Proceedings of International Seminar on Mathematics...

Proceedings of International Seminar on Mathematics Education and Graph Theory

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Mathematics Education

and Graph Theory

PROCEEDINGS OF INTERNATIONAL SEMINAR

ON MATHEMATICS EDUCATION AND GRAPH THEORY

Unit of Publication Department of Mathematics Education

Faculty of Teacher Training and Education Islamic University of Malang (UNISMA)

2014

Mathematics Education and Graph Theory

ii ISBN 978-602-71141-0-4

MATHEMATICS EDUCATION AND GRAPH THEORY Proceedings of International Seminar on Mathematics Education and Graph Theory © Department of Mathematics Education Faculty of Teacher Training and Education Islamic University of Malang, 2014

Reviewers Surahmat Supangken (UNISMA, Indonesia)Abdur�Rahman�As’ari�(UM,�Indonesia)Kiki Ariyanti Sugeng (UI, Indonesia)Sunismi (UNISMA, Indonesia)Akhsanul�In’am�(UMM,��Indonesia)

Editors Mustangin Abdul Halim Fathani

Layouter Teguh Permadi

First published, 2014 ISBN 978-602-71141-0-4

Published by

Unit of PublicationDepartment of Mathematics Education Faculty of Teacher Training and Education Islamic University of Malang (UNISMA) Malang, East Java, Indonesia Phone +62341- 551932, 551822. Fax +62341-552249 http://www.unisma.ac.id

These proceedings contain the full texts of paper and talks presentedin the International Seminar on Mathematics Education and Graph Theory on June 9, 2014

http://www.unisma.ac.id


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PREFACE

These proceedings contain the full text of papers and talks presented in the International Seminar on Mathematics Education and Graph Theory. This seminar was held in conjunction with the International Workshop on Graph Masters. The workshop was held on June 7–8, 2014, while the seminar was on June 9, 2014. These events were organized by Islamic University of Malang (Unisma) in cooperation with Indonesian Combinatorial Society (InaCombS).

The workshop and the seminar would not have been possible without the time and energy put forth by the invited speakers. The invited speakers of the workshop were: Mirka Miller, University of Newcastle, Australia; Joseph Miret, Universitat de Lleida, Spain; Christian Mauduit, Institut de Mathematiques de Luminy, France; Edy T. Baskoro, Bandung Institute of Technology, Indonesia; Surahmat Supangken, Islamic University of Malang, Indonesia; Tri Atmojo, State University of Semarang, Indonesia; and Purwanto, State University of Malang, Indonesia.

The invited speakers of the seminar were: Juddy Anne Osborn, University of Newcastle, Australia and Abdur� Rahman� As’ari, State University of Malang, Indonesia. The seminar was held on the area of mathematics education and graph theory. The main themes of the mathematics education seminar include topics within the following areas (but not limited to): philosophy of mathematics education, curriculum development, learning methods and strategies, learning media, development of teaching material, and assessment and evaluation of learning. The main themes covered in graph theory seminar include topics within the following areas (but not limited to): degree (diameter) problems, ramsey numbers, cycles in graphs, graph labeling, dimensions of graphs, graph coloring, algorithmic graph theory, and applications of graph theory in various fields.

We would like to thank you to the invited speakers and all presenters who have submitted papers, for their valuable and inspiring presentation. A special appreciation goes to: Surahmat Supangken, Rector of Unisma and Kiki Ariyanti Sugeng, the President of InaCombS, who have made a lot of efforts to prepare this seminar.

We also do not forget to express our gratitude to Islamic University of Malang (Unisma) for providing financial support, and to the Indonesian Combinatorial Society (InaCombS) for the support. We hope that you had a great time and valuable experience during the seminar in Malang.

Malang, July 22, 2014

Editors


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TABLE OF CONTENTS

Preface __ iii Table of Contents __ v

OPTIMIZING PROBLEM SOLVING ACTIVITY FOR TEACHING MATHEMATICAL THINKING

Abdur�Rahman�As’ari ......... 3–9

CONSTRUCTION THEORY OF CRITICAL THINKING AS PROCESS TOWARDS REFRACTION THINKING IN MATHEMATICS

Anton Prayitno, Akbar Sutawidjaja, Subanji, Makbul Muksar ......... 10–16

CONCEPT�IMAGE�AND�CONCEPT�DEFINITION�OF�A�STUDENT’S CONCEPT UNDERSTANDING

Budi Nurwahyu ......... 17–26

DEVELOPMENT OF TACIT KNOWLEDGE DIMENTION FOR MATHEMATICS NOVICE TEACHERS

Edy Bambang Irawan ......... 27–31

DEVELOPING��THE��STUDENT’S�MATHEMATICAL�REPRESENTATION�AND ABSTRACTION ABILITY THROUGH GUIDED DISCOVERY LEARNING

Eka Setyaningsih ......... 32–38

IDENTIFICATION THE UNI-CONNECTED MATHEMATICAL THINKING PROCESS IN MATH PROBLEM SOLVING

Elly Susanti, I Nengah Parta, Tjang Daniel Chandra ......... 39–49

BLENDED LEARNING AS A WAY TO OPTIMIZE SEMESTER CREDIT SYSTEM (SCS) Hapizah ......... 50–55

MAKING MATHEMATICAL CONNECTIONS IN SOLVING CONTEXTUAL MATHEMATICS PROBLEM: THEORETICAL REVIEW FROM THE PERSPECTIVE OF IQ AND GENDER

Karim ......... 56–64

THE ROLE OF LANGUAGE, LOGIC, AND MATH IN SCIENCE AND TECHNOLOGY DEVELOPMENT

M. Kharis ......... 65–68

REPRESENTATION OF MATHEMATICAL CONCEPT IN THE DIFFERENT PERSPECTIVE THEORY OF UNDERSTANDING

Mustangin ......... 69–77


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BUILD MATHEMATICAL KNOWLEDGE THROUGH PROBLEM-SOLVING STRATEGY

Saleh Haji ......... 78–81

NINE STRATEGIES OF CRITICAL THINKING DEVELOPMENT: A LITERATURE REVIEW

Slamet ......... 82–85

MATHEMATICS EDUCATION:

IMPLEMENTATION�OF�PROPS�“FASKAL”�TO�RAISE�ALGEBRA�FACTORIZATION�CONCEPT IN AISIYAH ORPHANAGE SUMBERSARI - JEMBER DISTRICT

Abi Suwito ......... 89–94

THE PROFILE OF PROBLEM SOLVING ON OPEN ENDED PROBLEMS OF STUDENTS WITH INTERMEDIATE LEVEL OF MATHEMATICS COMPETENCE

Agung Deddiliawan Ismail ......... 95–99

METACOGNITIVE AWARENESS ASPECTS IN SOLVING ALGEBRA Akhsanul In'am ......... 100–104

DEVELOPING SELF-RENEWAL CAPACITY SCALE BASED ON PACE MODEL Andri Suryana ......... 105–109

DIAGNOSTIC DIFFICULTY STUDENTS ORGANIZATION AT THE UNIVERSITY OF GUNUNG JATI IN PROVING USING MATHEMATICAL INDUCTION AND EFFORTS TO OVERCOME USING SCAFFOLDING

Azin Taufik ......... 110–116

CHARACTERISTICS OF THINKING PROCESSES OF ELEMENTARY SCHOOL STUDENTS WITH MODERATE ABILITY IN MATHEMATICS PROBLEMS SOLVING

Baiduri ......... 117–123

TEACHER’S�KNOWLEDGE�OF�CONTENTS�AND�STUDENTS�(KCS)�ON�QUADRILATERALS: CASE STUDY

Bettisari Napitupulu ......... 124–134

HOW TO�IMPROVE�STUDENTS’�ABILITY�IN�QUESTIONING BY ASKING THEIR RESPONSIBILITY OF WHAT THEY LEARN

Budi Mulyono ......... 135–139

TEACHING MATHEMATICS TO 0 – 1 YEAR OLD BABIES Christine Wulandari ......... 140–155

DEVELOPMENT OF MATHEMATICS LEARNING MATERIALS WITH ANCHORED INSTRUCTION MODEL FOR DISABILITY STUDENT IN INCLUSION CLASS

Dian�Kristanti,�Cholis�Sa’dijah,�Tjang�Daniel�Chandra ......... 156–175


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IMPLEMENTATION OF THE FIRST YEAR LESSON STUDY TO IMPROVE THE LEARNING QUALITY IN MATHEMATICS EDUCATION STUDY PROGRAM UNIVERSITY OF JEMBER

Dian Kurniati ......... 176–180

THE THINKING PROCESS OF JUNIOR HIGH SCHOOL STUDENTS ON THE CONCEPT OF RECTANGLE REVIEWED BY THEIR COGNITIVE STYLES

Endah Budi Rahaju ......... 181–190

BARRIERS TO STUDENT THINKING IN SOLVING THE PROBLEM OF FACTORING ALGEBRAIC FORM BASED ON THE LEVEL OF ALGEBRAIC THINKING AND SCAFFOLDING

Hairus Saleh, Ukhti Raudhatul Jannah ......... 191–207

LEARNING�MEDIA�“PUTRI”�IN�IMPROVING�TRIGONOMETRY LEARNING OUTCOMES AT SMK (VOCATIONAL SCHOOL)

Hastini Ratna Dewi, Wiwik Sugiarti ......... 208–217

METHOD OF INDUCTION IN THE PROCESS OF MASTERING CALCULUS IN MECHANICS

Hendra Gunawan ......... 218–223

THE REFLECTIVE THINKING STUDENT WITH LOGIC APPROACH IN PROBLEMS SOLVING OF SPEED, DISTANCE, AND TIME

Hery Suharna, Subanji, Toto Nusantara ......... 224–228

DEVELOPING RESEARCH INSTRUMENTS FOR PROFILING COGNITIVE PROCESSES OF STUDENT IN CONSTRUCTING MATHEMATICAL CONJECTURE FROM INDUCTION PROBLEMS

I Wayan Puja Astawa ......... 229–236

THE PROFILE OF REASONING SCHOOLGIRLS ELEMENTARY SCHOOL WHO HAVE HIGH MATHEMATICS ABILITY IN PROBLEMS SOLVING OF FRACTION

Iis Holisin ......... 237–246

A COGNITIVE LOAD OF THE SEVENTH GRADE STUDENTS IN TEACHING MATHEMATICS AT SMP NEGERI 3 MALANG

Isbadar Nursit ......... 247–253

THE DEVELOPMENT OF INSTRUMENTS TO IDENTIFIY CRITICAL THINKING SKILL OF JUNIOR HIGH SCHOOL STUDENTS IN SOLVING CRITICAL THINKING MATHEMATICS PROBLEMS

Ismail ......... 254–265

DEVELOPING A MATHEMATICS MODULE WITH CONTEXTUAL APPROACH AND ISLAMIC VALUES USING ADOBE FLASH CS3 AS A MATHEMATICS LEARNING RESOURCE FOR STUDENTS IN SMP/MTS

Mulin Nu'man, Ibrahim ......... 266–285

DESIGNING VIDEOS OBSERVATIONS PROJECT THROUGH SCIENTIFIC APPROACH�WITH�AUTHENTIC�ASSESSMENT�TO�INTEGRATE�STUDENTS’�KNOWLEDGES IN MATHEMATICS TEACHING AND LEARNING

Nurcholif Diah Sri Lestari ......... 286–295


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DEVELOPING SELF EFFICACY SCALE WITH THE ORIENTATION OF WEB-ASSISTED BRAIN-BASED LEARNING

Nuriana Rachmani Dewi (Nino Adhi) ......... 296–301

TEACHER’S�INTERACTION�PROCESS�IN�ASSISTING�STUDENTS�OF�SMAN�10�MALANG TO CONSTRUCT CONCEPT OF COMPREHENSION IN PROBABILITY MATERIAL

Ratna Widyastuti ......... 302–313

THE ANALYSIS OF STUDENTS DIFFICULTIES IN SOLVING PROBLEMS OF SET Rohana ......... 314–321

MATHEMATICS LEARNING BY MIND MAPPING METHOD Ryan Angga Pratama, Alhamidun ......... 322–327

CHARACTERIZATION OF ALGEBRAIC THINKING Process OF STUDENTSIN PATTERN GENERALIZING BASED APOS THEORY

Siti Inganah, Purwanto, Subanji, Swasono Rahardjo ......... 328–337

MATHEMATICAL COMMUNICATION PROFILE OF FEMALE-FIELD INDEPENDENT STUDENT OF JUNIOR HIGH SCHOOL IN SOLVING PROBLEM

Sudi Prayitno ......... 338–344

THE EFFECTS OF REALISTIC MATHEMATICS EDUCATION AND�STUDENTS’�COGNITIVE DEVELOPMENT LEVELS ON THE UNDERSTANDING OF CONCEPTS AND THE ABILITY IN SOLVING MATHEMATIC PROBLEMS BY JUNIOR HIGH SCHOOL STUDENTS

Sunismi ......... 345–359

THE IMPROVEMENT OF STUDENT UNDERSTANDS OF ADDITIONAND REDUCTION FRACTION CONCEPT THROUGH REALISTIC MATHEMATIC EDUCATION (RME) WITH MANIPULATIVE MATERIALS IN THE 4TH GRADE OF SDN GADANG I MALANG

Surya Sari Faradiba ......... 360–364

RESEARCH INSTRUMENT DEVELOPMENT�OF�STUDENTS’�REASONING�PROCESS�IN PROVING THEOREM

Susanah ......... 365–377

PROFILE OF STUDENT'S INTUITION IN THE ANALISYS OF THE VAN HIELE LEVEL IN GEOMETRY PROBLEM SOLVING

Susilo Bekti ......... 378–387

STUDENT’S�FOLDING�BACK�WHICH�HAS�A TENDENCY ON CONCEPTUAL KNOWLEDGE IN UNDERSTANDING LIMIT DEFINITION

Susiswo ......... 388–403

DEVELOPING TEACHING REALISTIC MATHEMATIC INTERACTIVE HANDBOOK ON STATISTICS SETTING ON ISLAMIC BOARDING SCHOOL OF IX GRADE MTs

Suwarno ......... 404–413


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INSTRUCTIONAL MATERIALS DEVELOPMENT ON CALCULUS II COURSE USING MATHEMATICS MOBILE LEARNING (MML) APPLICATION

Sunismi, Abdul Halim Fathani ......... 414–429

PATTERN AND STRUCTURE MATHEMATICS AWARENESS CONTRIBUTED TO NUMBER SENSE EARLY CHILDHOOD

Timbul Yuwono ......... 430–437

MIDDLE�SCHOOL�STUDENTS’�COVARIATIONAL�REASONING�IN�CONSTRUCTING�GRAPH OF FUNCTION

Ulumul Umah,�Abdur�Rahman�As’ari, I Made Sulandra ......... 438–448

THE�ANALYSIS�ON�STUDENTS’�ERRORS�IN�SOLVING�MATHEMATICAL WORD PROBLEMS OF CUBE AND BLOCK MATERIALS BASED ON THE STAGES OF NEWMAN’S�ERROR�ANALYSIS

Umi Farihah, Moh Nashihudin ......... 449–457

SIGNIFICANCE TRAINING OF PEDAGOGICAL AND PROFESSIONAL COMPETENCY THROUGH PEER LESSON METHOD IN DISCRETE MATHEMATICS SUBJECT

Wasilatul Murtafiah ......... 458–468

THE DEVELOPMENT OF MATHEMATICS E-PORTFOLIO ASSESSMENT MODEL FOR SENIOR HIGH SCHOOL

Zainal Abidin, Sikky El Walida ......... 469–476

PROCESS OF SPATIAL REASONING ON VOCATIONAL STUDENT HIGH ABILITY IN CONSTRUCTION CUBE (CASE STUDY ON STUDENTS WHO HAVING HIGH SPATIAL ABILITY)

Zuraidah ......... 477–488

GRACEFUL LABELING ON BAT GRAPH �� Annisa Dini Handayani, Kiki A Sugeng ......... 491–494

THE ANALYSIS OF AIR CIRCULATION ON COFFEE PLANTATION BASED ON THE LEVEL OF PLANTS ROUGHNESS AND DIAMOND LADDER GRAPHCROPPING PATTERN USING FINITE VOLUME METHOD

Arif Fatahillah, Dafik, Ervin Eka Riastutik, Susanto ......... 495–498

DEVELOPING MST, TSP, AND VRP APPLICATION Darmawan Satyananda ......... 499–508

SUPER (a,d)-VERTEX ANTIMAGIC TOTAL LABELING ON DIRECTED CYCLE GRAPHDevi Eka Wardani Meganingtyas, Dafik, Slamin ......... 509–515

SPECTRUM OF ANTIADJACENCY MATRIX OF SOME UNDIRECTED GRAPHS Fitri Alyani, Kiki A Sugeng ......... 516–519


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THE LOCATING CHROMATIC NUMBER OF STRONG PRODUCT OF COMPLETE GRAPH AND OTHER GRAPH

I.�A.�Purwasih,�E.�T.�Baskoro,�M.�Bača,�H.�Assiyatun,�D.�Suprijanto ......... 520–523

THE CHARACTERISTICS OF CRITICAL SET IN EDGE MAGIC TOTAL LABELING ON BANANA TREE GRAPH

Irham Taufiq, Triyani, Siti Rahmah Nurshiami ......... 524–527

VERTEX COLORING BY TOTAL LABELINGS OF SUN, WHEEL AND PRISM GRAPHS Isnaini Rosyida, Widodo, Ch. Rini Indrati, Kiki A. Sugeng ......... 528–533

ON DISCONNECTED RAMSEY (3K2,K3)-MINIMAL GRAPHSKristiana Wijaya, Edy Tri Baskoro, Hilda Assiyatun, Djoko Suprijanto ......... 534–537

TEACHING COMBINATORIAL GAMES INTERACTIVELY USING MAPLETS Loeky Haryanto, Arnensih Alimuddin ......... 538–546

THE AIR FLOW ANALYSIS OF COFFEE PLANTATION BASED ON CROPS PLANTING PATTERN OF THE TRIANGULAR GRID AND SHACKLE OF WHEEL GRAPHS BY USING A FINITE VOLUME METHOD

M. Nurrohim, Dafik, Arif Fatahillah, Moch. Avel Romanza P., Susanto ........ 547–550

ON THE DOMINATION NUMBER AND CHROMATIC NUMBER OF FLAKE GRAPHMohammad Nafie Jauhari ......... 551–554

SUPER (a,d) -EDGE ANTIMAGIC TOTAL LABELING OF SNAIL GRAPH Novian Riskiana Dewi, Dafik, Susi Setiawani ......... 555–458

TOTAL EDGE IRREGULARITY STRENGTH OF LAMPION GRAPH Nuris�Hisan�Nazula,�Slamin,�Dafik ......... 559–562

SUPER (a,d)-EDGE-ANTIMAGIC TOTAL LABELING OF UFO GRAPH Reni Umilasari, Dafik, Slamin ......... 563–568

CHARACTERISTIC STUDIES OF SOLUTION THE MULTIPLE TRIP VEHICLE ROUTING PROBLEM (MTVRP) AND ITS APPLICATION IN OPTIMIZATION OF DISTRIBUTION PROBLEM

Sapti Wahyuningsih, Darmawan Satyananda ......... 569–578

ONE TOUCH DRAWING FOR ANDROID-BASED GRAPH THEORY LEARNING Sikky El Walida ......... 579–582

VERTEX MAGIC TOTAL LABELING ON SUNS DIGRAPH Yuni Listiana, Darmaji, Slamin ......... 583–589

DEVELOPING MST, TSP, AND VRP APPLICATION

Darmawan Satyananda Department of Mathematics Education, State University of Malang

[email protected]

AbstractIn course of Graph Theory Graf and Application of Graph Theory in Mathematics Department State University of Malang, students learn some algorithms for optimization problem, such as for searching Minimum Spanning Tree (MST), Travelling Salesman Problem (TSP), and Vehicle Routing Problem (VRP). For helping solving problems, for example for research purposes or Field Work ( Praktek Kerja Lapangan), usually students use software for Graph application. Two of them are Grin and Giden. They have not been updated for a long time; meanwhile there are some new algorithms that are not contained in them. Algorithms studied in the course and are not contained in both software must be made as separate application. Skills developing the application is learned in Data Structure course, as discussed how structuring and operating data, with Graph and its application as one of the subject. An application is made to unify important algorithms studied in lectures and required in problem solving. The software implements algorithms grouped on problems of MST, TSP, and VRP. For data benchmark, TSPLIB dataset in XML format (Extensible Markup Language) is used.

Keywords: Graph Application, TSPLIB, XML

INTRODUCTION

In Mathematics Department, State University of Malang, Graph Theory subject is a mandatory course, and Graph Theory II(called Application of Graph Theory in Curriculum 2013 [hereinafter referred as AGT]) is an elective course. From the first time, interest of student who took AGT is quite high, approximately 30 of 60 students on each class.

AGT comprehends subject about application of several algorithms in Graph Theory for optimization problems, including finding of Minimum Spanning Tree (MST), Travelling Salesman Problem (TSP), Vehicle Routing Problem (VRP), Flow Problem, and Matching. In addition of studying the algorithm, there are assignments to the students in the form of real survey in company or institution to implement the specified algorithm according to the characteristics of the problem.

To date, students uses some software designed for the purpose, those are Giden (http://users.iems.northwestern.edu/~giden/), or Grin

(http://www.oocities.org/pechv_ru/main.html). Some algorithms studied in PTG are contained in it, for instance Nearest Neighbor or MTVRP, however some recent algorithm cannot be found as the software has not been updated.

To overcome the obstacles, students are encouraged to create the application (computer program) that implements certain algorithm. This accommodated through Data Structure, which is electional course. The course discusses about how the representation of logical and physical data, as well as how the use of data in particular algorithm. One of the topics covered in this course (as contained in various textbook of Data Structure) is Graph. This because graph can represents a wide range of optimization problem, has various forms of data structure, and has a lot of algorithm (http://interactivepython.org/courselib/static/pythonds/Graphs/graphintro.html)

In the course of Data Structure in Mathematics Department UM, there are little adjustment in presentation of the material (as compared to the one listed in the textbook or taught in Computer Science


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http://users.iems.northwestern.edu/~giden/)

http://www.oocities.org/pechv_ru/main.htm

http://interactivepython.org/courselib/static/

department) because in here we more focused on how the implementation of algorithms in Graf Theory. The impact of the combination between Graph Theory II / Application of Graph Theory with Data Structure are produced of some theses about creation of computer program on the implementation of algorithms in Graph Theory, in the topic about MST, TSP, VRP, coloring, matching, and network flow. Some of the products are utilized by students doing Job Training (Praktek Kerja Lapangan), for example in optimizing of distribution of goods.

However, because generally thesis only discusses about an algorithm, it is required some software products if they want to compare or use multiple algorithms at once. This is what underlies the need of integrated application that can accommodate various needs of users.

LITERATURE REVIEW

Data structure of graph A graph can be represented as an

adjacency matrix that records connection of two vertices (forming an edge). The connection of two vertices can be stated with a certain value, such as distance of two vertices (Euclidean, Manhattan, or something else), or as edge weight. If there is no such connection, the value can be 0 or infinite. Symmetric TSP has square and symmetric adjacency matrix (value of row iand column j is equal to value of row j and column i), it means that an edge is recorded twice. In the other side, asymmetric TSP has square adjacency matrix but it is not a symmetric matrix. Another representation of graph is using edge list that only records graph edges. An edge is only recorded once (edge uv or vu only).

Figure 1 shows example of a graph and the representation of its data structure in the form of adjacency matrix and edge list.

0

2

31

4 5

561

55

6 4

3 2

6

(a)

0 1 2 3 4 5 0 0 6 1 5 0 0 1 6 0 5 0 3 0 2 1 5 0 5 6 4 3 5 0 5 0 0 2 4 0 3 6 0 0 6 5 0 0 4 2 6 0

(b) v1 0 0 0 1 1 2 2 2 3 4 v2 1 2 3 2 4 3 4 5 5 5 c 6 1 5 5 3 5 6 4 2 6 0 1 2 3 4 5 6 7 8 9

(c) Figure 1. (a) Example of a graph, (b) Graph representation as an adjacency matrix, (c) Graph representation as an edge list

Figure 1.b represents graph of 6 vertices as the adjacency matrix, there are 36 elements whereas half part of the matrix equals to another half part (above and below its main diagonal) because the graph is undirected (assume there are no multiple edges). Non-existing edge represented as element valued as 0. In figure 1.c, the list has 10 elements where each element contains information about the edge, and distance or weight of the edge. Number of list elements is as much the edge of the graph, and it is able to represent multiple edges

Some comparisons between usage of matrix and list (Satyananda, 2012):


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� Matrix is more suitable if edge searching or edge updating is often performed because the access of elements is done randomly, contrary to the list which is done sequentially (must traversing from the beginning of the list until the edge is found or not at all). Two vertices that form an edge are used as identifier of matrix’� element� to� get� the� distance� or�weight of the edge.

� List is more suitable if the graph is not complete because there is no need to save unexisting edge as in the matrix with 0 or infinite value.

� List can be used to represent graph that has multiple edges between two vertices.

MST Problem of MST is about how to find

spanning tree in a graph (network), with least cost (Rosen, 2000). The spanning tree is a tree that connects all vertices in the graph. Some of the algorithms used in searching spanning tree are Kruskal (Rosen, 2000), Prim (Rosen, 2000), Boruvka (known also as Sollin algorithm) (Rosen, 2000), and Reverse-Delete (wikipedia).

TSP TSP is the problem of modeling a

salesman who departed from the place of origin, visited a number of other places exactly once, and must be returned to its original place, with the least possible cost or distance. When described, the route of the salesman will formed Hamilton cycle. For the purpose of this modeling, there are numerous algorithms that can be used, among of them are Nearest Neighbor Heuristic (Rosen, 2000), Nearest Insertion Heuristic (Rosen, 2000), Cheapest Link (Musser, 2007), and Genetic Algorithm (Mitchell, 1996, and Dwivedi, 2012).

VRPIn general, VRP problem is how to

determine the route to distribute goods from the depot the customers, with the least cost / time / distance, or the smallest number of vehicle. In VRP there are one or more depot,

one or more customer, one or more vehicle with specified capacity, distance / cost among depots and customers, service time on each customer, and time window as an overall time constraints that must be met (Rosen, 2000). Route generated by VRP could be more than one, depending on the constraints. VRP itself can be seen as set of TSP routes.

There are many variants of VRP in accordance with characteristic of problem. Kumar (2012) explained some of them: a. CVRP (Capacitated Vehicle Routing

Problem), where a vehicle with a limited capacity must get around to a number of customers who demand the goods in a certain amount.

b. VRPTW (Vehicle Routing Problem with Time Windows), where a vehicle has to serve a number of customers within a certain time limit at the least cost.

c. VRPBTW (Vehicle Routing Problem with Backhauls and Time Windows), in which the vehicle must deliver goods to the line-haul customer, while taking the goods from backhaul customers.

Benchmark data Some source provides standardized

test data used as a reference in a testing of the algorithm. One of them is TSPLIB that provides data for Symmetric TSP, Asymmetric TSP, Sequential Ordering Problem (SOP), and CVRP problems (http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/). The data from TSPLIB are in two kinds of format: TSPLIB standard format and XML format (to date, the XML format provided in TSPLIB website is only for TSP problem).

Here is sample for data in TSPLIB format (some data is removed, marked as ...).

NAME: berlin52 TYPE: TSP COMMENT: 52 locations in Berlin (Groetschel) DIMENSION: 52 EDGE_WEIGHT_TYPE: EUC_2D NODE_COORD_SECTION


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1 565.0 575.0 2 25.0 185.0 ... 51 1340.0 725.0 52 1740.0 245.0 EOF

In the file there are several descriptions, which are: � NAME: identity of data instance � TYPE: type of problem, in this case is

TSP problem � DIMENSION: number of vertices � EDGE_WEIGHT_TYPE: type of edge

weight, EUC_2D means Euclidean distance in 2-dimensonal plane (distance of 2 points). Other types are distance which is explicitly defined, Manhattan distance, and Geographical distance (Reinelt, 1995).

� NODE_COORD_SECTION: part shows data for every vertex 1 - n. If the TYPE is EUC_2D, then data for every vertex is coordinates of X-axis and Y-axis, and edge weight is calculated from the coordinates.

XML format is used as an alternative since it has more structured format than general format of TSPLIB, and it is easier to manage in Delphi or other recent programming language. In the original format of TSPLIB there are various representations of data (as an adjacency matrix, adjacency list, coordinate points to be calculated as a distance, directly as edge weight). In XML format there is only one type of data that is edge weight.

Here is sample of XML file for problem of TSP in the 5-vertices graph(some data is removed, marked as ...) <?xml version="1.0" encoding="UTF-8" standalone="ignoredDigitsno" ?> <travellingSalesmanProblemInstance> <name>Small 1</name> <source>Rostislav Stanek</source> <description>A small instance.</description>

<doublePrecision>15</doublePrecision> <ignoredDigits>5</ignoredDigits> <graph> <vertex> <edge cost="2.000000000000000e+00">1 </edge> <edge cost="1.000000000000000e+00">2 </edge> <edge cost="2.000000000000000e+00">3 </edge> <edge cost="1.000000000000000e+00">4 </edge> </vertex> ... <vertex> <edge cost="1.000000000000000e+00">0 </edge> <edge cost="2.000000000000000e+00">1 </edge> <edge cost="1.000000000000000e+00">2 </edge> <edge cost="2.000000000000000e+00">3 </edge> </vertex> </graph> </travellingSalesmanProblemInstance>

In the data file, all edge weight states as realnumber, although the actual value is integer.

The advantage of XML format is that user can define their structure of the XML document and all kinds of information contained in it (in the form of tag definition, attribute, value, and data type). The structure of an XML document are described in XML Schema files (it has file extension named. xsd), which also serves to validator, to check the validity of an XML document. The XML document in a specific structure is valid


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against a specific XML Schema file, so it will likely invalid when associated with different XML Schema file.

Delphi can generate unit XML data binding unit based on a specific XML Schema file for use in reading/writing XML documents that refer to the same XML Schema file (for XML document with different structure, need to be generated another XML data binding unit). This file is included in the main program as a Unit.

In the XML format above there are <graph> tags stating a graph. Inside are a number of <vertex> tags which symbolize one particular vertex. There are no attributes in this tag, so the vertex number implicitly indicated by the order of reading the file (starting from 0). Inside the <vertex> tag there are <edge> tags stating edge (arcs) formed by two vertices. This tag has�“cost”�attribute that states edge weight, and has a value that indicates vertex number that make up the edge of a particular <vertex>. When observed there is no edge between verticeswith the same vertex number, it can be considered that the weight is 0.

For VRP case, the data from the VRP web (http://www.bernabe.dorronsoro.es/vrp/) can be used as standard data. Here is sample of the data file (some data are omitted, marked as ...)

2 4 50 4 0 80 0 80 0 80 0 80 1 37 52 0 7 1 4 1 2 4 8 2 49 49 0 30 1 4 1 2 4 8 ... 49 48 28 0 18 1 4 1 2 4 8 50 56 37 0 10 1 4 1 2 4 8 51 20 20 0 0 0 0 52 30 40 0 0 0 0 53 50 30 0 0 0 0 54 60 50 0 0 0 0

According to the description in NEO website, the format comprises of:

- First line: VRP variant type, number of vehicle, number of customer, and the number of depot/day/vehicle

- Lines 2-(n+1) is time window of a route and capacity of each vehicle

- Lines (n+2) to (n+2+c) are for each customer, some of them are coordinatefor each customer, demand, and service time

- Last d lines state data of each depot, particularly is position coordinate.

Not all data provided are used; it depends on the need or the problem.

METHOD

Application in general Purpose of the application is to

combine many algorithms in a package, in order to solve a problem in many ways to get� the� “best”� result. There are three “groups”� of algorithms to be implemented: MST, TSP, and VRP. Algorithms implemented in MST are Prim, Kruskal, Boruvka, and Reverse Delete algorithm. In TSP, algorithms implemented are Nearest Neighbor, Repetitive Nearest Neighbor, Nearest Insertion Heuristic, Cheapest Link, and Genetic Algorithm. Finally, in VRP algorithm of Multiple Depot VehicleRouting Problem (MDVRP) is implemented.

Development steps General steps in developing the

application are: a. Designing the application

This step chooses algorithm to be implemented and its design. Requirement of the input and output is object to the algorithm and programming language used.

b. Implementing the design The design is implemented as an application (software) using Delphi programming language.

c. Testing the applicationApplication testing is done using user-defined data and standardized data from TSPLIB, the VRP web, and other reputable source.


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Designing the application a. Characteristic of the application

The application has input in the kind of� graph’s� vertices� and� edge� weight. The iinput is performed from the application itself or from external data file. The graph is graphically visualized to help user understanding Output of the application is shortest path (for TSP problems), spanning tree (for MST problems), and distribution route (for VRP problems), in textual andgraphical way.

b. Internal data structureData structure to represent graph

basically is an adjacency matrix with Single-type element, to enable value of edge weight in floating point.

type MatriksBobot=class private matriks:array of array of Single; ... end; type Graph=class private mtBobot:MatriksBobot; ... end;

On other cases that require finding edge which has least edge weight (edges are sorted by its weight) then structure in edge list is used as well.

type Sisi=class private u,v:SmallInt; bobot:Single; ... end; type ListSisi=class private lsSisi:TList; urut:Boolean; ... end;

c. Interface of the applicationUser interface uses 3 types of

component:(1) Image area to enter vertices of the

graph. Input of vertex is performed using mouse.

(2) Table (grid) sized n x n elements (n: number of vertex in a graph) to enter weight of edge uv. It is assumed that the graph is undirected, so that when value of line u column v (weight of edge uv) then value of line v column u will adjust. At the time a vertex is added to the image area, and then the number of rows / columns in the table is also increasing. Conversely when a value of edge weight is inputted then the value is shown in the graph in the image area. In the case of VRP, the grid is also used to enter data of customer requests

(3) Text area (memo) to display the result of calculation. The end result is also shown on the graph in different colors to indicate which side was selected as the spanning tree or Hamilton cycle or route of distribution.

The image area utilizes Image component and OnClick event to detect any click of mouse button in the image area. At the time click happens, then the coordinate position of the mouse pointer is recorded, and the number of rows/columns in a weight table (grid) increases. Vertex numbering starts at 0, but can be adjusted to start from 1 or by letter.

Weight table uses StringGrid component and onSetEditText event to detect any entry of edge weight. If the graph is undirected then value of weight entered to grid on row u column v will be copied to the row v column u. Weight of edges is not related to the actual distance between two vertices, but it is possible to automatically fill in the weights according to the distance between two vertices or as a random value. At the time an edge has weight then it is visualized on the image area.

Memo component is used to display textual results and other messages.


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d. Implementation of the algorithmThe algorithms implemented for the

MST problem in this application are Kruskal, Prim, Boruvka/Sollin, and Reverse-Delete. For the TSP problem, Nearest Neighbor Heuristic, Repetitive Nearest Neighbor Heuristic, Nearest Insertion Heuristic, Cheapest Link, and the Genetic Algorithm are implemented. Repetitive Nearest Neighbor is Nearest Neighbor algorithm using all possible starting vertex. For problem of VRP, it is implemented MDVRP algorithm assisted by Clarke-Wright algorithm for route grouping.

e. External dataThe graph in this application can be

saved in a text file to be opened in any time. The information stored is the number of vertices, the coordinates of the vertex position in the image area, and the weight of each edge (if there is no edge between two vertices then the weight is 0).

Data input for TSP and MST problems using data from TSPLIB, and data files from the results of the previous manual input. The case was taken from TSPLIB gr17 (17 points) and gr120 (120 points).

Data input to the VRP problem using the data from the VRP web, and data files from the results of the previous manual input. The case taken is p01 and p04. There are many variations MDVRP, so not all of the data available in the file is used. MDVRP algorithm in this application is using only the data: the number of depots, the number customers, demand of each customer, as well as the capacity of the vehicle. The number of vehicles is assumed to be enough (infinite and there is no time window and service time per customer).

RESULT

Here some screenshots of the application:

(a)

(b)

(c)

(d)

Figure 2. (a) Input of vertex data, (b) Weight table, (c) Output in textual form, (d) Table to

fill customer demand on VRP problems


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Result of the execution is displayed in Table 1 (routes are not shown).

MSTProblem /

Casegr17 gr120

Kruskal 1421 5805Prim 1421 (starting

vertex: 0)5805 (starting

vertex: 0)Boruvka 1421 5805Reverse-Delete

1166 5303

TSPProblem /

Casegr17 gr120

Nearest Neighbor Heuristic

2187 (starting vertex: 0)


Repetitive Nearest Neighbor Heuristic

2178 8438

Nearest Insertion Heuristic



Cheapest Link

2189 8241

Genetic Algorithm

3586 (Generation: 3, Population: 30,. Crossoverprob.: 0.7, Mutation prob.: 0.2, Elitism prob.:0.2)

43895 (Generation: 3, Population: 150, CrossoverProb.: 0.7, Mutation Prob.: 0.2, Elitism Prob.: 0.2)

VRPCase p01 p04

MDVRP Depot 0:- Vehicle 0 : distance: 57.78, capacity: 48.00- Vehicle 1 : distance: 51.17, capacity: 51.00- Vehicle 2 : distance: 53.52,

Depot: 0- Vehicle 0 : distance: 114.64, capacity: 95.00- Vehicle 1 : distance: 76.31, capacity: 100.00- Vehicle 2 : distance: 89.79,

capacity: 56.00- Vehicle 3 : distance: 26.96, capacity: 50.00Depot: 1- Vehicle 0 : distance: 91.09, capacity: 77.00- Vehicle 1 : distance: 77.72, capacity: 80.00- Vehicle 2 : distance: 39.75, capacity: 51.00- Vehicle 3 : distance: 23.50, capacity: 54.00Depot: 2- Vehicle 0 : distance: 66.03, capacity: 79.00- Vehicle 1 : distance: 38.36, capacity: 62.00- Vehicle 2 : distance: 24.14, capacity: 36.00Depot: 3- Vehicle 0 : distance: 58.18, capacity: 74.00- Vehicle 1 : distance: 57.40,

capacity: 96.00- Vehicle 3 : distance: 91.23, capacity: 96.00- Vehicle 4 : distance: 71.91,capacity: 90.00- Vehicle 5 : distance: 44.11, capacity: 94.00- Vehicle 6 : distance: 46.27, capacity: 87.00- Vehicle 7 : distance: 45.08, capacity: 97.00- Vehicle 8 : distance: 14.14, capacity: 23.00Depot: 1- Vehicle 0 : distance: 115.62, capacity: 97.00- Vehicle 1 : distance: 136.16, capacity: 94.00- Vehicle 2 : distance: 59.63, capacity: 91.00- Vehicle 3 : distance: 46.35, capacity: 59.00


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capacity: 59.00

Total: 665.6

- Vehicle 4 : distance: 49.10, capacity: 93.00- Vehicle 5 : distance: 59.72, capacity: 98.00- Vehicle 6 : distance: 75.58, capacity: 100.00- Vehicle 7 : distance: 57.57, capacity: 48.00

Total: 1193.21

Comparing to the best result from each case, Table 2 shows the comparison:

Table 2. The recorded best results Problem / case Best result

TSP (https://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/STSP.html)

gr17 2085gr120 6942

MDVRP (http://www.bernabe.dorronsoro.es/vrp/)

p01 576.87p04 1001.59

The best data for MST is not obtained because there is no record that provides the best results for the dataset from TSPLIB.

In general, the empirical results from the execution of the application against the test data is: (a) In the TSP problem of case gr17, three

algorithms (Nearest Neighbor, Repetitive Nearest Neighbor, Cheapest Link) give results that do not differ much, although the result was not any better than record in Table 2. Nearest Insertion Heuristic

algorithm gives best result, while the Genetic Algorithm gives the worst result

(b) In the TSP problem of case gr120, only two algorithms (Repetitive Nearest Neighbor and Cheapest Link) provide a worse result but not too much different, followed by Nearest Neighbor, and a Genetic Algorithm that gives the worst results. Nearest Insertion algorithm gives the best results.

(c) Genetic Algorithm gives the worst result, even tend to differ very much as number of vertex increases, presumably because it involves a random process at the time of the initial determination of the chromosome so that the results obtained can be uncertain. For TSP problem in graph with many vertices, Genetic Algorithm is discouraged because deviations can be even greater.

(d) In MDVRP problems, for both cases the results obtained are relatively closer to the best results, although with the larger value. Possible causes are due to differences in the determination of the customer bounded to a particular depot (clustering step) and the time saving for the determination of the initial route.

CLOSING

Applications created is a stub for the development of applications that can integrate various algorithms necessary and commonly used. The results still have not been able to approach the best results from standard data, but the differences that occur can be tolerated. The better algorithm is still searched, to get a more better result.

Suggestions for the development of such applications are: (a) Searching algorithm which can provide

relatively better results(b) Improving the implementation of the

algorithm in order to obtain a more efficient process

(c) Development of an application dinamically, in order to ease integration of a new algorithm into the application.


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REFERENCES

Dwivedi, Varshika, et.al. 2012. Travelling Salesman Problem using Genetic Algorithm. Proceeding of National Conference on Development of Reliable Information Systems, Techniques and Related Issues (DRISTI) 2012 (http://research.ijcaonline.org/dristi/number1/dristi1007.pdf, accessed in 1 Mei 2014)

Kumar, Suresh Nanda, and Ramasamy Panneerselvam. 2012. A Survey on the Vehicle Routing Problem and Its Variants. Intelligent Information Management, 2012, ed. 4, p. 66-74 (www.scirp.org/journal/PaperDownload.aspx?PaperID=19355, accessed in 20 Mei 2014)

Masruroh, Annisa. 2012. Algoritma Clark and Wright Pada Multi-Depot Vehicle Routing Problem (MDVRP). Universitas Negeri Malang, unpublished theses

Mitchell, Melanie. 1996. An Introduction to Genetic Algorithm. MIT Press.

Musser, Gary, dan Lynn Trimpe. 2007. A Mathematical View of Our World. Cengage Learning

(http://books.google.co.id/books?id=hplkLvGkuzwC&pg=PA389&dq=cheapest+link+algorithm&hl=en&sa=X&ei=PyVdU8f3IZGOiAfR-4HoBg&ved=0CC0Q6AEwAA#v=onepage&q=cheapest%20link%20algorithm&f=false, accessed in 1 Mei 2014)

Reinelt, Gerhard. 1995. TSPLIB 95. (http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/DOC.PS, accessed in 1 Mei 2014)

Rosen, Kenneth H, et.al (editor). 2000. Handbook Of Discrete And Combinatorial Mathematics. Florida: CRC Press.

Satynananda, Darmawan. 2012. Struktur Data. Universitas Negeri Malang, unpublished dictates

_____. TSPLIB-XML Description(http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/XML-TSPLIB/Description.pdf, accessed in 1 Mei 2014)

_____. Reverse-Delete Algorithm(http://en.wikipedia.org/wiki/Reverse-delete_algorithm, accessed in 1 Mei 2014)


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