# Modified Evolutionary Algorithm and Chaotic Search for Bilevel Programming Problems

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## Abstract

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## 1. Introduction

## 2. Bi-Level Programming Problems

## 3. Proposed Algorithm (CGA-BS)

#### 3.1. Basic Concepts of Genetic Algorithm

#### 3.2. Chaos Theory for Optimization Problems

#### 3.3. Modified Genetic Algorithm and Chaotic Search for BLPP (CGA-BS)

#### 3.3.1. Genetic Algorithm Based on BST

- Constrains handling: before starting the algorithm, constraints of the lower level problem is added to the constraints of the upper-level problem. The feasible solutions for the upper-level problem must be feasible and optimal solutions for the lower level problem. Therefore, constrains of both levels are considered as upper level constrains to guarantee the feasibility of solutions.
- Initial population: variables are randomly initialized within the search space bounds of upper-level bounds and lower-level bounds [50].
- Reference point: the repairing process needs one feasible point considered as a reference point to be entered and the algorithm procedure is completed.
- Repairing process: for constrained problems, some generated solutions don’t satisfy constraints and need to be repaired. The repairing process transforms the infeasible solutions to be feasible solutions [51].
- Evaluation: both upper-level objective function and lower-level objective function are used for evaluation of individuals and for determining the fitness of each solution to both upper-level problem and lower-level problem.
- Producing a new population: new selection technique, crossover operator and mutation operator have applied to produce the new population.

- 7.
- 8.
- GA termination: GA is terminated at the maximum number of generations or when population convergence occurs. If the termination condition doesn’t satisfy, return to 6.

#### 3.3.2. Chaotic Local Search

- Define a range of the upper-level variables for chaotic local search.
- Produce chaotic numbers using the logistic map: El-Shorbagy et al. in [46] presented a comparison between different chaotic maps for general nonlinear optimization problems and result that logistic map gives better performance than other chaotic maps and increase the solution quality rather than other chaotic maps. Therefore, in the proposed algorithm, we choose to apply the logistic map
- Generate the chaotic variables into the determined range.
- Find the best value: The produced chaotic variables are evaluated according to upper-level objective-function. The chaotic variables obtained best objective-function values are the best value.
- Update the local search boundary value for the best-obtained value.
- Stopping chaos search: chaotic local search is stopped at the specified iterations and put out the best solution as a local search solution.

#### 3.3.3. Algorithm Termination

Algorithm 1 Modified Genetic algorithm with chaotic search for BLPP (CGA-BS) |

1: Modified Genetic algorithm for upper level problem 2: Begin$k\leftarrow 1$ 3: Generate an initial population ${p}_{k}$ with size ${N}_{u}$; 4: Check feasibility and repair out unfeasible individuals of the population ${p}_{k}$; 5: While $k<Max\_ge{n}_{u}$Evaluate ${p}_{k}$ according to upper level objective function; Select ${p}_{ksu}$ from ${p}_{k}$ using selection operator; Evaluate ${p}_{ksu}$ according to lower level objective function; Select ${p}_{ksl}$ from ${p}_{ksu}$ using selection operator; Apply crossover operator with $CP$ and mutation operators with $MP$ on ${p}_{ksl}$ and produce children population ${p}_{k}{}^{\prime}$; |

Evaluate the fitness of the children ${{p}_{k}}^{\prime}$; Select ${p}_{ksl}$ from ${p}_{ksu}$ using selection operator; Apply crossover operator with $CP$ and mutation operators with $MP$ on ${p}_{ksl}$ and produce children population ${p}_{k}{}^{\prime}$; Evaluate the fitness of the children ${p}_{k}{}^{\prime}$; Keep the best from ${p}_{k}\cup {p}_{k}{}^{\prime}$ to form ${p}_{k+1}$ Archive ${p}_{Max\_Generation\_U.}={p}_{k+1}$; $k=k+1$ 6: End while7: Return to the individual $({x}^{*},{y}^{*})$ from ${p}_{Max\_Generation\_U.}$ with best fitness values and archive ${x}^{*}{}_{1}={x}^{*}$8: Chaotic local search around modified genetic algorithm solution $({x}^{*},{y}^{*})$9: Begin$i\leftarrow 1$ 10: While: $i<Max\_ge{n}_{c}$Define range of chaotic local search ${x}_{i}^{*}-\epsilon <{a}_{i},{x}_{i}^{*}+\epsilon {b}_{i}$ Generate ${z}^{k}$ using different logistic ma ${z}^{k+1}=\mu {z}^{k}(1-{z}^{k}),{z}^{0}\notin \left\{0.0,0.25,0.50,0.75,1.0\right\},k=1,2,\dots {N}_{c}$ ${x}_{i}^{k}={x}_{i}^{*}-\epsilon +2\epsilon {z}^{k}\forall i=1,\dots ,n$ If$f({x}^{k},{y}^{*})<f({x}^{*},{y}^{*})$ then ${x}^{*}={x}^{k}$ and archive ${x}^{*}{}_{2}={x}^{*}$Else if $f({x}^{k},{y}^{*})\ge f({x}^{*},{y}^{*})$ continue,End ifIf termination criteria satisfied,BreakEnd if$i\leftarrow i+1$ 11: End while12: Solve lower level problem using genetic algorithm for $x={x}^{*}{}_{1}$13: Begin$j\leftarrow 1$ 14: Generate an initial population ${p}_{j}$ with size ${N}_{l}$; 15: Check feasibility and repair out unfeasible individuals of the population ${p}_{j}$; 16: While $j<Max\_ge{n}_{l}$Evaluate ${p}_{j}$ according to lower level objective function; Select ${p}_{js}$ from ${p}_{j}$ using selection operator; Apply crossover operator with $CP$ and mutation operators with $MP$ on ${p}_{js}$ and produce children population |

Evaluate the fitness of the children ${p}_{j}{}^{\prime}$; Keep the best from ${p}_{j}\cup {p}_{j}{}^{\prime}$ to form ${p}_{j+1}$; Archive ${p}_{Max\_Generation\_l.}={p}_{j+1}$; $j=j+1$ 17: End while18: Return to the individual $({x}^{*}{}_{1},{y}^{*})$ from ${p}_{Max\_Generation\_l.}$ with best fitness values;19: Evaluate the fitness of $({x}^{*}{}_{1},{y}^{*})$ 20: Repeat from step 13 to step 18 for $x={x}^{*}{}_{2}$ 21: Evaluate the fitness of $({x}^{*}{}_{2},{y}^{*})$ 22: Return best evaluation in step 20 and 22 as algorithm solution |

## 4. Numerical Experiments

- GA generation: 200–1000
- Generation gap: 0.9
- Selection operator: Stochastic universal sampling
- Crossover operator: Single point
- Mutation operator: Real-value
- Crossover rate: 0.9
- Mutation rate: 0.07
- Number of Chaotic iterations: 10,000
- Chaotic rang: 10
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#### 4.1. Test Problems

#### 4.2. Results Analyses

#### 4.2.1. TP Test Set Results Analyses

#### 4.2.2. SMD Test Set Results Analyses

#### 4.2.3. P1–P24 Test Set Results Analyses

#### 4.2.4. Algorithm Performance Analyses with Bilevel Selection Technique

#### 4.2.5. Algorithm Performance Analyses with Chaos Search

#### 4.2.6. Computational Expense

## 5. Conclusions

- (a)
- Adapting CGA-BS to solve multi-level programming problems
- (b)
- Updating CGA-BS to solve the multi-objective multi-level programming problems.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviation

## Appendix A

Problem Formulation (n, m) | Problem Formulation (n, m) |
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P1$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=-{x}_{1}{}^{2}-3{x}_{2}{}^{2}-4{y}_{1}+{y}_{2}{}^{2}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=2{x}_{1}{}^{2}+{y}_{1}{}^{2}-5{y}_{2}\\ s.t\\ {x}_{1}{}^{2}-2{x}_{1}+{x}_{2}{}^{2}-2{y}_{1}+{y}_{2}\ge -3\\ {x}_{2}+3{y}_{1}-4{y}_{2}\ge 4\\ 0\le {y}_{i}i=1,2\end{array}\}\\ {x}_{1}{}^{2}+2{x}_{2}\le 4,\\ 0\le {x}_{i}i=1,2\end{array}$ | P2$(\mathit{n}=2,\mathit{m}=3)$$\begin{array}{l}MinimizeF(x,y)=-8{x}_{1}-4{x}_{2}+4{y}_{1}-40{y}_{2}-4{y}_{3}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={x}_{1}+2{x}_{2}+{y}_{1}+{y}_{2}+2{y}_{3}\\ s.t\\ {y}_{2}+{y}_{3}-{y}_{1}\le 1\\ 2{x}_{1}-{y}_{1}+2{y}_{2}-0.5{y}_{3}\le 1\\ 2{x}_{2}+2{y}_{1}-{y}_{2}-0.5{y}_{3}\le 1\\ 0\le {y}_{i}i=1,2\end{array}\}\\ 0\le {x}_{i}i=1,2\end{array}$ |

P3$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=rt(x)x-3{y}_{1}+4{y}_{2}+0.5t(y)y\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=0.5t(y)hy-t(b(x))y\\ s.t\\ -0.333{y}_{1}+{y}_{2}-2\le 0,{y}_{1}-0.333{y}_{2}-2\le 0\\ 0\le {y}_{i}i=1,2\end{array}\}\\ where,h=\left(\begin{array}{cc}1& -2\\ -2& 5\end{array}\right),b(x)=x,r=0.1\\ t(.)denotestransposeofavector\end{array}$ | P4$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=rt(x)x-3{y}_{1}+4{y}_{2}+0.5t(y)y\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=0.5t(y)hy-t(b(x))y\\ s.t\\ -0.333{y}_{1}+{y}_{2}-2\le 0,{y}_{1}-0.333{y}_{2}-2\le 0\\ 0\le {y}_{i}i=1,2\end{array}\}\\ where,h=\left(\begin{array}{cc}1& -2\\ -2& 5\end{array}\right),b(x)=x,r=1\\ t(.)denotestransposeofavector\end{array}$ |

P5$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=rt(x)x-3{y}_{1}+4{y}_{2}+0.5t(y)y\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=0.5t(y)hy-t(b(x))y\\ s.t\\ -0.333{y}_{1}+{y}_{2}-2\le 0,{y}_{1}-0.333{y}_{2}-2\le 0\\ 0\le {y}_{i}i=1,2\end{array}\}\\ where,h=\left(\begin{array}{cc}1& 3\\ 3& 10\end{array}\right),b(x)=x,r=0\\ t(.)denotestransposeofavector\end{array}$ | P6$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=rt(x)x-3{y}_{1}+4{y}_{2}+0.5t(y)y\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=0.5t(y)hy-t(b(x))y\\ s.t\\ -0.333{y}_{1}+{y}_{2}-2\le 0,{y}_{1}-0.333{y}_{2}-2\le 0\\ 0\le {y}_{i}i=1,2\end{array}\}\\ where,h=\left(\begin{array}{cc}1& 3\\ 3& 10\end{array}\right),b(x)=x,r=0.1\\ t(.)denotestransposeofavector\end{array}$ |

P7$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=rt(x)x-3{y}_{1}+4{y}_{2}+0.5t(y)y\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=0.5t(y)hy-t(b(x))y\\ s.t\\ -0.333{y}_{1}+{y}_{2}-2\le 0\\ {y}_{1}-0.333{y}_{2}-2\le 0\\ 0\le {y}_{i}i=1,2\end{array}\}\\ where,h=\left(\begin{array}{cc}1& 3\\ 3& 10\end{array}\right),b(x)=\left(\begin{array}{cc}-1& 2\\ 3& -3\end{array}\right)x,r=0.1\\ t(.)denotestransposeofavector\end{array}$ | P8$(\mathit{n}=1,\mathit{m}=2)$$\begin{array}{l}MaximizeF(x,y)=100x+1000{y}_{1}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Maximizef(x,y)={y}_{1}+{y}_{2}\\ s.t\\ x+{y}_{1}-{y}_{2}\le 1,{y}_{1}+{y}_{2}\le 1\end{array}\}\\ 0\le x\le 1\end{array}$ |

P9$(\mathit{n}=1,\mathit{m}=1)$$\begin{array}{l}MinimizeF(x,y)={x}^{2}+{(y-10)}^{2}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={(x+2y-30)}^{2}\\ s.t\\ x+y\le 20,0\le y\le 20\end{array}\}\\ -x+y\le 0,0\le \mathrm{x}\le 15\end{array}$ | P10$(\mathit{n}=1,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)={(x-1)}^{2}+2{y}_{1}-2x\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={(2{y}_{1}-4)}^{2}+{(2{y}_{2}-1)}^{2}\\ +x{y}_{1}\\ s.t\\ 4x+5{y}_{1}+4{y}_{2}\le 12,4{y}_{2}-4x-5{y}_{1}\le -4\\ 4x-4{y}_{1}+5{y}_{2}\le 4,4{y}_{1}-4x+5{y}_{2}\le 4\\ 0\le {y}_{i}i=1,2\end{array}\}\\ x\ge 0\end{array}$ |

P11$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MaximizeF(x,y)=\frac{({x}_{1}+{y}_{1})({x}_{2}+{y}_{2})}{1+{x}_{1}{y}_{1}+{x}_{2}{y}_{2}}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Maximizef(x,y)=-\frac{({x}_{1}+{y}_{1})({x}_{2}+{y}_{2})}{1+{x}_{1}{y}_{1}+{x}_{2}{y}_{2}}\\ s.t\\ 0\le {y}_{i}\le {x}_{i}i=1,2\end{array}\}\\ {x}_{1}{}^{2}+{x}_{2}{}^{2}\le 100,{x}_{1}-{x}_{2}\le 0\\ 0\le {x}_{i},i=1,2\end{array}$ | P12$(\mathit{n}=2,\mathit{m}=6)$$\begin{array}{l}MinimizeF(x,y)=-8{x}_{1}-4{x}_{2}+4{y}_{1}-40{y}_{2}-4{y}_{3}\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=\frac{1+{x}_{1}+{x}_{2}+2{y}_{1}-{y}_{2}+{y}_{3}}{6+2{x}_{1}+{y}_{1}+{y}_{2}-3{y}_{3}}\\ s.t\\ -{y}_{1}+{y}_{2}+{y}_{3}+{y}_{4}=1\\ 2{x}_{1}-{y}_{1}+2{y}_{2}-0.5{y}_{3}+{y}_{5}=1\\ 2{x}_{2}+2{y}_{1}-{y}_{2}-0.5{y}_{3}+{y}_{6}=1\\ {y}_{i}\ge 0i=1,2,\dots ,6\end{array}\}\\ {x}_{i}\ge 0,i=1,2\end{array}$ |

P13$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|2{x}_{1}+2{x}_{2}-3{y}_{1}-3{y}_{2}-60\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1}+20)}^{20}\\ +{({y}_{2}-{x}_{2}+20)}^{20}\\ s.t\\ {x}_{1}+{x}_{2}+{y}_{1}-2{y}_{2}\le 40,\\ -10\le {y}_{i}\le 20i=1,2\end{array}\}\\ 2{y}_{1}-{x}_{1}+10\le 0,2{y}_{2}-{x}_{2}+10\le 0\\ 0\le {x}_{i}\le 50i=1,2\end{array}$ | P14$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|\mathrm{sin}(2{x}_{1}+2{x}_{2}-3{y}_{1}-3{y}_{2}-60)\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1}+20)}^{2}\\ +{({y}_{2}-{x}_{2}+20)}^{2}\\ s.t\\ 2{y}_{1}-{x}_{1}+10\le 0,2{y}_{2}-{x}_{2}+10\le 0\\ -10\le {y}_{i}\le 20i=1,2\end{array}\}\\ {x}_{1}+{x}_{2}+{y}_{1}-2{y}_{2}\le 40\\ 0\le {x}_{i}\le 50,i=1,2\end{array}$ |

P15$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|tan(2{x}_{1}+2{x}_{2}-3{y}_{1}-3{y}_{2}-60)\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1}+20)}^{2}\\ +{({y}_{2}-{x}_{2}+20)}^{2}\\ s.t\\ 2{y}_{1}-{x}_{1}+10\le 0,2{y}_{2}-{x}_{2}+10\le 0\\ -10\le {y}_{i}\le 20i=1,2\end{array}\}\\ {x}_{1}+{x}_{2}+{y}_{1}-2{y}_{2}\le 40\\ 0\le {x}_{i}\le 50,i=1,2\end{array}$ | P16$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|\begin{array}{l}{({x}_{1}-30)}^{2}+{({x}_{2}-20)}^{2}-20{y}_{1}\\ +20{y}_{2}-225\end{array}\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1})}^{2}+{({y}_{2}-{x}_{2})}^{2}\\ s.t\\ 0\le {y}_{i}\le 10i=1,2\end{array}\}\\ 30-{x}_{1}-2{x}_{2}\le 0,{x}_{1}+{x}_{2}-25\le 0\\ {x}_{2}\le 15\end{array}$ |

P17$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|\begin{array}{l}sin({({x}_{1}-30)}^{2}+{({x}_{2}-20)}^{2}-20{y}_{1}\\ +20{y}_{2}-225)\end{array}\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1})}^{2}+{({y}_{2}-{x}_{2})}^{2}\\ s.t\\ 0\le {y}_{i}\le 10i=1,2\end{array}\}\\ 30-{x}_{1}-2{x}_{2}\le 0,{x}_{1}+{x}_{2}-25\le 0\\ {x}_{2}\le 15\end{array}$ | P18$(\mathit{n}=2,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|\begin{array}{l}tan({({x}_{1}-30)}^{2}+{({x}_{2}-20)}^{2}-20{y}_{1}\\ +20{y}_{2}-225)\end{array}\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={({y}_{1}-{x}_{1})}^{2}+{({y}_{2}-{x}_{2})}^{2}\\ s.t\\ 0\le {y}_{i}\le 10i=1,2\end{array}\}\\ 30-{x}_{1}-2{x}_{2}\le 0,{x}_{1}+{x}_{2}-25\le 0\\ {x}_{2}\le 15\end{array}$ |

P19$(\mathit{n}=1,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|{(x-1)}^{2}+2{y}_{1}-2x+1.2097\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={(2{y}_{1}-4)}^{2}+{(2{y}_{2}-1)}^{2}\\ +x{y}_{1}\\ s.t\\ 4x+5{y}_{1}+4{y}_{2}\le 12,4{y}_{2}-4x-5{y}_{1}\le -4\\ 4x-4{y}_{1}+5{y}_{2}\le 4,4{y}_{1}-4x+5{y}_{2}\le 4\\ 0\le {y}_{i}i=1,2\end{array}\}\\ 0\le x\end{array}$ | P20$(\mathit{n}=1,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|sin({(x-1)}^{2}+2{y}_{1}-2x+1.2097)\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={(2{y}_{1}-4)}^{2}+{(2{y}_{2}-1)}^{2}\\ +x{y}_{1}\\ s.t\\ 4x+5{y}_{1}+4{y}_{2}\le 12,4{y}_{2}-4x-5{y}_{1}\le -4\\ 4x-4{y}_{1}+5{y}_{2}\le 4,4{y}_{1}-4x+5{y}_{2}\le 4\\ 0\le {y}_{i}i=1,2\end{array}\}\\ 0\le x\end{array}$ |

P21$(\mathit{n}=1,\mathit{m}=2)$$\begin{array}{l}MinimizeF(x,y)=\left|tan({(x-1)}^{2}+2{y}_{1}-2x+1.2097)\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)={(2{y}_{1}-4)}^{2}+{(2{y}_{2}-1)}^{2}\\ +x{y}_{1}\\ s.t\\ 4x+5{y}_{1}+4{y}_{2}\le 12,4{y}_{2}-4x-5{y}_{1}\le -4\\ 4x-4{y}_{1}+5{y}_{2}\le 4,4{y}_{1}-4x+5{y}_{2}\le 4\\ 0\le {y}_{i}i=1,2\end{array}\}\\ 0\le x\end{array}$ | P22$(\mathit{n}=2,\mathit{m}=6)$$\begin{array}{l}MinimizeF(x,y)=\left|-8{x}_{1}-4{x}_{2}+4{y}_{1}-40{y}_{2}-4{y}_{3}+29.2\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=\frac{1+{x}_{1}+{x}_{2}+2{y}_{1}-{y}_{2}+{y}_{3}}{6+2{x}_{1}+{y}_{1}+{y}_{2}-3{y}_{3}}\\ s.t\\ -{y}_{1}+{y}_{2}+{y}_{3}+{y}_{4}=1\\ 2{x}_{1}-{y}_{1}+2{y}_{2}-0.5{y}_{3}+{y}_{5}=1\\ 2{x}_{2}+2{y}_{1}-{y}_{2}-0.5{y}_{3}+{y}_{6}=1\\ {y}_{i}\ge 0i=1,2,\dots ,6\end{array}\}\\ {x}_{i}\ge 0,i=1,2\end{array}$ |

P23$(\mathit{n}=2,\mathit{m}=6)$$\begin{array}{l}MinimizeF(x,y)=\left|\begin{array}{l}\mathrm{sin}(-8{x}_{1}-4{x}_{2}+4{y}_{1}-40{y}_{2}-4{y}_{3}\\ +29.2)\end{array}\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=\frac{1+{x}_{1}+{x}_{2}+2{y}_{1}-{y}_{2}+{y}_{3}}{6+2{x}_{1}+{y}_{1}+{y}_{2}-3{y}_{3}}\\ s.t\\ -{y}_{1}+{y}_{2}+{y}_{3}+{y}_{4}=1\\ 2{x}_{1}-{y}_{1}+2{y}_{2}-0.5{y}_{3}+{y}_{5}=1\\ 2{x}_{2}+2{y}_{1}-{y}_{2}-0.5{y}_{3}+{y}_{6}=1\\ {y}_{i}\ge 0i=1,2,\dots ,6\end{array}\}\\ {x}_{i}\ge 0,i=1,2\end{array}$ | P24$(\mathit{n}=2,\mathit{m}=6)$$\begin{array}{l}MinimizeF(x,y)=\left|\begin{array}{l}tan(-8{x}_{1}-4{x}_{2}+4{y}_{1}-40{y}_{2}-4{y}_{3}\\ +29.2)\end{array}\right|\\ s.t\\ y\in \underset{(y)}{\mathrm{arg}\mathrm{min}}\{\begin{array}{l}Minimizef(x,y)=\frac{1+{x}_{1}+{x}_{2}+2{y}_{1}-{y}_{2}+{y}_{3}}{6+2{x}_{1}+{y}_{1}+{y}_{2}-3{y}_{3}}\\ s.t\\ -{y}_{1}+{y}_{2}+{y}_{3}+{y}_{4}=1\\ 2{x}_{1}-{y}_{1}+2{y}_{2}-0.5{y}_{3}+{y}_{5}=1\\ 2{x}_{2}+2{y}_{1}-{y}_{2}-0.5{y}_{3}+{y}_{6}=1\\ {y}_{i}\ge 0i=1,2,\dots ,6\end{array}\}\\ {x}_{i}\ge 0,i=1,2\end{array}$ |

## Appendix B

**Table A2.**Ranks for CGA-BS, BLEAQ, NEA, NEA1, PSO-CST and BLEAQ and the corresponding reference in [36] according to upper level objective functions for TP1–TP10.

Problem | Rank According to Upper Level | |||||
---|---|---|---|---|---|---|

CGA-BS | BLEAQ [27] | NEA [41] | NEA1 [56] | PSO-CST [36] | Cor. Ref. [36] | |

TP1 | Rank1 | Rank1 | Rank1 | Rank1 | - | Rank1 |

TP2 | Rank1 | Rank2 | Rank1 | Rank1 | - | Rank3 |

TP3 | Rank1 | Rank2 | Rank4 | Rank4 | Rank3 | Rank4 |

TP4 | Rank1 | Rank4 | Rank3 | Rank3 | Rank2 | Rank3 |

TP5 | Rank2 | Rank3 | Rank1 | Rank1 | - | Rank4 |

TP6 | Rank1 | Rank2 | Rank2 | Rank2 | Rank3 | Rank4 |

TP7 | Rank2 | Rank3 | Rank1 | Rank1 | Rank1 | Rank1 |

TP8 | Rank2 | Rank3 | Rank1 | Rank1 | - | Rank3 |

TP9 | Rank1 | Rank2 | - | - | - | - |

TP10 | Rank2 | Rank1 | - | - | - | - |

**Table A3.**Ranks for CGA-BS, BLMA, NBLEA, BLEAQ, BIDE, FOA, PSO, LGMS-FOA algorithms according to upper level accuracy for SMD1–SMD6.

Problem | Rank According to Upper Level | |||||||
---|---|---|---|---|---|---|---|---|

CGA-BS | BLMA [57] | NBLEA [57] | BLEAQ [57] | BIDE [57] | FOA [58] | PSO [58] | LGMS-FOA [58] | |

SMD1 | Rank 1 | Rank 3 | Rank 5 | Rank 3 | Rank 4 | Rank 2 | Rank7 | Rank 6 |

SMD2 | Rank 4 | Rank2 | Rank 5 | Rank 6 | Rank 3 | Rank 1 | Rank 8 | Rank7 |

SMD3 | Rank 1 | Rank 3 | Rank 4 | Rank 6 | Rank 5 | Rank 2 | Rank 8 | Rank 7 |

SMD4 | Rank 2 | Rank 3 | Rank 5 | Rank 3 | Rank 4 | Rank 1 | Rank 7 | Rank 6 |

SMD5 | Rank 1 | Rank 3 | Rank 3 | Rank 3 | Rank 4 | Rank 2 | Rank 6 | Rank 5 |

SMD6 | Rank 1 | Rank 2 | Rank 2 | Rank 2 | Rank 3 | - | - | - |

**Table A4.**Ranks for CGA-BS, NEA1, PSO-CST algorithms according to upper level accuracy for SMD1–SMD6.

Problem | Rank According to Upper Level | ||
---|---|---|---|

CGA-BS | NEA1 [56] | PSO-CST [36] | |

P1 | Rank1 | Rank2 | Rank3 |

P2 | Rank1 | Rank2 | Rank3 |

P3 | Rank1 | Rank2 | - |

P4 | Rank2 | Rank1 | - |

P5 | Rank1 | Rank2 | - |

P6 | Rank2 | Rank1 | - |

P7 | Rank2 | Rank1 | - |

P8 | Rank1 | Rank2 | Rank3 |

P9 | Rank2 | Rank1 | Rank3 |

P10 | Rank1 | Rank2 | Rank3 |

P11 | Rank1 | Rank3 | Rank2 |

P12 | Rank1 | Rank2 | - |

P13 | Rank1 | Rank1 | - |

P14 | Rank1 | Rank1 | Rank2 |

P15 | Rank1 | Rank1 | Rank2 |

P16 | Rank2 | Rank1 | Rank3 |

P17 | Rank2 | Rank3 | Rank1 |

P18 | Rank2 | Rank1 | Rank3 |

P19 | Rank1 | Rank2 | Rank3 |

P20 | Rank1 | Rank2 | Rank3 |

P21 | Rank1 | Rank2 | Rank3 |

Problem | Statistics of Ranks | |||
---|---|---|---|---|

Rank1 | Rank2 | Rank3 | Rank4 | |

TP1 | 1 | 0 | 0 | 0 |

TP2 | 1 | 0 | 0 | 0 |

TP3 | 1 | 0 | 0 | 0 |

TP4 | 1 | 0 | 0 | 0 |

TP5 | 0 | 1 | 0 | 0 |

TP6 | 1 | 0 | 0 | 0 |

TP7 | 0 | 1 | 0 | 0 |

TP8 | 0 | 1 | 0 | 0 |

TP9 | 1 | 0 | 0 | 0 |

TP10 | 0 | 1 | 0 | 0 |

SMD1 | 1 | 0 | 0 | 0 |

SMD2 | 0 | 0 | 0 | 1 |

SMD3 | 1 | 0 | 0 | 0 |

SMD4 | 0 | 1 | 0 | 0 |

SMD5 | 1 | 0 | 0 | 0 |

SMD6 | 1 | 0 | 0 | 0 |

P1 | 1 | 0 | 0 | 0 |

P2 | 1 | 0 | 0 | 0 |

P3 | 1 | 0 | 0 | 0 |

P4 | 0 | 1 | 0 | 0 |

P5 | 1 | 0 | 0 | 0 |

P6 | 0 | 1 | 0 | 0 |

P7 | 0 | 1 | 0 | 0 |

P8 | 1 | 0 | 0 | 0 |

P9 | 0 | 1 | 0 | 0 |

P10 | 1 | 0 | 0 | 0 |

P11 | 1 | 0 | 0 | 0 |

P12 | 1 | 0 | 0 | 0 |

P13 | 1 | 0 | 0 | 0 |

P14 | 1 | 0 | 0 | 0 |

P15 | 1 | 0 | 0 | 0 |

P16 | 0 | 1 | 0 | 0 |

P17 | 0 | 1 | 0 | 0 |

P18 | 0 | 1 | 0 | 0 |

P19 | 1 | 0 | 0 | 0 |

P20 | 1 | 0 | 0 | 0 |

P21 | 1 | 0 | 0 | 0 |

**Table A6.**Ranks for CGA-BS, BLMA, NBLEA, BLEAQ, BIDE algorithms according to total function evaluations for SMD1–SMD6.

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**Figure 4.**Chaotic number produced using Chebyshev map, circle map, iterative map, logistic map, Singer map and Sinusoidal map.

**Figure 6.**Upper-level accuracy obtained with CGA-ES and three well-known algorithms for TP test problems.

**Figure 8.**Upper-level accuracy obtained with CGA-ES and five well-known algorithms for SMD1–SMD5 test problems.

**Figure 15.**Upper-level accuracy obtained by the proposed algorithm with and without algorithms bilevel selection technique for Tp1–Tp10.

**Figure 17.**Median total function evaluations for CGA-BS and four known algorithms for SMD test problem.

**Figure 18.**Rank percentages of the proposed algorithm according total function evaluations for SMD test problems.

**Table 1.**Upper level objective functions for CGA-ES, BLEAQ, NEA, NEA1, PSO-CST and BLEAQ and corresponding reference [36] in algorithms for TP1-TP10.

Problem | Upper Level Objective Function | |||||
---|---|---|---|---|---|---|

CGA-BS | BLEAQ [27] | NEA [41] | NEA1 [56] | PSO-CST [36] | Cor. Ref. [36] | |

TP1 | 225.00 | 225.00 | 225.00 | 225.00 | - | 225.00 |

TP2 | 0.00 | 1.27 × 10^{−2} | 0.00 | 0.00 | - | 5.00 |

TP3 | −18.94 | −18.68 | −12.68 | −12.68 | −14.78 | −12.68 |

TP4 | −29.35 | −29.16 | −29.20 | −29.20 | −29.21 | −29.20 |

TP5 | −3.90 | −3.59 | −3.92 | −3.92 | - | −3.15 |

TP6 | −1.25 | −1.21 | −1.21 | −1.21 | −1.17 | 3.57 |

TP7 | −1.96 | −1.87 | −1.98 | −1.98 | −1.98 | −1.98 |

TP8 | 8.75 × 10^{−5} | 1.8 × 10^{−3} | 0.00 | 0.00 | - | 5.00 |

TP9 | 8.68 × 10^{−6} | 1.20 × 10^{−5} | - | - | - | - |

TP10 | 1.24 × 10^{−4} | 1.03 × 10^{−4} | - | - | - | - |

**Table 2.**Lower level objective functions by CGA-ES, BLEAQ, NEA, NEA1, PSO-CST and BLEAQ and corresponding reference [36] algorithms for TP1-TP10.

Problem | Lower Level Objective Function | |||||
---|---|---|---|---|---|---|

CGA-BS | BLEAQ [27] | NEA [41] | NEA1 [56] | PSO-CST [36] | Cor. Ref. [36] | |

TP1 | 99.99 | 100.00 | 100.00 | 100.00 | - | 100.00 |

TP2 | 200.00 | 99.99 | 100.000 | 100.00 | - | 0.00 |

TP3 | −1.16 | −1.02 | −1.02 | −1.06 | −0.23 | −1.06 |

TP4 | 3.01 | 3.19 | 3.20 | 3.20 | −0.23 | 3.20 |

TP5 | −2.03 | −1.96 | −2.00 | −2.00 | - | −16.29 |

TP6 | 8.07 | 7.61 | 7.62 | 7.62 | 7.44 | 2.40 |

TP7 | 1.87 | 1.87 | 1.98 | 1.98 | 1.98 | 1.95 |

TP8 | 199.99 | 99.99 | 100.00 | 100.00 | - | 0.00 |

TP9 | 2.72 | 1.00 | - | - | - | - |

TP10 | 2.72 | 1.00 | - | - | - | - |

Problem | Optimal Solutions | CGA-BS Results | ||
---|---|---|---|---|

$\mathit{F}({\mathit{x}}^{*},{\mathit{y}}^{*})$ | $\mathit{f}({\mathit{x}}^{*},{\mathit{y}}^{*})$ | $\mathit{F}({\mathit{x}}^{\u2022},{\mathit{y}}^{\u2022})$ | $\mathit{f}({\mathit{x}}^{\u2022},{\mathit{y}}^{\u2022})$ | |

SMD1 | 0 | 0 | 0 | 0 |

SMD2 | 0 | 0 | 4.7659 × 10^{−}^{6} | 2.2190 × 10^{−}^{6} |

SMD3 | 0 | 0 | 0 | 0 |

SMD4 | 0 | 0 | 5.5692 × 10^{−}^{12} | 3.4094 × 10^{−}^{11} |

SMD5 | 0 | 0 | 1.1324 × 10^{−}^{9} | 1.1324 × 10^{−}^{9} |

SMD6 | 0 | 0 | 9.3428 × 10^{−}^{11} | 9.3428 × 10^{−}^{11} |

Problem | Upper Level Accuracy | |||||||
---|---|---|---|---|---|---|---|---|

CGA-BS | BLMA [57] | NBLEA [57] | BLEAQ [57] | BIDE [57] | FOA [58] | PSO [58] | LGMS-FOA [58] | |

SMD1 | 0 | 1 × 10^{−}^{6} | 5.03 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 3.41 × 10^{−}^{6} | 5.55 × 10^{−}^{10} | 1.79 | 3.14 × 10^{−}^{4} |

SMD2 | 2.22 × 10^{−6} | 1 × 10^{−}^{6} | 3.17 × 10^{−}^{6} | 5.44 × 10^{−}^{6} | 1.29 × 10^{−}^{6} | 8.60 × 10^{−}^{12} | 3.53 | 1.20 × 10^{1} |

SMD3 | 0 | 1 × 10^{−}^{6} | 1.37 × 10^{−}^{6} | 7.55 × 10^{−}^{6} | 4.10 × 10^{−}^{6} | 5.98 × 10^{−}^{10} | 3.05 | 8.58 × 10^{−}^{4} |

SMD4 | 3.41 × 10^{−11} | 1 × 10^{−}^{6} | 9.29 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 2.30 × 10^{−}^{6} | 1.3 × 10^{−}^{13} | 8.09 | 2.13 × 10^{−}^{2} |

SMD5 | 1.13 × 10^{−9} | 1 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 1.58 × 10^{−}^{6} | 9.4 × 10^{−}^{8} | 1.08 × 10^{2} | 1.49 |

SMD6 | 9.34 × 10^{−11} | 1 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 1.00 × 10^{−}^{6} | 3.47 × 10^{−}^{6} | - | - | - |

**Table 5.**CGA-BS results and best obtained solutions in [35] for P1–P24.

Problem | CGA-BS | NEA1 [56] | PSO-CST [36] | |||
---|---|---|---|---|---|---|

$\mathit{F}({\mathit{x}}^{\u2022},{\mathit{y}}^{\u2022})$ | $\mathit{f}({\mathit{x}}^{\u2022},{\mathit{y}}^{\u2022})$ | $\mathit{F}({\mathit{x}}^{\circ},{\mathit{y}}^{\circ})$ | $\mathit{F}({\mathit{x}}^{\circ},{\mathit{y}}^{\circ})$ | $\mathit{F}({\mathit{x}}^{\xb7},{\mathit{y}}^{\xb7})$ | $\mathit{f}({\mathit{x}}^{\xb7},{\mathit{y}}^{\xb7})$ | |

P1 | −18.5999 | −1.3902 | −12.6800 | 1.0160 | −14.7772 | −0.2316 |

P2 | −29.3529 | 3.0083 | −29.2000 | 3.2000 | −29.2064 | 2.3641 |

P3 | −8.9301 | −4.3844 | −8.9200 | −6.1400 | - | - |

P4 | −7.0717 | −0.5088 | −7.5800 | .05740 | - | - |

P5 | −12.2095 | −410.0104 | −11.9990 | 163.4200 | - | - |

P6 | −3.5990 | −1.9998 | −3.6000 | −2.0000 | - | - |

P7 | −3.9014 | −2.0302 | −3.9200 | −2.0000 | - | - |

P8(Max) | 1000.2 | 1.0518 | 1000 | 1 | 640.7139 | 0.9946 |

P9 | 100.0038 | 872.7460 | 100.0001 | 3.500 × 10^{−}^{11} | 100.0393 | 1.3500 × 10^{−}^{16} |

P10 | −1.2520 | 8.0708 | −1.2098 | 3.5700 | −1.1660 | 7.4441 |

P11(Max) | 2.0112 | −2.0112 | 1.9802 | −1.9802 | 1.9816 | −1.9816 |

P12 | −29.9602 | 0.5672 | −29.2000 | 0.3148 | - | - |

P13 | 0 | 200.0000 | 0 | 100.0000 | - | - |

P14 | 0 | 200.0000 | 0 | 100.0000 | 5.27 × 10^{−}^{2} | 1.0000 × 10^{−}^{8} |

P15 | 0 | 200.0000 | 0 | 100.0000 | 4.0000 × 10^{−}^{4} | 1.0000 × 10^{−}^{8} |

P16 | 1.2676 × 10^{−}^{5} | 38.6877 | 8.9900 × 10^{−}^{13} | 0.3148 | 7.5000 × 10^{−}^{3} | 125.08540 |

P17 | 1.9891 × 10^{−}^{11} | 1.3981 × 10^{−}^{10} | 6.2149 × 10^{−}^{4} | 1.0000 | 0.0000 | 84.2367 |

P18 | 6.2780 × 10^{−}^{7} | 23.9314 | 8.5086 × 10^{−}^{9} | 1.0000 | 1.0000 × 10^{−}^{4} | 25.6292 |

P19 | 3.3559 × 10^{−}^{8} | 6.9743 | 2.0325 × 10^{−}^{5} | 1.0000 | 8.2000 × 10^{−}^{3} | 2.5621 |

P20 | 1.6211 × 10^{−}^{9} | 13.8708 | 6.1725 × 10^{−}^{8} | 1.0000 | 3.7400 × 10^{−}^{2} | 2.6969 |

P21 | 3.4837 × 10^{−}^{9} | 7.5534 | 7.2265 × 10^{−}^{3} | 1.0000 | 3.3700 × 10^{−}^{2} | 2.7442 |

P22 | 1.8887 × 10^{−}^{7} | 0.3182 | - | - | - | - |

P23 | 2.6713 × 10^{−}^{6} | 0.2997 | - | - | - | - |

P24 | 1.6865 × 10^{−}^{6} | 0.6745 | - | - | - | - |

Problem | Upper Level F(x,y) | $\mathbf{Lower}\mathbf{Level}\mathit{f}(\mathit{x},\mathit{y})$ | ||
---|---|---|---|---|

CGA | CGA-BS | CGA | CGA-BS | |

TP1 | 229.3517 | 225.0001 | 178.9341 | 99.9999 |

TP2 | 0 | 0 | 200.0000 | 200.0000 |

TP3 | −18.9365 | −18.9365 | −1.1563 | −1.1563 |

TP4 | −29.3529 | −29.3529 | 3.0080 | 3.0080 |

TP5 | −1.0407 | −3.9014 | −1.0083 × 10^{−}^{5} | −2.0300 |

TP6 | −0.9609 | −1.2520 | 8.2634 | 8.0708 |

TP7 | −1.9600 | −1.9600 | 1.9660 | 1.9660 |

TP8 | 8.7468 × 10^{−}^{5} | 8.7468 × 10^{−}^{5} | 199.9989 | 199.9989 |

TP9 | 8.6845 × 10^{−}^{6} | 8.6845 × 10^{−}^{6} | 2.7183 | 2.7183 |

TP10 | 1.2379 × 10^{−}^{4} | 1.2379 × 10^{−}^{4} | 2.7183 | 2.7183 |

Problem | $\mathbf{Upper}\mathbf{Level}\mathit{F}({\mathit{x}}^{},{\mathit{y}}^{})$ | $\mathbf{Lower}\mathbf{Level}\mathit{f}({\mathit{x}}^{},{\mathit{y}}^{})$ | ||
---|---|---|---|---|

GA-BS | CGA-BS | GA-BS | CGA-BS | |

SMD1 | 1.0455 × 10^{−}^{5} | 0 | 3.3729 × 10^{−}^{6} | 0 |

SMD2 | 2.6470 × 10^{−}^{5} | 4.7659 × 10^{−}^{6} | 3.3886 × 10^{−}^{6} | 2.2190 × 10^{−}^{6} |

SMD3 | 3.5227 × 10^{−}^{5} | 0 | 1.2578 × 10^{−}^{5} | 0 |

SMD4 | 1.0295 × 10^{−}^{11} | 5.5692 × 10^{−}^{12} | 3.2578 × 10^{−}^{11} | 3.4094 × 10^{−}^{11} |

SMD5 | 5.5398 × 10^{−}^{6} | 1.1322 × 10^{−}^{9} | 5.5397 × 10^{−}^{6} | 1.1322 × 10^{−}^{9} |

SMD6 | 5.2723 × 10^{−}^{6} | 9.3428 × 10^{−}^{11} | 1.7905 × 10^{−}^{8} | 9.3428 × 10^{−}^{11} |

**Table 8.**Median upper level function evaluations for CGA-BS and four known algorithms for SMD test problem.

Problem | Median Upper Level Function Evaluations | ||||
---|---|---|---|---|---|

CGA-BS | BLMA [57] | NBLE [57] | BLEAQ [57] | BIDE [57] | |

SMD1 | 1.0101 × 10^{4} | 4.1200 × 10^{2} | 1.5200 × 10^{3} | 1.1900 × 10^{3} | 6.0000 × 10^{3} |

SMD2 | 5.0003 × 10^{4} | 4.2400 × 10^{2} | 1.5600 × 10^{3} | 1.2000 × 10^{3} | 6.0000 × 10^{3} |

SMD3 | 1.0002 × 10^{4} | 4.1200 × 10^{2} | 1.5600 × 10^{3} | 1.2900 × 10^{3} | 6.0000 × 10^{3} |

SMD4 | 1.25003 × 10^{5} | 5.5200 × 10^{2} | 1.5300 × 10^{3} | 1.3100 × 10^{3} | 6.0000 × 10^{3} |

SMD5 | 1.00003 × 10^{5} | 5.5200 × 10^{2} | 3.4000 × 10^{3} | 2.0600 × 10^{3} | 6.0000 × 10^{3} |

SMD6 | 1.37503 × 10^{5} | 4.8800 × 10^{2} | 4.0600 × 10^{3} | 4.0800 × 10^{3} | 6.0000 × 10^{3} |

**Table 9.**Median Lower level function evaluations for CGA-BS and four known algorithms for SMD test problem.

Problem | Median Lower Level Function Evaluations | ||||
---|---|---|---|---|---|

CGA-BS [57] | BLMA [57] | NBLE [57] | BLEAQ [57] | BIDE [57] | |

SMD1 | 1.50010 × 10^{4} | 3.05000 × 10^{5} | 9.52000 × 10^{5} | 2.37000 × 10^{5} | 1.80000 × 10^{7} |

SMD2 | 1.50398 × 10^{5} | 3.01000 × 10^{5} | 9.63000 × 10^{5} | 4.08000 × 10^{5} | 1.80000 × 10^{7} |

SMD3 | 2.0000 × 10^{4} | 3.09000 × 10^{5} | 1.04000 × 10^{6} | 3.02000 × 10^{5} | 1.80000 × 10^{7} |

SMD4 | 2.57998 × 10^{5} | 3.29000 × 10^{5} | 8.33000 × 10^{5} | 3.07000 × 10^{5} | 1.80000 × 10^{7} |

SMD5 | 2.5798E × 10^{5} | 3.28000 × 10^{5} | 2.22000 × 10^{6} | 8.42000 × 10^{5} | 1.80000 × 10^{7} |

SMD6 | 3.38598 × 10^{5} | 3.26000 × 10^{5} | 1.11000 × 10^{5} | 1.98000 × 10^{4} | 1.80000 × 10^{7} |

**Table 10.**Median total function evaluations for CGA-BS and four known algorithms for SMD test problem.

Problem | Median Total Function Evaluations | ||||
---|---|---|---|---|---|

CGA-BS | BLMA [57] | NBLEA [57] | BLEAQ [57] | BIDE [57] | |

SMD1 | 2.50010 × 10^{4} | 3.05412 × 10^{5} | 9.53520 × 10^{5} | 2.38190 × 10^{5} | 1.80060 × 10^{7} |

SMD2 | 2.00401 × 10^{5} | 3.01424 × 10^{5} | 9.64560 × 10^{5} | 4.09200 × 10^{5} | 1.80060 × 10^{7} |

SMD3 | 3.00020 × 10^{4} | 3.09412 × 10^{5} | 1.05560 × 10^{6} | 3.03290 × 10^{5} | 1.80060 × 10^{7} |

SMD4 | 3.83001 × 10^{5} | 3.29552 × 10^{5} | 8.34530 × 10^{5} | 3.08310 × 10^{5} | 1.80060 × 10^{7} |

SMD5 | 3.25801 × 10^{5} | 3.28552 × 10^{5} | 2.22540 × 10^{6} | 8.44060 × 10^{5} | 1.80060 × 10^{7} |

SMD6 | 4.76101 × 10^{5} | 3.26488 × 10^{5} | 1.115060 × 10^{5} | 2.38000 × 10^{4} | 1.80060 × 10^{7} |

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**MDPI and ACS Style**

Abo-Elnaga, Y.; Nasr, S.
Modified Evolutionary Algorithm and Chaotic Search for Bilevel Programming Problems. *Symmetry* **2020**, *12*, 767.
https://doi.org/10.3390/sym12050767

**AMA Style**

Abo-Elnaga Y, Nasr S.
Modified Evolutionary Algorithm and Chaotic Search for Bilevel Programming Problems. *Symmetry*. 2020; 12(5):767.
https://doi.org/10.3390/sym12050767

**Chicago/Turabian Style**

Abo-Elnaga, Yousria, and Sarah Nasr.
2020. "Modified Evolutionary Algorithm and Chaotic Search for Bilevel Programming Problems" *Symmetry* 12, no. 5: 767.
https://doi.org/10.3390/sym12050767