i

PROCEEDING

SOUTH EAST ASIAN CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS

SEACMA 2013

“Mathematical Harmony in Science and Technology”

Published by:

MATHEMATICS DEPARTMENT

INSTITUT TEKNOLOGI SEP ULUH NOPEMBER SURABAYA, INDONESIA

ii

PROCEEDING SOUTH EAST ASIAN CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS

SEACMA 2013

Editor : Prof. Dr. Basuki Widodo, M.Sc. Prof. Dr. M. Isa Irawan, MT. Dr. S ubiono Subchan, P h.D Dr. Darmaji Dr. Imam Mukhlas DR. Mahmud Yunus, M.Si

ISBN 978-979-96152-8-2

This Conference is held by cooperation with Mathematics Department, ITS

Secretariat : Mathematics Department Kampus ITS S ukolilo S urabaya 60111, Indonesia seacma@its.ac.id

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ORGANIZING COMMITTEE PATRONS : Rector of ITS

STEERING : Dean Faculty of Mathematics and Natural Sciences ITS

INTERNATIONAL SCIENTIFIC COMMITTEE

Prof. Basuki Widodo (ITS - Indones ia) Prof. M. Isa Irawan (ITS - Indonesia) Dr. Erna Apriliani (ITS - Indonesia) Dr. Subiono (ITS - Indonesia) Subchan, Ph. D (ITS - Indonesia) Dr. Abadi (Unesa-Indonesia) Dr. Miswanto (Unair-Indonesia) Dr. Eridani (Unair-Indonesia) Prof. Marjono (Universitas Brawijaya Indonesia) Prof. Toto Nusantara, M.Si (UM-Indonesia) Dr. Said Munzir, M.EngSc (Syiah Kuala University Banda Aceh-Indonesia) Dr. Nguyen Van Sanh (Mahidol University-Thaiand) Prof. dr. ir. Arnold W.Heemink (Delft Institute Applied Mthemathics-Netherland)

COMMITEE

Chairman : Subchan Ph. D Co-chairman : Dr. Imam Mukhlas Secretary : Soleha, S.Si, M.Si Tahyatul Asfihani S.Si, M.Si Finance : Dian Winda Setyawati, S.Si, M.Si

Technical Programme : Drs. Daryono Budi Utomo, M.Si. Drs. Lukman Hanafi, M.Sc. Drs. Bandung Arry Sanjoyo, M.T. Drs. I. G. Rai Usadha, M.Si Moh. Iqbal, S.Si, M.Si

Web and Publication : Dr. Budi Setiyono, S.Si, MT. Achmet Usman Ali

Transportation and Accommodation : Dr. Darmaji, S.Si, MT. Yunita HL

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Message

from the

Dean Faculty of Mathematics and Natural Sciences

Institut Teknologi Sepuluh Nopember

On behalf of the Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, it is a great honor and sincere to welcome all participants to the second of South East Asian Conference on Mathematics and Its Application (SEACMA 2013).

This year, Department of Mathematics, Institut Teknologi Sepuluh Nopember, have honor to organize this meaningful conference. I believe that the purpose of this conference is not only sharing knowledge among the mathematician and scholars in related fields but also to hearten new generation of expertise in mathematics to realize the science and technology advancement.

It is undeniable that there is mathematical harmony in science and technologies. Many disciplines like engineering, computer science, information technology, operational research, logistics management, risk management and many others are all the products of mathematics. Thus, it is essential that we must hold this annual conference as a stage for all scholars in finding new ideas and applications on Mathematics.

Greatly thank to all supportive session including organizing committee, keynote speakers, invited speakers, paper reviewers, participants and sponsors. This event will not achieve without you all. Finally, I hope that the outcome of SEACMA 2013 will be pleasing and most useful to everybody.

Sincerely yours,

Prof. Dr. R.Y. Perry Burhan Dean

v

Message from the Chairman

Organizing Committee

On behalf of the organizing committee it is my pleasure to welcome you to the South East Asian Conference on Mathematics and Its Applications. The conference aims to provide a forum for academics, researchers, and practitioners to exchange ideas and recent developments on mathematics. The conference is expected to faster networking, collaboration and joint effort among the conference participants to advance the theory and practice as well as to identify major trends in mathematics.

We are also very pleased to welcome keynote speakers of the conference: Prof. Dr. Ir. Arnold W. Heemink, from Delft Institute of Applied Mathematics, Netherlands, Prof. Dr. Muhammad Isa Irawan, MT, from Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia, and Dr. Said Munzir, M.EngSc, from Syiah Kuala University, Indonesia, Dr. Nguyen Van Sanh, from Department of Mathematics, Faculty of Science, Mahidol University, Thailand.

We will spend about a day together for the conference. This conference is attended delegates and contributors from Indonesia, Malaysia, Thailand, and Netherlands Such a spread of participation from around the world confirms the appropriateness of the “National” label of this conference. There are 35 papers to be presented orally. Papers presented in the conference will be included in the conference proceeding to be published at the conference.

The organizing committee would like to express our deepest appreciation to ITS Rector, keynote speakers, head of departments of Mathematics of ITS, and sponsors for the support, without all mentioned this conference may not be happened. Furthermore, my appreciation goes to the members of the committee for their hard work and cooperative teamwork in the preparation of the conference. Finally, we wish all participants enjoy a fruitful scientific and human discussion.

Subchan, PhD Conference Chairman

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CONTENTS Cover i Organizing Committee iii Message from Dean Faculty of Mathematics and Natural Sciences iv Message from Chairman of Organizing Committee v Contents vi Tentative Schedule ix Abstract of Keynote Speaker 1. Model Reduced Variational Data Assimilation: An Ensemble Approach To Model

Calibration (A.W. Heemink) 2. Computational Optimal Control: Past, Current and Future Trends

(Said Munzir) 3. An Introduction to Some Kinds of Radicals for Module Theory (Nguyen Van Sanh) 4. Optimization Combination of Number, Dimension, and Mooring Time of the Ships

in PT. (Persero) Pelabuhan Indonesia III Gresik Harbor Using Genetic Algorithms (M. Isa Irawan)

Mathematics Papers PM1 Partial Ordered Bilinear Form Semigroups in Term of Their Fuzzy Right

and Fuzzy Left Ideals (Karyati, Dhoriva Uwatul Wutsqa) PM2 ᶯ-Dual of Some New Double Sequence

(Sumardyono, Soepama D.W. and Supama) PM6 Construction Of Suboptimal Solution of the Stick and Rope Problem Using

Sequence of Semicircle (Abrari Noor Hasmi and Iwan Pranoto) PM7 A Laplace Transform DRBEM for Time-Dependent Infiltration from

Periodic Flat Channels (Imam Solekhudin) PM10 On The Total Irregularity Strength Of Gears

(R. Ramdani, A.N.M. Salman and H. Assiyatun) PM11 Construction Of Cone Difference Space-c0

(Sadjidon and Sunarsini) PM12 The Star Partition Dimention Of Corona Product Graph

(Darmaji) AM1 Measuring and Optimizing Conditional Value at Risk Using Copula

Simulation (Komang Dharmawan) AM4 Biomass Feedstock Optimization for the Production of Methane Gas from

Water Hyacinth, Elephant Grass and Rice Straw Waste (Suharmadi and Sumarno) AM5 Application of Disturbance Compensating Model Predictive Control

(DC-MPC) On Ship Control System (Sari Cahyaningtias, Tahiyatul Asfihani, Subchan)

vii

AM6 The Application of Ensemble Kalman Filter to Estimate the Debris Flow Distribution ( Ngatini, Erna Apriliani, Imam Mukhlash, Lukman Hanafi,and Soleha)

AM7 Modeling Consumer Prices Index Data in Yogyakarta using Truncated Polynomial Spline Regression ( Dhoriva Urwatul Wutsqa

and Yudhistirangga )

AM8 Flow Patterns Characteristics in Agitated Tank with Side Entering Impeller ( Sugeng Winardi, Tantular Nurtono, Widiyastuti, Siti Machmudah and Eka Lutfi Septiani)

AM10 Flood Gate Control of Barrage Using Ensemble Kalman Filter Based On Nonlinier Model Predictive Control (NMPC) (Evita Purnaningrum and Erna Apriliani)

AM15 Calculation of Moving Vehicles with Video Using Expectation Maximization Algorithm ( Budi Setiyono, R. Arif Firdaus Lazuardi and Shofwan Ali Fauji )

AM16 Application of fitting and Probabilistic Method to Characterize Physical &Mechanical Rock Properties Distribution ( S.A. Pramuditya, M.A. Azizi , S. Kramadibrata

and R.K. Wattimena)

AM17 Testing Proportion for Portfolio Profit of CAPM and LCAPM ( Retno Subekt, Rosita Kusumawati)

AM18 Numerical Simulation of Linear Water Waves Using Smoothed Particle Hydrodynamics (Rizal Dwi Prayogo and Leo Hari Wiryanto)

AM19 Mathematical Model of Drag Coeficient of Tandem Configuration on Re = 100 (Chairul Imorn, Suhariningsih, Basuki Widodo, and Triyogi Yuwono)

AM20 Performance Evaluation of Two-Microphone Separation with Convolutive Speech Mixtures ( Irwansyah and Dhany Arifianto)

AM22 Bayesian Analysis for Smoothing Spline in Semiparametric Multivariable Regression Model Using WinBUGS ( Rita Diana, I. Nyoman Budiantara, Purhadi

and Satwiko Darmesto)

AM23 Simulated Annealing Based Approach for the Dynamic Portfolio Selection of Stocks ( Novriana Sumarti, Agung Laksono and Viony A. Jantoro)

AM24 On the Text Documents Pattern Recognition Using Latent Semantic Analysis and Kolmogorov-Smirnov Test ( Soehardjoepri, Nur Iriawan, Brodjol Sutijo SU, Irhamah)

AM25 Study of Comparation Between Finite Difference and Crank- Nicholson Method in Heat Transfer (Lukman Hanafi and Durmin)

AM27 Multi-level Stock Investment Evaluation Using Fuzzy Logic (Mahmod Bin Othman) AM28 The Relationship Between Parvate-Gangal Mean Value Theorems and Holder

Continuous Function of Order 𝜶𝜶 ∈ (0,1) on The Cantor Set ( Supriyadi Wibowo and Pangadi) AM29 Calibration Of Rainfall Data by Using Ensemble Model Output Statistic

(EMOS) ( Dian Anggraeni, Heri Kuswanto, Irhamah and Waskitho Wibisono) AM30 Full Conditional Distributions of Bayesian Poisson-Lognormal 2-Level

Spatio-Temporal Extension for DHF Risk Analysis ( Mukhsar, Iriawan, N, Ulama, B. S. S, Sutikno and Kuswanto, H)

AM31 Fuzzy Inference System Implementation On Rainfall Events Prediction At North Surabaya (Farida Agustini Widjajati and Dynes Rizky Navianti)

viii

AM32 Monochromatic wave propagating over a step (L.H. Wiryanto and M. Jamhuri) AM33 The Normal Form For The Integral Of 3-Dimensional Maps Derived From A Sine-Gordon Equation (Zakaria L, Tuwankotta J.M., and Budhi M.W.S)

AM34 Stability Analisis of Lotka-Volterra Model with Holling Type II Functional Response ( Abadi, Dian Savitri, and Choirotul Ummah)

AM35 On The Construction of Prediction Interval for Double Seasonal Holt-Winters : a Simulation Study (Arinda R. Lailiya, Heri Kuswanto , Suhartono

and Mutiah Salamah)

AM36 Bayesian Neural Networks for Time Series Modeling (Fithriasari, K, Iriawan, N, Ulama, B. S

and Sutikno)

AM37 Upper Bound And Upper Limit of The Optimum Interpolation Function of Filter Bank (Dhany Arifianto)

ix

Tentative schedule Thursday, 14 November 2013

07.30 – 08.00 : Registration

08.00 – 08.30 : Opening Ceremony

08.30 – 09.00 : Keynote Session 1 by Prof. dr. ir. Arnold W.Heemink (Netherland)

09.00 – 09.30 : Keynote Session 2 by Dr. Said Munzir, M.EngSc (Indonesia)

09.30 – 10.00 : Discussion Session

10.00 – 10.15 : Break

10.15 – 10.45 : Keynote Session 3 by Dr. Nguyen Van Sanh(Thailand)

10.45 – 11.15 : Keynote Session 4 by Prof. M. Isa Irawan(Indonesia)

11.15 – 11.45 : Discussion Session

11.45 – 13.00 : Break

13.00 – 16.00 : Parallel Session

1 INTRODUCTION

Slope design has over the years become the domain of geotechnical engineers. While this has benefited the technology of theslope design processes, it has also separated the the mine design engineers from the responsibility of the risk versus reward relationship. Given the uncertainties that prevail within the geotechnical discipline, there always exists a probability that a slope may not perform aspredicted, and in the worst case can result in a catastrophic failure. The rock slope stability analysis affected by the random input parameters such as internal friction angle, cohesion, water pressure, seismic acceleration and density, which cannot be properly represented by a single value as input parameters in slope stability analysis. The uncertainty in the values of rock mass properties is a major factor in slope stability analysis.

Each input parameter has a certain distribution function. There are several methods used in characterizing a data distribution, namely: Chi-SquareMethod, Kolmogorov-Smirnov method, and the method of Anderson Darling. This paper uses the Kolmogorov-Smirnov method. The principle of this method is based on a comparison between the experimental cumulative distribution function and cumulative distribution theoretical assumptions.

The 4 assumptions of the cumulative distribution function will be compared with the empirical distribution functions, namely: normal, lognormal, beta, gamma.

The results of the characterization process will be used for the calculation of the safety factor and failure probability of slopes, and the distribution data are obtained from Tutupan Coal Mine, South Kalimantan, Indonesia.

2 BASIC THEORY 2.1 Probability Distribution Function

Probability distribution function describes the spread of a random variable that is used to estimate the probability of the occurrence of a parameter value. It has

Application of fitting and Probabilistic Method to Characterize Physical &Mechanical Rock Properties Distribution

S.A. Pramuditya1, M.A. Azizi 2, S. Kramadibrata3 and R.K. Wattimena4 1 Universitas Swadaya Gunung Jati, Indonesia

amamisurya@gmail.com, 2 Institut Teknologi Bandung, Indonesia

3 Institut Teknologi Bandung, Indonesia

4 Institut Teknologi Bandung, Indonesia

Abstract: Uncertainties of factors that influence slope stability have been acknowledged by many geotechnical engi-

neers, and these are drawn from the characterization of the geotechnical parameters of the slope. The characterization is associated with a process of identifying the distribution of random variable values used in the design of a single slope of an open mine, and also part of the probabilistic method which is an alternative approach in estimating the stability of a slope with a failure probability value (FP). This paper explains the characterization process using the Kolmogorov-Smirnov (K-S) method, which will determine the most appropriate function for the distribution of the random variable. The characterization results can be used in determining the Safety Factor including the level of con-fidence and Probability of Failure based on the best fitting function. Keyword: uncertainties, failure probability, slope stability, safety factor

Proceeding South East Asian Conference on Mathematics and its Application (SEACMA 2013) ISBN 978-979-96152-8-2

AM-16

typical and unique properties of distribution that make one function different from others. But this does not rule out the possibility that a distribution function is a derivative from another function. For example, the exponential distribution function is a special form of the gamma distribution function withshape parameter (a) is 1. Likewise,a uniform distribution function is a special form of the beta distribution function with shape parameter values α = 1 and β = 1. A triangular distribution function is also a special form of the beta distribution function with shape parameter values α1 = 1 and α2 = 2.

This paper employs 4 assumed continuous distribution functions, i.e. normal, lognormal, beta, and gamma. These are considered to represent the properties of the distribution of the random variables (see Table 1).

Table 1. Assumed Probability Density Function [8]

Distribution Probability Density Function Mean&Varians

Normal fx(x)=

1

σ√2πe-

12�x-μ

σ �2

−∞ < 𝑥 < ∞

Μ

σ2

Lognormal fx(x)=

1

xσ√2πe-

12�ln x-μ

σ �2

x>0

eμ+12σ2

e2μ+σ2�eσ2-1�

Beta f(x)=

xα1-1(1-x)α2-1

Β(α1,α2) ;0≤x≤1

f(x)=0, x others

α1

α1+α2

α1α2

(α1+α2)2(α1+α2+1)2

Gamma

f(x) = xα-1e-x/β

βαΓ(α); x > 0

f(x)=0, x others

αβ

αβ2

Probability Density

Function Cumulative Distribution

Function Figure 1. Probability Distribution Function

Figure 1 describes a Probability Density Function (PDF) and Cumulative

Distribution Function (CDF). Probability density functionsexplains the area in which the relative probability of a random number can be assumed to be a unique value compared to the other value. Failure probability (FP) is determined from the distribution of the safety factor. The area under the curve (FK <1) is defined as a failure probability value.

2.2 Fitting Distribution Function

If a theoretical distribution has been assumed, then the validity of the assumed distribution could be confirmed or refuted by a statistical test with goodness of fit. There are several methods to test the goodness of fit of some assumed distribution functions, i.e. Chi Square method, Kolmogrov-Smirnov (K-S) method, and Anderson-Darling method. This paper only discusses the K-S method. The basic procedure of this method includes a comparison between experimental and theoretical cumulative frequency distribution of the types of assumed distributions. If the difference of both frequencies is higher than those of a certain sample size, the assumed theoretical distribution will be rejected. The equation of this cumulative distribution function is as follows:

Sn(𝑥) = �0 𝑥 < 𝑥1

k

n 𝑥𝑘 ≤ 𝑥 < 𝑥𝑘+1

1 𝑥 ≥ 𝑥𝑛

(1)

𝑥1, 𝑥2, . . . . , 𝑥𝑛 R are the values of the sample data that is already set, and n is the sample size.

In the K-S test, the maximum difference between the Sn(𝑥)and 𝐹(𝑥) for all ranges of 𝑥 is a measure of the difference between assumed theoretical distributions and experimental data distribution. The maximum difference of both distributions can be written as follows:

𝐷𝑛 = 𝑚𝑎𝑥𝑥|𝐹(𝑥) − 𝑆𝑛(𝑥)| (2)

Theoretically Dn is a random variable whose distribution depends on n. To a certain significance level α, K-S test compares the maximum difference of the observations in above equation with the critical value Dn

α. Table 2 describes the critical value with a particular significance level α. Thus with a significance level α, the null hypothesis is rejected if Dn is higher than the critical value. To facilitate this method assisted with Matlab software Ver.7.12.0.

Table 2. Critical Value (Tse, 2009)

Significance level α 0.10 0.05 0.01

Critical value 1.22

√n

1.36

√n

1.63

√n

When two non-nested models are compared, the larger model with more pa-

rameters have the advantage of being able to fit the in-sample data with more flexible function and thus possibly a larger log-likelihood. To compare models on more equal terms, penalized log-likelihood may be adopted. The Akaike information criterion, denoted by AIC, proposes to penalize large models by subtracting the number of parameters in the model from its log-likelihood. Thus, AIC is defined as:

AIC = log L�θ�;x� -p (3)

Wherelog 𝐿�𝜃�;𝒙�is the log-likelihood and p is thenumber of estimated pa-rameters in the model. Based on this approach the model with the largest AIC is selected.

The above problem of the AIC can be corrected by imposing a different penalty on the log-likelihood. The Schwarz information criterion, also called the Bayesian information criterion, denoted by BIC, is defined as:

BIC = log L�θ�;x� -p

2log n (4)

The details of these methods can be seen in [8].

2.3 Monte Carlo Simulation

This method effectively simulates the response of the SF performance function to randomly select discrete values of the component variables. The process is repeated many times to obtain an approximate discrete PDF of the resulting SF values. The component random variables for each calculation are selected from a sample of random values that are based on the selected PDF of the random variables. Although the PDFs can take on any shape, the normal, lognormal, beta and gamma distributions are among the most favored for the analysis.

It should be noted that each Monte Carlo simulation will use a different sequence of random values, and the resulting probabilities, means, variances, and histograms will be slightly different. As the number of trials increases, the differences will become smaller.

Because the Monte Carlo simulation requires a large number of calculations, it often has been viewed as a less desirable option than the Taylor series or point estimation method. However, with the general availability of faster desktop computers, the Monte Carlo approach has gained wider acceptance due to its inherent simplicity and the ability to use existing computer programs with only minor modifications. Details of this method can be read in [1,3,8].

3 DATA COLLECTION & METHODOLOGY

Data gathering of the physical and mechanical properties of the lithology (coal, mudstone and sandstone) of Tutupan coal mines, South Kalimantan, Indonesia was carefully conducted. These properties include unit weight, cohesion, and internal friction angle.The number of data from each lithology consists of 10, 33 and 16for coal properties data, mudstone properties data, and sandstoneproperties data respectively.

Furthermore,the fitting test of K-S method was done using Matlab software Ver.7.12.0. By using equation [1], the empirical data representation can be done by sorting the data from smallest to highest value, which produces a histogram in the form of CDF (See Figure 2).

In the K-S method, the empirical distribution function was compared with the 4 assumptions in the theoretical distribution function,i.e. normal, lognormal, beta, and gamma. If the distribution function of the data is not actually Sn(x), then Dn has a considerable value. Thus, with a significance level α = 5%, the null hypothesis is rejected if Dn is greater than the critical value. The critical value for α = 5% is 1.36/√𝑛. For example, a fitting test was conducted on the cohesion distribution of mudstone (see Figure 3). The amount of data is 33 and α = 5%, the critical value is 0.237. To declare the assumptions of the functions fit and the hypothesis is accepted, the Dnvaluemust be smaller than the critical value. The results of the fitting test on cohesion distribution of mudstone indicates that the 4 assumed theoritical distribution functions have a Dn value smaller than the critical value, the hypothesis is then accepted.

Figure 2. EmpricalCDF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Data

Cum

ulat

ive

prob

abili

ty

KohMs data

0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Data

Cum

ulat

ive

prob

abili

ty

KohMs data

Normal

LognormalBeta

Gamma

Figure 3. Overlay Emprical and Assumed Theoritical CDF

Besides that, we also have to see the AIC and BIC values of the 4 assumed theoretical distribution functions. By using equation [3] and [4], AIC and BIC values can be determined. The largest value of that distribution is normal distribution, i.e. respectively 44.111 and 44.593.Thus, it can be said that the most appropriate distribution function is the normal distribution (see Table 2).

Table 3 presents the summary of the characterization test of rock properties distribution on 3 lithology of rock. Only the cohesion of coalthat has gamma distribution, while others have normal distribution.

Table 2. The results of Fitting Test on Cohesion of mudstone

Statistical Parameter

Normal Lognormal Beta Gamma

Mean 0.191 0.193 0.191 0.191

Dn 0.096 0.118 0.095 0.096

Critical value 0.237 0.237 0.237 0.237

Log-likelihood 46.111 42.632 45.543 45.147

AIC 44.111 40.632 43.543 43.147

BIC 44.593 41.113 44.024 43.629

Table 3. The result of Fitting Test in Rock Properties of Tutupan Coal Mine

Mudstone

Statistical Parameter

c (MPa)

φ (Deg.)

γ (kN/m2)

Mean 0.169 23.6 22.68 Variance 0.004 23.5 0.38

Std.Deviation 0.061 4.8 0.62 Dn 0.096 0.090 0.324

Critical Value 0.237 0.237 0.237 Log-likelihood 46.11 -98.38 -30.42

AIC 44.11 -100.38 -32.42 BIC 44.59 -99.82 -31.93

Distribution Normal Normal Normal Sandstone

Mean 0.191 27.8 21.48 Variance 0.003 33.1 -

Std.Deviation 0.057 5.8 - Dn 0.144 0.121 -

Critical Value 0.340 0.340 - Log-likelihood 23.54 -50.20 -

AIC 21.54 -52.20 - BIC 22.34 -51.40 -

Distribution Normal Normal - Coal

Mean 0.241 34.1 12.36 Variance 0.003 5.4 -

Std.Deviation 0.051 2.3 - Dn 0.190 0.120 -

Critical Value 0.430 0.430 - Log-likelihood 15.77 -22.13 -

AIC 13.77 -24.13 - BIC 14.77 -23.13 -

Distribution Gamma Normal -

4 DISCUSSION

Based on the characterization results using K-S method, each fitted distribution was included in the calculation of the safety factor. Given any variation in the random variable that is used in the determination of SF, then FPwas also be generated.

Characterization data can also be used to predict failure probability when the slope failed. This is possible when there is a strength decrease (see Table 4).

Table 4. The results of Failure Probability

Lithology Coal Mudstone Sandstone Slope Height (m) 15 15 15

Slope Angle (deg.) 70 70 70 SF Deterministic 5.5 2.4 2.8 SF Mean 5.5 2.5 2.8 Cohesion (kPa) 241 169 191 Cohesion (kPa) Back Analysis

57 68 66

Deterioration (%) 76.3 59.8 65.4 FP (%) 0 3.0 0 FP (%) Back Analysis

56.5 46.2 49.8

5 CONCLUSIONS

Several points can be concluded from the results of this study, and is as fol-lows: KS method is the most practical method to obtain the most suitable

distribution. The results of the characterization of the properties of 3 rock lithology

produce all types of variables tested had a normal distribution, except for the cohesion of coal that has a gamma distribution.

Characterization results are used to estimate the value of the safety factor following the data distribution, which would further strengthen the confidence level of the FK value.

6 AKNOWLEDGMENT

The authors would like to thank the committee which gives the opportunity to present this paper, and PT Adaro Indonesia for the support in providing data.

REFERENCES

1. Abramson, L.W., Lee, T.S., Sharma, S., and Boyce, G.M., 2002, Slope Stability and Stabi-lization Methods, second edition, John Wiley & Sons.

2. Ang, A.H-S., and Tang, W.H., 1984, Probability Concepts in Engineering Planning and Design – Vol.II Decision, Risk and ReliabilityJohn Wiley &Sons.

3. Baecher, G.B., and J.T. Christian. 2003. Reliability and statistics in geotechnical engineer-ing. Wiley, Chichester, U.K.

4. Hoek E., Factor of Safety and Probability of Failure, Chapter 8 - Rock Engineering. 5. M.A. Azizi, S.Kramadibrata, R.K. Wattimena, and I.Arif, 2010, Application of

Probabilistic Approach on Slope Stability Analysis, Annual Meeting of Indonesian Mining Professionals Association (PERHAPI) XIX 2010, Balikpapan.

6. M.A. Azizi, S.Kramadibrata, R.K. Wattimena, and I.Arif, 2011, Application of probabilistic approach on single slope stability analysis using limit equilibrium method, National conference of Statistical 2011, organized by Statistical Department, Faculty of Science, University of Diponegoro, Semarang, Indonesia.

7. M.A. Azizi, S.Kramadibrata, R.K. Wattimena, Suhedi, and S. Basuki, 2011, Probabilistic approach on single slope stability analysis using Bishop Method, National Conference of Mining 2011, organized by mining department, faculty of engineering, University of Lambung Mangkurat, South Kalimantan.

8. Tse, Y.K., 2009, Nonlife Actuarial Models, Theory, Methods and Evaluation, Cambridge University Press.