An Improved Artificial Electric Field Algorithm for MultiObjective Optimization
Abstract
:1. Introduction
2. Related Work
Our Contribution
3. Preliminaries and Background
3.1. MultiObjective Optimization
3.2. Artificial Electric Field Algorithm (AEFA)
 The electrostatic force exerted by the ${j}^{th}$ charged particle on the $i\mathrm{th}$ charged particle in the ${D}^{th}$ dimension at time $T$ is computed as:$${F}_{ij}^{D}\left(T\right)=K\left(t\right)\frac{\left({Q}_{i}\left(T\right)\ast {Q}_{j}\left(T\right)\right)\ast \left({P}_{j}^{D}\left(T\right){X}_{j}^{D}\left(T\right)\right)}{{R}_{ij}\left(T\right)+\epsilon}$$$${F}_{i}^{D}\left(T\right)={\displaystyle \sum}_{j=1,j\ne i}^{N}rand()\ast {F}_{ij}^{D}\left(T\right)$$
 The acceleration ${a}_{i}^{D}\left(T\right)$ of $i\mathrm{th}$ charged particle at time $T$ in ${D}^{th}$ dimension is computed using the Newton law of motion as follows:$${a}_{i}^{D}\left(T\right)=\frac{{Q}_{i}\left(T\right)\ast {E}_{i}^{D}\left(T\right)}{{M}_{i}^{D}\left(T\right)},{E}_{i}^{D}\left(T\right)=\frac{{F}_{i}^{D}\left(T\right)}{{Q}_{i}\left(T\right)}$$
3.3. ShiftBased Density Estimation
Algorithm 1 Density Estimation for NonDomination Solutions 
Input: Nondominated solutions $\left({P}_{{I}_{{c}_{1}}},{P}_{{I}_{{c}_{1}}},\dots .{P}_{{I}_{{c}_{\mathrm{n}}}}\right)$ Output: Density of each solution

3.4. Recombination and Mutation Operators
3.4.1. Bounded Exponential Crossover (BEX)
3.4.2. Polynomial Mutation Operator (PMO)
Algorithm 2 Bounded Exponential Crossover (BEX) 
Input: Parent solutions $x={x}_{1},{x}_{2},\dots .{x}_{m}$ and $y={y}_{1},{y}_{2},\dots .{y}_{m}and\mathrm{scaling}\mathrm{parameter}\mathsf{\lambda}0$ Output: Offspring solutions While $i\le \mathrm{m}\mathit{d}\mathit{o}$

4. Proposed Algorithm
4.1. Population Generation
Algorithm 3 Proposed MultiObjective Optimization Algorithm 
Input: Searching population of size $\left(n\right),$ External Population of size ($m)$ Output: Nondominated set of charged particles $\left(N{D}_{{C}_{p}}\right)$ Begin
For each $CP$ $\in {P}_{{I}_{c}}U\overline{{P}_{{I}_{c}}^{*}}$
For each $CP\in $ ${P}_{{I}_{c}}U\overline{{P}_{{I}_{c}}^{*}}$ do If $F\left(CP\right)<1$ then ${P}_{{I}_{c+1}}^{*}={P}_{{I}_{c+1}}^{*}U\left\{CP\right\}$ end if end for If $({P}_{{I}_{c+1}}^{*}<m)$ then $\overline{{P}_{{I}_{c}+1}}$ = $\overline{{P}_{{I}_{c}+1}}U\left(\left({P}_{{I}_{c}}U{P}_{{I}_{c}}^{*}\right)\left[1:m\left\overline{{P}_{{I}_{c}+1}}\right\right]\right)$ else delete nondominated solution from ${P}_{{I}_{c}}^{*}$ using Equation (17) as described in Section 4.3 end if
For each $CP$ $\in \overline{{P}_{t+1}}$ do If $CP$ is a nondominated solution then $N{D}_{{C}_{p}}$ = $N{D}_{{C}_{p}}*\left\{CP\right\}$; end if end for 
4.2. Fitness Evaluation
4.3. A FineGrained Elitism Selection Mechanism
 The distance between two adjacent particles ${P}_{{I}_{c}}$ and ${P}_{{I}_{C}+1}^{*}$ is computed as Algorithm 1.
 For each particle, an additional density estimation (shared crowding distance) is computed as follows:$${\sigma}_{crowd}^{}=\frac{SDE({P}_{{I}_{c}}^{*})}{\leftm\right}$$
5. Experimental Results and Discussion
5.1. Performance Comparison of the AEFA With Existing Evolutionary Approaches
5.1.1. Parameter Setting
5.1.2. Results and Discussion
5.2. Performance Comparison of the Proposed Algorithm with Existing MultiObjective Optimization Algorithms
5.2.1. Parameter Setting
5.2.2. Results and Discussion
5.3. Sensitivity Analysis of the Proposed Algorithm
Results and Discussion
6. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
References
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Author(s)  Objective/Work Done  Technique Proposed/Used  Performance Parameters  Research Gap(s) Identified 

Nobahari, et.al. [1]  Proposed a multiobjective gravitational search based on nondominated sorting for power transformer design  Nondominated sorting gravitational search algorithm (NSGSA)  Normalized arithmetic mean  The Algorithm lacks scalability when dealing with complex cases of power transformer design. 
Srinivas, and Deb [19]  Proposed a multiobjective genetic algorithm based on nondominated sorting  Nondominated sorting genetic algorithm (NSGA)  Chisquare test  The proposed algorithm shows a slow convergence rate. 
Deb and Jain [25]  Proposed an evolutionary approach for solving manyobjective optimization  Referencebased non dominated sorting approach (NSGA3) 
 Recombination operator can be improved to enhance population diversity 
Zhang and Li [28]  Proposed a multiobjective evolutionary algorithm based on decomposition  The multiobjective evolutionary algorithm based on decomposition (MOEA/D) 
 The penalty parameter used in the proposed algorithm is statically initialized. For an extremely lower or higher value, the performance of the penalty method decreases 
Martinez and Coello [30]  Proposed a decompositionbased multiobjective evolutionary algorithm (eMOEA/DDE) for constraint MOOP  A new selection mechanism based on $\mathsf{\epsilon}$constraint method 
 Constraint parameters $\left(\epsilon ,\delta \right)$ used in the algorithm are statically initialized. It can be initialized dynamically. 
Gu et.al. [31]  Proposed a multiobjective evolutionary algorithm based on the projection of the current nondominated solutions and equidistance interpolation  Dynamic weight design method with MOEA/D 
 The algorithm lacks efficacy to solve complex higherdimensional problems. 
Zhang, et al. [32]  Proposed a novel and computationally efficient approach to nondominated sorting  Efficient nondominated sort (ENS) 
 The efficiency of the algorithm decreases with an increase in the number of objectives 
Chong and Qiu [33]  Proposed MOO algorithm to solve multiobjective traveling salesman problem  Selfadaptive differential algorithm with a decompositionbased framework (DOSADE) 
 The algorithm becomes less effective as the number of salesmen increases 
Cheng et al. [34]  Proposed an evolutionary algorithm for manyobjective optimization  Reference vector guided evolutionary algorithm (RVEA) 
 The reference vector is static. The selection type of reference vector to be used in manyobjective optimization is not considered 
Hassanzade and Rouhani [35]  Proposed a multiobjective algorithm based on gravitational force  Multiobjective gravitational search algorithm (MOGSA) 
 The algorithm suffers premature convergence in solving complex higherdimensional problems. 
Yuan et al. [36]  Proposed a multiobjective gravitational search based on the concept of strength Pareto  Strength Pareto gravitational search (SPGSA) 
 Population diversity can be further improved. 
Symbol  Definition 

$n$  Initial searching population size 
${P}_{n}$  Searching population 
$m$  External population size 
${P}_{m}$  External population 
$C{P}_{i}^{D}\left(T\right)$  Position of $i\mathrm{th}$ charged particle (candidate solution) in ${D}^{th}$ dimension at time $T$ 
$ve{l}_{i}^{D}\left(T\right)$  velocity of $i\mathrm{th}$ charged particle in ${D}^{th}$ dimension at time $T$ 
$N{D}_{{C}_{p}}$  Nondominated set of charged particles (candidate solution) 
$Fitness\left(T\right)$  Objective fitness function 
$Best\left(T\right)$  Charged particle with best fitness at time $T$ 
$Worst\left(T\right)$  Charged particle with worst fitness at time $T$ 
$K$  Coulomb’s constant 
${Q}_{i}\left(T\right)$  Total charge on a $i\mathrm{th}$ charged particle at time $T$ 
${q}_{i}\left(T\right)$  Small charge of $i\mathrm{th}$ charged particle to determine the total charge acting on $i\mathrm{th}$ charged particle 
${F}_{ij}$  Force exerted by $j\mathrm{th}$ charge particle on $i\mathrm{th}$ charge particle 
$SDE$  Shift based density estimation 
${I}_{cmax}$  Maximum number of iterations 
${d}_{{P}_{I}{}_{{c}_{i}},{P}_{I}{}_{{c}_{i+1}}}$  Distance between two nondominated solutions (${P}_{I}{}_{{c}_{i}}$ and ${P}_{I}{}_{{c}_{i+1}}$) 
${\sigma}_{crowd}$  Shared crowding distance for each $CP$ 
${\sigma}_{k}{}_{crowd}$  Shared crowding distance of ${k}^{th}CP$ chosen for deletion from the external population set 
Benchmark Function  Type  Variable Bound  Objective Function  Dimension(s) 

F1  UM  $100\le {x}_{i}\le 100$  $F\left(x\right)={\sum}_{i=1}^{n}{x}_{i}^{2}$  30 
$i=1,2,3\dots n$  
F2  UM  $10\le {x}_{i}\le 10$  $F\left(x\right)=$${\sum}_{i=1}^{n}\left{x}_{i}\right+{\prod}_{i=1}^{n}{x}_{i}$  30 
$i=1,2\dots n$  
F3  UM  $100\le {x}_{i}\le 100$  $F\left(x\right)=\mathrm{max}\{\left{x}_{i}\right,1\le i\le n\}$  30 
$i=1,2\dots n$  
F4  LDMM  $0\le {x}_{i}\le 1$  $F\left(x\right)={\sum}_{i=1}^{4}{c}_{i}\mathrm{exp}({\sum}_{j=1}^{6}{a}_{ij}{\left({x}_{j}{p}_{ij}\right)}^{2})$  6 
$i=1,2\dots n$  
F5  LDMM  $0\le {x}_{i}\le 100$  $F\left(x\right)={\sum}_{i=1}^{7}{[\left(X{a}_{i}\right){\left(X{a}_{i}\right)}^{T}+{c}_{i}]}^{1}$  4 
$i=1,2\dots n$  
F6  LDMM  $0\le {x}_{i}\le 100$  $F\left(x\right)={\sum}_{i=1}^{10}{[\left(X{a}_{i}\right){\left(X{a}_{i}\right)}^{T}+{c}_{i}]}^{1}$  4 
$i=1,2\dots n$  
F7  HDMM  $500\le {x}_{i}\le 500$  $F\left(x\right)={\sum}_{i=1}^{n}{x}_{i}\mathrm{sin}\left(\sqrt{\left{x}_{i}\right}\right)$  30 
$i=1,2\dots n$  
F8  HDMM  $5.12\le {x}_{i}\le 5.12$  $F\left(x\right)={\sum}_{i=1}^{n}\left[{x}_{i}^{2}10cos\left(2\pi {x}_{i}\right)+10\right]$  30 
$i=1,2\dots n$  
F9  HDMM  $600\le {x}_{i}\le 600$  $F\left(x\right)=\frac{1}{40000}{\sum}_{i=1}^{n}{x}_{i}^{2}{\prod}_{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$  30 
$i=1,2\dots n$  
F10  HDMM  $50\le {x}_{i}\le 50$ $i=1,2\dots n$  $F\left(x\right)=0.1\{\mathrm{sin}\left(3\pi {x}_{i}\right)$+${\sum}_{i=1}^{n}{\left({x}_{i}1\right)}^{2}+1$ $+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)]$+${\left({x}_{n}1\right)}^{2}$ $\ast [1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)]\}$+${\sum}_{i=1}^{u}u\left({x}_{i},10,100,4\right)$, $u\left({x}_{i},a,k,m\right)=\{\begin{array}{c}k{\left({x}_{i}a\right)}^{m}{x}_{i}a\\ k{\left({x}_{i}a\right)}^{m}{x}_{i}a\end{array}$  30 
Description  Parameter  Value 

Population size  ${P}_{n},{P}_{m}$  50 
Initial value used in Coulomb’s constant  ${K}_{0}$  100 
Maximum number of iterations  ${I}_{cmax}$  300 for $F4F6$ and 1000 for the rest of the benchmark functions 
Benchmark  Optimization Algorithm  Statistical Performance Indices  

Average Best  Mean Best  Variance Best  Average Run Time  
F1  AEFA  $\mathbf{3.21}\times {\mathbf{10}}^{\mathbf{16}}$  $\mathbf{2.54}\times {\mathbf{10}}^{\mathbf{18}}$  $\mathbf{3.98}\times {\mathbf{10}}^{\mathbf{16}}$  $9.48\times {10}^{0}$ 
GSA  $3.33\times {10}^{14}$  $2.68\times {10}^{17}$  $4.79\times {10}^{14}$  $9.71\times {10}^{0}$  
ABC  $4.20\times {10}^{9}$  $1.52\times {10}^{9}$  $8.40\times {10}^{9}$  $1.63\times {10}^{1}$  
CK  $9.26\times {10}^{3}$  $9.44\times {10}^{3}$  $2.35\times {10}^{2}$  $3.88\times {10}^{0}$  
BSA  $4.87\times {10}^{3}$  $3.28\times {10}^{3}$  $4.53\times {10}^{3}$  $\mathbf{1.28}\times {\mathbf{10}}^{\mathbf{0}}$  
F2  AEFA  $\mathbf{1.26}\times {\mathbf{10}}^{\mathbf{8}}$  $\mathbf{2.00}\times {\mathbf{10}}^{\mathbf{8}}$  $\mathbf{1.10}\times {\mathbf{10}}^{\mathbf{8}}$  $9.72\times {10}^{0}$ 
GSA  $1.55\times {10}^{8}$  $2.08\times {10}^{8}$  $1.16\times {10}^{8}$  $9.95\times {10}^{0}$  
ABC  $3.22\times {10}^{6}$  $2.99\times {10}^{6}$  $1.35\times {10}^{6}$  $1.75\times {10}^{1}$  
CK  $1.62\times {10}^{0}$  $1.30\times {10}^{0}$  $8.15\times {10}^{1}$  $4.12\times {10}^{0}$  
BSA  $1.73\times {10}^{2}$  $1.49\times {10}^{2}$  $9.11\times {10}^{3}$  $\mathbf{1.98}\times {\mathbf{10}}^{\mathbf{0}}$  
F3  AEFA  $\mathbf{2.68}\times {\mathbf{10}}^{\mathbf{10}}$  $\mathbf{2.98}\times {\mathbf{10}}^{\mathbf{9}}$  $\mathbf{8.78}\times {\mathbf{10}}^{\mathbf{11}}$  $9.01\times {10}^{0}$ 
GSA  $3.34\times {10}^{9}$  $3.13\times {10}^{9}$  $9.05\times {10}^{10}$  $8.98\times {10}^{0}$  
ABC  $6.02\times {10}^{1}$  $6.15\times {10}^{1}$  $9.39\times {10}^{0}$  $1.71\times {10}^{1}$  
CK  $3.21\times {10}^{0}$  $3.22\times {10}^{0}$  $9.39\times {10}^{0}$  $3.92\times {10}^{0}$  
BSA  $4.52\times {10}^{0}$  $4.65\times {10}^{0}$  $1.05\times {10}^{0}$  $\mathbf{1.74}\times {\mathbf{10}}^{\mathbf{0}}$  
F4  AEFA  $\mathbf{3.30}\times {\mathbf{10}}^{\mathbf{0}}$  $\mathbf{3.30}\times {\mathbf{10}}^{\mathbf{0}}$  $\mathbf{4.38}\times {\mathbf{10}}^{\mathbf{16}}$  $2.44\times {10}^{0}$ 
GSA  $3.32\times {10}^{0}$  $3.32\times {10}^{0}$  $4.08\times {10}^{16}$  $2.24\times {10}^{0}$  
ABC  $3.32\times {10}^{0}$  $3.32\times {10}^{0}$  $2.33\times {10}^{9}$  $7.24\times {10}^{0}$  
CK  $3.32\times {10}^{0}$  $3.32\times {10}^{0}$  $4.80\times {10}^{5}$  $1.32\times {10}^{0}$  
BSA  $3.32\times {10}^{0}$  $3.32\times {10}^{0}$  $1.48\times {10}^{4}$  $\mathbf{6.74}\times {\mathbf{10}}^{\mathbf{1}}$  
F5  AEFA  $\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{2.68}\times {\mathbf{10}}^{\mathbf{16}}$  $2.34\times {10}^{0}$ 
GSA  $1.04\times {10}^{1}$  $1.04\times {10}^{1}$  $2.97\times {10}^{15}$  $2.29\times {10}^{0}$  
ABC  $1.04\times {10}^{1}$  $1.04\times {10}^{1}$  $1.38\times {10}^{3}$  $8.23\times {10}^{0}$  
CK  $1.04\times {10}^{1}$  $1.04\times {10}^{1}$  $2.69\times {10}^{3}$  $1.57\times {10}^{0}$  
BSA  $1.04\times {10}^{1}$  $1.04\times {10}^{1}$  $3.50\times {10}^{2}$  $\mathbf{1.04}\times {\mathbf{10}}^{\mathbf{0}}$  
F6  AEFA  $\mathbf{1.05}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{1.05}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{1.28}\times {\mathbf{10}}^{\mathbf{15}}$  $2.40\times {10}^{0}$ 
GSA  $1.05\times {10}^{1}$  $1.05\times {10}^{1}$  $1.47\times {10}^{15}$  $2.46\times {10}^{0}$  
ABC  $1.05\times {10}^{1}$  $1.05\times {10}^{1}$  $7.06\times {10}^{4}$  $9.34\times {10}^{0}$  
CK  $1.05\times {10}^{1}$  $1.05\times {10}^{1}$  $2.97\times {10}^{3}$  $2.97\times {10}^{3}$  
BSA  $1.05\times {10}^{1}$  $1.05\times {10}^{1}$  $6.17\times {10}^{2}$  $\mathbf{9.07}\times {\mathbf{10}}^{\mathbf{1}}$  
F7  AEFA  $2.79\times {10}^{3}$  $2.68\times {10}^{3}$  $4.18\times {10}^{2}$  $2.98\times {10}^{0}$ 
GSA  $2.84\times {10}^{3}$  $2.84\times {10}^{3}$  $4.22\times {10}^{2}$  $3.01\times {10}^{0}$  
ABC  $1.08\times {10}^{4}$  $1.07\times {10}^{4}$  $2.97\times {10}^{2}$  $5.86\times {10}^{0}$  
CK  $8.77\times {10}^{3}$  $8.79\times {10}^{3}$  $\mathbf{2.44}\times {\mathbf{10}}^{\mathbf{2}}$  $4.60\times {10}^{0}$  
BSA  $\mathbf{1.11}\times {\mathbf{10}}^{\mathbf{4}}$  $\mathbf{1.11}\times {\mathbf{10}}^{\mathbf{4}}$  $2.49\times {10}^{2}$  $\mathbf{2.57}\times {\mathbf{10}}^{\mathbf{0}}$  
F8  AEFA  $1.28\times {10}^{1}$  $1.28\times {10}^{1}$  $3.56\times {10}^{0}$  $1.06\times {10}^{1}$ 
GSA  $1.53\times {10}^{1}$  $1.54\times {10}^{1}$  $4.01\times {10}^{0}$  $1.10\times {10}^{1}$  
ABC  $\mathbf{1.06}\times {\mathbf{10}}^{\mathbf{1}}$  $\mathbf{2.24}\times {\mathbf{10}}^{\mathbf{6}}$  $\mathbf{3.05}\times {\mathbf{10}}^{\mathbf{1}}$  $1.89\times {10}^{1}$  
CK  $8.38\times {10}^{1}$  $8.40\times {10}^{1}$  $1.05\times {10}^{1}$  $4.80\times {10}^{0}$  
BSA  $2.84\times {10}^{1}$  $2.77\times {10}^{1}$  $4.18\times {10}^{0}$  $\mathbf{2.62}\times {\mathbf{10}}^{\mathbf{0}}$  
F9  AEFA  $4.08\times {10}^{0}$  $3.19\times {10}^{0}$  $1.80\times {10}^{0}$  $9.70\times {10}^{0}$ 
GSA  $4.15\times {10}^{0}$  $3.56\times {10}^{0}$  $1.81\times {10}^{0}$  $9.74\times {10}^{0}$  
ABC  $\mathbf{4.01}\times {\mathbf{10}}^{\mathbf{4}}$  $\mathbf{2.81}\times {\mathbf{10}}^{\mathbf{8}}$  $\mathbf{1.79}\times {\mathbf{10}}^{\mathbf{3}}$  $2.44\times {10}^{1}$  
CK  $1.03\times {10}^{1}$  $9.68\times {10}^{2}$  $9.6\times {10}^{2}$  $5.32\times {10}^{0}$  
BSA  $3.21\times {10}^{2}$  $1.87\times {10}^{2}$  $3.65\times {10}^{2}$  $\mathbf{2.42}\times {\mathbf{10}}^{\mathbf{0}}$  
F10  AEFA  $\mathbf{5.00}\times {\mathbf{10}}^{\mathbf{4}}$  $\mathbf{1.96}\times {\mathbf{10}}^{\mathbf{18}}$  $\mathbf{2.30}\times {\mathbf{10}}^{\mathbf{3}}$  $1.12\times {10}^{1}$ 
GSA  $5.49\times {10}^{4}$  $2.10\times {10}^{18}$  $2.46\times {10}^{3}$  $1.07\times {10}^{1}$  
ABC  $2.10\times {10}^{3}$  $1.17\times {10}^{3}$  $2.45\times {10}^{3}$  $1.27\times {10}^{1}$  
CK  $1.50\times {10}^{1}$  $1.49\times {10}^{1}$  $6.82\times {10}^{2}$  $7.82\times {10}^{0}$  
BSA  $4.98\times {10}^{4}$  $4.37\times {10}^{4}$  $3.60\times {10}^{4}$  $\mathbf{4.35}\times {\mathbf{10}}^{\mathbf{0}}$ 
Benchmark Function  Type  Variable Bound  Objective Function  Dimension(s) 

SCH [3]  BiObjective (Low dimension)  $3\le {x}_{i}\le 3$  Minimize ${F}_{1}\left(x\right)={x}^{2}$  1 
Minimize ${F}_{2}\left(x\right)={\left(x2\right)}^{2}$  
FON [3]  BiObjective (Low dimension)  $4\le {x}_{i}\le 4$ $i=1,2\dots n$  Minimize ${F}_{1}\left(x\right)=1\mathrm{exp}\left({\sum}_{i=1}^{n}{\left({x}_{i}\frac{1}{\sqrt{n}}\right)}^{2}\right)$  3 
Minimize ${F}_{2}\left(x\right)=1\mathrm{exp}\left({\sum}_{i=1}^{n}{\left({x}_{i}+\frac{1}{\sqrt{n}}\right)}^{2}\right)$  
ZDT1 [3]  BiObjective (High dimension)  $0\le {x}_{i}\le 1$ $i=1,2\dots n$  Minimize ${F}_{1}\left(x\right)={x}_{1}$  30 
Minimize ${F}_{2}\left(x\right)=g\left(1\sqrt{\frac{f1}{g}}\right)$, $g=1+9{\sum}_{i=2}^{n}\left(\frac{{x}_{i}}{n1}\right)$  
ZDT2 [3]  BiObjective (High dimension)  $0\le {x}_{i}\le 1$ $i=1,2\dots n$  Minimize ${F}_{1}\left(x\right)={x}_{1}$  30 
Minimize ${F}_{2}\left(x\right)=g\left(1{\left(\frac{f1}{g}\right)}^{2}\right)$, $g=1+9{\sum}_{i=2}^{n}{\left(\frac{{x}_{i}}{n1}\right)}^{}$  
MOP5 [5]  TriObjective (Low dimension)  $30\le x,y$<=30  Minimize ${F}_{1}\left(x,y\right)=\frac{1}{2}\left({x}^{2}+{y}^{2}\right)Sin\left({x}^{2}+{y}^{2}\right)$  2 
Minimize ${F}_{2}\left(x,y\right)=\frac{1}{8}\left(3x2y+4\right)+\frac{{\left(xy+1\right)}^{2}}{27}$ +15  
Minimize ${F}_{3}\left(x,y\right)=\frac{1}{{x}^{2}+{y}^{2}+1}1.1{e}^{}\left({x}^{2}+{y}^{2}\right)$  
MOP6 [5]  BiObjective (Low dimension)  $0\le x,y$<=1  Minimize ${F}_{1}\left(x,y\right)=x$  2 
Minimize ${F}_{2}\left(x,y\right)=\left(1+10y\right)$  
$\ast \left[1{\left(\frac{x}{1+10y}\right)}^{2}\left(\frac{x}{1+10y}\right)sin\left(8\pi x\right)\right]$  
DTLZT2 [13]  TriObjective (High dimension)  $0\le {x}_{i}\le 1$ $i=1,2\dots n$  Minimize ${F}_{1}\left(x\right)=Cos\left(\frac{\pi}{2}{x}_{1}\right)Cos\left(\frac{\pi}{2}{x}_{2}\right)\left(1+g\left(x\right)\right)$  12 
Minimize ${F}_{2}\left(x\right)=Cos\left(\frac{\pi}{2}{x}_{1}\right)Sin\left(\frac{\pi}{2}{x}_{2}\right)\left(1+g\left(x\right)\right)$  
Minimize ${F}_{3}\left(x\right)=Sin\left(\frac{\pi}{2}{x}_{2}\right)\left(1+g\left(x\right)\right)$, where $g={\sum}_{i=3}^{n}{\left({x}_{i}0.5\right)}^{2}$ 
Description  Parameter  Value 

Population (charged particles) size  ${P}_{n}$  100 
External Population size  ${P}_{m}$  100 for SCH, FON, ZDT 800 for MOP functions 
Initial value used in Coulomb’s constant  ${K}_{0}$  100 
The maximum number of iterations  ${I}_{cmax}$  100 for SCH and FON functions 250 for ZDT and MOP functions 
Initial crossover probability  ${P}_{co}$  1.0 
Final crossover probability  ${P}_{c1}$  0.0 
Initial mutation probability  ${P}_{m0}$  0.01 
Final mutation probability  ${P}_{m1}$  0.001 
Benchmark Function  Parameters  Algorithm  

Proposed Algorithm  SPGSA  BCMOA  NSGA II  NSPSO  
SCH  Mean  $\mathbf{1.48}\times {\mathbf{10}}^{\mathbf{1}}$  $1.65\times {10}^{1}$  $7.60\times {10}^{1}$  $3.8\times {10}^{1}$  $8.6\times {10}^{1}$ 
Variance  $1.22\times {10}^{4}$  $1.3\times {10}^{4}$  $1.00\times {10}^{3}$  $1.00\times {10}^{3}$  $8.60\times {10}^{4}$  
FON  Mean  $\mathbf{1.52}\times {\mathbf{10}}^{\mathbf{1}}$  $1.61\times {10}^{1}$  $4.85\times {10}^{1}$  $4.14\times {10}^{1}$  $5.81\times {10}^{1}$ 
Variance  $2.52\times {10}^{4}$  $2.4\times {10}^{4}$  $1.00\times {10}^{4}$  $9.80\times {10}^{4}$  $3.16\times {10}^{2}$  
ZDT1  Mean  $\mathbf{1.5}\times {\mathbf{10}}^{\mathbf{1}}$  $1.61\times {10}^{1}$  $5.98\times {10}^{1}$  $4.06\times {10}^{1}$  $6.38\times {10}^{1}$ 
Variance  $1.42\times {10}^{4}$  $1.4\times {10}^{4}$  $4.10\times {10}^{3}$  $1.26\times {10}^{3}$  $2.69\times {10}^{3}$  
ZDT2  Mean  $\mathbf{1.61}\times {\mathbf{10}}^{\mathbf{1}}$  $1.63\times {10}^{1}$  $6.89\times {10}^{1}$  $4.39\times {10}^{1}$  $5.80\times {10}^{1}$ 
Variance  $1.46\times {10}^{4}$  $1.5\times {10}^{4}$  $8.13\times {10}^{3}$  $1.19\times {10}^{3}$  $1.08\times {10}^{3}$ 
Benchmark Function  Parameters  Algorithm  

Proposed Algorithm  SPGSA  BCMOA  NSGA II  NSPSO  
SCH  Mean  $\mathbf{2.75}\times {\mathbf{10}}^{\mathbf{3}}$  $\mathbf{3.24}\times {\mathbf{10}}^{\mathbf{3}}$  $3.28\times {10}^{3}$  $3.14\times {10}^{3}$  $3.40\times {10}^{1}$ 
Variance  $1.89\times {10}^{8}$  $2.98\times {10}^{8}$  $2.16\times {10}^{8}$  $4.64\times {10}^{8}$  $7.40\times {10}^{4}$  
FON  Mean  $\mathbf{1.55}\times {\mathbf{10}}^{\mathbf{3}}$  $\mathbf{1.72}\times {\mathbf{10}}^{\mathbf{3}}$  $2.77\times {10}^{3}$  $2.36\times {10}^{3}$  $2.84\times {10}^{1}$ 
Variance  $1.30\times {10}^{8}$  $1.10\times {10}^{8}$  $3.41\times {10}^{8}$  $1.24\times {10}^{8}$  $9.04\times {10}^{2}$  
ZDT1  Mean  $\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{3}}$  $\mathbf{1.17}\times {\mathbf{10}}^{\mathbf{3}}$  $1.19\times {10}^{3}$  $4.02\times {10}^{3}$  $3.81\times {10}^{1}$ 
Variance  $1.98\times {10}^{9}$  $2.66\times {10}^{9}$  $3.41\times {10}^{8}$  $3.14\times {10}^{7}$  $3.22\times {10}^{3}$  
ZDT2  Mean  $\mathbf{7.90}\times {\mathbf{10}}^{\mathbf{5}}$  $\mathbf{8.06}\times {\mathbf{10}}^{\mathbf{4}}$  $8.37\times {10}^{4}$  $2.71\times {10}^{3}$  $4.59\times {10}^{1}$ 
Variance  $4.22\times {10}^{11}$  $6.37\times {10}^{10}$  $1.25\times {10}^{8}$  $7.69\times {10}^{8}$  $3.24\times {10}^{3}$ 
Benchmark Function  Parameters  Algorithm  

Proposed Algorithm  SPGSA  BCMOA  NSGA II  NSPSO  
SCH  Mean  $\mathbf{2.69}\times {\mathbf{10}}^{\mathbf{4}}$  $3.78\times {10}^{4}$  $3.78\times {10}^{4}$  $3.68\times {10}^{4}$  $4.5\times {10}^{4}$ 
Variance  $2.02\times {10}^{10}$  $3.06\times {10}^{10}$  $1.57\times {10}^{10}$  $3.4\times {10}^{10}$  $2.6\times {10}^{8}$  
FON  Mean  $\mathbf{4.05}\times {\mathbf{10}}^{\mathbf{5}}$  $2.13\times {10}^{4}$  $3.62\times {10}^{4}$  $2.94\times {10}^{4}$  $3.6\times {10}^{4}$ 
Variance  $3.12\times {10}^{11}$  $3.06\times {10}^{10}$  $8.03\times {10}^{10}$  $4.2\times {10}^{10}$  $1.8\times {10}^{9}$  
ZDT1  Mean  $\mathbf{3.12}\times {\mathbf{10}}^{\mathbf{5}}$  $2.4\times {10}^{4}$  $2.02\times {10}^{4}$  $5.56\times {10}^{4}$  $4.3\times {10}^{4}$ 
Variance  $4.37\times {10}^{11}$  $4.76\times {10}^{10}$  $3.64\times {10}^{9}$  $7.95\times {10}^{9}$  $3.4\times {10}^{8}$  
ZDT2  Mean  $\mathbf{6.16}\times {\mathbf{10}}^{\mathbf{5}}$  $9.71\times {10}^{5}$  $1.00\times {10}^{4}$  $4.18\times {10}^{4}$  $3.5\times {10}^{4}$ 
Variance  $1.14\times {10}^{11}$  $1.56\times {10}^{11}$  $1.97\times {10}^{10}$  $9.64\times {10}^{9}$  $1.6\times {10}^{8}$ 
Benchmark Function  Metric  Algorithm  

Proposed Algorithm  SPGSA  NSGSA  MOGSA  SMOPSO  MOGA II  
MOP5  GD  $\mathbf{2.86}\times {\mathbf{10}}^{\mathbf{6}}$  $3.9\times {10}^{5}$  $1.0\times {10}^{4}$  $1.2\times {10}^{3}$  $\mathbf{1.11}\times {\mathbf{10}}^{\mathbf{2}}$  $7.2\times {10}^{1}$ 
SM  $\mathbf{4.86}\times {\mathbf{10}}^{\mathbf{3}}$  $2.18\times {10}^{2}$  $3.4\times {10}^{2}$  $1.8\times {10}^{1}$  $3.96\times {10}^{1}$  $2.0\times {10}^{1}$  
MOP6  GD  $\mathbf{7.02}\times {\mathbf{10}}^{\mathbf{7}}$  $5.01\times {10}^{6}$  $3.0\times {10}^{5}$  $2.45\times {10}^{5}$  $\mathbf{2.98}\times {\mathbf{10}}^{\mathbf{4}}$  $1.00\times {10}^{3}$ 
SM  $\mathbf{4.86}\times {\mathbf{10}}^{\mathbf{3}}$  $5.34\times {10}^{2}$  $5.7\times {10}^{2}$  $1.80\times {10}^{1}$  $1.16\times {10}^{1}$  $9.42\times {10}^{1}$ 
Performance Metric  Benchmark Functions  Proposed Algorithm  $\mathbf{Proposed}\mathbf{Algorithm}(\mathbf{without}\mathit{B}\mathit{E}\mathit{X}$$\mathbf{and}\mathit{P}\mathit{M}\mathit{O}\mathbf{Operator})$  SPEA2  SPGSA  SPGSA_ES 

CM metric  SCH  $\mathbf{2.87}\times {\mathbf{10}}^{\mathbf{3}}$  $3.11\times {10}^{3}$  $3.31\times {10}^{3}$  $3.21\times {10}^{3}$  $3.23\times {10}^{3}$ 
FON  $\mathbf{1.45}\times {\mathbf{10}}^{\mathbf{3}}$  $1.58\times {10}^{3}$  $1.86\times {10}^{3}$  $1.60\times {10}^{3}$  $1.67\times {10}^{3}$  
ZDT1  $\mathbf{1.07}\times {\mathbf{10}}^{\mathbf{3}}$  $1.17\times {10}^{3}$  $1.31\times {10}^{3}$  $1.17\times {10}^{3}$  $1.43\times {10}^{3}$  
ZDT2  $\mathbf{5.08}\times {\mathbf{10}}^{\mathbf{4}}$  $7.28\times {10}^{4}$  $8.68\times {10}^{4}$  $7.84\times {10}^{4}$  $9.37\times {10}^{4}$  
MOP5  $\mathbf{4.18}\times {\mathbf{10}}^{\mathbf{3}}$  $5.11\times {10}^{3}$  $4.01\times {10}^{2}$  $3.65\times {10}^{2}$  $6.16\times {10}^{2}$  
MOP6  $\mathbf{6.18}\times {\mathbf{10}}^{\mathbf{6}}$  $4.11\times {10}^{5}$  $9.86\times {10}^{5}$  $9.84\times {10}^{5}$  $1.19\times {10}^{4}$  
DTLZ2  $\mathbf{1.88}\times {\mathbf{10}}^{\mathbf{4}}$  $5.01\times {10}^{3}$  $7.97\times {10}^{3}$  $8.01\times {10}^{3}$  $8.33\times {10}^{3}$  
GD metric  SCH  $\mathbf{4.08}\times {\mathbf{10}}^{\mathbf{5}}$  $4.98\times {10}^{5}$  $4.03\times {10}^{4}$  $3.76\times {10}^{4}$  $3.78\times {10}^{4}$ 
FON  $\mathbf{1.81}\times {\mathbf{10}}^{\mathbf{4}}$  $1.95\times {10}^{4}$  $2.37\times {10}^{4}$  $1.98\times {10}^{4}$  $2.01\times {10}^{4}$  
ZDT1  $\mathbf{2.31}\times {\mathbf{10}}^{\mathbf{4}}$  $2.43\times {10}^{4}$  $2.63\times {10}^{4}$  $2.50\times {10}^{4}$  $2.60\times {10}^{4}$  
ZDT2  $\mathbf{6.25}\times {\mathbf{10}}^{\mathbf{5}}$  $7.11\times {10}^{5}$  $9.41\times {10}^{5}$  $9.36\times {10}^{5}$  $9.78\times {10}^{5}$  
MOP5  $\mathbf{5.28}\times {\mathbf{10}}^{\mathbf{4}}$  $2.70\times {10}^{3}$  $1.90\times {10}^{2}$  $1.76\times {10}^{2}$  $2.13\times {10}^{2}$  
MOP6  $\mathbf{1.12}\times {\mathbf{10}}^{\mathbf{5}}$  $1.32\times {10}^{5}$  $1.32\times {10}^{5}$  $1.30\times {10}^{5}$  $1.55\times {10}^{5}$  
DTLZ2  $\mathbf{3.77}\times {\mathbf{10}}^{\mathbf{4}}$  $4.01\times {10}^{4}$  $1.09\times {10}^{3}$  $1.10\times {10}^{3}$  $1.18\times {10}^{3}$  
SM metric  SCH  $\mathbf{5.48}\times {\mathbf{10}}^{\mathbf{3}}$  $6.11\times {10}^{3}$  2$0.08\times {10}^{2}$  $1.36\times {10}^{2}$  $1.05\times {10}^{2}$ 
FON  $\mathbf{3.12}\times {\mathbf{10}}^{\mathbf{3}}$  $4.22\times {10}^{3}$  $3.66\times {10}^{3}$  $3.00\times {10}^{3}$  $3.51\times {10}^{3}$  
ZDT1  $\mathbf{7.81}\times {\mathbf{10}}^{\mathbf{4}}$  $8.11\times {10}^{4}$  $3.66\times {10}^{3}$  $3.18\times {10}^{3}$  $3.21\times {10}^{3}$  
ZDT2  $\mathbf{5.23}\times {\mathbf{10}}^{\mathbf{4}}$  $5.89\times {10}^{4}$  $3.61\times {10}^{3}$  $3.11\times {10}^{3}$  $3.58\times {10}^{3}$  
MOP5  $\mathbf{6.08}\times {\mathbf{10}}^{\mathbf{3}}$  $6.28\times {10}^{3}$  $6.09\times {10}^{1}$  $6.10\times {10}^{1}$  $6.15\times {10}^{1}$  
MOP6  $\mathbf{2.90}\times {\mathbf{10}}^{\mathbf{3}}$  $3.09\times {10}^{3}$  $6.01\times {10}^{3}$  $3.10\times {10}^{3}$  $3.47\times {10}^{3}$  
DTLZ2  $\mathbf{7.16}\times {\mathbf{10}}^{\mathbf{3}}$  $8.01\times {10}^{3}$  $5.74\times {10}^{2}$  $5.87\times {10}^{2}$  $2.3\times {10}^{2}$ 
Analysis  Sensitivity Analysis Performed on  Parameter Setting  Setting of Variable Parameters  Number of Parameter Setting Combinations 

Sensitivity Analysis 1 


 30 
Sensitivity Analysis 2  Initial population Size  Mutation probability = 0.001  Population Size = 50; 100; 200; 400; 500  20 
Maximum no. of Iterations  Crossover probability = 1.0  Maximum no. of Iterations = 500; 1000; 2000  50 
Performance Measures  Overall Performance of Parameter Setting Combinations  

Very Good  Good  Average  Poor  Very Poor  
Combined performance Score  5.0–4.0  4.0–3.0  3.0–2.0  2.0–1.0  1.0–0.0 
Value of Mutation Parameter  Value of Constant Parameters  Metrics  Combined Performance Score  Overall Performance  

GD  SM  
0.01 
 3899  24.6  2.9  Average 
0.07  67  32.3  3.6  Good  
0.09  86  33.5  3.0  Average  
0.001  135  28.4  1.8  Very Poor  
0.01 
 11,085  22  3.3  Good 
0.07  55  32  3.8  Good  
0.09  83  28.7  2.6  Average  
0.001  71  32  3.4  Good 
Value of Crossover Parameter  Value of Constant Parameters  Metrics  Combined Performance Score  Overall Performance  

GD  SM  
0.7 
 112  39  4.2  Very Good 
0.8  45  28  3.2  Good  
0.9  60  35  3.0  Average  
1.0  36  30  2.6  Average  
0.7 
 4454  26  3.8  Good 
0.8  10,121  23  3.6  Good  
0.9  3632  21  3.2  Good  
1.0  11,068  24  3  Good 
Initial Population Size  Value of Constant Parameters  Metrics  Combined Performance Score  Overall Performance  

GD  SM  
100 
 225  28  3.0  Average 
200  32  34  3.1  Good  
400  36  39  4.0  Very Good  
500  38  30  3.6  Good  
100 
 60  30  3.5  Good 
200  105  34  3.9  Good  
400  21  36  3.5  Good  
500  21  35  3.2  Good 
Maximum No of Iterations  Value of Constant Parameters  Metrics  Combined Performance Score  Overall Performance  

GD  SM  
500 
 65  30  3.4  Good 
1000  21  33  3.2  Good  
2000  30  34  3.2  Good  
500 
 50  32  3.3  Good 
1000  51  34  3.6  Good  
2000  24  34  3.8  Good 
Initial Value of Coulomb’s Constant  Value of Constant Parameters  Metrics  Combined Performance Score  Overall Performance  

GD  SM  
100 
 85  31  3.6  Good 
300  30  40  3.3  Good  
500  40  332  4.2  Very Good  
100 
 62  35  3.3  Good 
300  48  39  3.9  Good  
500  38  31  4.0  Very Good 
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Petwal, H.; Rani, R. An Improved Artificial Electric Field Algorithm for MultiObjective Optimization. Processes 2020, 8, 584. https://doi.org/10.3390/pr8050584
Petwal H, Rani R. An Improved Artificial Electric Field Algorithm for MultiObjective Optimization. Processes. 2020; 8(5):584. https://doi.org/10.3390/pr8050584
Chicago/Turabian StylePetwal, Hemant, and Rinkle Rani. 2020. "An Improved Artificial Electric Field Algorithm for MultiObjective Optimization" Processes 8, no. 5: 584. https://doi.org/10.3390/pr8050584