# A Customized Differential Evolutionary Algorithm for Bounded Constrained Optimization Problems

Authors: Wali Khan Mashwani; Zia Ur Rehman; Maharani A. Bakar; Ismail Koçak; Muhammad Fayaz
Journal: Complexity
 (2021)

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## Abstract

Bound-constrained optimization has wide applications in science and engineering. In the last two decades, various evolutionary algorithms (EAs) were developed under the umbrella of evolutionary computation for solving various bound-constrained benchmark functions and various real-world problems. In general, the developed evolutionary algorithms (EAs) belong to nature-inspired algorithms (NIAs) and swarm intelligence (SI) paradigms. Differential evolutionary algorithm is one of the most popular and well-known EAs and has secured top ranks in most of the EA competitions in the special session of the IEEE Congress on Evolutionary Computation. In this paper, a customized differential evolutionary algorithm is suggested and applied on twenty-nine large-scale bound-constrained benchmark functions. The suggested C-DE algorithm has obtained promising numerical results in its 51 independent runs of simulations. Most of the 2013 IEEE-CEC benchmark functions are tackled efficiently in terms of proximity and diversity.


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## Body

## 1. Introduction

Bound optimization is the mathematical process of optimizing an objective function in the presence of constraints imposed on the decision space. These variables in the decision space may be continuous, discrete, or mixed. The basic elements of optimization are decision variables, objective function, and constrained functions. The standard form for a single-objective, nonlinear, constrained optimization problem is described as follows [1]: (1)minimizefx,x=x1,x2,…,xn∈Ω⊆Rnsubjecttocix≤0,i=1,…,l,cjx=0,j=l+1,…,p,xkl≤xk≤xku,k=1,…,n,where x=x1,x2,…,xnT∈Ω is the candidate solution with n decision variables/real parameters, fx is the objective function, cix≤0 represents p inequality constrained functions, cjx=0 denotes q equality constrained functions, and xkl≤xk≤xku are bound constrained over the search space x∈Ω. An optimum solution x∗ that is optimal in some certain neighbourhood is called a local optimum. Mathematically, by a local optimum, we mean a solution x∗ such that fx∗≤fx for all x in some neighbourhood of x∗, Then, optimum solution x∗ is called global optimum solution if fx∗≤fx,∀x∈Ω⊆Rn.In the last two decades, various optimization methods were developed and are still developing for dealing with a diverse test suite of benchmark functions and real-world optimization problems [2–6]. These algorithms are inspired by Darwin’s theory of evolution with Darwinian’s principle “Survival of the fittest”. The main intrinsic operators in the framework of EAs to evolve population include mutation and crossover operators, by using mutation and crossover operator with elitism scheme [7]. EAs are nature-inspired-based techniques [8–10]. In general, EAs can be categorized into nine different groups including the biology-based [11, 12], physics-based [13, 14], social-based [15–18], music-based [19], chemical-based [20], sport-based [21], mathematics-based [22], swarm-based [23–29], and hybrid methods [30–37]. All these algorithms operate on a uniform and random set of solutions generated within the search domain of the given optimization and search problems. Among the evolutionary algorithms, the genetic algorithm (GA) [38], evolutionary strategy (ES) [39], evolutionary programming (EP) [5, 29], and genetic programming (GP) [40] are classical paradigms in evolutionary computing field [3]. Differential evolution (DE) [41], particle swarm optimization (PSO) [24], and ant colony optimization (ACO) [23, 42, 43], cuckoo search (CS) [44], and teaching learning-based optimization (TLBO) algorithm [45, 46] are comparatively recently developed EAs and successfully applied to various test suites of optimization problems and many real-world problems [18, 47–50].Differential evolution (DE) is one of the well-established population-based evolutionary algorithms, which was first introduced by Storn and Price for solving the Chebyshev polynomial fitting problems [51, 52]. DE is capable of handling nonlinear, linear, and multimodel objective functions and solving directional preferences over existing relays in the power system. The DE has also worked to train the weight of the neural network to deal with real-world problems. The basic idea of this technique is to avoid repetition of the existing set of solutions. Classic DE has two main control parameters F and CR that need to be settled efficiently. The settlement of these parameters including the scaling factor, F, and the probability of crossover,CR, is as discussed in [31, 41, 53–61]. Crossover is the main source of exploration, the mutation is utilized for exploitation purposes, and selection operators brought down the pressure for the survival of fittest individuals to evolve the population. The adjustment of various parameters used in search operator implementation is mainly problem-dependent, and their proper settings are quite difficult and time-consuming while performing trial and error experiments [62].Different search operators behave differently at different levels of the optimization search process while dealing with complicated optimization and search problems. It is quite difficult to determine that this particular crossover or mutation or selection operator is useful in this particular proposition optimization process. In this paper, a customized differential evolutionary algorithm is suggested for solving benchmark functions designed for the special session of the 2013 IEEE Congress of Evolutionary Computation (CEC’13) [63]. The proposed C-DE algorithm employs more than one mutation operator flowing by crossover to generate a new set of solutions for the next generation of the evolutionary search process. Experimental results demonstrated that the proposed C-DE algorithm is more promising as compared to the algorithm that employs a single mutation operator for population evolution.The rest of the paper is organized as follows. Section2 introduces the framework of the proposed customized differential evolutionary algorithm. Section 3 demonstrates the experimental results and characteristics of the used benchmark functions. Section 4 finally concludes this paper.


## 2. A Customized Differential Evolutionary Algorithm for Numerical Optimization Problems

Differential evolution (DE) is a stochastic algorithm that was first developed by Kenneth Price in 1994 [40]. DE is one of the most promising global search techniques that converge toward the known optimal solution of the problem at hand in a significant manner [60]. In general, DE has five different mutation strategies; among them, rand/1 and best/1rand/1 have performed better in our case. Among the five DE variants, DE rand/1 has faced premature convergence issues and slow convergence behaviour as compared to DE best/1rand/1. The proposed customized DE employs DE rand/1 and DE best/1rand/1 mutation strategies simultaneously to alleviate premature convergence of original differential evolution. The customized DE does the switching of aforementioned mutation strategies with linearly reducing weight parameter r from 0.25 to 0.01 during their whole course of optimization. (Figure 1).Algorithm 1:The pseudocode of DE rand/1 variant.
(1)
Set PopSize=nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin, upper scaling bound =  betamax, crossover probability =  pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
GenerateMutantPop from Pop using rand/1 mutation strategy;(9)
Use binomial crossover to generateNewPop;(10)
Evaluatefxit for each xit∈NewPop;(11)
SetFEVs=FEVs+nPop;(12)
UpdatePop by comparing Pop with NewPop through their fitness values;(13)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(14)
end whileAlgorithm 2:The pseudocode of DE rand/2 variant.
(1)
Set Pop Size =nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin, upper scaling bound =  betamax, crossover probability =  pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
GenerateMutantPop from Pop using rand/2 mutation strategy;(9)
Use binomial crossover to generateNewPop;(10)
Evaluatefxit for each xit∈NewPop;(11)
SetFEVs=FEVs+nPop;(12)
UpdatePop by comparing Pop with NewPop through their fitness values;(13)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(14)
end whileAlgorithm 3:The pseudocode of DE best/rand/2 variant.
(1)
Set Pop Size =nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin, upper scaling bound =  betamax, crossover probability =  pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
GenerateMutantPop from Pop using best/rand/2 strategy;(9)
Use binomial crossover to generateNewPop;(10)
Evaluatefxit for each xit∈NewPop;(11)
SetFEVs=FEVs+nPop;(12)
UpdatePop by comparing Pop with NewPop through their fitness values;(13)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(14)
end whileAlgorithm 4:The pseudocode of DE current/rand/2 variant.
(1)
Set Pop Size =nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin, upper scaling bound =  betamax, crossover probability =  pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
GenerateMutantPop from Pop using current/rand/2 mutation strategy;(9)
Use binomial crossover to generateNewPop;(10)
Evaluatefxit for each xit∈NewPop;(11)
SetFEVs=FEVs+nPop;(12)
UpdatePop by comparing Pop with NewPop through their fitness values;(13)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(14)
end whileAlgorithm 5:The pseudocode of DE best/1rand/1 variant.
(1)
Set Pop Size =nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin, upper scaling bound =  betamax, crossover probability =  pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
GenerateMutantPop from Pop using best/1rand/1 strategy;(9)
Use binomial crossover to generateNewPop;(10)
Evaluatefxit for each xit∈NewPop;(11)
SetFEVs=FEVs+nPop;(12)
UpdatePop by comparing Pop with NewPop through their fitness values;(13)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(14)
end whileFigure 1
Flowchart of the proposed customized DE (CDE) algorithm.Mutation strategies are the most distinctive components of differential evolution which are described as follows [64]:(1)
Rand/1: mathematically, it is formulated as follows:(2)yit=xr1t+F×xr2t−xr3t,wherer1≠r2≠r3,wherexr1t, xr2t, and xr3t are randomly taken distinct solutions from population other than xit (target solution). F∈0.3,0.7 is the scaling factor which is randomly generated. yit is the mutant vector. In Algorithm 1, the mutation strategy as described in equation (2) is used and solved the CEC-2013 unconstrained suite of benchmark functions. The abovementioned mutation strategy is good in exploration while comparatively weak in exploitation because of which it does not converge efficiently keeping in view the average value of most of the used benchmark functions (Algorithm 2).(2)
Best/rand/1: mathematically, it is formulated as follows:(3)yit=xbestt+F×xr1t−xr2t,wherer1≠r2,wherexbestt is the best solution in the current population, xr1t and xr2t are randomly chosen distinct solutions, and xit is the ith target solution. F∈0.3,0.7 is the scaling factor which is randomly generated. yit is the mutant vector. In Algorithm 3, the mutation strategy as described in equation (3) has shown better performance in exploitation and comparatively not good in exploration that induces a premature convergence.(3)
Current/rand/2: mathematically, it is formulated as follows:(4)yit=xit+F×xr2t−xr3t+F×xr4t−xr5t,wherer1≠r2≠r3≠r4≠r5,wherexit is the target solution, xr1t, xr2it, xr3t, and xr4t are randomly taken distinct solutions from population other than target solution. F∈0.3,0.7 is the scaling factor which is randomly generated. yit is the mutant vector. The mutation strategy as described in equation (4) is employed in Algorithm 4 for creating a new solution. The above-described mutation strategy is good in exploitation while comparatively weak in exploration.(4)
Rand/2: mathematically, it is formulated as follows:(5)yit=yr1t+F×xr2t−xr3t+F×xr4t−xr5t,wherexr1t, xr2t, xr3t, xr4t, and xr5t are randomly taken distinct solutions from population other than target solution. F∈0.3,0.7 is the scaling factor which is randomly generated. yit is the mutant vector. The mutation strategy as described in equation (5) has been used in Algorithm 4 and performed good in exploration while less effective in exploitation.(5)
Best/rand/2: mathematically, it is formulated as follows:(6)yit=xbestt+F×xr1t−xr2t+F×xr3t−xr4t,wherexbestt is the best solution in the current population and xr1t, xr2t, xr3t, and xr4t are taken randomly. F∈0.3,0.7 is the scaling factor which is randomly generated within the specified bound. yit is the mutant vector. The mutation strategy as described in equation (6) has been employed in Algorithm 4 to maintain exploitation versus exploration issue in order to avoid premature convergence during population evolution.(6)
Best/rand/1: mathematically, it is formulated as follows:(7)yit=xr1t+F×xr2t−xr3t+F×xbestt−xr1t,wherexr1t, xr2t, xr3t, xr4t, and xr5t are chosen randomly. The scaling factor, F∈0.3,0.7, is adjusted in all algorithms, and yit is the mutant vector as used in Algorithm 5.The proposed C-DE technique as outlined in Algorithm6 employs the mutation strategies as described in equations (2) and (6) along with crossover as described in equation (8) to create its new set of solutions for the next generation of population evolution.Algorithm 6:The pseudocode of the proposed customized DE algorithm.
(1)
Set Pop Size =nPop, Max Ftn Evaluations =  MaxFEVs, lower scaling bound =  betamin,upper scaling bound =  betamax,crossover probability = pCR;(2)
Randomly generate initialPop of size nPop;(3)
Set Function Evaluations Counter:FEVs=0;(4)
Evaluatefxit for each xit∈Pop;(5)
SetFEVs=FEVs+nPop;(6)
BestSol=x∗ such that x∗=minfx,∀x∈Pop;(7)
whileFEVs<MaxFEVsdo(8)
Taker randomly generated real number between 0 and 1;(9)
ifrand<Smthen(10)
GenerateMutantPop from Pop using rand/1 strategy;(11)
else(12)
GenerateMutantPop from Pop usingbest1/rand1 strategy;(13)
end if(14)
Use binomial crossover to generateNewPop;(15)
Evaluatefxit for each xit∈NewPop;(16)
SetFEVs=FEVs+nPop;(17)
UpdatePop by comparing Pop and NewPop through their fitness values;(18)
BestSol=x∗ such that fx∗=minfx,∀x∈Pop;(19)
end while


### 2.1. Crossover

In the current research, a binomial crossover operator is used. In the binomial crossover, every target vector is mixed with its corresponding mutant vector to get the trial vector as follows:(8)uit=yi,jt,ifrand≤CR orj=randi∈1:n;xi,jt,otherwise;where j is the coordinate counter, yit is the mutant vector, CR is the probability of crossover, randi is a randomly generated integer, n denotes the dimension of search space, xit is the target vector, and uit is the trial vector corresponding to xit.


### 2.2. Selection

The selection scheme for parent solution and their trial solution vector compete with each other based on their corresponding fitness values. The selection scheme is illustrated as follows:(9)xit+1=uit,iffuit<fyit;yit,otherwise;where uit is the trial vector, xit is the target vector, and fxit and fuit are the fitness values of the ith-trial and ith-target vector, respectively.


## 2.1. Crossover

In the current research, a binomial crossover operator is used. In the binomial crossover, every target vector is mixed with its corresponding mutant vector to get the trial vector as follows:(8)uit=yi,jt,ifrand≤CR orj=randi∈1:n;xi,jt,otherwise;where j is the coordinate counter, yit is the mutant vector, CR is the probability of crossover, randi is a randomly generated integer, n denotes the dimension of search space, xit is the target vector, and uit is the trial vector corresponding to xit.


## 2.2. Selection

The selection scheme for parent solution and their trial solution vector compete with each other based on their corresponding fitness values. The selection scheme is illustrated as follows:(9)xit+1=uit,iffuit<fyit;yit,otherwise;where uit is the trial vector, xit is the target vector, and fxit and fuit are the fitness values of the ith-trial and ith-target vector, respectively.


## 3. Simulation Results, Comparison, and Discussion

Due to the flurry of EAs recently developed, their performances are mainly analyzed by using different test suites of optimization and search problems. In the last few years, several unconstrained benchmark functions (i.e., bound-constrained) and constrained test problems are designed in special sessions of the IEEE Conference on Evolutionary Computation (IEEE-CEC) series for EA competition [63]. In a study of this paper, we have used twenty-eight different unconstrained continuous benchmark functions whose characteristics are given in Table 1. The first five test problems, F1−F5, are unimodal functions. Most of the evolutionary algorithms are getting trap in the local basin of attraction while solving these test problems. The problems from F6 to F20 are multimodal functions containing more than one local minima. These problems are much difficult to deal with as compared to unimodal functions. The functions F21 from F28 are composite functions.Table 1
The characteristics of the CEC’13 test function [63].
No.PropertiesFi∗=Fix∗UF1Sphere function−14002Rotated high conditioned elliptic function−13003Rotated bent cigar function−12004Rotated discus function−11005Different power function−1000MF6Rotated Rosenbrock’s function−9007Rotated Schaffers F7 function−8008Rotated Ackley’s function−7009Rotated Weierstrass function−60010Rotated Griewank’s function−50011Rastrigin’s function−40012Rotated Rastrigin’s function−30013Noncontinuous rotated Rastrigin’s function−20014Schwefel’s function−10015Rotated Schwefel’s function10016Rotated Katsuura function20017Lunacek bi-Rastrigin function30018Rotated Lunacek bi-Rastrigin function40019Expanded Griewank’s plus Rosenbrock’s function50020Expanded Scaffer’s F6 function600CF21Composition function 1 (n = 5, rotated)70022Composition function 2 (n = 3, unrotated)80023Composition function 3 (n = 3, rotated)90024Composition function 4 (n = 3, rotated)100025Composition function 5 (n = 3, rotated)110026Composition function 6 (n = 5, rotated)120027Composition function 7 (n = 5, rotated)130028Composition function 8 (n = 5, rotated)1400


### 3.1. PC Configuration and Simulation Environment

The simulations were conducted on a PC with 8 GB RAM and Intel(R)Core(TM)i3-7100U CPU @ 2.40 GHz processor. All simulations are done in MATLAB R2013a environment withpopsize=100. Each benchmark function has been solved in different dimensions including n=10,30,50 by executing 51 times independently. Maximum function evaluation was fixed to 10000×n. The algorithms for each problem are run 51 times independently to obtain the final best solution. Sm∈0,1 denotes the switching of mutation strategies.Tables2 and 3 present the numerical results of C-DE as outlined in Algorithm 6 in comparison with DE-1 as described in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F20 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 2. Similarly, the numerical results for the benchmark functions denoted by F21 to F28 are given in Table 3.Table 2
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F20 in n=10 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-15935834.56813e+062.28216e+062.11013e+06891922.510621DE-52494032.24398e+061.04699e+06987371426831.918032C-DE3105252.42089e+061.14756e+06996518512003.868545F03−1200DE-175046.14.66235e+061.55526e+061.1441e+061141062.835342DE-528606378126034587230535494580.836448C-DE−559.2967811665640518616.2131613.067626F04−1100DE-12663.8320828.91187612016.13735.150538DE-52661.713432.27847.187823.62023.165850C-DE2574.6611249.86114.036033.661982.673559F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−881.658−880.015−880.179−880.1580.213022DE-5−879.096−870.071−870.678−870.1871.999259C-DE−899.941−890.165−890.814−890.1862.263107F07−800DE-1−784.364−769.68−778.597−778.5473.576671DE-5−789.323−781.563−787.441−787.7211.694367C-DE−799.435−791.131−797.985−798.1811.262155F08−700DE-1−679.784−679.479−679.635−679.6360.068761DE-5−679.813−679.505−679.631−679.620.064325C-DE−679.819−679.523−679.657−679.6350.075517F09−600DE-1−588.254−585.522−586.761−586.7340.616831DE-5−591.155−587.466−588.721−588.5430.787579C-DE−596.588−593.206−594.598−594.4740.722405F10−500DE-1−488.93−487.95−488.6−488.6330.199213DE-5−491.612−491.105−491.349−491.3620.139360C-DE−499.622−499.119−499.386−499.3920.113407F11−400DE-1−400−400−400−4000.000000DE-5−400−400−400−4000.000000C-DE−400−400−400−4000.000000F12−300DE-1−281.09−263.199−269.691−269.3513.606150DE-5−284.128−270.525−277.174−276.6942.946380C-DE−292.656−280.911−286.92−286.9363.047453F13−200DE-1−183.951−166.48−174.742−174.2683.819038DE-5−188.699−174.674−181.719−182.0993.220455C-DE−193.282−181.128−186.944−186.9212.858693F14−100DE-1−94−93.9375−93.9988−940.008745DE-5−95.8751−89.0452−95.0246−95.68771.514591C-DE−99.9375−96.3352−99.5179−99.75020.905209F15100DE-1966.0641624.911345.421363.52141.570897DE-5850.8711581.881269.041266.33165.250631C-DE778.431452.191184.571212.61150.366735F16200DE-1200.555201.516201.119201.1370.196826DE-5200.51201.527201.122201.1270.223394C-DE200.649201.478201.098201.1090.203207F17300DE-1320.122320.122320.122320.1220.000001DE-5311.542318.288318.01318.1220.924608C-DE310.122310.286310.128310.1220.023545F18400DE-1433.641453.204444.794445.4143.694887DE-5429.56449.267437.735437.2573.894859C-DE421.706438.851429.938429.7533.321748F19500DE-1510.299510.76510.595510.5990.089374DE-5504.153504.684504.538504.5480.082422C-DE500.355500.752500.527500.5320.079466F20600DE-1602.688603.565603.182603.2010.210304DE-5602.184603.466602.781602.8150.299490C-DE602.163603.359602.785602.7730.276907Table 3
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF21 to F28 in n=10 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF21700DE-1966.5121200.191167.021200.1956.998745DE-51000.011200.191196.271200.1928.031778C-DE900.6241100.191092.951100.1936.339134F22800DE-1930.3451058.03987.216982.96229.935014DE-5907.2391106.44976.237997.54551.389783C-DE810.1531029.34899.141904.7453.322517F23900DE-11809.82484.842246.862289.89162.930066DE-51684.242609.92141.832145.54200.732186C-DE1619.942523.092093.092101.43198.583802F241000DE-11181.51238.291213.191212.2616.478379DE-51150.461238.261224.551231.5419.882749C-DE1141.111218.451208.851213.6415.683668F251100DE-11291.091334.341329.71331.297.180292DE-51266.161333.751320.131322.1611.632105C-DE1257.561314.851305.881308.810.946832F261200DE-11342.61394.561366.71365.1610.154832DE-51334.521425.071384.181387.636.985466C-DE1311.731400.071349.871341.5630.335554F271300DE-11703.241840.041770.841767.3934.869565DE-51620.081852.551650.551620.7164.877728C-DE1600.071824.261620.841600.4955.037487F281400DE-11716.318001797.84180012.230642DE-5155017501723.52175067.131550C-DE1500.6117001688.44170046.714278Tables4 and 5 demonstrate the numerical results in respect of the proposed C-DE as outlined in Algorithm 6 versus DE-1 as given in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F20 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 4. Similarly, the numerical results regarding the benchmark functions denoted by F21 to F28 are given in Table 5.Table 4
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F20 in n=30 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-11.36903e+082.08058e+081.77072e+081.76426e+0814956353.317846DE-51.19104e+081.54637e+081.33283e+081.33653e+087763339.936747C-DE1.47534e+076.42165e+073.17636e+073.12834e+078995583.667657F03−1200DE-11.07468e+091.30442e+091.13644e+091.12995e+0942393532.103937DE-51.05797e+081.76261e+081.20783e+081.1947e+0814758359.088589C-DE1036555.90805e+071.37612e+079.87793e+0613305275.710745F04−1100DE-1353547568157139.556623.99376.123241DE-518653.736440.725639.825351.44364.116408C-DE15683.932184.624428.424524.14004.135290F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−783.569−756.81−777.918−781.9848.219750DE-5−863.853−800.281−851.899−853.19912.856533C-DE−883.444−816.652−858.659−856.69212.862273F07−800DE-1−752.53−726.067−740.613−741.9176.037184DE-5−771.429−737.11−762.271−763.9587.427720C-DE−791.822−768.024−783.293−784.0845.870443F08−700DE-1−674.203−673.974−674.058−674.050.050295DE-5−669.238−668.974−669.045−669.0290.048735C-DE−679.167−678.961−679.048−679.040.043403F09−600DE-1−561.328−555.523−558.644−558.8261.427762DE-5−559.276−551.544−554.985−554.7611.682517C-DE−570.355−563.838−566.394−566.1981.325391F10−500DE-1−483.713−447.89−471.111−472.9887.436859DE-5−485.98−484.975−485.704−485.7540.222811C-DE−499.936−498.639−499.626−499.7190.269271F11−400DE-1−390−390−390−3900.000000DE-5−390−386.02−389.083−389.0051.011598C-DE−400−399.005−399.746−4000.437924F12−300DE-1−129.647−87.7556−105.127−104.3969.743013DE-5−181.252−111.655−139.279−138.23112.860856C-DE−189.566−130.269−150.997−149.48512.653504F13−200DE-1543.649606.651587.113588.02910.751144DE-5223.974285.604257.156257.86913.872433C-DE−76.0579−24.1726−46.6554−45.493310.970706F14−100DE-1958.6211163.131076.091073.9737.599756DE-51068.222325.261756.671732.49352.891847C-DE503.9071368.46870.066877.497193.156721F15100DE-16740.647841.497423.287454.15263.687213DE-56182.547797.067189.497202.74312.036005C-DE6651.87656.347232.337269.4287.708609F16200DE-1211.829212.918212.478212.5070.266427DE-5213.686214.943214.427214.4560.287333C-DE201.688202.936202.454202.4520.288254F17300DE-1351.495356.482353.565353.4931.325248DE-5360.308373.106367.184367.7472.924574C-DE347.203361.075355.355355.1482.813337F18400DE-1582.99647.337631.422632.87711.084542DE-5576.88621.743605.017606.25910.266823C-DE576.732615.316596.713597.6258.919994F19500DE-1515.067517.608516.468516.5010.599903DE-5524.311526.192525.428525.5090.411725C-DE503.979506.237505.276505.3860.497524F20600DE-1612.548613.437613.046613.0490.211167DE-5611.933613.14612.602612.6160.268802C-DE611.798613.08612.546612.5960.285701Table 5
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF21 to F28 in n=30 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF21700DE-1101011101083.33111040.626595DE-59501193.541055.47105084.276686C-DE9001143.54998.798100071.457633F22800DE-12410.213350.832881.942855.53227.697529DE-51997.43982.732865.522768.51410.507035C-DE1703.323285.892248.572238.3326.629221F23900DE-17761.698933.498467.148491.29276.464768DE-57610.148945.918282.958319.51315.448232C-DE7385.718893.868156.478238.79345.393637F241000DE-11285.611301.261295.221295.043.201388DE-51268.061328.191297.391298.5414.328935C-DE1212.51278.271248.741251.114.766463F251100DE-11442.271457.121451.451451.623.322013DE-51419.131445.031432.181432.25.749719C-DE1374.231395.21385.461386.975.218833F261200DE-11414.911425.421419.491419.412.455244DE-51411.071413.841411.941411.830.548844C-DE1401.051403.911402.061402.020.606003F271300DE-12512.92640.532579.622578.525.085286DE-52226.432554.192443.582456.8372.958778C-DE2129.692467.362361.892385.8873.692495F281400DE-118001800180018000.000000DE-517501750175017500.000000C-DE17001700170017000.000000Tables6 and 7 provide the experimental results approximated by the proposed C-DE herewith outlined in Algorithm 6 against DE-1 as given in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F19 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 6. Similarly, the numerical results regarding the benchmark functions denoted by F20 to F28 are given in Table 7.Table 6
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F19 in n=50 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-11.82669e+084.0298e+083.08217e+083.15245e+0840952346.718760DE-56.5501e+071.46685e+081.03416e+081.01884e+0818482876.197602C-DE5.44136e+071.46107e+081.04963e+081.05681e+0820121866.384027F03−1200DE-11.10012e+096.44772e+093.55854e+093.31787e+091263221747.238202DE-537540.45.7581e+089.78856e+073.8351e+07132237941.475719C-DE1.95311e+063.23647e+086.21553e+073.62992e+0770023673.941721F04−1100DE-169182.111915389629.890090.810106.552893DE-520820.941255.330832.530841.74612.877411C-DE20496.238856.929036.128704.63791.979814F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−836.081−834.721−835.527−835.5460.278565DE-5−835.493−780.995−832.512−833.4767.423864C-DE−855.053−850.805−852.701−852.7760.895707F07−800DE-1−659.767−624.433−642.8−642.1678.870652DE-5−738.024−694.803−716.596−716.0189.264348C-DE−788.119−756.809−770.008−769.7137.322152F08−700DE-1−676.032−675.804−675.865−675.8580.037308DE-5−676.982−676.812−676.87−676.860.041021C-DE−678.984−678.811−678.874−678.8670.042407F09−600DE-1−487.463−477.117−480.324−480.1721.930815DE-5−500.784−490.311−493.893−493.5122.256670C-DE−539.562−530.812−533.846−532.9592.223050F10−500DE-1−482.098−417.281−460.08−463.09116.289936DE-5−487.973−486.858−487.687−487.7290.227074C-DE−499.786−497.251−498.559−498.6320.544408F11−400DE-1−296.762−271.293−279.13−278.9524.588255DE-5−350−298.691−342.965−345.0258.544754C-DE−400−331.719−377.221−371.82720.613249F12−300DE-175.2158125.882103.524104.09613.276617DE-50.26887681.905849.921353.444518.892895C-DE−13.79377.658834.683236.796717.050881F13−200DE-1158.018226.329205.73207.24812.793081DE-5123.648181.726158.425159.41113.288146C-DE92.3484167.1142.689147.41317.611983F14−100DE-12757.282979.672825.252821.9345.565094DE-52303.214082.563069.743039.67409.883370C-DE1375.053542.592571.582608.89437.462372F15100DE-113765.61531914673.914734.7384.976680DE-51323715094.114454.514549.9399.510805C-DE12992.914685.414102.714161.2308.720813F16200DE-1209.112210.714210.335210.4030.284461DE-5212.755213.932213.389213.370.285913C-DE202.497203.836203.329203.3350.299347F17300DE-1441.674455.768449.039448.7782.675904DE-5498.309530.053516.227516.8176.054115C-DE418.995443.525431.271431.2216.529947F18400DE-1822.319876.283857.057857.38411.321625DE-5788.248868.463840.602843.17113.809299C-DE755.358809.619790.15791.58712.806493F19500DE-1524.859529.441527.534527.6030.879431DE-5522.265525.845524.077524.1950.823669C-DE510.223515.645513.479513.7431.248977Table 7
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF20 to F28 in n=50 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF20600DE-1622.329623.044622.787622.790.169152DE-5621.728622.563622.22622.2420.219888C-DE621.065622.68622.148622.210.296463F21700DE-119002822.1921981900.11404.866135DE-519002822.192518.612822.19398.516988C-DE9001822.191446.281536.44420.382683F22800DE-13505.935715.864396.734350.43493.522833DE-53843.149364.916308.586003.591222.434466C-DE2918.461614014.263946.52731.765773F23900DE-115239.116998.216361.216433377.677770DE-514668.217063.915928.816003.5537.803418C-DE13352.215814.815025.515024452.814685F241000DE-11456.811475.321467.971469.215.249819DE-51303.21449.871415.051423.0226.753472C-DE1219.051350.211307.841312.4327.412943F251100DE-11470.691495.41487.771488.685.477735DE-51463.791498.271482.961485.388.930093C-DE1460.91484.331473.41473.525.722868F261200DE-11434.881487.441455.531454.6212.242195DE-51410.061664.941457.741419.1187.860087C-DE1408.061658.641431.651416.6157.431295F271300DE-13228.463399.893324.783324.3338.861379DE-52950.163314.363157.773167.2979.562492C-DE2819.33270.463124.073138.6188.484672F281400DE-128002831.742800.6328004.443258DE-519105016.762150.741910833.557633C-DE18004911.422101.471800923.523003Figure2 displays the convergence graph of the F1 to F5 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 2 depicts the F1 to F5 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 2
Evolution in the best function values ofF1 to F5 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure3 displays the convergence graph of the F6 to F10 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 3 depicts the F6 to F10 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 3
Evolution in the best function values ofF6 to F10 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure4 displays the convergence graph of the F11 to F15 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 4 depicts the F11 to F15 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 4
Evolution in the best function values ofF11 to F15 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure5 displays the convergence graph of the F16 to F20 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 5 depicts the F16 to F20 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 5
Evolution in the best function values ofF16 to F20 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure6 displays the convergence graph of the F21 to F25 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 6 depicts the F21 to F25 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 6
Evolution in the best function values ofF21 to F25 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure7 displays the convergence graph of the F26 to F28 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 7 depicts the F26 to F28 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 7
Evolution in the best function values ofF26 to F28 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 3.1. PC Configuration and Simulation Environment

The simulations were conducted on a PC with 8 GB RAM and Intel(R)Core(TM)i3-7100U CPU @ 2.40 GHz processor. All simulations are done in MATLAB R2013a environment withpopsize=100. Each benchmark function has been solved in different dimensions including n=10,30,50 by executing 51 times independently. Maximum function evaluation was fixed to 10000×n. The algorithms for each problem are run 51 times independently to obtain the final best solution. Sm∈0,1 denotes the switching of mutation strategies.Tables2 and 3 present the numerical results of C-DE as outlined in Algorithm 6 in comparison with DE-1 as described in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F20 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 2. Similarly, the numerical results for the benchmark functions denoted by F21 to F28 are given in Table 3.Table 2
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F20 in n=10 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-15935834.56813e+062.28216e+062.11013e+06891922.510621DE-52494032.24398e+061.04699e+06987371426831.918032C-DE3105252.42089e+061.14756e+06996518512003.868545F03−1200DE-175046.14.66235e+061.55526e+061.1441e+061141062.835342DE-528606378126034587230535494580.836448C-DE−559.2967811665640518616.2131613.067626F04−1100DE-12663.8320828.91187612016.13735.150538DE-52661.713432.27847.187823.62023.165850C-DE2574.6611249.86114.036033.661982.673559F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−881.658−880.015−880.179−880.1580.213022DE-5−879.096−870.071−870.678−870.1871.999259C-DE−899.941−890.165−890.814−890.1862.263107F07−800DE-1−784.364−769.68−778.597−778.5473.576671DE-5−789.323−781.563−787.441−787.7211.694367C-DE−799.435−791.131−797.985−798.1811.262155F08−700DE-1−679.784−679.479−679.635−679.6360.068761DE-5−679.813−679.505−679.631−679.620.064325C-DE−679.819−679.523−679.657−679.6350.075517F09−600DE-1−588.254−585.522−586.761−586.7340.616831DE-5−591.155−587.466−588.721−588.5430.787579C-DE−596.588−593.206−594.598−594.4740.722405F10−500DE-1−488.93−487.95−488.6−488.6330.199213DE-5−491.612−491.105−491.349−491.3620.139360C-DE−499.622−499.119−499.386−499.3920.113407F11−400DE-1−400−400−400−4000.000000DE-5−400−400−400−4000.000000C-DE−400−400−400−4000.000000F12−300DE-1−281.09−263.199−269.691−269.3513.606150DE-5−284.128−270.525−277.174−276.6942.946380C-DE−292.656−280.911−286.92−286.9363.047453F13−200DE-1−183.951−166.48−174.742−174.2683.819038DE-5−188.699−174.674−181.719−182.0993.220455C-DE−193.282−181.128−186.944−186.9212.858693F14−100DE-1−94−93.9375−93.9988−940.008745DE-5−95.8751−89.0452−95.0246−95.68771.514591C-DE−99.9375−96.3352−99.5179−99.75020.905209F15100DE-1966.0641624.911345.421363.52141.570897DE-5850.8711581.881269.041266.33165.250631C-DE778.431452.191184.571212.61150.366735F16200DE-1200.555201.516201.119201.1370.196826DE-5200.51201.527201.122201.1270.223394C-DE200.649201.478201.098201.1090.203207F17300DE-1320.122320.122320.122320.1220.000001DE-5311.542318.288318.01318.1220.924608C-DE310.122310.286310.128310.1220.023545F18400DE-1433.641453.204444.794445.4143.694887DE-5429.56449.267437.735437.2573.894859C-DE421.706438.851429.938429.7533.321748F19500DE-1510.299510.76510.595510.5990.089374DE-5504.153504.684504.538504.5480.082422C-DE500.355500.752500.527500.5320.079466F20600DE-1602.688603.565603.182603.2010.210304DE-5602.184603.466602.781602.8150.299490C-DE602.163603.359602.785602.7730.276907Table 3
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF21 to F28 in n=10 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF21700DE-1966.5121200.191167.021200.1956.998745DE-51000.011200.191196.271200.1928.031778C-DE900.6241100.191092.951100.1936.339134F22800DE-1930.3451058.03987.216982.96229.935014DE-5907.2391106.44976.237997.54551.389783C-DE810.1531029.34899.141904.7453.322517F23900DE-11809.82484.842246.862289.89162.930066DE-51684.242609.92141.832145.54200.732186C-DE1619.942523.092093.092101.43198.583802F241000DE-11181.51238.291213.191212.2616.478379DE-51150.461238.261224.551231.5419.882749C-DE1141.111218.451208.851213.6415.683668F251100DE-11291.091334.341329.71331.297.180292DE-51266.161333.751320.131322.1611.632105C-DE1257.561314.851305.881308.810.946832F261200DE-11342.61394.561366.71365.1610.154832DE-51334.521425.071384.181387.636.985466C-DE1311.731400.071349.871341.5630.335554F271300DE-11703.241840.041770.841767.3934.869565DE-51620.081852.551650.551620.7164.877728C-DE1600.071824.261620.841600.4955.037487F281400DE-11716.318001797.84180012.230642DE-5155017501723.52175067.131550C-DE1500.6117001688.44170046.714278Tables4 and 5 demonstrate the numerical results in respect of the proposed C-DE as outlined in Algorithm 6 versus DE-1 as given in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F20 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 4. Similarly, the numerical results regarding the benchmark functions denoted by F21 to F28 are given in Table 5.Table 4
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F20 in n=30 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-11.36903e+082.08058e+081.77072e+081.76426e+0814956353.317846DE-51.19104e+081.54637e+081.33283e+081.33653e+087763339.936747C-DE1.47534e+076.42165e+073.17636e+073.12834e+078995583.667657F03−1200DE-11.07468e+091.30442e+091.13644e+091.12995e+0942393532.103937DE-51.05797e+081.76261e+081.20783e+081.1947e+0814758359.088589C-DE1036555.90805e+071.37612e+079.87793e+0613305275.710745F04−1100DE-1353547568157139.556623.99376.123241DE-518653.736440.725639.825351.44364.116408C-DE15683.932184.624428.424524.14004.135290F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−783.569−756.81−777.918−781.9848.219750DE-5−863.853−800.281−851.899−853.19912.856533C-DE−883.444−816.652−858.659−856.69212.862273F07−800DE-1−752.53−726.067−740.613−741.9176.037184DE-5−771.429−737.11−762.271−763.9587.427720C-DE−791.822−768.024−783.293−784.0845.870443F08−700DE-1−674.203−673.974−674.058−674.050.050295DE-5−669.238−668.974−669.045−669.0290.048735C-DE−679.167−678.961−679.048−679.040.043403F09−600DE-1−561.328−555.523−558.644−558.8261.427762DE-5−559.276−551.544−554.985−554.7611.682517C-DE−570.355−563.838−566.394−566.1981.325391F10−500DE-1−483.713−447.89−471.111−472.9887.436859DE-5−485.98−484.975−485.704−485.7540.222811C-DE−499.936−498.639−499.626−499.7190.269271F11−400DE-1−390−390−390−3900.000000DE-5−390−386.02−389.083−389.0051.011598C-DE−400−399.005−399.746−4000.437924F12−300DE-1−129.647−87.7556−105.127−104.3969.743013DE-5−181.252−111.655−139.279−138.23112.860856C-DE−189.566−130.269−150.997−149.48512.653504F13−200DE-1543.649606.651587.113588.02910.751144DE-5223.974285.604257.156257.86913.872433C-DE−76.0579−24.1726−46.6554−45.493310.970706F14−100DE-1958.6211163.131076.091073.9737.599756DE-51068.222325.261756.671732.49352.891847C-DE503.9071368.46870.066877.497193.156721F15100DE-16740.647841.497423.287454.15263.687213DE-56182.547797.067189.497202.74312.036005C-DE6651.87656.347232.337269.4287.708609F16200DE-1211.829212.918212.478212.5070.266427DE-5213.686214.943214.427214.4560.287333C-DE201.688202.936202.454202.4520.288254F17300DE-1351.495356.482353.565353.4931.325248DE-5360.308373.106367.184367.7472.924574C-DE347.203361.075355.355355.1482.813337F18400DE-1582.99647.337631.422632.87711.084542DE-5576.88621.743605.017606.25910.266823C-DE576.732615.316596.713597.6258.919994F19500DE-1515.067517.608516.468516.5010.599903DE-5524.311526.192525.428525.5090.411725C-DE503.979506.237505.276505.3860.497524F20600DE-1612.548613.437613.046613.0490.211167DE-5611.933613.14612.602612.6160.268802C-DE611.798613.08612.546612.5960.285701Table 5
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF21 to F28 in n=30 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF21700DE-1101011101083.33111040.626595DE-59501193.541055.47105084.276686C-DE9001143.54998.798100071.457633F22800DE-12410.213350.832881.942855.53227.697529DE-51997.43982.732865.522768.51410.507035C-DE1703.323285.892248.572238.3326.629221F23900DE-17761.698933.498467.148491.29276.464768DE-57610.148945.918282.958319.51315.448232C-DE7385.718893.868156.478238.79345.393637F241000DE-11285.611301.261295.221295.043.201388DE-51268.061328.191297.391298.5414.328935C-DE1212.51278.271248.741251.114.766463F251100DE-11442.271457.121451.451451.623.322013DE-51419.131445.031432.181432.25.749719C-DE1374.231395.21385.461386.975.218833F261200DE-11414.911425.421419.491419.412.455244DE-51411.071413.841411.941411.830.548844C-DE1401.051403.911402.061402.020.606003F271300DE-12512.92640.532579.622578.525.085286DE-52226.432554.192443.582456.8372.958778C-DE2129.692467.362361.892385.8873.692495F281400DE-118001800180018000.000000DE-517501750175017500.000000C-DE17001700170017000.000000Tables6 and 7 provide the experimental results approximated by the proposed C-DE herewith outlined in Algorithm 6 against DE-1 as given in Algorithm 1 and DE-5 hereby outlined in Algorithm 5. The numerical results for F1 to F19 benchmark functions in terms of minimum, maximum, average, median, and standard deviation in function values are summarized in Table 6. Similarly, the numerical results regarding the benchmark functions denoted by F20 to F28 are given in Table 7.Table 6
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF1 to F19 in n=50 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF01−1400DE-1−1400−1400−1400−14000.000000DE-5−1400−1400−1400−14000.000000C-DE−1400−1400−1400−14000.000000F02−1300DE-11.82669e+084.0298e+083.08217e+083.15245e+0840952346.718760DE-56.5501e+071.46685e+081.03416e+081.01884e+0818482876.197602C-DE5.44136e+071.46107e+081.04963e+081.05681e+0820121866.384027F03−1200DE-11.10012e+096.44772e+093.55854e+093.31787e+091263221747.238202DE-537540.45.7581e+089.78856e+073.8351e+07132237941.475719C-DE1.95311e+063.23647e+086.21553e+073.62992e+0770023673.941721F04−1100DE-169182.111915389629.890090.810106.552893DE-520820.941255.330832.530841.74612.877411C-DE20496.238856.929036.128704.63791.979814F05−1000DE-1−1000−1000−1000−10000.000000DE-5−1000−1000−1000−10000.000000C-DE−1000−1000−1000−10000.000000F06−900DE-1−836.081−834.721−835.527−835.5460.278565DE-5−835.493−780.995−832.512−833.4767.423864C-DE−855.053−850.805−852.701−852.7760.895707F07−800DE-1−659.767−624.433−642.8−642.1678.870652DE-5−738.024−694.803−716.596−716.0189.264348C-DE−788.119−756.809−770.008−769.7137.322152F08−700DE-1−676.032−675.804−675.865−675.8580.037308DE-5−676.982−676.812−676.87−676.860.041021C-DE−678.984−678.811−678.874−678.8670.042407F09−600DE-1−487.463−477.117−480.324−480.1721.930815DE-5−500.784−490.311−493.893−493.5122.256670C-DE−539.562−530.812−533.846−532.9592.223050F10−500DE-1−482.098−417.281−460.08−463.09116.289936DE-5−487.973−486.858−487.687−487.7290.227074C-DE−499.786−497.251−498.559−498.6320.544408F11−400DE-1−296.762−271.293−279.13−278.9524.588255DE-5−350−298.691−342.965−345.0258.544754C-DE−400−331.719−377.221−371.82720.613249F12−300DE-175.2158125.882103.524104.09613.276617DE-50.26887681.905849.921353.444518.892895C-DE−13.79377.658834.683236.796717.050881F13−200DE-1158.018226.329205.73207.24812.793081DE-5123.648181.726158.425159.41113.288146C-DE92.3484167.1142.689147.41317.611983F14−100DE-12757.282979.672825.252821.9345.565094DE-52303.214082.563069.743039.67409.883370C-DE1375.053542.592571.582608.89437.462372F15100DE-113765.61531914673.914734.7384.976680DE-51323715094.114454.514549.9399.510805C-DE12992.914685.414102.714161.2308.720813F16200DE-1209.112210.714210.335210.4030.284461DE-5212.755213.932213.389213.370.285913C-DE202.497203.836203.329203.3350.299347F17300DE-1441.674455.768449.039448.7782.675904DE-5498.309530.053516.227516.8176.054115C-DE418.995443.525431.271431.2216.529947F18400DE-1822.319876.283857.057857.38411.321625DE-5788.248868.463840.602843.17113.809299C-DE755.358809.619790.15791.58712.806493F19500DE-1524.859529.441527.534527.6030.879431DE-5522.265525.845524.077524.1950.823669C-DE510.223515.645513.479513.7431.248977Table 7
The experimental results obtained by DE-1 and DE-5 versus proposed C-DE by solving CEC’13 benchmark functionsF20 to F28 in n=50 dimensions.
P. noOptAlgorithmBestWorstMeanMedianStdF20600DE-1622.329623.044622.787622.790.169152DE-5621.728622.563622.22622.2420.219888C-DE621.065622.68622.148622.210.296463F21700DE-119002822.1921981900.11404.866135DE-519002822.192518.612822.19398.516988C-DE9001822.191446.281536.44420.382683F22800DE-13505.935715.864396.734350.43493.522833DE-53843.149364.916308.586003.591222.434466C-DE2918.461614014.263946.52731.765773F23900DE-115239.116998.216361.216433377.677770DE-514668.217063.915928.816003.5537.803418C-DE13352.215814.815025.515024452.814685F241000DE-11456.811475.321467.971469.215.249819DE-51303.21449.871415.051423.0226.753472C-DE1219.051350.211307.841312.4327.412943F251100DE-11470.691495.41487.771488.685.477735DE-51463.791498.271482.961485.388.930093C-DE1460.91484.331473.41473.525.722868F261200DE-11434.881487.441455.531454.6212.242195DE-51410.061664.941457.741419.1187.860087C-DE1408.061658.641431.651416.6157.431295F271300DE-13228.463399.893324.783324.3338.861379DE-52950.163314.363157.773167.2979.562492C-DE2819.33270.463124.073138.6188.484672F281400DE-128002831.742800.6328004.443258DE-519105016.762150.741910833.557633C-DE18004911.422101.471800923.523003Figure2 displays the convergence graph of the F1 to F5 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 2 depicts the F1 to F5 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 2
Evolution in the best function values ofF1 to F5 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure3 displays the convergence graph of the F6 to F10 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 3 depicts the F6 to F10 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 3
Evolution in the best function values ofF6 to F10 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure4 displays the convergence graph of the F11 to F15 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 4 depicts the F11 to F15 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 4
Evolution in the best function values ofF11 to F15 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure5 displays the convergence graph of the F16 to F20 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 5 depicts the F16 to F20 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 5
Evolution in the best function values ofF16 to F20 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure6 displays the convergence graph of the F21 to F25 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 6 depicts the F21 to F25 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 6
Evolution in the best function values ofF21 to F25 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)Figure7 displays the convergence graph of the F26 to F28 benchmark functions solved by the proposed C-DE algorithm versus DE-1 and DE-5 after 51 independent runs of simulation with different random seeds. The first panel of Figure 7 depicts the F26 to F28 benchmark functions as solved in ten dimensions, the second panel is for thirty dimensions, and the third panel is for fifty dimensions.Figure 7
Evolution in the best function values ofF26 to F28 solved with DE-1, DE-5, and C-DE in n=10, 30, and 50 dimensions.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 4. Conclusion

In this paper, three variants of differential evolutionary algorithms, namely, DE-1, DE-5, and C-DE, are suggested and their numerical results in terms ofbest, worst, mean, median, and Std are obtained for the unconstrained benchmark functions designed for the special session on evolutionary algorithm competition (the 2013 IEEE Conference on Evolutionary Computation). Among the three DE variants, C-DE as outlined in Algorithm 6 has obtained better experimental results as compared to DE-1 as reported in Algorithm 1 and DE-5 described in Algorithm 5. Based on comparative analysis, we conclude the following: Thebest fitness values of C-DE are much better for mostly test problems with n=10,30,50 subjects to the “No Free Lunch Theorem” concept in the parlance of evolutionary computing communities.Based on theworst objective function values, C-DE has again prominent position among the other two compared algorithms in order to solve most benchmark functions with dimensions keeping in view the criterion of “No Free Lunch Theorem” which guarantees the transparency of the manuscript.Themean objective function values indicate in respect of C-DE much better than the average function values of other DE variant algorithms.Themedian objective function values found by C-DE are much promising keeping in view the median values of the competing algorithms which reflect the fairness of the proposed algorithm.Competitors play a comparable role keeping in view the standard deviation values obtained for the used benchmark function.The efficiency and robustness of the proposed algorithm are satisfactory which can be observed from the error function values as plotted against the number of iterations.In future, we will be planning for the following tasks:To check the efficiency of C-DE on other existing unconstrained suites to more elaborate the importance of the proposed algorithmTo do parameter sensitivity to ensure the correct use of parameters on specific type problems is included in the future workFor checking the robustness and efficiency of the proposed algorithm, we are intended to compare it with the advanced version of DE such as JADE, SHADE, and IDEFor customization, the proposed algorithm with other advanced variants of DE can be come in count in future



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*Source: 5515701-2021-03-10.xml*