Two Strategies to Improve the Differential Evolution Algorithm for Optimizing Design of Truss Structures

Kao, Ching-Yun; Hung, Shih-Lin; Setiawan, Budy

doi:https://doi.org/10.1155/2020/8741862

Advances in Civil Engineering

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Research Article | Open Access

Volume 2020 | Article ID 8741862 | https://doi.org/10.1155/2020/8741862

Two Strategies to Improve the Differential Evolution Algorithm for Optimizing Design of Truss Structures

Ching-Yun Kao,¹Shih-Lin Hung,²and Budy Setiawan²

Academic Editor: Giuseppe Quaranta

Received08 Dec 2019

Accepted30 Apr 2020

Published20 Jul 2020

Abstract

The performance of differential evolution (DE) mostly depends on mutation operator. Inappropriate configurations of mutation strategies and control parameters can cause stagnation due to over exploration or premature convergence due to over exploitation. Balancing exploration and exploitation is crucial for an effective DE algorithm. This work presents an enhanced DE (EDE) for truss design that utilizes two new strategies, namely, integrated mutation and adaptive mutation factor strategies, to obtain a good balance between the exploration and exploitation of DE. Three mutation strategies (DE/rand/1, DE/best/2, and DE/rand-to-best/1) are combined in the integrated mutation strategy to increase the diversity of random search and avoid premature convergence to a local minimum. The adaptive mutation factor strategy systematically adapts the mutation factor from a large value to a small value to avoid premature convergence in the early searching period and to increase convergence to the global optimum solution in the later searching period. The outstanding performance of the proposed EDE is demonstrated through optimization of five truss structures.

1. Introduction

Structural design optimization is a critical and challenging topic and has attracted considerable attention in the last few decades. Optimization enables designers to generate many desirable designs while saving money and time [1].

Metaheuristic has been widely used over the last few decades to solve complex optimization problems because of its simplicity, ease of implementation, and ability to avoid local optima and it deals with derivative free problems [2, 3]. Some methods of metaheuristic have been extensively utilized to solve structural optimization problems. Such methods include genetic algorithm (GA) [4–6], ant colony optimization (ACO) [7–9], artificial bee colony (ABC) [10], particle swarm optimization (PSO) [11–19], big bang–big crunch (BB-BC) [20], and differential evolution (DE) algorithms [21, 22]. These methods belong to two most predominant and successful classes or directions in metaheuristic, swarm-based algorithms and evolutionary algorithms [2, 3, 23].

Swarm-based algorithms, such as ACO, ABC, and PSO algorithms, are inspired by collective behavior in animals. They encompass the implementation of collective intelligence of groups of simple agents that are based on the behavior of real-world insect swarms, as a problem solving tool. ACO proposed by Marco Dorigo [24] is a probabilistic technique used to solve computational problems that can be reduced to finding good paths through graphs. It is inspired by the behavior of ants in finding paths from the colony to food. It has strong robustness and good distributed calculative mechanism. Besides, it is easy to combine with other methods and shows good performance in resolving complex optimization problems. However, ACO has a longer search time than other methods and tends to terminate at a nonoptimal solution. ABC introduced by Karaboga [25] is an optimization algorithm motivated by the intelligent behavior of honey bees. It is simple and only uses common control parameters, such as colony size and maximum cycle number. However, it shows slow convergence speed during searching. The PSO algorithm, which is a probabilistic and iterative approach, finds the optimal position in the search space by simulating the behavior of a flock of foraging birds. The PSO algorithm has fewer parameters and is easier to implement than the GA. It also has a higher convergence rate than other evolutionary algorithms in solving several problems [26]. Although the PSO algorithm can rapidly converge in the early searching stage, premature convergence may cause particle searching to fall into a local optimum. Most modifications on the simple PSO have been made to improve its convergence rate and to increase the swarm diversity. He et al. [27] introduced the concept of passive congregation that affects the particle velocity in accordance with the positions of other randomly selected particles. A particle swarm optimizer with passive congregation (PSOPC) can improve the search efficiency and the probability of finding the optimal solution. Kaveh and Talatahari [28] presented a heuristic particle swarm ant colony optimization (HPSACO) for the optimum design of trusses. This algorithm is based on PSOPC, ACO, and harmony search scheme. Their comparison results showed that HPSACO has better efficiency and robustness than other PSO-based algorithms and has a higher convergence rate than PSO and PSOPC. Lu et al. [29] proposed an augmented PSO (AugPSO) algorithm with an increased convergence rate in early search and increased diversity that does not fall into a local optimum. The two major strategies used in the AugPSO algorithm are heuristic-inspired boundary shifting and mutation-like particle position resetting. The two strategies are inspired by a heuristic and mutation scheme in GAs. The boundary shifting approach forces particles to move to the boundary between feasible and infeasible regions for increasing the convergence rate in searching. The particle position resetting approach aims to increase the diversity of particles and to prevent the solution of particles from falling into local minima. Numerical analyses showed that the AugPSO algorithm is more robust than the PSO and PSOPC algorithms.

Evolutionary algorithms such as GA, BB-BC, and DE are inspired by natural evolution. They are population-based stochastic search algorithms performing with best-to-survive criteria. Each algorithm commences by creating an initial population of feasible solutions and iteratively evolves from generation to generation toward the best solution. In successive iterations of the algorithm, fitness-based selection occurs within the population of solutions. Better solutions are preferentially selected for survival into the next generation of solutions [23]. The GA proposed by Holland [30] is the most well-known branch among existing evolutionary algorithms. It follows the principles of the Charles Darwin’s theory of survival of the fittest. The three principal genetic operators in the GA involve selection, crossover, and mutation. Although a GA has many positive features, the convergence of binary GA is slow, and the result may not be the optimal solution. GAs are unsuitable for solving constraint optimization problems [23]. The BB-BC algorithm developed by Erol and Eksin [31] is inspired from evolution theories of the universe, namely, the big bang and big crunch theory. The BB-BC algorithm consists of two parts, namely, the big bang, where candidate solutions are randomly distributed over the search space, and the big crunch, where a contraction operation estimates a weighted average or center of mass for the population. The BB-BC algorithm has been applied to various optimization problems in different fields. However, the BB-BC algorithm is easily trapped in local optima, similar to most other heuristic optimization algorithms [32]. DE proposed by Storn and Price [33] has yielded promising results for solving complex optimization problems. It is well known for its simple structure, ease of application, quality of solution, and robustness [34]. Similar to other evolutionary algorithms, DE simulates the natural evolution via mutation, crossover, and selection to evolve a population of initially random solutions into an optimal solution [35]. The main difference between the GA and DE is that mutation is the result of small perturbations to the genes of an individual in the GA, whereas mutation is the result of arithmetic combinations of individuals in DE. Although DE is recognized for its simplicity and efficiency, Mohamed et al. [36] noted that DE has the following shortcomings: the convergence rate of DE is low; premature convergence in which the search process is trapped in a local optimum may cause DE to become progressively less diverse; there is a stagnation problem, in which the search process occasionally stops proceeding toward the global optimum although the population has not converged to any other point.

A metaheuristic algorithm globally explores the problem space and locally searches in the neighborhoods of the existing solutions to obtain new and better solutions. For the metaheuristic algorithm in solution space, balancing exploration and exploitation is crucial for an effective optimization algorithm. The former indicates the ability of the algorithm to discover new search areas, while the latter focuses on finding the best solution in a promising region of the search space. The performance of a metaheuristic algorithm is problem-dependent [2, 3].

This study focuses on performance improvement of DE for truss optimization problems. Mutation operator plays a crucial role in the DE algorithm [37]. However, inappropriate configurations of mutation strategies and control parameters (population size NP, mutation factor F, and crossover rate CR) can cause stagnation due to over exploration or premature convergence due to over exploitation [38]. Many researchers have suggested new techniques to improve the original DE [39–41]. These proposed modifications on DE are adjusting control parameters in an adaptive or self-adaptive manner and developing new mutations rule. In fact, the parameters of DE are problem-dependent and it is difficult to adjust them for different problems [42]. To improve the performance of original DE for truss optimization problem, an enhanced DE (EDE) algorithm is proposed in this work to obtain a good balance between the exploration and exploitation of DE. The two major strategies used in the EDE algorithm are integrated mutation and adaptive mutation factor strategies. Some of the developed mutation strategies are fit for global search with good exploration ability (e.g., mutation strategy DE/rand/1), and others are good at local search with good exploitation ability (e.g., mutation strategy DE/best/2) [43]. Three mutation strategies (DE/rand/1, DE/best/2, and DE/rand-to-best/1) are combined in the integrated mutationstrategy to increase the diversity of random search (the exploration ability in solution space) and avoid premature convergence to a local minimum. Each individual in the population uses one of the three mutation strategies. For gradient-based analytical approach, two steps are involved in each search step. The first step is finding an appropriate search direction, a step-descent gradient vector, in N-dimension solution space. Analogy in the DE algorithm, this step is finding an appropriate differential vector in mutation and crossover operators. The second step is a line search approach that finds a step length in that gradient direction (in a 1D problem). Similarly, this step sets a mutation factor for each individual in the DE algorithm. Inspired by gradient-based analytical approach, the adaptive mutation factor strategy systematically adapts the mutation factor from a large value to a small value on the basis of the typical convergence curves of truss optimization [4–22] to avoid premature convergence in the early searching period and to increase convergence to the global optimum solution in the later searching period. Five truss examples were presented to demonstrate the effectiveness and efficiency of the proposed EDE algorithm in this study. The performance of EDE was compared with that of original DE and other metaheuristic algorithms in the literature.

2. Problem Formulation

The optimization problem aims to optimize the design by minimizing the total weight of all members of a structure while satisfying the displacement and stress constraints on the design variables and structural responses. The general form is expressed as follows:where objective function O(X) is the total weight of the truss. Function h_j(X) and e_j are the jth inequality constraint function and its predefined specified threshold, respectively. Decision variable vector X is composed of N design variables, , which are the cross-sectional areas of the bar members of a truss. These design variables define the design space; the objective function is a “surface” of N dimensions embedded in a space of N + 1 dimensions. Design variable represents the cross-sectional area of the nth member of a truss. The lower and upper limits on design variable are and , respectively. An available solution region (problem search space) for the optimization problem is defined in terms of these limits. Inequality constraints can then be applied to reduce the size of the feasible region. In this work, these N design variables are clustered into NG groups, and each group may contain a variant number of members (_,, …, ). The bar members in the same group have identical cross-sectional areas. Thus, the number of design variables reduces from N to NG. Decision variable vector X can be rewritten as follows:

3. DE Algorithm

Storn and Price [33] coined the term DE. The basic idea that underlies DE is that the difference between two vectors yields a difference vector that can be utilized with a scaling factor to diversify the search space. The general DE procedure consists of four stages, namely, initialization, mutation, crossover, and selection.

3.1. Initialization

An initial population with NP individuals is generated through random sampling from the search space. Each individual is a vector containing NG design variables . To prevent the initial vectors from falling into an infeasible area, the initial population of cross-sectional areas of truss members is generated by selecting random values between the lower (half of the upper bound) and upper bounds:where rand[0, 1] represents a uniformly distributed random value between zero and one, and are the minimum and maximum cross-sectional areas of the jth group members, respectively, NP is the population size of DE, and NG is the number of element groups clustered from N design variables. Superscript (1) of denotes the first generation.

3.2. Mutation

For each vector , a mutated vector is generated:where and integers are randomly generated such that . F is the mutation factor, which controls the amplification of the difference vector . Superscript (G) denotes the Gth generation.

3.3. Crossover

To increase the diversity of parameter vectors, a trial vector is formed for each element :where i = 1, 2, …,NP; j = 1, 2, …, NG; CR is the crossover rate, which is drawn from the range of zero to one and preset by the user; is an integer selected from 1 to NG. The criterion that a random value is less than or equal to CR can trigger the crossover for each trial vector. The rate is randomly selected for each trial vector.

Before the selection step, each trial vector is checked to determine whether it violates the relevant constraints or not. If it does, then the trial vector will be rejected and replaced with the original vector of the current generation, andwhere i = 1, 2, …, NP; j = 1, 2, …, NG; and are the minimum and maximum allowable stresses of the jth member group, respectively; c = 1, 2, …, NC, and NC is the number of joints or connections; and are the minimum and maximum allowable displacements of the cth connection, respectively.

3.4. Selection

The performances of the trial and original vectors in the selection operation are compared. The better one is selected and passed to the next generation. The trial vector should satisfy the predefined constraints. The performance is determined using objective function O, which is the total weight of the structure under consideration. The new population for the next generation is formed aswhere and are the ith individuals of the population for the next generation (generation G + 1) and the current generation (generation G), respectively, and is the trial vector. The mutation, crossover, and selection steps are repeated until a specified number of generations is reached.

4. EDE Algorithm

4.1. Adaptive Mutation Factor

The DE algorithm with a fixed mutation factor has several shortcomings; for example, a low mutation factor yields a slow convergence rate, and a high mutation factor causes stagnation problem and fails to yield a good result because the search space is extremely large. The first strategy adaptively updates the mutation factor between the minimum and maximum mutation factors in each generation. In the early searching period, a large mutation factor is required to search with great diversity and to avoid premature convergence. In the later searching period, a low mutation factor is utilized to increase convergence for reaching the global optimum solution. The adaptive mutation factor function is developed on the basis of the typical convergence curves of truss optimization. Figure 1 plots the power function. For each generation G, a mutation factor F^(G) is generated using a power function with degree a and multiplier F_u:and the degree of power function a is given bywhere G = 1, 2, …, G_max, and G_max is the total number of generations. F_u and F_l are the user-defined maximum and minimum mutation factors, respectively, and their values are between zero and one.

4.2. Integrated Mutation Strategy

The integrated mutation strategy uses several mutation strategies to increase the convergence rate and search diversity in the parameter space. The three following mutation strategies are selected on the basis of the preliminary study:

(a)DE/rand/1(b)DE/best/2(c)DE/rand-to-best/1

where is the mutated vector, , , , and are random vectors, F^(G) is the adaptive mutation factor, is the best vector of the current generation, and is the ith vector of the current generation. The three strategies are chosen because each of them has unique characteristics and advantages.

According to Qiang and Mitchell [43], the three strategies are performed as follows:(a)DE/rand/1 is the original strategy and has a stronger exploration capability but converges more slowly than the strategies that use the best solution from the parent generation.(b)DE/best/2uses the best solution in the parent population and converges rapidly to the optimal solution, but it may encounter the stagnation problem, which refers to becoming stuck at a local minimum during multimodal function optimization.(c)DE/rand-to-best/1 compromises between the exploitation of the best solution and exploration of the random parameter space. It has highly diverse mutant vectors that are obtained from a randomly selected parent vector.

Ao and Chi [37] experimentally investigated differential evolution strategies (ESs). Their experimental results revealed that strategies that use the best vector typically have a high convergence speed but easily become stuck at a local optimum, whereas the DE/rand strategies converge faster to the global optimum. Therefore, the above strategies are combined to overcome the shortcomings of all types of strategy and to integrate their advantages. Each vector uses one of these strategies in the following sequence. For example, the first vector (i = 1) uses DE/rand/1, the second vector (i = 2) uses DE/best/2, and the third vector (i = 3) uses DE/rand-to-best/1. The fourth vector (i = 4) returns by applying DE/rand/1.

4.3. Optimization

Figure 2 presents the overall optimization process, which includes initialization, mutation, crossover, constraints handling, and selection and is implemented as follows:(a)Define the truss parameters, including joint coordinates, member connections, load cases, and constraints.(b)Define the DE parameters NG, NP, F, F_u, F_l, CR, and G_max.(c)Generate an matrix of the initial population by randomly generating vector elements between 0.5X^U and X^U.(d)Calculate degree of the power function a and adaptive mutation factor.(e)Define mutation strategies for each vector i on the basis of the integrated mutation strategy:(f)Generate NP mutated vectors and check if the cross-sectional areas in are beyond the boundary.(g)Crossover step: generate NP trial vectors.(h)Calculate stress and displacement by performing a matrix analysis of structures.(i)Check whether each of the trial vectors violates the constraints; if it does, then replace it by the original vector or the parent solution.(j)Selection step: select the best solution from the original and trial vectors on the basis of the objective function.(k)Sort the solutions and specify the best vector.(l)Repeat “step f” to “step k” for the next generation until the maximum number of generations is reached.(m)Generate the optimal solution.

The program is terminated when a defined number of generations G_max is reached and the final solution is generated. The final result is the total weight of the structure.

5. Numerical Study

Various benchmarks of truss structures were studied to evaluate the performance of the proposed EDE method: (1) a 10-bar truss, (2) a 25-bar truss, (3) a 72-bar truss, (4) a 120-bar truss, and (5) a 942-bar truss. The performance of the proposed method was compared with those of other methods in the literature to verify its feasibility. To test the individual effect of the two strategies, adaptive mutation factor strategy and integrated mutation strategy, used in the EDE algorithm to improve DE algorithm, the following versions of algorithm are also compared with EDE:(1)The same EDE version without adaptive mutation factor strategy, called EDE-1, is experimentally investigated to test the individual effect of adaptive mutation factor strategy on the performance of EDE algorithm.(2)The same EDE version without DE/rand/1 mutation, called EDE-2, is experimentally investigated to test the individual effect of integrated mutation strategy on the performance of EDE algorithm.(3)The same EDE version without DE/best/2 mutation, called EDE-3, is experimentally investigated to test the individual effect of integrated mutation strategy on the performance of EDE algorithm.(4)The same EDE version without DE/rand-to-best/1 mutation, called EDE-4, is experimentally investigated to test the individual effect of the proposed integrated mutation strategy on the performance of EDE algorithm.

5.1. Ten-Bar Truss

The truss is a cantilevered truss with a pinned support and a roller support. Figure 3 displays the configuration of the 10-bar truss and the loading condition. All members are assumed to be made from the same material with an elastic modulus of E = 10,000,000 psi and a density of 0.10 lb/in³. The design variables are the cross-sectional areas of all members. The minimum and maximum cross-sectional areas of the members are set to 0.1 and 35.0 in², respectively. The maximum deflection of any node in either direction must not exceed ±2 in, and the maximum allowable stress of all members is set to ±25,000 psi.

The DE parameters that need to be set are NP, F, CR, and G_max. In this study, NP is set to a value between 5 NG and 8 NG based on suggestions of Storn and Price [44] and Gamperle at al. [45], and NP is set to 50 in this example (NG is equal to 10 in this example). After determination of NP, G_max is set to a value making the number of analyses (the product of the number of population and the number of generations or ) of DE less than those of other methods in the literature for performance comparison. G_max is set to 200 in this example. After determination of NP and G_max, F and CR are determined by a two-step method. First, CR is set to 0.1 and F is from 0.1 to 1.0 every 0.1 to test DE to determine adequate F based on the suggestion of Storn and Price [44] that crossover rate CR = 0.1 is an initial good choice. DE with a certain F value is tested for five runs. Figure 4 shows the effects of various values of F on average weight of designs for the 10-bar truss using DE with CR = 0.1. Herein, mutation factor F is set to 0.5 since it provided minimum average weight (5209.233 lb). Second, CR is determined on the basis of predetermined F. F is the predetermined value and CR is from 0.1 to 1.0 every 0.1 to test DE to determine adequate CR. DE with a certain CRvalue is tested for five runs. Figure 5 shows the effects of various values of CR on average weight of designs for the 10-bar truss using DE with F = 0.5. Herein, CR is set to 0.9 since it provided minimum average weight (5063.911 lb). The EDE parameters that need to be set are NP, F_l, F_u, CR, and G_max. NP and G_max are the same as those of DE. F_l and F_u are determined based on the results of setting F in DE. Figure 4 shows that average weight is smaller when F is between 0.3 and 1.0. Therefore, F_u and F_l are set to 0.3 and 1.0, respectively. After determining F_l and F_u, CR is set to be from 0.1 to 1.0 every 0.1 to test EDE to determine adequate CR. EDE with a certain CR value is tested for five runs. Figure 6 shows the effects of various values of CR on average weight of designs for the 10-bar truss using EDE with F_l = 0.3 and F_u = 1.0. It shows that CR = 0.8 provided minimum average weight (5061.369 lb). Thus, CR is set to 0.8 in EDE. The EDE-1 parameters (NP, F, CR, and G_max) are the same as those of DE. The parameters of EDE-2, EDE-3, and EDE-4 (NP, F_l, F_u, CR, and G_max) are the same as those of EDE. The same method is also used to determine parameters (NP, F, F_l, F_u, CR, and G_max) in the following examples.

Table 1 compares designs for the 10-bar truss using the proposed algorithms with other optimization techniques. It is noted that the design runs of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 are set to 30 for all truss design examples in this study, since the normal approximation for design parameter (truss weight) will generally be good if the design runs are larger and equal to 30. Considering that the solution of HPSACO [28] violated the relevant constraints, the best truss design of EDE (5060.896 lb) is lighter than those of DE (5061.578 lb) and other four optimization techniques in the literature. The mean (5061.734 lb) of design weight of EDE is also smaller than those of DE and the other three optimization techniques in the literature (PSO, PSOPC, and AugPSO). The standard deviation of design weight of EDE (2.877 lb) is larger than that of DE but smaller than those of the other three optimization techniques in the literature (PSO, PSOPC, and AugPSO). Moreover, EDE required fewer analyses compared with GA [5]. Results revealed that EDE is more robust than other heuristic methods in solving this benchmark problem. Since the difference between mean (or median) weights of DE and EDE-1 is larger than that of EDE and EDE-1, the effect of integrated mutation strategy is larger than that of adaptive mutation factor strategy on the performance of EDE in this example. Comparing DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4, EDE provided minimum best design weight (5060.896 lb) and EDE-1 provided minimum median weight (5061.097 lb). In fact, the median weight of EDE (5061.098 lb) is almost the same as that of EDE-1. However, EDE-3 provided minimum mean (5061.517 lb) and standard deviation (0.500 lb) of design weight. Notably, comparing EDE, EDE-1, EDE-2, EDE-3, and EDE-4, the standard deviation of design weight of EDE-3 (0.500 lb) is minimum but the best design weight of EDE-3 (5060.987 lb) is maximum, which implies that a DE algorithm without using DE/best/2 strategy provides a more reliable solution but has less opportunity to search for the global optimum solution. Figure 7 compares convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4. EDE performed better and converged much faster than DE, and EDE-1 converged fastest in the early search period. The DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms required approximately 77, 56, 57, 66, 66, and 73 generations (3850, 2800, 2850, 3300, 3300, and 3650 analyses), respectively, to converge to a solution.

5.2. Twenty-Five-Bar Truss

The 25-bar transmission tower displayed in Figure 8 is one of the benchmarks utilized in structural optimization by various numerical techniques. The material density is 0.1 lb/in³, and modulus of elasticity is 10,000,000 psi. The 25 members are categorized into the following eight groups, namely, (1) A₁, (2) A₂‒A₅, (3) A₆‒A₉, (4) A₁₀‒A₁₁, (5) A₁₂‒A₁₃, (6) A₁₄‒A₁₇, (7) A₁₈‒A₂₁, and (8) A₂₂–A₂₅.

Table 2 presents the stress constraints on each group of elements. The structure is subjected to the loading condition specified in Table 3. Allowable displacements of each node in the x, y, and z directions are limited to a maximum value of 0.35 in. For each group of members, the minimum and maximum cross-sectional areas were set to 0.01 and 3.4 in², respectively.

In this example, the parameters of DE and EDE-1 are set to NP = 50, F = 0.6, CR = 0.9, and G_max = 160. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 50, F_u = 1.0, F_l = 0.3, CR = 0.9, and G_max = 160. Table 4 compares designs for the 25-bar truss using the proposed algorithms with other optimization techniques. The best truss design of EDE (545.163 lb) is lighter than those of DE (545.319 lb) and five other optimization techniques in the literature. The mean (545.166 lb) and standard deviation (0.007 lb) of design weight of EDE are also smaller than those of DE and other optimization techniques in the literature. Moreover, EDE required the fewest analyses compared with PSO [12], ACO [8], and BB-BC [20]. Results revealed that EDE is more robust than other heuristic methods in solving this benchmark problem. Since the difference between mean (or median) weights of DE and EDE-1 is smaller than that of EDE and EDE-1, the effect of integrated mutation strategy is smaller than that of adaptive mutation factor strategy on the performance of EDE in this example. Comparing DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4, minimum best design weight (545.163 lb) was provided by EDE, EDE-2, and EDE-4, and EDE provided minimum mean weight (545.166 lb), median weight (545.164 lb), and standard deviation of design weight (0.007 lb). The results showed that the EDE algorithm exhibited an improved computational efficiency and consistent performance compared with all other algorithms for this example. As presented in Figure 9, the optimization of 25-bar truss using EDE converged to the optimal solution more quickly than that using DE, and EDE-4 converged fastest in the early search period. The DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms required approximately 100, 80, 87, 80, 80, and 80 generations (5000, 4000, 4350, 4000, 4000, and 4000 analyses), respectively, to converge to a solution.

5.3. Seventy-Two-Bar Truss

The 72-bar truss is a benchmark problem in 3D truss optimization and has been tackled by several researchers using various methods. Figure 10 presents the configuration of the 72-bar space truss, including its nodes and the corresponding schemes for numbering groups of elements. The members of this truss are grouped into 16 categories. Table 5 presents the two independent loading conditions applied to the 72-bar space truss. All of the structural members were assumed to be made of the same material with an elastic modulus of 10⁷ psi and a density of 0.1 lb/in³. Displacements of the uppermost joints in the x and y directions are limited to be a maximum value of 0.25 in, satisfying the constraint in load case 1. Under load case 2, the maximum displacement of top floor in the z direction satisfies the displacement constraint. The maximum allowable stress is ±25000 psi. The minimum and maximum cross-sectional areas are set to 0.1 and 3.0 in², respectively.

In this example, the parameters of DE and EDE-1 are set to NP = 60, F = 0.4, CR = 0.9, and G_max = 200. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 60, F_u = 0.6, F_l = 0.3, CR = 0.9, and G_max = 200. Table 6 compares designs for the 72-bar truss using the proposed algorithms with other optimization techniques. The best truss design of EDE (379.645 lb) is lighter than those of DE (379.939 lb) and four other optimization techniques in the literature. The mean (379.807 lb) and standard deviation (0.184 lb) of design weight of EDE are also smaller than those of DE and other optimization techniques in the literature. Moreover, EDE required fewer analyses compared with GA [6], ACO [8], and BB-BC phase 1 [20]. Results revealed that EDE is more robust than other heuristic methods in solving this benchmark problem. Since the difference between mean (or median) weights of DE and EDE-1 is larger than that of EDE and EDE-1, the effect of integrated mutation strategy is larger than that of adaptive mutation factor strategy on the performance of EDE in this example. Comparing DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4, EDE provided minimum best design weight (379.645 lb) and mean weight (379.807 lb), and EDE-2 provided minimum median weight (379.734 lb). Although the standard deviation of design weight of EDE (0.184 lb) is a little larger than that of EDE-1 (0.108 lb), the results still showed that the EDE algorithm had excellent computational efficiency and consistency for this example. As presented in Figure 11, EDE optimized the 72-bar truss faster than DE, and EDE-2 converged fastest in the early search period. The DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms required approximately 126, 88, 92, 99, 114, and 83 generations (7560, 5280, 5520, 5940, 6840, and 4980 analyses), respectively, to converge to a solution.

5.4. One-Hundred-Twenty-Bar Truss

The 120-bar spatial truss is a dome-shaped truss and has become a benchmark in truss optimization. Figure 12 presents the configuration and group numbering schemes of the 120-bar truss. The structural members are symmetrically organized into seven groups of elements. All of the structural members were assumed to be made of the same material with an elastic modulus of 30,450,000 psi and a density of 0.288 lb/in³. The minimum and maximum cross-sectional areas of all members are set to 0.775 and 20.0 in², respectively. The yielding stress of steel is taken as 58,000 psi. The stress and displacement constraints are presented as follows:(a)According to the American Institute of Steel Construction’s allowable strength design [46], the allowable stress satisfies where is calculated using the slenderness ratio: where E is the modulus of elasticity, and F_y is the yielding stress of steel. The slenderness ratio that separates the elastic from inelastic buckling regions C_C is calculated: Slenderness ratio satisfies where k is the effective length factor, is the length of a member, the radius of gyration for the pipe sections, and is the cross-sectional area of the pipe.(b)Maximum displacement of all nodes in the x, y, and z directions is ±0.1969 in.

The 120-bar dome has 49 joints, with 37 unsupported joints and 12 hinged supports. The truss is subjected to vertical loading only at its unsupported joints in each layer ring. The loads are −13.49 kip at node 1, −6.744 kip at nodes 2 to 14, and −2.248 kip at other unsupported nodes (nodes 15 to 37).

In this example, the parameters of DE and EDE-1 are set to NP = 50, F = 0.4, CR = 0.9, and G_max = 200. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 50, F_u = 0.5, F_l = 0.4, CR = 0.9, and G_max = 200. Table 7 compares designs for the 120-bar truss using the proposed algorithms with other optimization techniques. The best truss design of EDE (20665.883 lb) is lighter than those of DE (20666.393 lb) and three other optimization techniques in the literature. The mean (20666.137 lb) and standard deviation (0.488 lb) of design weight of EDE are also smaller than those of DE and three other optimization techniques in the literature. Results revealed that EDE is more robust than other heuristic methods in solving this benchmark problem. Since the difference between mean (or median) weights of DE and EDE-1 is larger than that of EDE and EDE-1, the effect of integrated mutation strategy is larger than that of adaptive mutation factor strategy on the performance of EDE in this example. Comparing DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4, EDE-1 provided minimum best design weight (20665.825 lb), EDE provided minimum mean weight (20666.137 lb), EDE-1 provided minimum median weight (20665.986 lb), and EDE-3 provided minimum standard deviation (0.356 lb) of design weight. In fact, the median weight of EDE (20665.989 lb) is only a little larger than that of EDE-1, and the standard deviation of design weight of EDE (0.488 lb) is only a little larger than that of EDE-3 (0.356 lb); the results still showed that the EDE algorithm had excellent computational efficiency and consistency for this example. As presented in Figure 13, EDE optimized the 120-bar truss faster than DE, and EDE converged fastest in the early search period. The DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms required approximately 82, 60, 70, 66, 67, and 87 generations (4100, 3000, 3500, 3300, 3350, and 4350 analyses), respectively, to converge to a solution.

5.5. Nine-Hundred and Forty-Two-Bar Truss

The 942-bar spatial truss is a 26-story tower truss proposed by several researchers. Figures 14 and 15 display the geometry and element numbering of the truss. The allowable displacement is limited to ±15 in, and the allowable stress is limited to ±25 ksi. The allowable cross-sectional area of all members is between 1 and 200 in². The members of this truss are grouped into 59 categories. The tower is subject to a single loading condition consisting of horizontal and vertical loads, as follows: (i) the vertical loads in the z direction are −3.0, −6.0, and −9.0 kips at each node in the first, second, and third sections, respectively, (ii) the lateral loads in the y direction are 1.0 kip at all nodes of the tower, and (iii) the lateral loads in the x direction are 1.5 and 1.0 kips at each node on the left and right sides of the tower, respectively.

In this example, the parameters of DE and EDE-1 are set to NP = 200, F = 0.3, CR = 0.7, and G_max = 500. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 200, F_u = 0.7, F_l = 0.3, CR = 0.8, and G_max = 500. Table 8 compares designs for the 942-bar truss using the proposed algorithms with other optimization techniques. The best truss design of EDE (132441 lb) is lighter than those of DE (142436 lb) and Adaptive ESs [47]. The mean (133153 lb) and standard deviation (483.1 lb) of design weight of EDE are also smaller than those of DE. Moreover, EDE required fewer analyses compared with Adaptive ESs [47]. Results revealed that EDE is more robust than DE and Adaptive ESs. Since the difference between mean (or median) weights of DE and EDE-1 is larger than that of EDE and EDE-1, the effect of integrated mutation strategy is larger than that of adaptive mutation factor strategy on the performance of EDE in this example. Comparing DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4, EDE provided minimum best design weight (132441 lb), mean weight (133153 lb), and median weight (133018 lb). The standard deviation of design weight of EDE (483.1 lb) is only a little larger than that of EDE-3 (396.9 lb) and smaller than others; the results showed that the EDE algorithm had excellent computational efficiency and consistency for this example. As presented in Figure 16, EDE optimized the 120-bar truss faster in the later search period than DE, and EDE-1 converged fastest in the early search period. The DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms required approximately 400, 300, 260, 300, 300, and 400 generations (80000, 60000, 52000, 60000, 60000, and 80000 analyses), respectively, to converge to a solution.

5.6. Numerical Results and Statistical Comparisons of the Proposed Algorithms

In this section, the statistical analysis results of applying Friedman test [48–50] between DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 algorithms will be discussed. Table 9 lists the average ranks according to Friedman test for the six algorithms. The best ranks are shown in bold. From this table, we can see that values computed through Friedman test for all five truss problems are less than 0.05. Thus, it can be concluded that there is a significant difference between the performances of the algorithms. EDE ranked best in all five truss problems. Regarding mean ranking, EDE gets the first ranking followed by EDE-2, EDE-1, EDE-3, EDE-4, and DE. This observation confirms the positive effect of integrated mutation and adaptive mutation factor strategies on the EDE algorithm for truss optimization.

6. Conclusions

This work proposed an EDE algorithm for truss design, which improves the performance of the original DE by modifying the mutation operator using two new strategies, namely, adaptive mutation factor and integrated mutation strategies. Adaptive mutation factor strategy systematically adapts the mutation factor from a large value to a small value on the basis of the typical convergence curves of truss optimization to avoid premature convergence in the early searching period and to increase convergence to the global optimum solution in the later searching period. Integrated mutation strategy combined three mutation strategies, DE/rand/1, DE/best/2, and DE/rand-to-best/1, to increase the diversity of random search and avoid premature convergence to a local minimum. The effectiveness of the proposed EDE was demonstrated by using it to solve the 10-bar truss, 25-bar truss, 72-bar truss, 120-bar truss, and 942-bar truss optimization problems. The following important conclusions are drawn from the results.(1)EDE yielded results that competed favorably with those generated using original DE and other metaheuristic algorithms (the GA, PSO, PSOPC, AugPSO, ACO, BB-BC, and ABC algorithms) in the literature. Furthermore, EDE provided an extraordinary result within fewer analyses than required by other methods. EDE is highly competitive in terms of robustness, stability, and quality of the solution obtained.(2)Compared with EDE-1, EDE-2, EDE-3, and EDE4, EDE provided minimum best design weight and mean (or median) design weight, and the standard deviation of design weight of EDE is small for most truss optimization problems. Moreover, EDE gets the first ranking followed by EDE-2, EDE-1, EDE-3, and EDE-4 by the Friedman test. This observation confirms the positive effect of integrated mutation and adaptive mutation factor strategies on the EDE algorithm for truss optimization.(3)Mutation factor F and crossover rate CR greatly influence the effectiveness, efficiency, and robustness of the DE algorithm. The choice of F_l and F_u in the proposed EDE algorithm is subjective. Further research could investigate how to determine F_l and F_u more objectively. Moreover, EDE uses a constant CR in this study; the proposed EDE algorithm with an adaptive CR could be further researched.(4)This study focuses on comparing effectiveness of the proposed EDE algorithm with those of original DE and other metaheuristic algorithms in the literature for solving truss optimization problems. The effectiveness comparison of EDE and other advanced DE variants and application of EDE to solve other problems could be further researched.

Data Availability

The datasets generated and/or analysed during the current study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Ching-Yun Kao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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