- Original Article
- Open Access
Improved Differential Evolution with Shrinking Space Technique for Constrained Optimization
- Chunming FU^{1},
- Yadong XU^{2}Email author,
- Chao JIANG^{1},
- Xu HAN^{1} and
- Zhiliang HUANG^{1}
https://doi.org/10.1007/s10033-017-0130-4
© The Author(s) 2017
Received: 4 May 2016
Accepted: 2 April 2017
Published: 24 April 2017
Abstract
Most of the current evolutionary algorithms for constrained optimization algorithm are low computational efficiency. In order to improve efficiency, an improved differential evolution with shrinking space technique and adaptive trade-off model, named ATMDE, is proposed to solve constrained optimization problems. The proposed ATMDE algorithm employs an improved differential evolution as the search optimizer to generate new offspring individuals into evolutionary population. For the constraints, the adaptive trade-off model as one of the most important constraint-handling techniques is employed to select better individuals to retain into the next population, which could effectively handle multiple constraints. Then the shrinking space technique is designed to shrink the search region according to feedback information in order to improve computational efficiency without losing accuracy. The improved DE algorithm introduces three different mutant strategies to generate different offspring into evolutionary population. Moreover, a new mutant strategy called “DE/rand/best/1” is constructed to generate new individuals according to the feasibility proportion of current population. Finally, the effectiveness of the proposed method is verified by a suite of benchmark functions and practical engineering problems. This research presents a constrained evolutionary algorithm with high efficiency and accuracy for constrained optimization problems.
Keywords
- Constrained optimization
- Differential evolution
- Adaptive trade-off model
- Shrinking space technique
1 Introduction
Constrained optimization problems (COPs) widely exist in various scientific and engineering fields [1–3], such as mechanical design, path planning, etc. Perhaps it is not easy or difficult to obtain global optimal solutions by the traditional optimization techniques for some COPs involving nonlinear inequality or equality constraints, multi-modal and non-differential objective functions. Evolutionary algorithms (EAs) cooperated with constraint-handling techniques which have obtained more and more attention because of their flexibility, effectiveness and adaptability provide an effective and powerful avenue to cope with these COPs [4–6]. A large number of effective constrained optimization evolutionary algorithms (COEAs) have been proposed [7–9]. Recently, some representative constraint-handling techniques with EAs to solve COEAs have been summarized by COELLO [10]. The most general existing constraint-handling techniques are mainly categorized into three groups. Firstly, the method based on the penalty function aimed to transform a COP into an unconstrained one by adding a penalty term to the original objective function [11, 12]. Secondly, the approach based on the feasibility-based criterion preferred to select the feasible solutions rather than the infeasible solutions into the next evolutionary process [13, 14]. Thirdly, the method based on the multi-objective optimization technique aimed to transform the COPs into the unconstrained multi-objective optimization problems and utilized multi-objective optimization technique to deal with the converted problems [15, 16].
The performance of COEAs mainly depends on the constraint-handling techniques and EAs as the search optimizer. Differential evolution (DE) originally proposed by STORN and PRICE [17] was one of the most simple and powerful population-based evolutionary algorithms for global optimization. During the past two decades, different DE optimizers with constraint-handling techniques have been successfully developed to deal with different kinds of COPs. The first attempt was the constraint adaption with DE (CADE) algorithm which introduced multi-member individuals to generate more than one offspring by DE operators [18]. A cultural DE-based algorithm with the feasibility rule was proposed by LANDA and COELLO [19], which utilized different knowledge sources to influence the mutant operator in order to accelerate convergence. A multi-member diversity-based DE (MDDE) algorithm where each parent generated more than one offspring to enhance the diversity of population was presented by MEZURA-MONTES, et al [20] to solve COPs. The dynamic stochastic selection technique was put forward by ZHANG, et al [21] under the framework of multi-member DE. TESSEMA and YEN [11] designed an adaptive penalty formulation where the feasible proportion of the current population was utilized to tune the penalty factor. In order to combine the advantages of different constraint-handling techniques, MALLIPEDDI, et al [22] proposed an ensemble of constraint-handling techniques (ECHT) with DE and evolutionary programming optimizers for coping with COPs. ELSAYED, et al [23] introduced an algorithm framework to use multiple search operators in each generation with the feasibility rule for COPs. Each combination of search operators had its own sub-population, and the size of each sub-population varied adaptively during the progress of evolution depending on the reproductive success of the search operators. Subsequently, GONG, et al [24] developed a ranking-based mutation operator with an improved dynamic diversity mechanism for COPs. A modified differential evolution algorithm [25] was proposed to deal with the dimensional synthesis of the redundant parallel robot problem.
Recently, the adaptive trade-off model (ATM) [26] has been proposed to maintain a reasonable tradeoff to select better individuals to reserve into next generation between the feasible and infeasible individuals. The principal merit of ATM was that the promising infeasible individuals could be inherited into the next evolutionary process. The ATM with evolutionary strategy (ATMES) as the search optimizer has been utilized to solve COPs. In order to reduce the computational effort, the shrinking space technique introduced by AGUIRRE, et al [27] shrank the search region according to some feedback information and directed the search effort to the promising feasible region. Subsequently, WANG, et al [28] proposed a new method named AATM with high efficiency which benefited from the virtues of shrinking space technique and ATM. The performance of AATM algorithm could promptly converge to optimal results without loss of quality and precision.
Although AATM enhances the performance of ATMES by taking advantage of the shrinking space technique to address complicated COPs with multiple constraints, it still leaves a plenty of room to develop new approaches to solve COPs for improvement of accelerating the convergence rate and enhancing the quality of solutions within the limited time, especially for complicated engineering optimization problems. When using EAs to solve COPs, the search algorithm plays a crucial role on the performance of hybrid approaches as well as the constraint-handling techniques. Hence, this study employs an advanced search algorithm (i.e. an improved DE) to further improve the performance of AATM. The improved DE employs three different characteristic mutant strategies to generate different offspring into evolutionary population. Hence, combining the advantages of an improved differential evolution with adaptive trade-off model and shrinking space technique, called ATMDE, is proposed to deal with COPs. The remainders of this paper are organized as follows. In Section 2, the definitions of COP and some relevant concepts of multi-objective optimization are given, respectively. In Section 3, the basics of DE are briefly introduced. In Section 4, the proposed ATMDE algorithm is presented in detail. In Section 5, the performances of ATMDE are tested by 18 well-known benchmark test functions from the 2006 IEEE Congress on Evolutionary Computation (IEEE CEC2006) and several engineering optimization problems. Section 6 concludes this paper.
2 Statement of the Problem
Since the following method utilizes the concepts of the multi-objective optimization techniques to address constraints of COPs, some related multi-objective optimization concepts are introduced in advance.
Definition 1
Definition 2
Pareto optimality: u is said to be Pareto optimal only if vector v in the feasible region S doesn’t exist and v ≺ u, where v = f(v) = (f(v), G(v)) = (v _{1}, v _{2}), u = f(u) = (f(u), G(u)) = (u _{1}, u _{2}).
Definition 3
It should be noted that individuals in the Pareto optimal set are called non-dominated individuals.
Definition 4
3 Basics of Differential Evolution
After initialization, a mutant strategy is adopted to generate a mutant vector v _{ i } = (v _{ i,1}, v _{ i,2},···, v _{ i,n }) by its corresponding target vector x _{ i } = (x _{ i,1}, x _{ i,2},···, x _{ i,n }). There is a general nomenclature “DE/x/y” developed to denote the different DE mutant variants, where “DE” means differential evolution, “x” indicates which individual as the base vector is selected to be mutated, and “y” is the number of difference vectors chosen for perturbation of x. The following mutation strategies are most frequently used.
The above steps repeat generation by generation until the termination criterion is met.
4 Proposed Algorithm: ATMDE
The performance of COEAs mainly depends on the search ability of evolutionary algorithm and the effectiveness of constraint-handling technique. Hence, the proposed algorithm ATMDE utilizes an improved DE as search optimizer to reproduce offspring and introduces the adaptive trade-off model as the constraint-handling technique to select better individuals to retain into the next population. Furthermore, in order to reduce the redundant search region, the shrinking space technique is employed to enhance the convergence performance. This section will introduce the three core parts of ATMDE algorithm in detail, respectively.
4.1 Improved DE
The implementation of constructing “DE/rand/best/1” strategy is explained as follows. At the beginning, the “DE/rand/1” strategy is introduced to maintain the diversity of population in order to prevent the population from being stuck in a local optimum. This strategy has the ability to enhance the global search ability because the new individuals could learn the information from other individuals randomly chosen from the whole population. Then it is necessary to accelerate the convergence of the evolutionary population, so the “DE/best/1” strategy is employed to speed up convergence as the feasibility proportion of current population increases. The ‘‘DE/best/1’’ strategy utilizes the information of the best individual in the current population to generate new individual which can enhance the convergence speed. Hence, the proposed strategy “DE/rand/best/1” as shown in Algorithm 1 is constructed to balance diversity and convergence speed, which combines the “DE/rand/1” strategy and “DE/best/1” strategy through the feasibility proportion of current population. Specially, if a value randomly generated from [0, 1] is greater than the feasibility proportion of current population φ, the “DE/rand/1” strategy is adopted. Otherwise, the “DE/best/1” strategy is employed.
Algorithm 1 The “DE/rand/best/1” strategy
4.2 Adaptive Trade-Off Model
Generally, a constraint-handling technique to address constraints experiences three different situations in the whole evolutionary process: (1) the infeasible situation only includes the infeasible solutions; (2) the semi-feasible situation includes the feasible and infeasible individuals simultaneously; and (3) the infeasible situation only includes the infeasible individuals. The ATM strategy aims to construct an effective tradeoff scheme to address constraints for each situation according to their corresponding characteristics.
4.2.1 Infeasible Situation
In the infeasible situation, a hierarchical non-dominated selection strategy is introduced to choose individuals from Pareto front into the next population along with evolutionary process and is executed as follows: only the first half of non-dominated individuals with smaller constraint violations are selected to offspring population and are immediately eliminated from the parent population. This process repeats until the number of individuals reaches the size of the offspring population.
4.2.2 Semi-feasible Situation
The individuals are ranked based on the values of f _{fit}(·) in ascending order, and the individuals with smaller values are chosen to add into the offspring population until reaching its allowable size.
4.2.3 Feasible Situation
In this feasible situation, the constraint violations of COPs with zero are equivalent to be one of the unconstrained optimization problems because constraint violations of every individual are zero. Hence, only objective function is required to be considered, and Eq. (19) can be also used as a criterion to select better individuals because G _{nor}(·) is zero.
4.3 Shrinking Space Technique
The shrinking space technique is one of the most pivotal ingredients of IS-PAES [27] and AATM [28]. This technique aims to reduce the search region to focus the computational effort on the specific promising feasible. The main procedure of the shrinking space technique is carried out as Algorithm 2, where T denotes that the technique is performed at every T generations, α _{ i } is a threshold number, β is a reduced factor, and \(\bar{x}_{pob,i}\) and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{pob,i}\) denote the upper and lower bounds of the i-th variable in the selected offspring population, respectively. Afterward, the following specific operations are performed to shrink the search space around the promising individuals to determine the new boundaries for design variables.
4.4 Framework of ATMDE
5 Benchmark Test Functions
Details about 18 benchmark functions
Function | No. of variables n | Type of function | Ration ρ/ % | No. of constraints N |
---|---|---|---|---|
g01 | 13 | Quadratic | 0.01 | 9 |
g02 | 20 | Nonlinear | 99.9 | 2 |
g03 | 10 | Polynomial | 0.00 | 1 |
g04 | 5 | Quadratic | 52.1 | 6 |
g05 | 4 | Cubic | 0.00 | 5 |
g06 | 2 | Cubic | 0.01 | 2 |
g07 | 10 | Quadratic | 0.00 | 8 |
g08 | 2 | Nonlinear | 0.86 | 2 |
g09 | 7 | Polynomial | 0.51 | 4 |
g10 | 8 | Linear | 0.00 | 6 |
g11 | 2 | Quadratic | 0.00 | 1 |
g12 | 3 | Quadratic | 4.48 | 1 |
g14 | 10 | Nonlinear | 0.00 | 3 |
g15 | 3 | Quadratic | 0.00 | 2 |
g16 | 5 | Nonlinear | 0.02 | 38 |
g18 | 9 | Quadratic | 0.00 | 13 |
g19 | 15 | Nonlinear | 33.4 | 5 |
g24 | 2 | Linear | 79.6 | 2 |
5.1 Parameter Settings
To compare the robustness of different algorithms, benchmark functions are optimized at 30 independent runs. Then their statistical performances of the optimal solutions such as mean, standard deviation criteria are utilized to compare.
5.2 General Performance of ATMDE
Results obtained by ATMDE for 18 benchmark test function over 30 independent runs
Function | Optimal solution f ^{*} | Best solution f _{best} | Median solution f _{median} | Mean solution μ _{ f } | Worst solution f _{worst} | Standard deviation σ _{ f } |
---|---|---|---|---|---|---|
g01 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | 0 |
g02 | −0.803 619 | −0.803 617 | −0.803 617 | −0.803 617 | −0.803 610 | 1.238 9 × 10^{−6} |
g03 | −1.000 50 | −1.005 00 | −1.005 00 | −1.005 00 | −1.005 00 | 2.081 6 × 10^{−9} |
g04 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | −30 665.538 6 | 1.110 × 10^{−11} |
g05 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 5126.496 71 | 1.013 3 × 10^{−12} |
g06 | −6961.813 87 | −6961.813 87 | −6961.813 87 | −6961.813 87 | −6961.813 87 | 1.850 × 10^{−12} |
g07 | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 2.211 4 × 10^{−8} |
g08 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.564 1 × 10^{−17} |
g09 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.634 8 × 10^{−13} |
g10 | 7049.248 02 | 7049.248 02 | 7049.248 02 | 7049.248 02 | 7049.24802 | 8.700 × 10^{−7} |
g11 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.011 2 × 10^{−7} |
g12 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 |
g14 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 888 | 1.953 9 × 10^{−10} |
g15 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.937 8 × 10^{−13} |
g16 | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | 6.775 2 × 10^{−16} |
g18 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 7.454 9 × 10^{−10} |
g19 | 32.655 59 | 32.655 63 | 32.655 86 | 32.656 00 | 32.657 25 | 3.753 8 × 10^{−4} |
g24 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 3.735 5 × 10^{−15} |
5.3 ATMDE Compared with AATM
Comparison results of ATMDE and AATM on 18 benchmark test functions
Function | Best solution f _{best} | Mean solution μ _{ f } | Worst solution f _{worst} | Standard deviation σ _{ f } | ||||
---|---|---|---|---|---|---|---|---|
ATMDE | AATM | ATMDE | AATM | ATMDE | AATM | ATMDE | AATM | |
g01 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | −15.000 | 0 | 3.1 × 10^{−7} |
g02 | −0.803 617 | −0.803 38 | −0.803 617 | −0.791 21 | −0.803 61 | −0.767 | 1.2 × 10^{−6} | 8.6 × 10^{−3} |
g03 | −1.005 00 | −1.00 | −1.005 00 | −1.00 | −1.005 00 | −1.00 | 2.1 × 10^{−9} | 3.5 × 10^{−4} |
g04 | −30 665.539 | −30 665.5 | −30 665.539 | −30 665.5 | −30 665.5 | −30 665.5 | 1.1 × 10^{−11} | 1.0 × 10^{−4} |
g05 | 5 126.496 7 | 5 126.498 | 5 126.496 71 | 5 126.714 | 5 126.496 7 | 5 128.824 | 1.0 × 10^{−12} | 4.3 × 10^{−1} |
g06 | −6 961.814 | −6 961.81 | −6 961.814 | −6 961.81 | −6 961.81 | −6 961.81 | 1.6 × 10^{−12} | 7.1 × 10^{−12} |
g07 | 24.306 209 | 24.307 | 24.306 209 | 24.317 | 24.306 209 | 24.356 | 2.2 × 10^{−8} | 1.3 × 10^{−2} |
g08 | −0.095 825 | −0.095 82 | −0.095 825 | −0.095 82 | −0.095 82 | −0.095 82 | 2.6 × 10^{−17} | 5.8 × 10^{−} ^{18} |
g09 | 680.630 | 680.630 | 680.630 05 | 680.639 4 | 680.630 05 | 680.646 | 4.6 × 10^{-13} | 4.5 × 10^{−3} |
g10 | 7 049.248 | 7 049.603 | 7 049.2480 2 | 7 077.477 | 7 049.248 | 7 183.295 | 8.7 × 10^{−7} | 3.1 × 10^{1} |
g11 | 0.74990 | 0.75 | 0.7499 | 0.75 | 0.7499 | 0.75 | 1.0 × 10^{−7} | 3.8 × 10^{−6} |
g12 | −1.000 | −1.000 | −1.000 | −1.000 | −1.000 | −1.000 | 0 | 0 |
g14 | −47.764 888 | −47.762 | −47.764 888 | −47.750 | −47.764 8 | −47.712 | 1.9 × 10^{−10} | 1.0 × 10^{−2} |
g15 | 961.715 | 961.715 | 961.715 | 961.715 | 961.715 02 | 961.716 | 6.9 × 10^{−13} | 3.0 × 10^{−4} |
g16 | −1.905 155 | −1.905 15 | −1.905 155 | −1.905 15 | −1.905 15 | −1.905 15 | 6.8 × 10^{−16} | 2.4 × 10^{−14} |
g18 | −0.866 025 | −0.866 02 | −0.866 025 | −0.865 95 | −0.866 02 | −0.864 84 | 7.5 × 10^{−10} | 2.1 × 10^{−4} |
g19 | 32.655 63 | 32.725 | 32.655 86 | 32.952 | 32.657 25 | 33.243 | 3.8 × 10^{−4} | 1.4 × 10^{−1} |
g24 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | −5.508 01 | 3.7 × 10^{−15} | 1.8 × 10^{−15} |
w/t/l | 8/10/0 | 11/7/0 | 11/7/0 | 15/1/2 |
Furthermore, the computational cost of ATMDE and AATM both are relatively low compared with the IS-PAES algorithm [27], but the performance of ATMDE is better than those solved by AATM in terms of quality of results. It should be noted that comparison results between AATM and IS-PAES are shown in the reference [28] in which AATM with smaller fitness function evaluations (FFEs) has better performance than IS-PAES. Hence, ATMDE is an effective and efficient algorithm with limited FFEs for solving COPs.
5.4 Effectiveness of the “DE/rand/best/1” Strategy
Results obtained by ATMDE and ATMDE1 on 18 benchmark test functions
Function | Method | Best solution f _{best} | Median solution f _{median} | Mean solution μ _{ f } | Worst solution f _{worst} | Standard deviation σ _{ f } |
---|---|---|---|---|---|---|
g01 | ATMDE | −15.000 | −15.000 | −15.000 | −15.0000 | 0 |
ATMDE1 | −14.999 9 | −14.999 9 | −14.999 9 | −14.999 9 | 8.98 × 10^{−7} | |
g02 | ATMDE | −0.803 617 | −0.803 617 | −0.803 617 | −0.803 610 | 1.24 × 10^{−6} |
ATMDE1 | −0.802 125 | −0.802 125 | −0.802 124 | −0.802 092 | 6.14 × 10^{−6} | |
g03 | ATMDE | −1.005 00 | −1.005 00 | −1.005 00 | −1.005 00 | 2.08 × 10^{−9} |
ATMDE1 | −1.005 00 | −1.005 00 | −0.985 8 | −0.798 4 | 4.93 × 10^{−2} | |
g04 | ATMDE | −30 665.53 | −30 665.53 | −30 665.53 | −30 665.5 | 1.11 × 10^{−11} |
ATMDE1 | −30 665.53 | −30 665.53 | −30 665.53 | −30 665.53 | 1.85 × 10^{−11} | |
g05 | ATMDE | 5126.496 71 | 5 126.496 71 | 5126.496 71 | 5 126.496 71 | 1.01 × 10^{−12} |
ATMDE1 | 5126.496 71 | 5 126.496 71 | 5126.496 71 | 5 126.496 71 | 2.95 × 10^{−9} | |
g06 | ATMDE | −6961.813 | −6961.813 | −6961.813 | −6961.81 | 1.85 × 10^{−12} |
ATMDE1 | −6961.813 | −6961.813 | −6961.813 | −6961.81 | 2.78 × 10^{−12} | |
g07 | ATMDE | 24.306 209 | 24.306 209 | 24.306 209 | 24.306 209 | 2.21 × 10^{−8} |
ATMDE1 | 24.3062 497 | 24.306 2497 | 24.306 253 | 24.306 364 | 2.09 × 10^{−5} | |
g08 | ATMDE | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.56 × 10^{−17} |
ATMDE1 | −0.095 825 | −0.095 825 | −0.095 825 | −0.095 825 | 2.82 × 10^{−17} | |
g09 | ATMDE | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.63 × 10^{−13} |
ATMDE1 | 680.630 05 | 680.630 05 | 680.630 05 | 680.630 05 | 4.85 × 10^{−13} | |
g10 | ATMDE | 7 049.248 02 | 7 049.248 02 | 7 049.248 02 | 7 049.248 02 | 8.70 × 10^{−7} |
ATMDE1 | 7 049.339 29 | 7 049.699 7 | 7 049.800 6 | 7 051.246 3 | 4.59 × 10^{−1} | |
g11 | ATMDE | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.01 × 10^{−7} |
ATMDE1 | 0.749 90 | 0.749 90 | 0.749 90 | 0.749 90 | 1.12 × 10^{−16} | |
g12 | ATMDE | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 |
ATMDE1 | −1.000 00 | −1.000 00 | −1.000 00 | −1.000 00 | 0 | |
g14 | ATMDE | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 88 | 1.95 × 10^{−10} |
ATMDE1 | −47.764 888 | −47.764 888 | −47.764 888 | −47.764 88 | 1.67 × 10^{−8} | |
g15 | ATMDE | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.94 × 10^{−13} |
ATMDE1 | 961.715 022 | 961.715 022 | 961.715 022 | 961.715 022 | 6.94E × 10^{−13} | |
g16 | ATMDE | −1.905 155 | −1.905 155 | −1.905 155 | −1.905 155 | 6.78 × 10^{−16} |
ATMDE1 | −1.905 102 | −1.905 102 | −1.905 102 | −1.905 102 | 6.78 × 10^{−16} | |
g18 | ATMDE | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 7.45 × 10^{−10} |
ATMDE1 | −0.866 025 | −0.866 025 | −0.866 025 | −0.866 025 | 4.49 × 10^{−6} | |
g19 | ATMDE | 32.655 63 | 32.655 86 | 32.656 00 | 32.657 25 | 3.75 × 10^{−4} |
ATMDE1 | 32.676 38 | 32.702 88 | 32.704 75 | 32.774 96 | 2.16 × 10^{−2} | |
g24 | ATMDE | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 3.74 × 10^{−15} |
ATMDE1 | −5.508 013 | −5.508 013 | −5.508 013 | −5.508 013 | 4.52 × 10^{−15} | |
w/t/l | 6/12/0 | 6/12/0 | 7/11/0 | 7/11/0 | 14/3/1 |
5.5 Four Mechanical Benchmark Engineering Designs
Main features for each engineering design problem
Engineering benchmark | No. of variables n | Ration ρ/ % | No. of constrains N |
---|---|---|---|
Weld-beam design | 4 | 37.625 | 5 |
Spring design | 3 | 0.732 3 | 4 |
Speed reducer design | 7 | 23.015 2 | 11 |
Three-bar truss design | 2 | 21.870 6 | 3 |
Results about four benchmark engineering design problems
Engineering problems | Method | Best solution f _{best} | Mean solution μ _{ f } | Worst solution f _{worst} | Standard deviation σ _{ f } |
---|---|---|---|---|---|
Weld-beam design | ATMDE | 2.380 956 | 2.380 956 | 2.380 956 | 5.88 × 10^{−11} |
AATM | 2.382 326 | 2.386 976 | 2.391 592 | 2.20 × 10^{−3} | |
Spring design | ATMDE | 0.012 665 | 0.012 665 | 0.012 665 | 1.05 × 10^{−15} |
AATM | 0.012 668 | 0.012 708 | 0.012 861 37 | 4.50 × 10^{−5} | |
Speed reducer | ATMDE | 2994.473 6 | 2994.474 4 | 2994.474 45 | 1.18 × 10^{−5} |
AATM | 2994.516 7 | 2994.585 4 | 2994.659 79 | 3.30 × 10^{−2} | |
Three-bar truss design | ATMDE | 263.895 84 | 263.895 84 | 263.895 843 | 2.87 × 10^{−13} |
AATM | 263.895 84 | 263.896 6 | 263.900 41 | 1.10 × 10^{−3} |
Best design variables for four benchmark engineering design problems
Engineering problems | Method | Best design variable x _{best} | Best function values f _{best} |
---|---|---|---|
Weld-beam design | ATMDE | 0.244 368 975, 6.217 519 715, 8.291 471 390, 0.244 368 975 | 2.380 956 580 |
AATM | 0.244 106 586, 6.220 903 633, 8.298 161 229,0.244 382 231 | 2.382 326 | |
Spring design | ATMDE | 0.356 717 739, 0.051 689 061, 11.288 965 783 04 | 0.012 665 232 |
AATM | 0.359 690 411, 0.051 813 095, 11.119 252 680 | 0.012 668 261 | |
Speed reducer design | ATMDE | 3.50, 0.7, 17, 7.309 819 903, 7.715 173 384 44, 3.350 233 018 67, 5.286 521 228 48 | 2 994.473 624 |
AATM | 3.500 016 221, 0.700 001 177, 17.000 029 883, 7.300 297 290, 7.716 049 465, 3.350 239 798, 5.286 660 476 6 | 2 994.516 778 | |
Three-bar truss design | ATMDE | 0.788 675 135, 0.408 248 289 | 263.895 843 |
AATM | 0.788 681 755, 0.408 229 565 | 263.895 843 |
6 Engineering Applications
6.1 Vehicle Crashworthiness Problem
The finite element model (FEM) of the vehicle including 755 parts and 977 742 elements is established for the above objective and constraints. To improve efficiency, the response surfaces are established based on the samples by Latin hypercube sampling method. Then, the minimum mass M(x) solved by ATMDE algorithm is 10.53 kg and its corresponding five design variables are 2.00 mm, 2.50 mm, 2.50 mm, 2.76 mm and 1.68 mm, respectively. Under this circumstance, values of constraints are 0 g, −4208.33 J, − 60.74 mm, −0.0002 mm, respectively. Specifically, the mean value of integral acceleration \(\bar{a}({\varvec{x}})\) is 35 g, which can effectively protect passengers in the automobile when the collision inevitably occurs. Meanwhile, the inner and outer front rail can absorb 3908.33 J. In addition, the intrusions of upper and lower point at the engine are 289.26 mm and 200 mm, which can effectively reduce the occupants’ injuries to protect the passengers’ safety.
6.2 Structural Optimization Design of Tablet Computer
Four finite element models (FEM) are constructed for the above four performance constraints. To improve efficiency, the four corresponding response surfaces are established based on the given samples. Furthermore, the accuracy of the response surfaces is verified. Then the ATMDE algorithm is utilized to solve the tablet computer optimization problem. The structural thickness of optimized tablet computer is 6.42 mm which is a 31.7% reduction in compared with that of the original design (6.00 mm, 1.20 mm, 1.20 mm, 1.00 mm), and its design variables are 4.00 mm, 0.51 mm, 1.41 mm and 0.50 mm, respectively. Under this circumstance, the temperature of the chip is 62.05 °C and the temperature of shell surface is 37.66 °C which can ensure consumer daily-using comfortably. The thermal stress of battery is about 24 MPa, which can make sure the operating safety in daily. The maximal stress of touch screen is about 100 MPa, which can avoid the device broken during the collision of 0.5 m free fall. This optimized structural design is meaningful because the consumers are satisfied with the final design with better the appearance and portability for the tablet.
7 Conclusions
- (1)
An improved differential evolution with shrinking space technique and adaptive trade-off model, named ATMDE, is proposed to solve constrained optimization problems with high accuracy and robustness.
- (2)
The new “DE/rand/best/1” mutant strategy is constructed to generate offspring by the feasibility proportion of the current population, which could enhance performance of the ATMDE illustrated by results of test functions.
- (3)
In comparison with AATM algorithm, ATMDE achieves better performance verified by the simulation results of eighteen benchmark test functions from the IEEE CEC2006.
- (4)
The ATMDE is employed to optimize the structural optimization design of tablet computer, and the optimized thickness is a 31.7% reduction in compared with that of the original design.
Notes
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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