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

-e 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. -is 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.-ree mutation strategies (DE/rand/1,DE/best/2, andDE/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.-e 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. -e outstanding performance of the proposed EDE is demonstrated through optimization of five truss structures.


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].
Swarm-based algorithms, such as ACO, ABC, and PSO algorithms, are inspired by collective behavior in animals. ey 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. e 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.
e 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. is algorithm is based on PSOPC, ACO, and harmony search scheme.
eir comparison results showed that HPSACO has better efficiency and robustness than other PSObased 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. e two major strategies used in the AugPSO algorithm are heuristic-inspired boundary shifting and mutation-like particle position resetting. e two strategies are inspired by a heuristic and mutation scheme in GAs. e boundary shifting approach forces particles to move to the boundary between feasible and infeasible regions for increasing the convergence rate in searching. e 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. ey 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]. e 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. e 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]. e 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. e 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. e 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]. e 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. e 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. e performance of a metaheuristic algorithm is problem-dependent [2,3].
is 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][40][41]. ese 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 problemdependent 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. e 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]. ree mutation strategies (DE/rand/1, DE/best/2, and 2 Advances in Civil Engineering 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. e 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. e 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][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]

Problem Formulation
e 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. e 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, X � [x 1 x 2 . . . x N ] T , which are the crosssectional areas of the bar members of a truss. ese 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 x n represents the crosssectional area of the nth member of a truss. e lower and upper limits on design variable x n are x l n and x u n , 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 (g 1, g 2 , . . ., g NG ).
e bar members in the same group have identical cross-sectional areas. us, the number of design variables reduces from N to NG. Decision variable vector X can be rewritten as follows: (2)

DE Algorithm
Storn and Price [33] coined the term DE. e 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. e 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: for i � 1, 2, . . . , NP; j � 1, 2, . . . , NG, where rand[0, 1] represents a uniformly distributed random value between zero and one, X L j and X U j 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 X (1) j,i denotes the first generation.

Crossover.
To increase the diversity of parameter vectors, a trial vector u (G) j,i : 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 Advances in Civil Engineering by the user; j rand is an integer selected from 1 to NG. e criterion that a random value is less than or equal to CR can trigger the crossover for each trial vector. e rate is randomly selected for each trial vector. Before the selection step, each trial vector u (G) i 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, and where i = 1, 2, . . ., NP; j = 1, 2, . . ., NG; σ L j and σ U j 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; δ L c and δ U c are the minimum and maximum allowable displacements of the cth connection, respectively.

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

Adaptive Mutation Factor.
e 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. e 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. e 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 by where 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.

Integrated Mutation Strategy.
e integrated mutation strategy uses several mutation strategies to increase the convergence rate and search diversity in the parameter space. e three following mutation strategies are selected on the basis of the preliminary study: where r4 are random vectors, F (G) is the adaptive mutation factor, X (G) best is the best vector of the current generation, and X (G) i is the ith vector of the current generation. e 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: The adaptive mutation factor Generation (G) ... (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.
eir 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. erefore, the above strategies are combined to overcome the shortcomings of all types of strategy and to integrate their advantages. Each vector i � 1, 2, . . . , NP 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. e fourth vector (i = 4) returns by applying DE/rand/1. Figure 2 presents the overall optimization process, which includes initialization, mutation, crossover, constraints handling, and selection and is implemented as follows:

Optimization.
(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 NG × NP 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 factorF (G) � F u × G a . (e) Define mutation strategies for each vector i on the basis of the integrated mutation strategy:

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 120bar truss, and (5) a 942-bar truss. e 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) e 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.

Ten-Bar Truss.
e 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 Advances in Civil Engineering 0.10 lb/in 3 . e design variables are the cross-sectional areas of all members. e minimum and maximum cross-sectional areas of the members are set to 0.1 and 35.0 in 2 , respectively. e 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. e 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 N analyses (the product of the number of population and the number of generations or NP × G max ) 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 Start Define truss parmetrs (joints, members, loads, constraints, and E) and DE parameters (NP, F, F u , F l , CR, and G max ) Initialization (G = 1): generate an NG × NP matrix of initial population Mutation Adaptive mutation factor: Calculate the degree a = (ln F l -ln F u /ln G max ) and mutation factor F (G) = F u × G a (ii) Calculate stress σ j and displacement δ c Check constraints (feasible or not) for each trail vector Selection: select a better one between the original vector and the trial vector based on the performance Sort the solutions and assign the best vector X best for the next generation 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). e 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. erefore, 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). us, CR is set to 0.8 in EDE. e EDE-1 parameters (NP, F, CR, and G max ) are the same as those of DE. e 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. e 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. e 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). e 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,     Figure 8 is one of the benchmarks utilized in structural optimization by various numerical techniques. e material density is 0.1 lb/in 3 , and modulus of elasticity is 10,000,000 psi. e 25 members are categorized into the following eight groups, namely, (1) 25 . Table 2 presents the stress constraints on each group of elements. e 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 in 2 , respectively.

Twenty-Five-Bar Truss. e 25-bar transmission tower displayed in
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. e 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. e 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. e 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). e 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. e 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.

Seventy-Two-Bar Truss.
e 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. e members of this truss are grouped into 16 categories. Table 5 presents the two independent loading conditions applied to the 72bar space truss. All of the structural members were assumed to be made of the same material with an elastic modulus of 10 7 psi and a density of 0.1 lb/in 3 . 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. e maximum allowable stress is ±25000 psi. e minimum and maximum cross-sectional areas are set to 0.1 and 3.0 in 2 , respectively.

One-Hundred-Twenty-Bar
Truss. e 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 120bar truss. e 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/in 3 . e minimum and maximum cross-sectional areas of all members are set to 0.775 and 20.0 in 2 , respectively. e yielding stress of steel is taken  240in. 120 in. 60in. 60in. (1) (3) Advances in Civil Engineering 11 as 58,000 psi. e 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 σ − i is calculated using the slenderness ratio: where E is the modulus of elasticity, and F y is the yielding stress of steel. e slenderness ratio that separates the elastic from inelastic buckling regions C C is calculated:

Element group Member
Cross-sectional areas (in 2 ) Cao [6] Camp and Bichon [8] Camp [20] is work EDE-2 (best) EDE-3 (best) EDE-4 (best) Figure 11: Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 72-bar truss. 12 Advances in Civil Engineering Slenderness ratio λ i satisfies where k is the effective length factor, L i is the length of a member, the radius of gyration r i � 0.4993A 0.6777 i for the pipe sections, and A i is the cross-sectional area of the pipe.  Table 7 compares designs for the 120-bar truss using the proposed algorithms with other optimization techniques. e 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. e 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   ; 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 120bar truss faster than DE, and EDE converged fastest in the early search period. e 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.

Nine-Hundred and Forty-Two-Bar Truss.
e 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. e allowable displacement is limited to ±15 in, and the allowable stress is limited to ±25 ksi. e allowable cross-sectional area of all members is between 1 and 200 in 2 . e members of this truss are grouped into 59 categories. e 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. e 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. e best truss design of EDE (132441 lb) is lighter than those of DE (142436 lb) and Adaptive ESs [47]. e 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). e 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 120bar truss faster in the later search period than DE, and EDE-1 converged fastest in the early search period. e 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.  14 Advances in Civil Engineering   Figure 15: Element numbering and sizes of the 942-bar truss (reproduced from Hasancebi [47], under the Creative Commons Attribution License/public domain).   [48][49][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. e best ranks are shown in bold. From this table, we can see that P values computed through Friedman test for all five truss problems are less than 0.05. us, 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. is observation confirms the positive effect of integrated mutation and adaptive mutation factor strategies on the EDE algorithm for truss optimization.

Conclusions
is 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/randto-best/1, to increase the diversity of random search and avoid premature convergence to a local minimum. e 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 942bar truss optimization problems. e 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. is observation confirms the positive effect of integrated mutation and adaptive mutation factor strategies on the EDE algorithm for truss optimization.  Data Availability e datasets generated and/or analysed during the current study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.