
Citation: Shikai Wang, Heming Jia, Xiaoxu Peng. Modified salp swarm algorithm based multilevel thresholding for color image segmentation[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 700-724. doi: 10.3934/mbe.2020036
[1] | Bernardo D'Auria, José Antonio Salmerón . A note on Insider information and its relation with the arbitrage condition and the utility maximization problem. Mathematical Biosciences and Engineering, 2023, 20(5): 8305-8307. doi: 10.3934/mbe.2023362 |
[2] | Tao Li, Xin Xu, Kun Zhao, Chao Ma, Juan LG Guirao, Huatao Chen . Low-carbon strategies in dual-channel supply chain under risk aversion. Mathematical Biosciences and Engineering, 2022, 19(5): 4765-4793. doi: 10.3934/mbe.2022223 |
[3] | Zihan Chen, Minhui Yang, Yuhang Wen, Songyan Jiang, Wenjun Liu, Hui Huang . Prediction of atherosclerosis using machine learning based on operations research. Mathematical Biosciences and Engineering, 2022, 19(5): 4892-4910. doi: 10.3934/mbe.2022229 |
[4] | Min Zhu, Xiaofei Guo, Zhigui Lin . The risk index for an SIR epidemic model and spatial spreading of the infectious disease. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1565-1583. doi: 10.3934/mbe.2017081 |
[5] | Li Yang, Kai Zou, Kai Gao, Zhiyi Jiang . A fuzzy DRBFNN-based information security risk assessment method in improving the efficiency of urban development. Mathematical Biosciences and Engineering, 2022, 19(12): 14232-14250. doi: 10.3934/mbe.2022662 |
[6] | Yuan Yang, Lingshan Zhou, Xi Gou, Guozhi Wu, Ya Zheng, Min Liu, Zhaofeng Chen, Yuping Wang, Rui Ji, Qinghong Guo, Yongning Zhou . Comprehensive analysis to identify DNA damage response-related lncRNA pairs as a prognostic and therapeutic biomarker in gastric cancer. Mathematical Biosciences and Engineering, 2022, 19(1): 595-611. doi: 10.3934/mbe.2022026 |
[7] | Irfanullah Khan, Asif Iqbal Malik, Biswajit Sarkar . A distribution-free newsvendor model considering environmental impact and shortages with price-dependent stochastic demand. Mathematical Biosciences and Engineering, 2023, 20(2): 2459-2481. doi: 10.3934/mbe.2023115 |
[8] | Zhiqiang Li, Sheng Wang, Yupeng Cao, Ruosong Ding . Dynamic risk evaluation method of collapse in the whole construction of shallow buried tunnels and engineering application. Mathematical Biosciences and Engineering, 2022, 19(4): 4300-4319. doi: 10.3934/mbe.2022199 |
[9] | Vincenzo Luongo, Maria Rosaria Mattei, Luigi Frunzo, Berardino D'Acunto, Kunal Gupta, Shankararaman Chellam, Nick Cogan . A transient biological fouling model for constant flux microfiltration. Mathematical Biosciences and Engineering, 2023, 20(1): 1274-1296. doi: 10.3934/mbe.2023058 |
[10] | Shirin Ramezan Ghanbari, Behrouz Afshar-Nadjafi, Majid Sabzehparvar . Robust optimization of train scheduling with consideration of response actions to primary and secondary risks. Mathematical Biosciences and Engineering, 2023, 20(7): 13015-13035. doi: 10.3934/mbe.2023580 |
Abbreviations: (α, β, γ, η, θ, λ, μ, a, b): Model parameters; R (t): Reliability function; MTBF (τ): the average Lifetime; PV: Photovoltaic; a-Si: Amorphous silicon; pc-Si: Polycrystalline silicon; mc-Si: Single crystalline silicon
Many articles in literature have studied the degradation of photovoltaic modules when exposed to natural environments using accelerated tests to observe degradation in reality [1,2]. A study confirmed that after 20 years of continuous exposure a matrix of 70 polycrystalline silicon photovoltaic modules has undergone an average performance decay of 0.24% per year in a moderate subtropical climate environment [3,4]. Another study stated that after only one year of exposure in a tropical climate environment the electrical powers of two modules of type (a-Si) and (pc-Si) were degraded to 60% and 56% respectively of their initial values [5]. In addition to these results, another study has shown that some photovoltaic modules (mc-Si and pc-Si) had been degraded by 0.22% /year to 2.96% / year for the maximum power [6]. In the long run, the polycrystalline silicon modules have the best reliability with a degradation rate of 0.41% per year in a natural environment [7]. In a tropical environment (Ghana), the exposure of 14 polycrystalline silicon modules during a 19-year period recorded a degradation rate of 21% to 35% of nominal power [8]. The degradation is in the order of 1.2% per year for polycrystalline silicon modules and 0.8% per year for single crystalline silicon modules [9]. An important study that followed the degradation of 204 modules (123 mc-Si and 81 pc-Si) had revealed a degradation variance from 0 to 6% per year for exposure periods of 18 years to 24 years in a subtropical moderate environment [10]. In Saharan environment (southern Algeria for example) the degradation rate of polycrystalline silicon modules was very high ranging from 3.33% / year to 4.64% / year unlike mono-crystalline silicon modules which recorded a rate of 1.22% / year after 28 years of exposure [11,12]. Accelerated tests cannot evaluate totally the effect of natural environment on electrical and optical characteristics of a photovoltaic module [13]. But it is the only method to see the effect of single factor or limited number of climatic factors [13,14]. The return of experimental data within a period of operation in a natural environment allows to predict the lifetime and the degradation over the long term [15]. Our objective in this study is to search in the literature for an adequate model to simulate the reliability of photovoltaic modules (crystalline silicon) exposed in desert environments in order to probably estimate their degradation at any period of their operation. The method consists of using a genetic algorithm (artificial intelligence optimization method) to estimate the unknown parameters of the models and to check the competence of the simulation by comparing with feedbacks of experimental data.
Two kinds of methods in the literature are used to predict the duration of good operation and the reliability of a photovoltaic module exposed in a natural environment, the first that uses the return of experiments, or the second that utilizes accelerated tests [16]. In this study, we use the feedback data that are practically measured in the desert of California and in the Algerian Sahara (Adrar region), extracted from references [16,17,18] to estimate the lifespan of photovoltaic modules (single crystalline silicon) in these environments. In order to calculate the parameters of models we will use a genetic algorithm. The iterative stochastic genetic algorithm uses an initial population to reach an optimal solution of any problem [19]. The initially chosen population has evolved from generation to generation where the most suitable individuals have a great chance of breeding. This mechanism of intelligence is realized by respecting the following steps [20,21]:
1. Creation of an initial population
2. Assessment of individuals in population
3. Selection of adapted individuals
4. Reproduction by crossing and mutation
5. Formation of a new generation
This process is circulated until an optimal solution is obtained. Practically, we represent these steps according to the flowchart below (Figure 1):
Originally, the reliability concerned the high technology systems (nuclear, aerospace...) to guarantee their operational safety. Today, all areas are interested in the study of reliability to make decisions on ratio Cost / gain and to control the failure sources [22,23]. Reliability of a system is a quantity characterizing the safety of operation or measuring the probability of operation of an appliance according to prescribed standards (definition presented in 1962 by the Academy of Sciences). Reliability (or survival function) is expressed by:
R(t)=r−∫t0h(x)dx | (1) |
h(t): Instant failure rate (probability of seeing a failure in a short interval after instant t.
The average time of operation (lifetime) which is the Mean Time Before Failure (MTBF) is given by:
MTBF=∫+∞0R(t)dt | (2) |
According to their instantaneous rates of failure, the parametric reliability models are classified in the literature as follows [24,25,26]:
1. Models of constant rate: Exponential model.
2. Models of monotone rate: Weibul model, gamma model, Gompertz-Makeham model, exponential Weibul model, Mix of exponential models.
3. Models of rates of a bathtub shape: Modified Weibul model, exponential power model, quadratic model, and uniform model.
4. Models of rate in bell form: Generalized Weibul model, normal model, log-normal model, log logistic model, extreme values model.
The characteristics of chosen models are presented in the table 1 below:
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
After filtering we present only the cases where the calculated average error is less than 2%. This choice is solely made to limit the size of the study. The other cases are not interesting as our goal is to visualize the most adequate model having the least error.
Estimated parameters and simulated reliability graphs are shown below (Table 2, Figures 2 and 3):
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
After 20 years of operation in Adrar region we observe that the Weibul modified model predicts a 30% degradation of starting value of electrical power for this type of photovoltaic modules while the degradation is approximately of 38% in the desert of California.
In the Table 3 and Figures (4, 5) we present the estimated parameters and the simulated reliability:
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In this case the generalized Weibul model predicts a degradation of 35% in Adrar region after 20 years of operation while the degradation is approximately of 44% in the desert of California.
The Table 4 and the Figures 6, 7 show the estimated parameters and the simulated reliability of PV module:
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
By this model the degradation is about of 38% in Adrar region after 20 years of operation while the degradation is approximately of 46% in the desert of California.
Estimated parameters and simulated reliability graphs are shown below (Table 5, Figures 8 and 9):
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
By model of extreme values, the degradation is about of 22% in Adrar region after 20 years of operation while it is approximately of 37% in the desert of California.
In this case the estimated parameters and the simulated reliability are shown below (table 6, figure 10 and 11):
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Finally, the uniform model predicts a degradation of 24% in Adrar region After 20 years of operation while it is approximately of 41% in the desert of California.
These results are summarized in the following Table 7.
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |
The above results indicate that:
1. The experimental data used have guided us to predict the future of solar panels operating in desert environments. We therefore believe that more return data will give us confidence in models and methods.
2. The calculated mean error that present the average relative distance of the graph from the points of comparison is generally small (especially for the modified Weibul model). These reflect the skill of the optimization method used (the genetic algorithm).
3. Extrapolation of curves in longer durations allows informing on the reliability (outside of periods of real measurements).
4. The modified Weibul model is the most adequate of the models tested to simulate the reliability of photovoltaic modules (single crystalline silicon) and to estimate their lifetimes (MTBF) in the desert environments. It predicted a duration of nearly 30 years in the desert of California and of around 29 years for the Adrar area.
5. It should be noted that the degradation of electrical power of photovoltaic modules in Californian desert is significant compared to that of Adrar region in the first step (in the initial period of 30 years).
6. These obtained results are more or less comparable to those stated in references [11,12] (a degradation close to 1.53% /year in this study).
It has been confirmed in this article that the modified Weibul law is the most adequate model compared to other tested models to simulate the reliability function of photovoltaic modules and estimate their lifetime while operating in desert environments (California and Adrar). Using simulation findings, an average lifespan of about 30 years has been predicted for photovoltaic modules exposed in desert regions where the maximum power of the photovoltaic module is degraded to almost 46% of its initial value. The annual rate of degradation is in the order of 1.5% / year. This obtained result is more or less comparable to those presented in the literature. The prediction results must be taken into consideration for any study of construction of solar stations in the Saharan environments.
I thank my fellow researchers in Renewable Energy Research Unit in Saharan areas (URERMS) for all given help.
The authors declare there are no conflicts of interest in this paper.
[1] | Y. Feng, X. Shen, H. Chen, et al., Segmentation fusion based on neighboring information for MR brain images, Multimedia Tools Appl., 76 (2017), 23139-23161. |
[2] |
C. Wang, A. Y. Shi, X. Wang, et al., A novel multi-scale segmentation algorithm for high resolution remote sensing images based on wavelet transform and improved JSEG algorithm, Optik, 125 (2014), 5588-5595. doi: 10.1016/j.ijleo.2014.07.002
![]() |
[3] | R. Gao and H. Wu, Agricultural image target segmentation based on fuzzy set, Optik, 126 (2015), 5320-5324. |
[4] | K. Hammouche, M. Diaf and P. Siarry, A comparative study of various meta-heuristic techniques applied to the multilevel thresholding problem, Eng. Appl. Artif. Intell., 23 (2010), 676-688. |
[5] | M. Sezgin and B. Sankur, Survey over image thresholding techniques and quantitative performance evaluation, J. Electron. Imaging, 13 (2004), 146-165. |
[6] | A. García-Pedrero, C. Gonzalo-Martín and M. Lillo-Saavedra, A machine learning approach for agricultural parcel delineation through agglomerative segmentation, Int. J. Remote Sens., 38 (2017), 1809-1819. |
[7] | D. Li, G. Zhai, X. Yang, et al., Perceptual information hiding based on multi-channel visual masking, Neurocomputing, 269 (2017), 170-179. |
[8] | S. Yin, Y. Qian, and M. Gong, Unsupervised Hierarchical Image Segmentation through Fuzzy Entropy Maximization, Pattern Recognit., 68 (2017), 245-259. |
[9] | S. Kumar, P. Kumar, T. K. Sharma, et al., Bi-level thresholding using PSO, artificial bee colony and MRLDE embedded with Otsu method, Memetic Comput., 5 (2013), 323-334. |
[10] | A. Colorni, M. Dorigo and V. Maniezzo, Distributed optimization by ant colonies, Proceedings of the first European conference on artificial life, 1992, 134-142. Available from: https://zz.glgoo.top/books?hl=zh-CN&lr=&id=pWsNJkdZ4tgC&oi=fnd&pg=PA134&dq=Distributed+Optimization+by+Ant+Colonies&ots=86J4mUqQSC&sig=_d2DgNHGaDzWKRQuxcGEhpNKRaI#v=onepage&q=Distributed%20Optimization%20by%20Ant%20Colonies&f=false. |
[11] | W. Ding, C. Lin, S. Chen, et al., Multiagent-consensus-Map Reduce-based attribute reduction using co-evolutionary quantum PSO for big data applications, Neurocomputing, 272 (2018), 136-153. |
[12] | R. Eberhart and J. Kennedy, A new optimizer using particle swarm theory, Proceedings of the Sixth International Symposium on IEEE, 1995, 39-43. Available from: https://ieeexplore.ieee.org/document/494215. |
[13] | K. Mistry, L. Zhang, S. C. Neoh, et al., A Micro-GA Embedded PSO Feature Selection Approach to Intelligent Facial Emotion Recognition, IEEE Trans. Cybern., 47 (2017), 1496-1509. |
[14] | R. Dong, J. Xu and B. Lin, ROI-based study on impact factors of distributed PV projects by LSSVM-PSO, Energy, 124 (2017), 336-349. |
[15] | A. Fakhry, T. Zeng and S. Ji, Residual Deconvolutional Networks for Brain Electron Microscopy Image Segmentation, IEEE Trans. Med. Imaging, 36 (2017), 447-456. |
[16] | D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm, J. Global Optim., 39 (2007), 459-471. |
[17] | B. Jafrasteh and N. Fathianpour, Automatic extraction of geometrical characteristics hidden in ground-penetrating radar sectional images using simultaneous perturbation artificial bee colony algorithm, Geophys. Prospect., 65 (2017), 324-336. |
[18] | Y. Zhang and L. Wu, Optimal multi-level thresholding based on maximum Tsallis entropy via an artificial bee colony approach, Entropy., 13 (2011), 841-859. |
[19] | X. S. Yang, Firefly Algorithm, Lévy Flights and Global Optimization, Res. Dev. Intell. Syst. XXVI, 20 (2010), 209-218. |
[20] | M. F. P. Costa, A. M. A. C. Rocha, R. B. Francisco, et al., Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming, Optimization, 65 (2016), 1085-1104. |
[21] | O. P. Verma, D. Aggarwal and T. Patodi, Opposition and dimensional based modified firefly algorithm, Expert Syst. Appl., 44 (2016), 168-176. |
[22] | K. M. Sundaram, R. S. Kumar, C. Krishnakumar, et al., Fuzzy Logic and Firefly Algorithm based Hybrid System for Energy Efficient Operation of Three Phase Induction Motor Drives, Indian J. Sci. Technol., 9 (2016), 1-5. |
[23] | X. S. Yang, A New Metaheuristic Bat-Inspired Algorithm, Nat. Inspired Coop. Strategies Optim., 284 (2010), 65-74. |
[24] | H. Liang, Y. Liu, Y. Shen, et al., A Hybrid Bat Algorithm for Economic Dispatch with Random Wind Power, IEEE Trans. Power Syst., 99 (2018), 5052-5061. |
[25] | A. Mumtaz, R. Deo, N. Downs, et al., Multi-stage hybridized online sequential extreme learning machine integrated with Markov Chain Monte Carlo copula-Bat algorithm for rainfall forecasting, Atmos. Res., 213 (2018), 450-464. |
[26] |
Y. Yuan, X. Wu, P. Wang, et al., Application of improved bat algorithm in optimal power flow problem, Appl. Intell., 48 (2018), 2304-2314. doi: 10.1007/s10489-017-1081-2
![]() |
[27] | K. Kaced, C. Larbes, N. Ramzan, et al., Bat algorithm based maximum power point tracking for photovoltaic system under partial shading conditions, Sol. Energy, 158 (2017), 490-503. |
[28] | S. Mirjalili and A. Lewis, The Whale Optimization Algorithm, Adv. Eng. Software, 95 (2016), 51-67. |
[29] | R. Gupta, S. Ruosaari, S. Kulathinal, et al., Microarray image segmentation using additional dye-An experimental study, Mol. Cell. Probes, 21 (2007), 321-328. |
[30] | O. Diego, M. A. E. Aziz and A. E. Hassanien, Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm, Appl. Energy, 200 (2017), 141-154. |
[31] | N. Nahas, A. Khatab, D. Ait-Kadi, et al., Extended great deluge algorithm for the imperfect preventive maintenance optimization of multi-state systems, Reliab. Eng. Syst. Saf., 93 (2008), 1658-1672. |
[32] | D. H. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput., 1 (1997), 67-82. |
[33] | R. A. Ibrahim, A. A. Ewees, D. Oliva, et al., Improved salp swarm algorithm based on particle swarm optimization for feature selection, J. Ambient Intell. Humanized Comput., 10 (2019), 3155-3169. |
[34] | A. G. Hussien, A. E. Hassanien and E. H. Houssein, Swarming Behaviour of Salps Algorithm for Predicting Chemical Compound Activities, 2017 Eighth International Conference on Intelligent Computing and Information Systems, Egypt, 2018. Available from: https://ieeexplore_ieee.gg363.site/abstract/document/8260072. |
[35] | G. I. Sayed, G. Khoriba and M. H. Haggag, A novel chaotic salp swarm algorithm for global optimization and feature selection, Appl. Intell.,48 (2018), 3462-3481. |
[36] | M. H. Qais., H. M. Hasanien and S. Alghuwainem, Enhanced salp swarm algorithm: Application to variable speed wind generators, Eng. Appl. Artif. Intell., 80 (2019), 82-96. |
[37] | M. Sezgin and B. Sankur, Survey over image thresholding techniques and quantitative performance evaluation, J. Electron. Imaging, 13 (2004), 146-166. |
[38] | T. Pun, A New Method for Gray-Level Picture Thresholding Using the Entropy of the Histogram, Signal Process., 2 (1980), 223-237. |
[39] | N. Otsu, Threshold Selection Method from Gray-Level Histograms, IEEE Trans. Syst. Man Cybern., 9 (1979), 62-66. |
[40] | M. Subrahmanyam, Q. M. J. Wu, R. P. Maheshwari, et al., Modified color motif cooccurrence matrix for image indexing and retrieval, Comput. Electr. Eng., 39 (2013), 762-774. |
[41] | H. Gao, C. Pun and K. Sam, An efficient image segmentation method based on a hybrid particle swarm algorithm with learning strategy, Inf. Sci., 369 (2016), 500-521. |
[42] | U. Kandaswamy, D. A. Adjeroh and M. C. Lee, Efficient texture analysis of SAR imagery, IEEE Trans. Geosci. Remote Sens., 43 (2005), 2075-2083. |
[43] | K. S. Tan and N. A. M. Isa, Color image segmentation using histogram thresholding-Fuzzy C-means hybrid approach, Pattern Recognit., 44 (2011), 1-15. |
[44] | B. Akay, A study on particle swarm optimization and artificial bee colony algorithms for multilevel thresholding, Appl. Soft Comput., 13 (2013), 3066-3091. |
[45] | A. K. Bhandari, A. Kumar, S. Chaudhary, et al., A novel color image multilevel thresholding based segmentation using nature inspired optimization algorithms, Expert Syst. Appl., 63 (2016), 112-133. |
[46] | S. Pare, A. K. Bhandari, A. Kumar, et al., An optimal Color Image Multilevel Thresholding Technique using Grey-Level Co-occurrence Matrix, Expert Syst. Appl., 87 (2017), 335-362. |
[47] | A. K. Bhandari, V. K. Singh, A. Kumar, et al., Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur's entropy, Expert Syst. Appl., 41 (2014), 3538-3560. |
[48] | A. K. M. Khairuzzaman and S. Chaudhury, Multilevel thresholding using grey wolf optimizer for image segmentation, Expert Syst. Appl., 86 (2017), 64-76. |
[49] | H. Liang, H. Jia, Z. Xing, et al., Modified Grasshopper Algorithm-Based Multilevel Thresholding for Color Image Segmentation, IEEE Access, 7 (2019), 11258-11295. |
[50] | L. He and S. Huang, Modified firefly algorithm based multilevel thresholding for color image segmentation, Neurocomputing, 240 (2017), 152-174. |
[51] | Z. Xing and H. Jia, Multilevel Color Image Segmentation Based on GLCM and Improved Salp Swarm Algorithm, IEEE Access, 7 (2019), 37672-37690. |
[52] | S. Mishra and M. Panda, Bat Algorithm for Multilevel Colour Image Segmentation Using Entropy-Based Thresholding, Arabian J. Sci. Eng., 43 (2018), 1-30. |
[53] | M. Z. Ali, N. H. Awad., G. R. Robert, et al., A balanced Fuzzy Cultural Algorithm with a Modified Levy Flight Search for Real Parameter Optimization, Inf. Sci., 447 (2018), 12-35. |
[54] | A. A. Dubkov, B. Spagnolo and V. V. Uchaikin, Levy Flight Superdiffusion: An Introduction, Int. J. Bifurcation Chaos, 18 (2008), 2649-2672. |
[55] | R. Li and Y. Wang, Improved Particle Swarm Optimization Based on Lévy Flights, J. Syst. Simul., 29 (2017), 1685-1691. |
[56] |
A. Mesa, K. Castromayor, C. Garillos-Manliguez, et al., Cuckoo search via Levy flights applied to uncapacitated facility location problem, J. Ind. Eng. Int., 14 (2018), 585-592. doi: 10.1007/s40092-017-0248-0
![]() |
[57] | S. J. Mousavirad, H. Ebrahimpour-Komleh, Human mental search: A new population-based metaheuristic optimization algorithm, Appl. Intell., 47 (2017), 850-887. |
[58] | P. D. Sathya and R. Kayalvizhi, Modified bacterial foraging algorithm based mul-tilevel thresholding for image segmentation, Expert Syst. Appl., 24 (2011), 595-615. |
[59] | S. Mirjalili, A. H. Gandomi, S. Z. Mirjalili, et al., Salp Swarm Algorithm: A bio-inspired optimizer for engineering design problems, Adv. Eng. Software, 114 (2017), 13-48. |
[60] | I. Pavlyukevich, Lévy flights, non-local search and simulated annealing, J. Comput. Phys., 226 (2007), 1830-1844. |
[61] | P. Imkeller and I. Pavlyukevich, Lévy flights: Transitions and meta-stability, J. Phys. A Math. Gen., 39 (2006), 237-246. |
[62] | Z. Chen, T. J. Feng and Z. Houkes, Texture segmentation based on wavelet and Kohonen network for remotely sensed images, IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028), 1999, 816-821. Available from: https://ieeexplore_ieee.gg363.site/abstract/document/816656. |
[63] | M. Cuinin, Segmentation 3D des organes à risque du tronc masculin à partir d'images anatomiques TDM et IRM à l'aide de méthodes hybrids (in French), Normandie, 98 (2017), 1188. |
[64] | H. Jia, Z. Xing, W. Song, Three Dimensional Pulse Coupled Neural Network Based on Hybrid Optimization Algorithm for Oil Pollution Image Segmentation, Remote Sens., 11 (2019), 1046. |
[65] | F. Wilcoxon, Individual comparisons by ranking methods, Breakthroughs Stat., 1992 (1992), 196-202. |
1. | Bernardo D’Auria, Eduardo García-Portugués, Abel Guada, Discounted Optimal Stopping of a Brownian Bridge, with Application to American Options under Pinning, 2020, 8, 2227-7390, 1159, 10.3390/math8071159 | |
2. | Bernardo D’Auria, Jose A. Salmeron, Anticipative information in a Brownian−Poisson market, 2022, 0254-5330, 10.1007/s10479-022-05060-0 | |
3. | Bernardo D'Auria, José Antonio Salmerón, A note on Insider information and its relation with the arbitrage condition and the utility maximization problem, 2023, 20, 1551-0018, 8305, 10.3934/mbe.2023362 |
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |
Model | Model of two parameters | |
Reliability function | Average lifetime | |
Exponential model | R(t)=e−λtwithλ>0 | MTBF=1λ |
Weibul model | R(t)=e−(tη)β;β,η>0 | MTBF=ηΓ(1+1/β) |
Gamma model | R(t)=1−1Γ(μ)∫θt0xμ−1e−xdx;(μ,θ)>0 | MTBF=μθ |
Exponential power model | R(t)=e1−e(λt)αwithα>0;λ,α>0 | MTBF=∫+∞0e1−e(λt)αdt |
Normal model | R(t)=1−1√2πσ∫+∞0e−(x−μ)22σ2dx | MTBF=μ |
Log-normal model | R(t)=1−1√2πσ∫Int−∞e−(x−μ)22σ2dx | MTBF=eμ+σ22 |
Log logistic model | R(t)=αβαβ+tβorα>0;β>1 | MTBF=∫+∞0αβαβ+tβdt |
Uniform model | R(t)=b−tb−a;fort∈[a,b] | MTBF=∫bab−tb−adt |
Extreme values model | R(t)=e−α(eβt−1)withα>0,β>0 | MTBF=∫+∞0e−α(eβt−1)dt |
Gompertz-Makeham model | R(t)=e−at−bInc(ct−1) | MTBF=∫+∞0e−at−bInc(ct−1)dt |
exponential Weibul model | R(t)=1−{1−e−(tη)β}μorη,β,μ>0 | MTBF=∫+∞01−{1−e−(tη)β}μdt |
Mix of exponential models | R(t)=a1e−tθ1+(1−a1)e−tθ2;θ1,θ2>0;0<a1<1 | MTBF=a1θ1+(1−a1)θ2 |
Modified Weibul model | R(t)=e−(tη)βeμtwith(η;β;μ>0) | MTBF=∫+∞0e−(tη)βeμtdt |
Quadratic model | R(t)=e−(αt+β2t2+γ3t3)α,γ>0;−2√γα≤β≤0 | MTBF=∫+∞0e−(αt+β2t2+γ3t3)dt |
Generalized Weibul model | R(t)=e1−(1+(tη)β)1γwith(η,β,γ)>0 | MTBF=∫+∞0e1−(1+(tη)β)1γdt |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (71, 1.35, 0.03) | (49, 1.05, 0.01) |
Average lifetime MTBF (years) | 28.75 | 30.47 |
Uncertainty | 0.0039 | 0.0059 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, γ) | (38, 2.5, 0.5) | (77, 1.2, 0.4) |
Average lifetime MTBF (years) | 23.45 | 26.09 |
Uncertainty | 0.0184 | 0.0092 |
In Adrar Sahara | In the desert of California | |
Parameters model (η, β, μ) | (28, 4.05, 0.66) | (37, 2.65, 0.45) |
Average lifetime MTBF (years) | 22.57 | 23.40 |
Uncertainty | 0.0254 | 0.0028 |
In Adrar Sahara | In the desert of California | |
Parameters model (α, β) | (0.015, 0.68) | (0.02, 0.9) |
Average lifetime MTBF (years) | 50.09 | 31.99 |
Uncertainty | 0.0081 | 0.0227 |
In Adrar Sahara | In the desert of California | |
Parameters model (a, b) | (1.6, 77) | (0.4, 48) |
Average lifetime MTBF (years) | 37.70 | 23.80 |
Uncertainty | 0.0074 | 0.0139 |
Model of Reliability | Average life time (years) | Error Means (%) | |
in Adrar | in California | ||
modified Weibul model | 28.75 | 30.47 | 0.4 |
Uniform model | 37.70 | 23.80 | 1.1 |
Generalized Weibul model | 23.45 | 26.09 | 1.3 |
exponential Weibul model | 22.57 | 23.40 | 1.4 |
Extreme values model | 50.09 | 31.99 | 1.5 |
Average lifetime calculated | 32.51 | 27.15 | |
29.83 ≈ 30years |