A Novel DecompositionBased MultiObjective Evolutionary Algorithm with DualPopulation and Adaptive Weight Strategy
Abstract
:1. Introduction
2. Previous Knowledge
2.1. Problem Model
2.2. Dominance Relationship and Crowding Degree
2.3. Method of Decomposition
 Weighted Sum approach: The weight vector is used as a coefficient corresponding to the objective function one by one, and the mathematical formula is shown as below:$$\begin{array}{c}\hfill minimize\phantom{\rule{4pt}{0ex}}{g}^{ws}\left(x\right\lambda )=\sum _{i=1}^{m}{\lambda}_{i}{f}_{i}\left(x\right)\end{array}$$
 Tchebycheff approach: The decomposition method formula of this method is shown as below:$$\begin{array}{c}\hfill minimize\phantom{\rule{4pt}{0ex}}{g}^{te}\left(x\right\lambda ,{z}^{*})=ma{x}_{1\le i\le m}\{{\lambda}_{i}{f}_{i}\left(x\right){z}_{i}^{*}\left\right\}\end{array}$$
 Penaltybased Boundary Intersection approach: This method attempts to find the intersection point between a group of rays passing through the target space from an ideal point and the Pareto front. If these rays are uniformly distributed, then the intersection points found will be approximately uniformly distributed:$$\begin{array}{c}\hfill minimize\phantom{\rule{4pt}{0ex}}{g}^{pbi}\left(x\right\lambda ,{z}^{*})={d}_{1}+\theta {d}_{2}\end{array}$$$$\begin{array}{c}\hfill {d}_{1}=\frac{\u2225\begin{array}{c}{({z}^{*}F\left(x\right))}^{T}\lambda \end{array}\u2225}{\u2225\begin{array}{c}\lambda \end{array}\u2225}\end{array}$$$$\begin{array}{c}\hfill {d}_{2}=\Vert F\left(x\right)({z}^{*}{d}_{1}\frac{\lambda}{\u2225\begin{array}{c}\lambda \end{array}\u2225})\Vert \end{array}$$
3. Proposed Algorithm
3.1. Framework
Algorithm 1 MOEA/DDPAW 

3.2. Adaptive Weight Strategy
Algorithm 2 Adaptive Weight Strategy 

3.3. Enhanced Neighborhood Exploration Mechanism
Algorithm 3 Enhanced Individual Exploration 

3.4. Computational Complexity
4. Experiment and Analysis
4.1. Experimental Setup
4.2. Method of Comparison
 AGEMOEA [25]: A method based on nonEuclidean distance is used to estimate the geometric structure of the Pareto frontier, and the diversity and population density are dynamically adjusted to achieve a good convergence effect.
 MOEA/DURAW [26]: A variant of the MOEA/D algorithm, which uses a uniform random weight generation method and an adaptive weight method based on population sparsity to solve complex multiobjective optimization problems.
 NSGAIISDR [27]: A variant of the NSGAII algorithm, based on the angle between the candidate solutions, proposes an adaptive niche technique that identifies only the best convergent candidate solutions as nondominant solutions in each niche, thus better balancing the convergence and diversity of evolutionary multiobjective optimization.
 CMOPSO [28]: An improvement of the multiobjective particle swarm optimization algorithm, which uses a multiobjective particle swarm optimization algorithm based on competition mechanism. Particles are updated on the basis of each generation of population competition.
 MOEA/DDAE [29]: A variant of the MOEA/D algorithm, which uses the detection escape strategy to detect the algorithm stagnation state by using the feasible ratio and the overall constraint violation change rate, and then adjusts the constraint violation weight in time to guide the population search out of the stagnation state.
4.3. Performance Metric
4.4. Results and Analysis
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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AGEMOEA  MOEA/DURAW  NSGAIISDR  CMOPSO  MOEA/DDAE  MOEA/DDPAW  

ZDT1  $4.5566\times {10}^{3}$ $(6.87\times {10}^{5})$ −  $2.4524\times {10}^{2}$ $(1.77\times {10}^{2})$ −  $6.9117\times {10}^{3}$ $(5.58\times {10}^{4})$ −  $2.7065\times {10}^{3}$ $(5.07\times {10}^{5})$ −  $3.9702\times {10}^{3}$ $(5.41\times {10}^{5})$ −  $2.6074\times {10}^{3}$ $(2.84\times {10}^{5})$ 
ZDT2  $4.5403\times {10}^{3}$ $(8.37\times {10}^{5})$ −  $2.5001\times {10}^{2}$ $(2.13\times {10}^{2})$ −  $7.1562\times {10}^{3}$ $(6.65\times {10}^{4})$ −  $2.6063\times {10}^{3}$ $(2.18\times {10}^{5})$ +  $3.8912\times {10}^{3}$ $(4.80\times {10}^{5})$ −  $2.6713\times {10}^{3}$ $(3.75\times {10}^{5})$ 
ZDT3  $5.7301\times {10}^{3}$ $(1.25\times {10}^{4})$ −  $1.5227\times {10}^{2}$ $(5.57\times {10}^{3})$ −  $7.3570\times {10}^{3}$ $(5.29\times {10}^{4})$ −  $4.6740\times {10}^{3}$ $(5.97\times {10}^{5})$ −  $3.0474\times {10}^{3}$ $(2.20\times {10}^{5})$ =  $3.0440\times {10}^{3}$ $(3.07\times {10}^{5})$ 
ZDT4  $4.4869\times {10}^{3}$ $(7.35\times {10}^{5})$ −  $1.9056\times {10}^{1}$ $(1.17\times {10}^{1})$ −  $1.7007\times {10}^{0}$ $(9.77\times {10}^{1})$ −  $2.5600\times {10}^{3}$ $(1.87\times {10}^{5})$ =  $3.9801\times {10}^{3}$ $(1.43\times {10}^{4})$ −  $2.5614\times {10}^{3}$ $(1.80\times {10}^{5})$ 
ZDT6  $3.7375\times {10}^{3}$ $(1.06\times {10}^{4})$ −  $3.8146\times {10}^{2}$ $(1.13\times {10}^{2})$ −  $3.9215\times {10}^{3}$ $(2.32\times {10}^{4})$ −  $2.0549\times {10}^{3}$ $(1.12\times {10}^{5})$ +  $2.0538\times {10}^{3}$ $(9.57\times {10}^{6})$ +  $3.0974\times {10}^{3}$ $(1.80\times {10}^{5})$ 
UF1  $1.1084\times {10}^{1}$ $(3.08\times {10}^{2})$ −  $1.5812\times {10}^{1}$ $(6.28\times {10}^{2})$ −  $9.8125\times {10}^{2}$ $(3.07\times {10}^{2})$ −  $9.8281\times {10}^{2}$ $(2.75\times {10}^{2})$ −  $1.0932\times {10}^{1}$ $(2.46\times {10}^{2})$ −  $5.6112\times {10}^{2}$ $(2.53\times {10}^{2})$ 
UF2  $3.8838\times {10}^{2}$ $(1.12\times {10}^{2})$ −  $9.2979\times {10}^{2}$ $(4.17\times {10}^{2})$ −  $5.1230\times {10}^{2}$ $(4.79\times {10}^{3})$ −  $4.1317\times {10}^{2}$ $(1.83\times {10}^{2})$ −  $3.6030\times {10}^{2}$ $(8.57\times {10}^{3})$ −  $2.5671\times {10}^{2}$ $(6.07\times {10}^{3})$ 
UF3  $2.7212\times {10}^{1}$ $(5.31\times {10}^{2})$ −  $2.2370\times {10}^{1}$ $(2.86\times {10}^{2})$ =  $2.2401\times {10}^{1}$ $(3.91\times {10}^{2})$ =  $1.1325\times {10}^{1}$ $(1.83\times {10}^{2})$ +  $2.2560\times {10}^{1}$ $(4.50\times {10}^{2})$ =  $2.2389\times {10}^{1}$ $(5.22\times {10}^{2})$ 
UF4  $4.2069\times {10}^{2}$ $(1.11\times {10}^{3})$ =  $7.3622\times {10}^{2}$ $(2.48\times {10}^{3})$ −  $6.9834\times {10}^{2}$ $(3.86\times {10}^{3})$ −  $7.5978\times {10}^{2}$ $(5.37\times {10}^{3})$ −  $4.7901\times {10}^{2}$ $(2.65\times {10}^{3})$ −  $4.1958\times {10}^{2}$ $(1.37\times {10}^{3})$ 
UF5  $2.9557\times {10}^{1}$ $(5.91\times {10}^{2})$ +  $8.2835\times {10}^{1}$ $(1.68\times {10}^{1})$ −  $5.6403\times {10}^{1}$ $(2.35\times {10}^{1})$ −  $3.4625\times {10}^{1}$ $(1.17\times {10}^{1})$ =  $2.8012\times {10}^{1}$ $(6.97\times {10}^{2})$ +  $3.4526\times {10}^{1}$ $(1.06\times {10}^{1})$ 
UF6  $2.1861\times {10}^{1}$ $(1.29\times {10}^{1})$ −  $3.6294\times {10}^{1}$ $(8.79\times {10}^{2})$ −  $3.7128\times {10}^{1}$ $(1.03\times {10}^{1})$ −  $2.2384\times {10}^{1}$ $(1.36\times {10}^{1})$ −  $2.2171\times {10}^{1}$ $(1.16\times {10}^{1})$ −  $1.7370\times {10}^{1}$ $(1.07\times {10}^{1})$ 
UF7  $2.0336\times {10}^{1}$ $(1.68\times {10}^{1})$ −  $3.2097\times {10}^{1}$ $(1.35\times {10}^{1})$ −  $5.6390\times {10}^{2}$ $(6.95\times {10}^{3})$ +  $4.2142\times {10}^{2}$ $(8.18\times {10}^{2})$ +  $1.2998\times {10}^{1}$ $(1.37\times {10}^{1})$ =  $1.3069\times {10}^{1}$ $(1.65\times {10}^{1})$ 
+/−/≈  1/10/1  0/11/1  1/10/1  4/6/2  2/7/3 
AGEMOEA  MOEA/DURAW  NSGAIISDR  CMOPSO  MOEA/DDAE  MOEA/DDPAW  

ZDT1  $7.1987\times {10}^{1}$ $(7.12\times {10}^{5})$ −  $6.9618\times {10}^{1}$ $(1.11\times {10}^{2})$ −  $7.1439\times {10}^{1}$ $(9.20\times {10}^{4})$ −  $7.2132\times {10}^{1}$ $(1.10\times {10}^{4})$ =  $7.2031\times {10}^{1}$ $(8.91\times {10}^{5})$ =  $7.2179\times {10}^{1}$ $(4.37\times {10}^{5})$ 
ZDT2  $4.4430\times {10}^{1}$ $(8.86\times {10}^{5})$ −  $4.1205\times {10}^{1}$ $(2.49\times {10}^{2})$ −  $4.3866\times {10}^{1}$ $(1.07\times {10}^{3})$ −  $4.4593\times {10}^{1}$ $(9.49\times {10}^{5})$ =  $4.4497\times {10}^{1}$ $(6.43\times {10}^{5})$ =  $4.4635\times {10}^{1}$ $(4.39\times {10}^{5})$ 
ZDT3  $5.9919\times {10}^{1}$ $(3.56\times {10}^{5})$ −  $6.0067\times {10}^{1}$ $(9.53\times {10}^{3})$ =  $5.9570\times {10}^{1}$ $(9.80\times {10}^{4})$ −  $6.0025\times {10}^{1}$ $(2.00\times {10}^{5})$ =  $5.9972\times {10}^{1}$ $(3.38\times {10}^{5})$ =  $6.0025\times {10}^{1}$ $(2.86\times {10}^{5})$ 
ZDT4  $7.1982\times {10}^{1}$ $(1.51\times {10}^{4})$ =  $4.9527\times {10}^{1}$ $(1.25\times {10}^{1})$ −  $1.1887\times {10}^{2}$ $(3.91\times {10}^{2})$ −  $7.2080\times {10}^{1}$ $(3.77\times {10}^{5})$ =  $7.1990\times {10}^{1}$ $(5.28\times {10}^{4})$ =  $7.2085\times {10}^{1}$ $(3.91\times {10}^{5})$ 
ZDT6  $3.8827\times {10}^{1}$ $(9.93\times {10}^{5})$ −  $3.3722\times {10}^{1}$ $(1.49\times {10}^{2})$ −  $3.8797\times {10}^{1}$ $(2.28\times {10}^{4})$ −  $3.8990\times {10}^{1}$ $(1.42\times {10}^{5})$ =  $3.8894\times {10}^{1}$ $(2.64\times {10}^{5})$ =  $3.8989\times {10}^{1}$ $(2.05\times {10}^{5})$ 
UF1  $5.9141\times {10}^{1}$ $(3.37\times {10}^{2})$ =  $5.3667\times {10}^{1}$ $(4.43\times {10}^{2})$ −  $5.9203\times {10}^{1}$ $(3.11\times {10}^{2})$ =  $6.4258\times {10}^{1}$ $(2.07\times {10}^{2})$ +  $5.9444\times {10}^{1}$ $(3.18\times {10}^{2})$ +  $5.9283\times {10}^{1}$ $(2.48\times {10}^{2})$ 
UF2  $6.8149\times {10}^{1}$ $(7.27\times {10}^{3})$ −  $6.4148\times {10}^{1}$ $(2.14\times {10}^{2})$ −  $6.5972\times {10}^{1}$ $(5.73\times {10}^{3})$ −  $6.8410\times {10}^{1}$ $(5.82\times {10}^{3})$ −  $6.8280\times {10}^{1}$ $(9.50\times {10}^{3})$ −  $6.9356\times {10}^{1}$ $(4.70\times {10}^{3})$ 
UF3  $4.2156\times {10}^{1}$ $(4.92\times {10}^{2})$ −  $3.7971\times {10}^{1}$ $(2.83\times {10}^{2})$ −  $4.3565\times {10}^{1}$ $(4.95\times {10}^{2})$ =  $5.5462\times {10}^{1}$ $(2.84\times {10}^{2})$ +  $4.5738\times {10}^{1}$ $(4.15\times {10}^{2})$ +  $4.3776\times {10}^{1}$ $(4.94\times {10}^{2})$ 
UF4  $3.8987\times {10}^{1}$ $(9.88\times {10}^{4})$ =  $3.3750\times {10}^{1}$ $(4.01\times {10}^{3})$ −  $3.5040\times {10}^{1}$ $(4.93\times {10}^{3})$ −  $3.4091\times {10}^{1}$ $(6.96\times {10}^{3})$ −  $3.8255\times {10}^{1}$ $(2.35\times {10}^{3})$ −  $3.9087\times {10}^{1}$ $(1.46\times {10}^{3})$ 
UF5  $2.2734\times {10}^{1}$ $(6.03\times {10}^{2})$ −  $1.1752\times {10}^{2}$ $(3.67\times {10}^{2})$ −  $3.7383\times {10}^{2}$ $(4.51\times {10}^{2})$ −  $2.2329\times {10}^{1}$ $(7.19\times {10}^{2})$ −  $1.9594\times {10}^{1}$ $(8.21\times {10}^{2})$ −  $2.3272\times {10}^{1}$ $(5.43\times {10}^{2})$ 
UF6  $3.0167\times {10}^{1}$ $(7.36\times {10}^{2})$ −  $1.8115\times {10}^{1}$ $(5.64\times {10}^{2})$ −  $1.3430\times {10}^{1}$ $(7.64\times {10}^{2})$ −  $3.0416\times {10}^{1}$ $(6.17\times {10}^{2})$ −  $2.9014\times {10}^{1}$ $(8.92\times {10}^{2})$ −  $3.2208\times {10}^{1}$ $(6.49\times {10}^{2})$ 
UF7  $4.6429\times {10}^{1}$ $(1.17\times {10}^{1})$ =  $3.2311\times {10}^{1}$ $(8.29\times {10}^{2})$ −  $5.0512\times {10}^{1}$ $(1.25\times {10}^{2})$ +  $5.3911\times {10}^{1}$ $(6.15\times {10}^{2})$ +  $4.4741\times {10}^{1}$ $(1.18\times {10}^{1})$ −  $4.6793\times {10}^{1}$ $(9.88\times {10}^{2})$ 
+/−/≈  0/8/4  0/11/1  1/9/2  3/4/5  2/5/5 
AGEMOEA  MOEA/DURAW  NSGAIISDR  CMOPSO  MOEA/DDAE  MOEA/DDPAW  

DTLZ1  $2.0613\times {10}^{2}$ $(2.27\times {10}^{4})$ =  $1.5962\times {10}^{1}$ $(1.70\times {10}^{1})$ −  $1.5352\times {10}^{0}$ $(7.82\times {10}^{1})$ −  $1.4187\times {10}^{2}$ $(1.47\times {10}^{4})$ +  $1.4106\times {10}^{2}$ $(1.85\times {10}^{4})$ +  $2.0733\times {10}^{2}$ $(1.28\times {10}^{4})$ 
DTLZ2  $5.2410\times {10}^{2}$ $(1.20\times {10}^{4})$ −  $5.4875\times {10}^{2}$ $(1.16\times {10}^{3})$ −  $7.0669\times {10}^{2}$ $(2.59\times {10}^{3})$ −  $3.8143\times {10}^{2}$ $(4.62\times {10}^{4})$ =  $5.5726\times {10}^{2}$ $(4.67\times {10}^{4})$ −  $3.8036\times {10}^{2}$ $(2.77\times {10}^{4})$ 
DTLZ3  $5.2694\times {10}^{2}$ $(3.10\times {10}^{4})$ +  $8.2878\times {10}^{0}$ $(4.99\times {10}^{0})$ −  $6.8274\times {10}^{+1}$ $(2.11\times {10}^{+1})$ −  $5.8418\times {10}^{0}$ $(4.95\times {10}^{0})$ +  $4.0454\times {10}^{2}$ $(2.81\times {10}^{3})$ +  $6.3007\times {10}^{2}$ $(6.13\times {10}^{3})$ 
DTLZ4  $5.2463\times {10}^{2}$ $(1.25\times {10}^{4})$ −  $2.1671\times {10}^{1}$ $(2.33\times {10}^{1})$ −  $6.8917\times {10}^{2}$ $(2.04\times {10}^{3})$ −  $4.1291\times {10}^{2}$ $(7.46\times {10}^{4})$ −  $5.6860\times {10}^{2}$ $(9.63\times {10}^{4})$ −  $3.8195\times {10}^{2}$ $(3.76\times {10}^{4})$ 
DTLZ5  $5.1969\times {10}^{3}$ $(1.59\times {10}^{4})$ −  $4.4681\times {10}^{3}$ $(1.20\times {10}^{4})$ −  $6.1547\times {10}^{3}$ $(4.06\times {10}^{4})$ −  $2.8355\times {10}^{3}$ $(3.65\times {10}^{4})$ −  $4.9753\times {10}^{3}$ $(1.05\times {10}^{4})$ −  $2.2375\times {10}^{3}$ $(3.14\times {10}^{5})$ 
DTLZ6  $4.8476\times {10}^{3}$ $(6.32\times {10}^{5})$ −  $3.7324\times {10}^{2}$ $(1.70\times {10}^{1})$ −  $5.7994\times {10}^{3}$ $(3.30\times {10}^{4})$ −  $2.0859\times {10}^{3}$ $(1.65\times {10}^{5})$ +  $2.0784\times {10}^{3}$ $(1.44\times {10}^{5})$ +  $4.6200\times {10}^{3}$ $(1.03\times {10}^{4})$ 
DTLZ7  $1.1606\times {10}^{1}$ $(1.16\times {10}^{1})$ −  $7.4706\times {10}^{2}$ $(4.55\times {10}^{3})$ −  $8.3862\times {10}^{2}$ $(3.12\times {10}^{3})$ −  $4.2191\times {10}^{2}$ $(7.07\times {10}^{4})$ −  $6.1264\times {10}^{2}$ $(1.03\times {10}^{3})$ −  $3.8302\times {10}^{2}$ $(3.89\times {10}^{4})$ 
UF8  $2.4862\times {10}^{1}$ $(6.79\times {10}^{2})$ −  $3.0642\times {10}^{1}$ $(3.08\times {10}^{2})$ −  $3.2636\times {10}^{1}$ $(2.11\times {10}^{2})$ −  $5.4976\times {10}^{1}$ $(1.02\times {10}^{1})$ −  $2.4558\times {10}^{1}$ $(6.03\times {10}^{2})$ −  $1.4738\times {10}^{1}$ $(8.40\times {10}^{2})$ 
UF9  $1.6155\times {10}^{1}$ $(6.83\times {10}^{2})$ =  $4.0613\times {10}^{1}$ $(7.16\times {10}^{2})$ −  $4.8438\times {10}^{1}$ $(7.48\times {10}^{2})$ −  $8.3236\times {10}^{1}$ $(1.20\times {10}^{1})$ −  $1.9118\times {10}^{1}$ $(7.86\times {10}^{2})$ −  $1.5937\times {10}^{1}$ $(7.65\times {10}^{2})$ 
UF10  $3.6358\times {10}^{1}$ $(8.10\times {10}^{2})$ =  $6.4174\times {10}^{1}$ $(1.12\times {10}^{1})$ −  $1.3944\times {10}^{0}$ $(3.92\times {10}^{1})$ −  $3.3365\times {10}^{0}$ $(4.89\times {10}^{1})$ −  $3.6439\times {10}^{1}$ $(8.53\times {10}^{2})$ =  $3.6367\times {10}^{1}$ $(7.68\times {10}^{2})$ 
+/−/≈  1/6/3  0/10/0  0/10/0  3/6/1  3/6/1 
AGEMOEA  MOEA/DURAW  NSGAIISDR  CMOPSO  MOEA/DDAE  MOEA/DDPAW  

DTLZ1  $8.4252\times {10}^{1}$ $(3.47\times {10}^{4})$ +  $5.3287\times {10}^{1}$ $(3.11\times {10}^{1})$ −  $3.4133\times {10}^{3}$ $(1.68\times {10}^{2})$ −  $8.5111\times {10}^{1}$ $(5.83\times {10}^{4})$ +  $8.5268\times {10}^{1}$ $(2.90\times {10}^{4})$ +  $8.4117\times {10}^{1}$ $(3.90\times {10}^{4})$ 
DTLZ2  $5.6175\times {10}^{1}$ $(3.00\times {10}^{4})$ −  $5.5715\times {10}^{1}$ $(1.72\times {10}^{3})$ −  $5.2251\times {10}^{1}$ $(5.41\times {10}^{3})$ −  $5.6387\times {10}^{1}$ $(1.12\times {10}^{3})$ −  $5.5815\times {10}^{1}$ $(1.13\times {10}^{3})$ −  $5.7376\times {10}^{1}$ $(5.31\times {10}^{4})$ 
DTLZ3  $5.5835\times {10}^{1}$ $(2.08\times {10}^{3})$ +  $5.1647\times {10}^{1}$ $(4.31\times {10}^{3})$ −  $5.3619\times {10}^{1}$ $(3.27\times {10}^{2})$ −  $5.5981\times {10}^{2}$ $(1.71\times {10}^{1})$ −  $5.7501\times {10}^{1}$ $(8.29\times {10}^{4})$ +  $5.5038\times {10}^{1}$ $(7.16\times {10}^{3})$ 
DTLZ4  $5.6182\times {10}^{1}$ $(3.41\times {10}^{4})$ −  $4.8870\times {10}^{1}$ $(1.02\times {10}^{1})$ −  $5.2749\times {10}^{1}$ $(3.77\times {10}^{3})$ −  $5.5911\times {10}^{1}$ $(1.78\times {10}^{3})$ −  $5.5728\times {10}^{1}$ $(1.17\times {10}^{3})$ −  $5.7348\times {10}^{1}$ $(7.17\times {10}^{4})$ 
DTLZ5  $1.9943\times {10}^{1}$ $(1.98\times {10}^{4})$ =  $1.9954\times {10}^{1}$ $(2.00\times {10}^{4})$ =  $1.9831\times {10}^{1}$ $(3.61\times {10}^{4})$ −  $2.0056\times {10}^{1}$ $(2.48\times {10}^{4})$ +  $2.0125\times {10}^{1}$ $(3.16\times {10}^{5})$ +  $1.9946\times {10}^{1}$ $(1.10\times {10}^{4})$ 
DTLZ6  $1.9985\times {10}^{1}$ $(5.70\times {10}^{5})$ =  $1.9142\times {10}^{1}$ $(3.67\times {10}^{2})$ −  $1.9949\times {10}^{1}$ $(1.55\times {10}^{4})$ =  $2.0142\times {10}^{1}$ $(1.70\times {10}^{5})$ +  $2.0143\times {10}^{1}$ $(1.09\times {10}^{5})$ +  $1.9995\times {10}^{1}$ $(4.63\times {10}^{5})$ 
DTLZ7  $2.7206\times {10}^{1}$ $(1.44\times {10}^{2})$ −  $2.6880\times {10}^{1}$ $(2.11\times {10}^{3})$ −  $2.5826\times {10}^{1}$ $(2.08\times {10}^{3})$ −  $2.8016\times {10}^{1}$ $(7.92\times {10}^{4})$ +  $2.8487\times {10}^{1}$ $(3.11\times {10}^{4})$ +  $2.7742\times {10}^{1}$ $(8.43\times {10}^{4})$ 
UF8  $3.5375\times {10}^{1}$ $(4.38\times {10}^{2})$ −  $3.1125\times {10}^{1}$ $(3.26\times {10}^{2})$ −  $2.0278\times {10}^{1}$ $(2.86\times {10}^{2})$ −  $1.7675\times {10}^{2}$ $(1.73\times {10}^{2})$ −  $3.5435\times {10}^{1}$ $(4.49\times {10}^{2})$ −  $4.1812\times {10}^{1}$ $(6.58\times {10}^{2})$ 
UF9  $6.4945\times {10}^{1}$ $(6.08\times {10}^{2})$ =  $4.0670\times {10}^{1}$ $(5.48\times {10}^{2})$ −  $2.6278\times {10}^{1}$ $(7.22\times {10}^{2})$ −  $2.7339\times {10}^{2}$ $(3.36\times {10}^{2})$ −  $6.2154\times {10}^{1}$ $(7.11\times {10}^{2})$ −  $6.5909\times {10}^{1}$ $(6.73\times {10}^{2})$ 
UF10  $2.1485\times {10}^{1}$ $(7.62\times {10}^{2})$ +  $2.7731\times {10}^{2}$ $(3.04\times {10}^{2})$ −  $2.0316\times {10}^{1}$ $(5.27\times {10}^{2})$ +  $1.9435\times {10}^{1}$ $(7.34\times {10}^{2})$ −  $1.9891\times {10}^{1}$ $(6.79\times {10}^{2})$ =  $1.9983\times {10}^{1}$ $(7.43\times {10}^{2})$ 
+/−/≈  3/4/3  0/9/1  1/8/1  4/6/0  5/4/1 
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Ni, Q.; Kang, X. A Novel DecompositionBased MultiObjective Evolutionary Algorithm with DualPopulation and Adaptive Weight Strategy. Axioms 2023, 12, 100. https://doi.org/10.3390/axioms12020100
Ni Q, Kang X. A Novel DecompositionBased MultiObjective Evolutionary Algorithm with DualPopulation and Adaptive Weight Strategy. Axioms. 2023; 12(2):100. https://doi.org/10.3390/axioms12020100
Chicago/Turabian StyleNi, Qingjian, and Xuying Kang. 2023. "A Novel DecompositionBased MultiObjective Evolutionary Algorithm with DualPopulation and Adaptive Weight Strategy" Axioms 12, no. 2: 100. https://doi.org/10.3390/axioms12020100