# Enhanced Teaching–Learning-Based Optimization Algorithm for the Mobile Robot Path Planning Problem

^{*}

## Abstract

**:**

## 1. Introduction

- For the algorithm deployment, a divide-and-conquer design, coupled with the Dijkstra method, is developed to realize the problem transformation;
- The interpolation method is embedded into the proposed algorithm to smooth the traveling route as well as to reduce the problem dimensionality;
- An opposition-based learning strategy is utilized to modify the algorithm initialization process;
- To balance between exploitation and exploration, a novel, individual update method is established by hybridizing TLBO with DE.

## 2. Problem Formulation

#### 2.1. Problem Description

#### 2.2. Mathematical Model

## 3. TLBO and DE Algorithms

#### 3.1. TLBO

**Step 1.**Initialize the parameter settings, including the population size, $P$, maximum iteration, $G$, problem dimension, $D$, etc.;**Step 2.**Create initial individuals then calculate their objective values;**Step 3.**Employ the teacher phase to update the current population;**Step 4.**Employ the learner phase to update the current population;**Step 5.**If any algorithm termination criterion is satisfied, terminate the search process and output the current best individual; otherwise, go to Step 3.

- (1)
- Initial phase

- (2)
- Teacher phase

- (3)
- Learner phase

#### 3.2. DE

- (1)
- Mutation operation

- (2)
- Crossover operation

- (3)
- Selection operation

## 4. Proposed ETLBO for MRPP

#### 4.1. Divide-and-Conquer Design for MRPP

#### 4.2. Interpolation Application in ETLBO

#### 4.3. Pyramid of ETLBO

#### 4.3.1. Opposition-Based Learning Initialization Method

**Step 1.**Set $i\leftarrow 1$ and $d\leftarrow 1$. Then, go to step 2.**Step 2.**If condition $i\le P$ holds, go to step 3; otherwise, go to step 7;**Step 3.**If condition $d\le D$ holds, go to step 4; otherwise, go to step 5;**Step 4.**Set ${x}_{d}^{p,i}={x}_{d}^{l}+rand(0,1)\cdot ({x}_{d}^{u}-{x}_{d}^{l})$ and ${x}_{d}^{o,i}={x}_{d}^{l}+{x}_{d}^{u}-{x}_{d}^{p,i}$. Then, go to step 5;**Step 5.**Set $d\leftarrow d+1$, if condition $d>D$ holds, go to step 6; otherwise, go to step 3;**Step 6.**Set $i\leftarrow i+1$; then, go to step 2;**Step 7.**Select $P$ best individuals from $\{{x}^{p,1},\cdots ,{x}^{p,P},{x}^{o,1},\cdots ,{x}^{o,P}\}$ to form the initial population.

#### 4.3.2. Hybrid Offspring Generation Method

- (1)
- Teacher phase

- (2)
- Learner phase

#### 4.4. Framework of Proposed Methodology for MRPP

## 5. Experimental Studies

#### 5.1. Benchmark Function Problem

#### 5.1.1. Preliminaries

**A.****Unimodal functions**

**B.****Multimodal functions**

**C.****Evaluation metrics**

**D.****Non-parametric tests**

#### 5.1.2. Results and Discussion

**A.****Results of unimodal functions**

- ETLBO obtained the optimum for functions F2, F4, F6, and F9-F12 in all 50 runs; thus, the SD values were zero. In terms of functions F1, F3, and F7, ETLBO obtained near-optimal solutions and exceeded all other algorithms;
- For function 5, the optimum was obtained by PSO and DE. ETLBO performed third with respect to the previous two algorithms, and obtained a higher accuracy than the other three compared algorithms;
- For function F8, the DE algorithm yielded the best solution performance, followed by PSO. GA performed third with respect to the previous algorithms. ETLBO obtained more accurate solutions than TLBO and TLBO(S) in terms of the average value metric.

**B.****Results of multimodal functions**

- ETLBO obtained the optimum of functions F13 and F16 in all 50 independent runs; the resulting SD values were zero;
- For functions F15 and F19, ETLBO obtained near-optimal solutions and performed better than all compared algorithms. Regarding function F19, the result provided by ETLBO was a close approximation to the optimum and the corresponding SD value was very close to zero;
- The six algorithms all performed very well for function 14, and no significant difference was found regarding stability;
- For functions 18 and 20, TLBO(S) and DE yielded the best solution performances of all compared algorithms, respectively. Despite the fact that ETLBO failed to obtain the best solution performance, its metrics were acceptable in a relative manner since those values were located in an acceptable range;
- For function 17, ETLBO failed to obtain a good solution in 50 independent runs. The resulting metrics of ETLBO were higher when compared to the simulation results obtained by the compared algorithms, to a certain extent.

**C.****Nonparametric test results**

#### 5.2. MRPP Problem

#### 5.2.1. Preliminaries

**A.****Instance generation**

**B.****Evaluation metrics**

#### 5.2.2. Results and Discussion

**A.****Presentation of best routes**

- For instance 20 × 20, it can be seen from Figure 6 that ETLBO yielded the best solution performance of the three test algorithms, with the best route length of 151.92 (Figure 6d). ACO(G) performed second to ETLBO, with the best route length of 154.85 (Figure 6c). ACO(L) showed the weakest performance in the best route length, with an objective value of 156.07 (Figure 6b);
- For the 30 × 30 instance, ETLBO obtained more accurate solutions than ACO(L) and ACO(G) in terms of the best route length metric. The objective value of the proposed ETLBO amounted to 225.56 (see Figure 7d), which is superior to 232.63 (given by ACO(L) in Figure 7b) and 231.42 (provided by ACO(G) in Figure 7c);
- Similar conclusions can be drawn from Figure 8 and Figure 9, in which ETLBO yielded the best solution performance. The corresponding objective values for instances 40 × 40 and 50 × 50 were 291.27 (see Figure 8d) and 378.70 (see Figure 9d), respectively. ACO(G) performed second to ETLBO, with metric values of 291.27 (see Figure 8c) and 378.70 (see Figure 9c). ACO(L) ranked at the bottom, with an objective value of 302.99 (see Figure 8b) and 400.42 (see Figure 9b).
- The convergence curves illustrated in four figures (Figure 6a, Figure 7a, Figure 8a and Figure 9a) indicate that the proposed ETLBO not only had a faster convergence in solving all test instances when compared to the state-of-art ACO(L) and ACO(G) but the final convergence accuracy of the proposed approach was better than that of the compared algorithms.

**B.****Statistical comparison**

- In terms of the optimum, ETLBO was able to find the best solutions for all considered problem instances of different problem scales. However, ACO(L) and ACO(G) failed to obtain the best or near-optimal solutions especially for large-scale problem instances. In this regard, the values of the best deviation metric amounted to 5.73% (given by ACO(L)) and 4.51% (given by ACO(G));
- ETLBO yielded the best solution performance when it was applied to address different problem instances in terms of best-, worst-, and average-related metrics. ACO(G) performed second to ETLBO, while ACO(L) showed the weakest solution performance;
- The average metrics versus the problem size indicated that the increasing problem dimensionality raised the optimization difficulty. In this regard, the average deviations of ACO(L), ACO(G), and ETLBO for the 20 × 20 instance were 5.42%, 4.52%, and 3.10%, respectively. For the large-scale instance with a 50 × 50 size, the average deviations amounted to 8.81%, 8.20%, and 5.17%, respectively;
- Despite the diversity of problem instances, it is evident that all generated values from the proposed ETLBO were more acceptable when compared to the two other algorithms compared.

- For the 50 × 50 instance, the associated rank sum of ETLBO was 1275, and the resulting $p$ value was very close to zero. As is stated in previous subsections, the significance level was fixed at 0.05 for this study. Therefore, it can be deduced that ETLBO dominates the ACO(L) in all executions when they are adopted to this instance.
- For the instances 20 × 20, 30 × 30, and 40 × 40, the rank sum of ETLBO lay between the maximum and minimum rank threshold values, indicating that ETLBO and ACO(L) produced a mixed ranking. Nonetheless, the associated $p$ values were less than the predefined significance level. In other words, ETLBO performs equally or even better when compared to ACO(L).
- Despite the increasing problem dimensionality raising the optimization difficulty, the resulting $p$ values decreased with the enlargement of the problem scale. Thus, it can be deduced that ETLBO is more competitive than ACO(L) for a real-world-scale problem.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Label | Name | Formula | Range | Dimension | Optimum |
---|---|---|---|---|---|

F1 | Sphere | ${f}_{1}(x)={\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}$ | [−100, 100] | 100 | 0 |

F2 | Step | ${f}_{6}(x)={\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | [−100, 100] | 30 | 0 |

F3 | Quartic | ${f}_{7}(x)={\displaystyle \sum _{i=1}^{n}i{x}_{i}{}^{4}+rand\left(0,1\right)}$ | [−1.28, 1.28] | 30 | 0 |

F4 | Schwefel 2.21 | ${f}_{4}(x)=\mathrm{max}{x}_{i}\left\{\left|{x}_{i}\right|\right.,\left.1\le i\le n\right\}$ | [−100, 100] | 60 | 0 |

F5 | Booth | ${f}_{5}(x)={({x}_{1}+2{x}_{2}-7)}^{2}+{(2{x}_{1}+{x}_{2}-5)}^{2}$ | [−10, 10] | 2 | 0 |

F6 | Schwefel 2.22 | ${f}_{2}(x)={\displaystyle \sum _{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{n}\left|{x}_{i}\right|}$ | [−10, 10] | 40 | 0 |

F7 | Schwefel 1.2 | ${f}_{3}(x)={{\displaystyle \sum _{i=1}^{n}\left({\displaystyle \sum _{j-1}^{i}{x}_{j}}\right)}}^{2}$ | [−100, 100] | 50 | 0 |

F8 | Easom | ${f}_{8}(x)=-\mathrm{cos}({x}_{1})\mathrm{cos}({x}_{2})\mathrm{exp}(-{({x}_{1}-\pi )}^{2}-{({x}_{2}-\pi )}^{2})$ | [−100, 100] | 2 | −1 |

F9 | Bohachevsky | ${f}_{14}(x)={x}_{1}^{2}+2{x}_{2}^{2}-0.3\mathrm{cos}(3\pi {x}_{1})-0.4\mathrm{cos}(4\pi {x}_{2})+0.7$ | [−100, 100] | 2 | 0 |

F10 | Matyas | ${f}_{17}(x)=0.26\left({x}_{1}^{2}+{x}_{2}^{2}\right)-0.48{x}_{1}{x}_{2}$ | [−10, 10] | 2 | 0 |

F11 | Three-hump camel | ${f}_{18}(x)=2{x}^{2}-1.05{x}^{4}+\frac{{x}^{6}}{6}+xy+{y}^{2}$ | [−5, 5] | 2 | 0 |

F12 | Schaffer N.2 | ${f}_{20}(x)=0.5+\frac{{\mathrm{sin}}^{2}\left({x}^{2}-{y}^{2}\right)-0.5}{{\left[1+0.001\left({x}^{2}+{y}^{2}\right)\right]}^{2}}$ | [−100, 100] | 2 | 0 |

Label | Name | Formula | Range | Dimension | Optimum |
---|---|---|---|---|---|

F13 | Rastrigin | ${f}_{9}(x)={\displaystyle \sum _{i=1}^{n}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]}$ | [−5.12, 5.12] | 20 | 0 |

F14 | Branin | ${f}_{15}(x)={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{8}{\pi}\right)\mathrm{cos}{x}_{1}+10$ | [−5, 15] | 2 | 0.398 |

F15 | Ackley | ${f}_{10}(x)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)+20+e$ | [−32.32] | 30 | 0 |

F16 | Griewank | ${f}_{11}(x)=\frac{1}{4000}{\displaystyle \sum _{i=1}^{n}{x}_{i}^{2}}-{\displaystyle \prod _{i=1}^{n}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}+1$ | [−600, 600] | 30 | 0 |

F17 | Penalized | $\begin{array}{l}{f}_{12}(x)=\frac{\pi}{n}\left\{10{\mathrm{sin}}^{2}\left(\pi {y}_{i}\right)+{\displaystyle \sum _{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)+{\left({y}_{i}-1\right)}^{2}\right]}\right\}+{\displaystyle \sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)}\\ {y}_{i}=1+\frac{{x}_{i}+1}{4}\\ u\left({x}_{i}.a,k,m\right)=\left\{\begin{array}{l}k{\left({x}_{i}-a\right)}^{m};{x}_{i}>a\\ 0;-a<{x}_{i}<a\\ k{\left(-{x}_{i}-a\right)}^{m};{x}_{i}<-a\end{array}\right.\\ \end{array}$ | [−50, 50] | 50 | 0 |

F18 | Penalized2 | $\begin{array}{l}{f}_{13}(x)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum _{i=1}^{n-1}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i-1}\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{n}\right)\right]}\right\}+{\displaystyle \sum _{i=1}^{n}u\left({x}_{i},10,100,4\right)}\\ u\left({x}_{i}.a,k,m\right)=\left\{\begin{array}{l}k{\left({x}_{i}-a\right)}^{m};{x}_{i}>a\\ 0;-a<{x}_{i}<a\\ k{\left(-{x}_{i}-a\right)}^{m};{x}_{i}<-a\end{array}\right.\\ \end{array}$ | [−50, 50] | 30 | 0 |

F19 | Goldstein-Price | ${f}_{16}(x)=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]\ast \left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)\right]$ | [−2, 2] | 2 | 3 |

F20 | Levi | ${f}_{19}(x)={\mathrm{sin}}^{2}\left(3\pi x\right)+{\left(x-1\right)}^{2}\left(1+{\mathrm{sin}}^{2}\left(3\pi y\right)\right)+{\left(y-1\right)}^{2}\left(1+{\mathrm{sin}}^{2}\left(2\pi y\right)\right)$ | [−10, 10] | 2 | 0 |

Algorithm | Metric | ETLBO | TLBO | TLBO(S) | PSO | DE | GA |
---|---|---|---|---|---|---|---|

F1 | Best | 1.41 × 10^{−134} | 1.11 × 10^{−11} | 7.31 × 10^{−20} | 1.44 × 10^{01} | 4.61 × 10^{00} | 4.07 × 10^{02} |

Worst | 3.35 × 10^{−129} | 1.47 × 10^{−11} | 2.81 × 10^{−11} | 1.59 × 10^{02} | 6.83 × 10^{00} | 9.34 × 10^{02} | |

Median | 1.92 × 10^{−132} | 1.40 × 10^{−11} | 1.72 × 10^{−14} | 1.59 × 10^{01} | 5.73 × 10^{00} | 5.59 × 10^{02} | |

Average | 1.19 × 10^{−130} | 1.32 × 10^{−11} | 1.16 × 10^{−12} | 2.79 × 10^{01} | 5.72 × 10^{00} | 5.65 × 10^{02} | |

SD | 5.04 × 10^{−130} | 8.61 × 10^{−13} | 4.21 × 10^{−12} | 3.36 × 10^{01} | 5.11 × 10^{−01} | 9.93 × 10^{01} | |

F2 | Best | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 2.70 × 10^{−23} | 0.00 × 10^{00} | 0.00 × 10^{00} |

Worst | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 4.46 × 10^{−15} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

Median | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 1.90 × 10^{−20} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

Average | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 9.13 × 10^{−17} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

SD | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 6.23 × 10^{−10} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

F3 | Best | 1.29 × 10^{−07} | 4.80 × 10^{−01} | 7.70 × 10^{−05} | 2.45 × 10^{−02} | 2.50 × 10^{−02} | 1.43 × 10^{−02} |

Worst | 1.75 × 10^{−04} | 6.76 × 10^{−01} | 1.01 × 10^{−02} | 5.40 × 10^{−02} | 8.61 × 10^{−02} | 1.13 × 10^{−01} | |

Median | 2.95 × 10^{−05} | 6.17 × 10^{−01} | 1.41 × 10^{−03} | 5.23 × 10^{−05} | 5.20 × 10^{−02} | 3.90 × 10^{−02} | |

Average | 4.40 × 10^{−05} | 6.12 × 10^{−01} | 2.15 × 10^{−03} | 8.61 × 10^{−02} | 5.54 × 10^{−02} | 4.22 × 10^{−02} | |

SD | 3.85 × 10^{−05} | 3.66 × 10^{−02} | 1.86 × 10^{−03} | 1.30 × 10^{−02} | 1.48 × 10^{−02} | 1.84 × 10^{−02} | |

F4 | Best | 0.00 × 10^{00} | 4.72 × 10^{−09} | 1.00 × 10^{−16} | 2.19 × 10^{−01} | 1.00 × 10^{01} | 1.18 × 10^{01} |

Worst | 0.00 × 10^{00} | 6.11 × 10^{−09} | 2.62 × 10^{−13} | 1.56 × 10^{00} | 1.61 × 10^{01} | 3.07 × 10^{01} | |

Median | 0.00 × 10^{00} | 5.37 × 10^{−09} | 1.60 × 10^{−14} | 7.17 × 10^{−01} | 1.38 × 10^{01} | 1.88 × 10^{01} | |

Average | 0.00 × 10^{00} | 5.35 × 10^{−09} | 3.57 × 10^{−14} | 8.01 × 10^{−01} | 1.35 × 10^{01} | 1.97 × 10^{01} | |

SD | 0.00 × 10^{00} | 3.27 × 10^{−10} | 5.11 × 10^{−14} | 3.82 × 10^{−01} | 1.38 × 10^{00} | 3.76 × 10^{00} | |

F5 | Best | 0.00 × 10^{00} | 2.00 × 10^{00} | 8.37 × 10^{−08} | 0.00 × 10^{00} | 0.00 × 10^{00} | 2.07 × 10^{−03} |

Worst | 2.15 × 10^{−04} | 2.00 × 10^{00} | 2.30 × 10^{−01} | 0.00 × 10^{00} | 0.00 × 10^{00} | 1.91 × 10^{00} | |

Median | 3.74 × 10^{−16} | 2.00 × 10^{00} | 1.64 × 10^{−02} | 0.00 × 10^{00} | 0.00 × 10^{00} | 6.00 × 10^{−01} | |

Average | 8.05 × 10^{−06} | 2.00 × 10^{00} | 2.30 × 10^{−02} | 0.00 × 10^{00} | 0.00 × 10^{00} | 9.45 × 10^{−01} | |

SD | 3.00 × 10^{−05} | 4.35 × 10^{−09} | 4.97 × 10^{−02} | 0.00 × 10^{00} | 0.00 × 10^{00} | 8.29 × 10^{−01} | |

F6 | Best | 0.00 × 10^{00} | 9.76 × 10^{−03} | 1.31 × 10^{−08} | 5.01 × 10^{−06} | 1.72 × 10^{−03} | 1.39 × 10^{−02} |

Worst | 0.00 × 10^{00} | 3.79 × 10^{−01} | 1.19 × 10^{−05} | 1.36 × 10^{−01} | 3.69 × 10^{−03} | 9.86 × 10^{−02} | |

Median | 0.00 × 10^{00} | 1.35 × 10^{−01} | 6.33 × 10^{−07} | 4.43 × 10^{−04} | 2.57 × 10^{−03} | 3.73 × 10^{−02} | |

Average | 0.00 × 10^{00} | 1.60 × 10^{−01} | 2.01 × 10^{−06} | 4.13 × 10^{−03} | 2.69 × 10^{−03} | 4.04 × 10^{−02} | |

SD | 0.00 × 10^{00} | 8.19 × 10^{−02} | 2.71 × 10^{−06} | 1.81 × 10^{−02} | 4.34 × 10^{−04} | 1.72 × 10^{−02} | |

F7 | Best | 2.69 × 10^{−170} | 2.50 × 10^{−19} | 1.93 × 10^{−11} | 1.27 × 10^{03} | 7.34 × 10^{04} | 5.14 × 10^{03} |

Worst | 1.65 × 10^{−155} | 2.12 × 10^{−10} | 8.55 × 10^{−21} | 9.14 × 10^{03} | 1.25 × 10^{05} | 2.20 × 10^{04} | |

Median | 4.25 × 10^{−163} | 2.13 × 10^{−14} | 1.78 × 10^{−11} | 3.99 × 10^{03} | 1.08 × 10^{05} | 9.40 × 10^{03} | |

Average | 3.29 × 10^{−157} | 6.85 × 10^{−12} | 9.30 × 10^{−11} | 4.43 × 10^{03} | 1.03 × 10^{05} | 1.05 × 10^{04} | |

SD | 2.17 × 10^{−156} | 3.24 × 10^{−11} | 9.32 × 10^{−11} | 1.79 × 10^{03} | 1.24 × 10^{04} | 3.66 × 10^{03} | |

F8 | Best | −9.99 × 10^{−01} | −3.07 × 10^{−01} | 0.00 × 10^{00} | −9.99 × 10^{−01} | −1.00 × 10^{00} | −1.00 × 10^{00} |

Worst | −4.71 × 10^{−07} | −2.20 × 10^{−01} | 0.00 × 10^{00} | −2.46 × 10^{−08} | −1.00 × 10^{00} | −4.27 × 10^{−01} | |

Median | −8.43 × 10^{−01} | −2.48 × 10^{−01} | 0.00 × 10^{00} | −9.86 × 10^{−06} | −1.00 × 10^{00} | −1.00 × 10^{00} | |

Average | −4.79 × 10^{−01} | −2.51 × 10^{−01} | 0.00 × 10^{00} | −1.83 × 10^{−01} | −1.00 × 10^{00} | −9.88 × 10^{−01} | |

SD | 4.80 × 10^{−01} | 1.69 × 10^{−02} | 0.00 × 10^{00} | 3.77 × 10^{−01} | 0.00 × 10^{00} | 8.26 × 10^{−02} | |

F9 | Best | 0.00 × 10^{00} | 4.15 × 10^{−01} | 4.86 × 10^{−13} | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} |

Worst | 0.00 × 10^{00} | 4.22 × 10^{−01} | 1.10 × 10^{−07} | 0.00 × 10^{00} | 0.00 × 10^{00} | 4.80 × 10^{−01} | |

Median | 0.00 × 10^{00} | 6.08 × 10^{−01} | 1.45 × 10^{−09} | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

Average | 0.00 × 10^{00} | 4.13 × 10^{−01} | 7.18 × 10^{−09} | 0.00 × 10^{00} | 0.00 × 10^{00} | 9.38 × 10^{−03} | |

SD | 0.00 × 10^{00} | 2.82 × 10^{−02} | 1.80 × 10^{−08} | 0.00 × 10^{00} | 0.00 × 10^{00} | 6.72 × 10^{−02} | |

F10 | Best | 0.00 × 10^{00} | 1.64 × 10^{−10} | 8.19 × 10^{−22} | 3.09 × 10^{−135} | 3.95 × 10^{−101} | 1.24 × 10^{−52} |

Worst | 0.00 × 10^{00} | 1.08 × 10^{−03} | 3.72 × 10^{−13} | 8.89 × 10^{−109} | 1.36 × 10^{−93} | 1.18 × 10^{−04} | |

Median | 0.00 × 10^{00} | 3.20 × 10^{−04} | 9.11 × 10^{−15} | 1.23 × 10^{−118} | 1.24 × 10^{−97} | 8.97 × 10^{−11} | |

Average | 0.00 × 10^{00} | 3.75 × 10^{−04} | 3.75 × 10^{−14} | 2.00 × 10^{−110} | 4.04 × 10^{−95} | 5.00 × 10^{−06} | |

SD | 0.00 × 10^{00} | 3.01 × 10^{−04} | 6.93 × 10^{−14} | 1.27 × 10^{−109} | 2.05 × 10^{−94} | 1.95 × 10^{−05} | |

F11 | Best | 0.00 × 10^{00} | 3.83 × 10^{−09} | 7.28 × 10^{−19} | 3.70 × 10^{−245} | 4.13 × 10^{−101} | 3.20 × 10^{−78} |

Worst | 0.00 × 10^{00} | 9.47 × 10^{−04} | 3.08 × 10^{−12} | 5.38 × 10^{−232} | 1.40 × 10^{−93} | 8.07 × 10^{−06} | |

Median | 0.00 × 10^{00} | 7.89 × 10^{−05} | 4.01 × 10^{−14} | 2.71 × 10^{−240} | 1.26 × 10^{−97} | 1.37 × 10^{−21} | |

Average | 0.00 × 10^{00} | 2.17 × 10^{−04} | 3.06 × 10^{−13} | 1.13 × 10^{−233} | 4.01 × 10^{−95} | 1.67 × 10^{−07} | |

SD | 0.00 × 10^{00} | 2.51 × 10^{−04} | 6.46 × 10^{−13} | 0.00 × 10^{00} | 2.02 × 10^{−94} | 1.17 × 10^{−06} | |

F12 | Best | 0.00 × 10^{00} | 9.30 × 10^{−08} | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} |

Worst | 0.00 × 10^{00} | 3.44 × 10^{−01} | 2.01 × 10^{−13} | 0.00 × 10^{00} | 0.00 × 10^{00} | 2.99 × 10^{−03} | |

Median | 0.00 × 10^{00} | 1.07 × 10^{−01} | 6.55 × 10^{−15} | 0.00 × 10^{00} | 0.00 × 10^{00} | 0.00 × 10^{00} | |

Average | 0.00 × 10^{00} | 1.19 × 10^{−01} | 2.01 × 10^{−14} | 0.00 × 10^{00} | 0.00 × 10^{00} | 1.44 × 10^{−04} | |

SD | 0.00 × 10^{00} | 8.05 × 10^{−02} | 3.83 × 10^{−14} | 0.00 × 10^{00} | 0.00 × 10^{00} | 5.66 × 10^{−04} |

Function | Statistic | ETLBO | TLBO | TLBO(S) | PSO | DE | GA |
---|---|---|---|---|---|---|---|

F13 | Best | 0.00 × 10^{00} | 8.10 × 10^{−05} | 2.58 × 10^{−04} | 3.53 × 10^{−15} | 1.91 × 10^{−04} | 0.00 × 10^{00} |

Worst | 0.00 × 10^{00} | 5.01 × 10^{−02} | 5.46 × 10^{−02} | 2.22 × 10^{−03} | 2.35 × 10^{−02} | 2.40 × 10^{−05} | |

Median | 0.00 × 10^{00} | 8.70 × 10^{−03} | 6.52 × 10^{−03} | 1.27 × 10^{−06} | 3.67 × 10^{−03} | 0.00 × 10^{00} | |

Average | 0.00 × 10^{00} | 1.16 × 10^{−02} | 1.37 × 10^{−02} | 1.11 × 10^{−04} | 5.68 × 10^{−03} | 1.45 × 10^{−06} | |

SD | 0.00 × 10^{00} | 1.05 × 10^{−02} | 1.30 × 10^{−02} | 3.62 × 10^{−04} | 5.50 × 10^{−03} | 4.87 × 10^{−06} | |

F14 | Best | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} |

Worst | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | |

Median | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | |

Average | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | 3.98 × 10^{−01} | |

SD | 4.84 × 10^{−08} | 9.98 × 10^{−09} | 4.02 × 10^{−07} | 8.54 × 10^{−07} | 3.34 × 10^{−10} | 8.46 × 10^{−08} | |

F15 | Best | 8.83 × 10^{−16} | 2.04 × 10^{01} | 3.49 × 10^{−05} | 1.51 × 10^{−12} | 3.68 × 10^{−03} | 4.11 × 10^{00} |

Worst | 8.83 × 10^{−16} | 2.00 × 10^{01} | 3.63 × 10^{00} | 3.19 × 10^{00} | 1.01 × 10^{−02} | 9.22 × 10^{00} | |

Median | 8.83 × 10^{−16} | 2.04 × 10^{01} | 4.30 × 10^{−03} | 1.15 × 10^{00} | 6.10 × 10^{−03} | 7.36 × 10^{00} | |

Average | 8.83 × 10^{−16} | 2.00 × 10^{01} | 1.51 × 10^{00} | 1.00 × 10^{00} | 6.34 × 10^{−03} | 7.13 × 10^{00} | |

SD | 0.00 × 10^{00} | 2.14 × 10^{−01} | 1.72 × 10^{00} | 8.29 × 10^{−01} | 1.19 × 10^{−02} | 1.14 × 10^{00} | |

F16 | Best | 0.00 × 10^{00} | 1.00 × 10^{00} | 3.92 × 10^{−11} | 3.22 × 10^{−02} | 3.50 × 10^{00} | 3.08 × 10^{00} |

Worst | 0.00 × 10^{00} | 1.03 × 10^{00} | 2.64 × 10^{−07} | 2.23 × 10^{00} | 5.89 × 10^{00} | 2.06 × 10^{01} | |

Median | 0.00 × 10^{00} | 1.02 × 10^{00} | 6.76 × 10^{−09} | 6.09 × 10^{−01} | 4.33 × 10^{00} | 9.51 × 10^{00} | |

Average | 0.00 × 10^{00} | 1.02 × 10^{00} | 2.22 × 10^{−08} | 6.25 × 10^{−01} | 4.37 × 10^{00} | 1.05 × 10^{01} | |

SD | 0.00 × 10^{00} | 6.00 × 10^{−03} | 4.38 × 10^{−08} | 4.26 × 10^{−01} | 5.12 × 10^{−01} | 3.96 × 10^{00} | |

F17 | Best | 9.31 × 10^{−01} | 2.00 × 10^{−01} | 1.43 × 10^{−09} | 2.31 × 10^{−01} | 1.09 × 10^{00} | 5.49 × 10^{00} |

Worst | 1.34 × 10^{00} | 7.92 × 10^{−01} | 1.77 × 10^{−05} | 1.20 × 10^{01} | 2.91 × 10^{00} | 6.70 × 10^{02} | |

Median | 1.06 × 10^{00} | 5.14 × 10^{−01} | 8.22 × 10^{−07} | 3.15 × 10^{00} | 1.68 × 10^{00} | 1.25 × 10^{01} | |

Average | 1.11 × 10^{00} | 5.23 × 10^{−01} | 2.16 × 10^{−06} | 3.61 × 10^{00} | 1.73 × 10^{00} | 3.54 × 10^{01} | |

SD | 1.00 × 10^{−01} | 1.35 × 10^{−01} | 3.23 × 10^{−06} | 2.30 × 10^{00} | 4.33 × 10^{−01} | 1.08 × 10^{02} | |

F18 | Best | 2.59 × 10^{00} | 4.46 × 10^{01} | 7.64 × 10^{−11} | 3.44 × 10^{01} | 4.22 × 10^{01} | 1.91 × 10^{01} |

Worst | 2.98 × 10^{00} | 1.16 × 10^{01} | 1.62 × 10^{−05} | 1.17 × 10^{02} | 1.03 × 10^{02} | 1.24 × 10^{04} | |

Median | 2.98 × 10^{00} | 2.03 × 10^{01} | 1.51 × 10^{−07} | 5.93 × 10^{01} | 5.71 × 10^{01} | 5.52 × 10^{01} | |

Average | 2.75 × 10^{00} | 1.23 × 10^{01} | 1.53 × 10^{−06} | 6.09 × 10^{01} | 5.77 × 10^{01} | 9.39 × 10^{02} | |

SD | 4.42 × 10^{−02} | 3.89 × 10^{00} | 3.48 × 10^{−06} | 1.86 × 10^{01} | 1.12 × 10^{01} | 2.61 × 10^{03} | |

F19 | Best | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.28 × 10^{01} | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.00 × 10^{00} |

Worst | 3.00 × 10^{00} | 1.66 × 10^{01} | 3.27 × 10^{01} | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.00 × 10^{00} | |

Median | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.30 × 10^{01} | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.00 × 10^{00} | |

Average | 3.00 × 10^{00} | 3.30 × 10^{00} | 3.24 × 10^{01} | 3.00 × 10^{00} | 3.00 × 10^{00} | 3.00 × 10^{00} | |

SD | 1.41 × 10^{−14} | 1.95 × 10^{00} | 7.04 × 10^{−08} | 1.24 × 10^{−03} | 4.24 × 10^{−08} | 3.10 × 10^{−05} | |

F20 | Best | 3.02 × 10^{−01} | 1.67 × 10^{−03} | 1.24 × 10^{−03} | 1.33 × 10^{−31} | 1.30 × 10^{−31} | 1.35 × 10^{−31} |

Worst | 3.02 × 10^{−01} | 1.14 × 10^{01} | 1.00 × 10^{00} | 1.38 × 10^{−31} | 1.31 × 10^{−31} | 2.26 × 10^{−07} | |

Median | 3.02 × 10^{−01} | 1.52 × 10^{00} | 2.03 × 10^{−01} | 1.33 × 10^{−31} | 1.36 × 10^{−31} | 1.35 × 10^{−31} | |

Average | 3.02 × 10^{−01} | 2.25 × 10^{00} | 5.28 × 10^{−01} | 1.32 × 10^{−31} | 1.40 × 10^{−31} | 8.01 × 10^{−06} | |

SD | 4.20 × 10^{−05} | 2.43 × 10^{00} | 4.86 × 10^{−01} | 1.82 × 10^{−40} | 1.80 × 10^{−46} | 1.26 × 10^{−06} |

Function | TLBO | TLBO(S) | PSO | DE | GA | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | ${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | ${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | ${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | ${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | |

F1 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F2 | 2525 | 5.00 × 10^{−01} | = | 2525 | 5.00 × 10^{−01} | = | 1275 | 3.42 × 10^{−18} | √ | 2525 | 5.00 × 10^{−01} | = | 2525 | 5.00 × 10^{−01} | = |

F3 | 1275 | 3.42 × 10^{−18} | √ | 1342 | 1.74 × 10^{−16} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F4 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F5 | 1275 | 3.42 × 10^{−18} | √ | 1399 | 4.16 × 10^{−15} | √ | 3775 | 1.00 × 10^{00} | × | 3775 | 1.00 × 10^{00} | × | 1275 | 3.42 × 10^{−18} | √ |

F6 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F7 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F8 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 3580 | 1.00 × 10^{00} | × | 3775 | 1.00 × 10^{00} | × | 3731 | 1.00 × 10^{00} | × |

F9 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 2525 | 5.00 × 10^{−01} | = | 2525 | 5.00 × 10^{−01} | = | 1853 | 1.80 × 10^{−06} | √ |

F10 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F11 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F12 | 1275 | 3.42 × 10^{−18} | √ | 1399 | 4.16 × 10^{−15} | √ | 2525 | 5.00 × 10^{−01} | = | 2525 | 5.00 × 10^{−01} | = | 1300 | 1.52 × 10^{−17} | √ |

F13 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1342 | 1.74 × 10^{−16} | √ |

F14 | 2150 | 4.86 × 10^{−03} | √ | 2525 | 5.00 × 10^{−01} | = | 3580 | 1.00 × 10^{00} | × | 3580 | 1.00 × 10^{00} | × | 2525 | 5.00 × 10^{−01} | = |

F15 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F16 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F17 | 3775 | 1.00 × 10^{00} | × | 3775 | 1.00 × 10^{00} | × | 1300 | 1.52 × 10^{−17} | √ | 1399 | 4.16 × 10^{−15} | √ | 1275 | 3.42 × 10^{−18} | √ |

F18 | 1275 | 3.42 × 10^{−18} | √ | 3775 | 1.00 × 10^{00} | × | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ |

F19 | 1438 | 3.35 × 10^{−14} | √ | 1275 | 3.42 × 10^{−18} | √ | 1853 | 1.80 × 10^{−06} | √ | 3731 | 1.00 × 10^{00} | × | 3769 | 1.00 × 10^{00} | × |

F20 | 1275 | 3.42 × 10^{−18} | √ | 1275 | 3.42 × 10^{−18} | √ | 3775 | 1.00 × 10^{00} | × | 3775 | 1.00 × 10^{00} | × | 3775 | 1.00 × 10^{00} | × |

Instance | Algorithm | Best | Average | Worst | |||
---|---|---|---|---|---|---|---|

Length | Deviation (%) | Length | Deviation (%) | Length | Deviation (%) | ||

20 × 20 | ACO(L) | 156.07 | 2.73% | 160.16 | 5.42% | 163.64 | 7.71% |

ACO(G) | 154.85 | 1.93% | 158.78 | 4.52% | 161.92 | 6.58% | |

ETLBO | 151.92 | 0.00% | 156.64 | 3.10% | 160.71 | 5.78% | |

30 × 30 | ACO(L) | 232.63 | 3.13% | 240.06 | 6.43% | 245.56 | 8.87% |

ACO(G) | 231.42 | 2.60% | 238.56 | 5.76% | 244.35 | 8.33% | |

ETLBO | 225.56 | 0.00% | 233.78 | 3.64% | 240.21 | 6.49% | |

40 × 40 | ACO(L) | 302.99 | 4.02% | 313.52 | 7.64% | 322.99 | 10.89% |

ACO(G) | 301.27 | 3.43% | 309.56 | 6.28% | 318.85 | 9.47% | |

ETLBO | 291.27 | 0.00% | 303.87 | 4.33% | 312.49 | 7.28% | |

50 × 50 | ACO(L) | 400.42 | 5.73% | 412.07 | 8.81% | 421.13 | 11.20% |

ACO(G) | 395.77 | 4.51% | 409.77 | 8.20% | 418.70 | 10.56% | |

ETLBO | 378.70 | 0.00% | 398.27 | 5.17% | 410.42 | 8.37% |

Instance | ACO(L) | ACO(G) | ||||
---|---|---|---|---|---|---|

${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | ${\mathit{w}}_{\mathit{E}\mathit{T}\mathit{L}\mathit{B}\mathit{O}}$ | $\mathit{p}$ | Win | |

20 × 20 | 1590 | 5.75 × 10^{−11} | √ | 1631 | 3.56 × 10^{−10} | √ |

30 × 30 | 1399 | 4.16 × 10^{−15} | √ | 1575 | 2.89 × 10^{−11} | √ |

40 × 40 | 1300 | 1.52 × 10^{−17} | √ | 1438 | 3.35 × 10^{−14} | √ |

50 × 50 | 1275 | 3.42 × 10^{−18} | √ | 1342 | 1.74 × 10^{−16} | √ |

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## Share and Cite

**MDPI and ACS Style**

Lu, S.; Liu, D.; Li, D.; Shao, X.
Enhanced Teaching–Learning-Based Optimization Algorithm for the Mobile Robot Path Planning Problem. *Appl. Sci.* **2023**, *13*, 2291.
https://doi.org/10.3390/app13042291

**AMA Style**

Lu S, Liu D, Li D, Shao X.
Enhanced Teaching–Learning-Based Optimization Algorithm for the Mobile Robot Path Planning Problem. *Applied Sciences*. 2023; 13(4):2291.
https://doi.org/10.3390/app13042291

**Chicago/Turabian Style**

Lu, Shichang, Danyang Liu, Dan Li, and Xulun Shao.
2023. "Enhanced Teaching–Learning-Based Optimization Algorithm for the Mobile Robot Path Planning Problem" *Applied Sciences* 13, no. 4: 2291.
https://doi.org/10.3390/app13042291