J. Cent. South Univ. (2012) 19: 1953-1960 

DOI: 10.1007/s11771-012-1231-y

Integrating unascertained measurement and information entropy theory to assess blastability of rock mass

ZHOU Jian(周健), Li Xi-bing(李夕兵)

School of Resources and Safety Engineering, Central South University, Changsha 410083, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: Due to the complex features of rock mass blastability assessment systems, an evaluation model of rock mass blastability was established on the basis of the unascertained measurement (UM) theory and the actual characteristics of the project. Considering a comprehensive range of intact rock properties and discontinuous structures of rock mass, twelve main factors influencing the evaluation blastability of rock mass were taken into account in the UM model, and the blastability evaluation index system of rock mass was constructed. The unascertained evaluation indices corresponding to the selected factors for the actual situation were solved both qualitatively and quantitatively. Then, the UM function of each evaluation index was obtained based on the initial data for the analysis of the blastability of six rock mass at a highway improvement cutting site in North Wales. The index weights of the factors were calculated by entropy theory, and credible degree identification (CDI) criteria were established according to the UM theory. The results of rock mass blastability evaluation were obtained by the CDI criteria. The results show that the UM model assessment results agree well with the actual records, and are consistent with those of the fuzzy sets evaluation method. Meanwhile, the unascertained superiority degree of rock mass blastability of samples S1-S6 which can be calculated by scoring criteria are 3.428 5, 3.453 3,  4.058 7, 3.675 9, 3.516 7 and 3.289 7, respectively. Furthermore, the proposed method can take into account large amount of uncertain information in blastability evaluation, which can provide an effective, credible and feasible way for estimating the blastability of rock mass. Engineering practices show that it can complete the blastability assessment systematically and scientifically without any assumption by the proposed model, which can be applied to practical engineering.

Key words: rock mass; blastability; unascertained measurement (UM) model; information entropy; prediction

1 Introduction

Rock mass blastability indicates the ease with which a rock mass can be fragmented by blasting during the excavation process. It’s also the comprehensive reflection of the integrated physical and mechanical properties of rock under dynamic loading. Moreover, the classification of rock blastability not only was applied to estimating explosive consumption for blasting engineering, but also provided the basic parameters for the blasting design (BD). Therefore, it is very important to give a practical and reasonable way to classify the blastability of rock mass [1-3].

Several approaches have been used for estimating blastability [1, 3-6]. Some researchers have tried to correlate it with the data available from laboratory and field testing of rock parameters; some others have related it with rock and blast design parameters; yet some have tried to estimate blastability through approaches based on the drilling rates and/or blast performances in the field. The traditional rock mass classification methods, only focused on rock itself, used some isolated indexes which do not consider characteristics of blasting engineering but those of rock mass [1]. RAKISHEV [4] devised ?ve blastability classes according to the value of an index named critical fracture velocity. HEINEN and DIMOCK [5] proposed a graphical method for the assessment of blastability index according to seismic propagation velocity in rock mass. LILLY [6] developed an empirical method to assess rock mass blastability. In fact, many factors may affect the rock-blasting behavior besides the characteristic of rock mass. The latest improvements in computer methods have also opened up new vistas for the researchers to use various artificial intelligence algorithms for determination of blastability [1, 6-9]. FENG [7] established an arti?cial neural network (ANN) approach for comprehensive classification of rock stability, blastability and drillability. XUE et al [2] proposed the attribute recognition model (ARM) of assessment of classification for rock mass blastability in engineering blasting based on the theory of attribute mathematical theory. LATHAM and LU [3] proposed a blastability designation (BD), as a part of their developed energy-block transition (EBT) model, for prediction of blasted block size distribution (BBSD). AZIMI et al [8] investigated a fuzzy logic based blastability designation predictor model. A feed-forward back-propagation neural network was developed by HAN et al [9] to classify rock mass blastability. However, the developed ANN model has some limitations, such as black box approach, arriving at local minima, overfitting problem and absence of probabilistic output. Furthermore, there is not proper method to determine the number of hidden layers in the ANN model. The developed fuzzy logic model determines the fuzzy rules with difficulty.

Comprehensive classification of the blastability of rock is one of the most complex, knotty problems in rock mechanics and blast design. For this case, an effective method to assess the blastability of rock mass should be presented based on the unascertained mathematics theory in the current work. Unascertained mathematics has been proposed by WANG [10], which is a tool to describe subjective uncertainty quantitatively. It deals mainly with unascertained information which differs from stochastic information, fuzzy information, and grey information. Unascertained information refers to information demanded by decision-making over which the message itself has no uncertainty. Since the 1990s, LIU et al [11] and other scholars [12-15] have done much work and unascertained mathematics has been successfully used in many fields. It is, therefore, motivating to investigate the capability of UMM in rock mass blastability evaluation.

2 Details of unascertained measurement (UM) model

2.1 Determining evaluation space

Suppose that Γ has n objects to be assessed, and the assessment object space is Γ=(Γ₁, Γ₂, …, Γ_n). Each object of Γ_i (i=1, 2, …, n) has m evaluating indices, so the evaluating index space is Ψ=(Ψ₁, Ψ₂, … , Ψ_m). Then Γ_i can be denoted to be m-dimensional   where  is the measured value of evaluation object, R_i, with respect to evaluating index  For each  (i=1, 2, …, n; j=1, 2, …, m), we assume that there are p evaluation grades of (Δ₁, Δ₂, …, Δ_p).

The evaluation space is Θ, denoted as Θ=(Δ₁, Δ₂, …, Δ_p). Suppose Δ_k (k=1, 2, …, p) is the k-th evaluation grade, and the k-th grade is higher than that of the (k+1)-th one, denoted as Δ_k>Δ_k+1. If the grading rank (Δ₁, Δ₂, …, Δ_p) satisfies Δ₁>Δ₂>…>Δ_k+1 or Δ₁<Δ₂<…<Δ_k+1, (Δ₁, Δ₂, …, Δ_p) is called the ordered partition class of evaluation space Θ [11-15].

2.2 UM of single index

Denote the UM as = , where  is  the  degree of  belonging to the k-th evaluation grade of C_k , which satisfies the following principles [11-15]:

1) Nonnegative and limited principle

                            (1)

2) Convergent principle

                                (2)

3) Additive principle

           (3)

As an UM, these three principles must be met simultaneously; otherwise, it should be called as an estimate. The UM matrix of single index is constructed as

                  (4)

In practice, decision-maker should decide the specific structure based on the background, knowledge and prior knowledge [11-15]. At present, there are several commonly used methods to construct measure function, including straight-line method (SLM), quadratic curve method (QCM), sine curve method (SCM) and exponential curve method (ECM). Among them, the SLM function is the most widely used and the simplest UM function. The straight-line unascertained measure function is used in this work and the function graph is shown in Fig. 1.

Fig. 1 Straight-line UM function

The corresponding function expression of the above

UM function is

    (4)

2.3 Identification weight of index

When the membership of every index has been known, the “importance” of the weight has only one meaning, that is, the importance for the index classification data to determine the classification of the index. So, the index weight used in determining the composed membership by the single index is and only is the identification weight of the index. In this case, the common methods used to get the weight, such as Delphi method, Brainstorming, and Analytic hierarchy process (AHP) are helpless. The method of determining the index identification weight by using the information entropy is introduced as follows [2, 12].

SHANNON proposed the conception of information entropy, which is employed to measure the uncertainty roughly [12, 16]. Suppose that w_j is the relative important extent of measured index Ψ_j compared with

other indices. w_j satisfies 0≤w_j≤1, and , which

is called the weight of Ψ_j. Index weight vector w is characterized by w=(w₁, w₂, …, w_m). Then, w_j is given by

                         (5)

                              (6)

where 0≤χ_i≤1. The evaluation matrix of UM of single index is known, so w_j can be obtained by Eqs.(5) and (6).

2.4 Composite UM of multiple indices

Parameter υ_ik is denoted as the degree of the assessment object Γ_i belonging to the k-th evaluation grade of Δ_k. When υ_ik is equal to υ (Γ_i∈Δ_k), υ_ik is called the composite UM of multiple indices. After the single index matrix and the identification weight are derived, considering the meaning of the identification weight, the composite UM of multiple indices can be worked out. The comprehensive assessment vector is defined as

         (7)

where υ_ik satisfies 0≤υ_ik≤1 and  .

2.5 Principle of identification

Because the classification of the comment ranks (Δ₁, Δ₂, …, Δ_p,) is ordered, e.g., Δ_k is “better” than Δ_k+1, the maximum measure identification (MMI) principle is not available. The credible degree identification (CDI) principle is needed. Let the credible identification be λ, where 0.5≤λ≤1, and it is always 0.6 or 0.7 [11]. If the evaluation space meets Δ₁>Δ₂>…>Δ_p, Let

            (8)

Then, φ_i is the k₀ -section appraisal grade Δ_ki. The implication is as follows. The confidence that the grade of φ_i is not higher than k₀ is λ or the confidence that the grade of sample φ_i is higher than k₀ is 1- λ.

Suppose that the score value of C_t is I_t, J_Δi is given by

                              (9)

where is the unascertained superiority degree of evaluation object Δ_i, so J=(J_Δ1, J_Δ2, … J_Δi) is called the vector of unascertained superiority degree. The superiority degree of Δ_i is ordered according to the magnitude of J_Δi.

3 UM model for evaluating blastability of rock mass

The approach for the development of the UM-based assessment for the blastability of rock mass can be divided into seven stages: 1) Choice of the criterion indices for rock mass blastability classification; 2) Collection of data sets of rock mass; 3) Construction of UM function of single index; 4) Determination of the entropy weight of criterion indices; 5) Combined UM of multiple indices of rock mass blastability; 6) Establishment of the final UM model for evaluating the blastability of rock mass with the help of credible degree identification (CDI) principle; 7) Evaluation and validation of the UM model by evaluation with testing data and comparing it with literature correlations, as shown in Fig. 2.

3.1 System of assessment indices

Selecting the correct classification of rock blastability evaluation is to ensure that evaluation results are reasonable, economic and reliable in the pre-conditions. On the basis of the related studies [1-8], twelve factors affecting the blastability evaluation for rock mass are selected as the evaluation indices, which are strength, resistance to fracturing, sturdiness of rock, elasticity of rock, resistance to dynamic loading, hardness of rock, deformability, resistance to breaking, in-situ block sizes, fragility of rock mass, integrity of rock mass and discontinuity plane’s strength, designated as P1, P2, P3, P4, P5, P6, P7, P8, P9, P10, P11 and P12, respectively. The system of blastability assessment indexes is shown in Fig. 3.

Fig. 2 Principle flow chart for proposed UM-based approach for classification of blastability of rock mass

Fig. 3 Index systems of blastability comprehensive assessment of rock mass

Table 1 Suggested quantitative indications for classi?cation of blastability of rock mass associated with individual factor [3, 8]

And the classification standards for these indices are determined with reference to Refs. [3, 8], as shown in Table 1. The evaluation set is {C₁, C₂, C₃, C₄, C₅}, where designated classes C₁, C₂, C₃, C₄ and C₅ are denoted as very easy, easy, moderate, difficult, and very difficult, respectively.

3.2 UM model development and validation

The blastability assessment system developed above has been applied to a case study that assesses the blastability of the rock mass at a highway improvement cutting site in North Wales (hereafter referred to as the G cutting site) [3, 8]. Therefore, UM model was established using the theory discussed. The evaluation indexes of UM model for evaluating of classification of the blastability of rock mass was established based on the unascertained mathematics theory and the actual characteristics in this work. The evaluation indexes of the proposed model are as follows:

1) Construction of UM function of single index

The UM functions of single index were constructed to get the value of the evaluation factors, on the basis of the definition of the UM function and the classification in Table 1. The UM function of each index is illustrated in Fig. 4. Then, the evaluation matrix of UM of six rock mass sample could be obtained, according to the functions in Figs. 4(a)-(i) and values of the factors given in Table 2. Taking sample S1 for example, the values of the eight evaluation indices for S1 in Table 2 were substituted into the corresponding UM functions in  Figs. 4(a)-(i), respectively. Then, the evaluation matrix of UM of S1 was calculated as       

    (10)

2) Composite UM of multiple indices of rock mass blastability

The weights of the indices were determined by Eqs.(1)-(6). So, the weights of Γ_i denoted as w=(w₁, w₂,…, w₈) were (0.107 0, 0.100 7, 0.109 7, 0.101 2,  0.111 7, 0.176 9, 0.176 9, 0.115 7). Then, the composite UM of multiple indices of S1 were calculated as (0.145 0, 0.082 8, 0.197 3, 0.314 2, 0.260 7).

3) Evaluation results recognizing

In the present study, the grade of blastability of rock mass is divided into five grades: very easy, easy, moderate, difficult, and very difficult, therefore the credible degree identification (CDI) criteria were adopted to judge the grade of Γ_i instead of the maximum measurement identification rule, due to the sequence of the evaluating grade (C₁, C₂, C₃, C₄, C₅). And the confidence level λ is taken as 0.6. According to Eq. (7) of composite UM vector of multiple indices and Eq. (8) of CDI, k₀ =0.746 5, which is larger than λ in descending order, thus the blastability degree of S1 belongs to grade C₃. The final assessment results show that the sample S1 belongs to the third rank, which means “moderate”.

From the above, we can get that the several identified results are in accordance with score value criteria. Therefore, the superiority degree of R₁ is determined as grade C₃, that is, the blastability of sample S1 is moderate. In the same way, rock mass blastabilities of other samples are evaluated. The composite UM of multiple indices and the evaluated results are shown in Table 3 and Fig. 6. It can be concluded that blastabilities of sample S1-S6 are moderate, moderate, difficult, moderate, moderate, and moderate. This shows that the unascertained superiority degrees of sample S1-S6 are 3.428 5, 3.453 3, 4.058 7, 3.675 9, 3.516 7 and 3.289 7, respectively (Table 4), that is, S3>S4>S5>S2>S1>S6.

3.3 Discussion

In order to compare the difference between our methods and other evaluation methods, the evaluation results of blast design [3], LATHAM and LU method [3] and Fuzzy set model [8] are listed in Table 5. We can see from the UM model evaluation results and actual assessment results that the classification results of both Fuzzy models are accordant with the actual situation basically. Compared with conventional statistics, UM theory is an exclusive theory for the study of    machine learning law in the condition of small sample number.

The above-mentioned comparisons indicate that all four models are competitive with each other for blastability of rock mass, but the performance of the UM model is relatively superior to the others. Combining the theory with the actual case will verify the feasibility and practicality of the model. Employing the information entropy and credible identification principal, the shortcoming of fuzzy set assessment is overcome and the result becomes more objective. It is apt to realize scientific and rational decision-making. The reasons are that the UM pays more attention to the order of the assessment space and gives the rational rank and credible identification principles. All of those are not possessed by fuzzy set assessment. Besides, the computation process of the UM is more simple.

Fig. 4  UM functions of evaluation indices:   (a) Uniaxial compressive strength; (b) Point-load strength index; (c) Uniaxial tensile strength;   (d) Elasticity of rock; (e) Mean in-situ block sizes; (f) P-wave velocity; (g) Schmidt hardness value; (h) Density; (i) Fractal dimension of in-situ rock mass

Table 2 Test results of rock specimens data of evaluation indices (Data from Refs. [3, 8])

Table 3 Distinction results of rock mass samples using UMM with entropy

Fig. 6 Results of UM model by CDI criteria in descending order

Table 4 Score results of rock mass samples

Table 5 Comparasion results of different methods

4 Conclusions

1) An UM model of blastability of rock mass predictions is established based on uncertain mathematics theory. In the UM model approach, information entropy is used to determine the indice weights of the factors, and the field data sets are used to investigate the feasibility in the blastability prediction for rock mass. And then, the evaluation results are obtained according to the CDI criteria. the UM model underestimates the blastability of the studied rock masses (G cutting site) in comparison with the conventional methods and FIS model. The experimental results show that the classi?cation accuracies of UM model are superior to those of FIS model.

2) The evaluated blastability results of samples S1-S6 are moderate, moderate, difficult, moderate, moderate, moderate, respectively. It is shown that the unascertained superiority degrees of samples S1-S6 are 3.428 5, 3.453 3, 4.058 7, 3.675 9, 3.516 7 and 3.289 7, respectively, in the order of S3>S4>S5>S2>S1>S6.

3) Combining the theory with the actual case will verify the feasibility and practicality of the model. Employing the information entropy and credible identification principal, the shortcoming of fuzzy comprehensive assessment is overcome and the result becomes more objective. It is easy to realize scientific and rational decision-making. The application results show that it is easy to realize the assessment without any assumption. The UM approach will be used for prediction of mining, rock or geotechnique engineering in future research.

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(Edited by YANG Bing)

Foundation item: Project(50934006) supported by the National Natural Science Foundation of China; Project(2010CB732004) supported by the National Basic Research Program of China; Project(2009ssxt230) supported by the Central South University Innovation Fund, China; Project(CX2011B119) supported by the Graduated Students’ Research and Innovation Fund of Hunan Province, China

Received date: 2011-05-12; Accepted date: 2011-08-01

Corresponding author: ZHOU Jian, PhD; Tel: -86-13723887261; E-mail:csujzhou@126.com