-3_5. Publisher Full Text
57. Kaveh A, Ghazaan MI. Optimum design of skeletal structures using PSO-based algorithms. Period Polytech Civ Eng. 2017; 61:184-95. http://doi.org/10.3311/PPci.9614. Publisher Full Text
58. Degertekin SO. Optimum Design of Steel Frames via Harmony Search Algorithm. Stud. Comput. Intell., vol. 239, Springer; 2009, p. 51-78. http://doi.org/10.1007/978-3-642-03450-3_2.Publisher Full Text
59. Carbas S. Design optimization of steel frames using an enhanced firefly algorithm. Eng Optim. 2016; 48:2007-25. http://doi.org/10.1080/0305215X.2016.1145217. Publisher Full Text
60. Gholizadeh S, Razavi N, Shojaei E. Improved black hole and multiverse algorithms for discrete sizing optimization of planar structures. Eng Optim. 2019; 51:1645-67. http://doi.org/10.1080/0305215X.2018.1540697. Publisher Full Text
61. Kaveh A, Talatahari S, Alami MT. A newhybrid meta-heuristic for optimum design of frame structures. Asian J Civ Eng. 2012; 13:705-17.
62. Aydoğdu İ. Optimum design of 3-d irregular steel frames using ant colony optimization and harmony search algorithms. Middle East Technical University; 2010.
63. Javanmardi R, Ahmadi-Nedushan B. Cost Optimization of Steel-Concrete Composite I-Girder Bridges With Skew Angle and Longitudinal Slope, Using The Sm Toolbox and The Parallel Pattern Search Algorithm. vol. 11. 2021.
64. Salarnia AH, Ghasemi MR. Practical optimization of pedestrian bridges using grid search sensitivity based PSO. Int J Optim Civ Eng. 2021; 11:445-59.