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Structural and Multidisciplinary Optimization 41 2 , , Structural and Multidisciplinary Optimization 45 3 , , Neural Computing and Applications 28 6 , , Engineering and Applied Sciences Optimization, , Steel and Composite Structures 24 1 , , International Journal of Solids and Structures , Metaheuristics and Optimization in Civil Engineering, , GDE3 performs the sorting of the vector by calculating the crowding distance of the vector.
In the case of comparing feasible, incomparable, and nondominating solutions, both offspring and parent vectors are saved for the population of the next generation [ 4 ]. There is no need to remove elements, since the population size does not increase. GDE3 improves the ability to handle multiobjective optimization problems by giving a better distributed set of solutions and are less sensitive to the selection of control parameter values compared to the earlier GDE versions.
As a result, this procedure reduces the computational costs of the metaheuristic and improves its efficiency. Readers interested in GDE3 should refer to the texts by [ 30 , 31 ]. Different examples taken from several optimization literatures were used to show the performance of GDE3 metaheuristic. These examples have been previously solved using a variety of other techniques, which is useful to show the validity and effectiveness of the GDE3 metaheuristic. The optimal results were compared with data recently published in literatures. An experiment has been performed to determine the best values of F and CR for better performance in GDE3 metaheuristic.
For this purpose, both CR and F are varied from 0. The simulations were conducted for each value of F with respect to all values of CR. Hence, such simulations were conducted. The design variables of the optimization problem are the thickness of the beam b , the thickness of the weld h , the welded joint length l , and the beam width t. Figure 1 shows the welded beam design structure. Schematic of the welded beam design problem [ 1 ]. The values of l and h must be integer multiples of 0. The values of parameters involved in the formulation of the welded beam problem are also shown in Table 1.
Values of parameters involved in the formulation of the welded beam problem [ 1 ].
The results obtained by GDE3 are presented in Table 2. GDE3 found the global optimum requiring iterations i. Table 3 provides a comparison of this solution with the results of other optimization algorithms. It is apparent from the table that GDE3 metaheuristic finds a competitive solution using only 10, evaluations which is considerably lesser than those of other approaches. Further, a statistical evaluation of independent runs of the GDE3 metaheuristic is tabulated in Table 4 considering the best, worst, average, and the standard deviation std.
The ratio between the optimized costs corresponding to best and worst designs is 1.
Remarkably, GDE3 produced the overall best design result with a value of 1. For continuous optimization problem, [ 20 , 22 ] found a better design result with a value of 1. The pressure vessel problem is designed to minimize total cost which is comprised of the welding cost and forming material cost. The design variables of the optimization problem are the length of the cylindrical segment of the vessel L , the thickness of the cylindrical skin T s , the inner radius R , and the thickness of the spherical head T h.
The variables T s and T h are discrete values which are integer multiples of 0. Figure 2 shows the cylindrical pressure vessel capped at both ends by hemispherical heads.
Schematic of the pressure vessel design problem [ 1 ]. Unlike the usual limit of in considered in literatures, the upper bound of design variable L was increased to in to expand the search space. Optimization results are presented in Table 5. GDE3 produced a design result with a value of Table 6 compares the optimal design results produced by GDE3 with those reported in [ 1 , 17 , 20 , 21 , 24 , 32 ].
Further, a statistical evaluation of independent runs of the GDE3 metaheuristic is tabulated in Table 7 considering the best, worst, average, and the standard deviation std. The ratio between the optimized costs corresponding to worst and best designs is 1.
The best design result was produced by the Firefly algorithm. GDE3 metaheuristic produced the least performance compared to the other algorithms. The speed reducer design problem [ 25 ] is designed to minimize the weight of the speed reducer subjecting it to some constraints such as shaft stresses, surface stress, gear teeth bending stress, and shafts crosswise deflections.
The width of the gear face x 1 , teeth module x 2 , number of pinion teeth x 3 , first shaft length between bearings x 4 , second shaft length between bearings x 5 , the diameter of the first shaft x 6 , and diameter of the second shaft are the design variables of the optimization problem. Figure 3 shows the schematic of the speed reducer. Schematic of the speed reducer design problem [ 25 ]. The results obtained by GDE3 are presented in Table 8. GDE3 found the global optimum requiring iterations per optimization run.
Table 9 provides a comparison of this solution with the results of simple constrained particle swarm optimization. It is apparent from the table that GDE3 metaheuristic finds a competitive solution using only 10, objective function evaluations, which is considerably lesser than those of other approaches. Further, a statistical evaluation of independent runs of the GDE3 metaheuristic is tabulated in Table 10 considering the best, worst, average, and the standard deviation std.
Remarkably, GDE3 produced the overall best design result with a value of Speed reducer problem: comparison of generalized differential evolution 3 results with simple constrained particle swarm optimization.
The design variables are the number of active coils P , the diameter of the mean coil D , and the diameter of the wire d. The results obtained by GDE3 are presented in Table Table 12 provides a comparison of this solution with the results of simple constrained particle swarm optimization. Further, a statistical evaluation of independent runs of the GDE3 metaheuristic is tabulated in Table 13 considering the best, worst, average, and the standard deviation std. In the present study, the GDE3 algorithm is used as a simple and efficient optimization technique for handling engineering optimization problems.
http://wikbud.eu/wp-content/2019-10-05/6104.php The GDE3 algorithm also uses a very simple mechanism to deal with constrained functions and results generated by the algorithm indicate that such mechanism, despite its simplicity, is effective in practice. From this study, performance evaluation of the GDE3 algorithm through benchmark design optimization examples reveals the efficiency of this technique in solving practical optimization problems.
Although in the present study the algorithm is utilized only for solving engineering design optimization problems, GDE3 algorithm can easily be employed for solving other types of optimization problems as well. The authors declare that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v.
Published online Mar Author information Article notes Copyright and License information Disclaimer. Adekanmbi and P. 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. Abstract Metaheuristic algorithms are well-known optimization tools which have been employed for solving a wide range of optimization problems. Introduction In structural engineering, most design optimization problems are highly nonlinear consisting of different design variables and complex constraints such as displacements, geometrical configuration, stresses, and load carrying capability.
Generalized Differential Evolution Metaheuristic Several extensions of differential evolution [ 26 ] exist for solving constrained and nonconstrained multiobjective optimization problems [ 27 , 28 ]. The selection process in GDE3 is guided by these three rules: In a scenario where both the old vector and trial vector are infeasible, the old vector is selected if it dominates the trial vector, but if the trial vector weakly dominates the old vector, then the trial vector is selected.
Open in a separate window. Algorithm 1.
Editors: Yang, Xin-She, Bekdaş, Gebrail, Nigdeli, Sinan Melih (Eds.) This timely book deals with a current topic, i.e. the applications of metaheuristic algorithms, with a primary focus on optimization problems in civil engineering. The remaining chapters report on advanced studies. This timely book deals with a current topic, i.e. the applications of metaheuristic algorithms, with a primary focus on optimization problems in civil engineering.
The GDE3 algorithm [ 29 ]. Example 1 welded beam design optimization problem. We request your telephone number so we can contact you in the event we have difficulty reaching you via email. We aim to respond to all questions on the same business day.