Next Article in Journal
Using Real-Time Dynamic Prediction to Implement IoV-Based Collision Avoidance
Next Article in Special Issue
Development of Hybrid Machine Learning Models for Predicting the Critical Buckling Load of I-Shaped Cellular Beams
Previous Article in Journal
Deep Multiphysics and Particle–Neuron Duality: A Computational Framework Coupling (Discrete) Multiphysics and Deep Learning
Previous Article in Special Issue
A New Approach of Hybrid Bee Colony Optimized Neural Computing to Estimate the Soil Compression Coefficient for a Housing Construction Project
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Applied Sciences
Volume 9
Issue 24
10.3390/app9245372
applsci-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up
...
Endorse textsms
...
Comment
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Invasive Weed Optimization Technique-Based ANN to the Prediction of Rock Tensile Strength

by
Lei Huang
1,
Panagiotis G. Asteris
2,*,
Mohammadreza Koopialipoor
3,
Danial Jahed Armaghani
4,* and
M. M. Tahir
5
1
Water Conservancy and Civil Engineering College, Inner Mongolia Agricultural University, Hohhot 010018, China
2
Computational Mechanics Laboratory, School of Pedagogical and Technological Education, 14121 Athens, Greece
3
Faculty of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran 15914, Iran
4
Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam
5
UTM Construction Research Centre, Institute for Smart, Infrastructure and Innovative Construction (ISIIC), Faculty of Engineering, School of Civil Engineering, Universiti Teknologi Malaysia, Skudai 81310, Malaysia
*
Authors to whom correspondence should be addressed.
Appl. Sci. 2019, 9(24), 5372; https://doi.org/10.3390/app9245372
Submission received: 18 November 2019 / Revised: 3 December 2019 / Accepted: 6 December 2019 / Published: 9 December 2019
(This article belongs to the Special Issue Meta-heuristic Algorithms in Engineering)
Browse Figures
Versions Notes

Abstract

:
In many site investigation phases of civil and mining engineering projects, the tensile strength of the rocks is one of the most significant parameters that must be identified. This parameter can be determined directly through laboratory tests. However, conducting such laboratory tests is costly and time consuming. In this paper, a new artificial neural network (ANN)-based model is developed to predict rock tensile strength, using the invasive weed optimization (IWO) technique. Granite samples for the purpose of this research were selected from a tunnel located in Malaysia and underwent appropriate laboratory tests (i.e., Schmidt hammer, point load, dry density, as well as the Brazilian tensile strength (BTS) as system output). A simple regression analysis was carried out, and the obtained results confirmed the need for developing a model with multiple inputs, rather than one with only a single input, in order to predict BTS values. Aiming to highlight the capability of an IWO-ANN model in estimating BTS, artificial bee colony (ABC)-ANN and imperialism competitive algorithm (ICA)-ANN were also applied and developed. The parameters required for the ANN-based models were obtained using different parametric studies. According to calculated performance indices, a new hybrid IWO-ANN model can provide a higher accuracy level for the prediction of BTS compared to the ABC-ANN and ICA-ANN models. The results showed that the IWO-ANN model is a suitable alternative solution for a robust and reliable engineering design.

1. Introduction

Rock tensile strength (TS) is extensively used as a significant parameter when designing a geotechnical construction, such as a tunnel [1]. Therefore, literature is consisted of numerous methods attempting to predict the TS value, either directly or indirectly. In the direct approach, the researchers or practitioners have to either make use of the empirical equations already present in the relevant literature, or gather rock specimens and then test them in laboratory [1,2,3]. It should be noted that it is both time consuming and costly to prepare proper rock samples with the aim of performing TS tests in a laboratory [3]. On the other hand, in the indirect approach, the whole process can be made faster, simpler, and less costly by predicting the TS value with the help of other, less demanding laboratory experiments, e.g., the Schmidt hammer, P-wave velocity, density, and point load tests [1,4,5]. The list can be further elaborated with the Brazilian tensile strength (BTS) method, which has been proposed by the International Society for Rock Mechanics (ISRM) as an efficient method for determining TS [6].
Different researchers have suggested different empirical relations for the prediction of the TS value. Using statistical analyses, such as regression analysis and coefficients of variation, Kahraman et al. [7] attempted to determine the rock features with key impacts on the penetration rate of percussive drills. The properties they marked out as significant were the point load strength, uniaxial compressive strength (UCS), BTS, and Schmidt hammer rebound number. In addition, aiming to predict the TS value, Mishra and Basu [8] made use of two empirical methods: the block punch test and point load strength (Is50). Their findings indicated the superiority of the block punch index over Is50 regarding the estimation of BTS. Sheorey [9] reconfirmed the widely accepted idea that there is a correlation between BTS and UCS in rocks, and also indicated that the compressive strength of rock is roughly 10 times higher than the BTS level of the same rock. Nevertheless, rock has a site-specific behavior. Kahraman et al. [2] was mainly focused on how UCS and BTS are related to each other in various types of rock; the relative study resulted in finding a linear correlation between the two factors. Heidari et al. [3] compared all point load testing methods proposed in the literature in terms of their use in practical applications. They made use of the diametral, axial, and irregular methods for the purpose of predicting Is50. A comparison was made on the obtained results, and a number of equations were developed in a way to both practically and economically estimate the BTS value.
In addition, aiming to determine how directly TS and BTS are related to each other, Perras and Diederichs [10] analyzed rocks’ TS. They also made a comprehensive review of the methods used to measure direct TS, BTS, and alternative methods for the estimation of a rock’s TS. Their results showed that laboratory tests cannot be easily applied for the accurate prediction of rock TS. Armaghani et al. [11] carried out a number of laboratory tests on a total of 87 granite-type samples in order to predict their BTS value. They only made use of simple and multiple regressions for this purpose due to the high expense and time requirement of laboratory tests. Their findings showed a higher capability of multiple regression models compared to simple regression ones in terms of the accuracy level in the BTS prediction. The literature also contains great efforts for the development of intelligent systems aiming to be applicable to approximating science and to provide a solution to problems that may arise in engineering contexts [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56].
Furthermore, a number of significant studies have been carried out with similar objectives with the present paper. Both artificial neural networks (ANNs) and statistical methods were employed by Singh et al. [4] in order to predict TS in schistose rocks. They confirmed the higher accuracy level of ANNs compared to conventional methods regarding TS prediction. Likewise, ANNs are able to generalize results, while the conventional methods fail to do this. Baykasoğlu et al. [1] investigated the drawbacks of the artificial intelligent methods previously introduced in the literature, and for the first time applied various sets of genetic programming to experimental data aiming to predict limestone strength. In another project, a general prediction model was developed by Kumar et al. [57] in order to predict the values of UCS and BTS formed by rotary drilling into sedimentary rock. For the purpose of constructing and evaluating their proposed model, they utilized multiple regression analyses and prediction performance indices, respectively. They concluded that the model could be effectively applied to real-life situations. It is true that the ANN is a capable technique for solving/addressing engineering problems/issues; however, they are also subject to certain limitations, including low learning speed and an incapability to escape from local minima [58,59]. To remove these disadvantages, ANN can be integrated with optimization algorithms, such as artificial bee colony (ABC), imperialist competitive algorithm (ICA), invasive weed optimization (IWO), and particle swarm optimization (PSO). In this manner, ANNs can perform in their prediction tasks with a higher efficiency through the adjustment of their bias and weight. In recent years, such hybrid systems have been applied to a number of geotechnical problems (see [60,61,62]). Based on the above discussion, the present paper attempts to accomplish the following objectives:
  • To create an appropriate database applicable for the prediction of TS.
  • To develop a number of novel equations by means of simple regression analysis.
  • To design three hybrid intelligent models: IWO-ANN, ABC-ANN, and ICA-ANN.
  • To propose a hybrid intelligent model of the highest accuracy in predicting rock TS.

2. Laboratory Experiments and Regression Analysis

For the purpose of the research presented herein, a total of 100 granite samples were gathered from the Pahang Selangor Raw Water Tunnel (PSRWT) in Malaysia. Different rock index tests, i.e., the Schmidt hammer test (Rn), the dry density (DD) test, the point load test (Is50), as well as BTS were conducted in the laboratory, based on the ISRM standards [6]. In this research, a total of 80 datasets were provided in order to perform modeling and required analyses. Figure 1 summarizes the input and output datasets used. This figure also shows the relationship matrix between all data.
Simple regression models were utilized for the purpose of identifying the relationship between the model inputs (Is50, DD, and Rn) and the system output (BTS). A variety of equations, including power, linear, and exponential, were tested for the evaluation and selection of the most appropriate type of equation for the prediction of TS. The equations were evaluated with taking into consideration some prediction intervals (PIs) such as variance accounted for (VAF), root mean square error (RMSE), and R2, which have been recommended by lots of researchers such as [24,59,61,63]. In addition, the formulas in regard to such PIs were taken from Mohamad et al.’s [64] research. It is noted that the best fit is the one with a VAF of 100%, RMSE of 0, and R2 of 1. The equations used for the prediction of BTS, as well as their PIs, are listed in Figure 2. Selection of the equations was on the basis of their PIs results in comparison with other types of equations. For example, as can easily be noticed, R2 was calculated as 0.698, 0.674, and 0.676, for Rn, Is50, and DD, respectively. Figure 2 demonstrates the graphs of the equations that were developed for the TS prediction. The obtained results were found significant; however, aiming to achieve TS prediction results of a higher accuracy in real-life situations, it is necessary to develop a number of new models. Accordingly, three intelligent models, i.e., ABC integrated with ANN (ABC-ANN), IWO integrated with ANN (IWO-ANN), and ICA integrated with ANN (ICA-ANN) are constructed for the purpose of predicting the BTS of the granite type rock samples.

3. Methodology

3.1. Imperialist Competitive Algorithm

Atashpaz-Gargari and Lucas [65] pioneered the imperialist competitive algorithm (ICA) as an algorithm based on global search population and implemented to be effectively applicable for optimization problems. ICA starts with randomly generating the initial population denoted by countries. Thus, N number of countries (Ncountry) are produced. After that, the number of imperialists, i.e., Nimp, needs to be chosen as a certain number of the countries with minimum costs. The other countries (Ncol) in the system are employed as special functions amongst the other empires. Generally, the greater the number of colonies, the more powerful imperialists one will have. ICA is mainly consisted of three operators: assimilation, revolution, and competition [66,67,68]. A part of ICA is the colonies attracted similarly by existing imperialists, which means that the existing imperialists control the part of the initial colonies before any process. On the other hand, revolution makes big changes in the system. Through the competition procedure, the imperialists strive to acquire as many colonies as possible. Any empire that would be capable of achieving the predefined criteria is finally a winner. Such a process is iterated until the end of the desired benchmark. In ICA, the decades’ number has a process comparable to the number of particles in PSO and also that of generations in GA. To design them, it will be helpful to evaluate the outcomes obtained from the root mean square (RMSE). For a detailed explanation regarding ICA, one may refer to several studies in the literature [60,65,69]. An ICA structure is depicted in Figure 3.

3.2. Artificial Bee Colony

ABC is an optimization algorithm inspired by the bees’ social life and developed by Karaboga [70]. Each bee in this algorithm signifies a simple component. In case they form a bee colony, they will show a complex coherent behavior that will be capable of shaping an integrated system to discover and exploit the nectars of flowers. Each of the colonies contains three groups of bees, each of which has a certain task. The bees in the first group are scouts whose responsibility is to discover new sources of food. The scouts search in a random way throughout the outlying environment, and when they find a source, they will memorize the place. When each bee comes back to its hive, the bees existing within the hive share their information in regard to discovered sources with each other doing a waggle dance. Afterwards, several bees will be sent to exploit the sources. The bees within the second group are employed bees. Their responsibility is the exploitation of the predetermined food sources. The bees in the third group are onlooker bees. They stay in the hive waiting for the other bees; when information is exchanged with the other bees by waggle dance, they will select a resource in accordance with the fitness of the answer for exploitation.
ABC has the required capacity to solve numerous mathematical and engineering issues, e.g., optimizing the wells locations in oil basins [71], in cases where water is discharged from a dam [72], classification of data into clusters [73], and scheduling of machines [74], accidental failure of a nuclear power plant [75]. It can also be integrated with ANNs in order to predict the bottom pressure of wells along the network [76]. In the context of geotechnical engineering field, this algorithm has been merely applied to the prediction of the blast-induced back break [77]. Four steps are involved in ABC as follows [62,78,79].
First step: Initially, 50% of the bees living in the hive are considered as employed bees and the rest of them are non-employed ones. Just a bunch of the employed bees is assigned to each one of the explored food sources. That is, the employed bees equal the quantity of food sources. As a result, one employed bee is allocated to each of the available sources. In other words, inside the answer scope, existing food sources form the initial solution. When the initial solutions are created, each solution’s value needs to be computed considering the relationship of the problem.
Second step: For each solution, a new response can be formed using the following equation:
v i , j = x i . j + φ i . j ( x i . j − x k . j ) i ∈   { 1.   2. B N } j ∈   { 1.   2.   … D } k ∈   { 1.   2..   .   .   . B N }   &   k ≠ i φ ∈   [ − 1.1 ]
where
  • x i . j is the response i for the parameter j,
  • v i , j is parameter j in the new response,
  • i is the number of one to the number of solutions,
  • φ is a random number in the negative interval of 1-1,
  • k is a random number of one the answers or solusions,
  • BN is the number of initial solutions,
  • D is the number of optimization parameters.
When a new answer/response is formed, in case this answer’s value exceeds the former answer’s value, it will be substituted; if not, the answer will be forgotten.
Third step: The probability of bees coming from each of the defined sites will be computed using the equation below:
p i = f i t i ∑ n = 1 S N f i t n
where f i t i   signifies the source of the fitness of source i and p i denotes the choice probability of source i by the onlooker bees. With respect to each item’s fitness, some are assigned. All the bees in this step might be dedicated to a certain food site based on the basis of the fitness value. When each source’s value is computed by means of Equation (1), a fresh answer is created for the answers chosen. In case the value of this answer exceeds that of the former one, the former answer will be replaced with the new one; if not, it will be fined. The goal of the use of the fine is that a count is made for the number of failures in order to enhance the response; if the answer is not enhanced as desired, it will be added with one unit.
Step 4: In case the non-improvement answer counter reaches up to a predefined value (Cmax), the answer should be substituted with a randomly selected answer. In addition, the criteria for termination of the repetitions are also checked in this step, whether they have been met or not. If yes, the repetitions end; if not, it goes back to the second step. For more detailed information about ABC and the way it operates, one may refer to literature (e.g., [62,80]). A flowchart of ABC is depicted in Figure 4.

3.3. Invasive Weed Optimization (IWO)

As a population-based optimization technique, the invasive weed optimization (IWO) is able to satisfy the excellent performance of a mathematical equation in a way of both adaptation and randomization of a weed colony. As a fact, the great growth of the weeds (as very strong herbs) is considered as an important threat to plant products. Actually, their level of resistance against environmental and climate changes is high. So, according to its features, a robust optimization algorithm is achieved. In IWO, the weed community and their compatibility, resistance, and randomness try to solve a problem.
In respect to the IWO background, IWO was developed based on an agriculture phenomenon that is inspired by the invasive weed colonies. As explained before in the other way, weed (as a plant) is able to grow unintentionally. There are many benefits of weeds’ existence in urban spaces and else. However, if the unintentional grow of this plant includes some damages for human activities or the planet, it is considered as a “weed” [81]. Although IWO is a simple algorithm in terms of structure, concept, and implementation, it is a powerful optimization technique to solve almost all optimization problems. In order to have a better understanding about weeds and their habitat behavior, the following steps are important to explain:
(1) In the first step with name of “population initialization”, a number of seeds are partially spread in the search space.
(2) The second step or “reproduction” will start with the pouring of every plant into a flowering plant; then, the system is able to generate seeds that are worth their proportion. Subsequently, the number of plant seeds will linearly decrease from Smax to Smin using the following equation:
n ( w i ) = S max ( max f i t − f i t ( w i ) ) + S min ( f i t ( w i ) − min f i t ) max f i t − min f i t .
(3) The third stage is related to determinations of new positions of the seeds in the search space. In this step, the child’s seeds will be located around their parents.
(4) Step number 4 (or competitive elimination stage) is related to creation of the best seeds according to their merit. In fact, this will happen if the number of existing created seeds reach a certain level (Pmax).
(5) In the final stage, if the conditions are not fulfilled, in order to finish the process, the steps from the second stage will be repeated; otherwise, the execution of the algorithm ends. To have a better understanding of IWO step implementation, Figure 5 shows a general diagram for an invasive weed algorithm.

3.4. Hybrid Algorithms

The literature contains numerous studies conducted to improve ANN using optimization algorithms such as PSO, genetic algorithm (GA), ICA, and ABC (see [38,61,62,63,64,65]). The back-propagation (BP) does not act strongly in exploring the accurate global minimum; as a result, the ANN model might obtain unwanted results [82,83,84,85,86,87,88]. However, there is a higher probability for ANN to be trapped in local minima. In order to solve ANN problems and get a higher level of performance prediction, the weights and biases of ANNs should be optimized by implementing optimization algorithms such as ICA, ABC, and IWO. In particular, the hybrid ICA-ANN, ABC-ANN, and IWO-ANN models do not get caught in the local minima due to the robustness of the optimization algorithms. However, the power level of each hybrid algorithm depends on the power level of its optimization techniques i.e., ICA, ABC, and IWO. Therefore, the structure of optimization technique as well as their most effective parameters play an important role on results of hybrid ANN-based models. As shown in Figure 6, the most effective parameters on each optimization algorithm are presented. The present paper develops three hybrid models for the purpose of predicting TS: IWO-ANN, ICA-ANN, and ABC-ANN. In these hybrid models, IWO, ICA, and ABC are responsible for exploring global minimum; after that, ANN chooses it in a way to obtain the best results. Actually, the basic optimization algorithms were used to form new combinations, and they were connected to ANN. Then, different parametric studies were conducted to obtain the best combination for each hybrid model. It is important to mention that the source of optimization codes was taken from their original studies. Figure 6 shows three stages of analysis to predict the BTS of the rock samples. The first stage of the analysis is related to ANN structure selection. In the second stage, the best structures of optimization algorithms should be selected based on the most important mentioned parameters on them. Then, in the last stage, the hybrid ANN-based models were evaluated using the most important performance indices.

4. Model Development

Hybrid model processes are presented in this section. Three hybrid models that are based on the combination of the base model (ANN) with the algorithms of IWO, ABC, and ICA are discussed. To design the best model to predict the TS, each model is separately evaluated to identify the best performance of each.

4.1. ICA-ANN

To achieve an ICA-ANN model of the highest quality, the ICA parameters need to be completely investigated, but first, the architecture of ANN must be fully determined. The performance of the ANN models depends strongly on the suggested architecture of the network [89,90,91]. Therefore, determination of the optimal architecture is required to design an ANN model. The network architecture is defined as the number of hidden layer(s) and the number of nodes in each hidden layer(s). According to various researchers (e.g., [92,93,94]), one hidden layer can solve any complex function in a network. Hence, in this study, one hidden layer was selected to construct the ANN models. In addition, determining the neuron number(s) in the hidden layer is the most critical task in the ANN architecture, as mentioned by Sonmez et al. [95]. It was performed through a trial and error procedure, which finally resulted in the fact that an architecture of 3 × 5 × 1 (or a model with five hidden neurons) can offer optimum results. Thus, the above architecture was considered for the best ANN model as well as for all hybrid ANN-based models (as a confirmed initial ANN model) developed in this study. More information on obtaining ANN structures can be found in previous articles [96,97,98,99,100].
As noted previously, the key parameters in ICA are Ndecade, Ncountry, and Nimp. To set the Nimp value, lots of models were designed by means of different values of this parameter ranging from 15 to 40 with an incremental step of five. In these models, Ncountry was set to 200, and Ndecade was set to 150. The results obtained from such a parametric research confirmed that when Nimp is set to 25, a higher performance capacity can be achieved. On the other hand, for the purpose of choosing an optimal value for Ndecade, as can be seen in Figure 7, different models were created with Ncountry values ranging between 50 and 400, with an incremental step of 50. The models’ performance was then tested based on their RMSE. The RMSE results did not change any more after Ndecade = 550. Finally, with the two parameters of Nimp = 25 and Ndecade = 550, different numbers of countries were taken into account, and their ICA-ANN models were configured. Required evaluations were done on the models on the basis of the performance indices (PIs), i.e., RMSE values and the coefficient of determination (R2) (see Table 1). To select the hybrid models of the highest quality in this research, Zorlu et al.’s [101] ranking method was adopted.
This technique is fully explained by Zorlu et al. [101], based on which a rank value was allocated to each PI in its group (training and testing). For instance, values of 0.865, 0.871, 0.853, 0.899, 0.857, 0.861, 0.874, and 0.894 were obtained for the R2 values of training datasets of models 1 to 8, respectively, and values of 4, 5, 1, 8, 2, 3, 6, and 7 were set to their ranks, respectively. This process was also done in the case of the RMSE results. Afterwards, a summation value of rating of the R2 train, RMSE train, R2 test, and RMSE test was computed and allocated to each model, according to which model 4 (Ncountry = 200) with a total rank of 30 was found to be the best ICA-ANN model. An evaluation of the best ICA-ANN results is presented in the next sections. All the models were configured by means of a ratio of 80 to 20 for the training and testing datasets, respectively.
In this study, a certain color was dedicated to each row of the models. As the red color intensifies, it achieves a higher score than the rest of the scores within that column; on the other hand, the less intensified color shows that the parameter is lower than the other parameters in the column. For instance, model number 5 of ICA-ANN reflexes a high intensified red in the R2 column of the training section. Using such an approach, the best parameters were determined in each of the columns, and in the last one, an item with a collectively higher quality (concerning the red color intensity) was chosen. In Table 1, this system is observable. This way, the novel method is employed for the purpose of choosing the best models. To adopt this method in an effective way, we made use of coding in order to determine the colors’ intensity. It is worth mentioning that the use of the ranking system leads to the achievement of the same result. However, this innovative method of exploring the models of highest quality among others can be recognized as a novel and smart solution to the selection and categorization of models. Such a method of categorization is termed as a color intensity rating (CIR) system. This system was also applied to the results obtained from the other hybrid models developed in this paper. The application of this method can be found in some research [62,80].

4.2. ABC-ANN

To enhance the ANN performance, the ABC algorithm was also used. Generally, the BP algorithm is employed by researchers aiming at training ANN. BP suffers from a number of deficiencies that decreases the ANN performance quality. At the time the network is searching for the best system weights, errors must be minimized. In this regard, ABC searches for appropriate weights and optimizes them. An increase in the number of bees allows them to detect in a wider area. Such a process is continued until the algorithm can obtain minimum error. Each time the selection is done, the values remain constant, awaiting a better result to be achieved for the network weights. Under such conditions, various models were formed, each of which iterated its operation for 600 rounds in order to obtain results of higher reliability. Different bees can be observed in Figure 8.
As can be seen in Figure 8, the number of bees ranged between 5 and 40 in order to examine the effect of the number of bees. Better results are expected to be obtained with an increase in the number of bees, although when the operation of the algorithm is iterated for 500 times, almost all the answers coincide. It is because bees typically gather in spots wherein the best answer lays. It makes this algorithm faster than the other ones, and the resultant answer is typically of a higher performance quality.
Table 2 presents the values achieved from the ABC-ANN model in terms of the TS prediction in BTS. Based on the rating method, model number 8, which consisted of 40 bees, was the best one regarding the total R2 and RMSE scores. In case of the training and testing model number 8, the values of R2 and RMSE were 0.908, 0.0951, 0.904, and 0.0952, respectively.

4.3. IWO-ANN

In this section, the new IWO-ANN model is implemented. According to our literature review, there is no research yet that has combined these two algorithms. However, the implementation steps of this new model are similar to previous models. The IWO algorithm described above is known as a new optimization algorithm that is used for optimization in various issues. Various parameters are involved in this algorithm; some of its most important ones, include the number of iterations, the initial population of seeds, and so on. Some of these parameters have less impact on the results and can be obtained by trial and error methods. Therefore, these parameters are initially determined. The minimum and maximum number of seeds (Smin and Smax) were investigated from zero to 30. In this case, the Smin value is zero, and the best condition for the Smax is 20. The initial and final value of standard deviation parameter actually helps to knock off the selection, using the variance reduction exponent parameter. Values of 0.5, 0.001, and 4 were selected for these three parameters (the initial and final value of standard deviation and the variance reduction exponent parameters), respectively. Finally, the main parameters such as the number of iterations and the final number of plants were determined according to Figure 9. To compare the number of iterations, similar to those presented in the previous sections, 600 iterations were selected for all the models. As shown in Figure 9, the RMSE of the models changes up to 300 iterations, and after that, they reach a constant level. For this reason, 300 iterations is the optimum value for designing next models. Finally, different models of plant seeds were designed (5–40), the results of which are presented in Table 3. Using this table, the best model is selected. As can be obtained, model number 7 provides the best conditions for TS prediction.

5. Results and Discussion

Findings of the simple regression analysis carried out in this study demonstrated that there is a need for predictive models to make precise estimations regarding rock TS. To this end, the present paper presents three hybrid models. In the proposed models, three neurons were utilized within the input layer, i.e., Rn, DD, and Is50, while the BTS values were utilized in the output layer. Through a trial-and-error method, it was found that when the number of neurons within the hidden layer was set to five, the model performance was better than the other conditions. Aiming at choosing the best hybrid IWO-ANN, ABC-ANN, and ICA-ANN models, we developed numerous models on the basis of the results obtained from a number of parametric studies. At the final step, amongst the three hybrid models, the best one was chosen on the basis of the models’ system error performance. In addition, more evaluations were conducted on the chosen models through comparing between the other PIs, i.e., R2 and VAF. Table 4 presents the PI values of all hybrid models. Furthermore, Table 4 show the difference between the values achieved from the measured and predicted TS values of the rocks by means of IWO-ANN, ABC-ANN, and ICA-ANN, respectively. As can be concluded from Table 4, the IWO-ANN outperformed the other optimum hybrid models in terms of all the PIs. That is, IWO-ANN returned the most accurate values (i.e., the lowest RMSE with the maximum values of VAF and R2) for the training and testing datasets. For instance, in regard to testing the datasets, the IWO-ANN model with the RMSE of 0.0911, R2 of 0.917, and a VAF of 91.731 estimated the BTS more accurately compared to ABC-ANN with the RMSE of 0.0955, R2 of 0.904, and a VAF of 90.419. In addition, it presented a higher quality performance in comparison to ICA-ANN with an RMSE of 0.0978, R2 of 0.901, and VAF of 90.134. Therefore, although all of the above-mentioned models were found to be capable of predicting the BTS value with an acceptable level of precision, the IWO-ANN model performed the best. Obviously, the IWO-ANN model for the rock TS prediction can be taken into account as an innovative predictive model in engineering fields of study. Finally, Figure 10 shows the prediction results for the IWO-ANN model for all the samples. In this figure, given that all the laboratory samples are in the new IWO-ANN system, a careful evaluation of the final accuracy is obtained. A linear relationship between the actual (parameter x) and the predicted (parameter y) value is also given. As expected, the chosen model presents as a new solution to the tensile strength of the rock samples in future works according to defined conditions for system design.

6. Conclusions

The level of tensile strength (TS) of rock in various tunnel and civil engineering projects is considered as an important part in the design factor. However, performing actual tensile strength tests in a lab is sometimes costly and always complex. Therefore, recently, various techniques have been developed aiming to evaluate this parameter though other, more easily applicable, measurement techniques. The present study developed three hybrid intelligent predictive models, i.e., IWO-ANN, ICA-ANN, and ABC-ANN aiming at accurately predicting the tensile strength (TS) of the granitic rock samples based on the values of the Schmidt hammer test (Rn), the dry density (DD) test, and the point load test (Is50). First, a substantial relationship was explored between the inputs and output using simple regression analysis. The R2 results of roughly 0.7 for simple regression analyses showed the need for proposing models with multiple inputs, which resulted in creating the three models of IWO-ANN, ABC-ANN, and ICA-ANN, among which the IWO-ANN was finally found to be more successful in solving the problem at hand compared to the other two models. The results of the new IWO-ANN models for the training and testing sections were R2 = 0.928 and 0.917, respectively, which highlighted the high capabilities of this model for TS prediction. Hence, it can be used as a new solution for training intelligent systems on various issues.

Author Contributions

Formal analysis, M.K.; Conceptualization, D.J.A.; Supervision, P.G.A.; Writing—review & editing, L.H. and M.M.T. and P.G.A.

Funding

This research was supported by the National Natural Science Foundation of China (No. 51969023), High-level Talents Research Start-up Fund Project of Inner Mongolia Agricultural University (No. NDYB2018-60), Inner Mongolia Autonomous Region Natural Science Foundation of China (NO. 2018MS05005), and Inner Mongolia Autonomous Region Higher Education Science Research Project of China (No. NJZY18064).

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Baykasoğlu, A.; Güllü, H.; Çanakçı, H.; Özbakır, L. Prediction of compressive and tensile strength of limestone via genetic programming. Expert Syst. Appl. 2008, 35, 111–123. [Google Scholar] [CrossRef]
  2. Kahraman, S.; Fener, M.; Kozman, E. Predicting the compressive and tensile strength of rocks from indentation hardness index. J. S. Afr. Inst. Min. Metall. 2012, 112, 331–339. [Google Scholar]
  3. Heidari, M.; Khanlari, G.R.; Kaveh, M.T.; Kargarian, S. Predicting the Uniaxial Compressive and Tensile Strengths of Gypsum Rock by Point Load Testing. Rock Mech. Rock Eng. 2012, 45, 265–273. [Google Scholar] [CrossRef]
  4. Singh, V.; Singh, D.; Singh, T. Prediction of strength properties of some schistose rocks from petrographic properties using artificial neural networks. Int. J. Rock Mech. Min. Sci. 2001, 38, 269–284. [Google Scholar] [CrossRef]
  5. Nazir, R.; Momeni, E.; Armaghani, D.J.; Amin, M.F.M. Correlation between unconfined compressive strength and indirect tensile strength of limestone rock samples. Electron. J. Geotech. Eng. 2013, 18, 1737–1746. [Google Scholar]
  6. Ulusay, R.; Hudson, J.A.; ISRM. The Complete ISRM Suggested Methods for Rock Characterization, Testing and Monitoring: 1974–2006; Commission on Testing Methods, International Society for Rock Mechanics; ISRM Turkish Natl Group: Ankara, Turkey, 2007; 628 p. [Google Scholar]
  7. Kahraman, S.; Bilgin, N.; Feridunoglu, C. Dominant rock properties affecting the penetration rate of percussive drills. Int. J. Rock Mech. Min. Sci. 2003, 40, 711–723. [Google Scholar] [CrossRef]
  8. Mishra, D.A.; Basu, A. Use of the block punch test to predict the compressive and tensile strengths of rocks. Int. J. Rock Mech. Min. Sci. 2012, 51, 119–127. [Google Scholar] [CrossRef]
  9. Sheorey, P.R. Empirical Rock Failure Criteria; A A Balkema: Rotterdam, The Netherlands, 1997. [Google Scholar]
  10. Perras, M.A.; Diederichs, M.S. A Review of the Tensile Strength of Rock: Concepts and Testing. Geotech. Geol. Eng. 2014, 32, 525–546. [Google Scholar] [CrossRef]
  11. Armaghani, D.J.; Amin, M.F.M.; Yagiz, S.; Faradonbeh, R.S.; Abdullah, R.A. Prediction of the uniaxial compressive strength of sandstone using various modeling techniques. Int. J. Rock Mech. Min. Sci. 2016, 85, 174–186. [Google Scholar] [CrossRef]
  12. Koopialipoor, M.; Fahimifar, A.; Ghaleini, E.N.; Momenzadeh, M.; Armaghani, D.J. Development of a new hybrid ANN for solving a geotechnical problem related to tunnel boring machine performance. Eng. Comput. 2019, 1–13. [Google Scholar] [CrossRef]
  13. Zhou, J.; Aghili, N.; Ghaleini, E.N.; Bui, D.T.; Tahir, M.M.; Koopialipoor, M. A Monte Carlo simulation approach for effective assessment of flyrock based on intelligent system of neural network. Eng. Comput. 2019, 1–11. [Google Scholar] [CrossRef]
  14. Asteris, P.G.; Nozhati, S.; Nikoo, M.; Cavaleri, L.; Nikoo, M. Krill herd algorithm-based neural network in structural seismic reliability evaluation. Mech. Adv. Mater. Struct. 2019, 26, 1146–1153. [Google Scholar] [CrossRef]
  15. Chahnasir, E.S.; Zandi, Y.; Shariati, M.; Dehghani, E.; Toghroli, A.; Mohamad, E.T.; Shariati, A.; Safa, M.; Wakil, K.; Khorami, M. Application of support vector machine with firefly algorithm for investigation of the factors affecting the shear strength of angle shear connectors. SMART Struct. Syst. 2018, 22, 413–424. [Google Scholar]
  16. Mohammadhassani, M.; Nezamabadi-Pour, H.; Suhatril, M.; Shariati, M. Identification of a suitable ANN architecture in predicting strain in tie section of concrete deep beams. Struct. Eng. Mech. 2013, 46, 853–868. [Google Scholar] [CrossRef]
  17. Toghroli, A.; Suhatril, M.; Ibrahim, Z.; Safa, M.; Shariati, M.; Shamshirband, S. Potential of soft computing approach for evaluating the factors affecting the capacity of steel–concrete composite beam. J. Intell. Manuf. 2018, 29, 1793–1801. [Google Scholar] [CrossRef]
  18. Safa, M.; Shariati, M.; Ibrahim, Z.; Toghroli, A.; Baharom, S.B.; Nor, N.M.; Petkovic, D. Potential of adaptive neuro fuzzy inference system for evaluating the factors affecting steel-concrete composite beam’s shear strength. Steel Compos. Struct. 2016, 21, 679–688. [Google Scholar] [CrossRef]
  19. Hasanipanah, M.; Amnieh, H.B.; Khamesi, H.; Armaghani, D.J.; Golzar, S.B.; Shahnazar, A. Prediction of an environmental issue of mine blasting: An imperialistic competitive algorithm-based fuzzy system. Int. J. Environ. Sci. Technol. 2018, 15, 551–560. [Google Scholar] [CrossRef]
  20. Jiang, W.; Arslan, C.A.; Tehrani, M.S.; Khorami, M.; Hasanipanah, M. Simulating the peak particle velocity in rock blasting projects using a neuro-fuzzy inference system. Eng. Comput. 2019, 35, 1203–1211. [Google Scholar] [CrossRef]
  21. Rad, H.N.; Hasanipanah, M.; Rezaei, M.; Eghlim, A.L. Developing a least squares support vector machine for estimating the blast-induced flyrock. Eng. Comput. 2018, 34, 709–717. [Google Scholar] [CrossRef]
  22. Zhou, J.; Li, X.; Mitri, H.S. Classification of rockburst in underground projects: Comparison of ten supervised learning methods. J. Comput. Civ. Eng. 2016, 30, 4016003. [Google Scholar] [CrossRef]
  23. Zhou, J.; Shi, X.; Du, K.; Qiu, X.; Li, X.; Mitri, H.S. Feasibility of random-forest approach for prediction of ground settlements induced by the construction of a shield-driven tunnel. Int. J. Geomech. 2016, 17, 4016129. [Google Scholar] [CrossRef]
  24. Liao, X.; Khandelwal, M.; Yang, H.; Koopialipoor, M.; Murlidhar, B.R. Effects of a proper feature selection on prediction and optimization of drilling rate using intelligent techniques. Eng. Comput. 2019, 1–12. [Google Scholar] [CrossRef]
  25. Zhou, J.; Li, E.; Wang, M.; Chen, X.; Shi, X.; Jiang, L. Feasibility of Stochastic Gradient Boosting Approach for Evaluating Seismic Liquefaction Potential Based on SPT and CPT Case Histories. J. Perform. Constr. Facil. 2019, 33, 4019024. [Google Scholar] [CrossRef]
  26. Zhou, J.; Shi, X.; Li, X. Utilizing gradient boosted machine for the prediction of damage to residential structures owing to blasting vibrations of open pit mining. J. Vib. Control 2016, 22, 3986–3997. [Google Scholar] [CrossRef]
  27. Zhou, J.; Li, X.; Mitri, H.S. Evaluation method of rockburst: State-of-the-art literature review. Tunn. Undergr. Sp. Technol. 2018, 81, 632–659. [Google Scholar] [CrossRef]
  28. Monjezi, M.; Dehghani, H. Evaluation of effect of blasting pattern parameters on back break using neural networks. Int. J. Rock Mech. Min. Sci. 2008, 45, 1446–1453. [Google Scholar] [CrossRef]
  29. Khandelwal, M.; Armaghani, D.J.; Faradonbeh, R.S.; Ranjith, P.G.; Ghoraba, S. A new model based on gene expression programming to estimate air flow in a single rock joint. Environ. Earth Sci. 2016, 75, 739. [Google Scholar] [CrossRef]
  30. Manoj, K.; Monjezi, M. Prediction of flyrock in open pit blasting operation using machine learning method. Int. J. Min. Sci. Technol. 2013, 23, 313–316. [Google Scholar] [CrossRef]
  31. Khandelwal, M.; Singh, T.N. Correlating static properties of coal measures rocks with P-wave velocity. Int. J. Coal Geol. 2009, 79, 55–60. [Google Scholar] [CrossRef]
  32. Harandizadeh, H.; Toufigh, M.M.; Toufigh, V. Application of improved ANFIS approaches to estimate bearing capacity of piles. Soft Comput. 2019, 23, 9537–9547. [Google Scholar] [CrossRef]
  33. Asteris, P.G.; Kolovos, K.G.; Douvika, M.G.; Roinos, K. Prediction of self-compacting concrete strength using artificial neural networks. Eur. J. Environ. Civ. Eng. 2016, 20, s102–s122. [Google Scholar] [CrossRef]
  34. Apostolopoulou, M.; Armaghani, D.J.; Bakolas, A.; Douvika, M.G.; Moropoulou, A.; Asteris, P.G. Compressive strength of natural hydraulic lime mortars using soft computing techniques. Procedia Struct. Integr. 2019, 17, 914–923. [Google Scholar] [CrossRef]
  35. Armaghani, D.J.; Koopialipoor, M.; Marto, A.; Yagiz, S. Application of several optimization techniques for estimating TBM advance rate in granitic rocks. J. Rock Mech. Geotech. Eng. 2019, 11, 779–789. [Google Scholar] [CrossRef]
  36. Xu, H.; Zhou, J.; GAsteris, P.; Jahed Armaghani, D.; Tahir, M.M. Supervised Machine Learning Techniques to the Prediction of Tunnel Boring Machine Penetration Rate. Appl. Sci. 2019, 9, 3715. [Google Scholar] [CrossRef] [Green Version]
  37. Armaghani, D.J.; Hatzigeorgiou, G.D.; Karamani, C.; Skentou, A.; Zoumpoulaki, I.; Asteris, P.G. Soft computing-based techniques for concrete beams shear strength. Procedia Struct. Integr. 2019, 17, 924–933. [Google Scholar] [CrossRef]
  38. Asteris, P.G.; Nikoo, M. Artificial bee colony-based neural network for the prediction of the fundamental period of infilled frame structures. Neural Comput. Appl. 2019, 1–11. [Google Scholar] [CrossRef]
  39. Asteris, P.G.; Moropoulou, A.; Skentou, A.D.; Apostolopoulou, M.; Mohebkhah, A.; Cavaleri, L.; Rodrigues, H.; Varum, H. Stochastic Vulnerability Assessment of Masonry Structures: Concepts, Modeling and Restoration Aspects. Appl. Sci. 2019, 9, 243. [Google Scholar] [CrossRef] [Green Version]
  40. Hajihassani, M.; Abdullah, S.S.; Asteris, P.G.; Armaghani, D.J. A Gene Expression Programming Model for Predicting Tunnel Convergence. Appl. Sci. 2019, 9, 4650. [Google Scholar] [CrossRef] [Green Version]
  41. Asteris, P.G.; Apostolopoulou, M.; Skentou, A.D.; Moropoulou, A. Application of artificial neural networks for the prediction of the compressive strength of cement-based mortars. Comput. Concr. 2019, 24, 329–345. [Google Scholar]
  42. Cavaleri, L.; Asteris, P.G.; Psyllaki, P.P.; Douvika, M.G.; Skentou, A.D.; Vaxevanidis, N.M. Prediction of Surface Treatment Effects on the Tribological Performance of Tool Steels Using Artificial Neural Networks. Appl. Sci. 2019, 9, 2788. [Google Scholar] [CrossRef] [Green Version]
  43. Asteris, P.G.; Tsaris, A.K.; Cavaleri, L.; Repapis, C.C.; Papalou, A.; Di Trapani, F.; Karypidis, D.F. Prediction of the fundamental period of infilled RC frame structures using artificial neural networks. Comput. Intell. Neurosci. 2016, 2016, 20. [Google Scholar] [CrossRef] [Green Version]
  44. Psyllaki, P.; Stamatiou, K.; Iliadis, I.; Mourlas, A.; Asteris, P.; Vaxevanidis, N. Surface treatment of tool steels against galling failure. In MATEC Web of Conferences. EDP Sci. 2018, 188, 4024. [Google Scholar]
  45. Asteris, P.G.; Ashrafian, A.; Rezaie-Balf, M. Prediction of the compressive strength of self-compacting concrete using surrogate models. Comput. Concr. 2019, 24, 137–150. [Google Scholar]
  46. Koopialipoor, M.; Tootoonchi, H.; Armaghani, D.J.; Mohamad, E.T.; Hedayat, A. Application of deep neural networks in predicting the penetration rate of tunnel boring machines. Bull. Eng. Geol. Environ. 2019, 1–14. [Google Scholar] [CrossRef]
  47. Yang, H.Q.; Li, Z.; Jie, T.Q.; Zhang, Z.Q. Effects of joints on the cutting behavior of disc cutter running on the jointed rock mass. Tunn. Undergr. Sp. Technol. 2018, 81, 112–120. [Google Scholar] [CrossRef]
  48. Yang, H.Q.; Xing, S.G.; Wang, Q.; Li, Z. Model test on the entrainment phenomenon and energy conversion mechanism of flow-like landslides. Eng. Geol. 2018, 239, 119–125. [Google Scholar] [CrossRef]
  49. Yang, H.; Liu, J.; Liu, B. Investigation on the cracking character of jointed rock mass beneath TBM disc cutter. Rock Mech. Rock Eng. 2018, 51, 1263–1277. [Google Scholar] [CrossRef]
  50. Zhou, J.; Koopialipoor, M.; Murlidhar, B.R.; Fatemi, S.A.; Tahir, M.M.; Armaghani, D.J.; Li, C. Use of Intelligent Methods to Design Effective Pattern Parameters of Mine Blasting to Minimize Flyrock Distance. Nat. Resour. Res. 2019, 1–15. [Google Scholar] [CrossRef]
  51. Guo, H.; Zhou, J.; Koopialipoor, M.; Armaghani, D.J.; Tahir, M.M. Deep neural network and whale optimization algorithm to assess flyrock induced by blasting. Eng. Comput. 2019, 1–14. [Google Scholar] [CrossRef]
  52. Koopialipoor, M.; Nikouei, S.S.; Marto, A.; Fahimifar, A.; Armaghani, D.J.; Mohamad, E.T. Predicting tunnel boring machine performance through a new model based on the group method of data handling. Bull. Eng. Geol Environ. 2018, 78, 3799–3813. [Google Scholar] [CrossRef]
  53. Asteris, P.; Roussis, P.; Douvika, M. Feed-forward neural network prediction of the mechanical properties of sandcrete materials. Sensors 2017, 17, 1344. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  54. Cavaleri, L.; Chatzarakis, G.E.; Di Trapani, F.; Douvika, M.G.; Roinos, K.; Vaxevanidis, N.M.; Asteris, P.G. Modeling of Surface Roughness in Electro-Discharge Machining Using Artificial Neural Networks. Adv. Mater. Res. 2017, 6, 169–184. [Google Scholar]
  55. Sarir, P.; Chen, J.; Asteris, P.G.; Armaghani, D.J.; Tahir, M.M. Developing GEP tree-based, neuro-swarm, and whale optimization models for evaluation of bearing capacity of concrete-filled steel tube columns. Eng. Comput. 2019, 1–19. [Google Scholar] [CrossRef]
  56. Chen, H.; Asteris, P.G.; Jahed Armaghani, D.; Gordan, B.; Pham, B.T. Assessing Dynamic Conditions of the Retaining Wall: Developing Two Hybrid Intelligent Models. Appl. Sci. 2019, 9, 1042. [Google Scholar] [CrossRef] [Green Version]
  57. Kumar, B.R.; Vardhan, H.; Govindaraj, M. Prediction of Uniaxial Compressive Strength, Tensile Strength and Porosity of Sedimentary Rocks Using Sound Level Produced During Rotary Drilling. Rock Mech. Rock Eng. 2011, 44, 613–620. [Google Scholar] [CrossRef]
  58. Koopialipoor, M.; Ghaleini, E.N.; Haghighi, M.; Kanagarajan, S.; Maarefvand, P.; Mohamad, E.T. Overbreak prediction and optimization in tunnel using neural network and bee colony techniques. Eng. Comput. 2019, 35, 1191–1202. [Google Scholar] [CrossRef]
  59. Zhao, Y.; Noorbakhsh, A.; Koopialipoor, M.; Azizi, A.; Tahir, M.M. A new methodology for optimization and prediction of rate of penetration during drilling operations. Eng. Comput. 2019, 1–9. [Google Scholar] [CrossRef]
  60. Koopialipoor, M.; Fallah, A.; Armaghani, D.J.; Azizi, A.; Mohamad, E.T. Three hybrid intelligent models in estimating flyrock distance resulting from blasting. Eng. Comput. 2019, 35, 243–256. [Google Scholar] [CrossRef] [Green Version]
  61. Ghaleini, E.N.; Koopialipoor, M.; Momenzadeh, M.; Sarafraz, M.E.; Mohamad, E.T.; Gordan, B. A combination of artificial bee colony and neural network for approximating the safety factor of retaining walls. Eng. Comput. 2018, 35, 647–658. [Google Scholar] [CrossRef]
  62. Koopialipoor, M.; Armaghani, D.J.; Hedayat, A.; Marto, A.; Gordan, B. Applying various hybrid intelligent systems to evaluate and predict slope stability under static and dynamic conditions. Soft Comput. 2019, 23, 5913–5929. [Google Scholar] [CrossRef]
  63. Koopialipoor, M.; Armaghani, D.J.; Haghighi, M.; Ghaleini, E.N. A neuro-genetic predictive model to approximate overbreak induced by drilling and blasting operation in tunnels. Bull. Eng. Geol Environ. 2019, 78, 981–990. [Google Scholar] [CrossRef]
  64. Mohamad, E.T.; Armaghani, D.J.; Momeni, E.; Yazdavar, A.H.; Ebrahimi, M. Rock strength estimation: A PSO-based BP approach. Neural Comput. Appl. 2018, 30, 1635–1646. [Google Scholar] [CrossRef]
  65. Atashpaz-Gargari, E.; Lucas, C. Imperialist competitive algorithm: An algorithm for optimization inspired by imperialistic competition. In Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2007), Singapore, 25–28 September 2007; pp. 4661–4667. [Google Scholar]
  66. Hajihassani, M.; Jahed Armaghani, D.; Marto, A.; Tonnizam Mohamad, E. Ground vibration prediction in quarry blasting through an artificial neural network optimized by imperialist competitive algorithm. Bull. Eng. Geol Environ. 2014, 74, 873–886. [Google Scholar] [CrossRef]
  67. Armaghani, D.J.; Hasanipanah, M.; Mohamad, E.T. A combination of the ICA-ANN model to predict air-overpressure resulting from blasting. Eng. Comput. 2016, 32, 155–171. [Google Scholar] [CrossRef]
  68. Gordan, B.; Koopialipoor, M.; Clementking, A.; Tootoonchi, H.; Mohamad, E.T. Estimating and optimizing safety factors of retaining wall through neural network and bee colony techniques. Eng Comput. 2019, 35, 945–954. [Google Scholar] [CrossRef]
  69. Armaghani, D.J.; Mohamad, E.T.; Narayanasamy, M.S.; Narita, N.; Yagiz, S. Development of hybrid intelligent models for predicting TBM penetration rate in hard rock condition. Tunn. Undergr. Sp. Technol. 2017, 63, 29–43. [Google Scholar] [CrossRef]
  70. Karaboga, D. An Idea Based on Honey Bee Swarm for Numerical Optimization; Technical Report-tr06; Erciyes University, Engineering Faculty, Computer Engineering Department: Kayseri, Turkey, 2005. [Google Scholar]
  71. Nozohour-leilabady, B.; Fazelabdolabadi, B. On the application of artificial bee colony (ABC) algorithm for optimization of well placements in fractured reservoirs; efficiency comparison with the particle swarm optimization (PSO) methodology. Petroleum 2016, 2, 79–89. [Google Scholar] [CrossRef] [Green Version]
  72. Ahmad, A.; Razali, S.F.M.; Mohamed, Z.S.; El-shafie, A. The Application of Artificial Bee Colony and Gravitational Search Algorithm in Reservoir Optimization. Water Resour. Manag. 2016, 30, 2497–2516. [Google Scholar] [CrossRef]
  73. Zhang, C.; Ouyang, D.; Ning, J. An artificial bee colony approach for clustering. Expert Syst. Appl. 2010, 37, 4761–4767. [Google Scholar] [CrossRef]
  74. Rodriguez, F.J.; García-Martínez, C.; Blum, C.; Lozano, M. An artificial bee colony algorithm for the unrelated parallel machines scheduling problem. In International Conference on Parallel Problem Solving from Nature; Springer: Berlin/Heidelberg, Germany, 2012; pp. 143–152. [Google Scholar]
  75. Oliveira, I.M.S.D.; Schirru, R.; de Medeiros, J. On the performance of an artificial bee colony optimization algorithm applied to the accident diagnosis in a pwr nuclear power plant. In Proceedings of the 2009 international nuclear Atlantic conference (INAC 2009), Rio de Janeiro, Brazil, 27 September–2 October 2009. [Google Scholar]
  76. Irani, R.; Nasimi, R. Application of artificial bee colony-based neural network in bottom hole pressure prediction in underbalanced drilling. J. Pet. Sci. Eng. 2011, 78, 6–12. [Google Scholar] [CrossRef]
  77. Ebrahimi, E.; Monjezi, M.; Khalesi, M.R.; Armaghani, D.J. Prediction and optimization of back-break and rock fragmentation using an artificial neural network and a bee colony algorithm. Bull. Eng. Geol. Environ. 2016, 75, 27–36. [Google Scholar] [CrossRef]
  78. Karaboga, D.; Basturk, B. A powerful and efficient algorithm for numerical function optimization: Artificial bee colony (ABC) algorithm. J. Glob. Optim. 2007, 39, 459–471. [Google Scholar] [CrossRef]
  79. Sun, L.; Koopialipoor, M.; Armaghani, D.J.; Tarinejad, R.; Tahir, M.M. Applying a meta-heuristic algorithm to predict and optimize compressive strength of concrete samples. Eng. Comput. 2019, 1–13. [Google Scholar] [CrossRef]
  80. Koopialipoor, M.; Ghaleini, E.N.; Tootoonchi, H.; Armaghani, D.J.; Haghighi, M.; Hedayat, A. Developing a new intelligent technique to predict overbreak in tunnels using an artificial bee colony-based ANN. Environ. Earth Sci. 2019, 78, 165. [Google Scholar] [CrossRef]
  81. Mehrabian, A.R.; Lucas, C. A novel numerical optimization algorithm inspired from weed colonization. Ecol. Inform. 2006, 1, 355–366. [Google Scholar] [CrossRef]
  82. Bashir, Z.A.; El-Hawary, M.E. Applying wavelets to short-term load forecasting using PSO-based neural networks. IEEE Trans. Power Syst. 2009, 24, 20–27. [Google Scholar] [CrossRef]
  83. Lin, C.-J.; Hsieh, M.-H. Classification of mental task from EEG data using neural networks based on particle swarm optimization. Neurocomputing 2009, 72, 1121–1130. [Google Scholar] [CrossRef]
  84. Taghavifar, H.; Mardani, A.; Taghavifar, L. A hybridized artificial neural network and imperialist competitive algorithm optimization approach for prediction of soil compaction in soil bin facility. Measurement 2013, 46, 2288–2299. [Google Scholar] [CrossRef]
  85. Yagiz, S.; Karahan, H. Prediction of hard rock TBM penetration rate using particle swarm optimization. Int. J. Rock Mech. Min. Sci. 2011, 48, 427–433. [Google Scholar] [CrossRef]
  86. Armaghani, D.J.; Hasanipanah, M.; Mahdiyar, A.; Majid, M.Z.A.; Amnieh, H.B.; Tahir, M.M. Airblast prediction through a hybrid genetic algorithm-ANN model. Neural Comput. Appl. 2018, 29, 619–629. [Google Scholar] [CrossRef]
  87. Momeni, E.; Nazir, R.; Armaghani, D.J.; Maizir, H. Prediction of pile bearing capacity using a hybrid genetic algorithm-based ANN. Measurement 2014, 57, 122–131. [Google Scholar] [CrossRef]
  88. Liou, S.-W.; Wang, C.-M.; Huang, Y.-F. Integrative Discovery of Multifaceted Sequence Patterns by Frame-Relayed Search and Hybrid PSO-ANN. J. UCS 2009, 15, 742–764. [Google Scholar]
  89. Asteris, P.G.; Armaghani, D.J.; Hatzigeorgiou, G.D.; Karayannis, C.G.; Pilakoutas, K. Predicting the shear strength of reinforced concrete beams using Artificial Neural Networks. Comput. Concr. 2019, 24, 469–488. [Google Scholar]
  90. Asteris, P.G.; Kolovos, K.G. Self-compacting concrete strength prediction using surrogate models. Neural Comput. Appl. 2019, 31, 409–424. [Google Scholar] [CrossRef]
  91. Armaghani, D.; Mohamad, E.; Hajihassani, M. Evaluation and prediction of flyrock resulting from blasting operations using empirical and computational methods. Eng. Comput. 2016, 32, 109–121. [Google Scholar] [CrossRef]
  92. Hornik, K.; Stinchcombe, M.; White, H. Multilayer feedforward networks are universal approximators. Neural Netw. 1989, 2, 359–366. [Google Scholar] [CrossRef]
  93. Mohamad, E.T.; Faradonbeh, R.S.; Armaghani, D.J.; Monjezi, M.; Majid, M.Z.A. An optimized ANN model based on genetic algorithm for predicting ripping production. Neural Comput. Appl. 2017, 28, 393–406. [Google Scholar] [CrossRef]
  94. Armaghani, D.J.; Hajihassani, M.; Sohaei, H.; Mohamad, E.T.; Marto, A.; Motaghedi, H.; Moghaddam, M.R. Neuro-fuzzy technique to predict air-overpressure induced by blasting. Arab. J. Geosci. 2015, 8, 10937–10950. [Google Scholar] [CrossRef]
  95. Sonmez, H.; Gokceoglu, C.; Nefeslioglu, H.A.; Kayabasi, A. Estimation of rock modulus: For intact rocks with an artificial neural network and for rock masses with a new empirical equation. Int. J. Rock Mech. Min. Sci. 2006, 43, 224–235. [Google Scholar] [CrossRef]
  96. Yang, H.; Koopialipoor, M.; Armaghani, D.J.; Gordan, B.; Khorami, M.; Tahir, M.M. Intelligent design of retaining wall structures under dynamic conditions. Steel Compos. Struct. 2019, 31, 629–640. [Google Scholar]
  97. Koopialipoor, M.; Noorbakhsh, A.; Noroozi Ghaleini, E.; Jahed Armaghani, D.; Yagiz, S. A new approach for estimation of rock brittleness based on non-destructive tests. Nondestruct. Test. Eval. 2019, 1–22. [Google Scholar] [CrossRef]
  98. Xu, C.; Gordan, B.; Koopialipoor, M.; Armaghani, D.J.; Tahir, M.M.; Zhang, X. Improving Performance of Retaining Walls Under Dynamic Conditions Developing an Optimized ANN Based on Ant Colony Optimization Technique. IEEE Access 2019, 7, 94692–94700. [Google Scholar] [CrossRef]
  99. Mohamad, E.T.; Koopialipoor, M.; Murlidhar, B.R.; Rashiddel, A.; Hedayat, A.; Armaghani, D.J. A new hybrid method for predicting ripping production in different weathering zones through in-situ tests. Measurement 2019, 147, 106826. [Google Scholar] [CrossRef]
  100. Koopialipoor, M.; Murlidhar, B.R.; Hedayat, A.; Armaghani, D.J.; Gordan, B.; Mohamad, E.T. The use of new intelligent techniques in designing retaining walls. Eng. Comput. 2019, 1–2. [Google Scholar] [CrossRef]
  101. Zorlu, K.; Gokceoglu, C.; Ocakoglu, F.; Nefeslioglu, H.A.; Acikalin, S. Prediction of uniaxial compressive strength of sandstones using petrography-based models. Eng. Geol. 2008, 96, 141–158. [Google Scholar] [CrossRef]
Applsci 09 05372 g001 550
Figure 1. General information related to data.
Figure 1. General information related to data.
Applsci 09 05372 g001
Applsci 09 05372 g002 550
Figure 2. The best simple regression based on the (a) Schmidt hammer test (Rn), (b) point load test (Is50) and (c) dry density (DD).
Figure 2. The best simple regression based on the (a) Schmidt hammer test (Rn), (b) point load test (Is50) and (c) dry density (DD).
Applsci 09 05372 g002
Applsci 09 05372 g003 550
Figure 3. Imperialism competitive algorithm (ICA) flowchart.
Figure 3. Imperialism competitive algorithm (ICA) flowchart.
Applsci 09 05372 g003
Applsci 09 05372 g004 550
Figure 4. Artificial bee colony (ABC) flowchart.
Figure 4. Artificial bee colony (ABC) flowchart.
Applsci 09 05372 g004
Applsci 09 05372 g005 550
Figure 5. Invasive weed optimization (IWO) flowchart.
Figure 5. Invasive weed optimization (IWO) flowchart.
Applsci 09 05372 g005
Applsci 09 05372 g006 550
Figure 6. Modeling procedures of hybrid artificial neural network (ANN)-based techniques in predicting Brazilian tensile strength (BTS).
Figure 6. Modeling procedures of hybrid artificial neural network (ANN)-based techniques in predicting Brazilian tensile strength (BTS).
Applsci 09 05372 g006
Applsci 09 05372 g007 550
Figure 7. Hybrid models of ICA with various Ncountry numbers.
Figure 7. Hybrid models of ICA with various Ncountry numbers.
Applsci 09 05372 g007
Applsci 09 05372 g008 550
Figure 8. Hybrid models of artificial bee colony (ABC) with various bee sizes.
Figure 8. Hybrid models of artificial bee colony (ABC) with various bee sizes.
Applsci 09 05372 g008
Applsci 09 05372 g009 550
Figure 9. Hybrid models of invasive weed optimization (IWO) with various seed numbers.
Figure 9. Hybrid models of invasive weed optimization (IWO) with various seed numbers.
Applsci 09 05372 g009
Applsci 09 05372 g010 550
Figure 10. Prediction results of the best IWO-ANN model for all samples.
Figure 10. Prediction results of the best IWO-ANN model for all samples.
Applsci 09 05372 g010
Table
Table 1. Investigating the effects of various Ncountry in predicting BTS.
Table 1. Investigating the effects of various Ncountry in predicting BTS.
Model No.NcountryNetwork ResultRankingTotal Rank
TR TSTR TS
R2RMSE R2RMSER2RMSE R2RMSE
1500.8650.1019 0.8770.110444 3314
21000.8710.1012 0.8820.100155 4418
31500.8530.1136 0.8630.118711 114
42000.8990.096 0.9010.097888 7730
52500.8570.1113 0.8710.111922 228
63000.8610.1049 0.9020.097133 8824
73500.8740.1003 0.8990.098366 6624
84000.8980.0964 0.8850.099777 5524
TR: Training, TS: Testing.
Table
Table 2. Investigating effects of various bees in predicting BTS.
Table 2. Investigating effects of various bees in predicting BTS.
Model No.Bees No.Network ResultRankingTotal Rank
TR TSTR TS
R2RMSE R2RMSER2RMSE R2RMSE
150.8430.1295 0.8540.126411 114
2100.8570.1122 0.8670.109122 228
3150.8960.0987 0.8990.097155 6622
4200.8950.0986 0.8790.099544 3314
5250.9060.0959 0.9020.096377 7728
6300.9010.097 0.8940.097866 5522
7350.8750.1001 0.8910.098833 4414
8400.9080.0946 0.9040.095588 8832
TR: Training, TS: Testing.
Table
Table 3. Investigating effects of various seeds in predicting BTS.
Table 3. Investigating effects of various seeds in predicting BTS.
Model No.Seeds No.Network ResultRankingTotal Rank
TR TSTR TS
R2RMSE R2RMSER2RMSE R2RMSE
150.8980.0964 0.8890.097155 4418
2100.8690.0998 0.8810.098822 3310
3150.8610.1018 0.8770.100411 226
4200.8740.0987 0.8690.102833 118
5250.9230.0918 0.9090.094377 6626
6300.9090.0935 0.9220.092766 8828
7350.9280.0911 0.9170.093688 7730
8400.8870.0979 0.8940.095944 5518
TR: Training, TS: Testing.
Table
Table 4. Results of hybrid models. RMSE: root mean square error, VAF: variance accounted for.
Table 4. Results of hybrid models. RMSE: root mean square error, VAF: variance accounted for.
Hybrid ModelR2VAFRMSE
TRTSTRTSTRTS
IWO-ANN0.9280.91792.87291.7310.09110.0936
ABC-ANN0.9080.90490.81690.4190.09460.0955
ICA-ANN0.8990.90189.88990.1340.09600.0978
TR: Training, TS: Testing.

Share and Cite

MDPI and ACS Style

Huang, L.; Asteris, P.G.; Koopialipoor, M.; Armaghani, D.J.; Tahir, M.M. Invasive Weed Optimization Technique-Based ANN to the Prediction of Rock Tensile Strength. Appl. Sci. 2019, 9, 5372. https://doi.org/10.3390/app9245372

AMA Style

Huang L, Asteris PG, Koopialipoor M, Armaghani DJ, Tahir MM. Invasive Weed Optimization Technique-Based ANN to the Prediction of Rock Tensile Strength. Applied Sciences. 2019; 9(24):5372. https://doi.org/10.3390/app9245372

Chicago/Turabian Style

Huang, Lei, Panagiotis G. Asteris, Mohammadreza Koopialipoor, Danial Jahed Armaghani, and M. M. Tahir. 2019. "Invasive Weed Optimization Technique-Based ANN to the Prediction of Rock Tensile Strength" Applied Sciences 9, no. 24: 5372. https://doi.org/10.3390/app9245372

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop