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

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{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

## References

- García, J.; Altimiras, F.; Peña, A.; Astorga, G.; Peredo, O. A binary cuckoo search big data algorithm applied to large-scale crew scheduling problems. Complexity
**2018**, 2018. [Google Scholar] [CrossRef] - García, J.; Moraga, P.; Valenzuela, M.; Crawford, B.; Soto, R.; Pinto, H.; Peña, A.; Altimiras, F.; Astorga, G. A Db-Scan Binarization Algorithm Applied to Matrix Covering Problems. Comput. Intell. Neurosci.
**2019**, 2019. [Google Scholar] [CrossRef] [PubMed][Green Version] - Al-Madi, N.; Faris, H.; Mirjalili, S. Binary multi-verse optimization algorithm for global optimization and discrete problems. Int. J. Mach. Learn. Cybern.
**2019**, 10, 3445–3465. [Google Scholar] [CrossRef] - Kim, M.; Chae, J. Monarch Butterfly Optimization for Facility Layout Design Based on a Single Loop Material Handling Path. Mathematics
**2019**, 7, 154. [Google Scholar] [CrossRef][Green Version] - García, J.; Crawford, B.; Soto, R.; Astorga, G. A clustering algorithm applied to the binarization of Swarm intelligence continuous metaheuristics. Swarm Evol. Comput.
**2019**, 44, 646–664. [Google Scholar] [CrossRef] - García, J.; Lalla-Ruiz, E.; Voß, S.; Droguett, E.L. Enhancing a machine learning binarization framework by perturbation operators: Analysis on the multidimensional knapsack problem. Int. J. Mach. Learn. Cybern.
**2020**. [Google Scholar] [CrossRef] - García, J.; Moraga, P.; Valenzuela, M.; Pinto, H. A db-Scan Hybrid Algorithm: An Application to the Multidimensional Knapsack Problem. Mathematics
**2020**, 8, 507. [Google Scholar] [CrossRef][Green Version] - Saeheaw, T.; Charoenchai, N. A comparative study among different parallel hybrid artificial intelligent approaches to solve the capacitated vehicle routing problem. Int. J. Bio-Inspir. Comput.
**2018**, 11, 171–191. [Google Scholar] [CrossRef] - Crawford, B.; Soto, R.; Astorga, G.; García, J. Constructive metaheuristics for the set covering problem. In International Conference on Bioinspired Methods and Their Applications; Springer: Berlin, Germany, 2018; pp. 88–99. [Google Scholar]
- Valdez, F.; Castillo, O.; Jain, A.; Jana, D.K. Nature-inspired optimization algorithms for neuro-fuzzy models in real-world control and robotics applications. Comput. Intell. Neurosci.
**2019**, 2019, 9128451. [Google Scholar] [CrossRef] - Penadés-Plà, V.; García-Segura, T.; Yepes, V. Robust Design Optimization for Low-Cost Concrete Box-Girder Bridge. Mathematics
**2020**, 8, 398. [Google Scholar] [CrossRef][Green Version] - García-Segura, T.; Yepes, V.; Frangopol, D.M.; Yang, D.Y. Lifetime reliability-based optimization of post-tensioned box-girder bridges. Eng. Struct.
**2017**, 145, 381–391. [Google Scholar] [CrossRef] - Yepes, V.; Martí, J.V.; García, J. Black Hole Algorithm for Sustainable Design of Counterfort Retaining Walls. Sustainability
**2020**, 12, 2767. [Google Scholar] [CrossRef][Green Version] - Marti-Vargas, J.R.; Ferri, F.J.; Yepes, V. Prediction of the transfer length of prestressing strands with neural networks. Comput. Concr.
**2013**, 12, 187–209. [Google Scholar] [CrossRef] - Fu, W.; Tan, J.; Zhang, X.; Chen, T.; Wang, K. Blind parameter identification of MAR model and mutation hybrid GWO-SCA optimized SVM for fault diagnosis of rotating machinery. Complexity
**2019**, 2019, 3264969. [Google Scholar] [CrossRef] - Sierra, L.A.; Yepes, V.; García-Segura, T.; Pellicer, E. Bayesian network method for decision-making about the social sustainability of infrastructure projects. J. Clean. Prod.
**2018**, 176, 521–534. [Google Scholar] [CrossRef] - Crawford, B.; Soto, R.; Astorga, G.; García, J.; Castro, C.; Paredes, F. Putting continuous metaheuristics to work in binary search spaces. Complexity
**2017**, 2017, 8404231. [Google Scholar] [CrossRef][Green Version] - Shi, Y. Particle swarm optimization: Developments, applications and resources. In Proceedings of the 2001 Congress on Evolutionary Computation, Seoul, Korea, 27–30 May 2001; Volume 1, pp. 81–86. [Google Scholar]
- Hatamlou, A. Black hole: A new heuristic optimization approach for data clustering. Inf. Sci.
**2013**, 222, 175–184. [Google Scholar] [CrossRef] - Yang, X.S.; Deb, S. Cuckoo search via Lévy flights. In Proceedings of the 2009 World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India, 9–11 December 2009; pp. 210–214. [Google Scholar]
- Yang, X.S. A new metaheuristic bat-inspired algorithm. In Nature Inspired Cooperative Strategies for Optimization (NICSO 2010); Springer: Berlin, Germany, 2010; pp. 65–74. [Google Scholar]
- Yang, X.S. Firefly algorithms for multimodal optimization. In International Symposium on Stochastic Algorithms; Springer: Berlin, Germany, 2009; pp. 169–178. [Google Scholar]
- Pan, W.T. A new fruit fly optimization algorithm: Taking the financial distress model as an example. Knowl.-Based Syst.
**2012**, 26, 69–74. [Google Scholar] [CrossRef] - Li, X.L.; Shao, Z.J.; Qian, J.X. An optimizing method based on autonomous animats: Fish-swarm algorithm. Syst. Eng. Theory Pract.
**2002**, 22, 32–38. [Google Scholar] - Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A gravitational search algorithm. Inf. Sci.
**2009**, 179, 2232–2248. [Google Scholar] [CrossRef] - Calvet, L.; de Armas, J.; Masip, D.; Juan, A.A. Learnheuristics: Hybridizing metaheuristics with machine learning for optimization with dynamic inputs. Open Math.
**2017**, 15, 261–280. [Google Scholar] [CrossRef] - Caserta, M.; Voß, S. Matheuristics: Hybridizing Metaheuristics and Mathematical Programming. In Metaheuristics: Intelligent Problem Solving; Springer: Berlin, Germany, 2009; pp. 1–38. [Google Scholar]
- Talbi, E.G. Combining metaheuristics with mathematical programming, constraint programming and machine learning. Ann. Oper. Res.
**2016**, 240, 171–215. [Google Scholar] [CrossRef] - Juan, A.A.; Faulin, J.; Grasman, S.E.; Rabe, M.; Figueira, G. A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems. Oper. Res. Perspect.
**2015**, 2, 62–72. [Google Scholar] [CrossRef][Green Version] - Chou, J.S.; Nguyen, T.K. Forward Forecast of Stock Price Using Sliding-Window Metaheuristic-Optimized Machine-Learning Regression. IEEE Trans. Ind. Inform.
**2018**, 14, 3132–3142. [Google Scholar] [CrossRef] - Sayed, G.I.; Tharwat, A.; Hassanien, A.E. Chaotic dragonfly algorithm: An improved metaheuristic algorithm for feature selection. Appl. Intell.
**2019**, 49, 188–205. [Google Scholar] [CrossRef] - De León, A.D.; Lalla-Ruiz, E.; Melián-Batista, B.; Moreno-Vega, J.M. A Machine Learning-based system for berth scheduling at bulk terminals. Expert Syst. Appl.
**2017**, 87, 170–182. [Google Scholar] [CrossRef] - García, J.; Crawford, B.; Soto, R.; Castro, C.; Paredes, F. A k-means binarization framework applied to multidimensional knapsack problem. Appl. Intell.
**2018**, 48, 357–380. [Google Scholar] [CrossRef] - Molina-Moreno, F.; Martí, J.V.; Yepes, V. Carbon embodied optimization for buttressed earth-retaining walls: Implications for low-carbon conceptual designs. J. Clean. Prod.
**2017**, 164, 872–884. [Google Scholar] [CrossRef] - Voß, S. Meta-heuristics: The state of the art. In Workshop on Local Search for Planning and Scheduling; Springer: Berlin, Germany, 2000; pp. 1–23. [Google Scholar]
- Bishop, C.M. Pattern Recognition and Machine Learning; Springer: Berlin, Germany, 2006. [Google Scholar]
- Asta, S.; Özcan, E.; Curtois, T. A tensor based hyper-heuristic for nurse rostering. Knowl.-Based Syst.
**2016**, 98, 185–199. [Google Scholar] [CrossRef][Green Version] - Martin, S.; Ouelhadj, D.; Beullens, P.; Ozcan, E.; Juan, A.A.; Burke, E.K. A multi-agent based cooperative approach to scheduling and routing. Eur. J. Oper. Res.
**2016**, 254, 169–178. [Google Scholar] [CrossRef][Green Version] - García, J.; Crawford, B.; Soto, R.; Astorga, G. Astorga, G. A percentile transition ranking algorithm applied to binarization of continuous swarm intelligence metaheuristics. In International Conference on Soft Computing and Data Mining; Springer: Johor, Malaysia, 2018; pp. 3–13. [Google Scholar] [CrossRef]
- Vecek, N.; Mernik, M.; Filipic, B.; Xrepinsek, M. Parameter tuning with Chess Rating System (CRS-Tuning) for meta-heuristic algorithms. Inf. Sci.
**2016**, 372, 446–469. [Google Scholar] [CrossRef] - Ries, J.; Beullens, P. A semi-automated design of instance-based fuzzy parameter tuning for metaheuristics based on decision tree induction. J. Oper. Res. Soc.
**2015**, 66, 782–793. [Google Scholar] [CrossRef][Green Version] - Li, Z.Q.; Zhang, H.L.; Zheng, J.H.; Dong, M.J.; Xie, Y.F.; Tian, Z.J. Heuristic evolutionary approach for weighted circles layout. In International Symposium on Information and Automation; Springer: Berlin, Germany, 2010; pp. 324–331. [Google Scholar]
- Yalcinoz, T.; Altun, H. Power economic dispatch using a hybrid genetic algorithm. IEEE Power Eng. Rev.
**2001**, 21, 59–60. [Google Scholar] [CrossRef] - Kaur, H.; Virmani, J.; Thakur, S. A genetic algorithm-based metaheuristic approach to customize a computer-aided classification system for enhanced screen film mammograms. In U-Healthcare Monitoring Systems; Advances in Ubiquitous Sensing Applications for Healthcare; Dey, N., Ashour, A.S., Fong, S.J., Borra, S., Eds.; Academic Press: Cambridge, MA, USA, 2019; pp. 217–259. [Google Scholar] [CrossRef]
- Faris, H.; Hassonah, M.A.; Ala’M, A.Z.; Mirjalili, S.; Aljarah, I. A multi-verse optimizer approach for feature selection and optimizing SVM parameters based on a robust system architecture. Neural Comput. Appl.
**2018**, 30, 2355–2369. [Google Scholar] [CrossRef] - Faris, H.; Aljarah, I.; Mirjalili, S. Improved monarch butterfly optimization for unconstrained global search and neural network training. Appl. Intell.
**2018**, 48, 445–464. [Google Scholar] [CrossRef] - Chou, J.S.; Thedja, J.P.P. Metaheuristic optimization within machine learning-based classification system for early warnings related to geotechnical problems. Autom. Constr.
**2016**, 68, 65–80. [Google Scholar] [CrossRef] - Pham, A.D.; Hoang, N.D.; Nguyen, Q.T. Predicting compressive strength of high-performance concrete using metaheuristic-optimized least squares support vector regression. J. Comput. Civ. Eng.
**2015**, 30, 06015002. [Google Scholar] [CrossRef] - Göçken, M.; Özçalıcı, M.; Boru, A.; Dosdoğru, A.T. Integrating metaheuristics and artificial neural networks for improved stock price prediction. Expert Syst. Appl.
**2016**, 44, 320–331. [Google Scholar] [CrossRef] - Chou, J.S.; Pham, A.D. Nature-inspired metaheuristic optimization in least squares support vector regression for obtaining bridge scour information. Inf. Sci.
**2017**, 399, 64–80. [Google Scholar] [CrossRef] - Kuo, R.; Lin, T.; Zulvia, F.; Tsai, C. A hybrid metaheuristic and kernel intuitionistic fuzzy c-means algorithm for cluster analysis. Appl. Soft Comput.
**2018**, 67, 299–308. [Google Scholar] [CrossRef] - Mann, P.S.; Singh, S. Energy efficient clustering protocol based on improved metaheuristic in wireless sensor networks. J. Netw. Comput. Appl.
**2017**, 83, 40–52. [Google Scholar] [CrossRef] - De Alvarenga Rosa, R.; Machado, A.M.; Ribeiro, G.M.; Mauri, G.R. A mathematical model and a Clustering Search metaheuristic for planning the helicopter transportation of employees to the production platforms of oil and gas. Comput. Ind. Eng.
**2016**, 101, 303–312. [Google Scholar] [CrossRef] - Faris, H.; Mirjalili, S.; Aljarah, I. Automatic selection of hidden neurons and weights in neural networks using grey wolf optimizer based on a hybrid encoding scheme. Int. J. Mach. Learn. Cybern.
**2019**, 10, 2901–2920. [Google Scholar] [CrossRef] - De Rosa, G.H.; Papa, J.P.; Yang, X.S. Handling dropout probability estimation in convolution neural networks using meta-heuristics. Soft Comput.
**2018**, 22, 6147–6156. [Google Scholar] [CrossRef][Green Version] - Tuba, M.; Alihodzic, A.; Bacanin, N. Cuckoo search and bat algorithm applied to training feed-forward neural networks. In Recent Advances in Swarm Intelligence and Evolutionary Computation; Springer: Berlin, Germany, 2015; pp. 139–162. [Google Scholar]
- Rere, L.; Fanany, M.I.; Arymurthy, A.M. Metaheuristic algorithms for convolution neural network. Comput. Intell. Neurosci.
**2016**, 2016, 1537325. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rashid, T.A.; Hassan, M.K.; Mohammadi, M.; Fraser, K. Improvement of variant adaptable LSTM trained with metaheuristic algorithms for healthcare analysis. In Advanced Classification Techniques for Healthcare Analysis; IGI Global: Hershey, PA, USA, 2019; pp. 111–131. [Google Scholar]
- Jothi, R.; Mohanty, S.K.; Ojha, A. DK-means: A deterministic k-means clustering algorithm for gene expression analysis. Pattern Anal. Appl.
**2019**, 22, 649–667. [Google Scholar] [CrossRef] - García, J.; Pope, C.; Altimiras, F. A Distributed-Means Segmentation Algorithm Applied to Lobesia botrana Recognition. Complexity
**2017**, 2017, 5137317. [Google Scholar] [CrossRef][Green Version] - Arunkumar, N.; Mohammed, M.A.; Ghani, M.K.A.; Ibrahim, D.A.; Abdulhay, E.; Ramirez-Gonzalez, G.; de Albuquerque, V.H.C. K-means clustering and neural network for object detecting and identifying abnormality of brain tumor. Soft Comput.
**2019**, 23, 9083–9096. [Google Scholar] [CrossRef] - Abdel-Basset, M.; Wang, G.G.; Sangaiah, A.K.; Rushdy, E. Krill herd algorithm based on cuckoo search for solving engineering optimization problems. Multimed. Tools Appl.
**2019**, 78, 3861–3884. [Google Scholar] [CrossRef] - Chi, R.; Su, Y.X.; Zhang, D.H.; Chi, X.X.; Zhang, H.J. A hybridization of cuckoo search and particle swarm optimization for solving optimization problems. Neural Comput. Appl.
**2019**, 31, 653–670. [Google Scholar] [CrossRef] - Li, J.; Xiao, D.D.; Lei, H.; Zhang, T.; Tian, T. Using Cuckoo Search Algorithm with Q-Learning and Genetic Operation to Solve the Problem of Logistics Distribution Center Location. Mathematics
**2020**, 8, 149. [Google Scholar] [CrossRef][Green Version] - Pan, J.S.; Song, P.C.; Chu, S.C.; Peng, Y.J. Improved Compact Cuckoo Search Algorithm Applied to Location of Drone Logistics Hub. Mathematics
**2020**, 8, 333. [Google Scholar] [CrossRef] - Yepes, V.; Alcala, J.; Perea, C.; González-Vidosa, F. A parametric study of optimum earth-retaining walls by simulated annealing. Eng. Struct.
**2008**, 30, 821–830. [Google Scholar] [CrossRef] - Molina-Moreno, F.; García-Segura, T.; Martí, J.V.; Yepes, V. Optimization of buttressed earth-retaining walls using hybrid harmony search algorithms. Eng. Struct.
**2017**, 134, 205–216. [Google Scholar] [CrossRef] - Ministerio de Fomento. EHE: Code of Structural Concrete; Ministerio de Fomento: Madrid, Spain, 2008. [Google Scholar]
- Ministerio de Fomento. CTE. DB-SE. Structural Safety: Foundations; Ministerio de Fomento: Madrid, Spain, 2008. (In Spanish)
- Huntington, W.C. Earth Pressures and Retaining Walls; Literary Licensing, LLC: Whitefish, MT, USA, 1957. [Google Scholar]
- Calavera, J. Muros de Contención y Muros de Sótano; INTEMAC: Madrid, Spain, 2001. (In Spanish) [Google Scholar]
- CEB-FIB. Model Code. Design Code; Thomas Telford Services Ltd.: London, UK, 2008. [Google Scholar]
- Hays, W.L.; Winkler, R.L. Statistics: Probability, Inference, and Decision; Holt, Rinehart, and Winston: New York, NY, USA, 1971. [Google Scholar]
- Wilcoxon, F. Individual comparisons by ranking methods. In Breakthroughs in Statistics; Springer: Berlin, Germany, 1992; pp. 196–202. [Google Scholar]

**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} |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## 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