Minimum Safety Factor Evaluation of Slopes Using Hybrid Chaotic Sand Cat and Pattern Search Approach
Abstract
:1. Introduction
- An efficient hybrid metaheuristic approach based on chaotic sand cat and pattern search techniques, known as CSCPS, was developed.
- The performance of CSCPS for numerical function optimization was assessed on thirteen common benchmarking functions, and the results were compared with existing commonly used optimization algorithms.
- To demonstrate the efficiency of the proposed technique in solving real-world problems, the new method was used to determine the minimum factor of safety of earth slopes.
- The objective function of a slope stability problem was modeled by using the Morgenstern–Price limit equilibrium approach for the slip surface’s overall shape.
- The efficiency of the proposed CSCPS for slope stability assessment was investigated by using two benchmark problems based on the literature, and the obtained results were compared with those evaluated previously by the other techniques.
2. Safety Factor Formulation
- Step 1.
- Create a trial slip surface and divide it into n vertical segments.
- Step 2.
- Determine Ri and Ti by using the equations below:
- Step 3.
- Pick the function for inter-slice forces; f(x) = 1 is assumed in this study.
- Step 4.
- Consider FOS and λ initial amounts according to the following criteria:The λ and FOS might be set to 0 and 1, respectively, as their initial values [64].
- Step 5.
- Evaluate Φi and i−1 based on Equations (6) and (7).
- Step 6.
- Calculate FOS based on Equation (8):
- Step 7.
- Compute Φi and i−1 and calculate FOS once more by performing Steps 5 and 6 again.
- Step 8.
- Define Ei and λ based on the Equations (9) and (10).
- Step 9.
- Reevaluate FOS by using the obtained λ, and this process ends when the distinction between the found FOS and λ falls below 0.005 and 0.01, respectively. For the analysis in this paper, the generation failure surface approach developed by Cheng et al. [65] was implemented for the general form of the failure surface, which divides the assumed failure soil mass into some vertical sections of equal width.
3. Proposed Chaotic Sand Cat and Pattern Search
3.1. Sand Cat Optimization
Algorithm 1. Sand cat optimization algorithm. |
Initialize the population Calculate the fitness function based on the objective function Initialize the r, rG, R While (t ≤ tMax) For each sand cat Get a random angle θ ( If (|R| ≤ 1) Update the search agent based on exploitation part of Equation (17); Else Update the search agent based on the exploration part of Equation (17); End End t = t+1 End |
3.2. Chaotic Sand Cat Optimization
3.3. Pattern Search (PS)
3.4. Chaotic Sand Cat–Pattern Search
4. Model Verification
5. Model Application
5.1. A Uniform Soil Slope
5.2. Slope in a Multi-Layered Soil
6. Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Author, Year | Reference | Optimization Method | Application |
---|---|---|---|
Goh, 2000 | [28] | Genetic algorithm | Locate the critical circular slip surface in slope stability analysis |
Zolfaghari, Heath, and McCombie, 2005 | [6] | Genetic algorithm | In slope stability, look for critical non-circular failure surfaces. |
Cheng et al., 2007 | [29] | Particle swarm optimization | Two-dimensional slope stability analysis |
Cheng et al., 2008 | [16] | Improved harmony search algorithm | Slope stability analysis |
Chan, Zhang, and Ng, 2009 | [30] | Hybrid genetic algorithms | Optimization of pile groups |
Kahatadeniya, Nanakorn, and Neaupane, 2009 | [31] | Ant colony optimization | Determination of the critical failure surface of earth slope |
Khajehzadeh et al., 2011 | [32] | Modified particle swarm optimization | Optimum design of spread footing and retaining wall |
Camp and Akin, 2012 | [33] | Big bang–big crunch optimization | Optimum design of retaining wall |
Camp and Assadollahi, 2013 | [34] | Hybrid big bang–big crunch algorithm | Optimizing the cost and CO2 of concrete footings |
Khajehzadeh, Taha, and Eslami, 2013 | [35] | hybrid firefly algorithm | Minimize of total cost and CO2 emissions of the foundation subjected to geotechnical and structural requirements. |
Kang, Li, and Ma, 2013 | [36] | Artificial bee colony algorithm | In slope stability, finding the critical slip surface |
Khajehzadeh, Taha, and Eslami, 2014 | [37] | Adaptive gravitational search algorithm | Multi-objective optimization of foundation |
Kashani, Gandomi, and Mousavi, 2016 | [9] | Imperialistic competitive algorithm | Locating the critical slip surface of earth slope |
Gordan et al., 2016 | [38] | Particle swarm optimization and neural network | Prediction of seismic slope stability |
Gandomi and Kashani, 2017 | [39] | Accelerated particle swarm optimization, firefly algorithm, levy-flight krill herd, whale optimization algorithm, ant lion optimizer, grey wolf optimizer, moth–flame optimization algorithm, and teaching-learning-based optimization algorithm | Construction cost minimization of shallow foundation |
Aydogdu, 2017 | [40] | Biogeography-based optimization algorithm | Cost optimization of retaining wall |
Gandomi et al., 2017 | [10] | Genetic algorithm, differential evolution, evolutionary strategy, and biogeography-based optimization | Slope stability analysis |
Mahdiyar et al., 2017 | [41] | Monte Carlo simulation technique | Safety assessment of slope |
Gandomi, Kashani, and Zeighami, 2017 | [42] | Interior search algorithm | Retaining wall optimization |
Chen et al., 2019 | [43] | Hybrid imperialist competitive algorithm and artificial neural network | Prediction of safety factor values of retaining walls |
Koopialipoor et al., 2019 | [44] | Imperialist competitive algorithm, genetic algorithm, particle swarm optimization, and artificial bee colony combined with artificial neural network | Predict the slope safety exposed to static and dynamic conditions |
Yang et al., 2019 | [45] | Neural network system | Design retaining wall structures based on smart and optimal systems |
Xu et al., 2019 | [46] | Hybrid artificial neural network and ant colony optimization | Dynamic conditions of retaining wall structures |
Himanshu and Burman, 2019 | [11] | Particle swarm optimization | Determination of critical failure surface considering seepage and seismic loading |
Kalemci et al., 2020 | [47] | Grey wolf optimization algorithm | Optimization of retaining walls |
Kaveh, Hamedani, and Bakhshpoori, 2020 | [48] | Eleven meta-heuristic algorithms | Optimal design of cantilever retaining walls |
Kashani et al., 2020 | [49] | Differential algorithm, evolution strategy, and biogeography-based optimization algorithm | Optimum design of shallow foundation |
Wang et al., 2020 | [50] | Extreme gradient boosting method | Evaluating the earth dam slopefailure probability. |
Moayedi et al., 2021 | [51] | Harris hawks’ optimization | Predicting the factor of safety in the presence of rigid foundations |
Sharma, Saha, and Lohar, 2021 | [52] | Hybrid butterfly and symbiosis organism search algorithm | Optimization of retaining wall |
Kaveh and Seddighian, 2021 | [53] | Black hole mechanics optimization, firefly algorithm, evolution strategy, sine cosine algorithm | Slope critical surfaces optimization with seepage and seismic effects |
Temur, 2021 | [54] | Teaching-learning based optimization | Optimization of retaining wall |
Li and Wu, 2021 | [55] | Improved salp swarm algorithm | Locating critical slip surface of slopes |
Arabali et al., 2022 | [56] | Adaptive tunicate swarm algorithm | Optimization of construction cost and CO2 emissions of shallow foundation |
Khajehzadeh, Keawsawasvong, and Nehdi, 2022 | [57] | Artificial neural networks combined with rat swarm optimization | Prediction of the ultimate bearing capacity of shallow foundations and their optimum design |
Khajehzadeh, Kalhor, et al., 2022 | [58] | Adaptive sperm swarm optimization | Optimum design of retaining structures under seismic load |
Kashani et al., 2022 | [59] | Multi-objective particle swarm optimization, multi-objective multi-verse optimization, and Pareto envelope-based selection algorithm | Multi-objective optimization of mechanically stabilized earth retaining wall |
Function | Range | n (Dim) | |
---|---|---|---|
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 |
Function | Range | n (Dim) | |
---|---|---|---|
428.9829 × n | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 | ||
0 | 30 |
F | Index | CSCPS | SCO | PSO | FA | MVO | SSA | TSA |
---|---|---|---|---|---|---|---|---|
F1 | Mean | 0.00 | 2.42 × 10−97 | 4.98 × 10−9 | 7.11 × 10−3 | 2.81 × 10−1 | 3.29 × 10−7 | 8.31 × 10−56 |
SD | 0.00 | 7.22 × 10−97 | 1.40 × 10−8 | 3.21 × 10−3 | 1.11 × 10−1 | 5.92 × 10−7 | 1.02 × 10−58 | |
F2 | Mean | 0.00 | 1.16 × 10−52 | 7.29 × 10−4 | 4.34 × 10−1 | 3.96 × 10−1 | 1.9111 | 8.36 × 10−35 |
SD | 0.00 | 2.55 × 10−52 | 1.84 × 10−3 | 1.84 × 10−1 | 1.41 × 10−1 | 1.6142 | 9.86 × 10−35 | |
F3 | Mean | 0.00 | 7.84 × 10−81 | 1.40 × 10 | 1.66 × 103 | 4.31 × 10 | 1.50 × 103 | 1.51 × 10−14 |
SD | 0.00 | 3.49 × 10−80 | 7.13 | 6.72 × 102 | 8.97 | 707.05 | 6.55 × 10−14 | |
F4 | Mean | 0.00 | 4.57 × 10−46 | 6.00 × 10−1 | 1.11 × 10−1 | 8.80 × 10−1 | 2.44 × 10−5 | 1.95 × 10−5 |
SD | 0.00 | 9.98 × 10−46 | 1.72 × 10−1 | 4.75 × 10−2 | 2.50 × 10−1 | 1.89 × 10−5 | 4.49 × 10−4 | |
F5 | Mean | 7.22 × 10−8 | 2.80 × 10 | 4.93 × 10 | 7.97 × 10 | 1.18 × 102 | 136.56 | 28.4 |
SD | 4.78 × 10−9 | 8.73 × 10−1 | 3.89 × 10 | 7.39 × 10 | 1.43 × 102 | 154.00 | 0.842 | |
F6 | Mean | 0.00 | 2.15 | 6.92 × 10−2 | 6.94 × 10−3 | 2.02 × 10−2 | 5.72 × 10−7 | 3.67 |
SD | 0.00 | 4.47 × 10−1 | 2.87 × 10−2 | 3.61 × 10−3 | 7.43 × 10−3 | 2.44 × 10−7 | 0.3353 | |
F7 | Mean | 1.39 × 10−5 | 1.51 × 10−4 | 8.94 × 10−2 | 6.62 × 10−2 | 5.24 × 10−2 | 8.82 × 10−5 | 0.0018 |
SD | 2.65 × 10−5 | 1.33 × 10−4 | 0.0206 | 4.23 × 10−2 | 1.37 × 10−2 | 6.94 × 10−5 | 4.62 × 10−4 |
F | Index | CSCPS | SCO | PSO | FA | MVO | SSA | TSA |
---|---|---|---|---|---|---|---|---|
F8 | Mean | −1.25 × 104 | −1.01 × 104 | −6.01 × 103 | −5.85 × 103 | −6.92 × 103 | −7.46 × 103 | −7.89 × 103 |
SD | 0.00 | 1.70 × 103 | 1.30 × 103 | 1.61 × 103 | 9.19 × 102 | 634.67 | 599.26 | |
F9 | Mean | 0.00 | 0.00 | 4.72 × 10 | 1.51 × 10 | 1.01 × 102 | 55.45 | 151.45 |
SD | 0.00 | 0.00 | 1.03 × 10 | 1.25 × 10 | 1.89 × 10 | 18.27 | 35.87 | |
F10 | Mean | 8.88 × 10−16 | 8.77 × 10−16 | 3.86 × 10−2 | 4.58 × 10−2 | 1.15 | 2.84 | 2.409 |
SD | 0.00 | 0.00 | 2.11 × 10−1 | 1.20 × 10−2 | 7.87 × 10−1 | 6.58 × 10−1 | 1.392 | |
F11 | Mean | 0.00 | 0.00 | 5.50 × 10−3 | 4.23 × 10−3 | 5.74 × 10−1 | 2.29 × 10−1 | 0.0077 |
SD | 0.00 | 0.00 | 7.39 × 10−3 | 1.29 × 10−3 | 1.12 × 10−1 | 1.29 × 10−1 | 0.0057 | |
F12 | Mean | 1.57 × 10−32 | 1.25 × 10−1 | 1.05 × 10−2 | 3.13 × 10−4 | 1.27 | 6.82 | 6.373 |
SD | 2.88 × 10−48 | 5.41 × 10−2 | 2.06 × 10−2 | 1.76 × 10−4 | 1.02 | 2.72 | 3.458 | |
F13 | Mean | 1.35 × 10−32 | 1.99 | 4.03 × 10−1 | 2.08 × 10−3 | 6.60 × 10−2 | 21.31 | 2.897 |
SD | 2.95 × 10−48 | 2.51 × 10−1 | 5.39 × 10−1 | 9.62 × 10−4 | 4.33 × 10−2 | 16.99 | 0.643 |
Optimization Method | Analysis Method | Number of Slices | Minimum FOS | ||
---|---|---|---|---|---|
Kh = 0.0 | Kh = 0.1 | Kh = 0.2 | |||
GA [6] | Morgenstern–Price | - | 1.75 | _ | _ |
SA [8] | Spencer’s method | 40 | 1.7267 | _ | _ |
HS [8] | Spencer’s method | 40 | 1.7264 | _ | _ |
PSO [11] | Bishop’s method | 40 | 1.7195 | _ | _ |
SCO (current study) | Morgenstern–Price | 40 | 1.7275 | 1.4012 | 1.1897 |
CSCPS (current study) | Morgenstern–Price | 40 | 1.7132 | 1.3803 | 1.1424 |
Soil Properties | Unit | Layer Number | |||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
Unit weight, γ | kN/m3 | 18.63 | 18.63 | 18.63 | 18.63 |
Cohesion, c′ | kPa | 14.7 | 16.7 | 4.9 | 34.3 |
Friction angle, ϕ′ | Degree | 20 | 21 | 10 | 28 |
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Iraji, A.; Karimi, J.; Keawsawasvong, S.; Nehdi, M.L. Minimum Safety Factor Evaluation of Slopes Using Hybrid Chaotic Sand Cat and Pattern Search Approach. Sustainability 2022, 14, 8097. https://doi.org/10.3390/su14138097
Iraji A, Karimi J, Keawsawasvong S, Nehdi ML. Minimum Safety Factor Evaluation of Slopes Using Hybrid Chaotic Sand Cat and Pattern Search Approach. Sustainability. 2022; 14(13):8097. https://doi.org/10.3390/su14138097
Chicago/Turabian StyleIraji, Amin, Javad Karimi, Suraparb Keawsawasvong, and Moncef L. Nehdi. 2022. "Minimum Safety Factor Evaluation of Slopes Using Hybrid Chaotic Sand Cat and Pattern Search Approach" Sustainability 14, no. 13: 8097. https://doi.org/10.3390/su14138097