1 Introduction

Nowadays, circular diamond saw blades are widely used to processing the dimension stone in stone processing plant and quarry. The performance and life of these tools represent a major factor in the cost of stone industry. Therefore, predicting the sawing performance for a particular stone is important. Sawing performance is affected by many factors. One of the most important factors that affect sawing performance is the physical and mechanical properties of stone. In this regard, many studies have been performed. Some of these studies are listed in Table 1.

Table 1 Literature review of sawability studies
Full size table

This paper describes the application of meta-heuristic algorithm and fuzzy C-mean to predict the sawability of dimension stone based on some important physical and mechanical properties of rock. To achieve this goal, in the first step, four major properties of dimension stone including the uniaxial compressive strength, Schmiazek F-abrasivity, Mohs hardness and Young’s modulus were tested in rock mechanics laboratory. During the research process, two groups of dimension stones were selected and analyzed. The dimension stone block samples were collected from a number of Iranian quarries. In the second step, the studied dimension stones were classified by imperialist competitive algorithm. The objectives are to minimize distance among members of a cluster and to maximize the distance of cluster centers from each other. The ICT is an evolutionary algorithm to solve optimization complex problems. This method was created by the simulation of the sociopolitically processes, and fuzzy C-mean was considered as a flexible clustering method [24]. To show the efficiency of the proposed methods, the classification results were validated by test results. Finally, comparing the results of the proposed two methods with test results, it can be derived that the proposed methods can find an appropriate solution. Hence, the scope of this paper is the applications of ICA technique and fuzzy logic in mining engineering with focus on optimization of classification.

2 Imperialist competitive algorithm

Nowadays, meta-heuristic algorithms are widely used for optimization and solving engineering problems in engineering sciences. Several evolutionary algorithms are provided for optimization such that it can be noted to the genetic algorithm (GA) [25], particle swarm optimization algorithm (PSO) [26] and bat algorithm [27]. The imperialist competitive algorithm is one of the most effective methods to find optimal solution of different problems among meta-heuristic algorithms. This algorithm is for solving nonlinear integer programming such as dynamic programming (DP) [28]. This algorithm inspired from a sociopolitical phenomenon proposed by Atashpaz-Gargari and Lucas [29]. The ICA is a great algorithm in the field of evolutionary computation based on the sociopolitical evolution of humans. This algorithm has an application effective in various fields of sciences in optimization of complex issues. In general, the imperialist competitive algorithm can be used in any optimization problem without limitation. This matter has caused which this algorithm is used for solving lots of problems in the field of electrical engineering, mechanics engineering, industry, management, civil engineering and artificial intelligence, etc. [30–34]. The ICA established based on three main principles, which include assimilation policy, imperialistic competition and revolution. Assimilation policy is moving colonies toward their relevant imperialist in a randomly deviated direction. Imperialistic competition and revolution are competition of take possession of colonies from other empires and random changes in the characteristics of some countries, respectively. Finally, algorithm continues with these three main principles until a stop condition is satisfied.

Like other meta-heuristic algorithms, process begins with an initial population in this algorithm, each of them being called a country. The countries have two types: imperialists or colonies. In fact, the country is an array of variable values like to chromosome in GA terminology. Each empire gets and controls one or more of the colonies of the weakest imperialist based their capacity and ability. Some of them that have the lowest costs of the fitness function selected as imperialist and also remaining population considered are as a colony. At first, normalized cost is calculated for per empire by Eq. (1), where the C n is the normalized cost of empire, max{c i } is the highest cost among imperialists and c n is the cost of nth imperialist.

$$C_{\text{n}} = \,c_{\text{n}} - \,\,\,\,\mathop {\hbox{max} }\limits_{i} \left\{ {c_{i} } \right\}$$
(1)

Then, with having cost of normalized, the normalized strength of each imperialist (P n) is calculated by Eq. (2) which based on this process, colonies divided among the empire:

$$P_{\text{n}} = \,\left| {\frac{{C_{\text{n}} }}{{\sum\limits_{i = 1}^{{N_{\text{imp}} }} {C_{i} } }}} \right|$$
(2)

Therefore, the number of initial colonies obtained by Eq. (3) in which NC n and N col are the initial number of colonies of nth imperialist and the number of total colonies, respectively:

$$NC_{\text{n}} = \,\text{round}\left\{ {P_{{{\text{n}}.(N_{col} )}} } \right\}$$
(3)

Figure 1 shows moving colonies toward their relevant imperialist.

Fig. 1
figure 1

Moving colonies toward their relevant imperialist in a randomly deviated direction

Full size image

According to Fig. 1, the value of X is defined by Eq. (4) that (d) and (β) are distance between colony and imperialist and a parameter greater than one, respectively. The β > 1 causes to get closer to the imperialist from both sides. Also, for increasing search area around of the imperialist, this absorption occurs with a random amount of deviation in absorption direction of colony by empire. This deviation is shown with \(\theta\). This value is random number that selected randomly and uniform distribution according to Eq. (5).

$$x\, \sim \,U(0,\beta \times d)$$
(4)
$$\theta \, \sim \,U( - \gamma ,\gamma )$$
(5)

γ is a parameter that controls the angle of deviation range and is considered equal to \(\frac{\pi }{4}\) (Rad). At the final step, time of empire competitive is among the imperialists. At this stage, each colony separated from the weakest empire and joins to a stronger colonial. Whatever colonial is stronger and it is more likely to be selected by colony. Finally, each empire that lost their colony becomes as a new colony. This process will continue to establish the stop condition in the algorithm [35].

3 Fuzzy C-mean

Fuzzy C-means clustering is an evolutionary model for creating the hard C-means clustering. In the meantime, application of the fuzzy sets theory in the clustering process is very effective and beneficial. In other words, the fuzzy clustering technique is one of the most effective methods in the determination of the set member’s membership degree. Fuzzy clustering is viewed as a flexible clustering method. This method was proposed by Bezdek et al. [36] in 4 steps based on the iteration optimization [37].

In the first step, the number of clusters represented by the symbol c is determined (2 ≤ c ≤ n) and a value is also determined for the weighting parameter of m′. This parameter plays a key role in the clustering process’s fuzziness. Then, a primary partitioning matrix of U(0) is calculated, which will be updated per iteration of the algorithm. Each algorithm’s iteration is determined with an r value where r = 0, 1, 2, 3 and …

In the next step, the centroid of clusters, {Vi(r)}, is calculated per iteration.

Using Eqs. (6)–(10), the partition matrix for the rth iteration will be updated as U (r). μ (r+1) ik is the degree of membership of the kth data in the ith cluster for the algorithm’s (r + 1) th iteration in the third step. dik represents the Euclidean distance between kth data and ith cluster center. In Eq. (6), φ indicates the equation hardness.

$$\mu_{ik}^{(r + 1)} = \left[ {\sum\limits_{j = 1}^{c} {\left( {\frac{{d_{ik}^{(r)} }}{{d_{jk}^{(r)} }}} \right)^{{\frac{2}{(m' - 1)}}} } } \right]^{ - 1}\quad {\rm for}\,\,I_{k} = \varphi$$
(6)
$$\mu_{ik}^{(r + 1)} = 0\quad \text{for all classes}\,\,\,i\,\,\,\text{where}\,\,i\, \in \tilde{I}_{k}$$
(7)
$$I_{k} = \left\{ {i|2 \le C < n;\,d_{ik}^{(r)} = 0} \right\}$$
(8)
$$\tilde{I}_{k} = \{ 1,2, \ldots ,c\} - I_{k}$$
(9)
$$\sum\limits_{{i \in I_{k} }} {\mu_{ik}^{(r + 1)} } = 1$$
(10)

In the last step, the clustering’s precision level is controlled by Eq. (11), where ε L is the precision level, i.e., calculations have reached the desired precision level of ε L and clustering has led to a proper optimization, and therefore, the calculations end here. Otherwise, we should return to the second step and repeat the calculations. This loop is repeated until the desired precision level is obtained. Due to the complexity and heavy calculations and the risk of error in manual computations, a MATLAB code is used for calculations.

$$\parallel \tilde{U}^{(r + 1)} - \tilde{U}^{(r)} \parallel \,= \varepsilon_{L}$$
(11)

4 Studied quarries

Iran is among the countries that possess great potentials and capabilities in the production of dimension stones. There is a wide range of sedimentary, igneous and metamorphic dimension stone resources in Iran. Due to the high quality, beautiful color and texture of dimension stones of Iran, these resources are distinguished and superior as compared their compared to their counterparts from other parts of the word, and in some cases, they are one of a kind in market. Therefore, the dimension stones industry can be accounted as a strong point of mining sector from different aspects such as exploitation, domestic production, export and foreign exchange revenue. The exploitable dimension stone deposits of Iran can be summarized as travertine about 59 million tons, marble about 500 million tons and granite about 60 million tons [38]. In this paper, 12 quarries with famous dimension stones as follows: Red Granite (A1), Black Granite (A2), White Granite (A3), Chocolate Granite Khoramdeh (A4), Pearl Granite (A5), Cream Marble Harsin (A6), Pink Marble Anarak (A7), Red Travertine (A8), Travertine Hajiabad (A9), Travertine Darebokhari (A10), Marble Salsali (A11) and Pink Marble Haftoman (A12) are considered and studied. The location, geological and discontinuities of studied quarries are given in Fig. 2 and Table 2. The samples of stones which have been studied were collected from these mines and transferred to laboratory.

Fig. 2
figure 2

Location of studied quarries

Full size image
Table 2 Geology and discontinuities characteristics of studied quarries
Full size table

5 Laboratory studies

For laboratory tests, sample blocks were collected from the studied quarries. Standard tests were done to measure the four major of physical and mechanical properties of rock were chosen for assessing the sawability of dimension stone. These parameters are listed as follows:

  1. (a)

    Schmiazek abrasivity factor (SF-a)

  2. (b)

    Uniaxial compressive strength (UCS)

  3. (c)

    Mohs hardness (MH)

  4. (d)

    Young’s modulus (YM)

The reasons of the selections of these parameters are given below.

5.1 Schmiazek F-abrasivity factor (SF-a)

Abrasiveness influences the wear of sawing tools. Abrasiveness is mainly affected by various factors such as mineral composition, the hardness of mineral constituents and grain size, grain shape and grain angularity [39]. Schimazek’s F-abrasiveness factor is the most common factor for evaluation of rock abrasivity. It is depended on textural and mechanical properties. The Schimazek’s F-abrasiveness factor is defined as

$${\text{SFa}} = \frac{{{\text{EQC}} \times {\text{Gs}} \times {\text{BTS}}}}{100}$$
(12)

where F is the Schimazek’s wear factor (N/mm), EQC is the equivalent quartz content percentage, Gs is the median grain size (mm) and BTS is the indirect Brazilian tensile strength.

The Schimazek’s F-abrasiveness factor was selected as an abrasiveness index of rock in clustering system. The results of laboratory studies to determine this index are given in Table 3.

Table 3 Result of laboratory studies to determine the SF-a
Full size table

5.2 Uniaxial compressive strength (UCS)

Uniaxial compressive strength is an important mechanical parameter to present the engineering property of rock. Rock material strength is used as an important parameter in many rock mass classification systems [40]. Uniaxial compressive strength test can be considered as characteristic of rock strength, density, weathering, texture and matrix type. Therefore, using this parameter in this study to predict the sawability of dimension stone is necessary. To determine the uniaxial compressive strength of studied dimension stones, the 5 standard NX core samples with a length-to-diameter ratio of 2.5:1 were taken using a diamond rotating drill from block sample. A circular diamond saw blade was used to cut the specimens to their final lengths. After that, the surfaces of specimens were grinded using a grinding machine to achieve a high-quality surface for the axial loading. The mechanical tests were carried out by a servo-controlled testing machine designed for rock test. The standard uniaxial compressive strength test of core samples was carried out under a loading rate of 1 MPa/s [41]. Finally, the average uniaxial compressive strength was calculated for each studied dimension stones. The results are given in Table 4.

Table 4 Result of laboratory studies
Full size table

5.3 Mohs hardness scale (MH)

Hardness can be defined as the rock’s resistance to fracture or plastic deformation due to scratching and cracking from circular diamond saw. The factors that affect rock hardness are the hardness of the constitutive minerals, cohesion forces, homogeneity and the water content of rock [42]. Hardness is a good index of all above given parameters of rock material. Mohs hardness scale is the common and useful index to evaluate the hardness of rock. It was selected as hardness index in the clustering system. The mean hardness of each dimension stone is calculated based on the hardness of contained minerals using the following equation:

$${\text{Mean}}\,\,\,{\text{Hardness}} = \sum\limits_{i = 1}^{n} {M_{i} } \times H_{i}$$
(13)

where M i is mineral amount (%), Hi is Mohs hardness and n is the total number of minerals in the dimension stone. The mean Mohs hardness of each studied dimension stone is given in Table 4.

5.4 Young’s modulus (YM)

According to rock behavior during the fracture process, especially in sawing, the way that rocks reach the failure point has a great influence on sawability. The best scale for rock elasticity is Young’s modulus. Based on ISRM suggested methods [41], the tangent Young’s modulus at a stress level equal to 50 % of the ultimate uniaxial compressive strength is used in this clustering system. The results are given in Table 4.

Then, the pseudo-code must be written for the fitness function and the other parts of the algorithm in MATLAB. Afterward, for more evaluation and data analysis the samples are evaluated in 2, 3, 4 and 5 clusters. The results of classification were not agreeable with properties of the studied dimension stones for 2, 4 and 5 clusters. The lowest and the most appropriate cost function achieved for 3 clusters.

6 Application of imperialist competitive algorithm in dimension stone sawability prediction

One of the most impressive applications of ICA is data clustering of information. The objective function is one of the most significant parts in meta-heuristic algorithms. In fact, the fitness function of an optimization problem is to maximize the distance of cluster centers from each other and to minimize distance among members of a cluster. The Lloyd’s algorithm (k-means clustering) is the fitness function in imperialist competitive algorithm. It shows in Eq. (14):

$${\text{Obj}} . {\text{Function}} = \sum\limits_{i = 1}^{n} {\mathop {\hbox{min} }\limits_{1 \le j \le k} \,{\text{d}}(x_{i} ,m_{j} )}$$
(14)

In fact, for the optimization of data clustering, Lloyd’s algorithm (k-means clustering) is used based on the imperialist competitive algorithm in this research and the main results of optimization obtained from ICA.

6.1 Data clustering by 3 clusters

In data mining for the total sample dimension stones, the considered limits in the clustering are maximum iteration of 150, minimum acceptance precision of ε L  = 0.0001 and 75 population as initial population for ICA. The Lloyd’s algorithm conducted an optimization as the fitness function and found to maximize the distance of cluster centers from each other and to minimize distance among samples of a cluster based on the considered limits of ICA and four important physical and mechanical properties of samples. The algorithm process stops after the calculations. For fuzzy C-mean, considered limits in the clustering are maximum iteration of 100, minimum acceptance precision of εL = 0.00001 and the weighting parameter of m′ = 2. Distance of clusters’ centers from criteria and classification of samples for each method are given in Tables 5 and 6, respectively.

Table 5 Distance of clusters’ centers from criteria by ICA and fuzzy C-mean for 3 clustering
Full size table
Table 6 Optimization and classification of samples of ICA and fuzzy C-mean
Full size table

Table 7 represents the difference between the results of step 120 and 121 in which it is less than the minimum acceptance precision of 0.0001 and this process is fixed from 121th to 150th iteration; therefore, the computing stops at the 200th iteration for ICA. Furthermore, as listed in Table 7, the difference between the results of stages 47 and 48 is less than the minimum acceptance precision of 0.00001; therefore, the calculations stop at the 17th iteration.

Table 7 Precision level and calculation termination of ICA and fuzzy C-mean
Full size table

Figure 3 shows data clustering based on 3 clusters, and Fig. 4 shows minimum cost per iteration in which the 121th iteration is reached to the optimal point.

Fig. 3
figure 3

Position of samples in clustering

Full size image
Fig. 4
figure 4

Minimum cost per iteration in ICA

Full size image

7 Validation of results

Validation of results is an important phase of search. For validation of the applied algorithm and its results, field studies were carried out. The production rate of each studied dimension stones was used as a criterion for evaluating and validating the results. For this purpose, some stone factories in Shamsabad located in Tehran were selected, and the performance of the diamond circular saw was measured in term of hourly production rates (P h). The production rate and sawability classification of the studied dimension stones are given in Table 8.

Table 8 Production rate and sawability classification of the studied dimension stones
Full size table

According to Table 5, the applied algorithm is almost able to classify the studied dimension stone into two geological groups based on physical and mechanical properties. The first group (Class 1) is hard dimension stone and the second group (Class 3) is soft dimension stone such as marble and travertine. In addition, the applied algorithm can be used to predict and classify the dimension stone sawability or the performance of circular diamond saw. The sawability of studied dimension stones was classified into two major groups. The first group (Class 1) has a poor sawability with a P h less than 6.5 m2/h, and the second group (Class 3) has a good sawability with a P h greater than 8.5 m2/h. The results show the capability of the imperialist competitive algorithm (ICA) and fuzzy clustering technique for solving optimal classification in stone industry.

8 Conclusion

The circular saw machine is one of the most important machines used in the stone processing factories. The performance prediction of circular diamond saw blade is very important in the cost estimation and the planning of the factories. In this paper, it was aim to evaluate and classify the dimension stone by imperialist competitive algorithm and fuzzy C-mean based on some important physical and mechanical properties of dimension stone. To achieve this goal, two groups of Iranian famous dimension stones were studied. The studied dimension stones were classified by applied algorithms, and the classification results were validated by the performance of the diamond circular saw was measured in term of hourly production rates (P h) at some stone factories in Shamsabad. According to the results of this study, the applied algorithms were almost able to classify the studied dimension stones into two geological groups such as hard dimension stone (Class 1) and soft dimension stone (Class 3). Moreover, the sawability results of studied dimension stones confirm the capability of applied algorithm. The first class has a poor sawability, and the third class 3 has a good sawability. The results of ICA algorithm showed that the identified samples are similar to that obtained from fuzzy C-mean. In addition, based on optimal and comparative results and prediction of the dimension stone sawability, ICA and fuzzy C-mean make it possible to evaluate the hurly production rate of circular diamond saw although ICA is superior to fuzzy C-mean based on the results of optimum partition. Finally, the results showed that the sawability of dimension stone can reliably be classified from some important physical and mechanical properties of dimension stone such as uniaxial compressive strength, Schmiazek F-abrasivity, Mohs hardness scale and Young’s modulus by using imperialist competitive algorithm and fuzzy clustering technique.