Research Article Rock Mass Blastability Classification ...

Research ArticleRock Mass Blastability Classification Using Fuzzy PatternRecognition and the Combination Weight Method

Shuangshuang Xiao Kemin Li Xiaohua Ding and Tong Liu

State Key Laboratory of Coal Resources and Safe Mining School of Mines China University of Mining and TechnologyXuzhou Jiangsu 221116 China

Correspondence should be addressed to Kemin Li likemin515hotmailcom

Received 27 January 2015 Accepted 27 May 2015

Academic Editor Rama S R Gorla

Copyright copy 2015 Shuangshuang Xiao et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Rock mass blastability classification provides a theoretical basis for rock mass blasting design which is used to select blastingexplosives to estimate the unit explosive consumption and to determine blasting design parameters The primary factors thataffect rock mass blastability were analyzed by selecting five indexes for rock mass blastability classification that is the rockProtodyakonov coefficient rock tensile strength rock density rock wave impedance and integrity coefficient of rock mass andby identifying standards for the rock mass blastability classification and a method for testing the blasting classification indexesThe index weights were calculated using the combination weight method which is based on game theory A model for rockmass blastability classification was developed in combination with a fuzzy pattern recognition method This classification methodwas applied to a Heidaigou open-pit coal mine where mudstone fine sandstone medium sandstone and coarse sandstone weredetermined to have a blastability degree of II which corresponds to a blastability characterization of ldquoeasyrdquo and the unit explosiveconsumption of mudstone fine sandstone medium sandstone and coarse sandstone was determined to be 044 042 040 and036 kgm3 respectively These results were used to develop a loose blasting design that was effective for loose blasting

1 Introduction

Rock mass blastability is a measure of the resistance of a rockmass to blasting and crushing The physical and mechanicalproperties and structural characteristics of rocks synchronizeto various extents and in different ways to impede blastingand crushing under blasting loading [1] Thus the rockmass blastability is also a comprehensive indicator of severalinherent properties of a rock mass under dynamic loadingThe rock mass blastability reflects the degree of difficulty inrock blasting [2] An understanding of rock mass blastabilityand systematic rock mass blastability classifications form thetheoretical basis of blasting optimization design [3]The rockmass blastability can be used to select suitable explosivesestimate the explosive unit consumption and determinereasonable blasting parameters which can reduce blastingcosts and improve labor productivity by ensuring predictableblasting characteristics

Foreign and Chinese scholars have conducted numer-ous research studies on rock mass blastability classification

using different methods from various perspectives and havedeveloped a variety of indexes and methods for rock massblastability classification [4]There are currently twomethodsfor rock mass blastability classification In the first methodthe analysis and calculation involve one or more parametersand a numerical value such as a blastability index or thecrushing energy is chosen as a measure of the blastability ofthe rock mass [5] In the second method various parametersare chosen to describe the rock mass and the rock massblastability classification is performed using statistical math-ematics fuzzy mathematics or other mathematical methodsThe characteristics and internal mechanisms that affect rockmass blastability can be identified more accurately usingseveral indexes to systematically evaluate the rock massblastability Therefore many classification schemes and eval-uation algorithms have been applied to rock mass blastabilityclassification including neural networks [6ndash8] projectionpursuit [9] genetic algorithms [10] fuzzy set theory [11ndash13] cluster analysis [14] and attribute recognition [15] Each

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 724619 11 pageshttpdxdoiorg1011552015724619

2 Mathematical Problems in Engineering

algorithm has its advantages and disadvantages For examplea neural network has considerable fault-tolerance abilityand a rapid evaluation speed but requires a representativelearning sample In addition the learning parameters andnumber of hidden layers are difficult to identify and thenumber of hidden layers affects the convergence rate theconvergence properties of the network and its applicabilityto nonlinear problems The index weights do not need tobe identified when the rock mass blastability is classifiedusing the projection pursuit algorithm thereby ensuring thatthe classification is objective However when optimizing theprojection direction this scheme can easily converge to alocal optimum which results in early maturing or earlyconvergence among other problems Genetic algorithms canbe used to accurately classify the respective categories but hasadditional parameter requirements such as gene variablesand genetic generations The challenge encountered in usingcluster analysis and attribute recognition is to determinereasonable index weights

There are three essential requirements for developing arock mass blastability classification model First the mostrepresentative characteristic must be chosen as the classifi-cation index and classification standards must be developedSecond each index should be assigned a reasonable weightFinally a suitable evaluating algorithm should be chosenTherock blasting mechanism and the factors affecting rock massblastability for the aforementioned research scenario wereused to identify the classification indexes and classificationstandards for rock mass blastability The blastability of arock mass was described using the following values ldquoeasyrdquoldquomoderaterdquo ldquodifficultrdquo and other fuzzy values depending onpractical production requirements The indices of two rocksamples typically have similar values but are characterized bydifferent rating categories by observation This result is notreasonableThus rock blastability can be characterized usingtransitional values that lie in between different levels that isthe values are fuzzy There is no distinct boundary betweendifferent levels The same rock mass could be assigned to dif-ferent classifications by different people or based on differentsituations Thus it is more suitable to use fuzzy mathematicsto classify rock mass blastability Rock mass classification canthen be based on this developed rockmass rating and the rockcharacteristics that is rock mass blastability classificationis a pattern recognition problem Therefore fuzzy patternrecognition was used to develop a rock mass blastabilityclassification model However the weights of the indexesare not considered in pattern recognition which preventsthe application of this method to cases with unequal indexweightsThe combinationweightmethodwas used to identifythe index weights to reduce the effects of subjective factorsand avoid irrelevant factors

2 Indexes and Standards for theClassification of Rock Mass Blastability

21 Selection of Classification Index Explosive blasting canfracture a rock mass in two ways First the cohesive forcebetween rock granules can be overcome thereby rupturingthe internal rock structure and producing a new fracture

surface Second primary and secondary fractures can beexacerbated via further expansion Therefore the primaryinfluential factors of rock mass blastability are the physicaland mechanical properties of the rock and the structuralcharacteristics of the rock mass [16] Typical indexes forclassifying rock mass blastability include the rock densityrock wave impedance rock tensile strength integrity coef-ficient of the rock mass and the mean crack interval of therock mass These indexes reflect different aspects of rockmass blastability However to simplify rock mass blastabilityclassification and enable its practical application all of theindexes are not used The characteristics of a rock massmust be considered when choosing indexes for rock massblastability classification Minor representative indexes cancomprehensively reflect different aspects of the rock massblastability There should be little or no correlation betweenthe indexes The chosen indexes should be easy to obtainusing various methods such as experiments and field mea-surements The aforementioned considerations were used toselect the following final indexes for when considering rockmass blastability

211 Protodyakonov Coefficient and Tensile Strength of RockThe shock wave and detonation gas produced by explosiveblasting can typically rupture a rock mass through pullingand pressing Therefore the Protodyakonov coefficient andtensile strength of the rock are important parameters inrock mass blastability During blasting the rock is subjectto temporary impact loading for which the rock dynamicloading strength is clearly higher than the rock static loadingstrength Therefore the rock mass blastability can be accu-rately measured by indexes for the dynamic loading strengththat are affected by the triaxial effect of the rock Howeverthe dynamic loading strength of rock is difficult to measureand exhibits a strong linear correlationwith the uniaxial staticloading compression strength and tensile strength [17] Thusthe static loading strength is chosen as one of the indexesfor rock mass blastability classification The Protodyakonovcoefficient of the rock which is determined from the uniaxialcompressive strength of the rock (1) is an objective measureof rock fastness that is widely applied in China Thereforethe Protodyakonov coefficient and compressive strength ofthe rock are chosen as indexes for rock mass blastabilityclassification

119891 =120590119901

119870 (1)

In the equation above 119891 is the Protodyakonov coefficientof the rock 120590

119901is the rockrsquos uniaxial compressive strength in

MPa and 119870 is a constant equal to 10MPa

212 Rock Density The energy produced from rock blastingis transferred into kinetic energy in the rock block whichcan result in the displacement or thrusting of the rock blockA higher rock density causes more of the energy producedin rock blasting to be consumed by the displacement andthrusting of the rock Therefore the amount of energyconsumed is indicative of the difficulty of the rock blast thatis the rock mass blastability decreases with increasing rock

Mathematical Problems in Engineering 3

Table 1 Classification standards of rock mass blastability

Blastability class 120590119905

(MPa) 119891119875

(tm3) 119870V119885

(106 kgm3 timesms) Characterization of blastability

I le15 le25 le20 le015 le3 Very easyII 15ndash3 25ndash6 20ndash24 015ndash035 3ndash6 EasyIII 3ndash6 6ndash10 24ndash275 035ndash055 6ndash9 ModerateIV 6ndash12 10ndash18 275ndash30 055ndash075 9ndash12 DifficultV ge12 ge18 ge30 ge075 ge12 Very difficult

density Therefore the rock density is generally used as anindex for rock mass blastability classification

213 Rock Wave Impedance The dynamic Poissonrsquos ratiodynamic elastic modulus bulk modulus and Lame param-eter for rock can be derived from the P- and S-wave velocitiesof the rock All of the physical property indexes of rocks suchas the mineral composition porosity water-bearing andweathering degree are captured in the P-wave velocity of therockThe P-wave velocity of the rock can be easily measuredThe rock wave impedance can be obtained by multiplyingthe P-wave velocity of the rock by the rock density (2)The impedance is a measure of the force of the disturbancerequired to produce a unit speed of a moving rock particleduring the transmission of a stress wave in the rock and is ameasure of the resistance of the rock to momentum transferTherefore the rock wave impedance is chosen as one of theindexes for the rock mass blastability classification

119911 = 120588Vpr (2)

In the equation above 119911denotes the rockwave impedance(106 kgm3 times ms) 120588 denotes the rock density (kgm3) andVpr denotes the P-wave velocity of the rock (ms)

214 Integrity Coefficient of Rock Mass The geological prop-erties of a rock mass such as the integrity fissure and degreeof development of a joint fissure are captured in the P-wave velocity of the rock mass A fast wave propagationvelocity in a rock mass typically corresponds to mild rockdensification hardness integrity andweathering In contrasta slow wave propagation velocity corresponds to severe rockporosity weakness fragmentation structural developmentand weathering The integrity coefficient of a rock mass isgiven by the square of the ratio of the P-wave velocity of arockmass to the P-wave velocity of the rock (3) which reflectsthe extent of fracturing for a geological discontinuity such asa joint fissure A rock mass with a small integrity coefficientis susceptible to a large amount of rock mass crushing andthe rock mass can be easily blasted Therefore the integritycoefficient of the rock mass is chosen as one of the indexesfor the rock mass blastability classification

119870119881= (

VpmVpr

)

2

(3)

In the equation above119870119881denotes the integrity coefficient

of the rock mass Vpm denotes the P-wave velocity of the rock

h

Explosive Rock mass

Drill hole

1 2

Vibrationpick-up 1 pick-up 2

Vibration

s1

t1 t2

s2

Figure 1 Schematic for testing the P-wave velocity of the rockmass

mass (ms) and Vpr denotes the P-wave velocity of the rock(ms)

In conclusion five indexes were chosen for the rockmass blastability classification the Protodyakonov coefficientand tensile strength of the rock rock density rock waveimpedance and the integrity coefficient of the rock massAmong these indexes the Protodyakonov coefficient andtensile strength of the rock are mechanical property indexesof the rock the rock density and rock wave impedanceare physical property indexes of the rock and the integritycoefficient of the rock mass is a measure of the geologicalproperties of the rock mass These five indexes primarilyreflect the relevant physical and mechanical properties andcharacteristics of the geological structure of a rock mass andblasting and can be easily obtained by fieldmeasurements andexperiments

22 Determination of the Standards for Rock Mass BlastabilityClassification Thevalue selection and designation of classifi-cation standards play an important role in the development ofmodels for rock mass blastability classification In the litera-ture the standards for classification indexes are determinedusing five ranks for rock mass blastability very easy easymoderate difficult and very difficult These classificationstandards for rock mass blastability are shown in Table 1 [18]

23 Measurements and Results for the Indexes

231 P-Wave Velocity of the Rock Mass Figure 1 illustratesa hole that is drilled by a geological drilling rig to measurea rock mass A small quantity of explosive is placed at thebottom of the hole and two vibration pick-up instrumentsare placed around the hole These two instruments should be

4 Mathematical Problems in Engineering

Figure 2 Vibration pick-up

placed as far apart as possible and in a straight line with thehole Detonating the explosive in the hole produces a surfacewave and volume wave in the rock mass where the volumewave consists of P-wave and S-wave componentsThe velocityof the P-wave is higher than that of the S-wave and thusthe P-wave reaches the measuring point before the S-waveWhen the P-wave reaches measuring point 1 it triggers theldquovibration pick-up 1rdquo which records the time 119905

1at which the

P-wave reaches point 1 The ldquovibration pick-up 2rdquo recordsthe time 119905

2at which the P-wave reaches measuring point

2 The P-wave velocity of the rock mass is calculated usingthe difference between the times that the P-wave reachesmeasuring points 1 and 2 and the distances between the twomeasuring points and the center of the explosive cartridgeEquation (4) shows the formula used to calculate the P-wavevelocity of the rock mass for the geometry shown in Figure 1[19]

Vpm =radicℎ2 + (1199041 + 1199042)

2minus radicℎ2 + 1199041

2

1199052 minus 1199051 (4)

In the equation above ℎ denotes the depth of the hole (inm) 1199041denotes the distance betweenmeasuring point 1 and the

hole (in m) 1199042denotes the distance between the two points

(in m) 1199051denotes the time at which the P-wave reaches point

1 (in s) and 1199052denotes the time at which the P-wave reaches

point 2 (in s)Following the aforementioned principles field tests were

performed at different locations in the Heidaigou open-castcoal mine where the drilling hole diameter was 127mmand the hole depth was between 5 and 17m A vibrationvelocity transducer (model CD-21 manufactured by BeijingInstrument Industry Group Co Ltd in Beijing China) wasused as the vibration pick-up instrument and a blastingvibration instrument (model EXP3850 Chengdu VIDTSDynamic Instrument Co Ltd in Chengdu China) was usedas the blasting vibration recording instrument as shown inFigures 2 and 3 respectively

The bench consisted primarily of medium sandstone anda hole was drilled in the bench at an altitude of 1185m asshown in Figure 4 The depth of the hole was 163m and thedistances 119904

1and 1199042were 71 and 167m respectively

Figure 3 Blasting vibration recorder

Figure 4 Photograph of the rock mass

000 002 004 006 008 010 012

0000000400080012001600200024

Am

p (c

ms

)

Time (s)minus006 minus004

minus0004minus0008minus0012minus0016minus0020minus0024

minus002

Figure 5 Test pattern of the P-wave velocity of the rock mass

The two times that were obtained from the test 1199051=

minus2365ms and 1199052= minus1375ms are shown in Figure 5 The P-

wave velocity of the medium sandstone mass was calculatedto be vpm = 1118ms using (4)

Similarly the measurements indicated that the wavevelocity in the mudstone mass was 1273ms the wavevelocity in the sandstone mass was 944ms and the wave

Mathematical Problems in Engineering 5

Figure 6 Rock samples

Table 2 Test results for the rock mass blastability indexes

Rock mass 120590119905

120590119901

120588 120590pm Vpr(MPa) (MPa) (tm3) (ms) (ms)

Mudstone 219 276 268 127 3 374 4Fine sandstone 272 436 241 957 322 6Medium sandstone 226 273 236 1118 272 4Coarse sandstone 150 243 223 796 236 0

velocity in the coarse sandstone rockmass was 796ms in theHeidaigou open-cast coal mine

232 P-Wave Velocity of the Rock Mass Different types ofrocks were collected from the Heidaigou open-cast coal mineand processed into standard samples with specifications of12060150mm times 100mm as shown in Figure 6 A TICO ultrasonicdetector was used to determine the P-wave velocity of therock The TICO detector had two sensors a sender and areceiver The distance l between the two sensors and thepropagation time 119905 of the test sound wave to travel betweenthe two sensors were used to calculate the wave velocity Vpr =119897119905 The wave velocities were as follows 3744ms for themudstone 3226ms for the fine sandstone 2724ms for themedium sandstone and 2360ms for the coarse sandstone

233 Rock Density Tensile Strength and CompressiveStrength The rock density was determined by weighing thesamples and calculating the sample volumes

The tensile strength and uniaxial compressive strengthof the rock were measured using microcomputer controlelectron universal testingmachines (modelWDW300KeXinTesting Machine Co Ltd in Changchun China) The spec-ifications for the test samples for the uniaxial compressivestrength measurements were 12060150mm times 100mm The Brazilsplitting method was used to measure the tensile strength ofthe rock with specifications of 12060150mm times 25mm

The test results for the rock density rock tensile strengthand rock compressive strength are provided in Table 2

Table 3 Rock mass blastability classification index values

Rock mass 120590119905 119891

119875119870V

119885

(MPa) (tm3) (106 kgm3 timesms)Mudstone 219 276 268 012 1003Fine sandstone 272 436 241 009 777Medium sandstone 226 273 236 017 643Coarse sandstone 150 243 223 011 526

Table 4 Judgment matrix 119875

Variables 120590119905

119891 120588 119870V 119911

120590119905

1 15 13 14 12119891 5 1 3 2 4119875 3 13 1 12 2119870V 4 12 2 1 3119911 2 14 12 13 1

The classification index values of the rock masses werecalculated and are shown in Table 3

3 Determination of the Index Weights Usingthe Combination Weight Method

To circumvent the deficiencies of the subjective weightmethod and the objective weight method the index weightwas systematically determined using the analysis hierarchyprocess (AHP) to calculate the subjective weight the entropymethod (EM) was used to calculate the objective weight andthe combination weight method (CWM) which is based ongame theory (GT) was used to determine the index weightsfor the rock mass blastability classification

31 Determination of Subjective Weight Using the AnalysisHierarchy Process AHP is a systematic hierarchical analysismethod that incorporates both qualitative and quantitativeanalyses thus AHP is simple and practical and avoids theuncertainties and errors that can arise when determining theindexweightsThemethodology used to calculate theweightsusing the AHP is detailed in Saatyrsquos papers [20 21]

A pairwise comparison of the Protodyakonov coefficienttensile strength density wave impedance and integritycoefficient of the rock mass was performed to construct ajudgment matrix as shown in Table 4

The calculation produced the eigenvector W = (006

042 016 026 010)T maximum eigenvalue 120582max = 5068and the consistency index CI = 0017 For 119899 = 5 elementsthe random index was found to be RI = 112 which was usedto calculate the consistency ratio CR = 0017112 = 0015 lt01 Therefore the judgment matrix satisfied the consistencycheck The consistency ratio weight vector was determinedusing AHP to beW1 = (006 042 016 026 010)

32 Determination of the Objective Weight Using the EntropyMethod Information theory states that the informationentropy is a measure of the degree of disorder in a system

6 Mathematical Problems in Engineering

Lower information entropy indicates a lower degree ofdisorder and a higher utility value of the information theopposite holds true for higher information entropy [22]An indicator that does not produce different effects fordifferent levels of rock mass blastability is not useful forrock mass blastability classification Thus if an indicatorproduces a smaller difference for different levels of rockmass blastability the effect of the indicator on the rock massblastability classification is smaller and the correspondinginformation entropy is higher the opposite holds true foran indicator that produces a larger difference for differentlevels of rock mass blastability That is the difference degreeof the indicator for the rock mass blastability classification isinversely proportional to the information entropyThereforethe index weight was determined from the difference degreeof the rock mass blastability classification indicator using theEMThe calculation procedure is detailed in the papers of Pei-Yue et al [23] and Zou et al [24]

Using the sample data for mudstone fine sandstonemedium sandstone and coarse sandstone that are givenin Table 3 the entropy of each indicator was calculated tobe H = (0984 8 0979 6 0998 4 0979 9 0979 5) whichyielded the consistency ratio weight vectorW2 = (020 026002 026 026)

33 Combination Weight from Game Theory The weightsobtained from different weight methods may not be consis-tent with each other GT can be used to minimize the sum ofthe differences between the final determined weight and theweight determined from each method [25] The procedurefor determining the GT-based combination weight is givenbelow [26]

Assume that 119871methods are used to weight the indexes ofthe rock mass blastability classification where the 119871 weightvectors are given byW

119896= (1205961198961 1205961198962 120596

119896119899) (where 1 le 119896 le

119871 and 119899 indicates the number of indexes of the rock massblastability classification) A combination weight vector canbe constructed from a random linear combination of the 119871weight vectors as follows

W119888=119871

sum119896=1

120572119896sdotWT119896 (5)

In the equation above W119888denotes the combination

weight vector and 120572119896denotes the coefficient of the linear

combination where 120572119896gt 0

The most satisfactory combination weight vector isobtained by minimizing the deviation between the com-bination weight vector W

119888and each weight vector W

119896by

optimizing the coefficient 120572119896of the 119871 linear combinations

The game model is given as

min1003817100381710038171003817100381710038171003817100381710038171003817

119871

sum119896=1

120572119896sdotWT119896minusWT119897

10038171003817100381710038171003817100381710038171003817100381710038172

(119897 = 1 2 119871) (6)

The differential attribute of the matrix is used to calculatethe optimum first derivative condition in (6) which can bewritten in the following matrix form

[[[[[[[

[

W1 sdotWT1 W1 sdotWT

2 sdot sdot sdot W1 sdotWT119871

W2 sdotWT1 W2 sdotWT

2 sdot sdot sdot W2 sdotWT119871

d

W119871sdotWT

1 W119871sdotWT

2 sdot sdot sdot W119871sdotWT119871

]]]]]]]

]

[[[[[[

[

1205721

1205722

120572119871

]]]]]]

]

=

[[[[[[[

[

W1 sdotWT1

W2 sdotWT2

W119871sdotWT119871

]]]]]]]

]

(7)

Thematrix is used to calculate the normalization process-ing of (120572

1 1205722 120572

119871) as shown in (8) to obtain the optimal

linear combining coefficient 120572lowast119896(1 le 119896 le 119871) as follows

120572lowast119896=

120572119896

sum119871

119896=1120572119896

(8)

Therefore the optimal combination weight vector is

Wlowast =119871

sum119896=1

120572lowast119896sdotW119896 (9)

In the equation aboveWlowast denotes the optimal combina-tion weight vector Wlowast = (120596lowast

119895)1times119899

where 120596lowast119895is the optimal

combination weight vector for the 119895th index with sum119899119894=1120596lowast119895=

1The calculated subjective and objective weights were used

to determine 120572lowast1= 0691 2 and 120572lowast

2= 0308 8 using (7)

and (8) andWlowast = (010 037 012 026 015)was calculatedusing (9) The index weights that were determined using theAHP EM and CWM are shown in Figure 7

Figure 7 illustrates that the index weights that were deter-mined using the combination weight method lay betweenthose calculated using the AHP and those calculated usingEM The CWM balanced and coordinated the impacts thatthe subjective and objective methods exerted on the weightsthus overcoming the one-sidedness of each method thefinalized CWM produced more realistic results than theindividual methods

4 Fuzzy Pattern Recognition Model forRock Mass Blastability

41 Model Principles We assume that the rock mass blasta-bility can be divided into 119898 levels for 119899 indexes where eachblastability has its own grading standards (the value range ofeach index) and that the119898 grading standards of the rockmassblastability can serve as the 119898 fuzzy subsets 119860

1 1198602 119860

119898

that constitute a standard sample database 1198601 1198602 119860

119898

The value of the 119895th grading standard of the rock mass

Mathematical Problems in Engineering 7

0

01

02

03

04

05

Indexes of blastability classification

Wei

ght

f z

AHPEMCWM

120590t Kv120588

Figure 7 Comparison of weights

blastability classification is denoted by 119909119895(119895 = 1 2 119899)

and the universe 119880 = 119906 | 119906 = (1199091 1199092 119909

119899) namely 119906

denotes the set of 119899 indexes for a certain rock mass sampleThe membership functions 120583

119894(119909119895) and 120583

119894(119906) are con-

structed for a particular fuzzy subset 119860119894 For these functions

119894 (119894 = 1 2 119898) denotes the number of the rock massblastability classification for a standard sample 119895 (119895 =1 2 119899) denotes the index number and 119909

119895indicates the

value of the 119895th index for a specific rock mass sample 120583119894(119909119895)

denotes the membership degree of the 119895th index of a specificrock mass sample that relatively belongs to the blastabilitydegree of the 119894th rock mass and 120583

119894(119906) is the membership

degree of a specific rock mass sample that relatively belongsto the blastability degree of the 119894th rock mass

If 1199060isin 119880 and 119896 isin 1 2 119898 exist such that 120583

119896(1199060) =

max1205831(1199060) 1205832(1199060) 119906

119898(1199090) 1199060is considered to belong

to 119860119896 which means that the blastability degree of rock mass

sample 1199060belongs to degree 119896

42 Development of the Membership Function To establisha fuzzy relation between the classification indexes and thestandard samples the membership functions must first bedeveloped between each index and each standard sample[27] The fuzziest principle and clearest principle must beobeyed while formulating the membership function That isthe membership degree is 05 at the endpoint of the intervalfor the fuzziest state and the membership degree is 1 at themidpoint of the interval for the clearest state Moreover thesum of the membership degrees at any point is 1 [28]

Commonly used membership functions include triangu-lar trapezoidal normal distribution and mountain-shapedmembership functions [29] Mikkili and Panda concludedthat there are no considerable differences among the mem-bership degrees that correspond to different membership

Table 5 Value range of the index

Indexnumber Degree 1 Degree 2 sdot sdot sdot Degree119898 minus 1 Degree119898

119895 le1198861119895

1198861119895sim1198862119895

sdot sdot sdot 119886(119898minus2)119895

sim119886(119898minus1)119895

ge119886(119898minus1)119895

functions and the analysis results are consistent [30] There-fore a trapezoidal membership function was used to formu-late the fuzzy assessment matrix Based on the principle usedto identify the membership function the trapezoidal mem-bership function degenerated into a triangular membershipfunction

Table 1 illustrates that the values of the classificationindexes that were chosen in this study increased with thedegree of rock mass blastability namely the incrementalindex The assumed value range of the 119895th index for the 119894thdegree is provided in Table 5

Let us consider the trapezoidal membership function asan example where the membership functions for each indexof each degree are given as follows

1205831 (119909119895) =

1 119909119895lt 1198871119895

119909119895+ 1198871119895 minus 21198861119895

2 (1198871119895 minus 1198861119895)1198871119895 le 119909119895 lt 1198861119895

1198872119895 minus 119909119895

2 (1198872119895 minus 1198861119895)1198861119895 le 119909119895 lt 1198872119895

0 119909119895ge 1198872119895

120583119894(119909119895)

=

0 119909119895lt 119887(119894minus1)119895 119909119895 ge 119887(119894+1)119895

119909119895minus 119887(119894minus1)119895

2 (119886(119894minus1)119895 minus 119887(119894minus1)119895)

119887(119894minus1)119895 le 119909119895 lt 119886(119894minus1)119895

119909119895+ 119887119894119895minus 2119886(119894minus1)119895

2 (119887119894119895minus 119886(119894minus1)119895)

119886(119894minus1)119895 le 119909119895 lt 119887119894119895

119887119894119895minus 2119886119894119895minus 119909119895

2 (119886119894119895minus 119887119894119895)

119887119894119895le 119909119895lt 119886119894119895

119887(119894+1)119895 minus 119909119895

2 (119887(119894+1)119895 minus 119886119894119895)

119886119894119895le 119909119895lt 119887(119894+1)119895

120583119898(119909119895) =

0 119909119895lt 119887(119898minus1)119895

119909119895minus 119887(119898minus1)119895

119887119898119895minus 119887(119898minus1)119895

119887(119898minus1)119895 le 119909119895 lt 119887119898119895

1 119909119895ge 119887119898119895

(10)

In the equations above 119909119895denotes the value of the 119895th

index for a given rock sample 120583119894(119909119895) denotes themembership

degree of index 119909119895that relatively belongs to the 119894th blastability

degree and 119887119898119895= 2119886(119898minus1)119895

minus 119887(119898minus1)119895

where 1198871119895= max2119886

1119895minus

1198872119895 11988611198952 and 119887

119894119895= (119886(119894minus1)119895

+ 119886119894119895)2 (2 le 119894 le 119898 minus 1)

8 Mathematical Problems in Engineering

0

05

10

0

1 2 3 m

b1j a1j b2j a2j b3j bmj

xj

a3j a(mminus1)j

middot middot middot

middot middot middot

120583i(xj)

Figure 8 Pictorial representation of trapezoidal membership func-tions

Building theclassification system

Combinationweight

Fuzzy patternrecognition

Determining classification standards

Calculating weight of each index

Determining membership matrix

The principle of maximum membership

degree

Classification result

Calculating

Calculating

Calculating

subjective weightusing AHP

objective weightusing EM

combination weightusing GT

Figure 9 Rock mass blastability classification procedure

Equations (10) correspond to themembership function ofthe incremental index which is shown in Figure 8Themem-bership function of the descending index can be determinedusing the same method and is thus not presented here

43 Identification of the Rock Mass Blastability Degree Theprocedure for the rock mass blastability classification issummarized in Figure 9 The membership degree of everyblastability degree can be calculated using (11) togetherwith the classification index and the classification standardprovided in section two the combination weight providedin section three and the membership function provided insection four for any type of rock as long as the values ofthe n indexes are known and the blastability degree of therock can be determined using the principle of the maximummembership degree Consider

120583119894(119906) =

1119899

119899

sum119895=1120596lowast119895120583119894(119909119895) 1 le 119894 le 119898 (11)

Table 6 Membership degree for each index for mudstone

Blastability class 120590119905

119891 120588 119870V 119911

I 004 043 0 073 0II 096 057 0 027 0III 0 0 070 0 016IV 0 0 030 0 084V 0 0 0 0 0

Table 7 Membership degree of each rock mass

Rock mass Blastability classI II III IV V

Mudstone 035 038 011 016 000Fine sandstone 024 051 024 001 000Medium sandstone 027 059 014 000 000Coarse sandstone 044 051 005 000 000

In the equation above 120596lowast119895denotes the satisfactory combi-

nation weight of the 119895th index

5 Project Application

51 Rock Mass Blastability Classification The identified clas-sification index provided in Table 1 was used to calculatethe specific membership function using (10) Let us considerthe rock mass blastability classification of mudstone fromthe Heidaigou open-cast coal mine as an example Themembership degree between each index and each rock massblastability classification was calculated by substituting therelevant indexes of the rock mass of mudstone into themembership function in Table 3 and is shown in Table 6

Equation (11) was used to calculate the following values1205831(119906) = 035 120583

2(119906) = 038 120583

3(119906) = 011 120583

4(119906) = 016 and

1205835(119906) = 0The principle of themaximummembership degree

was used to determine that the largest rock mass blastabilitydegree ofmudstone was 120583

2(119906) which corresponds to a degree

of 2 the ldquoeasyrdquo blasted rock massThe membership degree between each index for the rock

masses of fine sandstone medium sandstone and coarsesandstone and the rock mass blastability classification wassimilarly confirmed and the membership degree betweeneach rock mass and the rock mass blastability classificationwas calculated as shown in Table 7 and Figure 10 Theprinciple of the maximummembership degree indicated thatthe rock masses of fine sandstone medium sandstone andcoarse sandstone could all be categorized as easy blasted rockmasses

Figure 10 illustrates that although each rock mass had amaximum membership degree for a blastability of II whichbelonged to the easy blasted rock mass the mudstone andcoarse sandstone rock mass also had a large membershipdegree for a blastability of I indicating that the degreefor mudstone and coarse sandstone was between the veryeasy blasted and easy blasted rock masses Although thefine sandstone and coarse sandstone had equal membershipdegrees for a blastability of II the coarse sandstone had a

Mathematical Problems in Engineering 9

I II III IV V0

01

02

03

04

05

06

07

Blastability class

Mem

bers

hip

degr

ee

MudstoneFine sandstoneMedium sandstoneCoarse sandstone

Figure 10 Membership degree of each rock mass

Table 8 Unit explosive consumption of rock mass of each classifi-cation

Blastability class Unit explosive consumption (kgm3)I le035II 035ndash045III 045ndash065IV 065ndash090V ge090

larger membership degree for a blastability of I and a smallermembership degree for a blastability of III illustrating thatcoarse sandstone was more explosive than fine sandstonedespite both sandstones being categorized as easy blastedrock mass

52 Unit Explosive Consumption The relevant literature andmaterial and blasting experience from open-cast mines wasused to determine the unit explosive consumption (ANFO)for the loose blasting of a rock mass for each classification asshown in Table 8

Let 1199022 1199023 and 119902

4denote the midpoints of the interval

of the unit explosive consumption for degrees II III and IVin Table 8 respectively where 119902

1= 035 minus (119902

2minus 035) and

1199025= 090 + (090 minus 119902

4) form the vector 119876 = (119902

119894)1times5

=(030 040 055 078 102)

The membership matrixU = (120583119894(119906))1times5

between the rockmass and the rockmass blastability degreewas calculatedTheunit explosive consumption for the loose blasting of the rockmass was then calculated using

119902 =

le 03 1205831 (119906) = 1

ge 102 1205835 (119906) = 1

119876 sdot 119880T 1205831 (119906) = 1 1205835 (119906) = 1

(12)

Consider the mudstone rock mass in the Heidaigouopen-cast coal mine as an example where the unit explosiveconsumption for loose blasting was calculated to be 119902mud =(030 040 055 078 102) sdot (035 038 011 016 0) =044 kgm3 Similarly the unit explosive consumption valuesfor fine sandstone medium sandstone and coarse sandstonewere determined to be 119902

119891= 042 kgm3 119902med = 040 kgm

3and 119902119888= 036 kgm3 respectively

When there are several types of rock masses in a blastingarea the unit explosive consumption can be approximatedusing

119902119886=sum119899

119894=1119902119898119894119881119898119894

sum119899

119894=1119881119898119894

(13)

In the equation above 119902119886denotes the average unit

explosive consumption (in kgm3) for loose blasting in theblasting area 119902

119898119894denotes the unit explosive consumption (in

kgm3) for loose blasting of the rock mass 119894 119881119898119894

denotes thevolume (in m3) of the rock mass 119894 and 119899 denotes the numberof types of rock masses in the blasting area

When there are several rock masses that are approxi-mately level or when there is a gentle incline from the top tothe bottom of the blasting area 119881

119898119894can be replaced by ℎ

119898119894in

(13) thus ℎ119898119894

denotes the thickness of the rock mass 119894 (m)

6 Conclusions and Further Research

(1) There are no clear boundaries between differentclassifications for the blastability of a rock mass thatis there is fuzziness in the classification problemThus a fuzzy pattern recognition method was usedto develop a model for rock mass blastability clas-sification The classification results from the modelwere obtained as a vector which providedmore infor-mation than a point value Thus a simple algorithmwas used to make full use of the information and theclassification procedure was easy to understand

(2) The relevant indexes of each rock mass that wastested in the Heidaigou open-cast coal mine wereused with the developed rock mass blastability clas-sification model to demonstrate that mudstone finesandstone medium sandstone and coarse sandstonein the Heidaigou open-cast coal mine could all beconsidered easy blasted rock massThe unit explosiveconsumption of mudstone fine sandstone mediumsandstone and coarse sandstone was determined tobe 044 042 040 and 036 kgm3 respectively Blast-ing experiments in theHeidaigou open-cast coalminedemonstrated the accuracy of the aforementionedconclusions These conclusions were used to developa loose blasting design which yielded good results

(3) The standards for the blastability classification of therock masses were determined from the literatureThese standards can be applied to actual situationsbut are not based on sufficiently large amounts ofsample data therefore the scientific basis of thesestandards has not been confirmed In addition a

10 Mathematical Problems in Engineering

sufficiently large amount of sample data was notused with the objective weight method Thus thisinsufficient sample size likely affected the weights thatwere determined by the objective weight method tosome extent Therefore the next step is to collecta large amount of sample data of rock masses intypical areas for use in a more accurate calculationof the weight of each classification index and a moreaccurate calculation and analysis of the standards forthe blastability classification of rock masses

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Financial support for this work provided by the NationalHigh Technology Research and Development Program ofChina (no 2012AA062002) is gratefully acknowledged

References

[1] J Zhou and X Li ldquoIntegrating unascertainedmeasurement andinformation entropy theory to assess blastability of rock massrdquoJournal of Central South University vol 19 no 7 pp 1953ndash19602012

[2] Q Y Ma and Z H Zhang ldquoClassification of frozen soilblastability by using perception neural networkrdquo Journal of CoalScience amp Engineering vol 8 no 1 pp 54ndash58 2002

[3] J-L Shang J-H Hu R-S Mo X-W Luo and K-P ZhouldquoPredication model of game theory-matter-element extensionfor blastability classification and its applicationrdquo Journal ofMining amp Safety Engineering vol 30 no 1 pp 86ndash92 2013

[4] J-P Latham and P Lu ldquoDevelopment of an assessment systemfor the blastability of rockmassesrdquo International Journal of RockMechanics and Mining Sciences vol 36 no 1 pp 41ndash55 1999

[5] S R Chen S Q Xie Y L Li Y X Yu and P Z WuldquoExperimental study of physical properties and drillability andblastability of special ore-bodiesrdquo Journal of Central SouthUniversity of Technology vol 35 no 4 pp 667ndash669 2004

[6] Y D Cai and L S Yao ldquoArtificial neural network approach ofdetermining the grade of the blasting classification of rocksrdquoBlasting vol 10 no 2 pp 50ndash52 1993

[7] X T Feng ldquoA study on neural network on rock blastabilityrdquoExplosion and Shock Waves vol 14 no 4 pp 298ndash306 1994

[8] X T Feng ldquoA neural network approach to comprehensive clas-sification of rock stability blastability and drillabilityrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 9 no 2 pp 57ndash62 1995

[9] C Fang X-G Zhang and Z-H Dai ldquoProjection pursuitregression method of rock blastability classification based onartificial fish-swarm algorithmrdquo Blasting vol 26 no 3 pp 14ndash17 2009

[10] Y D Cai ldquoApplication of genetic programming in determiningthe blasting classification of rocksrdquo Explosion and Shock Wavesvol 15 no 4 pp 329ndash334 1995

[11] P P Huang ldquoRock blastability classification with fuzzy synthe-sisrdquo Quarterly of the Changsha Institute of Mining Research vol9 no 4 pp 63ndash72 1989

[12] Y Azimi M Osanloo M Aakbarpour-Shirazi and A AghajaniBazzazi ldquoPrediction of the blastability designation of rockmasses using fuzzy setsrdquo International Journal of Rock Mechan-ics and Mining Sciences vol 47 no 7 pp 1126ndash1140 2010

[13] A Aydin ldquoFuzzy set approaches to classification of rockmassesrdquoEngineeringGeology vol 74 no 3-4 pp 227ndash245 2004

[14] S J Qu S L Mao W S Lu M Y Xin Y J Gong and XY Jin ldquoA method for rock mass blastability classifcation basedon weighted clustering analysisrdquo Journal of University of Scienceand Technology Beijing vol 28 no 4 pp 324ndash329 2006

[15] J-G Xue J Zhou X-Z Shi H-Y Wang and H-Y HuldquoAssessment of classification for rock mass blastability based onentropy coefficient of attribute recognition modelrdquo Journal ofCentral South University (Science and Technology) vol 41 no 1pp 251ndash256 2010

[16] Y L Yu D S Wang and S J Qu ldquoZoning of the blastingcompliance of rocks in shuichang open pitrdquo Chinese Journal ofRock Mechanics and Engineering vol 9 no 3 pp 195ndash201 1990

[17] S-J Qu M-Y Xin S-L Mao et al ldquoCorrelation analyses ofblastability indexes for rock massrdquo Chinese Journal of RockMechanics and Engineering vol 24 no 3 pp 468ndash473 2005

[18] P Lu The characterisation and analysis of in-situ and blastedblock size distribution and the blastability of rock masses [PhDthesis] University of London London UK 1997

[19] K M Li Z H Guo and Y Zhang Research and Applicationof Casting Blast Technologies in Opencast Mine China CoalIndustry Publishing House Beijing China 2011

[20] T L Saaty ldquoHow to make a decision the analytic hierarchyprocessrdquo European Journal of Operational Research vol 48 no1 pp 9ndash26 1990

[21] T L Saaty ldquoDecision making with the analytic hierarchyprocessrdquo International Journal of Services Sciences vol 1 no 1pp 83ndash98 2008

[22] J Wu J Sun L Liang and Y Zha ldquoDetermination of weightsfor ultimate cross efficiency using Shannon entropyrdquo ExpertSystems with Applications vol 38 no 5 pp 5162ndash5165 2011

[23] L Pei-Yue Q Hui and W Jian-Hua ldquoApplication of set pairanalysis method based on entropy weight in groundwaterquality assessmentmdasha case study in dongsheng city northwestChinardquo E-Journal of Chemistry vol 8 no 2 pp 851ndash858 2011

[24] Z H Zou Y Yun and J N Sun ldquoEntropy method fordetermination of weight of evaluating indicators in fuzzysynthetic evaluation for water quality assessmentrdquo Journal ofEnvironmental Sciences vol 18 no 5 pp 1020ndash1023 2006

[25] W Zhang J-P Chen Q Wang et al ldquoSusceptibility analysis oflarge-scale debris flows based on combination weighting andextension methodsrdquo Natural Hazards vol 66 no 2 pp 1073ndash1100 2013

[26] D C Chi TMa and S Li ldquoApplication of extension assessmentmethod based on game theory to evaluate the running condi-tion of irrigation areasrdquo Transactions of the CSAE vol 24 no 8pp 36ndash39 2008

[27] H Karimnia and H Bagloo ldquoOptimum mining method selec-tion using fuzzy analytical hierarchy processndashQapiliq salt mineIranrdquo International Journal of Mining Science and Technologyvol 25 no 2 pp 225ndash230 2015

[28] N A Bahri F M A Ebrahimi and S G Reza ldquoA fuzzy logicmodel to predict the out-of-seam dilution in longwall miningrdquo

Mathematical Problems in Engineering 11

International Journal of Mining Science and Technology vol 25no 1 pp 91ndash98 2015

[29] Y H Su M C He and X M Sun ldquoEquivalent characteristicof membership function type in rock mass fuzzy classificationrdquoJournal of University of Science and Technology Beijing vol 29no 7 pp 670ndash675 2007

[30] S Mikkili and A K Panda ldquoSimulation and real-time imple-mentation of shunt active filter id-iq control strategy formitiga-tion of harmonics with different fuzzy membership functionsrdquoIET Power Electronics vol 5 no 9 pp 1856ndash1872 2012

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