# A Hybrid k-Means Cuckoo Search Algorithm Applied to the Counterfort Retaining Walls Problem

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## Abstract

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_{2}emissions are important. The first is relevant in the competitiveness and efficiency of the company, the second in environmental impact. From the point of view of computational complexity, the problem is challenging due to the large number of possible combinations in the solution space. In this article, a k-means cuckoo search hybrid algorithm is proposed where the cuckoo search metaheuristic is used as an optimization mechanism in continuous spaces and the unsupervised k-means learning technique to discretize the solutions. A random operator is designed to determine the contribution of the k-means operator in the optimization process. The best values, the averages, and the interquartile ranges of the obtained distributions are compared. The hybrid algorithm was later compared to a version of harmony search that also solved the problem. The results show that the k-mean operator contributes significantly to the quality of the solutions and that our algorithm is highly competitive, surpassing the results obtained by harmony search.

## 1. Introduction

- A machine-learning algorithm is proposed to allow metaheuristics commonly defined and used in continuous optimization addressing discrete optimization problems simply and effectively. To perform this process, the algorithm uses k-means. This clustering technique has been selected because it has solved other binary combinatorial problems efficiently [5,33]. The selected metaheuristic is CS. Its selection is because it has been frequently used in solving continuous optimization problems and its tuning is relatively simple, which allows focusing on the discretization process.
- A random operator is designed to study the contribution of the k-means operator in the discretization process.
- This hybrid algorithm is applied to the design of the buttresses retaining wall problem. Optimization is carried out for cost and emissions of CO
_{2}. A comparison is made between the proposed hybrid algorithm and an adaptation of the harmony search (HS) proposed in [34]. The design of the counterfort retaining walls will be detailed in Section 3.

## 2. Hybridizing Metaheuristics with Machine Learning

## 3. Problem Definition

#### 3.1. Optimization Problem

_{2}equivalent emissions units (${e}_{i}$). These construction units correspond to formwork, materials, excavation and earth-fill. The cost and emission functions are based on a 1 m wide strip [34]. The emission and cost values were obtained from [34,66] and are shown in Table 1. Then, in a general way, our optimization problem is defined according to Equation (1).

#### 3.2. Problem Design Variables

#### 3.3. Problem Design Parameters

#### 3.4. Problem Constraints

## 4. The K-Means Discrete Algorithm

#### 4.1. Cuckoo Search Algorithm

- Each cuckoo lays one egg at a time and deposits its egg in a randomly chosen nest.
- The nests with the best results, i.e., with high-quality eggs, will be considered in the next generation.
- The number of nests available is a fixed parameter. The egg laid by a cuckoo can be discovered by the host bird with a probability ${p}_{a}\in (0,1)$

Algorithm 1 Cuckoo search algorithm |

1: Objective function f(x) |

2: Generate initial solutions of n host nests. |

3: while stop criterion are meet do |

4: Get a cuckoo randomly and replace using Lévy flights. |

5: Evaluate the fitness. |

6: Choose in a random way a nest j among n: |

7: if${f}_{i}$ > ${f}_{j}$ then |

8: replace the solution. |

9: end if |

10: portion ${p}_{a}$ of the worst nests are eliminated and new ones are created. |

11: keep best solutions. |

12: find the current best |

13: end while |

#### 4.2. k-Means Operator

Algorithm 2 k-means operator |

1: Function kmeanOp($lx\left(t\right)$, $lx(t+1)$) |

2: Input $x\left(t\right)$, $x(t+1)$ |

3: Output lTranProb(t+1) |

4: $l{\Delta}^{i}\left(x(t+1)\right)\leftarrow $ getDelta($lx\left(t\right)$, $lx(t+1)$) |

5: $Clust\leftarrow $ getClusters($l{\Delta}^{i}\left(x\left(t\right)\right)$,k) |

6: $lTranProb(t+1)\leftarrow $ getTranProb($Clust$, $lx\left(t\right)$)–Equation (11) |

7: return ($lTranProb(t+1)$) |

#### 4.3. Discretization Operator

Algorithm 3 Discretization operator. |

1: Function DiscOp($lTranProb(t+1)$, $lx\left(t\right)$) |

2: Input $lTranProb(t+1)$ |

3: Output $x(t+1)$–Where x(t+1) is discrete. |

4: movement = 0 |

5: for ${x}^{i}\in x\left(t\right)\in lx\left(t\right)$ do |

6: if ${r}_{1}>0.5$ then |

7: movement = 1 |

8: else |

9: movement = −1 |

10: end if |

11: ${x}^{i}$ = max(1,min(${x}_{best}^{i}$, ${x}^{i}$ + movement)) |

12: end for |

## 5. Results and Discussion

#### 5.1. Parameter Settings

- The deviation of the best local value obtained in five executions compared with the best global value:$$bSolution=1-\frac{BestGlobalValue-BestlocalValue}{BestGlobalValue}$$
- The deviation of the worst value obtained in five executions compared with the best global value:$$wSolution=1-\frac{BestGlobalValue-WorstLocalValue}{BestGlobalValue}$$
- The deviation of the average value obtained in five executions compared with the best global value:$$aSolution=1-\frac{BestGlobalValue---AverageLocalValue}{BestGlobalValue}$$
- The convergence time for the average value in each experiment is normalized according to Equation (15).$$nTime=1-\frac{convergenceLocalTime-minGlobalTime}{maxGlobalTime-minGlobalTime}$$

#### 5.2. Random Operator

_{2}emissions, when analyzing the best value indicator, we observe that the k-means operator obtains better results at all heights. However, the maximum difference from the random operator does not exceed 10%. The average indicator is again higher in the case of k-means, the Wilcoxon test indicating that the difference is significant. When analyzing the violin plots of the emissions shown in Figure 9, we see that the interquartile range obtains better quality results in the case of the k-means discretization. However, the result is not as remarkable as in the case of cost optimization.

#### 5.3. Comparisons

_{2}, the curves behave similarly to that of the cost optimization case. For small values of wall height, very similar values are obtained. As the height of the wall increases, the quality of the k-means solutions improves compared to the HS. this is seen in Figure 12.

## 6. Conclusions

_{2}as objective functions. The cuckoo search optimization algorithm was used to be discretized. Additionally, a random operator was constructed to determine the contribution of the k-mean operator in the optimization process. It was concluded that k-means produces better results than the random operator and in many cases this does it systematically, thus reducing the dispersion of the solutions. In addition, when we compare k-means with HS, we observe that as we increase the height, where the optimization problem becomes more difficult because it is more difficult to obtain stability of the wall with respect to overturning and sliding, k-means is more robust than HS reaching the height of 14 (m) at a difference of 4.76% in favor of k-means in optimizing emissions and 4.87% in minimizing costs. On the other hand, when we analyze the dispersion of the set of solutions, we see that k-means once again perform better than HS, especially for heights greater than 12 (m).

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 11.**Comparison between the best solutions obtained by the k-means and HS algorithms in cost optimization.

**Figure 12.**Comparison between the best solutions obtained by the k-means and HS algorithms in emission optimization.

Unit | Emissions (CO_{2}-eq) | Cost (€) |
---|---|---|

kg of steel B400 | 3.02 | 0.56 |

kg of steel B500 | 2.82 | 0.58 |

m^{3} of concrete HA-25 in stem | 224.34 | 56.66 |

m^{3} of concrete HA-30 in stem | 224.94 | 60.80 |

m^{3} of concrete HA-35 in stem | 265.28 | 65.32 |

m^{3} of concrete HA-40 in stem | 265.28 | 70.41 |

m^{3} of concrete HA-45 in stem | 265.91 | 75.22 |

m^{3} of concrete HA-50 in stem | 265.95 | 80.03 |

m^{2} stem formwork | 1.92 | 21.61 |

m^{3} of backfill | 28.79 | 5.56 |

m^{3} of concrete HA-25 in foundation | 224.34 | 50.65 |

m^{3} of concrete HA-30 in foundation | 224.94 | 54.79 |

m^{3} of concrete HA-35 in foundation | 265.28 | 59.31 |

m^{3} of concrete HA-40 in foundation | 265.28 | 64.40 |

m^{3} of concrete HA-45 in foundation | 265.91 | 69.21 |

m^{3} of concrete HA-50 in foundation | 265.95 | 74.02 |

Variables | Lower Bound | Increment | Upper Bound | N of Values |
---|---|---|---|---|

c | H/20 | 5 cm | H/5 | f(H) ^{1} |

b | 25 cm | 2.5 cm | 122.5 | 40 |

p | 20 cm | 10 cm | 610 | 60 |

t | 20 cm | 15 cm | 905 | 60 |

${e}_{c}$ | 25 cm | 2.5 cm | 122.5 | 40 |

d | H/5 cm | 5 cm | 2H/3 | f(H) ^{1} |

${f}_{ck}$ | 25, 20, 25, 40, 45, 50 | 7 | ||

${f}_{yk}$ | 400, 500 | 2 | ||

${A}_{1}$ to ${A}_{10}$ | 6, 8, 10, 12, 16, 20, 25, 32 | 8 | ||

1 steel rebar | 2 rebars | 12 rebars | 6 | |

${A}_{11}$ to ${A}_{12}$ | 6, 8, 10, 12, 16, 20, 25, 32 | 8 | ||

1 steel rebar | 4 rebars | 10 rebars | 7 |

Parameter Considered | Value |
---|---|

Bearing capacity | 0.3 MPa |

Fill slope | 0 |

Foundation depth, H2 | 2 m |

Uniform load on top of the fill, $\gamma $ | 10 kN/m^{2} |

Wall-fill friction angle, $\delta $ | 0° |

Base-friction coefficient, $\mu $ | tg 30° |

Safety coefficient against sliding, ${\gamma}_{fs}$ | 1.5 |

Safety coefficient against overturning, ${\gamma}_{fo}$ | 1.8 |

EHE safety coefficient for loading | Normal |

ULS safety coefficient of concrete | 1.5 |

ULS safety coefficient of steel | 1.15 |

EHE ambient exposure | IIa |

Parameters | Description | Value | Range |
---|---|---|---|

N | Number of Nest | 5 | [5, 10, 15] |

k | Number of transition groups K-means Operator | 5 | [4, 5, 6] |

$\gamma $ | Step Length | 0.01 | 0.01 |

$\kappa $ | Lévy distribution parameter | 1.5 | 1.5 |

Iteration Number | Maximum iterations | 800 | [800] |

Height (m) | Best Value | Avg | Best Value | Avg | Best Value | Avg Random |
---|---|---|---|---|---|---|

k-Means | k-Means | Random | Random | HS | HS | |

6 | 591 | 595.5 | 600 | 621.3 | 595 | 600.15 |

7 | 678 | 682.8 | 687 | 721.4 | 689 | 694.98 |

8 | 775 | 778.9 | 785 | 851.7 | 784 | 788.38 |

9 | 911 | 922.3 | 981 | 1094.3 | 934 | 941.29 |

10 | 1095 | 1127.0 | 1184 | 1296.7 | 1130 | 1143.64 |

11 | 1302 | 1384.5 | 1545 | 1704.5 | 1354 | 1381.50 |

12 | 1528 | 1608.9 | 1839 | 1994.3 | 1590 | 1707.24 |

13 | 1775 | 1905.3 | 2241 | 2510.8 | 1840 | 2067.37 |

14 | 2049 | 2301.6 | 2775 | 3267.1 | 2154 | 2348.71 |

Wilcoxon p-value | 1.6 × 10^{−5} | 1.31 × 10^{−3} |

Height (m) | Best Value | Avg | Best Value | Avg | Best Value | Avg Random |
---|---|---|---|---|---|---|

k-Means | k-Means | Random | Random | HS | HS | |

6 | 1242 | 1251.2 | 1274 | 1304.3 | 1250 | 1289.14 |

7 | 1440 | 1458.5 | 1467 | 1501.3 | 1478 | 1511.53 |

8 | 1659 | 1683.2 | 1696 | 1788.6 | 1699 | 1731.54 |

9 | 1997 | 2111.1 | 2180 | 2214.6 | 2050 | 2097.45 |

10 | 2470 | 2572.1 | 2975 | 3132.7 | 2560 | 2617.81 |

11 | 3061 | 3206.6 | 3540 | 3801.7 | 3124 | 3201.45 |

12 | 3715 | 3921.5 | 4168 | 4555.7 | 3865 | 4046.95 |

13 | 4470 | 4676.4 | 5039 | 5484.5 | 4650 | 4955.95 |

14 | 5294 | 5621.2 | 5877 | 6319.1 | 5550 | 6241.00 |

Wilcoxon p-value | 1.2 × 10^{−7} | 2.51 × 10^{−4} |

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## Share and Cite

**MDPI and ACS Style**

García, J.; Yepes, V.; Martí, J.V.
A Hybrid k-Means Cuckoo Search Algorithm Applied to the Counterfort Retaining Walls Problem. *Mathematics* **2020**, *8*, 555.
https://doi.org/10.3390/math8040555

**AMA Style**

García J, Yepes V, Martí JV.
A Hybrid k-Means Cuckoo Search Algorithm Applied to the Counterfort Retaining Walls Problem. *Mathematics*. 2020; 8(4):555.
https://doi.org/10.3390/math8040555

**Chicago/Turabian Style**

García, José, Victor Yepes, and José V. Martí.
2020. "A Hybrid k-Means Cuckoo Search Algorithm Applied to the Counterfort Retaining Walls Problem" *Mathematics* 8, no. 4: 555.
https://doi.org/10.3390/math8040555