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Aimed at improving the insufficient search ability of constraint differential evo lution with single constraint handling technique when solving complex optim ization prob lem, this paper proposes a constraint differential evolution algorithm based on ensemble of constraint handling techniques and multi-population framework, called ECMPDE. First, handling three improved variants of differential evolution algorithms are dynamically matched with two constraint handling techniques through the constraint allocation mechanism. Each combination includes three variants with corresponding constraint handling technique and these combinations are in the set. Second, the population is divided into three smaller subpopulations and one larger reward subpopulation. Then a combination with three constraint algorithms is randomly selected from the set, and the three constraint algorithms are run in three sub-populations re spectively. According to the improvement of fitness value, the optimal constraint algorithm is selected to run on the reward sub-population , which can share in formation and close cooperation among populations. In order to verify the effectiveness of the proposed algorithm, 12 standard constraint optimization problems and 10 engineering constraint optimization problems are tested. The experimenta l results show that ECMPDE is an effective algorithm for solving constraint optimization problems.

Most of the decision-making, engineering application, and control problems can be summed up as optimization problems, which are widely used in real life or production, such as engineering design [

The commonly used constraint handling techniques include penalty function method [

Differential Evolution (DE) [

Many scholars introduce differential evolution algorithm into constraint handling technique to solve constraint optimization problems. In 2012, Wang and Cai [

According to the principle of free lunch [

The structure of this paper is organized as follows. Section 2 introduces the preliminary knowledge. Section 3 introduces the proposed method in detail, including its constraint allocation mechanism, adaptive value calculation and multi-population framework. Extensive experiments and discussions are carried out in Section 4. Section 5 concludes this paper.

A) Constraint optimization problems

Without loss of generality, a constraint optimization problems (minimization problem) can be defined as:

min f ( x ) , x = ( x 1 , ⋯ , x D ) ∈ S s .t . g j ( x ) ≤ 0 , j = 1 , 2 , ⋯ , m ; h j ( x ) = 0 , j = m + 1 , m + 2 , ⋯ , n ; (1)

where f ( x ) is the objective function, x = ( x 1 , ⋯ , x D ) ∈ Ω ⊆ S represents the D-dimension decision vector . The feasible region Ω is restricted by n linear or nonlinear constraints:

Ω = { x ∈ S | g j ( x ) ≤ 0 , j = 1 , ⋯ , m ; h j ( x ) = 0 , j = m + 1 , ⋯ , n } (2)

where S is decision space with upper boundary L i and lower boundary U i :

L i ≤ x i ≤ U i , 1 ≤ i ≤ D (3)

g j ( X ) ( j = 1 , 2 , ⋯ , m ) and h j ( X ) ( j = m + 1 , ⋯ , n ) are the jth inequality constraint and the n-jth equality constraint, respectively.

For COPs, the total constraint violation degree of the solution x is defined as:

G ( x ) = ∑ j = 1 n G j ( x ) (4)

where G j ( x ) is the degree of constraint violation on the jth constraint, it can be expressed as follows:

G j ( X ) = { max { g j ( X ) , 0 } , 1 ≤ j ≤ m max { | h ( X ) j − δ | , 0 } , m + 1 ≤ j ≤ n (5)

in (5), δ is a positive tolerance value in equality constraints and usually set to 0.0001. The solution x is a feasible solution when G ( x ) = 0 .

B) Constraint handling techniques

1) Feasibility rule method

Feasibility Rule [

a) If G ( X 1 ) < G ( X 2 ) , X 1 is better than X 2 .

b) When G ( X 1 ) = G ( X 2 ) = 0 , if f ( X 1 ) < f ( X 2 ) , X 1 is better than X 2 .

c) When G ( X 1 ) > 0 and G ( X 2 ) > 0 , if G ( X 1 ) < G ( X 2 ) , X 1 is better than X 2 .

In the feasibility rule, in the first stage, the infeasible solution with low violation of the overall constraint is selected. In the second stage, we first select all the feasible solutions, and then select the infeasible solutions with low degree of overall constraint default. In the third stage, only the feasible scheme with the best target value is selected. The above criteria (1) and criterion (3) are for constraints, the two infeasible solutions push the infeasible solutions to the feasible region through the comparison of the total constraint violation degree, while the criterion (2) is aimed at the objective. The comparison of the two feasible solutions on the target value improves the global solution.

Feasible rules were first used in genetic algorithms by Ded. Mezura-Montes [

2) ε-constraint handling method

The ε-constraint handling method [

{ f ( x i ) < f ( x j ) , if G ( x j ) ≤ ε ∧ G ( x j ) ≤ ε ; f ( x i ) < f ( x j ) , if G ( x i ) = G ( x j ) ; G ( x i ) < G ( x j ) , otherwise . (6)

where ε is decreaing as its iterative generation grows, and its calculation formula is as follows:

ε = { ε 0 ( 1 − t T ) c p , if t T ≤ p ; 0 , otherwise . (7)

c p = − log ε 0 + λ log ( 1 − p ) (8)

where ε represents intialized as ε 0 = V ( X 0.2 ∗ P S ) , if ε = 0 , ε -constraint handling method is feasible rule method. X 0.2 * P S is the top 0.2 * PS th individual according to vilation degree, and λ is 6. The control parameter ranges are T ∈ [ 0.1 ∗ T max , 0.8 ∗ T max ] and c p ∈ [ 2 , 10 ] representive.

Takahama [

C) Differential evolution

Differential evolution algorithm is a real-coded parallel search algorithm. Its initial iteration mainly includes four operation steps: initialization, mutation, crossover and selection. After initializing the population, the algorithm generates a new generation of population through mutation, crossover and selection to guide the search process to approach the global optimal solution.

Firstly, an initial population containing NP objective vectors (NP individuals) is randomly generated in the decision space. After G generation, X i , G = [ x i , 1 G , x i , 2 G , ⋯ , x i , D G ] , i = 1 , 2 , ⋯ , N P is expressed as the i vector, where D represents the dimension of the optimization problem to be solved. Let V denote the mutant vector, and the six commonly used mutation operators are enumerated as follows:

DE\rand\1 [

DE\rand\2 [

DE\best\1 [

DE\best\2 [

DE\current-to-best\1 [

DE\current-to-pbest\1 [

where F is a scaling factor with values between 0 and 1. where r 1 , r 2 , r 3 , r 4 and r 5 are mutually different integers randomly chosen from [1, NP], “best” is the individual with the best fitness in the population, “current” represents the best individual in the current population, and “pbest” represents p individuals with the best fitness in the population.

In the crossover stage, a crossover operator is conducted on each pair of X i , d and V i , d to produce a trial vector. This operation can further enhance the population diversity. The commonly used binomial crossover operators are defined as:

U i , d = { V i , d , if r a n d ( 0 , 1 ) < C r or j = j r a n d ; X i , d , otherwise . (15)

where j r a n d ∈ { 1 , 2 , 3 , ⋯ , D } represents a random integer uniformly generated between 1 and D, The cross strength is controlled by a parameter C r ∈ [ 0 , 1 ] .

When the trial vector is generated, the selection operator will select the best individual as t + 1 generation in the pairwise comparison between U i , d and X i , d :

X i , d t + 1 = { U i , d t , if f ( U i , d t ) ≤ f ( X i , d t ) ; X i , d t , otherwise . (16)

JADE is a simple and efficient variant of DE. In JADE, a new “current-to-pbest/1” mutation strategy is adopted as follows:

V = X c u r r e n t + F ∗ ( X p b e s t − X r 1 ) + F ∗ ( X r 2 − X ˜ r 3 ) (17)

JADE’s mutation strategy “current-to-pbest/1” is incorporated with an archive A which contains some recently declared inferior solutions. Let P denote the current population, and X ˜ r 3 is from the union of A and P.

The algorithm refers to a complex parameter adaptation mechanism. The parameters F and Cr are generated by the normal distribution and Cauchy distribution of each objective vector, respectively. Better control parameter values are passed to subsequent generation by recording promising F and Cr values and using them for new parameter generation.

CoDE proposed by Wang et al. is a DE variant with compound trial vector generation strategy and control parameters. These strategies and parameters have obvious advantages, and it can be regarded as a low-level integrated DE of variation strategy and control parameters. Three kinds of trial vector generation strategies namely “ rand/1/bin”, “rand/2/bin” and “current-rand/1” and three combinations of control parameters are adopted in CoDE, i.e. [F = 1.0, Cr = 0.9], [F = 1.0, Cr = 0.1] and [F = 0.8, Cr = 0.2]. In each generation, the experimental vector generation strategy and control parameter settings are randomly selected from the policy candidate pool and the parameter candidate pool, respectively.

EPSDE, originally designed by Mallipeddi et al., is a popular DE variant that integrates mutation strategies and parameters. In EPSDE, the variation strategy pool and parameter pool are constructed respectively. “DE/best/2/bin”, “DE/rand/1/bin” and “DE/current-to-rand/1/bin” are included in the policy pool. The pool of CR values is taken in the range 0.1 - 0.9 in steps of 0.1, and the pool of F values is taken in the range 0.4 - 0.9 in steps of 0.1. Each individual in the initial population is a signed variation strategy and parameter values randomly selected from each pool. Variation strategies and parameter values that produce better offspring survive, while those that do not produce better offspring are reinitialized.

Experimental research shows that JADE can effectively solve unimodal optimization problems, CoDE shows good performance in solving various optimization problems, especially in dealing with some multimodal optimization problems, while EPSDE is particularly effective in solving some highly complex problems.

In the process of global searching of the constraint optimization problem, the population is generally clarified into three cases: 1) there are infeasible solutions in the population, and the main purpose is to find the feasible region and increase the proportion of feasible solutions; 2) there are partial feasible solutions in the population. When solving the problem, we not only make full use of the information of the feasible solution, but also consider the information of the infeasible solution to enhance the diversity of the solution; 3) when all the solutions are feasible, the optimal solution is found in the feasible region.

Different constraint handling methods can be used handling in different situations of the search process. How to select different algorithms and handling constraint handling techniques according to different solving stages of constraint optimization problems plays an important role in the search process. Based on the above consideration, this paper proposes a constraint differential evolution algorithm based on ensemble of constraint handling techniques and multi-population framework, which is denoted as ECMPDE. The algorithm uses three complementary DE variants, namely JADE, jDE and EPSDE, and combines these DE variants with two constraint handlinghandling techniques one-to-one. In the three combinations, the computing resources are allocated to the optimal collocation by improving the proportion of their respective fitness values. First, the population is divided into three small-scale equal subpopulations and larger reward subpopulations, and a subpopulation is randomly assigned to the three DE variants. Second, the constraint handling technique is combined with DE variant by the constraint allocation mechanism, and the combined constraint handling mechanism is used to compare the target vector with the experimental vector to generate a new population and calculate the improved value of the fitness value. Third, the best combination is determined according to the cumulative improved fitness value and the proportion of cumulative consumption, and then the reward subpopulation is assigned to the combination; finally, when running to the next generation, if the optimal collocation of the previous generation is smaller than the solution obtained by the current optimal collocation, it will be added to the constraint mechanism pool to increase its selection probability.

A) Constraint allocation mechanism

Different problems have different characteristics. According to the theory of free lunch, single constraint handling technique can easily lead to weak performance in solving some problems. In this paper, feasibility rules and constraint handling methods are put into the constraint mechanism pool, and a constraint optimization algorithm allocation mechanism is designed. In the constraint handling mechanism, there are 8 combinations according to the combination principle.

Suppose the feasibility rule is A, the constraint handling method is B, and the combined pool P is as follows:

the combined pool P = { P 1 = { p o p 1 ( A J A D E ) , p o p 2 ( A j D E ) , p o p 3 ( A E P S D E ) } P 2 = { p o p 1 ( A J A D E ) , p o p 2 ( B j D E ) , p o p 3 ( A E P S D E ) } P 3 = { p o p 1 ( A J A D E ) , p o p 2 ( A j D E ) , p o p 3 ( B E P S D E ) } P 4 = { p o p 1 ( A J A D E ) , p o p 2 ( B j D E ) , p o p 3 ( B E P S D E ) } P 5 = { p o p 1 ( B J A D E ) , p o p 2 ( A j D E ) , p o p 3 ( A E P S D E ) } P 6 = { p o p 1 ( B J A D E ) , p o p 2 ( A j D E ) , p o p 3 ( B E P S D E ) } P 7 = { p o p 1 ( B J A D E ) , p o p 2 ( B j D E ) , p o p 3 ( A E P S D E ) } P 8 = { p o p 1 ( B J A D E ) , p o p 2 ( B j D E ) , p o p 3 ( B E P S D E ) } (18)

When each generation triggers the constraint allocation mechanism, a collocation is randomly selected from the constraint mechanism pool and assigned to the corresponding three DE variants, each with a population, for example, Suppose P 2 is randomly selected, JADE algorithm and feasibility rules are run in p o p 1 , jDE algorithm and constraint handling methods are run in p o p 2 , and EPSDE algorithm and feasibility rules are run in p o p 3 .

B) Fitness calculation

In 2009 [

F F ( x i ) = { f n o r m ( x i ) , if R F = 1 ; G n o r m ( x i ) , if R F = 0; f n o r m ( x i ) 2 + G n o r m ( x i ) 2 , otherwise . (19)

In the semi-feasible case, the population contains both unfeasible individuals and feasible individuals. For constraint optimization, how to deal with the unfeasible individuals in the population is a very important problem. Since some information carried by some infeasible individuals may be very important to find the optimal solution, it is unreasonable to exclude all infeasible individuals in the semi-feasible case.

C) Multi-population based ensemble framework

In this paper, similar to EDEV [

p o p = ∪ i = 1 , 2 , 3 , 4 p o p i (20)

Suppose that NP is the size of the population and N P i is the size of p o p i . λ i denotes the proportion between subpopulations and population. Thus we have

N P i = λ i ⋅ N P (21)

∑ i = 1 , 2 , 3 , 4 λ i = 1 (22)

Here we just let λ 1 = λ 2 = λ 3 , Each subpopulation is randomly assigned to a combination of DE variants and constraint handling techniques, while the rewarded subpopulation is assigned to the combination with the best results of the three populations. A population division operation is performed in each generation. With the progress of the algorithm, we determine the most effective DE variant ( i b e s t ) in the last period of time according to the ratio between the cumulative fitness improvement and the function evaluation of consumption after each generation.

Δ f i = ∑ ( F F t − F F t + 1 ) (23)

i b e s t = arg max i = 1 , 2 , 3 ( Δ f i Δ f e s i ) (24)

where Δ f i represents the accumulated function fitness improvements during the last ng generations attributed by the ith constituent DE variant and Δ f e s i is the consumed number of function evaluations. arg max i = 1 , 2 , 3 represents the maximum value selected in Δ f i / Δ f e s i , ( i = 1 , 2 , 3 ) .

The population division used in this article is different from ECHDE and EDEV. In ECHDE, the population is divided into four populations of the same size. The four constraint handling mechanisms each have a population. In EDEV, the population is divided into three equal subpopulations and a larger reward subpopulation, which is given to the best DE variants after accumulating a certain algebra. The best combination is selected directly from the three combinations and then given to the reward subpopulation to run.

D) Algorithm framework

In this paper, a new constraint differential evolution algorithm is proposed through the methods of multi-population, multi-DE variants and ensemble of constraint handling techniques. The ECMPDE framework is as follows (Algorithm 1):

The ECMPDE framework is as follows (Algorithm 1):

In this paper, ECMPDE is compared with CCoDE [

In this paper, the population size of the ECMPDE algorithm is set as 100 for the experiment, and the parameters in the variant are set as shown in

Algorithm | parameter settings |
---|---|

ECMPDE | λ 1 = λ 2 = λ 3 = 0.1 , λ 4 = 0.7 , n g = 20 , N P = 100 |

JADE | p = 0.05 , c = 0.1 , N P = 100 |

jDE | τ 1 = τ 2 = 0.1 , F l = 0.1 , F u = 0.9 , N P = 100 |

EPSDE | C R ∈ [ 0.1 , 0.9 ] , F ∈ [ 0.4 , 0.9 ] , N P = 100 |

A) Result Analysis of Standard COPs

The MaxFEs of 12 standard test problems is set as 500,000. The population size of CCoDE, ECHTEP, FROFI, DyHF, jDE and JADE was set as 100. The test results of ECMPDE algorithm and six comparison algorithms are shown in

As can be seen from

Algorithm | statistics | G01 | G02 | G03 | G04 | G05 | G06 |
---|---|---|---|---|---|---|---|

ECMPDE | Minbest | −15 | −0.80362 | −0.95559 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.80325 | −0.77313 | −30665.5 | 5126.497 | −6961.81 | |

Std | 0 | 0.000976 | 0.00179 | 1.09E−11 | 9.09E−13 | 1.82E−12 | |

CCoDE | Minbest | −15 | −0.80362 | −1.0005 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.80181 | −1.0005 | −30665.5 | 5126.497 | −6961.81 | |

Std | 0 | 0.005241 | 2.47E−16 | 1.09E−11 | 9.09E−13 | 1.82E−12 | |

DyHF | Minbest | −15 | −0.80362 | −1.0005 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.80314 | −1.0005 | −30665.5 | 5126.497 | −6961.81 | |

Std | 0 | 0.001852 | 5.32E−16 | 1.09E−11 | 9.09E−13 | 1.82E−12 | |

ECHTDE | Minbest | −15 | −0.80362 | −1.0005 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.80185 | −1.0005 | −30665.5 | 5126.497 | −6961.81 | |

Std | 2.47E−07 | 0.003963 | 5.75E−16 | 1.09E−11 | 9.09E−13 | 1.82E−12 | |

FROFI | Minbest | −15 | −0.80362 | −1.0005 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.8031 | −1.0005 | −30665.5 | 5126.497 | −6961.81 | |

Std | 0 | 0.002783 | 4.78E−16 | 1.09E−11 | 9.09E−13 | 1.82E−12 | |

jDE | Minbest | −15 | −0.80362 | −0.7137 | −30665.5 | 5126.497 | −6961.81 |

Minmean | −15 | −0.80252 | −0.51277 | −30665.5 | 5162.889 | −6961.81 | |

Std | 0 | 0.003303 | 0.096372 | 1.09E−11 | 61.37129 | 1.82E−12 | |

JADE | Minbest | −15 | −0.78594 | −0.78462 | −30665.5 | 65535 | −6961.81 |

Minmean | −15 | −0.76934 | −0.49245 | −30665.5 | 65535 | −6961.81 | |

Std | 2.59E−09 | 0.006689 | 0.106397 | 0.012892 | NAN | 1.82E−12 |

Algorithm | statistics | G07 | G08 | G09 | G10 | G11 | G12 |
---|---|---|---|---|---|---|---|

ECMPDE | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 1.23E−14 | 2.09E−17 | 3.09E−13 | 2.12E−14 | 1.11E−16 | 0 | |

CCoDE | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 6.83E−15 | 2.78E−17 | 4.06E−13 | 4.55E−12 | 1.11E−16 | 0 | |

DyHF | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 5.84E−15 | 2.78E−17 | 3.41E−13 | 4.49E−12 | 1.11E−16 | 0 | |

ECHTDE | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 8.45E−12 | 2.83E−17 | 3.98E−13 | 1.17E−11 | 1.11E−16 | 0 | |

FROFI | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 7.81E−15 | 2.78E−17 | 4.53E−13 | 4.12E−12 | 1.11E−16 | 0 | |

jDE | Minbest | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 |

Minmean | 24.30621 | −0.09583 | 680.6301 | 7049.248 | 0.7499 | −1 | |

Std | 6.14E−07 | 2.78E−17 | 4.24E−13 | 6.64E−06 | 1.11E−16 | 0 | |

JADE | Minbest | 24.44074 | −0.09583 | 680.6731 | 7095.689 | 0.749901 | −1 |

Minmean | 24.8193 | −0.09583 | 680.8112 | 7160.225 | 0.749929 | −1 | |

Std | 0.256666 | 2.78E−17 | 0.082211 | 29.72393 | 3.01E−05 | 0 |

Statistic Test | Symbol | ECMPDE | CCoDE | DyHF | ECHTDE | FROFI | jDE | JADE |
---|---|---|---|---|---|---|---|---|

Mann-Whitney | Win | + | 3 | 3 | 6 | 3 | 6 | 9 |

Tied | ≈ | 7 | 7 | 5 | 7 | 6 | 3 | |

Lose | − | 2 | 2 | 1 | 2 | 0 | 0 | |

Iman-Davenport | MinMean’s Rank | 2.956 | 3.47 | 3.01 | 3.89 | 3.41 | 4.32 | 4.52 |

MinBest’s Rank | 2.81 | 3.02 | 2.98 | 3.57 | 3.33 | 4.12 | 4.25 |

From the Iman-Davenport test in

B) Comparisons of Engineering COPs

In this paper, ECMPDE algorithm and CCoDE, ECHTEP, FROFI, DyHF, jDE and other five algorithms are compared and tested on 10 engineering constraint optimization problems to further verify the effectiveness of ECMPDE algorithm in practical problems. The group size setting of CCoDE, ECHTEP, FROFI, DyHF, jDE algorithm is the same as that of ECMPDE algorithm, which is set to 100. In order to facilitate the comparison of the results, the maximum evaluation times of all algorithms for solving engineering optimization problems MaxFEs is set to 500,000 times, the number of runs is 30 times, and the parameter setting is the same as that of IV. The test results are shown in

Similar to the previous section, the test results of the three statistical analysis methods are compared in

From the comparison of the results of Mann-Whitney rank sum test in

Algorithm | Statistics | Tension pressure spring design | Welded beam design | Pressure vessel design | Reducer design | Design of three-bar truss |
---|---|---|---|---|---|---|

ECMPDE | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 3.94E−18 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 0 | |

CCoDE | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 6.43E−18 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 0 | |

DyHF | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 2.01E−16 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 4.3E−12 | |

ECHTDE | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 7.2E−18 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 0 | |

FROFI | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 5.09E−18 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 0 | |

jDE | Minbest | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 |

Minmean | 0.012665 | 1.724852 | 5885.333 | 2994.471 | 263.8958 | |

Std | 1.38E−12 | 1.11E−15 | 9.09E−13 | 1.82E−12 | 0 |

Algorithm | Statistics | Design of hydrodynamic thrust bearing | Conical wheel design | Rolling bearing design | Butterfly spring design | Design of gear train |
---|---|---|---|---|---|---|

ECMPDE | Minbest | 1625.443 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1625.443 | 14.67253 | −79216.2 | 1.979675 | 5.42E−12 | |

Std | 4.55E−13 | 9.4E−15 | 0.00045 | 1.33E−15 | 6.93E−12 | |

CCoDE | Minbest | 1625.443 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1667.956 | 14.67253 | −81845.3 | 1.979675 | 4.74E−12 | |

Std | 228.9411 | 3.55E−15 | 0 | 1.33E−15 | 6.11E−12 | |

DyHF | Minbest | 1625.443 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1888.634 | 14.67253 | −81678 | 1.983829 | 2.7E−12 | |

Std | 181.67 | 1.04E−14 | 900.7875 | 0.02237 | 1.21E−27 | |

ECHTDE | Minbest | 1625.443 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1625.443 | 14.67253 | −81844.9 | 1.992137 | 7.86E−12 | |

Std | 4.55E−13 | 3.55E−15 | 1.357644 | 0.037387 | 2.08E−11 | |

FROFI | Minbest | 1654.128 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1725.325 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 | |

Std | 33.08269 | 3.55E−15 | 0 | 1.33E−15 | 1.21E−27 | |

jDE | Minbest | 1625.443 | 14.67253 | −81845.3 | 1.979675 | 2.7E−12 |

Minmean | 1625.443 | 14.67253 | −79216.2 | 1.979675 | 4.06E−12 | |

Std | 4.55E−13 | 1.04E−14 | 9837.045 | 1.42E−15 | 5.08E−11 |

Statistic Test | Symbol | ECMPDE vs | CCoDE | ECHTEP | FROFI | DyHF | jDE |
---|---|---|---|---|---|---|---|

Mann-Whitney | Win | + | 4 | 6 | 2 | 5 | 4 |

Tied | ≈ | 10 | 9 | 11 | 9 | 10 | |

Lose | − | 1 | 0 | 2 | 1 | 1 | |

Iman-Davenport | MinMean’s Rank | 3 | 3.36 | 3.76 | 3.23 | 4.03 | 3.6 |

MinBest’s Rank | 3.06 | 3.56 | 3.56 | 3.6 | 3.53 | 3.3 | |

Wilcoxon | Signed Rank | R − | 59.5 | 59.5 | 58.5 | 59.5 | 56 |

R + | 60.5 | 60.5 | 61.5 | 60.5 | 64 |

the rank of ECMPDE algorithm is the smallest which is 3.06. In these comparisons, the performance of ECMPDE algorithm is better than that of other comparison algorithms. In the comparison results of Wilcoxon symbolic rank test, the symbolic rank sum is less than that, indicating that the performance of ECMPDE algorithm is the best.

Based on the analysis of the above results, we can know that the overall solution effect of ECMPDE algorithm is better than that of CCoDE, ECHTEP, FROFI, DyHF and jDE in 15 engineering constraint optimization problems, and the performance of the algorithm is the best.

In this paper, we use two constraint handling mechanisms to combine multiple DE variants and combine multiple group frameworks to overcome the shortcomings of weak constraint solving ability and low search ability. In this paper, ECMPDE combines DE variants such as JADE, jDE and EPSDE. The whole population of the algorithm is divided into two categories, namely, three subpopulations and one reward subpopulation. Each constituent DE variant has a subpopulation, and each variant is assigned a constraint handling mechanism through the constraint allocation mechanism, while the reward subpopulation is dynamically assigned to the combination of the best-performing DE variants and constraint handling mechanisms. The improved population division occurs in each generation, and the redistribution of reward subpopulations is triggered by each generation. Through the comparison of the best collocation of the previous generation and the current best collocation, the excellent collocation is saved in the constraint mechanism pool to improve the probability of selection. According to the analysis of the simulation results, ECMPDE is better than DyHF, ECHTEP, FROFI, jDE and JADE in solving test problems, and its performance is similar to that of CCoDE. This algorithm can improve the searching ability of the algorithm, improve the ability to solve constraints, and successfully inherit the advantages of its DE variant, showing strong robustness and high accuracy. In the future research, it is hoped that there will be excellent constraint handling mechanisms to further improve ECMPDE.

The author would like to thank the area editor and anonymous reviewers for their valuable comments and suggestions for this paper. This work is proudly supported in part by National Natural Science Foundation of China (No. 61763008, 11661030, 11401357, 11861027), Natural Science Foundation of Guangxi (No. 2018GXNSFAA138095), Fangchenggang Scientific research and technology development plan 2014 No. 42, and Project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([

The authors declare that there is no conflict of interests regarding the publication of this article.

Wei, Y.T., Feng, Q.X. and Yuan, S.N. (2020) Differential Evolution Algorithm Based on Ensemble of Constraint Handling Techniques and Multi-Population Framework. International Journal of Intelligence Science, 10, 22-40. https://doi.org/10.4236/ijis.2020.102003