Research ArticleA Fuzzy Comprehensive Assessment Approach and Applicationof Rock Mass Cavability in Block Caving Mining

Rongxing He, Huan Liu , Fengyu Ren, Guanghui Li, and Jing Zhang

School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China

Correspondence should be addressed to Huan Liu; gyliuhuan@163.com

Received 28 April 2019; Revised 23 May 2019; Accepted 27 June 2019; Published 7 July 2019

Academic Editor: Roberto Fedele

Copyright © 2019 Rongxing He et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Cavability assessment is an important subject during the feasibility stages before determining whether to use block caving mining.This paper provides a fuzzy comprehensive assessment (FCA) approach based on the cavability assessment approaches and itsinfluencing factors, which are all fuzzy. This approach combines the cavability influencing factors with engineering empiricalapproaches by fuzzy mathematics, which improves the applicability of the cavability assessment results. This approach is appliedto assess the cavability via cores in the Luoboling copper molybdenum mine. The spatial distribution of the rock mass cavabilityat different depths of the borehole is obtained. The cavability ranks of various rocks are determined in different locations. Theseassessment results can provide a basis for demonstrating the feasibility of block cavingmining in the Luoboling coppermolybdenummine. The study can also provide a basis for the design of mining engineering.

1. Introduction

Block caving mining refers to all mining operations inwhich the ore body caves naturally after undercutting andthe caved material is recovered through drawpoints [1].As a special mechanism in mechanics and technology, animportant subject is to assess the rock mass cavability duringthe feasibility stages before determining whether to use thismining method [2]. The rock mass cavability has a majorinfluence on the mining block heights, production rates,undercutting orientations, undercutting areas, draw controls,mining rates, preconditioning engineering, and so on and isan important guarantee for a mine to achieve the expectedeconomic benefits. Cavability assessment is conducted byclassifying the rock mass cavability and determining theranks according to a given geological environment. And themine determines whether to use block caving mining at thecurrent industrial level. Cavability assessment is also a multi-index and nonlinear complex system engineering of the rockmass.

In the rock mass cavability field, most approaches ofcavability assessment are based on numerical modelling [3,4], mathematical models [2, 5–7], and geomechanical classi-fications [1, 8–16]. In addition, geomechanical classifications

have been widely used in cavability assessment [13, 17].Geomechanical classifications seem to be in tune with cav-ability assessment in describing the same problems of arock mass [17]. Because geomechanical classifications useengineering empirical assessments of the rock mass strengthin relation to the existing stresses and measures of therock structure, the classifications are fuzzy, such as rockquality designation (RQD) [12], rock mass rating (RMR)[8, 9, 16], mining rock mass rating (MRMR) [1, 11], rockmass quality Q-classification (Q) [10, 14], and rock massbasic quality (BQ) [15]. The selection of the influencingfactors and the determination of the rock mass ratingsare both fuzzy because different approaches use differentinfluencing factors as indicators. In addition, the influencingfactors of cavability are interrelated with each other andpresent great complexities, which lead to different cavabilityassessment results. Therefore, the cavability assessment andits influencing factors are both fuzzy, and fuzzy mathematicscan accurately describe and address the fuzzy phenomena.Therefore, we will adopt fuzzy mathematics to assess thecavability of the rock mass.

Fuzzy mathematics has been applied to predict petro-physical rock parameters [18] and mechanical rock param-eters [19–22] and to analyse various properties [2, 23–27]

HindawiMathematical Problems in EngineeringVolume 2019, Article ID 2063640, 13 pageshttps://doi.org/10.1155/2019/2063640

2 Mathematical Problems in Engineering

Table 1: The rock cavability classification based on the influencing factors.

Influencing factors The ranks of cavabilityI II III IV V

𝐼𝑠(50) >10 MPa 4∼10 MPa 2∼4 MPa 1∼2 MPa 0∼1 MPaRQD 90∼100% 75∼90% 50∼75% 25∼50% 0∼25%𝐽r Very rough Rough Slightly rough Smooth Slickenside𝐽a 0 <0.1 mm 0.1∼1 mm 1∼5 mm >5 mm

𝐽f None Hard filling Hard filling Soft filling Soft filling< 5 mm > 5 mm < 5 mm >5 mm

𝑊c Dry Damp Wet Dripping FlowingIss >0.40 0.31∼0.40 0.22∼0.31 0.13∼0.22 0.00∼0.13𝑄v 0.9 0.7 0.5 0.3 0.1𝑄r 0.8∼1.0 0.6∼0.8 0.4∼0.6 0.2∼0.4 0∼0.2

and phenomena [28, 29] of the rock mass. In a rock masscavability study, Rafiee et al. [2] designed a fuzzy expertsemiquantitative coding methodology to assess the cavabilityof the rock mass, and Rafiee et al. [7] applied the fuzzyrock engineering systems method to account for the intricateinteractions that exist among parameters in real projects.Shaoyong et al. [6] combined fuzzy mathematics and thematter element analysis method and established a modelof cavability of the rock mass in terms of complex fuzzymatter element analysis. Although many researchers haveapplied fuzzy mathematics to assess the cavability of a rockmass, the researchers conducted cavability modelling basedon influencing factors. The influencing factors of cavabilityare not very clear in the current understanding. Under thesecircumstances, engineering empirical approaches are stillimportant.

In this paper, we combine influencing factors with engi-neering empirical approaches by fuzzy mathematics andcarry out a fuzzy comprehensive assessment (FCA). Thisapproach improves the applicability of the assessment resultsof cavability. The approach is applied to assess the cavabilityof cores in a mine, and we obtain the spatial distribution ofthe rock mass cavability at different depths of the borehole.The cavability ranks of the various rocks in the hangingwall, ore body, and rocks in the ore body and footwall aredetermined. The assessment results provide a reference andbasis to decide whether to adopt the block caving miningmethod and determine the mining engineering design.

2. Fuzzy Assessment (FA) of Rock MassCavability Based on Influencing Factors

In the rock mass cavability field, the approaches of cavabilityassessmentwere dependent on influencing factors. It was nec-essary to take certain influencing factors into consideration inthe FA of rock mass cavability.

2.1. Determination of the Influencing Factors and AssessmentRanks. To date, studies on the influencing factors of cav-ability have been presented in the literature [1–17]. Theseinfluencing factors can be summarized as rock strength,discontinuity properties, water, and in situ stress. In the

assessment of rock mass cavability, the appropriate selectionof influencing factors was critical to the reliability of theassessment results. When relatively few influencing factorswere selected, these factors could not fully reflect the rockmass cavability and even lead to incorrect results of thecavability assessment. When too many influencing factorswere selected and these factors connected to each other, thissituation might exaggerate the influence of a certain factor onthe rock mass cavability and lead to incorrect results. Theseincorrect results were due to the influencing factors beinginterrelated with each other and subjectivity in determiningthe factors. Therefore, we analysed the relationship anddifference among the influencing factors of cavability anddetermined the influencing factors based on the presentstudies.

In the approaches of cavability assessment, the indicesof the rock strength were the uniaxial compressive strength(UCS) or point load strength index (Is(50)). Determiningthe Is(50) was a more practical, time-saving, and economicalmethod compared to determining the UCS [30]. A largenumber of studies [31–34] have shown that the Is(50) hasa good correlation with the UCS. Therefore, the Is(50) waschosen to represent the rock strength in this paper. It wasclear that the cavability of a rock mass decreases whenthe rock strength increases [7]. To quantify the Is(50), theIs(50) of intact rock, based on the RMR classification, wassubdivided into five ranks (the results of the ranks are listedin Table 1). The discontinuity properties were some of themost important influencing factors on the cavability of therock mass. The most important of these properties used todescribe the discontinuities were the RQD, joint spacing,intactness index of the rock mass, volumetric joint countof the rock mass, joint roughness (𝐽r), joint aperture (𝐽a),and joint filling (𝐽f). It was clear that the RQD had a goodcorrelationwith the joint spacing, intactness index of the rockmass, and volumetric joint count of the rock mass. At thesame time, the RQD was most commonly used in cavabilityassessments.Therefore, these factorswere chosen to representthe discontinuity properties, including the RQD, 𝐽r, 𝐽a, and 𝐽f .These factors were also subdivided into five ranks based onthe RMR classification (the results of the ranks are listed inTable 1). Water was usually described qualitatively. The water

Mathematical Problems in Engineering 3

condition (𝑊c) in this study, as in the RMR classification, wasdivided into five ranks (the results of the ranks are listed inTable 1). Lastly, the ratio of the UCS value to the in situ stressvalue was chosen to represent the in situ stress [7]. This ratiocould be translated into the ratio of the Is(50) value to the insitu stress value according to the relationship between theIs(50) and the UCS (UCS=22.8 Is(50) [15]). The ratio of Is(50)value to in situ stress value is abbreviated to Iss. The resultsof the Iss ranks are listed in Table 1. The influencing factorset of cavability was determined as U={𝑢1, u2, u3, u4, u5, u6,𝑢7}={𝐼𝑠(50), RQD, 𝐽r, 𝐽a, 𝐽f , 𝑊c, 𝐼𝑠𝑠}.

The cavability assessment ranks were generally dividedinto five ranks, including extremely difficult caving I, difficultcaving II, fair caving III, easy caving IV, and extremely easycavingV. For the convenience of establishing themembershipfunction during a follow-up operation, the ranks of cavabilityassessment were quantified (including quantitative value𝑄v and quantitative range 𝑄r). The quantitative results ofthe ranks are listed in Table 1. The rank set of cavabilityassessment was determined as V= {V1, v2, v3, v4, V5}={I, II,III, IV, V}.2.2. FuzzyAssessment Matrix and ItsMembership Function. Itwas important for the fuzzy assessment matrix to determinethe membership of each influencing factor.These factors canbe divided into qualitative indices and quantitative indicesaccording to Table 1. The memberships of the qualitativeindices could be determined by counting the assessmentfrequencies from several surveyors. The memberships of thequantitative indices could be determined by membershipfunctions.

Before establishing the membership functions of thequantitative indices, it was necessary to convert the measuredvalue into a value within the quantitative range 𝑄r for eachquantitative index. This method was convenient for estab-lishing the membership function and assessing the rockmasscavability. The converted map function 𝑓(𝑢𝑖) is established asfollows:

𝑓 (𝑢𝑖) = 𝑞𝑖min + 𝑞𝑖max − 𝑞𝑖min𝑝𝑖max − 𝑝𝑖min

(𝑢𝑖 − 𝑝𝑖min) (1)

𝑓 (𝑢𝑖) = 𝑞𝑖max − 𝑞𝑖max − 𝑞𝑖min𝑝𝑖max − 𝑝𝑖min

(𝑢𝑖 − 𝑝𝑖min) (2)

where 𝑢𝑖 is the measured value of the quantitative index. In(1), the cavability of the rock mass decreases with increasingmeasured value, and i is 1, 2, or 7. In (2), the cavabilityof the rock mass increases with increasing measured value,and i is 4. 𝑝𝑖max and 𝑝𝑖min are the upper range value andthe lower range value of the classification range based onmeasured value 𝑢𝑖, respectively. When 𝑝𝑖max tends to infinityat the boundary, the value should be limited according tothe measured value and empirical value. 𝑞𝑖max and 𝑞𝑖min arethe upper range value and the lower range value of thequantitative range𝑄r, respectively, and the values correspondto 𝑢𝑖. The converted map function uses linear transformationand does not change the assessment results.

After establishing the converted map function, it wasnecessary to establish the membership functions of the

quantitative indices. In fuzzy set theory, the membershipfunction of an index might contain some uncertainty, so themembership is expressed as a degree of belonging to a set[29]. Different people might establish different membershipfunctions for the same fuzzy set because of the limitationsof human understanding. However, Yonghua et al. [35]proved that different membership functions had equiva-lent characteristics in rock mass engineering. Triangularand trapezoidal shapes were the most common types ofmembership functions in rock or rock mass engineering[18, 22, 23, 26, 28, 29].

Therefore, we adopted the inference method of fuzzyset characteristics to establish the membership function andcombined the triangular and trapezoidal shapes of the mem-bership function. The inference method entailed looking forthe special elements in the quantitative range, such as thevalue of membership being equal to 0, 0.5, or 1. The totalvalue of membership was 1 for each influencing factor inthe five ranks. In rock or rock mass engineering [6, 18, 22,23, 26, 28, 29], the membership function usually adoptedan intermediate type for each quantitative index. That is,the value of membership was 0.5 at the endpoint for eachquantitative range. The value of membership was 1 in themiddle range for each quantitative range. In the middle rangefor each quantitative range, the value of membership was 0for the neighbourhood range. Eventually, the membershipfunction Aji=Aj(f (ui)) is established based on the aboveprinciples as follows:

𝐴5 (𝑓 (𝑢𝑖))

={{{{{{{{{{{

1, 𝑓 (𝑢𝑖) ≤ 0.1 + 𝛿𝑓 (𝑢𝑖)

2𝛿 − 0.2 + 𝛿 − 0.32𝛿 − 0.2 , 0.1 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.3 − 𝛿

0, 𝑓 (𝑢𝑖) > 0.3 − 𝛿

(3)

𝐴4 (𝑓 (𝑢𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑢𝑖) ≤ 0.1 + 𝛿𝑓 (𝑢𝑖)

0.2 − 2𝛿 − 𝛿 + 0.10.2 − 2𝛿, 0.1 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.3 − 𝛿

1, 0.3 − 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.3 + 𝛿𝑓 (𝑢𝑖)

2𝛿 − 0.2 + 𝛿 − 0.52𝛿 − 0.2 , 0.3 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.5 − 𝛿

0, 𝑓 (𝑢𝑖) > 0.5 − 𝛿

(4)

𝐴3 (𝑓 (𝑢𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑢𝑖) ≤ 0.3 + 𝛿𝑓 (𝑢𝑖)

0.2 − 2𝛿 − 𝛿 + 0.30.2 − 2𝛿, 0.3 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.5 − 𝛿

1, 0.5 − 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.5 + 𝛿𝑓 (𝑢𝑖)

2𝛿 − 0.2 + 𝛿 − 0.72𝛿 − 0.2 , 0.5 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.7 − 𝛿

0, 𝑓 (𝑢𝑖) > 0.7 − 𝛿

(5)

4 Mathematical Problems in Engineering

𝐴2 (𝑓 (𝑢𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑢𝑖) ≤ 0.5 + 𝛿𝑓 (𝑢𝑖)

0.2 − 2𝛿 − 𝛿 + 0.50.2 − 2𝛿 , 0.5 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.7 − 𝛿

1, 0.7 − 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.7 + 𝛿𝑓 (𝑢𝑖)

2𝛿 − 0.2 + 𝛿 − 0.92𝛿 − 0.2 , 0.7 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.9 − 𝛿

0, 𝑓 (𝑢𝑖) > 0.9 − 𝛿

(6)

𝐴1 (𝑓 (𝑢𝑖))

={{{{{{{{{{{

0, 𝑓 (𝑢𝑖) ≤ 0.7 + 𝛿𝑓 (𝑢𝑖)

0.2 − 2𝛿 − 𝛿 + 0.70.2 − 2𝛿 , 0.7 + 𝛿 < 𝑓 (𝑢𝑖) ≤ 0.9 − 𝛿

1, 0.9 − 𝛿 < 𝑓 (𝑢𝑖)

(7)

where 𝛿 is the neighbourhood value centred on the midpointof each quantitative range.The default value of 𝛿 is 0.05 in thispaper.

According to (1) to (7), the memberships of the quantita-tive indices can be calculated. Combined with the member-ships of the qualitative indices, the fuzzy assessment matrix Rcan be established as follows:

𝑅

=

I II III IV V

[[[[[[[[[[[[[[[

𝐴11𝐴12𝐴13𝐴14𝐴15𝐴16𝐴17

𝐴21𝐴22𝐴23𝐴24𝐴25𝐴26𝐴27

𝐴31𝐴32𝐴33𝐴34𝐴35𝐴36𝐴37

𝐴41𝐴42𝐴43𝐴44𝐴45𝐴46𝐴47

𝐴51𝐴52𝐴53𝐴54𝐴55𝐴56𝐴57

]]]]]]]]]]]]]]]

𝐼𝑠(50)𝑅𝑄𝐷𝐽r𝐽a𝐽f𝑊u

𝐼𝑠𝑠

(8)

where Aji is the value of membership and Aji means thatinfluencing factor ui has a membership of rank vj in thecavability assessment.

2.3. Determination of the Fuzzy Relative Weight Based on theAnalytic Hierarchy Process. Because the extent of influencewas different for each influencing factor in the cavabilityassessment, it was necessary to determine the fuzzy relativeweight of the factor. Among the approaches for determiningthe weight, the analytic hierarchy process has been widelyapplied due to its simplicity, scalability, and pairwise andeasy comparison of variables by assigning weights [36].The analytic hierarchy process has been extensively usedin complex decision making with fuzzy mathematics [36–39]. The analytic hierarchy process refers to a multicriteriadecision-making approach in which factors are arranged in ahierarchic structure [40]. The analytic hierarchy process cancombine qualitative analysis with quantitative analysis in the

process of cavability assessment. The determination processis as follows.

The first step was the structuring of the rock mass cav-ability as a hierarchy. The judgement factors that contributeto the cavability were determined. That is, the judgementfactors were the influencing factor set U={𝑢1, u2, u3, u4, u5,u6, 𝑢7}={𝐼𝑠(50), RQD, 𝐽r, 𝐽a, 𝐽f , 𝑊c, 𝐼𝑠𝑠}.

The second step was the elicitation of the pairwisecomparison judgements and establishing a judgementmatrix.The elements were arranged into a matrix and judgementswere elicited from the people who had difficulties about therelative importance of the elements with respect to the rockmass cavability. The scale to use in making the judgementswas 1∼9 and the reciprocal [40]. In evaluating the judgementfactors relative to the rock mass cavability, the evaluation wasconducted according to the present research findings on therelative importance of the judgement factors with respect tothe rock mass cavability. This paper mainly combines theRMR with the Q-classification method, and the judgementmatrix P can be established as follows:

𝑃

=

𝐼𝑠(50) 𝑅𝑄𝐷 𝐽r 𝐽a 𝐽f 𝑊u 𝐼𝑠𝑠

[[[[[[[[[[[[[[[

14222

0.330.2

0.251

0.330.330.50.20.2

0.531222

0.33

0.530.5110.50.33

0.520.5110.50.2

350.5221

0.33

5533531

]]]]]]]]]]]]]]]

𝐼𝑠(50)𝑅𝑄𝐷𝐽r𝐽a𝐽f𝑊u

𝐼𝑠𝑠

(9)

The third step was to calculate the order of the relativeimportance. When calculating the maximum eigenvalue of Pwith 𝜆pmax=7.51, the eigenvector 𝑋p is as follows:

𝑋p

= 𝐼𝑠(50) 𝑅𝑄𝐷 𝐽r 𝐽a 𝐽f 𝑊u 𝐼𝑠𝑠[ 0.26 0.76 0.23 0.35 0.39 0.20 0.08]

(10)

The eigenvector𝑋p is the order of the relative importance.The fuzzy relative weight coefficient of each index wasobtained by normalizing the eigenvector 𝑋p. The weightvectors are 𝑐p=[𝑐p1, 𝑐p2, 𝑐p3, 𝑐p4, 𝑐p5, 𝑐p6, 𝑐p7]=[0.11, 0.34, 0.10,0.15, 0.17, 0.09, 0.04].

The fourth step was the consistency check. The weightcoefficient 𝑐p of each factor was obtained. It was necessary tocheck whether the distribution of the weight coefficients wasreasonable. The formula of the consistency index (CI) is asfollows:

𝐶𝐼p = 𝜆pmax − 𝑛p𝑛p − 1 = 7.51 − 7

7 − 1 = 0.085 (11)

where 𝑛p is the number of judgement factors and 𝑛p=7.

Mathematical Problems in Engineering 5

The CI is compared with the average random consistencyindex RI (𝑛p=7, RI7=1.35). The consistency ratio, CR, can beobtained as follows:

𝐶𝑅𝑝 =𝐶𝐼p𝑅𝐼7 = 0.085

1.35 = 0.063 < 0.10 (12)

According to (12), the judgement matrix meets the con-sistency check. That is, the distribution of the fuzzy relativeweight coefficients is reasonable.

2.4. Fuzzy Mapping and Fuzzy Assessment Based on theInfluencing Factors. After determining the weight vector 𝑐pand the fuzzy assessment matrix R, fuzzy subset B can beobtained by fuzzy linear variation. The fuzzy subset B is asfollows:

𝐵 = 𝑐p ∘ 𝑅 = I II III IV V[𝑏1 𝑏2 𝑏3 𝑏4 𝑏5]

(13)

where “∘” is a synthetic operator.Theweighted average modelis adopted because all kinds of influencing factors affect therock mass cavability.

The FA method was based on the influencing factorsinvolved in calculating the quantitative FA value. The calcu-lation method considered that membership 𝑏i of quantitativevalue 𝑄vi was a weight coefficient, and the weighted averagevalue of each 𝑄vi was taken as a quantitative value of the FA.Thequantitative FA value was compared with the quantitativerange 𝑄r, and the rank of cavability was obtained. The FAvalue is calculated as follows:

𝐹𝐴 = ∑5𝑖=1 𝑏𝑖𝑄v𝑖

∑5𝑖=1 𝑏𝑖(14)

3. Fuzzy Comprehensive Assessment (FCA) ofthe Rock Mass Cavability

Atpresent, there aremany assessment approaches for the rockmass cavability. However, these approaches were proposedbased on certain geological conditions or on a given engineer-ing background. For example, RMR was based on experiencegained in numerous visits to construction sites abroad andin South Africa [8, 9, 16], MRMR was combined RMR withmining engineering [1, 11], and Q was originally developed toassist in the empirical design of tunnel and cavern reinforce-ment and support [10, 14].There might be certain limitationsor inadaptability in applying these approaches directly. If twoormore approaches were applied at the same time, the resultswere often different. However, these approaches were basedon a large number of engineering practices and engineeringexperiences. The approaches had a strong reference valuein specific practical projects. Therefore, it was necessaryto synthesize these assessment approaches according to thespecific mining geology. Furthermore, more objective andreasonable results of the cavability were obtained. The resultsprovided a strong reference and basis to decide whether toadopt the block caving mining and determine the miningengineering design. Fuzzymathematics provided themethod

for synthesizing these different assessment approaches. Themethod was fuzzy mapping and fuzzy comprehensive assess-ment, which was based on fuzzy comprehensive assessmentmatrix and fuzzy relative weight.

3.1. Selection of the Assessment Approaches and AssessmentRanks. Among the assessment approaches, the most widelyused approaches include rock quality designation RQD, rockmass rating RMR, mining rock mass ratingMRMR, and rockmass quality Q-classification Q. The MRMR was proposedfor mining but was most affected by engineering experiencein the assessment process. Because the mine was in thefeasibility stage, it was not put into production and no rockmass excavation engineering occurred. There was hardly anyengineering experience that could be referenced, and it wasimpossible to revise the parameters in theMRMR.Therefore,theMRMRwasnot selected as an index for theFCA approach.At the same time, the FA of the rock mass cavability basedon the influencing factors was introduced. The final selectionof the assessment approach set was determined as Z={𝑧1,z2, z3, z4, 𝑧5}={𝑅𝑄𝐷, RMR, Q, BQ, 𝐹𝐴}. The rock cavabilityclassifications based on the assessment approaches are listedin Table 2.

The ranks of cavability assessment were also divided intofive ranks, including extremely difficult caving I, difficultcaving II, fair caving III, easy caving IV, and extremely easycaving V. The rank set of the cavability assessment wasdetermined as V= {V1, v2, v3, v4, V5}={I, II, III, IV, V}. Theranks of cavability are listed in Table 2.

3.2. Fuzzy Comprehensive Assessment Matrix and Its Member-ship Function. As shown in Table 2, the indices (assessmentapproaches) are quantitative indices in the FCA. The mem-berships of the quantitative indices could be determined bymembership functions. Before establishing the membershipfunction, it was necessary to convert the measured value intoa value within the quantitative range 𝑄r for each quantitativeindex. This method was also convenient for establishing themembership function and assessing the rock mass cavability.The converted map function f (zi) is established as follows:

𝑓 (𝑧𝑖) = 𝑞𝑖min + 𝑞𝑖max − 𝑞𝑖min𝑝𝑖max − 𝑝𝑖min

(z𝑖 − 𝑝𝑖min) (15)

where zi is the calculated value of the quantitative indexand i is 1, 2, 3, 4, 5. Here, 𝑝𝑖max and 𝑝𝑖min are the upperrange value and the lower range value of the classificationrange based on calculated value zi, respectively. When 𝑝𝑖maxtends to infinity at the boundary, the value should be limitedaccording to the calculated value and empirical value. 𝑞𝑖maxand 𝑞𝑖min are the upper range value and the lower range valueof the quantitative range 𝑄r, respectively, and correspond tozi.

After establishing the converted map function, it wasnecessary to establish the membership functions of the quan-titative indices. The inference method was also adopted toestablish the membership function. The established methodwas similar to the FA method. The membership functionZji=Zj(f (zi)) is established as follows:

6 Mathematical Problems in Engineering

Table 2: The rock cavability classification based on the assessment approaches.

Assessment approaches The ranks of cavabilityI II III IV V

RQD 90∼100% 75∼90% 50∼75% 25∼50% 0∼25%RMR 81∼100 61∼80 41∼60 21∼40 0∼20Q >40 10∼40 4∼10 1∼4 0∼1BQ >550 451∼550 351∼450 251∼350 0∼250FA 0.8∼1.0 0.6∼0.8 0.4∼0.6 0.2∼0.4 0∼0.2𝑄v 0.9 0.7 0.5 0.3 0.1𝑄r 0.8∼1.0 0.6∼0.8 0.4∼0.6 0.2∼0.4 0∼0.2

𝑍5 (𝑓 (𝑧𝑖))

={{{{{{{{{{{

1, 𝑓 (𝑧𝑖) ≤ 0.1 + 𝛿𝑓 (𝑧𝑖)

2𝛿 − 0.2 + 𝛿 − 0.32𝛿 − 0.2 , 0.1 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.3 − 𝛿

0, 𝑓 (𝑧𝑖) > 0.3 − 𝛿

(16)

𝑍4 (𝑓 (𝑧𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑧𝑖) ≤ 0.1 + 𝛿𝑓 (𝑧𝑖)

0.2 − 2𝛿 − 𝛿 + 0.10.2 − 2𝛿, 0.1 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.3 − 𝛿

1, 0.3 − 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.3 + 𝛿𝑓 (𝑧𝑖)

2𝛿 − 0.2 + 𝛿 − 0.52𝛿 − 0.2 , 0.3 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.5 − 𝛿

0, 𝑓 (𝑧𝑖) > 0.5 − 𝛿

(17)

𝑍3 (𝑓 (𝑧𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑧𝑖) ≤ 0.3 + 𝛿𝑓 (𝑧𝑖)

0.2 − 2𝛿 − 𝛿 + 0.30.2 − 2𝛿, 0.3 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.5 − 𝛿

1, 0.5 − 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.5 + 𝛿𝑓 (𝑧𝑖)

2𝛿 − 0.2 + 𝛿 − 0.72𝛿 − 0.2 , 0.5 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.7 − 𝛿

0, 𝑓 (𝑧𝑖) > 0.7 − 𝛿

(18)

𝑍2 (𝑓 (𝑧𝑖))

=

{{{{{{{{{{{{{{{{{{{{{{{{{

0, 𝑓 (𝑧𝑖) ≤ 0.5 + 𝛿𝑓 (𝑧𝑖)

0.2 − 2𝛿 − 𝛿 + 0.50.2 − 2𝛿, 0.5 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.7 − 𝛿

1, 0.7 − 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.7 + 𝛿𝑓 (𝑧𝑖)

2𝛿 − 0.2 + 𝛿 − 0.92𝛿 − 0.2 , 0.7 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.9 − 𝛿

0, 𝑓 (𝑧𝑖) > 0.9 − 𝛿

(19)

𝑍1 (𝑓 (𝑧𝑖))

={{{{{{{{{{{

0, 𝑓 (𝑧𝑖) ≤ 0.7 + 𝛿𝑓 (𝑧𝑖)

0.2 − 2𝛿 − 𝛿 + 0.70.2 − 2𝛿, 0.7 + 𝛿 < 𝑓 (𝑧𝑖) ≤ 0.9 − 𝛿

1, 0.9 − 𝛿 < 𝑓 (𝑧𝑖)

(20)

where 𝛿 is the neighbourhood value centred on the midpointof each quantitative range.The default value of 𝛿 is 0.05 in thispaper.

According to (15) to (20), the memberships of the quan-titative indices can be calculated. The fuzzy comprehensiveassessment matrix 𝑅z can be established:

𝑅z

=

I II III IV V

[[[[[[[[[

𝑍11𝑍12𝑍13𝑍14𝑍15

𝑍21𝑍22𝑍23𝑍24𝑍25

𝑍31𝑍32𝑍33𝑍34𝑍35

𝑍41𝑍42𝑍43𝑍44𝑍45

𝑍51𝑍52𝑍53𝑍54𝑍55

]]]]]]]]]

𝑅𝑄𝐷𝑅𝑀𝑅𝑄𝐵𝑄𝐹

(21)

where Zji is the value of membership and Zji means thatassessment approach zi has a membership of rank vj of thecavability assessment.

3.3. Determination of the Fuzzy Relative Weight Basedon the Analytic Hierarchy Process. Because the extent ofthe influence was different for each assessment approachin the cavability assessment, it was necessary to deter-mine the fuzzy relative weight. The analytic hierarchy pro-cess was also adopted. The determination process was asfollows.

The first step was the structuring of the rock mass cav-ability as a hierarchy. The judgement factors that contributeto the cavability were determined. That is, the assessmentapproach set constituted the judgement factors, Z={𝑧1, z2, z3,z4, 𝑧5}={𝑅𝑄𝐷, RMR, Q, BQ, 𝐹𝐴}.

The second step was the elicitation of pairwise com-parison judgements and establishing a judgement matrix.The established method was similar to the FA method. Inmaking the judgements of the assessment approaches relativeto the rock mass cavability, the process was done accordingto the applicable conditions, engineering backgrounds, andapplication statuses of these assessment approaches. Thispaper mainly relies on the specific mining geology of theLuoboling copper-molybdenummine.The judgementmatrix𝑃z can be established as follows:

Mathematical Problems in Engineering 7

𝑃z

=

𝑅𝑄𝐷 𝑅𝑀𝑅 𝑄 𝐵𝑄 𝐹𝐴

[[[[[[[[[

15346

0.210.50.52

0.33210.53

0.252214

0.170.50.330.251

]]]]]]]]]

𝑅𝑄𝐷𝑅𝑀𝑅𝑄𝐵𝑄𝐹𝐴

(22)

The third step was to calculate the order of the relativeimportance.When calculating themaximum eigenvalue of𝑃zwith 𝜆zmax=5.003, the eigenvector 𝑋z is as follows:

𝑋z = 𝑅𝑄𝐷 𝑅𝑀𝑅 𝑄 𝐵𝑄 𝐹𝐴[0.10 0.51 0.33 0.25 0.75] (23)

The eigenvector 𝑋z was the order of the relative impor-tance.The fuzzy relative weight coefficient of each assessmentapproach was obtained by normalizing the eigenvector 𝑋z.The weight vector is 𝑐z = [𝑐z1, 𝑐z2, 𝑐z3, 𝑐z4, 𝑐z5]=[0.05, 0.26, 0.17,0.13, 0.39].

The fourth step was the consistency check. The weightcoefficient 𝑐z of each factor was obtained. It was necessary tocheck whether the distribution of the weight coefficients wasreasonable. The formula of the CI is as follows:

𝐶𝐼z = 𝜆zmax − 𝑛z𝑛z − 1 = 5.003 − 5

5 − 1 = 0.00075 (24)

where 𝑛z is the number of judgement factors and 𝑛z=5.Compare the CI with the average RI (𝑛z=5, RI5=1.12).The

consistency ratio, 𝐶𝑅z, can be obtained as follows:

𝐶𝑅z = 𝐶𝐼z𝑅𝐼5 = 0.00075

1.12 = 0.0007 < 0.10 (25)

According to (25), the judgement matrix meets the con-sistency check. That is, the distribution of the fuzzy relativeweight coefficients is reasonable.

3.4. Fuzzy Mapping and Fuzzy Comprehensive Assessment.After determining the weight vector 𝑐z and fuzzy assessmentmatrix 𝑅z, the fuzzy subset 𝐵z can be obtained by fuzzy linearvariation. The fuzzy subset 𝐵z is as follows:

𝐵𝑧 = 𝑐𝑧 ∘ 𝑅𝑧 =I II III IV V

[𝑏𝑧1 𝑏𝑧2 𝑏𝑧3 𝑏𝑧4 𝑏𝑧5](26)

where “∘” is a synthetic operator.Theweighted average modelis adopted because all kinds of assessment approaches affectthe rock mass cavability.

The FCA method was used to calculate the quantitativeFCA value. The calculation method considered that mem-bership 𝑏zi of quantitative value 𝑄vi was a weight coefficient,and the weighted average value of each 𝑄vi was taken as aquantitative value of the FCA. The quantitative value of theFCA was compared with the quantitative range 𝑄r, and therank of cavability was obtained. The FCA value is calculatedas follows:

𝐹𝐶𝐴 = ∑5𝑖=1 𝑏z𝑖𝑄v𝑖

∑5𝑖=1 𝑏z𝑖(27)

4. Practical Application in the LuobolingCopper-Molybdenum Mine

The Luoboling copper-molybdenum mine belongs to a por-phyry deposit. The characteristics of the ore body includedeep burial, large distribution area, large thickness, largedip change, low grade, large reserves, and complex shape.According to the characteristics of the ore body, blockcaving mining was determined during the feasibility stages.Therefore, it was a crucial step to assess the rock masscavability and obtain the spatial distribution maps of thecavability. Cavability assessment was beneficial for the engi-neering layout, the stope structure parameter selection, anddetermining whether to use block caving mining.

However, there was no mining excavation engineer-ing during the feasibility stages for the Luoboling copper-molybdenum mine, and the characteristic parameters of therock mass could not be obtained. This situation was also thesame for other mines during the feasibility stages. However,a total of 176 boreholes were completed in the explorationstage of the Luoboling copper-molybdenum mine. The totalfootage was 130761.23 m, and the controlled area was 6.77km2. A large number of cores were retained. Therefore,this paper determined the rock mass cavability throughcores.

First, 25 boreholes were determined from the 176 bore-holes according to the spatial position relationship betweenthe borehole and the ore body (as shown in Figure 1). Otherconsiderations included the spacing of borehole, the volumeof work, and the shape of ore body. After that, each boreholecore was divided into several groups in the vertical directionaccording to lithology and RQD value. If the lithology wasconsistent and the RQD value was close within a verticaldistance, the cores were divided into a group. The rock masscavability was assessed according to the groups.

4.1. Fuzzy Assessment (FA) of the Rock Mass Cavability.According to the groups, the measured values of the seveninfluencing factors were determined. The methods of mea-surement included mechanics experiments, field surveys,and measurements, which referred to the hydrogeology andengineering geology of the boreholes. The FA approach wasadopted to assess the rock mass cavability. The result of theassessment is shown in Figure 2. As can be seen fromFigure 2,the general trend of cavability is from easy caving IV todifficult caving II in the vertical direction of the boreholes.

4.2. Fuzzy Comprehensive Assessment (FCA) of the Rock MassCavability. The rock mass cavability was assessed by theapproaches of RQD, RMR, Q, and BQ. The results of theassessment are shown in Figures 3–6.

It can be seen fromFigures 2–6 that the trend of cavabilityranged from extremely easy caving V to extremely difficultcaving I in the vertical direction of the boreholes. However,the results of the assessment were quite different for the

8 Mathematical Problems in Engineering

xy

z

No. 1 ore body No. 2 ore body No. 3 ore body No. 4 ore body Boreholes

Figure 1: The spatial position relationship between the measured boreholes and the ore body.

x

0.8~1FA-value 0.6~0.8 0.4~0.6 0.2~0.4 0~0.2

y

z

Figure 2:The spatial distribution of the FA value in the vertical direction of the boreholes.

different approaches. Therefore, the FCA approach was alsonecessary for the cavability assessment.

The comprehensive approach of the FCA was adopted toassess the rock mass cavability. The result of the assessmentis shown in Figure 7. As is seen from Figure 7, the generaltrend of cavability is from easy caving IV to difficult cavingII in the vertical direction of the boreholes. In the boreholesas a whole, the upper part belongs to easy caving IV, and thelower part belongs to fair caving III and difficult caving II.

4.3. The Assessment Results of the Rock Mass Cavability. Weobtained the spatial distribution of the rock mass cavabilityat different depths of the borehole. The advantage of theassessment cavability for cores was that we combined thecavability with the geological information from the boreholes(such as the lithology and location). We could count thelengths of the cores that had the same ranks of cavability andsame locations. In this way, it was beneficial for analysing the

rock mass cavability in different locations. We obtained thecavability ranks of the various rocks in the hanging wall, orebody, and rocks in the ore body and footwall (as shown inFigure 8), which were based on the spatial distribution of theFCA value in the vertical direction of the boreholes (Figure 7).

It can be seen from (a) of Figure 8 that the rock masscavability of the rock in the hangingwall ismainly easy cavingIV, a small portion of the rock is fair caving III, and a verysmall amount of the rock is difficult caving II or extremelyeasy caving V. It can be seen from (b) of Figure 8 that the rockmass cavability of the ore in the ore body is difficult caving II,fair caving III, and easy caving IV, and a very small amountof the rock is extremely easy caving V; fair caving III and easycaving IV account for approximately 70%. It can be seen from(c) of Figure 8 that the rock mass cavability of the rock inthe ore body is difficult caving II, fair caving III, and easycaving IV; a very small amount of the rock is extremely easycaving V; and fair caving III and easy caving IV account for

Mathematical Problems in Engineering 9

x

90~100RQD-value (%) 75~90 50~75 25~50 0~25

y

z

Figure 3: The spatial distribution of the RQD value in the vertical direction of the boreholes.

x

81~100RMR-value 61~80 41~60 21~40 0~20

y

z

Figure 4: The spatial distribution of the RMR value in the vertical direction of the boreholes.

approximately 65%. It can be seen from (d) of Figure 8 that therock mass cavability of the rock in the footwall is mainly faircaving III, a small amount of the rock is difficult caving II, anda small amount of the rock is easy caving IV.On thewhole, themine has hardly any rock mass of extremely difficult caving Iand extremely easy caving V. And the rock mass cavability ismainly fair caving III and easy caving IV. The borehole corescan verify these conclusions during the feasibility stages. Asshown in Figure 9, some typical cores of mine are presented.It can be seen from (a) of Figure 9 that a part of the rock isfragmented at the top of the borehole. As the depth increasesthe cores are relatively intact (as shown in (b) and (c) ofFigure 9), but the cores have many discontinuities and helpthe rockmass naturally cave.Therefore, the assessment results

of mine are correct. These assessment results can provide abasis for demonstrating the feasibility of block caving miningin the Luoboling copper-molybdenum mine. The study canalso provide a basis for designing the mining engineeringnext.

5. Conclusions

In this study, a fuzzy comprehensive assessment (FCA)approachwas provided (as shown in Figure 10) that was basedon cavability assessment and its influencing factors, whichwere fuzzy. For this purpose, we determined the influencingfactors and assessment approaches of cavability, establishedthe converted map functions and membership functions,

10 Mathematical Problems in Engineering

x

40~100Q-value 10~40 4~10 1~10 0~10

y

z

Figure 5: The spatial distribution of the Q value in the vertical direction of the boreholes.

x

550~700BQ-value 450~550 350~450 250~350 0~250

y

z

Figure 6: The spatial distribution of the BQ value in the vertical direction of the boreholes.

and adopted the analytic hierarchy process to determine thefuzzy relative weights.This approach combined the cavabilityinfluencing factors with engineering empirical approaches byfuzzymathematics. Themethod improved the applicability ofthe assessment results of cavability. Finally, the FCA approachwas applied to assess the cavability of cores in the Luobolingcopper–molybdenum mine, which, according to the minestage, has no mining excavation engineering at present. Thespatial distribution of the rock mass cavability at differentdepths of the borehole was obtained. We combined cavabilitywith the geological information from the boreholes anddetermined the cavability ranks of the various rocks in thehanging wall, ore body, and rocks in the ore body andfootwall. The assessment results provided a reference andbasis to decide whether to adopt the block caving miningmethod and to design the mining engineering.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Rongxing He and Huan Liu contributed to the formulationof the overarching research goals and aims and conductedthe FCA; Rongxing He and Fengyu Ren determined theinfluencing factors and assessment approaches of cavability;Fengyu Ren determined the measured boreholes in the

Mathematical Problems in Engineering 11

x

0.8~1FCA-value 0.6~0.8 0.4~0.6 0.2~0.4 0~0.2

y

z

Figure 7: The spatial distribution of the FCA value in the vertical direction of the boreholes.

GranodioriteGranodiorite porphyry

Quartz orthophyre

0

1000

2000

3000

4000

5000

6000

Leng

th (m

)

I II VIII IVThe rank of cavability

(a) Rock in the hanging wall

Copper–molybdenum oreCopper ore

Molybdenum ore

0

1000

2000

3000

4000

5000Le

ngth

(m)

I II VIII IVThe rank of cavability

(b) Ore

GranodioriteGranodiorite porphyry

Quartz orthophyre

0

100

200

300

400

500

Leng

th (m

)

I II VIII IVThe rank of cavability

(c) Rock in the ore body

GranodioriteGranodiorite porphyry

0

100

200

300

400

500

600

700

800

Leng

th (m

)

I II VIII IVThe rank of cavability

(d) Rock in the footwall

Figure 8: The statistical results of the rock mass cavability in the different locations.

12 Mathematical Problems in Engineering

(a) The top of boreholecores

(b) The middle of boreholecores

(c) The bottom of boreholecores

Figure 9: The borehole cores at different positions.

FCA

RQD RMR Q BQ FA

Assessment approaches

Influencing factors

Is(50) RQD JＬ J； J＠ W＝ Iss

Figure 10: The fuzzy comprehensive assessment (FCA) approach.

Luoboling copper-molybdenum mine; Huan Liu carried outthe FCA and wrote the paper; and Huan Liu, Guanghui Li,and Jing Zhang completed the mechanics experiments, fieldsurveys, andmeasurements and referred to the hydrogeologyand engineering geology of the boreholes in the Luobolingcopper,molybdenum mine.

Acknowledgments

The study is jointly supported by grants from the NationalKey Research and Development Program of China (Grantno. 2016YFC0801604) and the Key Program of the NationalNatural Science Foundation of China (Grant no. 51534003).The authors are grateful for the support.

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