# KUMPULAN ABSTRAK PUBLIKASI

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```					                      KUMPULAN ABSTRAK PUBLIKASI

1. Terbit di Proceeding of the International Modelling and Simulation (MODSIM),
page 161-2166, Tahun 2001.

Heuristics Approach for The Degree Constrained
Minimum Spanning Tree Problems

L. Caccetta and Wamiliana
Dept. of Mathematics and Statistics,
Curtin University of Technology, Perth, Western Australia
caccetta@maths.curtin.edu.au

Abstract The Degree Constrained Minimum Spanning Tree Problem (DCMST) arises naturally in
communication networks where the degree of a vertex represents the number of line interfaces available at a
terminal (center). Because of its NP-completeness, a number of heuristics have been proposed. In this paper we
propose two new heuristics: one based on the method of Tabu Search (CW1) and other based on a penalty
function approach. Our heuristics are implemented and extensively tested on 1200 simulated problems. The
computational results support our methods.

Keywords: Minimum spanning tree; Tabu search; Degree constrained

2. Terbit di Jurnal Sains dan Teknologi Volume 8, hal 1 – 12, 2002

Modified Penalty Algorithms for The Degree Constrained
Minimum Spanning Tree Problems

Wamiliana
Dept. of Mathematics, Faculty of Mathematics and Natural sciences
The University of Lampung, Bandarlampung, Indonesia
wamiliana@mail.com

Abstract The Degree Constrained Minimum Spanning Tree Problem is concerned with finding, in a given edge
weighted graph G (all weights are non-negative), the minimum weight spanning tree T satisfying specified
degree restrictions on the vertices. This problem arises naturally in communication networks where the degree
of a vertex represents the number of line interfaces available at a terminal (center). Since, apart from some
trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been
proposed. In this paper we propose a heuristic based on a penalty function approach. Our heuristic is
implemented and extensively tested on simulated problems. The computational results will be discussed.

Keywords: Minimum spanning tree; Iterative method ; Degree constrained

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3. Disampaikan pada Australian Symposium and Optimization Day, Perth , 2002.

Heuristics Approach for The Degree Constrained
Minimum Spanning Tree Problems

L. Caccetta and Wamiliana
Dept. of Mathematics and Statistics,
Curtin University of Technology, Perth, Western Australia

Abstract The Degree Constrained Minimum Spanning Tree Problem is concerned with finding, in a given edge
weighted graph G (all weights are non-negative), the minimum weight spanning tree T satisfying specified
degree restrictions on the vertices. This problem arises naturally in communication networks where the degree
of a vertex represents the number of line interfaces available at a terminal (center). Since, apart from some
trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been
proposed. In this paper we propose two new heuristics: one based on the method of Tabu search and other based
on a penalty function approach. For comparative analysis, we test our methods on some benchmark problems.
The computational results support our methods.

Keywords: Minimum spanning tree; Tabu search; Degree constrained

4. Terbit di Jurnal Penelitian Sains No. 14, halaman 44 – 59, tahun 2003

Penentuan Facet Suatu Polytope Knapsack DenganMinimal Cover

Wamiliana
Jurusan Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam
Universitas Lampung

Abstrak. Dalam representasi suatu polytop, facet memainkan peranan yang penting sebab
karakretisasi dari facet akan merepresentasikan polytop. Facet adalah bentuk persamaan dari
pertidaksamaan-pertidaksamaan yang sah (valid inequalities) dari suatu polytop, sedangkan
dari valid inequalities tersebut dapat ditentukan ‘cover’ dari polytop tersebut yang berupa
‘strong cover’ maupun ‘minimal cover’. Setiap strong cover mendefinisikan facet, tetapi
tidak demikian halnya dengan minimal cover. Dalam tulisan ini akan diberikan
bagaimanakan cara agar suatu minimal cover dari polytop P dapat menjadi facet dari polytop
tersebut.

Kata kunci: polyhedron, polytop, knapsack, valid inequalities, facet, cover

5. Terbit di Journal Of Quantitative Methods, Volume 1, page 20-32, 2004

A First Level Tabu Search Heuristic

2
for the Degree Constrained Minimum Spanning Tree Problem

L. Caccetta , and Wamiliana1
1)   Dept. Math and Stat, Curtin University of Technology, Australia
2)   Department of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University

Abstract

In this paper we proposed a new heuristics based on Tabu search Method
that incorporated only short term memory for solving the Degree
Constrained Minimum Spanning Tree Problem. The Degree Constrained
Minimum Spanning Tree Problem is concerned with finding a minimum
weight spanning tree T in a weighted graph G (all weights are non negative)
with the requirement that the vertices satisfy a prescribed degree restriction
in T. This problem arises naturally in communications networks where the
degree restriction represents the number of line interfaces available at a
terminal. We implement and test our heuristic extensively on 2160
simulated problems and on some benchmark problems. Our computational
results support to the use of the method.

6. Terbit di Procceding of South East Asia Mathematics Society (SEAMS) , page
133- 140 , Tahun 2004

Tabu Search Based Heuristics for the Degree Constrained
Minimum Spanning Tree Problem
Wamiliana
Dept. of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University,
Indonesia
wamiliana@mail.com

Louis Caccetta
Dept. of Mathematics and Statistics, Curtin University of Technology, Perth, Western Australia.

Abstract

The Degree Constrained Minimum Spanning Tree Problem is concerned with finding a
minimum weight spanning tree T in a weighted graph G (all weights are non negative) with the
requirement that the vertices satisfy a prescribed degree restriction in T. This problem arises
naturally in communications networks where the degree restriction represents the number of line
interfaces available at a terminal. Since, apart from some trivial cases, the problem is
computational difficult (NP-complete), a number of heuristics have been proposed. In this paper
we propose new heuristics based on the Tabu search method. Our heuristic are implemented and
extensively tested on 2160 simulated problems and on some benchmark problems. Our
computational results support to the use of the methods.

3
Keywords: minimum spanning tree, tabu search, degree constrained, network design

7. Terbit di Jurnal Sains dan teknologi Volume 10 No 2, page 18 – 24

Some Tree Representations for Solving The Degree Constrained Minimum
Spanning Tree Problems
Wamiliana
Dept. of Mathematics, Faculty of Mathematics and Natural sciences
The University of Lampung, Bandarlampung, Indonesia
wamiliana@mail.com

Abstract To solve the Degree Constrained Minimum Spanning Tree (DCMST) Problem
using computer we need to represent the tree in the most efficient way in order to reduce the
processing time, especially the initialization process. The DCMST Problem is concerned
with finding, in a given edge weighted graph G (all weights are non-negative), the minimum
weight spanning tree T satisfying specified degree restrictions on the vertices. This problem
arises naturally in communication networks where the degree of a vertex represents the
number of line interfaces available at a terminal (center). In this paper we will discuss some
tree representations that can be used for solving the DCMST Problem.

Keywords: Minimum spanning tree; degree constrained; tree representation.

8. Terbit di Jurnal Teknik Industri , Volume 6 No. 1, halaman 1 – 9, tahun 2004.
http://puslit.petra.ac.id/journals/industrial

SOLVING THE DEGREE CONSTRAINED
MINIMUM SPANNING TREE PROBLEM
USING TABU AND MODIFIED PENALTY SEARCH METHODS
Wamiliana
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University
E-mail: wamiliana@mail.com

ABSTRACT
In this paper we consider the Degree Constrained Minimum Spanning Tree Problem. This problem is
concerned with finding, in a given edge weighted graph G (all weights are nonnegative), the
minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem
arises naturally in communication networks where the degree of a vertex represents the number of
line interfaces available at a center. Because of its NP-completeness, a number of heuristics have been
proposed. In this paper we propose two new search methods: one based on the method of Tabu search
and the other based on a penalty function approach. For comparative analysis, we test our methods on
some benchmark problems. The computational results support our methods.

4
Keywords: minimum spanning tree, tabu search, degree constrained.

9. Terbit di Jurnal Sains dan Teknologi, Volume 11 No. 2, page 93 – 96, tahun 2005

The Design of Greedy Algorithm for Solving The Multi Period Degree
Constrained Minimum Spanning Tree Problem
Wamiliana1 , Dwi Sakethi 1, Akmal Junaidi 1, and Edy Tri Baskoro2
1. Dept. of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University
2. Dept of Mathematics, Bandung Institute of Technology

Abstract
With the limitation fund in hand, the process of installations of some links to the available
networks has to be determined in such a way so that the important links that should be installed
in certain period, must be installed in that period or before with the minimum cost. Besides, the
network itself has a degree restriction in every node which limits the number of links that
incident to. This problem is referred to as the Multi Period Degree Constrained Minimum
Spanning Tree Problems (MPDCMST). In this paper we present the design of the simple
greedy algorithm that we use to solve that problem.

Keywords: multi period, degree constrained, minimum spanning tree

10. Terbit di Jurnal Quantitative Methods, Volume 2 No. 2, page 10 – 16 , tahun 2006.

COMPUTATIONAL ASPECTS OF THE MODIFIED PENALTY METHODS FOR
SOLVING THE DEGREE CONSTRAINED MINIMUM SPANNING TREE PROBLEMS

Wamiliana1 and L. Caccetta2
1
Department of Mathematics, Faculty of Mathematics and Natural Sciences ,
University of Lampung, Indonesia
2
Department of Mathematics and Statistics, Curtin University of Technology,
Perth, Westren Australia

Abstract. The Degree Constrained Minimum Spanning Tree Problem is concerned with finding, in a given
edge weighted graph G (all weights are non-negative), the minimum weight spanning tree T satisfying specified
degree restrictions on the vertices. This problem arises naturally in communication networks where the degree
of a vertex represents the number of line interfaces available at a terminal (center). Since, apart from some
trivial cases, the problem is computationally difficult (NP-complete), a number of heuristics have been
proposed. In this paper we will discuss the computational aspects of the heuristics we have developed that based
on a penalty function approach.

Keywords: Minimum spanning tree; Iterative method ; Degree constrained

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11. Terbit di Jurnal Sains MIPA, Volume 13, No. 1 , page 61 -65, tahun 2007

Tabu Search’s Diversification Strategy for the Degree
Constrained Minimum Spanning Tree Problem

Wamiliana
Dept of Mathematics, Faculty of mathematics and Natural Sciences, Lampung University
Email : wamiliana@mail.com

Abstract
(HARDCOPYNYA ADA DI RUMAH PAK, SOFTCOPYNYA GAK KETEMU)

12. Terbit di Prosiding Seminar nasional Metode Kuantitatif, halaman 251 – 259, Tahun
2007

Riset Operasi Untuk Memajukan Perusahaan Anda

Wamiliana
Jurusan Matematika, FMIPA Universitas Lampung

Abstract
In recent years, operations research software for mainframes and microcomputers has become widely
available, especially to help finding optimization problems. However, it is useless unless the user
understands its application and purpose. Users must ensure that mathematical input accurately reflects
the real-life problems to be solved and that the numerical results are correctly applied to solve them.
In this paper will be discussed the components in decision making, steps of solving the problems,
some of optimization techniques and some problems to illustrate how operations research can be used
to improve the quality, profit and performance of the company.
Keywords: operations research, optimization problems.

13. Terbit di Prosiding Seminar nasional Metode Kuantitatif, halaman 22 - 34, Tahun
2007

TEKNIK PENJADWALAN KULIAH MODEL CLASS – TEACHER
DENGAN MENGGUNAKAN PEWARNAAN EDGE

Oleh
Anita Sari dan Wamiliana
Jurusan Matematika dan Ilmu Pengetahuan Alam FMIPA Unila

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Timetabling is one of the most important issues in running the activities in academic
institutions such as schools, colleges and universities. Given m teachers, n courses, t
times and r rooms, we are suppose to assign every teacher a specific course, time and
rooms which are conflict one to the other. In this paper we will discuss two types of
assigning that issue, especially related to the teacher-class model and generalized model
using graph theoretic concepts, in this case using edge coloring. We also will give some
examples to illustrate how the model can be adopted into the problem.
Keywords : timetabling, edge coloring.

14. Terbit di Prosiding Seminar Nasional Metode Kuantitatif, halaman 66 - 75, Tahun
2007

GOAL PROGRAMMING SEBAGAI SALAH SATU ALAT PENGAMBILAN KEPUTUSAN
DALAM OPERASIONAL SUATU PERUSAHAAN

Oleh
Fitriani1 dan Wamiliana1

Abstrak
Dalam suatu perusahaan sering terjadi masalah dalam pengambilan keputusan untuk memenuhi target
yang diharapkan. Sangat mungkin terjadi pemimpin perusahaaan mempunyai tujuan yang lebih dari
satu dan ia ingin memperoleh hasil yang optimal yaitu suatu hasil untuk mencapai semua tujuan yang
ditentukan dengan cara yang paling baik diantara semua alternative yang mungkin. Goal
programming yang merupakan bentuk khusus dari program linear memberikan satu cara mengubah
multi tujuan menjadi tujuan tunggal, dan memberikan solusi yang dapat memenuhi target yang telah
ditetapkan’ walaupun memungkinkan salah satu tujuan tidak tercapai. Dalam makalah ini akan
didiskusikan tentang non preemptive goal programming dengan menganggap semua tujuan sama
penting dan tidak ada tujuan (goal) yang diprioritaskan dengan memberikan beberapa contoh kasus.

Keyword : Goal programming, non preemptive

15. Terbit di Jurnal sains MIPA, Edisi Khusus, Halaman 6 – 9, tahun 2008

COMB INEQUALITIES FOR THE DEGREE CONSTRAINED
MINIMUM SPANNING TREE PROBLEM

Wamiliana, Admi Syarif, Akmal Junaidi, and Fitriani

7
Abstract

The Degree Constrained Minimum Spanning Tree Problem (DCMST) is a problem of
finding the minimum spanning tree in a given weighted graph whilst also satisfies degree
requirement in every vertex. Since the DCMST can be formulated as MILP, then all
constraints are valid inequalities, and among those constraints some are facets defining.
In this paper we will discuss how to find constraints that constitute comb inequalities for
the DCMST for vertex order 5 to 15 with increments 1.

16. Terbit di Jurnal sains MIPA, Edisi Khusus, Halaman 1 – 5, tahun 2008

Computational aspects of Greedy Algorithm for Solving The Multi Period
Degree Constrained Minimum Spanning Tree Problem
Akmal Junaidi1, Wamiliana1 , Dwi Sakethi 1, and Edy Tri Baskoro2
1. Dept. of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University
2. Dept of Mathematics, Bandung Institute of Technology

Abstract

Given center already set, The Multi Period Degree Constrained Minimum Spanning Tree
Problem (MPDCMST) is a problem of determining how many vertices (can be computers,
cities, and so on) should be installed in a certain period in such a way so that the cost of
installation is minimum. After all the periods done, all of the vertices must be in the network,
and still the cost of installation must be the minimum. In addition, the network itself has a
degree restriction in every vertex which limits the number of links that incident to. In this
paper we will discuss the algorithm we have developed and give results on 600 random table
data and 3 benchmark problems taken from TSPLIB.

17. Terbit di Prosiding Seminar Nasional Sains dan Teknologi I, 2007

Konstruksi Polyomino Tiga Dimensi Dengan Luas Minimal
Muryaniningsih1 , Wamiliana2, and Akmal Junaidi2
1.     Mahasiswa Jurusan Matematika, FMIPA Universitas Lampung
2.     Staf pengajar Jurusan Matematika, FMIPA Universitas Lampung

Abstrak

Secara umum, polyomino adalah domino yang dihimpun dari n persegi dengan ukuran yang
sama, yang disusun dengan tepat, sedangkan dalam bidang tiga dimensi, polyomino
didefinisikan sebagai domino yang dihimpun dari n kubus dengan ukuran yang sama.

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Permasalahan yang timbul dari penyusunan polyomino tiga dimensi ini adalah bagaimanakah
cara yang paling tepat untuk menyusun kubus-kubus tersebut menjadi sebuah polyomino tiga
dimensi dengan volume tetap namun memiliki luas permukaan yang paling minimal. Dari
penelitian yang telah dilakukan dapat disimpulkan bahwa polyomino tiga dimensi dengan
luas minimal memiliki bentuk j   j      j    3  l   l     2 k  yaitu jumlah dari quasicube,
quasisquare,             dan          bar,dimana  ,  ,  0,1 , 0  k  l   , l  l     k   j    j    dengan
volume n  j  j    j     l  l     k dan
luas 2  j  j     j  j      j    j     2  2l     2 1k 0 .
Kata kunci : polyomino, bar, quasisquare, quasicube,

18. Terbit di Prosiding Seminar Nasional Sains dan Teknologi I, 2007

KONSTRUKSI PENGGAMBARAN ORTOGONAL GRAF LINGKARAN DENGAN
JUMLAH LENGKUNGAN YANG MINIMUM
Ridho Setiawan Rahman1, Wamiliana2, Akmal Junaidi2
1. Mahasiswa Jurusan Matematika, FMIPA Universitas Lampung
2. Staf pengajar Jurusan Matematika, FMIPA Universitas Lampung

Abstrak
Suatu penggambaran ortogonal dari suatu graf bidang G adalah suatu penggambaran dengan diberi
peletakkan yang masing – masing vertex dipetakan pada setiap titik dan masing – masing edge digambar
sebagai urutan garis horizontal dan vertikal. Suatu lengkungan adalah suatu kondisi di mana suatu edge
berganti arah di dalam penggambarannya. Permasalah yang timbul dari penggambaran ortogonal adalah
bagaimana mencari jumlah lengkungan yang minimum dengan tetap menjaga keterhubungan pada graf
semula dan mendiskusikan syarat perlu dan cukup agar graf bidang berderajat maksimum 3 mempunyai
penggambaran ortogonal tanpa lengkungan. Dalam tulisan ini akan didiskusikan syarat yang harus
dipenuhi oleh suatu graf bidang lingkaran mempunyai penggambaran orthogonal, dan jika graf tersebut
mempunyai penggambaran ortogonal apakah penggambaran tersebut sudah mempunyai lengkungan
yang minimum.
Kata kunci : ortogonal, lengkungan, graf bidang lingkaran.

19. Terbit di Prosiding Seminar Nasional Sains dan Teknologi II, 2008

Representing Degree Restricted Tree Using Augmented Adjacency Matrix

Wamiliana, Admi Syarif, and Didik Kurniawan
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University

9
Abstract
In graph theory, tree plays an important role due to its structure that can be used to
represent many problems such as in desain the telecommunication networks,
distribution networks and so on. There are some type of tree representations such as
vector representation, adjacency list, permutation, and Prufer’s numbers. For
problems that use tree as the key structure, like the DCMST problem, the tree
representation by Prufer number is an advantage because in the Prufer number
representation we can get information about the degree of the vertex in the tree. The
vertex in the tree that has degree r, in the Prufer number representation will appear
r-1 times. In this paper we propose an alternative method for representing degree
restricted tree, that is the augmented adjacency matrix.

Keywords : degree restricted tree, adjacency matrix, augmented

20. Terbit di Prosiding Seminar Nasional Sains dan Teknologi II, 2008

Determine the Balance Interval for Two Sets of points
Attiya Yuliana1 dan Wamiliana2
1
Student at the Depatment of Mathematics, FMIPA Lampung University
2
Senior Lecture at the Depatment of Mathematics, Lampung University

ABSTRACT

Given n,m numbers of red and blue points on the plane. Let point P on the plane
and integer k, 1≤ k ≤ n,so that there exist tow rays r1 and r2 originated from
point P so that they form convex region that consists exactly k red points and
k  ble points. Next, the points are arranged to determine if there exists a
balance interval for those two sets of points in general position. This
arrangement is done by using orthogonal projection on the line. In order to find
the interval, we need to know the necessary and sufficient condition for the one
to be exist. In this paper we discuss those conditions.

Kata kunci : interval seimbang, posisi umum, proyeksi ortogonal

21. Terbit di Prosiding Hasil-Hasil Penelitian Universitas Lampung, 2008

THE TREE Tn,k CONFIGURATIONS USING POLYA’S POLYNOMIAL
OF ORDER 2 AND 3

Wamiliana and Asmiati

1
0
Dept. of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University
Jl. Soemantri Brojonegoro No.1, Bandar Lampung
wamiliana@unila.ac.id

Abstract
In graph theory, tree plays an important role because it used to represent many
problems such as in desain the telecommunication networks, distribution networks and so
on. The Tn,k , is the tree with n vertex and k branches and Polya’s polynomial is a
polynomial in the form of cycle index. In this paper we will discuss the configuration of Tn,k
using Polya’s polynomial for order 2 and 3.

Keywords: Tree Tn,k, cycle index, Polya’s polynomial

22. Terbit di Proceeding of International Conference Quantitative Methods Used in
Economics and Business , page 353 –357, Tahun 2008

Queeing Networks and Applications

Wamiliana
Dept. of Mathematics, Faculty of Mathematics and Natural Sciences, Lampung University
wamiliana@unila.ac.id

1
1

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