^{1}

^{2}

^{2}

^{1}

^{2}

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,

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 [

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 [

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 [

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 [

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 [

This study focuses on performance improvement of DE for truss optimization problems. Mutation operator plays a crucial role in the DE algorithm [

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:_{j}(_{j} are the _{,}

Storn and Price [

An initial population with NP individuals is generated through random sampling from the search space. Each individual is a vector containing NG design variables

For each vector

To increase the diversity of parameter vectors, a trial vector

Before the selection step, each trial vector

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

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 ^{(G)} is generated using a power function with degree _{u}:_{max}, and _{max} is the total number of generations. _{u} and _{l} are the user-defined maximum and minimum mutation factors, respectively, and their values are between zero and one.

Adaptive mutation factor curve.

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:

DE/rand/1

DE/best/2

DE/rand-to-best/1

where ^{(G)} is the adaptive mutation factor,

According to Qiang and Mitchell [

Ao and Chi [

Figure

Define the truss parameters, including joint coordinates, member connections, load cases, and constraints.

Define the DE parameters NG, NP, _{u}, _{l}, CR, and _{max}.

Generate an ^{U} and ^{U}.

Calculate degree of the power function

Define mutation strategies for each vector

Generate NP mutated vectors

Crossover step: generate NP trial vectors.

Calculate stress and displacement by performing a matrix analysis of structures.

Check whether each of the trial vectors violates the constraints; if it does, then replace it by the original vector or the parent solution.

Selection step: select the best solution from the original and trial vectors on the basis of the objective function.

Sort the solutions and specify the best vector.

Repeat “step f” to “step k” for the next generation until the maximum number of generations is reached.

Generate the optimal solution.

Flow chart of the optimization process.

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

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,

The same EDE version without

The same EDE version without

The same EDE version without

The same EDE version without

The truss is a cantilevered truss with a pinned support and a roller support. Figure ^{3}. 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 in^{2}, 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.

Configuration of the 10-bar truss (reproduced from Lu et al. [

The DE parameters that need to be set are NP, _{max}. In this study, NP is set to a value between 5 NG and 8 NG based on suggestions of Storn and Price [_{max} is set to a value making the number of analyses _{max} is set to 200 in this example. After determination of NP and _{max}, _{l}, _{u}, CR, and _{max}. NP and _{max} are the same as those of DE. _{l} and _{u} are determined based on the results of setting _{u} and _{l} are set to 0.3 and 1.0, respectively. After determining _{l} and _{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 _{l} = 0.3 and _{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, _{max}) are the same as those of DE. The parameters of EDE-2, EDE-3, and EDE-4 (NP, _{l}, _{u}, CR, and _{max}) are the same as those of EDE. The same method is also used to determine parameters (NP, _{l}, _{u}, CR, and _{max}) in the following examples.

Effects of various values of

Effects of various values of CR on average weight of designs for the 10-bar truss using DE with

Effects of various values of CR on average weight of designs for the 10-bar truss using EDE with _{l} = 0.3 and _{u} = 1.0.

Table

Performance comparison for the 10-bar truss.

Element group | Member | Cross-sectional areas (in^{2}) | ||||||||||

Camp et al. [ | Kaveh and Talatahari [ | Lu et al. [ | This work | |||||||||

GA | HPSACO | PSO | PSOPC | AugPSO | DE | EDE | EDE-1 | EDE-2 | EDE-3 | EDE-4 | ||

1 | 1 | 28.92 | 30.307 | 20.149 | 25.923 | 30.457 | 30.272 | 30.542 | 30.559 | 30.501 | 30.721 | 30.570 |

2 | 2 | 0.1 | 0.1 | 0.1 | 0.39 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

3 | 3 | 24.07 | 23.434 | 32.233 | 23.247 | 23.584 | 23.425 | 23.203 | 23.317 | 23.304 | 23.175 | 23.232 |

4 | 4 | 13.96 | 15.505 | 14.831 | 18.208 | 15.029 | 14.842 | 15.233 | 15.193 | 15.261 | 15.295 | 15.172 |

5 | 5 | 0.1 | 0.1 | 0.1 | 0.108 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

6 | 6 | 0.56 | 0.5241 | 0.116 | 0.1 | 0.564 | 0.559 | 0.530 | 0.526 | 0.547 | 0.551 | 0.556 |

7 | 7 | 7.69 | 7.4365 | 8.349 | 9.007 | 7.42 | 7.445 | 7.456 | 7.448 | 7.439 | 7.466 | 7.448 |

8 | 8 | 21.95 | 21.079 | 28.039 | 26.629 | 20.987 | 21.128 | 21.019 | 21.020 | 20.961 | 21.023 | 20.969 |

9 | 9 | 22.09 | 21.229 | 22.909 | 18.736 | 21.524 | 21.745 | 21.539 | 21.486 | 21.540 | 21.361 | 21.582 |

10 | 10 | 0.1 | 0.1 | 3.066 | 0.196 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

Weight_{best} (lb) | 5076.31 | 5,606.036 | 5,225.282 | 5,061.209 | 5061.578 | 5060.896 | 5060.951 | 5060.899 | 5060.987 | 5060.900 | ||

Weight_{worst} (lb) | — | — | 7,008.815 | 5,854.163 | 5,179.516 | 5066.422 | 5076.892 | 5076.791 | 5076.687 | 5062.822 | 5076.907 | |

Weight_{mean} (lb) | — | — | 6,256.895 | 5,425.074 | 5,103.484 | 5063.490 | 5061.734 | 5061.719 | 5061.878 | 5061.517 | 5061.930 | |

Weight_{median} (lb) | — | — | — | — | — | 5062.980 | 5061.098 | 5061.097 | 5061.120 | 5061.342 | 5061.207 | |

Weight_{stdev} (lb) | — | — | 370.071 | 175.764 | 31.755 | 1.310 | 2.877 | 2.864 | 2.885 | 0.500 | 2.901 | |

_{analyses} | 15000 | — | — | — | — | 10000 | 10000 | 10000 | 10000 | 10000 | 10000 |

Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 10-bar truss.

The 25-bar transmission tower displayed in Figure ^{3}, and modulus of elasticity is 10,000,000 psi. The 25 members are categorized into the following eight groups, namely, (1) _{1}, (2) _{2}‒_{5}, (3) _{6}‒_{9}, (4) _{10}‒_{11}, (5) _{12}‒_{13}, (6) _{14}‒_{17}, (7) _{18}‒_{21}, and (8) _{22}–_{25}.

Configuration of the 25-bar truss (reproduced from Sonmez [

Table ^{2}, respectively.

Allowable stresses for the 25-bar truss.

Member group | Members | Compression (psi) | Tension (psi) |

1 | 1 | 35092 | 35000 |

2 | 2, 3, 4, 5 | 11590 | 35000 |

3 | 6, 7, 8, 9 | 17305 | 35000 |

4 | 10, 11 | 35092 | 35000 |

5 | 12, 13 | 35092 | 35000 |

6 | 14, 15, 16, 17 | 6759 | 35000 |

7 | 18, 19, 20, 21 | 6959 | 35000 |

8 | 22, 23, 24, 25 | 11082 | 35000 |

The loading condition of the 25-bar truss.

Node | Case 1 | Case 2 | ||||

_{x} (kip) | _{y} (kip) | _{z} (kip) | _{x} (kip) | _{y} (kip) | _{z} (kip) | |

1 | 0 | 20 | −5 | 1 | −10 | −10 |

2 | 0 | −20 | −5 | 0 | −10 | −10 |

3 | 0 | 0 | 0 | 0.5 | 0 | 0 |

6 | 0 | 0 | 0 | 0.6 | 0 | 0 |

In this example, the parameters of DE and EDE-1 are set to NP = 50, _{max} = 160. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 50, _{u} = 1.0, _{l} = 0.3, CR = 0.9, and _{max} = 160. Table

Performance comparison for the 25-bar truss under the two load cases.

Element group | Member | Cross-sectional areas (in^{2}) | ||||||||||

Schutte and Groenwold [ | Camp and Bichon [ | Camp [ | Sonmez [ | This work | ||||||||

PSO | ACO | BB-BC phase 1 | BB-BC phase 2 | ABC | DE | EDE | EDE-1 | EDE-2 | EDE-3 | EDE-4 | ||

1 | 1 | 0.01 | 0.01 | 0.01 | 0.01 | 0.011 | 0.01 | 0.01 | 0.011 | 0.01 | 0.01 | 0.01 |

2 | 2,3,4,5 | 2.122 | 2 | 2.092 | 2.092 | 1.979 | 1.969 | 1.986 | 1.972 | 1.986 | 1.987 | 1.987 |

3 | 6,7,8,9 | 2.893 | 2.966 | 2.964 | 2.964 | 3.003 | 3.015 | 2.995 | 3.003 | 2.995 | 2.994 | 2.994 |

4 | 10,11 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.011 | 0.01 | 0.01 | 0.01 |

5 | 12,13 | 0.01 | 0.012 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

6 | 14,15,16,17 | 0.671 | 0.689 | 0.689 | 0.689 | 0.69 | 0.697 | 0.684 | 0.690 | 0.684 | 0.683 | 0.684 |

7 | 18,19,20,21 | 1.611 | 1.679 | 1.601 | 1.601 | 1.679 | 1.682 | 1.677 | 1.686 | 1.677 | 1.677 | 1.677 |

8 | 22,23,24,25 | 2.717 | 2.668 | 2.686 | 2.686 | 2.652 | 2.641 | 2.662 | 2.650 | 2.662 | 2.663 | 2.662 |

Weight_{best} (lb) | 545.21 | 545.53 | 545.48 | 545.38 | 545.19 | 545.319 | 545.163 | 545.245 | 545.163 | 545.164 | 545.163 | |

Weight_{worst} (lb) | — | — | — | — | — | 546.096 | 545.196 | 545.570 | 546.906 | 545.210 | 545.441 | |

Weight_{mean} (lb) | 546.84 | 546.34 | 546.4 | 545.78 | — | 545.514 | 545.166 | 545.386 | 545.224 | 545.169 | 545.180 | |

Weight_{median} (lb) | — | — | — | — | — | 545.462 | 545.164 | 545.362 | 545.164 | 545.167 | 545.167 | |

Weight_{stdev}(lb) | 1.478 | 0.94 | 0.653 | 0.491 | — | 0.204 | 0.007 | 0.092 | 0.318 | 0.009 | 0.051 | |

_{analyses} | 9596 | 16500 | 9746 | 10820 | — | 8000 | 8000 | 8000 | 8000 | 8000 | 8000 |

Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 25-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 ^{7} psi and a density of 0.1 lb/in^{3}. Displacements of the uppermost joints in the ^{2}, respectively.

Configuration of the 72-bar truss (reproduced from Zeng and Li [

The loading condition of the 72-bar truss.

Node | Case 1 | Case 2 | ||||

_{x} (kip) | _{y} (kip) | _{z} (kip) | _{x} (kip) | _{y} (kip) | _{z} (kip) | |

17 | 5 | 5 | −5 | 0 | 0 | −5 |

18 | 0 | 0 | 0 | 0 | 0 | −5 |

19 | 0 | 0 | 0 | 0 | 0 | −5 |

20 | 0 | 0 | 0 | 0 | 0 | −5 |

In this example, the parameters of DE and EDE-1 are set to NP = 60, _{max} = 200. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 60, _{u} = 0.6, _{l} = 0.3, CR = 0.9, and _{max} = 200. Table

Performance comparison of the 72-bar truss.

Element group | Member | Cross-sectional areas (in^{2}) | |||||||||

Cao [ | Camp and Bichon [ | Camp [ | This work | ||||||||

GA | ACO | BB-BC phase 1 | BB-BC phase 2 | DE | EDE | EDE-1 | EDE-2 | EDE-3 | EDE-4 | ||

1 | 1–4 | 1.8562 | 1.948 | 1.9004 | 1.8577 | 1.9296 | 1.8770 | 1.9129 | 1.8924 | 1.8718 | 1.9469 |

2 | 5–12 | 0.4933 | 0.508 | 0.5252 | 0.5059 | 0.4989 | 0.5190 | 0.5153 | 0.5073 | 0.5114 | 0.5146 |

3 | 13–16 | 0.1 | 0.101 | 0.1 | 0.1 | 0.1021 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

4 | 17, 18 | 0.1 | 0.102 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1001 | 0.1 | 0.1003 | 0.1 |

5 | 19–22 | 1.283 | 1.303 | 1.3134 | 1.2476 | 1.2747 | 1.2641 | 1.2698 | 1.2800 | 1.2692 | 1.2468 |

6 | 23–30 | 0.5028 | 0.511 | 0.4801 | 0.5269 | 0.5153 | 0.5055 | 0.5069 | 0.5167 | 0.5063 | 0.5092 |

7 | 31–34 | 0.1 | 0.101 | 0.1 | 0.1 | 0.1001 | 0.1 | 0.1 | 0.1 | 0.1001 | 0.1001 |

8 | 35, 36 | 0.1 | 0.1 | 0.1 | 0.1012 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 |

9 | 37–40 | 0.5177 | 0.561 | 0.5254 | 0.5209 | 0.5406 | 0.5201 | 0.5260 | 0.5054 | 0.5468 | 0.5251 |

10 | 41–48 | 0.5227 | 0.492 | 0.5267 | 0.5172 | 0.5061 | 0.5163 | 0.5138 | 0.5165 | 0.5217 | 0.5128 |

11 | 49–52 | 0.1 | 0.1 | 0.1016 | 0.1004 | 0.1008 | 0.1 | 0.1001 | 0.1 | 0.1004 | 0.1001 |

12 | 53, 54 | 0.1049 | 0.107 | 0.1253 | 0.1005 | 0.1018 | 0.1002 | 0.1001 | 0.1 | 0.1004 | 0.1 |

13 | 55–58 | 0.1557 | 0.156 | 0.1558 | 0.1565 | 0.1561 | 0.1564 | 0.1564 | 0.1568 | 0.1563 | 0.1565 |

14 | 59–66 | 0.5501 | 0.55 | 0.5456 | 0.5507 | 0.5477 | 0.5488 | 0.5488 | 0.5442 | 0.5445 | 0.5457 |

15 | 67–70 | 0.3981 | 0.39 | 0.4314 | 0.3922 | 0.4199 | 0.4170 | 0.4007 | 0.4089 | 0.4086 | 0.3997 |

16 | 71, 72 | 0.6749 | 0.592 | 0.5231 | 0.5922 | 0.5718 | 0.5640 | 0.5685 | 0.5791 | 0.5756 | 0.5721 |

Weight_{best} (lb) | 380.32 | 380.24 | 380.46 | 379.85 | 379.939 | 379.645 | 379.656 | 379.652 | 379.708 | 379.694 | |

Weight_{worst} (lb) | − | − | − | − | 386.970 | 380.476 | 380.063 | 383.078 | 380.356 | 381.039 | |

Weight_{mean} (lb) | − | 383.16 | 384.75 | 382.08 | 380.618 | 379.807 | 379.808 | 379.910 | 379.954 | 380.008 | |

Weight_{median} (lb) | − | − | − | − | 380.342 | 379.764 | 379.774 | 379.734 | 379.894 | 379.916 | |

Weight_{stdev} (lb) | − | 3.66 | 2.434 | 1.912 | 1.251 | 0.184 | 0.108 | 0.618 | 0.159 | 0.332 | |

_{analyses} | 15000 | 18500 | 12679 | 6942 | 12000 | 12000 | 12000 | 12000 | 12000 | 12000 |

Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 72-bar truss.

The 120-bar spatial truss is a dome-shaped truss and has become a benchmark in truss optimization. Figure ^{3}. The minimum and maximum cross-sectional areas of all members are set to 0.775 and 20.0 in^{2}, respectively. The yielding stress of steel is taken as 58,000 psi. The stress and displacement constraints are presented as follows:

According to the American Institute of Steel Construction’s allowable strength design [

where

where _{y} is the yielding stress of steel. The slenderness ratio that separates the elastic from inelastic buckling regions _{C} is calculated:

Slenderness ratio

where

Maximum displacement of all nodes in the

Configuration of the 120-bar truss (reproduced from Lee and Geem [

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, _{max} = 200. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 50, _{u} = 0.5, _{l} = 0.4, CR = 0.9, and _{max} = 200. Table

Performance comparison of the 120-bar truss.

Element group | Cross-sectional areas (in^{2}) | ||||||||

Lu et al. [ | This work | ||||||||

PSO | PSOPC | AugPSO | DE | EDE | EDE-1 | EDE-2 | EDE-3 | EDE-4 | |

1 | 3.331 | 3.325 | 3.287 | 3.287 | 3.288 | 3.288 | 3.288 | 3.288 | 3.288 |

2 | 5.397 | 5.972 | 3.486 | 3.572 | 3.545 | 3.549 | 3.556 | 3.560 | 3.555 |

3 | 4.513 | 4.505 | 4.256 | 4.255 | 4.254 | 4.254 | 4.254 | 4.254 | 4.254 |

4 | 3.272 | 3.026 | 2.752 | 2.754 | 2.753 | 2.753 | 2.753 | 2.753 | 2.753 |

5 | 1.674 | 0.914 | 1.353 | 1.301 | 1.317 | 1.315 | 1.311 | 1.309 | 1.311 |

6 | 6.459 | 4.153 | 3.507 | 3.508 | 3.507 | 3.507 | 3.507 | 3.507 | 3.507 |

7 | 3.548 | 2.415 | 2.411 | 2.404 | 2.406 | 2.405 | 2.405 | 2.405 | 2.405 |

Weight_{best} (lb) | 26728.980 | 22654.325 | 20675.545 | 20666.393 | 20665.883 | 20665.825 | 20665.861 | 20665.944 | 20665.884 |

Weight_{worst} (lb) | 53777.510 | 38471.212 | 21678.621 | 20716.242 | 20668.554 | 20673.359 | 20669.575 | 20667.394 | 20671.517 |

Weight_{mean} (lb) | 31401.182 | 26424.703 | 21175.514 | 20671.274 | 20666.137 | 20666.462 | 20666.391 | 20666.309 | 20666.580 |

Weight_{median} (lb) | — | — | — | 20667.828 | 20665.989 | 20665.986 | 20666.074 | 20666.153 | 20666.231 |

Weight_{stdev} (lb) | 5653.241 | 4103.610 | 259.105 | 10.423 | 0.488 | 1.400 | 0.882 | 0.356 | 1.235 |

_{analyses} | — | — | — | 10000 | 10000 | 10000 | 10000 | 10000 | 10000 |

Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 120-bar truss.

The 942-bar spatial truss is a 26-story tower truss proposed by several researchers. Figures ^{2}. 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

Layout of the 942-bar truss (reproduced from Hasancebi [

Element numbering and sizes of the 942-bar truss (reproduced from Hasancebi [

In this example, the parameters of DE and EDE-1 are set to NP = 200, _{max} = 500. The parameters of EDE, EDE-2, EDE-3, and EDE-4 are set to NP = 200, _{u} = 0.7, _{l} = 0.3, CR = 0.8, and _{max} = 500. Table

Performance comparison of the 942-bar truss.

Element group | Members | Cross–sectional areas (in^{2}) | ||||||
---|---|---|---|---|---|---|---|---|

Hasancebi [ | This work | |||||||

Adaptive ESs | DE | EDE | EDE-1 | EDE-2 | EDE-3 | EDE-4 | ||

1 | 1, 2 | 1.020 | 5.503 | 1.867 | 1.177 | 1.805 | 1.527 | 1.351 |

2 | 3–10 | 1.037 | 3.807 | 0.325 | 0.108 | 0.130 | 0.246 | 0.239 |

3 | 11–18 | 2.943 | 3.738 | 3.533 | 4.120 | 3.730 | 3.657 | 2.752 |

4 | 19–34 | 1.920 | 2.442 | 1.982 | 2.248 | 1.870 | 1.870 | 1.923 |

5 | 35–46 | 1.025 | 11.269 | 10.283 | 9.062 | 10.473 | 11.110 | 11.668 |

6 | 47–58 | 14.961 | 10.952 | 10.717 | 10.379 | 10.717 | 10.027 | 10.362 |

7 | 59–82 | 3.074 | 10.436 | 9.427 | 9.878 | 9.378 | 10.401 | 10.465 |

8 | 83–86 | 6.780 | 8.689 | 4.620 | 4.464 | 2.708 | 2.808 | 5.141 |

9 | 87–94 | 18.580 | 6.797 | 8.100 | 9.911 | 7.250 | 6.362 | 6.107 |

10 | 95–98 | 2.415 | 23.975 | 23.242 | 22.531 | 23.263 | 21.361 | 29.080 |

11 | 99–106 | 6.584 | 8.455 | 7.191 | 5.441 | 7.568 | 7.317 | 7.038 |

12 | 107–122 | 6.291 | 4.776 | 5.839 | 5.761 | 5.615 | 5.001 | 4.661 |

13 | 123–130 | 15.383 | 12.470 | 15.171 | 16.285 | 16.669 | 17.421 | 17.639 |

14 | 131–162 | 2.100 | 1.998 | 2.056 | 2.139 | 2.038 | 2.128 | 2.139 |

15 | 163–170 | 6.021 | 2.646 | 4.374 | 3.360 | 3.977 | 4.023 | 3.667 |

16 | 171–186 | 1.022 | 0.100 | 0.104 | 0.222 | 0.100 | 0.363 | 0.524 |

17 | 187–194 | 23.099 | 21.578 | 22.630 | 21.533 | 22.433 | 21.486 | 21.802 |

18 | 195–226 | 2.889 | 2.529 | 2.323 | 2.528 | 2.515 | 2.486 | 2.331 |

19 | 227–234 | 7.960 | 6.817 | 7.478 | 8.255 | 7.111 | 7.515 | 7.643 |

20 | 235–258 | 1.008 | 0.644 | 0.551 | 0.779 | 0.448 | 0.543 | 0.379 |

21 | 259–270 | 28.548 | 32.349 | 27.738 | 26.512 | 28.049 | 27.327 | 29.069 |

22 | 271–318 | 3.349 | 3.304 | 2.959 | 2.931 | 2.776 | 2.894 | 2.895 |

23 | 319–330 | 16.144 | 17.146 | 15.878 | 15.792 | 15.216 | 15.574 | 16.259 |

24 | 331–338 | 24.822 | 22.083 | 24.093 | 23.860 | 22.865 | 21.512 | 22.837 |

25 | 339–342 | 38.401 | 52.242 | 35.819 | 34.526 | 34.530 | 34.837 | 29.940 |

26 | 343–350 | 3.787 | 1.786 | 1.717 | 3.466 | 4.145 | 1.520 | 2.794 |

27 | 351–358 | 12.320 | 19.338 | 11.374 | 9.421 | 9.966 | 12.545 | 9.453 |

28 | 359–366 | 17.036 | 13.275 | 14.507 | 12.802 | 13.606 | 15.593 | 13.035 |

29 | 367–382 | 14.733 | 15.048 | 12.021 | 13.493 | 13.284 | 13.351 | 13.338 |

30 | 383–390 | 15.031 | 12.601 | 15.345 | 14.397 | 14.915 | 14.688 | 15.111 |

31 | 391–398 | 38.597 | 26.285 | 34.848 | 35.191 | 36.275 | 36.871 | 36.072 |

32 | 399–430 | 3.511 | 3.564 | 3.145 | 3.099 | 2.947 | 3.235 | 2.926 |

33 | 431–446 | 2.997 | 5.679 | 2.083 | 1.803 | 2.368 | 2.216 | 1.968 |

34 | 447–462 | 3.060 | 5.278 | 2.340 | 2.349 | 2.565 | 2.539 | 2.410 |

35 | 463–486 | 1.086 | 1.602 | 0.123 | 0.185 | 0.100 | 0.190 | 0.227 |

36 | 487–498 | 1.462 | 0.990 | 0.213 | 0.356 | 0.421 | 0.100 | 0.117 |

37 | 499–510 | 59.433 | 52.600 | 55.823 | 56.099 | 52.331 | 52.739 | 53.967 |

38 | 511–558 | 3.632 | 2.594 | 2.963 | 2.928 | 3.036 | 3.266 | 3.046 |

39 | 559–582 | 1.887 | 2.196 | 2.320 | 2.323 | 2.188 | 2.317 | 2.320 |

40 | 583–606 | 4.072 | 3.954 | 2.892 | 2.854 | 2.953 | 2.615 | 2.787 |

41 | 607–630 | 1.595 | 0.979 | 0.244 | 0.121 | 0.267 | 0.257 | 0.104 |

42 | 631–642 | 3.671 | 0.911 | 1.134 | 1.841 | 1.382 | 1.284 | 0.815 |

43 | 643–654 | 79.511 | 74.845 | 73.370 | 76.727 | 73.712 | 72.484 | 72.741 |

44 | 655–702 | 3.394 | 3.735 | 3.076 | 3.041 | 3.114 | 2.884 | 2.907 |

45 | 703–726 | 1.581 | 1.843 | 2.076 | 2.093 | 1.946 | 2.006 | 2.079 |

46 | 727–750 | 4.204 | 2.666 | 3.074 | 3.079 | 3.313 | 3.419 | 3.300 |

47 | 751–774 | 1.329 | 1.317 | 0.308 | 0.725 | 0.417 | 0.416 | 0.483 |

48 | 775–786 | 2.242 | 0.100 | 0.277 | 0.451 | 0.577 | 0.553 | 0.256 |

49 | 787–798 | 96.886 | 78.724 | 86.644 | 86.760 | 84.354 | 86.335 | 88.369 |

50 | 799–846 | 3.710 | 3.145 | 3.107 | 3.195 | 3.130 | 3.142 | 3.619 |

51 | 847–870 | 1.055 | 8.557 | 2.075 | 0.658 | 2.384 | 2.469 | 2.599 |

52 | 871–894 | 4.566 | 4.121 | 3.539 | 2.949 | 3.452 | 3.830 | 3.869 |

53 | 895–902 | 9.606 | 30.069 | 9.719 | 7.361 | 12.883 | 9.837 | 10.473 |

54 | 903–906 | 2.984 | 12.180 | 8.382 | 7.519 | 8.500 | 8.553 | 10.070 |

55 | 907–910 | 45.917 | 54.295 | 40.526 | 40.681 | 40.973 | 43.239 | 39.841 |

56 | 911–918 | 1.000 | 0.535 | 0.129 | 0.201 | 0.271 | 0.171 | 0.132 |

57 | 919–926 | 62.426 | 31.736 | 56.362 | 59.574 | 55.438 | 53.716 | 52.565 |

58 | 927–934 | 2.977 | 4.960 | 4.725 | 2.888 | 3.893 | 5.330 | 4.996 |

59 | 935–942 | 1.000 | 8.553 | 0.122 | 0.225 | 2.893 | 0.763 | 0.335 |

Weight_{best} (lb) | 141241 | 142,436 | 132,441 | 132,529 | 132,713 | 132,658 | 132,905 | |

Weight_{worst} (lb) | — | 151,351 | 134,404 | 135,764 | 135,694 | 134,170 | 135,632 | |

Weight_{mean} (lb) | — | 145,986 | 133,153 | 133,386 | 133,436 | 133,243 | 133,942 | |

Weight_{median} (lb) | — | 145,639 | 133,018 | 133,279 | 133,280 | 133,161 | 133,698 | |

Weight_{stdev} (lb) | — | 2292.7 | 483.1 | 613.0 | 709.5 | 396.9 | 652.3 | |

_{analyses} | 150,000 | 100,000 | 100,000 | 100,000 | 100,000 | 100,000 | 100,000 |

Convergence rates of DE, EDE, EDE-1, EDE-2, EDE-3, and EDE-4 for the 942-bar truss.

In this section, the statistical analysis results of applying Friedman test [

Average ranks for different algorithms across all five truss problems.

Algorithm | 10-bar truss | 25-bar truss | 72-bar truss | 120-bar truss | 942-bar truss | Mean ranking | Rank |

DE | 5.73 | 5.70 | 5.73 | 5.80 | 6 | 5.79 | 6 |

EDE | |||||||

EDE-1 | 2.80 | 5.20 | 2.67 | 2.77 | 3.17 | 3.32 | 3 |

EDE-2 | 2.90 | 2.27 | 2.53 | 2.87 | 3.03 | 2.72 | 2 |

EDE-3 | 3.77 | 3.03 | 4.03 | 3.70 | 2.47 | 3.40 | 4 |

EDE-4 | 3.17 | 2.60 | 3.67 | 3.47 | 4.17 | 3.42 | 5 |

Friedman | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

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,

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.

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

Mutation factor _{l} and _{u} in the proposed EDE algorithm is subjective. Further research could investigate how to determine _{l} and _{u} more objectively. Moreover, EDE uses a constant CR in this study; the proposed EDE algorithm with an adaptive CR could be further researched.

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.

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

The authors declare that they have no conflicts of interest.