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SEAMS - GMU 2011

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Proceedings of the 6 SEAMS-GMU International Conference on Mathematics and Its Applications

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Mathem

atics and Its Applications in the

Developm

ent of Sciences and Technology.

Proceedings of the 6 S

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-GM

U International

Conference on M

athematics and Its A

pplications

th

Department of Mathematics

Faculty of Mathematics & Natural Sciences

Universitas Gadjah Mada

Sekip Utara Yogyakarta - INDONESIA 55281

Phone : +62 - 274 - 552243 ; 7104933

Fax. : +62 - 274 555131

MATHEMATICS AND ITS APPLICATIONS IN THE DEVELOPMENT OF SCIENCES AND TECHNOLOGY

International Conference on Mathematics and Its Applications

TheISBN 978-979-17979-3-1

PROCEEDINGS OF THE 6TH

SOUTHEAST ASIAN MATHEMATICAL SOCIETY

GADJAH MADA UNIVERSITY

INTERNATIONAL CONFERENCE ON MATHEMATICS

AND ITS APPLICATIONS 2011

Yogyakarta, Indonesia, 12th – 15th July 2011

DEPARTMENT OF MATHEMATICS

FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITAS GADJAH MADA

YOGYAKARTA, INDONESIA

2012

Published by

Department of Mathematics

Faculty of Mathematics and Natural Sciences

Universitas Gadjah Mada

Sekip Utara, Yogyakarta, Indonesia

Telp. +62 (274) 7104933, 552243

Fax. +62 (274) 555131

PROCEEDINGS OF

THE 6TH SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA UNIVERSITY

INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS 2011

Copyright @ 2012 by Department of Mathematics, Faculty of Mathematics and

Natural Sciences, Universitas Gadjah Mada, Yogyakarta, Indonesia

ISBN 978-979-17979-3-1

PROCEEDINGS OF THE 6TH

SOUTHEAST ASIAN MATHEMATICAL SOCIETY-GADJAH MADA

UNIVERSITY

INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS

APPLICATIONS 2011

Chief Editor:

Sri Wahyuni

Managing Editor :

Indah Emilia Wijayanti Dedi Rosadi

Managing Team :

Ch. Rini Indrati Irwan Endrayanto A. Herni Utami Dewi Kartika Sari

Nur Khusnussa’adah Indarsih Noorma Yulia Megawati Rianti Siswi Utami

Hadrian Andradi

Supporting Team :

Parjilan Warjinah Siti Aisyah Emiliana Sunaryani Yuniastuti

Susiana Karyati Tutik Kristiastuti Sudarmanto

Tri Wiyanto Wira Kurniawan Sukir Widodo Sumardi

EDITORIAL BOARDS

Algebra, Graph and Combinatorics

Budi Surodjo

Ari Suparwanto

Analysis

Supama

Atok Zulijanto

Applied Mathematics

Fajar Adi Kusumo

Salmah

Computer Science

Edi Winarko

MHD. Reza M.I. Pulungan

Statistics and Finance

Subanar

Abdurakhman

LIST OF REVIEWERS

Abdurakhman

Universitas Gadjah Mada, Indonesia

Insap Santosa


Achmad Muchlis

Institut Teknologi Bandung, Indonesia

Intan Muchtadi-Alamsyah


Adhitya Ronnie Effendi


Irawati


Agus Buwono

Institut Pertanian Bogor, Indonesia

Irwan Endrayanto A.


Agus Maman Abadi

Universitas Negeri Yogyakarta, Indonesia

Jailani


Agus Yodi Gunawan


Janson Naiborhu


Ari Suparwanto


Joko Lianto Buliali

Institut Teknologi Sepuluh Nopember,

Indonesia

Asep. K. Supriatna

Universitas Padjadjaran, Indonesia

Khreshna Imaduddin Ahmad S.


Atok Zulijanto


Kiki Ariyanti Sugeng

Universitas Indonesia

Azhari SN


Lina Aryati


Budi Nurani


M. Farchani Rosyid


Budi Santosa

Institut Teknologi Sepuluh Nopember, Indonesia

Mardiyana

Universitas Negeri Surakarta, Indonesia

Budi Surodjo


MHD. Reza M. I. Pulungan


Cecilia Esti Nugraheni

Universitas Katolik Parahyangan, Indonesia

Miswanto

Universitas Airlangga, Indonesia

Ch. Rini Indrati


Netty Hernawati

Universitas Lampung, Indonesia

Chan Basarrudin


Noor Akhmad Setiawan


Danardono


Nuning Nuraini


Dedi Rosadi


Rieske Hadianti


Deni Saepudin

Institut Teknologi Telkom, Indonesia

Roberd Saragih


Diah Chaerani


Salmah


Edy Soewono


Siti Fatimah

Universitas Pendidikan Indonesia

Edy Tri Baskoro


Soeparna Darmawijaya


Edi Winarko


Sri Haryatmi


Endar H Nugrahani


Sri Wahyuni


Endra Joelianto


Subanar


Eridani

Universitas Airlangga, Indonesia

Supama


Fajar Adi Kusumo


Suryanto


Frans Susilo

Universitas Sanata Dharma, Indonesia

Suyono

Universitas Negeri Jakarta, Indonesia

Gunardi


Tony Bahtiar


Hani Garminia


Wayan Somayasa

Universitas Haluoleo, Indonesia

Hartono


Widodo Priyodiprojo


Hengki Tasman


Wono Setyo Budhi


I Wayan Mangku


Yudi Soeharyadi


Indah Emilia Wijayanti


PREFACE

It is an honor and great pleasure for the Department of Mathematics –

Universitas Gadjah Mada, Yogyakarta – INDONESIA, to be entrusted by the

Southeast Asian Mathematical Society (SEAMS) to organize an international

conference every four years. Appreciation goes to those who have developed and

established this tradition of the successful series of conferences. The SEAMS -

Gadjah Mada University (SEAMS-GMU) 2011 International Conference on

Mathematics and Its Applications took place in the Faculty of Mathematics and

Natural Sciences of Universitas Gadjah Mada on July 12th – 15th, 2011. The

conference was the follow up of the successful series of events which have been

held in 1989, 1995, 1999, 2003 and 2007.

The conference has achieved its main purposes of promoting the exchange

of ideas and presentation of recent development, particularly in the areas of pure,

applied, and computational mathematics which are represented in Southeast Asian

Countries. The conference has also provided a forum of researchers, developers,

and practitioners to exchange ideas and to discuss future direction of research.

Moreover, it has enhanced collaboration between researchers from countries in the

region and those from outside.

More than 250 participants from over the world attended the conference.

They come from USA, Austria, The Netherlands, Australia, Russia, South Africa,

Taiwan, Iran, Singapore, The Philippines, Thailand, Malaysia, India, Pakistan,

Mongolia, Saudi Arabia, Nigeria, Mexico and Indonesia. During the four days

conference, there were 16 plenary lectures and 217 contributed short

communication papers. The plenary lectures were delivered by Halina France-

Jackson (South Africa), Jawad Y. Abuihlail (Saudi Arabia), Andreas Rauber (Austria),

Svetlana Borovkova (The Netherlands), Murk J. Bottema (Australia), Ang Keng Cheng

(Singapore), Peter Filzmoser (Austria), Sergey Kryzhevich (Russia), Intan Muchtadi-

Alamsyah (Indonesia), Reza Pulungan (Indonesia), Salmah (Indonesia), Yudi

Soeharyadi (Indonesia), Subanar (Indonesia) Supama (Indonesia), Asep K. Supriatna

(Indonesia) and Indah Emilia Wijayanti (Indonesia). Most of the contributed papers

were delivered by mathematicians from Asia.

We would like to sincerely thank all plenary and invited speakers who

warmly accepted our invitation to come to the Conference and the paper

contributors for their overwhelming response to our call for short presentations.

Moreover, we are very grateful for the financial assistance and support that we

received from Universitas Gadjah Mada, the Faculty of Mathematics and Natural

Sciences, the Department of Mathematics, the Southeast Asian Mathematical

Society, and UNESCO.

We would like also to extend our appreciation and deepest gratitude to all

invited speakers, all participants, and referees for the wonderful cooperation, the

great coordination, and the fascinating efforts. Appreciation and special thanks are

addressed to our colleagues and staffs who help in editing process. Finally, we

acknowledge and express our thanks to all friends, colleagues, and staffs of the

Department of Mathematics UGM for their help and support in the preparation

during the conference.

The Editors

October, 2012

CONTENTS

Title i Publisher and Copyright ii Managerial Boards iii Editorial Boards iv List of Reviewers v Preface vii Paper of Invited Speakers On Things You Can’t Find : Retrievability Measures and What to do with Them ……............... 1 Andreas Rauber and Shariq Bashir

A Quasi-Stochastic Diffusion-Reaction Dynamic Model for Tumour Growth ..……................... 9 Ang Keng Cheng

*-Rings in Radical Theory ...………………………………………………..................................................... 19 H. France-Jackson

Clean Rings and Clean Modules ...………………………………………................................................... 29 Indah Emilia Wijayanti

Research on Nakayama Algebras ……...…………………………………................................................. 41 Intan Muchtadi-Alamsyah

Mathematics in Medical Image Analysis: A Focus on Mammography ...…............................... 51 Murk J. Bottema, Mariusz Bajger, Kenny MA, Simon Williams

The Order of Phase-Type Distributions ..………………………………….............................................. 65 Reza Pulungan

The Linear Quadratic Optimal Regulator Problem of Dynamic Game for Descriptor System… 79 Salmah

Chaotic Dynamics and Bifurcations in Impact Systems ………………........................................... 89 Sergey Kryzhevich

Contribution of Fuzzy Systems for Time Series Analysis ……………….......................................... 121 Subanar and Agus Maman Abadi

Contributed Papers Algebra

Degenerations for Finite Dimensional Representations of Quivers ……................................... 137 Darmajid and Intan Muchtadi-Alamsyah

On Sets Related to Clones of Quasilinear Operations …...……………………………………………………. 145 Denecke, K. and Susanti, Y.

Normalized H Coprime Factorization for Infinite-Dimensional Systems …………………………… 159 Fatmawati, Roberd Saragih, Yudi Soeharyadi

Construction of a Complete Heyting Algebra for Any Lattice ………………………………………………. 169 Harina O.L. Monim, Indah Emilia Wijayanti, Sri Wahyuni

The Fuzzy Regularity of Bilinear Form Semigroups …………………………………………………..…………. 175 Karyati, Sri Wahyuni, Budi Surodjo, Setiadji

The Cuntz-Krieger Uniqueness Theorem of Leavitt Path Algebras ………………………………………. 183 Khurul Wardati, Indah Emilia Wijayanti, Sri Wahyuni

Application of Fuzzy Number Max-Plus Algebra to Closed Serial Queuing Network with

Fuzzy Activitiy Time ………………………………………………………………………………………………..…………… 193 M. Andy Rudhito, Sri Wahyuni, Ari Suparwanto, F. Susilo

Enumerating of Star-Magic Coverings and Critical Sets on Complete Bipartite Graphs………… 205 M. Roswitha, E. T. Baskoro, H. Assiyatun, T. S. Martini, N. A. Sudibyo

Construction of Rate s/2s Convolutional Codes with Large Free Distance via Linear System

Approach ………………………………………………………………………………………………........…………………….. 213 Ricky Aditya and Ari Suparwanto

Characteristics of IBN, Rank Condition, and Stably Finite Rings ………....................................... 223 Samsul Arifin and Indah Emilia Wijayanti

The Eccentric Digraph of n mP P Graph ………………………………………………………………….….……… 233 Sri Kuntarti and Tri Atmojo Kusmayadi

On M -Linearly Independent Modules ……………………………………………………………………….......... 241 Suprapto, Sri Wahyuni, Indah Emilia Wijayanti, Irawati

The Existence of Moore Penrose Inverse in Rings with Involution …........................................ 249 Titi Udjiani SRRM, Sri Wahyuni, Budi Surodjo

Analysis

An Application of Zero Index to Sequences of Baire-1 Functions ………….................................. 259 Atok Zulijanto

Regulated Functions in the n-Dimensional Space ……………………....................….....................… 267 Ch. Rini Indrati

Compactness Space Which is Induced by Symmetric Gauge ……..........................………………... 275 Dewi Kartika Sari and Ch. Rini Indrati

A Continuous Linear Representation of a Topological Quotient Group …................................ 281 Diah Junia Eksi Palupi, Soeparna Darmawijaya, Setiadji, Ch. Rini Indrati

On Necessary and Sufficient Conditions for L into 1 Superposition Operator ……...………... 289 Elvina Herawaty, Supama, Indah Emilia Wijayanti

A DRBEM for Steady Infiltration from Periodic Flat Channels with Root Water Uptake ……….. 297 Imam Solekhudin and Keng-Cheng Ang

Boundedness of the Bimaximal Operator and Bifractional Integral Operators in Generalized

Morrey Spaces …………………………………...................................................................................... 309 Wono Setya Budhi and Janny Lindiarni

Applied Mathematics

A Lepskij-Type Stopping-Rule for Simplified Iteratively Regularized Gauss-Newton Method.. 317 Agah D. Garnadi

Asymptotically Autonomous Subsystems Applied to the Analysis of a Two-Predator One-

Prey Population Model ............................................................................................................. 323 Alexis Erich S. Almocera, Lorna S. Almocera, Polly W.Sy

Sequence Analysis of DNA H1N1 Virus Using Super Pair Wise Alignment .............................. 331 Alfi Yusrotis Zakiyyah, M. Isa Irawan, Maya Shovitri

Optimization Problem in Inverted Pendulum System with Oblique Track ……………………………. 339 Bambang Edisusanto, Toni Bakhtiar, Ali Kusnanto

Existence of Traveling Wave Solutions for Time-Delayed Lattice Reaction-Diffusion Systems 347 Cheng-Hsiung Hsu, Jian-Jhong Lin, Ting-Hui Yang

Effect of Rainfall and Global Radiation on Oil Palm Yield in Two Contrasted Regions of Sumatera, Riau and Lampung, Using Transfer Function ..........................................................

365

Divo D. Silalahi, J.P. Caliman, Yong Yit Yuan

Continuously Translated Framelet............................................................................................ 379 Dylmoon Hidayat

Multilane Kinetic Model of Vehicular Traffic System ............................................................. 386 Endar H. Nugrahani

Analysis of a Higher Dimensional Singularly Perturbed Conservative System: the Basic

Properties ….............................................................................................................................. 395 Fajar Adi Kusumo

A Mathematical Model of Periodic Maintence Policy based on the Number of Failures for

Two-Dimensional Warranted Product ..…................................................................................. 403 Hennie Husniah, Udjianna S. Pasaribu, A.H. Halim

The Existence of Periodic Solution on STN Neuron Model in Basal Ganglia …………………………. 413 I Made Eka Dwipayana

Optimum Locations of Multi-Providers Joint Base Station by Using Set-Covering Integer

Programming: Modeling & Simulation ….................................................................................. 419 I Wayan Suletra, Widodo, Subanar

Expected Value Approach for Solving Multi-Objective Linear Programming with Fuzzy

Random Parameters …..................................................................................................... ......... 427 Indarsih, Widodo, Ch. Rini Indrati

Chaotic S-Box with Piecewise Linear Chaotic Map (PLCM) ...................................................... 435 Jenny Irna Eva Sari and Bety Hayat Susanti

Model of Predator-Prey with Infected Prey in Toxic Environment .......................................... 449 Lina Aryati and Zenith Purisha

On the Mechanical Systems with Nonholonomic Constraints: The Motion of a Snakeboard

on a Spherical Arena ………………………………………..……………………................................................ 459 Muharani Asnal and Muhammad Farchani Rosyid

Safety Analysis of Timed Automata Hybrid Systems with SOS for Complex Eigenvalues …….. 471 Noorma Yulia Megawati, Salmah, Indah Emilia Wijayanti

Global Asymptotic Stability of Virus Dynamics Models and the Effects of CTL and Antibody Responses ………………………………………………………………………………….……………………………………….. 481

Nughtoth Arfawi Kurdhi and Lina Aryati

A Simple Diffusion Model of Plasma Leakage in Dengue Infection …………………………..………… 499 Nuning Nuraini, Dinnar Rachmi Pasya, Edy Soewono

The Sequences Comparison of DNA H5N1 Virus on Human and Avian Host Using Tree

Diagram Method …………………………………..……………………………………………………………..…………….. 505 Siti Fauziyah, M. Isa Irawan, Maya Shovitri

Fuzzy Controller Design on Model of Motion System of the Satellite Based on Linear Matrix

Inequality …………………………………………………………………….…………….…………..…….……………………..

515 Solikhatun and Salmah

Unsteady Heat and Mass Transfer from a Stretching Suface Embedded in a Porous Medium

with Suction/injection and Thermal Radiation Effects…………………………………………………………. 529 Stanford Shateyi and Sandile S Motsa

Level-Set-Like Method for Computing Multi-Valued Solutions to Nonlinear Two Channels

Dissipation Model ……………………………………………………………………………….....………………………….. 547 Sumardi, Soeparna Darmawijaya, Lina Aryati, F.P.H. Van Beckum

Nonhomogeneous Abstract Degenerate Cauchy Problem: The Bounded Operator on the

Nonhomogen Term ………………………………………………………………………..…………………..……………… 559 Susilo Hariyanto, Lina Aryati,Widodo

Stability Analysis and Optimal Harvesting of Predator-Prey Population Model with Time

Delay and Constant Effort of Harvesting ………………………………………………………..……………………. 567 Syamsuddin Toaha

Dynamic Analysis of Ethanol, Glucose, and Saccharomyces for Batch Fermentation ………….. 579 Widowati, Nurhayati, Sutimin, Laylatusysyarifah

Computer Science, Graph and Combinatorics

Survey of Methods for Monitoring Association Rule Behav ior ............................. 589 Ani Dijah Rahajoe and Edi Winarko

A Comparison Framework for F ingerprint Recognition Methods . . . . . . . . . . . . . . . . . . . . 601 Ary Noviyanto and Reza Pulungan

The Global Behav ior of Certa in Turing System …………………. ..................................... 615 Janpou Nee

Logic Approach Towards Formal Verif ication of Cryptographic Protocol . . . . . . . . . 621 D.L. Crispina Pardede, Maukar, Sulistyo Puspitodjati

A Framework for an LTS Semantics for Promela ….. . . . . . . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . 631 Suprapto and Reza Pulungan

Mathematics Education

Modelling On Lecturers’ Performance with Hotteling-Harmonic-Fuzzy …................................ 647 H. A. Parhusip and A. Setiawan

Differences in Creativity Qualities Between Reflective and Impulsive Students in Solving

Mathematics …………………......................................……...................…………………........................ 659 Warli

Statistics and Finance

Two-Dimensional Warranty Policies Using Copula ...…………...................………………………………. 671 Adhitya Ronnie Effendie

Consistency of the Bootstrap Estimator for Mean Under Kolmogorov Metric and Its

Implementation on Delta Method ……................................………………….................................. 679 Bambang Suprihatin, Suryo Guritno, Sri Haryatmi

Multivariate Time Series Analysis Using RcmdrPlugin.Econometrics and Its Application for

Finance ..................................................................................................................................... 689 Dedi Rosadi

Unified Structural Models and Reduced-Form Models in Credit Risk by the Yield Spreads …. 697 Di Asih I Maruddani, Dedi Rosadi, Gunardi, Abdurakhman

The Effect of Changing Measure in Interest Rate Models …….…....................…....................... 705 Dina Indarti, Bevina D. Handari, Ias Sri Wahyuni

New Weighted High Order Fuzzy Time Seriesfor Inflation Prediction ……................................ 715 Dwi Ayu Lusia and Suhartono

Detecting Outlier in Hyperspectral Imaging UsingMultivariate Statistical Modeling and

Numerical Optimization ………………………………………...........................................……………......... 729 Edisanter Lo

Prediction the Cause of Network Congestion Using Bayesian Probabilities ............................. 737 Erwin Harapap, M. Yusuf Fajar, Hiroaki Nishi

Solving Black-Scholes Equation by Using Interpolation Method with Estimated Volatility……… 751 F. Dastmalchisaei, M. Jahangir Hossein Pour, S. Yaghoubi

Artificial Ensemble Forecasts: A New Perspective of Weather Forecast in Indonesia ............... 763 Heri Kuswanto

Second Order Least Square for ARCH Model …………………………....................………………............. 773 Herni Utami, Subanar, Dedi Rosadi, Liqun Wang

Two Dimensional Weibull Failure Modeling ……………………..…..................……….......................... 781 Indira P. Kinasih and Udjianna S. Pasaribu

Simulation Study of MLE on Multivariate Probit Models …......................................................... 791 Jaka Nugraha

Clustering of Dichotomous Variables and Its Application for Simplifying Dimension of

Quality Variables of Building Reconstruction Process ............................................................. 801 Kariyam

Valuing Employee Stock Options Using Monte Carlo Method ……………................................. 813 Kuntjoro Adji Sidarto and Dila Puspita

Classification of Epileptic Data Using Fuzzy Clustering .......................................................... 821 Nazihah Ahmad, Sharmila Karim, Hawa Ibrahim, Azizan Saaban, Kamarun Hizam

Mansor

Recommendation Analysis Based on Soft Set for Purchasing Products ................................. 831 R.B. Fajriya Hakim, Subanar, Edi Winarko

Heteroscedastic Time Series Model by Wavelet Transform ................................................. 849 Rukun Santoso, Subanar, Dedi Rosadi, Suhartono

Parallel Nonparametric Regression Curves ............................................................................ 859 Sri Haryatmi Kartiko

Ordering Dually in Triangles (Ordit) and Hotspot Detection in Generalized Linear Model for

Poverty and Infant Health in East Java ……................................…………………………………………. 865 Yekti Widyaningsih, Asep Saefuddin, Khairil Anwar Notodiputro, Aji Hamim Wigena

Empirical Properties and Mixture of Distributions: Evidence from Bursa Malaysia Stock

Market Indices …………………....................................................................................................... 879 Zetty Ain Kamaruzzaman, Zaidi Isa, Mohd Tahir Ismail An Improved Model of Tumour-Immune System Interactions …………………………………………….. 895 Trisilowati, Scott W. Mccue, Dann Mallet

Paper of Invited

Speakers

2010 Mathematics Subject Classification : 03B50, 03C65 , 62H25.

647

Proceedings of ”The 6th SEAMS-UGM Conference 2011” Mathematics Education, pp. 647 - 658.

MODELLING ON LECTURERS’ PERFORMANCE WITH

HOTTELING-HARMONIC-FUZZY

H.A PARHUSIP AND A. SETIAWAN

Abstract. This paper is a brief research on modelling of lecturers' performance with modified

Hotelling-Fuzzy. The observation data is considered to be as a fuzzy set which is obtained from

students survey at Mathematics Department of Science Mathematics Faculty in SWCU in the year

2008-2009. The modified Hotelling relies on a harmonic mean instead of an arithmetic mean which is normally used in a literature. The used charaterization function is an exponential function which

identifies the fuzziness. The result will be a general measurement of lecturers' performance. Based on

the 4 variables used in the analysis (Utilizes content scope and sequence planning, Clearness of assignments and evaluations, Systematical in lecturing, and Encourages students attendance) we

obtain that lecturers' performance are fair, poor, fair, and poor respectively. Keywords and Phrases : Hotelling, harmonic mean, fuzzy, exponential function, characteristic function.

1. INTRODUCTION

University is one of post investments for human development in education which most

activities based on the learning process during in a university. This process relies on

interaction between lecturers and students. After some period of time, students may adopt

some knowledge which may determine their futures. Therefore lecturers require to transfer

their knowledge such that students capable to reproduce their knowledge obtained in their

universities as a support for themselves in their working places.

Since a lecturer becomes an important agent in the effort of human development in a

university, some necessary conditions must be satisfied as a lecturer. This paper will propose

some results on lecturers' performance based on students survey. Students evaluate a lecturer

by some questions (some are adopted from a literature given in a web which are listed on

Table 1 and these are considered the same items in Indonesian that have been used as

648 H.A. PARHUSIP AND A. SETIAWAN

instruments in the observation for this paper.

According to Yusrizal [1], the items are validated which is not done in this paper. We

have used these items to evaluate lecturers‟ performance. Each item will be quantified as 0,

1, 2 or 3 which means poor, fair, satisfactory and good respectively. Since all students did not

meet with all lecturers in 1 semester then the research is simply made as a general evaluation

for all lecturers in the

Department.

The used data here are obtained from students survey in Mathematics Department in

Science and Mathematics Faculty, Satya Wacana Christian University in the year 2008-2009.

Furthermore, the lecturers‟ performance can not be exactly in one of the values 0,1, 2 or 3.

Therefore data can be considered nonprecise and hence we refer to fuzzy in presenting the

analysis.

The remaining paper is organized as follows. The used theoretical background is

shown in Section 2. Procedures to analyze the obtained data are shown on Section 3. The

analysis is then shown in Section 4 and finally some conclusion and remark are written in the

last Section of this paper.

TABLE I

QUESTIONS FOR STUDENTS OBSERVATION TO EVALUATE A

LECTURER HELD DURING IN A CLASS

No Questions

1 Utilizes content scope and sequence planning

2 Clearness of assignments and evaluations

3 Systematical in lecturing

4 Encourages students attendance

5 Demonstrates accurate and current knowledge in subject field

6 Creates and maintains an environment that supports learning

7 Clearness to answer the question from a student

8 Attractiveness of the lecture

9 Clearness of purpose on each assignment

10 Lecturer‟s competence on stimulating students excitement

11 Lecturer‟s competence on delivering idea

12

Does teacher have available time to help students

for the related topic outside the class?

13 Effectiveness of time used in the class

14 Provides relevant examples and demonstrations to illustrate concepts and skills

15 The quality of lecturer‟s feedback due to the given assignment.

16 Score the overall the lecturer‟s performance

maxwell, G = gauss, Oe = oersted; Wb = weber, V = volt, s = second,

T = tesla, m = meter, A = ampere, J = joule, kg = kilogram, H = henry.

Modelling On Lecturers‟ Performance with Hotteling-Harmonic-Fuzzy 649

2. MODELLING HOTELLING-HARMONIC-FUZZY

ON LECTURERS’ PERFORMANCE

2.1 Review Stage. Some authors have proposed a quality evaluation based on some

mathematical approaches such as using fuzzy set. Teacher performance is studied with fuzzy

system [2] to generate an assesment criteria. In complex system such as cooling process of a

metal, the combined PCA (Principal Component Analysis) algorithm and 2T -Hotteling are

used to define quality index after all data are normalized in a range -1 and 1 [3]. The other

instrument for lecturers‟ performance have been developed and validitated using variable

analysis . Lecturers‟ strictness are also estimated using intuitionistic fuzzy [4]. There are

several standards can also be used to indicate a quality of a teacher which are more a

qualitative approach which are easily found through internet. In a strategic planning, SWOT

is one of the most pointed references. In SWOT analysis some uncertainties can be

encountered and therefore using fuzzy will renew the classical SWOT by presenting the

internal and external variables in the sense of fuzzy as shown by Ghazinoory, et.all[5].

It is assumed that the given data contain non-precision. Thus the representation data

will be as a fuzzy set [6]. The fuzzy data are indicated by a characteristic function. There are

some well-known characteristic functions. In this research, we apply an exponential

characteristic function ,i.e

2

21

1

),(

,1

),(

)(

mxxR

mxm

mxxL

x with

1

1

1exp)(

q

a

mxxL

and

2

2

2exp)(

q

a

mxxR . (1)

There are 6 parameters in Eq.(1), i.e

111 ,, qam and 222 ,, qam that must be determined. An

illustration of this characteristic function is depicted in Figure 1.

Figure 1 shows us that a single value x*=0.5 can be considered that the value may vary

in the interval 5.1*25.0 x . This allows us to tolerate that the given value is not exactly 0.5

but it can be in this interval which indicates its non-precision. However, if this tolerance is

too big, one introduces a parameter which varies in 10 such that our tolerance can

be chosen freely. For example, if 2.0 then the tolerance for saying 5.1*25.0 x is

reduced as illustrated in Figure 1. We may say that 85.0*35.0 x .

Figure 1. )(x with the given data is 0.5 (denoted by star on the peak) , and the value of

each parameter 111 ,, qam is 0.4980, 0.1 , 2 respectively and the value of each parameter

222 ,, qam is 0.2, 0.502 , 0.8 respectively. The tolerance for x* is reduced. The

corresponding cut is also shown as a horizontal line for 2.0 .

A problem appears here that the determination of parameters becomes time consuming


TABLE II

STUDENTS SURVEY FOR 6 LECTURERS BASED ON 4 VARIABLES

THE SYMBOL jD INDICATES THE NAME OF J-TH LECTURER.

No Questions D1 D2 D3 D4 D5 D6

Row

Harmonic

Average

1

Utilizes content

scope and sequence

planning 3.0 1.67 1.89 1.71 1.62 2.2

1.92

2

Clearness of

assignments and

evaluations 3.0 1.51 2.00 1.57 1.58 2.05 1.84

3 Systematical in lecturing 3.0 1.72 1.11 1.29 2.01 1.83 1.65

4

Encourages students

attendance 2.0 1.41 1.78 1.14 2.48 1.83 1.69

Column-Harmonic

Average 2.75 1.57 1.61 1.39 1.86 1.97

1.77

TOTAL

AVERAGE

for each value. Up to now, there exists no optimization procedure to find the best values of

parameters here. Therefore we may vary parameters based on the given data. Let us consider

by this following example.

Example 1: Let the given data be a vector x

[0.5 0.67 0.83 1. 1.17 1.33 1.5 1.67

1.83 2]T

. Let the vector x

be an element in the observation space XM . We want to

express this vector in the sense of fuzzy. By trial and error we try to use the value of each

parameter. This means that each value in the vector has its own characteristic function. We

will also define the average value of this vector by the harmonic mean and the characteristic

function of this average value. The used parameters for all characteristic functions are

2,1.0 211 qqa , 2.02 a . These parameters are chosen freely. The illustration of

characteristic functions is depicted in Figure 2a. The values of 1m and 2m are taken from

the given vector. The characteristic function of the harmonic average is obtained as

309.1),(

039.1,1

039.1),(

)(

xxR

x

xxL

x with

2

1.0

039.1exp)(

xxL and

2

2.0

039.1exp)(

xxR .

Thus the characteristic function is defined after the harmonic mean is obtained. The statistical

test for multivariate data is based on the 2T named as Hotelling [7] in a classical sense that

)( 0

1

0

2

XSXnT

(2)

whereT

pxxxX )......,,,( 21 is a vector of an arithmetic mean of each variable,

n

i

kik xn

x1

1 is an arithmetic mean for the k-th item. It is also known from Wikipedia or [7]


that 2

1,

2 ~ npT or pnpF

np

pn

,

2 ~)1( . (3)

Figure 2 The illustration of all characteristic functions with 11 ,qa are 0.1 and 2. and

22 , qa are 0.2 and 2.The value of 1m and 2m are taken from the given vector.

Since 2 is distributed as

)()(

)1(, pnpF

pn

pn

, then 2 can be used for hypotheses. A test of

hypothesis: 00 :

H versus

01 :

H at the level of significance, reject 0H in favor of

1H if [7] )(

)(

)1(,

2 pnpFpn

pn

. (4)

Thus the control limit of the 2 control chart can be formed as [8]

)()(

)1(, pnpF

pn

pnUCL

and LCL=0. (5)

There is a reason proposed in [8] for LCL=0 , but it ignores here. It is convenient to refer to

the simultaneous intervals for the confident interval of each j , j=1,…,p as ([7](page 193))

jxn

sF

pn

pnx

n

sF

pn

pn jj

pnpjj

jj

pnp )()(

)1()(

)(

)1(,,

, j=1,…,p.

Equation (5) allows us to simplify the statement above as

n

sUCLx

n

sUCLx

jj

jj

jj

j , j =1,…,p. (6)

In this paper, we will replace an arithmetic mean by a harmonic mean in the sense of

fuzzy. The harmonic mean is used to increase the precision of average quantity (since we

know that arithmetic mean ≥ harmonic mean) and the harmonic mean is formulated as

n

i i

H

x

nx

1

1

The harmonic mean will be useful if we have highly oscillated data. Additionally, there exists

no literature so far in doing this way which shows an originality of this paper though the

number of the used data is considerable small. The used covariance matrix S is in the usual

way, i.e

pppp

p

p

SSS

SSS

SSS

S

,2,1,

,22221

,11211

....

...............

...

... with

n

i

qqippipq xxxxn

S1

))((1


for p, q = 1, 2, ...., M.

The example 2 shows the idea of Hotteling-harmonic-Fuzzy. Note that the number of samples

for each item (variable) is 6.

Example 2. Suppose the given data are shown in Table 2 with each number in the entry is an

average from the jn -number of students for each lecturer. We assume that the result is

independent with the number of students in each class.

If each number in the Table 2 presented as a characteristic function then there will be

many parameters must be determined. Hence we define the characteristic function of the

harmonic mean of each item. The resulting of characteristic-harmonic mean is depicted in

Figure 2.

Furthermore, the harmonic mean of each lecturer in the whole items and for each item

for all lecturers are shown in Table 2. Based on the average quantity, one has 1.77 which is

not precisely in the original values (0, 1, 2, and 3). Since it closes to 2, the lecturers‟

performance is considered satisfactory. This paper will evaluate into more rigorously.

The covariance matrix in the classical sense can be obtained by function cov in

MATLAB, one yields

0.2422 0.2008 0.1185 0.0939

0.2008 0.4447 0.2759 0.2748

0.1185 0.2759 0.3180 0.2921

0.0939 0.2748 0.2921 0.2776

S

. Furthermore, using harmonic mean, the matrix

covariance sample hS is obtained as

0.2157 0.1881 0.1114 0.0889

0.1881 0.4015 0.2488 0.2449

0.1114 0.2488 0.2765 0.2531

0.0889 0.2449 0.2531 0.2396

hS

. In the rest paragraphs

we will compute all related formulas using harmonic mean.

One may also observe that covariance matrix hS is positive definite by computing its

eigenvalues which are all positive (0.0015, 0.0717, 0.1710 and 0.8891).We have also

compute the invers matrix of hS as

2192.199823.157142.540196.67

9823.154292.203587.533350.71

7142.543587.536743.2744680.324

0196.67335.714680.3240665.395

1

hS. Finally one

needs to consider the Hotelling to involve in this paper which mostly taken from literature

(Johnson and Wichern, 2007) and reformulate it using harmonic mean.

The hypothesis test with :0H T1,2,2,30 against :1H T1,2,2,31 at level

significance 10.0 is employed here. The observed 2 is

2 =238.9666. Comparing the

observed 2 =238.9666 with the critical value (using Eq.4)

)10.0(

2

)4(5)10.0(

)(

)1(2,4, FF

pn

pnpnp

10(9.2434)=92.434.

Thus we get 2 > 92.434 and consequently we reject 0H at the 10% level of significance.


For completeness, The value of 10.0(2,4F ) is computed using function in MATLAB and type

as finv(0.90,4,2). Finally, we try fuzziness takes place in this hypothesis.

2.2 Hotelling-fuzzy. We agree that hypothesis is proposed by assuming that the given

0 is known. In this case we should present the example T1,2,2,30

with its

characteristics and using the previous procedures shown in Example 1, we directly obtain the

characteristics as depicted in Figure 3. Since we allow 0

as in the Figure 3, then we will

also have 2 as many as the number of points to define each characteristic function. Thus we

have a vector value of 2 which is shown in Figure 3.

Figure 3 The illustration of all characteristic functions of 0

with 2,1.0 11 qa and

2,2.0 22 qa . The values of 1m and 2m are taken from the harmonic mean of each item on Table

2. Note that there exists 4 characteristic functions but the second and the third are in the same curves.

Be aware that the given 0

in the sense in fuzzy means that all points in horizontal axis

(x) with )(x > 0. Practically means that we need to search the index of each characteristics

function (since each function is discritized) and hence find the corresponded value of x. What

will be a problem here ?. We need always a set of 0

(with the number of element is the

same as the predicted one) . Let us study for )(x = 0.6 for one of possible memberships of

0

. We draw a horizontal line that denotes )(x = 0.6 to find intersection points such that we

can draw the vertical line (denoted by an arrow in Figure 4).

The vertical lines denote the possible new sets of 0

and we always have two values of

each given characteristic value (membership) that may act as the bounds of each fuzzy-0

.

Let us denote these intervals as joI ,

where index j denotes the j-th vector of 0

. Thus to

make the computation simpler, we choose the inf( joI ,

) and sup( joI ,

) to continue our

hypothesis. The used number of points will influence the obtained fuzzy-0

.


Figure 4 The illustration of all

0

taken from the characteristic function with 2,1.0 11 qa and

2,2.0 22 qa . The values of 1m and 2m are taken from the harmonic mean of each item on Table

2.

Due to numerical techniques, the idea to use the inf( joI ,

) and sup( joI ,

) still can not

be implemented. Instead, we can do manually, but it will be time consuming if we have a

large number of points. Thus we are left this problem for the next research. In this paper we

apply 0

in the hypothesis as the usual procedure in a classical sense.

Example 3. Suppose that we have T1,2,2,30 and the resulting characteristic function is

shown in Figure 4. If )(x =0.6, we get intervals of each value of T1,2,2,30 as shown in

Figure 4. Practically, there exists no )(x =0.6 precisely since its value is constructed by

Equation 1. Manually, by drawing the horizontal line and the vertical line as shown in

Figure 4, we have approximations of inf( joI ,

) and sup( joI ,

). We propose that

inf( joI ,

)=[0.8, 1.9, 1.9, 2.9] and sup( joI ,

).=[1.1, 2.1, 2.1, 3.2]. These results lead to the

same conclusion that :0H T1,2,2,30 is again rejected at the 10% level of significance.

On the other hand, one may determine all sets of 0

to find a 100%(1- ) confidence

region for the mean of a p-dimensional normal distribution such that

)( 0

1

0

2

XSXnT

)()(

)1(, pnpF

pn

pn

(7)

provided the positive definite matrix covariance S. The complete observation will be

examined in Section 4. One may introduce Principal Component Analysis (PCA) of Variable

Analysis to reduce the number of variables if necessary.

3. RESEARCH METHOD

The used data are taken from the students‟ survey. We assume that the observations

are independent with the given lecturers and the number of students on each class. Table 1

contains questions which are used to evaluate lecturers‟ performance. The result is shown in

Table 3.


3.2 Calculate harmonic mean of variable

The harmonic mean of each variable is computed using formula in Section 2.

3.3 Covariance matrix is computed which its average formula is taken from the harmonic

mean. If the covariance matrix is singular, one needs to reduce the number of variables by

principal component analysis.

3.4 Given the predicted 0

then define the characteristic function of each element of 0

.

3.5 Compute the value of 2 for each set of 0

taken from the characteristic function

.

4. RESULT AND ANALYSIS

Evaluation is based on Table 2 which means that we have p=16 as the number of

variables. Unfortunately, the covariance matrix is almost singular that causes the invers

matrix is badly obtained. One may observe by computing its determinant which tends to 0

(O( 17010 )). According to [7](page 110), this is caused by the number of samples is less than

the number of variables which happens in any samples. This means that some variables

should be removed from the study. The corresponding reduced data matrix will then lead to a

covariance matrix of full rank and nonzero generalized variance. Thus as mention in Section

TABLE III

STUDENTS SURVEY FOR 6 LECTURERS BASED ON 16 VARIABLES

THE SYMBOL jD INDICATES THE NAME OF J-TH LECTURER.

No D1 D2 D3 D4 D5 D6

1 4.00 2.67 2.89 2.71 2.62 3.20

2 4.00 2.51 3.00 2.57 2.58 3.05

3 4.00 2.72 2.11 2.29 3.01 2.83

4 3.20 2.41 2.78 2.14 3.48 2.83

5 4.00 2.59 2.11 2.29 3.14 2.88

6 3.20 2.79 2.89 2.29 3.48 2.85

7 4.00 2.41 3.11 2.43 3.35 3.12

8 3.73 2.33 2.33 2.14 2.88 2.67

9 4.00 2.21 3.00 2.71 2.75 3.02

10 3.73 2.33 2.89 2.29 2.67 3.03

11 3.73 2.62 3.00 2.71 3.40 2.92

12 4.00 2.31 2.56 2.86 3.18 2.43

13 3.47 2.51 3.56 2.86 3.05 3.38

14 3.47 2.23 3.22 2.71 3.05 3.20

15 3.73 2.36 3.11 2.29 2.92 2.92

16 3.73 2.64 2.78 2.57 3.35 3.12


2, we need to use principal components analysis (PCA).

The standard PCA was employed, we compute the proportion of total variance

accounted by the first principal component is

p

j

j

1

1

= 0.7943. Continuing, the first two

components accounted for a proportion

p

j

j

1

21

= 0.8933 of the sample variance. After

the sixth eigenvalue, we observe that only six variables with nonzero proportions. Thus we

will only consider the 6 variables in the analysis. Additionally, we also observe that

0j ,j=7,…,p whereas 1 3.1702, 2 0.3948, 3 0.2859, 4 0.1110,

5 0.0274, and 6 0.0016. It is reasonable to choose the first six variables.

Unfortunately, the singular covariance matrix still exists. To handle this problem, the pseudo-

invers matrix is applied. Again, one needs to concern Equation (2) that the number of

samples is not allowed to be the same as the number of variables which leads to the division

by zero. Since the eigenvalues represent variables variances, we select only the first four

variables such that n > p. In this case, we are led to the result in Section 2.

The control limit of the 2 control chart can be obtained by using Eq.(4), one yields

)10.0()46(

4)12(2,4FUCL

= 92.4342 and LCL=0.

Section 2 suggests us to compute the confidence intervals of the reduced data and we have the

following results

8552.40453.1 1 ; 9181.48337.0 2 ;

1428.52776.0 3 ; 5201.49465.0 4 .

Up to now, the conclusion for lecturers‟ performance qualitatively is not yet concluded

which is practically useful for application. These intervals show us that all observed values

are in the intervals. Finally, we suggest that the evaluation is proceeded as follows :

1. Compute the UCL= )()(

)1(, pnpF

pn

pn

, n= the number of the observed lecturers, p =

the number of variables used in the evaluation (the number of items used to survey).

2. Compute the harmonic mean of each variable.

3. Compute the interval confidence of harmonic mean of each variable using the

Equation (6).

4. Divide each interval by the number of categories of qualities as the number of

subintervals which each subinterval corresponds to each quality. These subintervals

can be presented their characteristics functions as shown in Figure (5). Since we

have 4 categories for qualities (poor, fair, satisfactory and good) then each interval

will be divided into four subintervals. Thus we have categories for each variable in

the intervals given in Table 4.


TABLE IV

INTERVAL CONFIDENCE FOR EACH VARIABLE

VS EACH QUALITY OF LECTURERS‟ PERFORMANCE

TABLE V

QUALITY OF LECTURERS‟ PERFORMANCE

BASED ON THE DATA SURVEY 2008-2009.

Name

of parameter

Utilizes

content scope

and

sequence

planning

(2.75)

Clearness

of assignments

and

evaluations

(1.57)

Systematical

in lecturing (1.61)

Encourages

students attendance

(1.39)

Fair Poor Fair Poor

Fig 5 Characteristic function of each subinterval of 1 .


5. CONCLUSION

This paper proposes an evaluation of lecturers‟ performance by Hotelling-harmonic-

fuzzy. The harmonic mean is used to increase the precision of averaging and replace the

arithmetic mean in the covariance matrix. The Hotelling is employed here to find the

confidence interval of mean value of each variable. Using 16 variables, we obtain singular

covariance matrix such that only 4 variables must be considered in the analysis through

principal component analysis and we must have the number of variables (p) must be less than

the number of observations (n).

The studied data are the students‟ survey in Mathematics Department of Science and

Mathematics Faculty, Satya Wacana Christian University in the year 2008-2009. Based on

the 4 variables used in the analysis, we obtain that lecturers‟ performance are minimum (

fair).

Acknowledgement. The authors gratefully acknowledge T. Mahatma,SPd, M.Kom for

assistance with English in this manuscript, and the Research Center in Satya Wacana

Christian University for financial support of this work.

References

[1] YUSRIZAL AND HALIM, A.,”Development and Validation of an Intrument to Access the Lecturers‟ Performance

in the Education and Teaching Duties”, Jurnal Pendidikan Malaysia 34(2) 33-47,2009.

[2] JIAN MA AND ZHOU, D. „‟Fuzzy Set Approach to the Assessment of Student-Centered Learning‟‟, IEEE

Transaction on Education, Vo. 43, No. 2,2000.

[3] BOUHOUCHE, S., BOUHOUCHE, M. LAHRECHE, , A. MOUSSAOUI, AND J. BAST, „‟Quality Monitoring Using

Principal Component Analysis and Fuzzy Logic Application in Continuous Casting Process‟‟, American

Journal of Applied Sciences, 4 (9):637-644,ISSN 1546-9239, 2007.

[4] SHANNON, A, E. SOTIROVA, K. ATANASSOV, M. KRAWCZAK, P.M, PINTO, T. KIM, „‟Generalized Net Model of

Lecturers‟s Evaluation of Student Work With Intuitionistic Fuzzy Estimations‟‟, Second International

Workshop on IFSs Banska Bystrica, Slovakia, NIFS 12,4,22-28,2006.

[5] GHAZINOORY, A., ZADEH, A.E., “Memariani, Fuzzy SWOT analysis, Journal of Intelligent & Fuzzy Systems”

18, 99-108, IOP Press, 2007.

[6] VIERTL, R., “Statistical Methods for Non-Precise Data”, CRC Press,Tokyo, 1996.

[7] JOHNSON, R.A. AND WICHERN,D.W Applied Multivariate Statistical Analysis, 6th ed. Prentice Hall, ISBN 0-13-

187715-1, 2007.

[8] KHALIDI, M.S.A, „‟Multivariate Quality Control, Statistical Performance and Economic Feasibility‟‟,

Dissertation, Wichita State University, Wichita-Kansas ,2007.

HANNA ARINI PARHUSIP

FSM-Satya Wacana Christian University.

e-mail: [email protected]

ADI SETIAWAN

FSM-Satya Wacana Christian University.

e-mail: [email protected]

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