A Minimum Cost Design Approach for Steel Frames Based on a Parallelized Firefly Algorithm and Parameter Control

Abstract

In this work, the applicability of a Firefly Algorithm (FA) to the real problem of the minimum cost of a detailed design for steel frames is studied. To reduce the calculation time, which is a common problem of meta-heuristic algorithms when they are used to solve real design cases, and to better suit the characteristics of the algorithm, a parallel migration strategy has been implemented and tested. As it is well known that the performance of any metaheuristic algorithm depends on the chosen value of its parameters, an extensive sensitivity analysis has been carried out. This not only serves to improve performance but also provides information on how it depends on the values of these parameters. With the information obtained from this analysis, and in order to achieve the robust behavior of the algorithm, a parameter control strategy has also been implemented and tested. Finally, a study demonstrating the close dependence between one of the parameters and the number of variables considered in the examples has been carried out. As a result of this final study, a simple expression is proposed that provides the minimum necessary population based on the number of variables in the problem.

1. Introduction

In the last three decades, a notable effort has been made to propose new design methods in the fields of meta-heuristic optimization and artificial intelligence. In these fields, biologically inspired algorithms, such as swarm-based methods, have proliferated [1]. These algorithms are currently being successfully applied to optimal design problems, image processing, business management, machine learning, and other areas. After the proposal of these new methods, initially tested on complex but idealized problems, other researchers have worked on adapting them to real engineering problems. This second line of research became necessary after the discovery of “no free lunch” (NFL) theorems [2,3], which established that no algorithm is better than another to solve a specific problem. The best algorithms are those that perform well in a specific case after adjusting their approach.

In addition to these works on the proposal and development of new algorithms and their applications and adjustments to specific cases, other authors have focused on studying the differences in their performance or new strategies to improve it. Joshi and Bansal [4] propose a generalized strategy to find the most suitable value of any parameter for a meta-heuristic algorithm. Loubière et al. [5] improve an Artificial Bee Colony (ABC) optimization algorithm using a sensitivity analysis method. Younis and Yang [6] propose two tightly coupled hybrid metaheuristic schedulers. The first scheduler combines Ant Colony Optimization (ACO) and Variable Neighborhood Search (VNS), in which ACO acts as the main algorithm that calls VNS as a supporting algorithm during execution. The second scheduler merges the Genetic Algorithm (GA) with VNS in the same way. The FA, which is a swarm-based method, has been shown to be more efficient than GA and Particle Swarm Optimization (PSO) [7]. A simple analysis of FA shows that Differential Evolution (DE), Accelerated Particle Swarm Optimization (APSO), Simulated Annealing (SA), and Harmony Search (HS) are variants of FA [1]. This has increased FA’s popularity, and it has been used to solve not only theoretical problems but also real cases [8].

Moreover, FA can manage a population divided into subpopulations simply. Each population can swarm around a local optimum, allowing the set of subpopulations to yield information in the entire design space. Finally, the global optimum can be detected among all the local optima. If the attractiveness parameter is properly adjusted, the algorithm evolves, causing the visibility among subpopulations to increase as the process progresses. This facilitates exploration at the beginning and exploitation at the end of the process [9].

This strategy of dividing the total population into subpopulations also facilitates the parallelization of the process to reduce calculation times. Some previous studies suggested that parallelization seemed to be the only general strategy capable of reducing computation times in its applicability to real engineering cases. Leite and Topping [10] evaluate parallelization strategies applied to engineering problems, where design spaces can be very complex and highly constrained, with function evaluations ranging from medium to high cost. Thierauf and Cai [11] present a method for solving mixed-discrete structural optimization problems based on a two-level parallel evolution strategy. Later, Gomes and Esposito [12], Hasançebi et al. [13], and Truong et al. [14] proposed parallelization techniques applied to real steel structures that ensured precise optimal solutions with clearly reduced calculation times. The strategy of parallelizing the process has been extended not only to structures but also to other fields of engineering [15,16,17].

For all these reasons, FA seems a good algorithm candidate for solving complex, real design problems. Gandomi et al. [18] use an FA to solve classical mixed continuous/discrete structural optimization problems. Talatahari et al. [19] present an adaptive FA, which uses a feasibility-based method for handling constraints to determine the optimal design of tower-shaped structures. Talbi [20] provides comprehensive information on meta-heuristic algorithms, including FA, to solve complex optimization problems in a wide range of applications, from networking and bioinformatics to engineering design, routing, and scheduling.

An example of this type of problem is the optimal design of steel structures, where the number of variables is usually high, and the objective function and constraints are often nonlinear. Once the algorithm has been selected as a synthesis tool, it must be defined to be efficient and robust. This involves not only establishing a simple and consistent formulation for the objective function and the constraints that can guarantee its good performance [21] but also tuning the algorithm’s parameters to ensure solutions close to the global optimum [22].

This paper studies the applicability of an FA to the real problem of the minimum-cost, detailed design of steel frames. Our main objective is to achieve a robust and efficient design method. The concept of robustness is linked to stability. The concept of efficiency is linked to reducing computational cost and calculation times.

A parallelization strategy, based on dividing the total population of fireflies into subpopulations, has been used to reduce computation time. The random migration of some fireflies among subpopulations allows them to exchange information. This parallelization–migration strategy has already been successfully tested in other problems, such as in the design of concrete sections subjected to biaxial bending [23].

A sensitivity analysis has been performed, and appropriate algorithm parameter values have been established. A new parameter control strategy has been implemented and tested with the information obtained in the sensitivity analysis to achieve an algorithm that behaves robustly. This new strategy takes advantage of subpopulations to explore different landscapes and thus allows us to obtain the different best values of the same parameter in each of the subpopulations.

Finally, a study demonstrating the close dependence between one of the parameters, which greatly affects the computational cost of the proposed numerical procedure, and the number of variables considered in the examples, has been carried out. As a result of this final study, a simple expression is proposed that is easy to memorize and provides the minimum necessary population based on the number of variables in the problem. This expression allows the designer to considerably reduce calculation time by testing different values of this minimum necessary population to achieve adequate performance.

Two steel frame design problems demonstrate the effectiveness and robustness of the proposed design method.

2. Optimum Design Problem

2.1. General Optimal Design Problem

Generally speaking, an optimal design problem, or optimization problem, is one that seeks the values of design variables that: (i) provide the minimum (or maximum) value for an objective function and (ii) provide valid (or not violated) values for a set of design constraints.

The design variables of a structure can be properties of the cross-section of the elements (areas, thicknesses, moments of inertia, etc.), geometric parameters, topological parameters, and material properties. The weight of the structure is the most common objective function. Other objective functions, such as cost, reliability, stiffness, etc., are used in other applications. Constraints are the conditions the design must meet to be valid according to design codes or standards.

The problem of calculating the minimum cost design of a steel frame can be generally stated as follows:

to find the design variable vector

X (X₁, X₂, …, X_nv)

(1)

to minimize the objective function (total cost of the structure)

f(X)

(2)

subject to the constraints

g_c(X) ≥ 0; c = 1, 2, …, nc

(3)

X_v^L ≤ X_v ≤ X_v^U; v = 1, 2, …, nv

(4)

where X is the nv-dimensional design variable vector, f(X) is the objective function, g_c(X) is inequality constraint c, nc is the total number of constraints, X_v^L is the lower bound for variable v, X_v is variable v, X_v^U is the upper bound for variable v, and nv is the number of design variables.

The most complex optimization problems, including the two examples presented in this work, are those in which the following applies:

• There are a large number of variables (1) and large scales (X_v^U − X_v^L) (4), creating an extensive design space.
• The variables are discrete or a mixture of continuous/discrete variables.
• The objective function (2) is non-linear, and the constraints (3) are many and non-linear. Cases in which derivatives of the objective function and the constraints appear at some point in the design space with values close to zero (functions insensitive to variables) or very high (jumps) are especially difficult to solve.

One reason meta-heuristic algorithms have become popular is because they can successfully address these problems. The disadvantage they present is their high computational cost. Therefore, it is essential to improve the algorithm’s performance by tuning its parameters [1,20].

2.2. The Firefly Algorithm

Meta-heuristic methods can be used to solve this problem. Among these methods is the FA, which belongs to the group of so-called nature-inspired algorithms. The FA is a swarm-based method whose search strategy depends on two sequential phases: exploration and exploitation. The first focuses on finding the global optimum throughout the design space. The second focuses on selecting the best design found up to that point among several options. During the process, the algorithm parameters change their values to encourage one or the other of these two phases [1].

The FA reproduces the social behavior of fireflies in a simple and idealized way. Fireflies communicate with each other, search for prey, and find a mate using different patterns of bioluminescent flashes [1]. The characteristics of these flashes have been idealized to develop this algorithm. Similar to real fireflies flying through 3D space and changing their spatial coordinates, the fireflies fly through nv-dimensional space, where the coordinates are now the variables of the design problem (1). The best design corresponds to a firefly i that flashes more than the others because it is in a position ${\mathbf{X}}_i^{best}$ that minimizes a function. This is called the fitness function. The fitness function takes into account the objective function (2) and inequality constraints (3) that are violated (g_c(X) < 0). During the process, the algorithm ensures that the variables (1) are within the limits (4) that define the design space.

The FA is guided by two basic rules:

• A firefly’s brightness is inversely proportional to the fitness function, so the firefly in the population with the lowest fitness function value (lowest cost and fewest constraint violations) shines brighter than the rest of the population. Fireflies are ordered by their brightness from the brightest to the dimmest in the population.
• Fireflies have no gender, so any one can be attracted to another in the population. The attraction between two fireflies, i and j, in the population decreases with the distance between them. The dimmer firefly i moves toward the brighter one j. The attraction ${\beta }_{ij}$ between them is defined as

$${\beta }_{ij}\left(r_{ij}\right)=\raisebox{1ex}{${\beta }_0$}\!\left/ \!\raisebox{-1ex}{$(1+\gamma ·r_{ij}^m)$}\right.$$

(5)

where γ (parameter) is the absorption coefficient for a given medium, m (parameter) is an integer so that m ≥ 1, r_ij is the distance between these two fireflies, and ${\beta }_0$ (parameter) is the attraction at r_ij = 0.

The position of a firefly i attracted to another, brighter firefly j in the population is obtained with the following expression:

$${\mathbf{X}}_i^{}={\mathbf{X}}_i^{}+\left({\mathbf{X}}_j^{}-{\mathbf{X}}_i^{}\right)·\raisebox{1ex}{${\beta }_0$}\!\left/ \!\raisebox{-1ex}{$(1+\gamma ·r_{ij}^m)$}\right.$$

(6)

where ${\mathbf{X}}_i^{}$ is the nv-dimensional vector of variables (or coordinates) of the i-th firefly, ${\mathbf{X}}_j^{}$ is the nv-dimensional vector of variables of the j-th firefly, and r_ij is the Cartesian distance in the nv-dimensional space obtained as follows:

$$r_{ij}=‖{\mathbf{X}}_i^{}-{\mathbf{X}}_j^{}‖$$

(7)

2.3. Variables

The total population of fireflies (nt) is established by means of an array, which is a set of data organized in rows, columns, and planes. The data stored in the array are the values of the variables (1). In each plane of the array, we have a matrix representing a subpopulation of fireflies, so we will have as many planes (np) as there are subpopulations. Each matrix has as many rows as there are fireflies in each subpopulation (n) and as many columns as variables considered in the problem (nv). Thus, each firefly is assigned a set of variable values representing a particular design X (1). At the beginning of the process, the values of the variables stored in the array are randomly determined within the bounds (4) of the design space.

In each particular design X, there are variables assigned to elements (beams or columns of the structure) and variables assigned to the semi-rigid connections (end-plates and bolts) between the elements. The variables assigned to the elements are the areas of the section. The variables assigned to the semi-rigid connections are the thickness of the end-plates and the diameter of the bolts used. Therefore, a particular design X includes as many area values as there are steel shapes for the elements and as many values of end-plate thicknesses and bolt diameters as there are semi-rigid connections.

To find the discrete values (DV) of the variables, either those associated with commercial steel shapes (IPE or HEB), the thicknesses of commercial steel end-plates, or the diameters of commercial steel bolts, a randomized rounding approach has been established. This approach compares the continuous value (CV) of a variable obtained during the procedure with a value obtained at random (RV) between two limits: (i) the commercial value immediately below the CV of the variable (lower commercial value or LCV) and (ii) the commercial value immediately above the CV of the variable (upper commercial value or UCV). If the CV is less than the RV, the DV adopted for the variable is the LCV. If the CV is greater than the RV, the DV adopted for the variable is the UCV (Figure 1).

This rounding procedure favors adopting the value of the closest limit (LCV or UCV) for the DV of the variable. However, if the CV of the variable is halfway between both limits (close to the mean), neither of them is favored since the probability of adopting either is similar (Figure 2).

This procedure of rounding continuous variables to discrete values avoids creating a bias in the process as much as possible so that during the exploration phase, there is a similar probability of choosing the LCV or the UCV, and, during the exploitation phase, of choosing the value of the LCV or the UCV closest to the CV of the variables.

As the variables used for elements are areas, and the variables for semi-rigid connections are plate thicknesses and bolt diameters, the orders of magnitude of these variables may be somewhat different. For example, using International System units, it is possible to have an area of 0.001643 m² (IPE140), an end-plate with a thickness of 0.018 m, and bolts with a diameter of 0.020 m. This can reduce the effectiveness of the algorithm. It is only necessary to divide the value of the variables, called design variables X, by their upper limits to eliminate this problem. These new variables P are called optimization variables or coordinates because these coordinates are used by the algorithm to obtain new coordinates for the fireflies. For example, if an area of 0.0156 m² (IPE600), an end-plate with a thickness of 0.040 m, and a bolt with a diameter of 0.036 m are chosen as upper limits, the previous design variables (0.001643, 0.018, and 0.020) correspond to the optimization variables 0.1053, 0.4500, and 0.5556, respectively.

The design variables X can be obtained at any time from the optimization variables P by multiplying their values by those of their upper limits, for example, after obtaining new firefly coordinates. But since the values of the coordinates P are continuous, the values of the design variables X will also be continuous (CV). Therefore, they must be rounded randomly, as explained above, to obtain commercial values (DV). Only then will realistic designs X and realistic values of the objective function f(X) and constraints g_c(X) be obtained.

2.4. Fitness Function

A detailed description of the fitness function can be found in the previous work by Sánchez-Olivares and Tomás [21]. To facilitate reading, a brief description is included here. The fitness function is defined as

$$F\left({\mathbf{X}}_i\right)=\frac{f\left({\mathbf{X}}_{\mathrm{i}}\right)}{\frac{1}{n}{\sum }_{i=1}^{n}f\left({\mathbf{X}}_{\mathrm{i}}^0\right)}\left(1-{\displaystyle \sum _{c=1}^{nvc}}{g}_c\left({\mathbf{X}}_i\right)\right)-{\displaystyle \sum _{c=1}^{nvc}}{g}_c\left({\mathbf{X}}_{\mathrm{i}}\right)$$

(8)

where X_i = (X_i1, X_i2, …, X_inv) is the nv-dimensional design variable vector obtained by the algorithm for firefly i in the current iteration, f(X_i) is the total cost of design X_i, ${\mathbf{X}}_i^0$ is the nv-dimensional design variable vector randomly obtained for firefly i in the first iteration, f(${\mathbf{X}}_i^0$) is the total cost of design ${\mathbf{X}}_i^0$, and nvc is the number of violated constraints (g_c(X_i) < 0) for firefly i in the current iteration.

2.5. Constraints

The following expressions of the constraints have been established following the requirements of Eurocode 3 [24]. A detailed description of the constraints can be found in the previous work by Sánchez-Olivares and Tomás [21]. A brief description is included here.

2.5.1. Story Lateral Sway Constraints

The normalized form of a frame story lateral sway constraint is

$$g_m^{sls}=1-\frac{D_m}{D_m^{\max }}\ge 0$$

(9)

where D_m is the displacement, as an absolute value, of the degree of freedom m and $D_m^{\max }$ is the maximum displacement allowed for the degree of freedom m.

2.5.2. Beam Deflection Constraints

The normalized form of a frame beam bending deflection constraint is

$$g_l^{bd}=1-\frac{v_l}{v_l^{\max }}\ge 0$$

(10)

where v_l is the flexural deflection, as an absolute value, of beam l, and $v_l^{\max }$ is the maximum deflection allowed for beam l.

2.5.3. Slenderness Constraints

The normalized form of a frame member (beam or column) slenderness constraint is

$$g_l^{m\lambda }=1-\frac{\left(\frac{K_l{L}_l}{i_l}\cdot \frac{1}{\pi \sqrt{E/f_y}}\right)}{{\overline{\lambda }}_l^{\max }}\ge 0$$

(11)

where K_l is the effective column-length factor of member l; L_l is the length of member l; i_l is the radius of gyration around the relevant axis of member l; E is Young’s modulus of steel; f_y is the specified yield strength of steel; and ${\overline{\lambda }}_l^{\max }$ is the maximum in-plane non-dimensional slenderness allowed for member l.

2.5.4. Buckling Constraints

The normalized form of a frame member buckling constraint is as follows:

• for Class 1 and Class 2 cross-sections

$$g_{l,y}^{mb}=1-\frac{P_l}{{\chi }_{l,y}{A}_l\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,y}\frac{M_l}{{\chi }_{l,LT}{W}_{l,pl}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(12)

$$g_{l,z}^{mb}=1-\frac{P_l}{{\chi }_{l,z}{A}_l\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,z}\frac{M_l}{{\chi }_{l,LT}{W}_{l,pl}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(13)

• for Class 3 cross-sections

$$g_{l,y}^{mb}=1-\frac{P_l}{{\chi }_{l,y}{A}_l\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,y}\frac{M_l}{{\chi }_{l,LT}{W}_{l,el}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(14)

$$g_{l,z}^{mb}=1-\frac{P_l}{{\chi }_{l,z}{A}_l\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,z}\frac{M_l}{{\chi }_{l,LT}{W}_{l,el}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(15)

• for Class 4 cross-sections

$$g_{l,y}^{mb}=1-\frac{P_l}{{\chi }_{l,y}{A}_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,y}\frac{\left(M_l+P_l{e}_{l,P}\right)}{{\chi }_{l,LT}{W}_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(16)

$$g_{l,z}^{mb}=1-\frac{P_l}{{\chi }_{l,z}{A}_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}-k_{l,z}\frac{\left(M_l+P_l{e}_{l,P}\right)}{{\chi }_{l,LT}{W}_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(17)

where P_l and M_l are the factored axial force and the factored bending moment of member l, respectively; A_l is the cross-sectional area of member l; A_l,eff is the effective cross-sectional area of member l; W_l,pl is the plastic section modulus dependant on the relevant cross-section of member l; W_l,el is the elastic section modulus dependant on the relevant cross-section of member l; W_l,eff is the effective section modulus dependant on the relevant cross-section of member l; e_l,P is the shift of the relevant centroidal axis of member l; χ_l,y and χ_l,z are the y-axis and z-axis reduction factors for the relevant flexural buckling curve of member l, respectively; χ_l,LT is a reduction factor for the relevant lateral torsional buckling curve of member l; k_l,y and k_l,z are interaction factors of member l; and γ_m is the safety factor corresponding to the strength of the material.

2.5.5. Strength Constraints

The normalized form of a frame member strength constraint is as follows:

• for Class 1 and Class 2 cross-sections

$$g_l^{ms}=1-\frac{M_l}{M_{l,P,Rd}}\ge 0$$

(18)

• for Class 3 cross-sections

$$g_l^{ms}=1-\frac{P_l}{A_l\left(\frac{f_y}{{\gamma }_m}\right)}-\frac{M_l}{W_{l,el}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(19)

• for Class 4 cross-sections

$$g_l^{ms}=1-\frac{P_l}{A_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}-\frac{\left(M_l+P_l{e}_{l,P}\right)}{W_{l,eff}\left(\frac{f_y}{{\gamma }_m}\right)}\ge 0$$

(20)

where $M_{l,P,Rd}$ is the design plastic moment strength reduced due to axial force P_l and the shear force of member l.

2.5.6. Semi-Rigid Connection Strength Constraints

The normalized form of a frame semi-rigid strength constraint is

$$g_l^{srcs}=1-\frac{M_{lk}}{M_{lk,Rd}}\ge 0$$

(21)

where M_lk is the factored bending moment of member l and end k, and M_lk,Rd is the design moment resistance of the semi-rigid connection of member l and end k.

2.5.7. Semi-Rigid Connection Assembly Constraints

The normalized form of a frame semi-rigid connection assembly constraint is

$$g_l^{srca}=1-\frac{d_{b,lk}}{d_{b,lk}^{max}}$$

(22)

where $d_{b,lk}$ is the bolt diameter adopted for the semi-rigid connection of member l and end k, and $d_{b,lk}^{max}$ is the maximum bolt diameter allowed for the semi-rigid connection of member l and end k.

3. Optimization Methodology

3.1. Parallel Firefly Algorithm with Migration (PFAM)

The PFAM reproduces the complex social behavior of real fireflies in a simplified way. This simple or idealized behavior is guided by four basic rules:

• The population is divided into subpopulations that extend the search throughout the design space. This division into subpopulations favors finding local optima in the exploration phase. From among all the local optima found, the best local optimum is selected at the end of the process in the exploitation phase. This becomes a candidate for the global optimum, as previously mentioned [9]. Dividing the population into subpopulations also makes it easier to parallelize the algorithm [23]. A MATLAB^® code has been defined to implement this parallel strategy using the MATLAB^® parfor command.
• A firefly’s brightness is inversely proportional to the fitness function, so the firefly in the subpopulation with the lowest fitness function value (lowest cost and fewest constraint violations) shines brighter than the rest of the subpopulation. Fireflies are ordered by their brightness from the brightest to the dimmest in each subpopulation.
• Fireflies have no gender, so any one can be attracted to another in a subpopulation. The position of a firefly i in a subpopulation, which is attracted to another, brighter firefly j in the same subpopulation, is obtained using the following expression

$${\mathbf{P}}_i^k={\mathbf{P}}_i^q+\left({\mathbf{P}}_j^{k-1}-{\mathbf{P}}_i^q\right)·\raisebox{1ex}{${\beta }_0$}\!\left/ \!\raisebox{-1ex}{$(1+\gamma ·r_{ij}^m)$}\right.+\alpha ·C^q·\mathrm{L}\mathrm{é}\mathrm{v}\mathrm{y}\left(w\right)·\left({\mathbf{P}}^U-{\mathbf{P}}^L\right)$$

(23)

where k is the current iteration, ${\mathbf{P}}_i^q$ is the nv-dimensional vector of optimization variables (or coordinates) of the i-th firefly at the q-th iteration, ${\mathbf{P}}_j^{k-1}$ is the nv-dimensional vector of optimization variables of the j-th firefly at the (k − 1)-th iteration, α is the randomization parameter [18,25], Lévy (w) is a random Lévy distribution where w is a parameter in which 0 ˂ w ˂ 2 [26,27], ${\mathbf{P}}_{}^U$ is the upper bound vector for the optimization variables, ${\mathbf{P}}_{}^L$ is the lower bound vector for the optimization variables, and r_ij is the Cartesian distance in the nv-dimensional space obtained as follows:

$$r_{ij}=‖{\mathbf{P}}_i^q-{\mathbf{P}}_j^{k-1}‖$$

(24)

And C^q is a simplified form of a sinusoidal chaotic variable at the q-th iteration in which 0 ˂ C^q ˂ 1, which is defined as follows [27]:

$$C^q=\sin \left(\pi C^{q-1}\right);{\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}C}^0=0.7$$

(25)

iv.

Some fireflies from the subpopulation containing the brightest firefly of all the subpopulations (origin subpopulation) are randomly selected and sent to other randomly selected subpopulations. This process of migration allows the fireflies to share information. As the origin subpopulation is the one yielding information, there is a risk that the search for the other subpopulations might be conditioned to include the local optimum. This can result in a biased search, with the search for local optima favored over a global optimum. To reduce this potential bias, a simple strategy consisting of perturbing the best design in the origin population and in each subpopulation has been included in the algorithm [23]. Therefore, if these best designs in each subpopulation are local optima, the perturbation eliminates this status, at least during the next iteration. The perturbation of the brightest firefly (i = 1) in a subpopulation is defined using the following expression:

$${\overline{\mathbf{P}}}_1^k={\mathbf{P}}_1^k+{\alpha }^2(\mathrm{rand}-\frac{1}{2})\left({\mathbf{P}}^U-{\mathbf{P}}^L\right)$$

(26)

where ${\overline{\mathbf{P}}}_1^k$ is the perturbed value of the nv-dimensional vector of the brightest firefly’s optimization variables, ${\mathbf{P}}_1^k$ is the nv-dimensional vector of the optimization variables obtained as defined in (23), and rand is a random number generator uniformly distributed in [0, 1].

3.2. Parameter Tuning in PFAM

Selecting the parameters is vital to achieving good results from the algorithm. However, this is complicated by the nonlinear interdependence among the parameters and because their value also depends on the fitness landscape. Furthermore, a parameter setting that provides good results in some cases may be inappropriate in others. When the parameter is inappropriate, the problem of robustness appears. This adjustment should provide at least acceptable results for a wide range of problems [28].

Parameter tuning can be considered from two different perspectives [22]:

• configuring an algorithm by choosing parameter values that optimize its performance; or
• analyzing an algorithm by studying how its performance depends on the values of its parameters.

The first perspective involves finding the optimal configuration of the algorithm. Extensive knowledge about the information the algorithm can find while traversing the design space is not essential. The second perspective involves finding more information that reveals the robustness of the algorithm, the distribution of the quality of the solutions, the sensitivity, etc. Both perspectives have been considered in this work, but the second one is more closely related to the concept of robustness. According to Eiben and Smit [22], many definitions of robustness can be obtained by extending the two previous perspectives with the following:

iii.

analyzing an algorithm to see how its performance varies when different problems are considered; and

iv.

analyzing an algorithm to see how its performance changes when it is executed several times independently.

Therefore, different problems must be analyzed to see how different executions of the algorithm applied to each of these problems provide information about its stability. These issues are addressed in this paper, but before establishing a study strategy to clarify them, some relevant information on FA parameters should be considered.

Some studies deal with the problem of establishing FA parameter values [9,18,27]. In general, the FA is very efficient, but an oscillatory behavior problem can appear [7,9]. This problem is avoided by reducing the randomization parameter as the search process progresses. The work of the authors of reference [18] indicates that, in most cases, a value of $\alpha \in \left[0.01,1\right]$ is adequate.

Another FA parameter that greatly affects its behavior is the absorption coefficient γ [1]. This parameter influences attractiveness. Its value is more or less influential depending on whether we are in the exploration phase or the exploitation phase. In the exploitation phase, it determines convergence speed since, in expression (23), the third term has less weight than the second. When γ is small, the attractiveness is constant, so all the fireflies can be seen in a transparent medium. However, when γ is large, the attractiveness decreases, and the fireflies fly in a foggy space, making it difficult for them to see each other. When this happens, the fireflies move almost randomly, which entails a random search. As a result, the behavior of the FA is halfway between these two extreme situations. Some authors have determined that, in practice, this parameter takes values from the interval [0.01, 10] [8,12].

The implemented MATLAB^® code allows the randomization parameter α [7,9] and the absorption coefficient γ to be reduced as the process progresses [25]. This strategy makes it possible to gradually move from the exploration phase to the exploitation phase. If k_max is the maximum number of iterations and k is the current iteration, the variable t = k / k_max can define the function (Figure 3).

$$s\left(t\right)=\left\{\begin{array}{cc}1& 0<t\le t_c\\ \frac{{\left(t-1\right)}^2}{{\left(t_c-1\right)}^2}& t_c<t\le 1\end{array}\right.$$

(27)

And the new randomization parameter α(t) and absorption coefficient γ(t) are

$$\alpha \left(t\right)={\alpha }_0·s\left(t\right)$$

(28)

$$\gamma \left(t\right)={\gamma }_0·s\left(t\right)$$

(29)

The simplest strategy is to set $\gamma =1/\sqrt{L}$ [7], where L is the typical length of the design variables, and relate the randomization parameter α to the actual scale of each design variable [18]. For this reason, and because the optimization variables are normalized in this work, the values of α₀ and γ₀ throughout the unit can be adopted in expressions (28) and (29).

These basic operations were implemented in MATLAB^® and are summarized in the Appendix A.

According to expression (5), parameters β₀ and m also affect attractiveness, with β₀ being that with the most direct and simple effect [1]. Parameter m only has a minor influence. A value of m ⸦ [1,3] can be taken [1]. A constant value of β₀ = 1 can be adopted, since this value has been proven suitable in most of the design examples studied in the literature [1,18,29].

The parameters of Yang’s original FA [1,7] and PFAM are included in

Table 1. FA and PFAM parameters to be tuned.

Algorithm	Parameters
FA	β0	m	γ	α	nt	kmax
PFAM	β0	m	γ0	α0	nt	kmax	tc

.

This study strategy is based on the concept of robustness and how this concept is associated with different design problems, parametric values, and independent executions [22]. Design problems of the same structural typology widely used in construction have been considered. In each of these problems, the behavior of the PFAM with different parametric values has been studied, and several independent executions have been performed. A value of β₀ = 1 has been adopted following studies by previously mentioned authors, and changes have been considered in the rest of the parameters included in Table 1.

3.3. Parameter Control in PFAM

An extensive classification and review of tuning and parameter control methods can be found in the work of Eiben and Smit [22]. According to these authors, there are two possible ways to proceed with parameter tuning: (i) exploiting knowledge or finding the set of parameters that provides the best performance, and (ii) exploring parameter space, which involves searching for the most information about the robustness of the process. Tuning methods generally balance exploitation and exploration and establish different categories of robustness. In the classification by Eiben and Smit [22], there are four main approaches: one based on only finding the best set of parameters (meta-EA), another based only on searching for information (sampling), and two that are a mixture of exploitation and exploration (screening and model-based).

The screening approach provides information on the influence of a model’s parametric values through an efficient exploration of multidimensional space [5]. One screening method is Morris’ Elementary Effect (EE) [30]. This method can detect the parameters that influence the model as well as their interactions. A sensitivity analysis allows the influence of an input II parameter π_z on a function F to be detected and quantified. The elementary effect (EE_z) of π_z on F can be obtained using the expression (30)

$${EE}_z\left(\mathsf{\Pi }\right)=\frac{F\left({\pi }_1,\dots ,{\pi }_z+∆,\dots ,{\pi }_D\right)-F\left(\mathsf{\Pi }\right)}{∆};\mathsf{\Pi }=({\pi }_1,\dots ,{\pi }_z,\dots ,{\pi }_D)$$

(30)

where ∆ is an offset and D is the number of parameters.

A variant of this screening method was proposed by Joshi and Bansal [4]. The method proposed by these authors is a parameter control method applied to a metaheuristic algorithm (Gravitational Search Algorithm, GSA). The same authors commented that the screening method included in GSA (Parameter Tuning in Gravitational Search Algorithm, PTGSA) could be applied to other metaheuristic algorithms apart from GSA. In their work, the authors checked the robustness of the PTGSA method by applying it to 15 unconstrained continuous test functions of the CEC 2015 test suite [31]. Additionally, they compared the PTGSA method with other metaheuristic algorithms to evaluate its performance and robustness.

A parameter control method adapted to the proposed algorithm PFAM (Parameter Control in Parallel Firefly Algorithm with Migration, PCPFAM) was applied to solve the design problem of steel structures (expressions 5 to 19). The design problems previously used to verify the robustness of PFAM were considered to verify the efficiency of PCPFAM.

In this work, the elementary effects ${\mathbf{E}\mathbf{E}}_z^k$ are calculated from the differences in the value of the fitness function F when a k-th iteration is performed in the process, and the value of the z-th parameter π_z is modified, according to the expression (31)

$${\mathbf{E}\mathbf{E}}_z^k=\left\{\frac{\left[F\left({\mathsf{\Pi }}^{k-1},{\mathbf{P}}^{k-1}\right)-F\left({\mathsf{\Pi }}_z^k,{\mathbf{P}}_z^k\right)\right]}{\left[F\left({\mathsf{\Pi }}^{k-1},{\mathbf{P}}^{k-1}\right)+F\left({\mathsf{\Pi }}_z^k,{\mathbf{P}}_z^k\right)\right]}\right\}/\left({\mathbf{P}}_z^k-{\mathbf{P}}^{k-1}\right)$$

(31)

where

$${\mathsf{\Pi }}_z^k=({\pi }_1^{k-1},{\pi }_2^{k-1},\dots ,{\pi }_{z-1}^{k-1},{\pi }_z^k,{\pi }_{z+1}^{k-1},\dots ,{\pi }_{D-1}^{k-1},{\pi }_D^{k-1})$$

(32)

$${\mathsf{\Pi }}^{k-1}=({\pi }_1^{k-1},{\pi }_2^{k-1},\dots ,{\pi }_{z-1}^{k-1},{\pi }_z^{k-1},{\pi }_{z+1}^{k-1},\dots ,{\pi }_{D-1}^{k-1},{\pi }_D^{k-1})$$

(33)

$${\mathbf{P}}_z^k=\left[\begin{array}{c}{\overline{\mathbf{P}}}_{z,1}^k\\ ⋮\\ {\mathbf{P}}_{z,n}^k\end{array}\right]$$

(34)

$${\mathbf{P}}^{k-1}=\left[\begin{array}{c}{\overline{\mathbf{P}}}_1^{k-1}\\ ⋮\\ {\mathbf{P}}_n^{k-1}\end{array}\right]$$

(35)

Therefore, according to (31), the elementary effect ${EE}_{z,i}^k$ for firefly i in the k-th iteration, when the value of the z-th parameter π_z is modified, can be obtained using the following expression

$${EE}_{z,i}^k=\left\{\frac{\left[F\left({\mathsf{\Pi }}^{k-1},{\mathbf{P}}_i^{k-1}\right)-F\left({\mathsf{\Pi }}_z^k,{\mathbf{P}}_{z,i}^k\right)\right]}{\left[F\left({\mathsf{\Pi }}^{k-1},{\mathbf{P}}_i^{k-1}\right)+F\left({\mathsf{\Pi }}_z^k,{\mathbf{P}}_{z,i}^k\right)\right]}\right\}/‖{\mathbf{P}}_{z,i}^k-{\mathbf{P}}_i^{k-1}‖$$

(36)

Or

$${EE}_{z,i}^k=\left[\frac{\left(F_i^{k-1}-F_{z,i}^k\right)}{\left(F_i^{k-1}+F_{z,i}^k\right)}\right]/{∆}_{z,i}^k=\left\{\frac{1}{2}\frac{\left(F_i^{k-1}-F_{z,i}^k\right)}{\left[\raisebox{1ex}{$\left(F_i^{k-1}+F_{z,i}^k\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right]}\right\}/{∆}_{z,i}^k$$

(37)

In addition, comparing expressions (31) and (36), it is evident that the following expression holds:

$${\mathbf{E}\mathbf{E}}_z^k=\left[\begin{array}{c}{EE}_{z,1}^k\\ ⋮\\ {EE}_{z,n}^k\end{array}\right]$$

(38)

Figure 4 shows a hypothetical evolution of an F_i-landscape between two consecutive iterations, k − 1 and k, and between two consecutive m − 1 and m iterations.

In other words, the elementary effect ${EE}_{z,i}^k$ gives us a measure of the change in F_i due to a change in the position of firefly i under the influence of the change in the value of parameter π_z at the k-th iteration.

If the following expressions hold

$$\left|F_i^{k-1}-F_{z,i}^k\right|=\left|F_i^{m-1}-F_{z,i}^m\right|$$

(39)

$${∆}_{z,i}^k={∆}_{z,i}^m$$

(40)

Then

$$\left|{EE}_{z,i}^k\right|<\left|{EE}_{z,i}^m\right|$$

(41)

Being that

$$\raisebox{1ex}{$\left(F_i^{m-1}+F_{z,i}^m\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.<\raisebox{1ex}{$\left(F_i^{k-1}+F_{z,i}^k\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\to \left(F_i^{m-1}+F_{z,i}^m\right)<\left(F_i^{k-1}+F_{z,i}^k\right)$$

(42)

So the ${EE}_{z,i}$ is larger when changes in the position of firefly i occur near a global optimum. Thus, the larger the change in F_i and the closer firefly i is to a global optimum, the larger the value of ${EE}_{z,i}$.

As ${EE}_{z,i}^k<0,\forall i\in \left[\begin{array}{ccc}1& \cdots & n\end{array}\right]$ rarely occurs, weights $w_{z,i}^k$ calculated from the ${EE}_{z,i}^k$ are adopted here as below:

$$w_{z,i}^k=\left[\frac{\left|F_i^{k-1}-F_{z,i}^k\right|}{\left(F_i^{k-1}+F_{z,i}^k\right)}\right]/{∆}_{z,i}^k=\left|{EE}_{z,i}^k\right|$$

(43)

Once the values of $w_{z,i}^k$ for the n fireflies of a subpopulation have been calculated at the k-th iteration, they are stored in an array that has as many dimensions as parameters to tune, plus an additional dimension for the n fireflies in the subpopulation. For example, if there are n fireflies and two parameters to tune with dimensions D₁ and D₂

$${\mathsf{\Pi }}_1^k=({\pi }_{\mathrm{1,1}}^{k-1},{\pi }_{\mathrm{1,2}}^{k-1},\dots ,{\pi }_{1,z_1-1}^{k-1},{\pi }_{1,z_1}^k,{\pi }_{1,z_1+1}^{k-1},\dots ,{\pi }_{1,D_1-1}^{k-1},{\pi }_{1,D_1}^{k-1})$$

(44)

$${\mathsf{\Pi }}_2^k=({\pi }_{\mathrm{2,1}}^{k-1},{\pi }_{\mathrm{2,2}}^{k-1},\dots ,{\pi }_{2,z_2-1}^{k-1},{\pi }_{2,z_2}^k,{\pi }_{2,z_2+1}^{k-1},\dots ,{\pi }_{2,D_2-1}^{k-1},{\pi }_{2,D_2}^{k-1})$$

(45)

The values of the weights

$$w_{z_1,z_2,i}^k;\forall i=1,2,\cdots ,n$$

(46)

are stored at the k-th iteration in an array of D₁ rows, D₂ columns, and n planes, as follows:

$${\mathbf{W}}_{D1,D2,n}^k=\left\{\begin{array}{cc}\begin{array}{c}\leftarrow D_{2}columns\to \\ \begin{array}{c}\left[\begin{array}{ccc}{w}_{\mathrm{1,1},1}^{k-1}& \cdots & w_{1,D_2,1}^{k-1}\\ ⋮& w_{z_1,z_2,1}^k& ⋮\\ w_{D_1,\mathrm{1,1}}^{k-1}& \cdots & w_{D_1,D_2,1}^{k-1}\end{array}\right]\\ \begin{array}{c}⋮\\ \left[\begin{array}{ccc}{w}_{\mathrm{1,1},i}^{k-1}& \cdots & w_{1,D_2,i}^{k-1}\\ ⋮& w_{z_1,z_2,i}^k& ⋮\\ w_{D_1,1,i}^{k-1}& \cdots & w_{D_1,D_2,i}^{k-1}\end{array}\right]\\ ⋮\end{array}\\ \left[\begin{array}{ccc}{w}_{\mathrm{1,1},n}^{k-1}& \cdots & w_{1,D_2,n}^{k-1}\\ ⋮& w_{z_1,z_2,n}^k& ⋮\\ w_{D_1,1,n}^{k-1}& \cdots & w_{D_1,D_2,n}^{k-1}\end{array}\right]\end{array}\end{array}& \begin{array}{c}\uparrow \\ D_{1}rows\\ \downarrow \end{array}\end{array}\right\}\begin{array}{c}\begin{array}{c}\uparrow \\ \\ \end{array}\\ \begin{array}{c}\\ nplanes\\ \end{array}\\ \begin{array}{c}\\ \\ \downarrow \end{array}\end{array}$$

(47)

It is possible to obtain a measure of the impact or influence that the value of parameter π_z has on the entire subpopulation when the process advances one iteration k. To achieve this, the mean of $w_{z,i}^k$ is calculated using the following expression:

$${\mu }_z^k=\frac{1}{n}\sum _{i=1}^n{w}_{z,i}^k;\forall z=1,2,\dots ,D$$

(48)

which has been adopted.

In this work, the mean value ${\mu }_z^k$ is stored in an array assigned to each subpopulation. This array has as many dimensions as there are parameters to tune. For example, if there are two parameters to tune with dimensions D₁ and D₂, the values of the means

$${\mu }_{z_1,z_2}^k=\frac{1}{n}\sum _{i=1}^n{w}_{z_1,z_2,i}^k;\forall z_1=1,2,\dots ,D_1;\forall z_2=1,2,\dots ,D_2$$

(49)

are stored in an array of D₁ rows and D₂ columns as follows:

$${\mathbf{M}}_{D_1,D_2}^k=\begin{array}{cc}\begin{array}{c}\leftarrow D_{2}columns\to \\ \left[\begin{array}{ccc}{\mu }_{\mathrm{1,1}}^{k-1}& \cdots & {\mu }_{1,D_2}^{k-1}\\ ⋮& {\mu }_{z_1,z_2}^k& ⋮\\ {\mu }_{D_1,1}^{k-1}& \cdots & {\mu }_{D_1,D_2}^{k-1}\end{array}\right]\end{array}& \begin{array}{c}\uparrow \\ D_{1}rows\\ \downarrow \end{array}\end{array}$$

(50)

Once array ${\mathbf{M}}_{D_1,D_2}^k$ has been updated, the algorithm selects the maximum value stored in it. The values of parameters ${\pi }_{1,z_1^{\prime }}^{k+1}$ and ${\pi }_{2,z_2^{\prime }}^{k+1}$ to be used in the next iteration k+1 correspond to the ${z\prime }_1$-row and ${z\prime }_2$-column of ${\mathbf{M}}_{D_1,D_2}^k$ where the maximum value

$${\mu }_{z_1^{\prime },z_2^{\prime }}^k=\mathrm{m}\mathrm{a}\mathrm{x}\left({\mathbf{M}}_{D_1,D_2}^k\right)$$

(51)

is stored as follows:

$${\mathbf{M}}_{D_1,D_2}^k=\begin{array}{cc}\begin{array}{c}{z\prime }_2-column\\ \left[\begin{array}{ccc}{\mu }_{\mathrm{1,1}}^{k-1}& \begin{array}{ccc}& \cdots & \end{array}& {\mu }_{1,D_2}^{k-1}\\ ⋮& {\mu }_{z_1^{\prime },z_2^{\prime }}^k& ⋮\\ {\mu }_{D_1,1}^{k-1}& \begin{array}{ccc}& \cdots & \end{array}& {\mu }_{D_1,D_2}^{k-1}\end{array}\right]\end{array}& \begin{array}{c}\\ \begin{array}{c}\\ {z\prime }_1-row\\ \end{array}\end{array}\end{array}$$

(52)

$${{\pi }_{1,z_1^{\prime }}^{k+1}=\mathsf{\Pi }}_1^k\left({z\prime }_1\right)$$

(53)

$${{\pi }_{2,z_2^{\prime }}^{k+1}=\mathsf{\Pi }}_2^k\left({z\prime }_2\right)$$

(54)

With this procedure, it is possible to improve the performance of the algorithm by selecting the ${\pi }_{1,z_1^{\prime }}^{k+1}$ and ${\pi }_{2,z_2^{\prime }}^{k+1}$ parameters in ${\mathsf{\Pi }}_1^k$ and in ${\mathsf{\Pi }}_2^k$, respectively, that maximize the changes in the given landscape F_i, especially those produced in coordinates close to a global optimum.

There are several possibilities for how the algorithm selects the specific value of a parameter ${\pi }_{z^{\prime }}^{k+1}$ from among the D values included in ${\mathsf{\Pi }}^k$. A simple way to proceed is to directly assign D values $\left[\begin{array}{ccc}{\pi }_1^{k-1}& \begin{array}{ccc}\cdots & {\pi }_z^k& \cdots \end{array}& {\pi }_D^{k-1}\end{array}\right]$ for ${\mathsf{\Pi }}^k$ that fall within a range $\left[\begin{array}{ccc}{\pi }_{minimum}& \cdots & {\pi }_{maximum}\end{array}\right]$ so that there is a correspondence between each value ${\pi }_z^k$ in ${\mathsf{\Pi }}^k$ and each mean ${\mu }_z^{k-1}$ in ${\mathbf{M}}_D^k$. However, the question arises as to how many D-values to consider in ${\mathsf{\Pi }}^k$. If D is a large value, there will be many values not only in ${\mathsf{\Pi }}^k$ but also in ${\mathbf{M}}_D^k$, which makes it more expensive to select ${\mu }_{z^{\prime }}^k$ and ${\pi }_{z^{\prime }}^{k+1}$. Another possibility is to obtain an estimate of ${\pi }_{z^{\prime }}^{k+1}$ from the values stored in ${\mathbf{M}}_D^k$. In this work, this second possibility was chosen, considering three unique values

$${\mathsf{\Pi }}^k=\left[\begin{array}{ccc}{\pi }_{minimum}& {\pi }_{intermediate}^k& {\pi }_{maximum}\end{array}\right]=\left[\begin{array}{ccc}{\pi }_{min}& {\pi }_{int}^k& {\pi }_{max}\end{array}\right]$$

(55)

with

$${\mathsf{\Pi }}^0=\left[\begin{array}{ccc}{\pi }_{min}& \frac{\left({\pi }_{min}+{\pi }_{max}\right)}{2}& {\pi }_{max}\end{array}\right]$$

(56)

regardless of the number of parameters to consider. If, for example, two parameters are considered

$${\mathsf{\Pi }}_1^k=\left(\begin{array}{ccc}{\pi }_{1,min}& {\pi }_{1,int}^k& {\pi }_{1,max}\end{array}\right)$$

(57)

$${\mathsf{\Pi }}_2^k=\left(\begin{array}{ccc}{\pi }_{2,min}& {\pi }_{2,int}^k& {\pi }_{2,max}\end{array}\right)$$

(58)

then ${\mathbf{M}}_D^k$ is a matrix with three rows and three columns with the following configuration:

$${\mathbf{M}}_{\mathrm{3,3}}^k=\left[\begin{array}{ccc}{\mu }_{\mathrm{1,1}}^{k-1}& {\mu }_{\mathrm{1,2}}^{k-1}& {\mu }_{\mathrm{1,3}}^{k-1}\\ {\mu }_{\mathrm{2,1}}^{k-1}& {\mu }_{2^{\prime },2^{\prime }}^k& {\mu }_{\mathrm{2,3}}^{k-1}\\ {\mu }_{\mathrm{3,1}}^{k-1}& {\mu }_{\mathrm{3,2}}^{k-1}& {\mu }_{\mathrm{3,3}}^{k-1}\end{array}\right]$$

(59)

The estimation of ${{\pi }_{z^{\prime }}^k=\pi }_{int}^k$ was performed in this work using the Lagrange polynomials $L_z^k\left({\pi }_z\right)$. With three unique values for ${\mathsf{\Pi }}^k$ according to expression (56), the three corresponding Lagrange’s polynomials are as follows:

$$L_{min}^k\left({\pi }_z\right)=\frac{\left({\pi }_z-{\pi }_{int}^k\right)·\left({\pi }_z-{\pi }_{max}\right)}{\left({\pi }_{min}-{\pi }_{int}^k\right)·\left({\pi }_{min}-{\pi }_{max}\right)}$$

(60)

$$L_{int}^k\left({\pi }_z\right)=\frac{\left({\pi }_z-{\pi }_{min}\right)·\left({\pi }_z-{\pi }_{max}\right)}{\left({\pi }_{int}^k-{\pi }_{min}\right)·\left({\pi }_{int}^k-{\pi }_{max}\right)}$$

(61)

$$L_{max}^k\left({\pi }_z\right)=\frac{\left({\pi }_z-{\pi }_{min}\right)·\left({\pi }_z-{\pi }_{int}^k\right)}{\left({\pi }_{max}-{\pi }_{min}\right)·\left({\pi }_{max}-{\pi }_{int}^k\right)}$$

(62)

from which function ${\mu }^{k+1}\left({\pi }_z\right)$ can be defined as follows:

$${\mu }^{k+1}\left({\pi }_z\right)=L_{min}^k\left({\pi }_z\right)·{\mu }_1^{k-1}+L_{int}^k\left({\pi }_z\right)·{\mu }_{2^{\prime }}^k+L_{max}^k\left({\pi }_z\right)·{\mu }_3^{k-1}$$

(63)

Function ${\mu }^{k+1}\left({\pi }_z\right)$ is represented in Figure 5. This function ${\mu }^{k+1}\left({\pi }_z\right)$ allows the maximum estimated value ${\mu }_{2^{\prime }}^{k+1}$ and parameter ${\pi }_{int}^{k+1}$ to be obtained.

This parameter control procedure was integrated into the PFAM in MATLAB^®, obtaining the PCPFAM algorithm, whose basic operations are summarized in the Appendix A.

The performance of the PCPFAM algorithm was verified using two design examples, which have been extensively studied by several authors and are presented below.

4. Examples

4.1. Three-Bay Two-Story Framework

The three-bay two-story structure shown in Figure 6, Figure 7 and Figure 8 has been previously studied by Cabrero and Bayo [32], Ali et al. [33], and Sánchez-Olivares and Tomás [21]. Figure 6 shows the geometry of the structure and the loads considered. The load factors that have been considered are γ_G = 1.3 for self-weight loads (q_sw) and γ_Q = 1.5 for imposed (q_il) and wind (q_w) loads.

Figure 7 shows the numbering used for the nodes and members. The expressions from 6 to 19 have been considered following the requirements of Eurocode 3 [24]. The columns and beams have HEB and IPE steel shapes, respectively.

The material considered for the steel shapes in members and steel end-plates in semi-rigid connections was steel S275 with Young’s modulus E = 210 GPa, f_y = 275 MPa yield strength, and a resistance factor of γ_m = 1.0. The ultimate tensile strength and the resistance factor considered for bolts in the semi-rigid connections were f_ub = 800 MPa and γ_m = 1.25, respectively. The material cost of the steel considered was C = 1.6 EUR/kg. The cost of the rigid connections in the frame was calculated assuming a fixity factor of 0.964 [21]. To model the behavior of the extended end-plates with stiffened column flange semi-rigid connections (EEP1) and extended end-plates without stiffened column flange semi-rigid connections (EEP2), the Component Method was used.

Figure 8 shows the numbering used for the design variables. The design variables are included in

Table 2. Design variables for the three-bay two-story framework.

Members	Design Variable	Lower Bound	Upper Bound
columns	X1 (area), X2 (area)	2.604 mm2 (HEB100)	40.000 mm2 (HEB1000)
beams	X3 (area), X4 (area)	764 mm2 (IPE80)	15.600 mm2 (IPE600)
beams	X5 (db), X7 (db), X9 (db), X11 (db)	10 mm	36 mm
beams	X6 (tep), X8 (tep), X10 (tep), X12 (tep)	8 mm	25 mm
db = bolt diameter; tep = end-plate thickness

.

The PFAM algorithm was run 100 times using different random seeds to guarantee that the best-generated solution would be considered the optimal design solution. Ten subpopulations were considered in all the runs. The values of the parameters in the PFAM algorithm were the following: β₀ = 1, m = 2, γ₀ = 10, α₀ = 1, nt = 200, k_max = 100, and t_c = 0.1. Figure 9 shows the evolutions of the objective functions obtained in one of the 100 executions considering rigid connections and both EEP1 and EEP2 semi-rigid connections. The optimal design solutions obtained by Cabrero and Bayo [32], Ali et al. [33], and the PFAM algorithm proposed in this work are included in

Table 3. Final designs for the three-bay two-story framework.

Variable	Cabrero and Bayo [32]			Ali et al. [33]		PFAM
	Rigid	Semi-rigid	Pinned	EEP1	EEP2	Rigid	EEP1	EEP2
X1	HEB140	HEB140	HEB120	HEB140	HEB140	HEB180	HEB120	HEB120
X2	HEB160	HEB160	HEB140	HEB140	HEB140	HEB140	HEB160	HEB160
X3	IPE270	IPE270	IPE330	IPE240	IPE240	IPE240	IPE300	IPE300
X4	IPE200	IPE200	IPE240	IPE200	IPE180	IPE180	IPE240	IPE240
X5 (db)	-	22	-			-	16	16
X6 (tep)	-	15	-			-	8	15
X7 (db)	-	22	-			-	20	20
X8 (tep)	-	14	-			-	15	12
X9 (db)	-	20	-			-	12	10
X10 (tep)	-	10	-			-	12	8
X11 (db)	-	16	-			-	16	16
X12 (tep)	-	12	-			-	10	10
Total cost (EUR)	5440.9	4124.6	4545.4	8434.2	6878.6	6202.3	5841.2	4804.8
	(132%)	(100%)	(110%)	(123%)	(100%)	(129%)	(122%)	(100%)

.

4.1.1. Parameter Tuning in PFAM for the Three-Bay Two-Story Framework

To study the performance of PFAM, a simple sensitivity analysis was performed considering different values for a pair of parameters, for example:

$${\pi }_{1,z_1};\forall z_1\in \left[\begin{array}{ccc}1& \cdots & D_1\end{array}\right]$$

(64)

$${\pi }_{2,z_2};\forall z_2\in \left[\begin{array}{ccc}1& \cdots & D_2\end{array}\right]$$

(65)

maintaining the rest of the parameters ${\pi }_3,{\pi }_4,\cdots$ with constant values. With each pair of concrete values $\left(\begin{array}{cc}{\pi }_{1,z_1}& {\pi }_{2,z_2}\end{array}\right)$, PFAM was executed a sequence of times with randomly generated initial independent populations. After these independent executions of the algorithm, a measure of its performance was obtained using two metrics [12]:

• Accuracy, calculated as the ratio between the cost of the optimal design solution (Table 3) and the mean value of the cost obtained from the independent executions, which measures the efficiency of the algorithm; and
• Standard Deviation of the cost obtained from the independent executions, which measures the dispersion or robustness of the results obtained using the algorithm.

Once the parameters with the best metrics were obtained (for example, $\left(\begin{array}{cc}{\pi }_{1,best}& {\pi }_{2,best}\end{array}\right)$), they were selected and became part of the group of parameters that were maintained at a constant value, and the process was repeated, taking into account different values of another pair of parameters (for instance, $\left(\begin{array}{cc}{\pi }_3& {\pi }_4\end{array}\right)$).

With EEP2 semi-rigid connections and the initial constant values β₀ = 1, m = 2, γ₀ = 10, α₀ = 1, and t_c = 0.1, the values of the nt and k_max parameters were modified. As this was the first sensitivity calculation performed, extensive ranges of both parameters were considered. There were 50 independent runs. The accuracy results are shown in

Table 4. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for nt and k_max. Accuracy.

		kmax
		100	200	300	400	500	600	700	800	900	1000
nt	50	0.981796	0.990914	0.990803	0.997039	0.997004	0.999103	0.998948	0.999067	0.999034	0.998969
	100	0.999115	0.999042	0.999084	0.999165	0.999109	0.998943	0.999144	0.999165	0.999358	0.999089
	150	0.999397	0.998991	0.999293	0.999117	0.999282	0.999038	0.999264	0.999216	0.999222	0.999495
	200	0.999395	0.999179	0.999267	0.999370	0.999269	0.999276	0.999370	0.999211	0.999275	0.999354
	250	0.999448	0.999395	0.999275	0.999203	0.999431	0.999379	0.999275	0.999249	0.999325	0.999213
	300	0.999280	0.999265	0.999174	0.999227	0.999295	0.999218	0.999327	0.999195	0.999250	0.999414

, and the standard deviation results are shown in

Table 5. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for nt and k_max. Standard deviation.

		kmax
		100	200	300	400	500	600	700	800	900	1000
nt	50	156.87	78.49	67.06	42.47	42.42	1.82	1.68	1.56	1.84	1.64
	100	2.12	1.51	1.48	1.79	1.80	1.67	1.75	1.75	1.89	1.72
	150	1.91	1.38	1.83	2.09	2.08	1.56	2.13	1.84	1.58	2.36
	200	1.34	1.46	1.85	1.80	1.88	1.72	1.82	1.88	1.68	2.31
	250	1.89	2.01	1.93	1.80	1.86	1.78	1.71	1.79	1.94	2.24
	300	1.98	2.07	1.47	2.06	1.55	2.19	1.90	1.67	1.87	1.44

.

In Table 4, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation (below 1.5) in Table 5. The shaded areas in both tables correspond to the lowest values of efficiency and robustness. The values of parameters nt = 250 and k_max = 100 gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation.

With EEP2 semi-rigid connections and the constant values β₀ = 1, nt = 250, k_max = 100, γ₀ = 10, and α₀ = 1, the values of the t_c and m parameters were modified. As was carried out previously with nt and k_max, and because it was the first sensitivity calculation performed with parameters t_c and m, wide ranges of both parameters were considered. There were 50 independent runs. The accuracy results are shown in

Table 6. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for t_c and m. Accuracy.

		m
		0.1	1	2	3	4
tc	0.1	0.999225	0.999392	0.999354	0.999191	0.999156
	0.3	0.999084	0.999235	0.999058	0.999128	0.999107
	0.5	0.999025	0.999019	0.999127	0.999075	0.999005
	0.7	0.997381	0.996625	0.997035	0.995176	0.991253
	0.9	0.861724	0.910561	0.925047	0.975938	0.994787

, and the standard deviation results are shown in

Table 7. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for t_c and m. Standard deviation.

		m
		0.1	1	2	3	4
tc	0.1	1.28	1.44	1.42	2.09	1.81
	0.3	1.55	1.97	1.39	2.19	1.50
	0.5	1.61	1.96	1.62	2.39	2.65
	0.7	33.54	30.80	40.10	57.53	77.53
	0.9	266.93	245.71	161.07	119.86	51.44

.

In Table 6, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation (below 1.5) in Table 7. The values of parameters t_c = 0.1 and m = 1, or m = 2 gave good, similar results, so they were chosen for the following sensitivity calculation.

With EEP2 semi-rigid connections and the constant values β₀ = 1, nt = 250, k_max = 100, t_c = 0.1, and m = 1, the values of the α₀ and γ₀ parameters were modified. As was previously with nt, k_max, t_c, and m, and because it was the first sensitivity calculation performed with parameters α₀ and γ₀, wide ranges of both parameters were considered. There were 50 independent runs. The accuracy results are shown in

Table 8. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0
		0.01	0.1	1	10	100
α0	0.3	0.986493	0.982331	0.995845	0.999307	0.999308
	0.4	0.985612	0.983652	0.991808	0.999444	0.999239
	0.5	0.979556	0.991716	0.999222	0.999210	0.999225
	0.6	0.976846	0.988560	0.999291	0.999396	0.999380
	0.7	0.986101	0.993526	0.999215	0.999200	0.999343
	0.8	0.995233	0.989365	0.999282	0.999277	0.999291
	0.9	0.987816	0.993768	0.999153	0.999281	0.999218
	1.0	0.982179	0.997174	0.999199	0.999186	0.999426

, and the standard deviation results are shown in

Table 9. Three-bay two-story framework with EEP2 semi-rigid connections. Sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0
		0.01	0.1	1	10	100
α0	0.3	101.94	111.36	51.51	2.05	2.00
	0.4	110.38	112.20	73.18	1.71	1.79
	0.5	160.42	89.93	1.97	1.64	2.11
	0.6	132.91	101.12	2.33	1.78	1.91
	0.7	130.83	64.21	2.01	1.44	1.75
	0.8	56.41	84.66	1.80	1.47	1.65
	0.9	85.31	65.29	1.84	1.54	2.01
	1.0	128.09	42.95	1.85	1.60	1.44

.

In Table 8, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation (below 1.5) in Table 9. The values of parameter γ₀ below 1 produced poor results regardless of the value of parameter α₀.

In summary, after performing these sensitivity analyses, the following was observed:

(i)

the selected values of parameters nt = 250 and k_max = 100 produced good metric results with low computational cost;

(ii)

parameter t_c affected the exploration and exploitation phases of the algorithm, obtaining good metric values when its values were reduced to around t_c = 0.1;

(iii)

the values of m close to the unit produced good metric values for low values of t_c; and

(iv)

the values of parameter γ₀ greater than the unit produced good metric values regardless of the parameter α₀ value adopted.

After these first results, the influence of the algorithm’s number of independent runs on the metrics was studied. Ten cases were considered, which are included in

Table 10. Three-bay two-story framework with EEP2 semi-rigid connections. Influence of the number of independent runs on the metrics.

Number of Independent Runs	Accuracy	Standard Deviation
10	0.999274	1.39
20	0.999326	1.91
30	0.999280	1.37
40	0.999139	1.77
50	0.999263	1.55
60	0.999248	1.41
70	0.999234	1.97
80	0.999204	1.77
90	0.999263	1.69
100	0.999225	1.66

. The parameter values adopted in all the cases were the following: β₀ = 1, nt = 250, k_max = 100, t_c = 0.1, α₀ = 1.0, γ₀ = 10, and m = 2.

Table 10 shows that the number of independent runs had very little effect on the results of the metrics, so 50 independent runs were carried out in the following sensitivity analyses.

To better understand the behavior of the metrics with more detailed variations of parameters α₀ and γ₀, an additional sensitivity analysis was performed with the constant values of the parameters β₀ = 1, nt = 250, k_max = 100, t_c = 0.1, and m = 1. The accuracy results are shown in

Table 11. Three-bay two-story framework with EEP2 semi-rigid connections. Second sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0					Mean
		1	26	51	76	101
α0	0.3	0.996939	0.999256	0.999294	0.999330	0.999348	0.999307
	0.4	0.997757	0.999354	0.999314	0.999322	0.999274	0.999316
	0.5	0.998507	0.999251	0.999334	0.999361	0.999267	0.999303
	0.6	0.998454	0.999238	0.999315	0.999303	0.999261	0.999279
	0.7	0.999295	0.999246	0.999285	0.999346	0.999270	0.999287
	0.8	0.999253	0.999226	0.999315	0.999302	0.999335	0.999295
	0.9	0.999262	0.999278	0.999283	0.999333	0.999336	0.999308
	1.0	0.999244	0.999220	0.999310	0.999303	0.999338	0.999293
mean		0.998589	0.999259	0.999306	0.999325	0.999304

, and the standard deviation results are shown in

Table 12. Three-bay two-story framework with EEP2 semi-rigid connections. Second sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0					Mean
		1	26	51	76	101
α0	0.3	43.83	1.69	1.65	1.68	1.86	1.72
	0.4	34.53	1.79	1.63	1.89	1.97	1.82
	0.5	25.59	1.67	1.76	1.83	1.77	1.76
	0.6	25.90	1.71	1.57	1.82	1.84	1.73
	0.7	2.06	1.68	1.84	1.87	1.82	1.80
	0.8	1.75	1.73	1.44	1.97	1.86	1.75
	0.9	1.96	1.59	1.61	1.68	1.79	1.67
	1.0	1.81	1.88	1.63	1.89	1.73	1.78
mean		17.18	1.72	1.64	1.83	1.83

.

Table 11 shows the values of the accuracy metric, and its mean values calculated, using rows and columns. The maximum values have been highlighted in bold. Table 12 shows the standard deviation metric, and its mean values calculated, using rows and columns. The minimum values have been highlighted in bold. The metric values included in the tables are good and very similar to each other, except for those corresponding to the values of parameters γ₀ = 1 and α₀ belonging to the range [0.3, 0.6]. Figure 10, Figure 11, Figure 12 and Figure 13 include the graphs of the accuracy and standard deviation means. These graphs show the sensitivity of both metrics against parameters γ₀ and α₀. Some of the graphs also include linear regression analysis on the data, which helps detect trends in both metrics against changes in the parameter values.

Figure 10 shows how the accuracy metric improves slightly when parameter α₀ decreases. Figure 12 shows something similar occurring with the standard deviation metric, which does not exceed values of 1.83. Therefore, the PFAM is efficient and robust for values of α₀ between 0.3 and 1.0.

Figure 11 and Figure 13 show that, for values of parameter γ₀ above unity, the accuracy and standard deviation metrics give good and very similar values. Table 8 and Table 9 show that this effect appears with γ₀ values above 10. Therefore, PFAM is efficient and robust for γ₀ values between 10 and 100.

To better understand the behavior of the metrics with more detailed variations of parameters t_c and m, an additional sensitivity analysis was performed with constant values of parameters β₀ = 1, nt = 250, k_max = 100, α₀ = 0.3, and γ₀ = 76. The accuracy results are shown in

Table 13. Three-bay two-story framework with EEP2 semi-rigid connections. Second sensitivity analysis for t_c and m. Accuracy.

		m
		0.1	1	2	3	4
tc	0.1	0.999232	0.999384	0.999448	0.998102	0.995946
	0.3	0.999262	0.999339	0.998328	0.998743	0.998043
	0.5	0.997609	0.997444	0.999372	0.998379	0.999071
	0.7	0.978781	0.979813	0.991873	0.997080	0.998367
	0.9	0.887117	0.896482	0.914793	0.982355	0.999070

, and the standard deviation results are shown in

Table 14. Three-bay two-story framework with EEP2 semi-rigid connections. Second sensitivity analysis for t_c and m. Standard deviation.

		m
		0.1	1	2	3	4
tc	0.1	1.82	1.94	2.12	31.45	58.40
	0.3	1.79	1.80	40.93	23.67	31.90
	0.5	33.30	57.67	41.78	28.40	15.16
	0.7	152.44	133.41	89.46	40.12	30.48
	0.9	299.68	292.25	279.46	135.48	18.96

.

If Table 6 and Table 7 (obtained with α₀ = 1.0, γ₀ = 10) are compared with Table 13 and Table 14 (obtained with α₀ = 0.3, γ₀ = 76), we can see that the shaded areas, with poor values of efficiency and robustness, are more extensive in Table 13 and Table 14. This demonstrates the interdependence among the parameters. Therefore, any sensitivity study needs to take this effect into account.

From the sensitivity analyses performed in this example, PFAM initially seems efficient and robust for the parameter values in

Table 15. Three-bay two-story framework with EEP2 semi-rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.3]	[0.3, 1.0]	[10, 100]	[1, 2]	250	100

. Nevertheless, these values were checked in another problem to study PFAM’s behavior and establish another level of robustness, namely that of the algorithm against changes in the problem.

Furthermore, parameters nt and k_max in Table 15 clearly depend on the problem. This is because each problem has a different number of variables, which affects the size of the design space. If the design space is large, sufficiently high nt values should be considered to ensure that the fireflies can capture enough information in the exploration phase. Likewise, sufficiently high k_max values must be taken into account so the algorithm can evolve from the exploration phase to the exploitation phase. The problem in which the connections between members of the structure are rigid, using only the necessary design variables X₁, X₂, X₃, and X₄ (see Table 2 and Table 3), has been considered in the same example to illustrate these ideas.

With rigid connections and initial constant values β₀ = 1$,$ m = 2, γ₀ = 10, α₀ = 1, and t_c = 0.1, the values of the nt and k_max parameters were modified. The accuracy results are shown in

Table 16. Three-bay two-story framework with rigid connections. Sensitivity analysis for nt and k_max. Accuracy.

		kmax
		100	200	300	400
nt	20	0.285222	0.247785	0.261767	0.259278
	30	1.000000	0.963353	1.000000	1.000000
	40	0.973976	0.981145	0.974113	0.958629
	50	0.978822	0.973921	0.900450	0.977659
	60	1.000000	1.000000	0.903428	1.000000
	70	1.000000	1.000000	0.978588	0.975847

, and the standard deviation results are shown in

Table 17. Three-bay two-story framework with rigid connections. Sensitivity analysis for nt and k_max. Standard deviation.

		kmax
		100	200	300	400
nt	20	7397.80	8252.30	7543.02	8751.77
	30	0.00	1055.17	0.00	0.00
	40	741.13	533.04	737.13	1197.06
	50	600.14	742.73	1770.93	633.85
	60	0.00	0.00	1848.24	0.00
	70	0.00	0.00	606.91	473.55

.

In Table 16, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 17. The shaded areas in both figures correspond to the lowest efficiency (accuracy below 0.9) and robustness (standard deviation above 1000) values. The values of parameters nt = 30 and k_max = 100 gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation.

With rigid connections and constant values β₀ = 1, nt = 30, k_max = 100, γ₀ = 10, and α₀ = 1, the values of the t_c and m parameters were modified. As previously carried out with nt and k_max, and because it was the first sensitivity calculation performed with parameters t_c and m, wide ranges of both parameters were considered. The accuracy results are shown in

Table 18. Three-bay two-story framework with rigid connections. Sensitivity analysis for t_c and m. Accuracy.

		m					Mean
		0.1	1	2	3	4
tc	0.1	0.939981	0.952263	0.907926	0.906337	0.997757	0.940853
	0.3	0.892027	0.952263	1.000000	1.000000	0.972434	0.963345
	0.5	0.991727	0.968677	0.991667	0.949523	0.865564	0.953432
	0.7	0.849647	0.989311	0.962938	0.930684	0.935820	0.933680
	0.9	0.773004	0.783040	0.802276	0.852616	0.928368	0.827861
mean		0.889277	0.929111	0.932961	0.927832	0.939988

, and the standard deviation results are shown in

Table 19. Three-bay two-story framework with rigid connections. Sensitivity analysis for t_c and m. Standard deviation.

		m					Mean
		0.1	1	2	3	4
tc	0.1	1706.55	1326.28	2747.97	1933.51	62.20	1555.30
	0.3	3168.78	3407.01	0.00	0.00	672.47	1449.65
	0.5	106.54	896.74	107.80	1357.73	2315.51	956.86
	0.7	2880.12	119.48	813.47	1813.61	1268.89	1379.11
	0.9	2499.57	2696.02	2340.14	2202.24	405.41	2028.68
mean		2072.31	1689.11	1201.87	1461.42	944.90

.

In Table 18, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 19. Figure 14, Figure 15, Figure 16 and Figure 17 include the graphs of the accuracy and standard deviation means. These graphs show the sensitivity of both metrics to parameters m and t_c. Some of the graphs also include linear regression analyses of the data, which helps detect trends in both metrics against changes in the value of the parameters.

Table 18 and Table 19 show that good and similar results were obtained for the metrics of parameter values t_c = 0.1 to 0.7 and m = 1 to 4. However, Figure 14, Figure 15, Figure 16 and Figure 17 show that the means of the metrics improve for high values of parameter m and low values of parameter t_c, the latter having the most influence on the results. For this reason, two sensitivity analyses were performed: (i) with parameter values t_c = 0.3 and m = 2, and (ii) with parameter values t_c = 0.1 and m = 4.

• First sensitivity analysis: with rigid connections and constant values β₀ = 1, nt = 30, k_max = 100, t_c = 0.3, and m = 2, the values of the α₀ and γ₀ parameters were modified. As previously carried out, and because it was the first sensitivity calculation performed with parameters α₀ and γ₀, wide ranges of both parameters were considered. The accuracy results are shown in Table 20, and the standard deviation results are shown in Table 21.

Table 20. Three-bay two-story framework with rigid connections. First sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	1.000000	0.946488	0.992191	0.910663	0.891969	0.758022	0.668462	0.881113
	0.4	1.000000	0.967716	0.993883	0.882127	0.832489	0.786748	0.786933	0.892842
	0.5	1.000000	0.971335	0.992453	0.943934	0.753382	0.838944	0.921210	0.917323
	0.6	0.971560	0.952837	0.898089	0.844372	0.708637	0.773464	0.598494	0.821065
	0.7	0.997763	0.993318	0.933460	0.909359	0.779504	0.788907	0.662411	0.866389
	0.8	0.997763	0.973706	0.922725	0.923220	0.887281	0.861669	0.754091	0.902922
	0.9	0.970565	0.966832	0.956470	0.974998	0.883387	0.700241	0.827291	0.897112
	1.0	0.939673	0.952912	0.955308	0.911318	0.808622	0.770159	0.680694	0.859812
mean		0.984665	0.965643	0.955572	0.912499	0.818159	0.784769	0.737448

Table 20. Three-bay two-story framework with rigid connections. First sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	1.000000	0.946488	0.992191	0.910663	0.891969	0.758022	0.668462	0.881113
	0.4	1.000000	0.967716	0.993883	0.882127	0.832489	0.786748	0.786933	0.892842
	0.5	1.000000	0.971335	0.992453	0.943934	0.753382	0.838944	0.921210	0.917323
	0.6	0.971560	0.952837	0.898089	0.844372	0.708637	0.773464	0.598494	0.821065
	0.7	0.997763	0.993318	0.933460	0.909359	0.779504	0.788907	0.662411	0.866389
	0.8	0.997763	0.973706	0.922725	0.923220	0.887281	0.861669	0.754091	0.902922
	0.9	0.970565	0.966832	0.956470	0.974998	0.883387	0.700241	0.827291	0.897112
	1.0	0.939673	0.952912	0.955308	0.911318	0.808622	0.770159	0.680694	0.859812
mean		0.984665	0.965643	0.955572	0.912499	0.818159	0.784769	0.737448

Table 21. Three-bay two-story framework with rigid connections. First sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	0.00	1128.45	131.15	1532.62	1684.33	2990.43	3459.81	1560.97
	0.4	0.00	868.15	94.18	1763.05	2588.30	4499.15	3664.25	1925.30
	0.5	0.00	761.60	97.77	1141.50	3045.85	2183.85	638.44	1124.15
	0.6	749.05	1244.95	2289.27	3165.81	4131.11	2993.27	5184.88	2822.62
	0.7	62.20	101.90	1747.11	1915.66	3609.77	2567.04	4039.21	2006.13
	0.8	62.20	686.37	2161.60	1170.22	1555.39	1829.78	3129.82	1513.62
	0.9	683.55	856.40	1024.49	231.44	1627.07	3962.12	2520.01	1557.87
	1.0	1491.07	1062.74	849.50	1365.04	2745.92	3131.86	3486.90	2019.00
mean		381.01	838.82	1049.38	1535.67	2623.47	3019.69	3265.41

Table 21. Three-bay two-story framework with rigid connections. First sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	0.00	1128.45	131.15	1532.62	1684.33	2990.43	3459.81	1560.97
	0.4	0.00	868.15	94.18	1763.05	2588.30	4499.15	3664.25	1925.30
	0.5	0.00	761.60	97.77	1141.50	3045.85	2183.85	638.44	1124.15
	0.6	749.05	1244.95	2289.27	3165.81	4131.11	2993.27	5184.88	2822.62
	0.7	62.20	101.90	1747.11	1915.66	3609.77	2567.04	4039.21	2006.13
	0.8	62.20	686.37	2161.60	1170.22	1555.39	1829.78	3129.82	1513.62
	0.9	683.55	856.40	1024.49	231.44	1627.07	3962.12	2520.01	1557.87
	1.0	1491.07	1062.74	849.50	1365.04	2745.92	3131.86	3486.90	2019.00
mean		381.01	838.82	1049.38	1535.67	2623.47	3019.69	3265.41

In Table 20, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 21. Parameter γ₀ values above 10 produce poor results regardless of the value of parameter α₀. The best results were obtained in the γ₀ range [0.1, 10] and the α₀ range [0.3, 0.5]. Figure 18, Figure 19, Figure 20 and Figure 21 include the graphs of the accuracy and standard deviation means. These graphs show the sensitivity of both metrics to parameters γ₀ and α₀. Some of the graphs also include linear regression analyses on the data, which helps detect trends in both metrics against changes in the value of the parameters.

Figure 18 shows how the accuracy metric improves slightly when parameter α₀ decreases. Figure 20 shows similar behavior with the standard deviation metric. Figure 19 and Figure 21 show that the accuracy and standard deviation values are reasonably good and quite similar for values of parameter γ₀ below 25.

ii.

Second sensitivity analysis: with rigid connections and constant values β₀ = 1, nt = 30, k_max = 100, t_c = 0.1, and m = 4, the values of the α₀ and γ₀ parameters were modified. As previously carried out, wide ranges of parameters α₀ and γ₀ were considered. The accuracy results are shown in

Table 22. Three-bay two-story framework with rigid connections. Second sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	0.959600	0.997763	1.000000	0.981479	0.882398	0.985085	0.965603	0.967418
	0.4	0.900188	0.952283	0.893250	0.944367	0.916266	0.896737	0.878108	0.911600
	0.5	0.967273	0.935148	0.995535	1.000000	0.944180	0.963209	0.901178	0.958075
	0.6	0.866837	0.900200	0.852008	0.973318	0.917225	0.929504	0.927877	0.909567
	0.7	0.953037	0.971434	0.997763	0.995754	0.808664	0.774844	0.813296	0.902113
	0.8	0.939030	0.997763	0.980223	0.959263	0.867033	0.866746	0.877740	0.926828
	0.9	0.931758	0.974162	0.843241	0.921092	0.948576	0.834133	0.828258	0.897317
	1.0	1.000000	0.958216	0.995945	0.831920	0.935031	0.826498	0.885162	0.918967
mean		0.939715	0.960871	0.944746	0.950899	0.902422	0.884594	0.884653

, and the standard deviation results are shown in

Table 23. Three-bay two-story framework with rigid connections. Second sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0							Mean
		0.1	1	10	25	50	75	100
α0	0.3	1167.79	62.20	0.00	362.37	2676.21	119.23	456.98	692.11
	0.4	2183.26	1389.88	2287.82	1347.20	1618.20	2351.95	2448.71	1946.72
	0.5	875.22	1923.60	85.61	0.00	1467.76	454.00	1717.72	931.99
	0.6	4261.05	2170.59	3074.81	646.22	1542.44	1558.64	1775.64	2147.05
	0.7	1366.83	752.74	62.20	81.53	3724.06	3934.32	3824.41	1963.72
	0.8	1305.55	62.20	559.64	1061.93	2821.36	2937.13	2327.36	1582.17
	0.9	2031.52	673.09	3783.01	2022.11	1291.52	2975.81	3209.71	2283.82
	1.0	0.00	1045.56	78.17	3015.74	1637.17	4034.38	2069.16	1697.17
mean		1648.90	1009.98	1241.41	1067.14	2097.34	2295.68	2228.71

.

In Table 22, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 23. In terms of parameter γ₀, values above 75 and in the α₀ range of [0.7, 1.0] produce poor results. For the rest of the α₀ and γ₀ values, a dispersion of poor metric values was observed. Figure 22, Figure 23, Figure 24 and Figure 25 include the graphs of the accuracy and standard deviation means. These graphs show the sensitivity of both metrics to parameters γ₀ and α₀. Some of the graphs also include linear regression analyses on the data, which helps detect trends in both metrics against changes in the value of the parameters.

Figure 22 shows how the accuracy metric improves when parameter α₀ decreases. Figure 24 shows similar behavior with the standard deviation metric. Figure 23 and Figure 25 show that the accuracy and standard deviation metrics are reasonably good and have very similar values for parameter γ₀ in the range [1,25].

Comparing Figure 18, Figure 20, Figure 22 and Figure 24, showing the evolution of the accuracy and standard deviation metrics with α₀, we can see that the sensitivity of both metrics with α₀ increases when the value of t_c is reduced. Better results are also obtained from the metrics for values of t_c = 0.1. Finally, the best results are obtained for α₀ values in the range [0.3, 0.5] for both t_c = 0.1 and t_c = 0.3.

Comparing Figure 19, Figure 21, Figure 23 and Figure 25, showing the evolution of the accuracy and standard deviation metrics with γ₀, we can see that the sensitivity of both metrics with γ₀ increases when the value of t_c is reduced. Better results are also obtained from the metrics for values of t_c = 0.1. Finally, the best results are obtained for γ₀ values in the range [1,25] for t_c = 0.1 and for γ₀ values in the range [0.1, 10] for t_c = 0.3.

To better understand the behavior of the metrics with more detailed variations of parameters t_c and m, an additional sensitivity analysis was conducted with constant values for parameters β₀ = 1, nt = 30, k_max = 100, α₀ = 0.3, and γ₀ = 0.1. The accuracy results are shown in

Table 24. Three-bay two-story framework with rigid connections. Second sensitivity analysis for t_c and m. Accuracy.

		m					Mean
		0.1	1	2	3	4
tc	0.1	0.966977	0.997763	0.965874	0.997763	1.000000	0.985675
	0.3	0.937299	1.000000	1.000000	0.971894	0.949063	0.971651
	0.5	0.997763	1.000000	1.000000	0.925724	1.000000	0.984697
	0.7	1.000000	0.957633	1.000000	0.944917	0.930939	0.966698
	0.9	1.000000	0.969958	0.973759	0.998333	1.000000	0.988410
mean		0.980408	0.985071	0.987927	0.967726	0.976001

, and the standard deviation results are shown in

Table 25. Three-bay two-story framework with rigid connections. Second sensitivity analysis for t_c and m. Standard deviation.

		m					Mean
		0.1	1	2	3	4
tc	0.1	947.27	62.20	859.38	62.20	0.00	386.21
	0.3	1855.53	0.00	0.00	802.13	1488.70	829.27
	0.5	62.20	0.00	0.00	1531.86	0.00	318.81
	0.7	0.00	1227.17	0.00	1616.95	2057.68	980.36
	0.9	0.00	859.11	700.23	46.33	0.00	321.13
mean		573.00	429.69	311.92	811.89	709.28

.

If Table 18 and Table 19 (obtained with α₀ = 1.0, γ₀ = 10) are compared with Table 24 and Table 25 (obtained with α₀ = 0.3, γ₀ = 0.1), we can see that the shaded areas, corresponding to poor values of efficiency and robustness, are more extensive in Table 18 and Table 19. This, again, demonstrates the interdependence among the parameters.

As a result of the sensitivity analyses performed in this example, PFAM initially seems efficient and robust for the parameter values in

Table 26. Three-bay two-story framework with rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.5]	[0.3, 0.5]	[0.1, 25]	[1, 4]	30	100

.

If we take into account the parameter values included in Table 15 (EEP2 semi-rigid connections) and Table 26 (rigid connections) and look for the intersection or common values in these tables for a three-bay two-story framework, PFAM seems efficient and robust for the parameter values included in

Table 27. Three-bay two-story framework with rigid or EEP2 semi-rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.3]	[0.3, 0.5]	[10, 25]	[1, 2]	30, 250	100

.

4.1.2. Parameter Control for the Three-Bay Two-Story Framework

As parameters t_c and α₀ are closely related, and both define a strategy for moving from the exploration phase to the exploitation phase that is unique from start to finish, they were considered constant in the PCPFAM procedure proposed in this work. The value of t_c = 0.1 was adopted to guarantee a sufficient margin of passage from the exploration phase to the exploitation phase. The value of α₀ = 1.0 was also chosen so the subpopulations could initially explore the entire design space in the exploration phase regardless of the values of the other parameters. Furthermore, the value of 2 was given to parameter m, as it was found to have little effect on the results. However, parameter nt depends on the number of variables considered in each problem, affecting the entire process as t_c and α₀ do, so the value found in the previous sensitivity analysis was used. Finally, the proposed PCPFAM procedure was applied to the γ parameter, which greatly affected the efficiency and robustness results.

Considering both semi-rigid connections EEP2 and rigid connections, and constant values β₀ = 1, k_max = 100, t_c = 0.1, α₀ = 1.0, and m = 2, the PCPFAM procedure was applied to parameter γ. The range of values for γ was [1, 50]. The evolution of parameter γ in each subpopulation for a run is shown in Figure 26. The accuracy and standard deviation results are shown in

Table 28. Three-bay two-story framework. PCPFAM. Accuracy and Standard deviation.

PCPFAM
EEP2 Semi-Rigid Connections		Rigid Connections
Accuracy	Standard Deviation	Accuracy	Standard Deviation
0.999336	2.08	1.000000	0.00

, and there were 50 independent runs.

4.2. Three-Bay Ten-Story Framework

The three-bay ten-story structure shown in Figure 27 and Figure 28 has been previously studied by Xu and Grierson [34], Foley and Schinler [35], Kameshki and Saka [36], Ali et al. [33], and Sánchez-Olivares and Tomás [21]. Figure 27 shows the geometry of the structure and the loads considered: floor dead load (D_f); floor live load (L_f); roof dead load (D_r); roof live load (L_r); and wind load (W). Taking these loads into account, two load combinations were considered:

I: (1.0D_f + 1.0D_r + 0.4L_f + 0.4L_r + 0.5W),

II: (1.2D_f + 1.2D_r + 0.5L_f + 0.5L_r + 1.3W).

Figure 28 shows the numbering used for the nodes, members, and design variables. The expressions from 6 to 19 were considered following the requirements of Eurocode 3 [24]. The columns and beams have HEB and IPE steel shapes, respectively, and the design variables are included in

Table 29. Design variables for the three-bay ten-story framework.

Members	Variable	Lower Bound	Upper Bound
columns	X1 (area) to X10 (area)	HEB100	HEB1000
beams	X11 (area) to X20 (area)	IPE80	IPE600
beams	X21 (db), X23 (db), X25 (db), X27 (db), X29 (db), X31 (db), X33 (db), X35 (db), X37 (db), X39 (db)	10 mm	36 mm
beams	X22 (tep), X24 (tep), X26 (tep), X28 (tep), X30 (tep), X32 (tep), X34 (tep), X36 (tep), X38 (tep), X40 (tep)	8 mm	25 mm

.

Steel S235 with Young’s modulus E = 210 GPa, f_y = 235 MPa yield strength, and a resistance factor of γ_m = 1.0 was the material chosen for the steel shapes in the members and the steel plates in the semi-rigid connections. The ultimate tensile strength and resistance factor for the bolts in the semi-rigid connections were f_ub = 800 MPa and γ_m = 1.25, respectively. The material cost of the steel was set at C = 1.6 EUR/kg. The cost of the rigid connections in the frame was calculated assuming a fixity factor of 0.964 [21]. To model the behavior of extended end-plates with stiffened column flange semi-rigid connections (EEP1) and extended end-plates without stiffened column flange semi-rigid connections (EEP2), the Component Method was used.

The PFAM algorithm was run 100 times using different random seeds so the best-generated solution would be chosen as the optimal design solution. Ten subpopulations were considered in all the runs. The values of the parameters used in the PFAM algorithm were the following: β₀ = 1$,$ m = 2, γ₀ = 10, α₀ = 1, nt = 1000, k_max = 100, and t_c = 0.1. Figure 29 shows the evolutions of the objective functions in one of the 100 executions with rigid connections and EEP1 and EEP2 semi-rigid connections. The optimal design solutions obtained by Ali et al. [33] and using the PFAM algorithm proposed in this work are included in

Table 30. Final designs for the three-bay ten-story framework.

Variable	Ali et al. [33]		PFAM
	EEP1	EEP2	Rigid	EEP1	EEP2
X1	HEB450	HEB400	HEB320	HEB320	HEB320
X2	HEB450	HEB500	HEB900	HEB800	HEB800
X3	HEB300	HEB300	HEB360	HEB300	HEB300
X4	HEB360	HEB360	HEB400	HEB450	HEB450
X5	HEB280	HEB260	HEB260	HEB260	HEB280
X6	HEB300	HEB300	HEB340	HEB340	HEB340
X7	HEB240	HEB240	HEB240	HEB240	HEB240
X8	HEB240	HEB260	HEB260	HEB260	HEB260
X9	HEB200	HEB200	HEB220	HEB180	HEB180
X10	HEB200	HEB220	HEB180	HEB180	HEB180
X11	IPE450	IPE450	IPE360	IPE400	IPE400
X12	IPE450	IPE550	IPE400	IPE400	IPE400
X13	IPE450	IPE450	IPE400	IPE450	IPE450
X14	IPE450	IPE500	IPE400	IPE450	IPE500
X15	IPE450	IPE400	IPE400	IPE400	IPE450
X16	IPE450	IPE450	IPE400	IPE450	IPE500
X17	IPE450	IPE500	IPE360	IPE400	IPE450
X18	IPE400	IPE500	IPE360	IPE450	IPE500
X19	IPE500	IPE400	IPE330	IPE360	IPE400
X20	IPE400	IPE330	IPE270	IPE300	IPE300
X21 (db)			-	22	24
X22 (tep)			-	25	25
X23 (db)			-	24	24
X24 (tep)			-	25	25
X25 (db)			-	24	24
X26 (tep)			-	25	25
X27 (db)			-	24	27
X28 (tep)			-	25	25
X29 (db)			-	24	24
X30 (tep)			-	25	25
X31 (db)			-	22	24
X32 (tep)			-	20	20
X33 (db)			-	22	20
X34 (tep)			-	20	20
X35 (db)			-	22	20
X36 (tep)			-	15	12
X37 (db)			-	22	20
X38 (tep)			-	20	15
X39 (db)			-	16	20
X40 (tep)			-	15	12
Total cost (EUR)	107,314.3 (113%)	95,193.1 (100%)	84,904.3 (153%)	59,864.4 (108%)	55,630.2 (100%)

.

4.2.1. Parameter Tuning in PFAM for the Three-Bay Ten-Story Framework

To study PFAM’s performance for the three-bay ten-story framework, a similar sensitivity analysis to the previous one was performed considering different values for a pair of parameters.

With EEP2 semi-rigid connections, and initial constant values β₀ = 1, m = 2, γ₀ = 10, α₀ = 1, and t_c = 0.1, the values of the nt and k_max parameters were modified. Based on the results from the previous example, a wide range of values for the nt parameter and only three values for the k_max parameter were considered. There were 50 independent runs. The accuracy results are shown in

Table 31. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for nt and k_max. Accuracy.

		kmax
		100	200	300
nt	500	0.970802	0.985971	0.988744
	600	0.983134	0.986557	0.991328
	700	0.989667	0.988853	0.991794
	800	0.983148	0.986735	0.991775
	900	0.985811	0.991861	0.993167
	1000	0.989935	0.990955	0.990948

, and the standard deviation results are shown in

Table 32. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for nt and k_max. Standard deviation.

		kmax
		100	200	300
nt	500	901.14	667.36	339.27
	600	610.27	326.69	120.05
	700	173.28	183.04	141.15
	800	739.94	189.09	243.24
	900	234.04	141.30	265.04
	1000	154.01	273.08	274.00

.

In Table 31, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 32. The values of parameters nt = 1000 and k_max = 100 gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation.

With EEP2 semi-rigid connections, and constant values β₀ = 1, nt = 1000, k_max = 100, γ₀ = 10, and α₀ = 1, the values of the t_c and m parameters were modified. Taking the information above into account, sufficiently wide ranges of the t_c and m parameters were considered. The accuracy results are shown in

Table 33. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for t_c and m. Accuracy.

		m			Mean
		1	2	3
tc	0.1	0.988397	0.987065	0.967465	0.980976
	0.3	0.987863	0.975870	0.955743	0.973159
	0.5	0.974107	0.964007	0.951925	0.963346
mean		0.9835	0.9756	0.9584

, and the standard deviation results are shown in

Table 34. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for t_c and m. Standard deviation.

		m			Mean
		1	2	3
tc	0.1	275.28	213.71	1498.46	662.49
	0.3	326.75	1018.38	1162.37	835.83
	0.5	674.17	1015.90	1019.74	903.27
mean		425.40	749.33	1226.86

.

In Table 33, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 34. Parameter values t_c = 0.1 and m = 1, or m = 2 had good and similar results, so they were chosen for the following sensitivity calculation.

With EEP2 semi-rigid connections, and constant values β₀ = 1, nt = 1000, k_max = 100, t_c = 0.1, and m = 1, the values of the α₀ and γ₀ parameters were modified. Taking this above information into account, sufficiently wide ranges of the α₀ and γ₀ parameters were considered. The accuracy results are shown in

Table 35. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0				Mean
		0.1	1	10	25
α0	0.3	0.968874	0.977775	0.994244	0.988987	0.982470
	0.4	0.944331	0.980527	0.993107	0.993071	0.977759
	0.5	0.968911	0.982921	0.993255	0.992099	0.984297
	0.6	0.949196	0.979579	0.990534	0.992214	0.977881
	0.7	0.909226	0.986223	0.990505	0.989142	0.968774
	0.8	0.909226	0.983006	0.987852	0.987241	0.966831
	0.9	0.966962	0.983008	0.989730	0.984751	0.981113
	1.0	0.969974	0.982574	0.986272	0.988684	0.981876
mean		0.948337	0.981952	0.990688	0.989524

, and the standard deviation results are shown in

Table 36. Three-bay ten-story framework with EEP2 semi-rigid connections. Sensitivity analysis for α₀ and γ₀ Standard deviation.

		γ0				Mean
		0.1	1	10	25
α0	0.3	695.53	1006.48	51.63	578.95	583.15
	0.4	5032.57	801.17	176.01	185.58	1548.83
	0.5	1127.87	470.94	186.29	176.27	490.34
	0.6	3371.61	693.22	121.17	273.67	1114.92
	0.7	8076.86	285.18	151.78	196.01	2177.46
	0.8	2910.99	876.56	666.37	792.41	1311.58
	0.9	1196.02	1019.10	258.01	716.03	797.29
	1.0	1076.03	352.63	669.74	116.06	553.62
mean		2935.94	688.16	285.13	379.37

.

In Table 35, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 36. Parameter γ₀ values below 1 produced poor results regardless of the value of parameter α₀.

In summary, after performing these sensitivity analyses, the following was observed:

(i)

the selected values of parameters nt = 1000 and k_max = 100 produced good metric results with low computational cost;

(ii)

parameter t_c affected the exploration and exploitation phases of the algorithm, obtaining good metric values when values of parameter t_c of around 0.1 were adopted;

(iii)

m values close to the unit produced good metric values for low t_c values; and

(iv)

the values of parameter γ₀ greater than the unit produced good metric values, with α₀ in the range [0.3, 1.0].

As a result of the sensitivity analyses performed in this example, PFAM initially seems efficient and robust for the parameter values in

Table 37. Three-bay ten-story framework with EEP2 semi-rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.3]	[0.3, 1.0]	[10,25]	[1, 2]	1000	100

. However, these values have been checked in another problem to study PFAM’s behavior and establish another level of robustness, namely that of the algorithm against changes in the problem.

As in the previous example, a problem with rigid connections between members of the structure has been considered, for which only the design variables X₁ to X₂₀ are necessary (see Table 29 and Table 30).

With rigid connections, and the initial constant values β₀ = 1$,$ m = 2, γ₀ = 10, α₀ = 1, and t_c = 0.1, the nt and k_max parameter values were modified. The accuracy results are shown in

Table 38. Three-bay ten-story framework with rigid connections. Sensitivity analysis for nt and k_max. Accuracy.

		kmax
		100	200	300
nt	250	0.991133	0.994741	0.993758
	350	0.990264	0.993835	0.991943
	450	0.995536	0.993890	0.995469
	550	0.995994	0.997256	0.996817
	650	0.995159	0.994355	0.996200
	750	0.995524	0.996871	0.997500

, and the standard deviation results are shown in

Table 39. Three-bay ten-story framework with rigid connections. Sensitivity analysis for nt and k_max. Standard deviation.

		kmax
		100	200	300
nt	250	1062.06	418.96	464.40
	350	855.65	482.26	730.82
	450	421.07	890.25	414.91
	550	358.55	248.38	227.97
	650	422.30	480.08	345.64
	750	466.57	219.60	256.73

.

In Table 38, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 39. The shaded areas in both tables correspond to the lowest values of efficiency and robustness. The values of parameters nt = 450 and k_max = 100 gave good results and had a low computational cost, so they were chosen for the following sensitivity calculation.

Considering rigid connections, and constant values β₀ = 1, nt = 450, k_max = 100, γ₀ = 10, and α₀ = 1, the t_c and m parameter values were modified. The accuracy results are shown in

Table 40. Three-bay ten-story framework with rigid connections. Sensitivity analysis for t_c and m. Accuracy.

		m			Mean
		1	2	3
tc	0.1	0.995677	0.988884	0.993793	0.992785
	0.3	0.993986	0.995758	0.991494	0.993746
	0.5	0.990757	0.990991	0.991238	0.990996
mean		0.993473	0.991878	0.992175

, and the standard deviation results are shown in

Table 41. Three-bay ten-story framework with rigid connections. Sensitivity analysis for t_c and m. Standard deviation.

		m			Mean
		1	2	3
tc	0.1	325.31	1225.13	696.53	748.99
	0.3	478.43	429.79	525.33	477.85
	0.5	443.69	696.55	539.24	559.83
mean		415.81	783.82	587.04

.

In Table 40, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 41. The ranges of both parameters t_c and m produced good and similar results. Figure 30 and Figure 31 include the graphs of the accuracy and standard deviation means. These graphs show the sensitivity of both metrics to parameters m and t_c. Some of the graphs also include linear regression analyses of the data, which helps detect trends in both metrics against changes in the value of the parameters.

Table 40 and Table 41 and Figure 30 and Figure 31 show that the parameter values t_c = 0.3 and m = 1 gave the best results, so they were chosen for the following sensitivity calculation.

Considering rigid connections, and the constant values β₀ = 1, nt = 450, k_max = 100, t_c = 0.3, and m = 1, the values of the α₀ and γ₀ parameters were modified. The results obtained for accuracy are shown in

Table 42. Three-bay ten-story framework with rigid connections. Sensitivity analysis for α₀ and γ₀. Accuracy.

		γ0					Mean
		0.1	1	10	25	50
α0	0.3	0.993540	0.994849	0.997128	0.994266	0.991229	0.994202
	0.4	0.995109	0.994394	0.996860	0.992294	0.992450	0.994221
	0.5	0.993989	0.995789	0.995578	0.993692	0.992928	0.994395
	0.6	0.993387	0.996551	0.996689	0.994053	0.991264	0.994389
mean		0.994006	0.995396	0.996564	0.993576	0.991968

. The results obtained for standard deviation are shown in

Table 43. Three-bay ten-story framework with rigid connections. Sensitivity analysis for α₀ and γ₀. Standard deviation.

		γ0					Mean
		0.1	1	10	25	50
α0	0.3	633.18	526.82	272.53	404.81	675.01	502.47
	0.4	680.30	478.73	252.84	607.21	453.03	494.42
	0.5	574.85	348.84	380.81	405.75	421.31	426.31
	0.6	810.14	98.94	125.15	400.91	396.91	366.41
mean		674.62	363.33	257.83	454.67	486.57

.

In Table 42, the maximum accuracy values have been highlighted in bold, as have the minimum values of standard deviation in Table 43. The ranges of both parameters α₀ and γ₀ produced good and similar results.

As a result of the sensitivity analyses performed in this example, PFAM initially seems efficient and robust for the parameter values in

Table 44. Three-bay ten-story framework with rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.5]	[0.3, 0.6]	[0.1, 50]	[1, 3]	450	100

.

If we take into account the values of the parameters in Table 37 (EEP2 semi-rigid connections) and Table 44 (rigid connections), and look for the intersection or common values of both tables for the three-bay two-story framework, we can conclude that PFAM seems efficient and robust for the parameter values in

Table 45. Three-bay ten-story framework with rigid or EEP2 semi-rigid connections. Optimum values of PFAM parameters.

Parameter	β0	tc	α0	γ0	m	nt	kmax
Values	1	[0.1, 0.3]	[0.3, 0.6]	[10, 25]	[1, 2]	450, 1000	100

.

4.2.2. Parameter Control for the Three-Bay Ten-Story Framework

As in the previous example, parameters t_c, α₀, and m were considered constants in the parameter control procedure (PCPFAM) proposed in this work. The proposed PCPFAM procedure on γ was taken into account.

Considering both semi-rigid connections EEP2 and rigid connections, and the constant values β₀ = 1, k_max = 100, t_c = 0.1, α₀ = 1.0, and m = 2, the PCPFAM procedure was applied to parameter γ. The range of values for γ was [1, 50]. The evolution of parameter γ in each subpopulation for a run is shown in Figure 32. The accuracy and standard deviation results are shown in

Table 46. Three-bay ten-story framework. PCPFAM. Accuracy and Standard deviation.

PCPFAM
EEP2 Semi-Rigid Connections		Rigid Connections
Accuracy	Standard Deviation	Accuracy	Standard Deviation
0.992651	207.15	0.995769	389.17

. There were 50 independent runs.

4.3. Analysis and Discussion of the Results

In the first example, an extensive sensitivity analysis was carried out using PFAM. Different values were considered for a couple of parameters within extensive ranges, establishing those obtaining better metrics as tuned values. With tuned values, another pair of parameters were modified. This procedure was carried out several times. As expected, a strong interdependence among the parameter values was observed because the tuned values for a couple of them depended on the constant values chosen for the rest of the parameters.

Therefore, adjusting parameters through simple procedures, such as the one shown here, is an arduous task that requires time and has a high computational cost.

In the second example, a less extensive sensitivity analysis was carried out, as it was only intended to check the robustness of PFAM when different design cases are considered.

At the end of the process, small differences were found in the tuned values of parameters t_c, α₀, and m when comparing the results obtained in both examples with rigid and semi-rigid connections. However, greater differences were found in the tuned values of parameters γ₀ and nt. This situation could create uncertainty for future users when applying PFAM to the design of a new structure that differs considerably from previously designed structures.

The PCPFAM selects the values of one or several parameters that maximize changes in landscape F_i during the design process. However, the parameters can be classified into two groups: those that can change during the process (β₀, α, γ, m) and those chosen at the beginning of the process that do not change during the process (t_c, nt, k_max). PCPFAM can only act on the former.

The uncertainty in the choice of γ was resolved using PCPFAM. After having carried out the different sensitivity analyses with PFAM, some tuned and untuned values of the rest of the parameters were adopted in PCPFAM with a constant value during the process. The results showed that the algorithm obtained good metrics when parameter γ was changeable during the design process, which made the algorithm more robust.

However, there is still uncertainty in the choice of parameter nt. The adjusted values obtained through the sensitivity analyses carried out with PFAM show that nt depends on nv, with very different values in each of the examples.

To eliminate the uncertainty in the choice of parameter nt, its dependence on nv was studied. First, the total number of possible designs (TNPD) in the examples was calculated as variations with repetitions of ne elements taken from nve to nve. In the two examples, 24 (ne = 24) HEB steel shapes were considered for the columns and 18 (ne = 18) IPE steel shapes for the beams. When there were semi-rigid connections in the examples, 10 (ne = 10) different types of steel bolts and 12 (ne = 12) different types of steel end-plates were also considered. In the examples, nve is the number of variables used for the HEB and IPE shapes and the steel bolts and steel end-plates (see Table 2, Table 3, Table 29 and Table 30). Thus, the TNPD in all the examples are included in

Table 47. TNPD and be.

Case	TNPD	be
Three-bay two-story framework with rigid connections	(ne = 18) (nve = 2) ∙ (ne = 24) (nve = 2) = = 182 ∙ 242 = 186,624 ≈ 20.7846096914 = = (be = 20.784609691) (nve = 2+2)	20.784609691
Three-bay two-story framework with semi-rigid connections	182 ∙ 242 ∙ 104 ∙ 124 = = 3.86983526 ∙ 1013 ≈ ≈ 13.56149931912	13.561499319
Three-bay ten-story framework with rigid connections	1810 ∙ 2410 = = 2.26379694 ∙ 1026 ≈ ≈ 20.78460969120	20.784609691
Three-bay two-story framework with semi-rigid connections	1810 ∙ 2410 ∙ 1010 ∙ 1210 = = 1.40168340 ∙ 1047 ≈ ≈ 15.08920115640	15.089201156

. As can be seen, ne does not vary in the different cases, and nve does. Furthermore, the TNPD is a large number in all the cases. Therefore, it seems reasonable to use logarithms, although doubts arise about the base to use. In Table 47, base be gives the same TNPD previously obtained when it is raised to the number of variables in each case.

Thus, when calculating the logarithms, a single value, which was the average of the four values of be included in Table 47 and is equal to 17.554979964, was used for the base of be.

Table 48. Calculation of logarithms over TNPD, nv, and minpop.

Case	TNPD	log17.554979964(TNPD)	nv	Minpop
Three-bay two-story framework with rigid connections	186,624	4.24	4	30
Three-bay two-story framework with semi-rigid connections	3.86983526 · 1013	10.92	12	250
Three-bay ten-story framework with rigid connections	2.26379694 ∙ 1026	21.18	20	450
Three-bay two-story framework with semi-rigid connections	1.40168340 ∙ 1047	37.89	40	1000

includes the minimum populations nt (minpop) that yielded the most precise results with minimal computational cost with k_max = 100 for each case (see Table 4, Table 5, Table 16, Table 17, Table 31, Table 32, Table 38 and Table 39). Likewise, the TNPD logarithms have been included for each case using the same base be = 17.554979964.

The third column of Table 48 shows that the logarithm calculated in each case was close to the number of variables nv considered (4, 12, 20, and 40). This is evident from looking at the calculations performed. Figure 33 shows minpop on the ordinate axis and nv on the abscissa axis. Likewise, the Least-Square Regression Line is included.

Figure 33 shows the good fit obtained (R² = 0.9997496). The calculations carried out using logarithms remind us of the concept of entropy S, which is a measure of the disorder or randomness of the system. In certain fields of knowledge, when information I is extracted from a system, entropy S is reduced [37,38,39]. Thus, the entropy S of a design space with volume TNPD, and the information I that PFAM and PCPFAM extract from it, are related (S ≥ S − I ≥ 0). In this context, minpop could be defined as the minimum or sufficient number of fireflies that PFAM or PCPFAM requires in the design process to provide enough information for a valid design. Since an algorithm can be defined as an information management tool, if the information minpop that the algorithm works with is sufficient, the difference S − I is a small enough value to obtain good metrics.

Finally, using the formula of the Least-Square Regression Line included in Figure 33, a simple expression resolving the uncertainty in the choice of nt is proposed. The estimated minimum population estminpop, as a function of nv, can be obtained using the following expression:

$$estminpop=3^3·\left(nv-3\right);\forall nv>3$$

(66)

Expression (66), applied to values of nv = 4, 12, 20, and 40, provides the values of estminpop included in Figure 34. In addition, Figure 34 includes values of minpop and the Least-Square Regression Line.

5. Conclusions

In this work, a numerical method for the optimal detailed design of steel frames based on a FA has been studied and proposed. The main objective has been to make the method sufficiently efficient and robust. The algorithm has been parallelized to reduce execution times, and a migration strategy has been implemented to allow information to be transferred among subpopulations, avoiding biases. An extensive sensitivity analysis of the algorithm has allowed knowledge about its performance to be extracted. Additionally, a variant of the algorithm has been proposed that implements a parameter control strategy. This variant has been tested on the parameter that most affects performance based on the information from the previous sensitivity analysis. Finally, a study demonstrating the near dependence between one of the parameters and the number of variables considered in the examples has been carried out. Two examples of steel structures, which have been extensively studied by other authors, have served to verify the efficiency and robustness of the optimal design method proposed.

The following conclusions can be drawn:

• The sensitivity of the PFAM algorithm is higher for changes in parameters t_c, γ₀, and nt than for changes in parameters α₀, m, and k_max.
• As expected, a close dependence between parameters t_c, α₀, γ₀, and m has been detected. Changes in the value of one produce changes in the optimal tuned values of the rest of them.
• The optimal tuned values of the parameters are different when considering EEP2 semi-rigid connections (SRC) or rigid connections (RC). The optimal tuned values somewhat affected are those related to parameters t_c (SRC: [0.1, 0.3]; RC: [0.1, 0.5]), α₀ (SRC: [0.3, 1.0]; RC: [0.3, 0.5]), γ₀ (SRC: [10, 25]; RC: [0.1, 25]), and m (SRC: [1, 2]; RC: [1, 3]). The most affected optimal tuned values are those related to parameter nt (SRC: [250, 1000]; RC: [30, 450]).
• If the intersection between the above somewhat affected optimal tuned values is considered, it is concluded that the optimal tuned values are the following: t_c ⸦ [0.1, 0.3]; α₀ ⸦ [0.3, 0.5]; γ₀ ⸦ [10, 25]; and m ⸦ [1, 2].
• The value of parameter k_max = 100 produces quite satisfactory results with a lower computational cost in all the cases considered.
• The sensitivity study does not allow an interval of optimal tuned values for the parameter nt independent of the problem to be determined.
• Considering different cases with constant values of some parameters, β₀ = 1, k_max = 100, t_c = 0.1, α₀ = 1.0, and m = 2, all of them non-optimal and with a range of values for γ of the interval [1, 50], the PCPFAM procedure produces results as satisfactory as those obtained with the PFAM procedure and with optimal tuned values of the parameters. The fact that parameter γ varies in the interval [1, 50], maximizing the fitness landscape, compensates for not choosing optimal tuned values of other parameters, which makes PCPFAM more robust than PFAM.
• PCPFAM can only act on parameters β₀, α, γ, and m, which may change during the design process, but not on parameters t_c, nt, and k_max, which remain constant. Among the latter, nt is the one that most affects the algorithm’s performance. Therefore, PCPFAM cannot eliminate the uncertainty in the choice of parameter nt.
• The optimal tuned values of the nt parameter (minpop), found in the sensitivity analysis with PFAM, were very different and depended greatly on the case considered. The study was able to determine that parameter nt depends mainly on the number of variables nv considered in each case.
• Once the dependence of parameter nt, with respect to the number of variables nv, was detected, a Least-Square regression adjustment was made with the optimal tuned values minpop of parameter nt and the number of variables nv. A simple expression, which is easy to remember, removes the uncertainty for nt, giving it a value (estminpop) that is a function of the number of variables nv in the problem.

The proposed method can be used in engineering offices to obtain optimal detailed designs in a reasonable time. To achieve this, simple models can first be defined, such as those that consider rigid connections. After analysis with commercial software, the combinations of decisive loads in the design in ultimate limit states and serviceability limit states are detected. Finally, these actions are included in the proposed method, and the detailed optimal design is obtained. This procedure is necessary, since only one combination of loads has been considered in this study.

We should highlight that the proposed method is limited to planar structures. Therefore, it does not include the effect of node rotations on elements connected perpendicularly to the plane of the structure. Furthermore, it only considers static loads. These limitations serve to define future improvements to the proposed method.