The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
DEVELOPMENT OF A SPECIAL MATRIX TECHNIC FOR
ROAD NETWORK ANALYSIS
Case of Identifying Un-Connected and Miss-connected Node
Hitapriya Suprayitno
Lecturer-Researcher - Civil Engineering Department
Institute of Technology Sepuluh Nopember (ITS) - ITS Campuss - Surabaya
HP : 0811.327.813, e-mail : [email protected]
Abstract
Network analysis is needed in a lot in transportation research and transportation study. This needs a computer
aided calculation. As spread-sheet is the most versatile for this purpose, to maximize its advantages network
data must be presented in a matrix form equipped with special matrix operation. A special matrix technics
need to be developed. The special matrix technics has been developed successfully. It consists of matrix
convention and several matrix operation formulae, which is needed to identify un-connected and missconnected node, has been developed. An example of calculation method trial is also presented.
Keywords: road network analysis, special matrix technic.
INTRODUCTION
We face a lot of Network Analysis in Transportation. We can mention, for example : Transport
Modelling, Shortest Path, Public Transport Planning, Road Network Planning, Traffic Assignment
and others.
Road Network Calculation, in a lot of cases, deals with a vaste network with a lot of nodes, links
and of course data. Therefore, a computer aided calculation is inevitable. Simple and powerful
software, very well known and every body can use it, is a spread-sheet type software. Nowadays,
we know the very popular Microsoft Excell. It is very handy, practical and powerfull. In order that
the spread-sheet powerfull facilities can be used in maximum, the network data must be presented
in a matrix form and the calculation must be in a form of matrix calculation (Suprayitno 2014).
A general matrix form and its calculation method needs to be developed. This must be oriented to
be used under spread-sheet programme and can be programmmed easily by using a programming
language. The first step is to define the special matrix form, followed by a review of related
mathematics knowledge – trying to identify existing mathematical calculation method which can be
used and ended by developing several calculation method, matrix operation, which are needed to
identify un-connected and miss-connected node.
This paper present the basic result of an effort to build the “Special Matrix Technic for
Transportation Network Analysis”. The research produce several calculation methods (Suprayitno
2008, Suprayitno 2009, Suprayitno 2010, Suprayitno 2014, Suprayitno et al 2014, Suprayitno et al
2014a). But since the paper is restricted in length, only basic result is presented here.
LITERATURE REVIEW
Network Model
Road Transportation Network Model are developped in various forms depend on the object
modeled and its purpose. Major models are e.g simple node-links model, 4 steps transportation
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
model, traffic flow network model, cell transmission model, etc. (Binning & Crabtree 2002;
Daganzo 1994; INRO 1999; Ortuzar 1998; Tamin 2008).
Mathematics
Graph Theory
Graph Theory is branch of mathematics, specially designated for graph (network) analysis, which
is already very well developed. It has several main characteritics. A graph consists of several edges
connected each other at vertexes, the vertex coordinates are not considered at all. The graph is
grouped into planar graph and and non planar graph. Those two have different characteristics.
Graph data can be presented as a list of edges data, an incidence matrix of edge data and an
adjacency matrix. In adjacency matrix, all cell are used for presenting edge data (Bondy & Murty
1982).
Network Analysis
Network Analysis is a branch of Mathematics, specially designated to discuss network analysis. It
is actually a Graph Theory with different name (Brandes & Erlebach 1998).
Operation Research – Network Case
Operation Research is a mathematic branch dealing with optimization. It is already very well
developed, either in deterministic or stochastic problem, continue or discrete problem. Network
optimization is also treated for the following case, e.g : critical path method, minimum shortest
path, optimum spanning tree, optimum flow, etc. The minimum shortest path algorythm, started
from Dijkstra algorythm to Kruskal-Wallis algorythm and the others, based their calculation on
data in a list form with heuristic type and enumeration type calculation (Goldberg 2000; Hillier &
Lieberman 1990; Dimyati & Dimyati 1994; Suprayitno 2008; Taha 1992).
Discrete Mathematics – Network Case
Special mathematics designated to deal with discrete phenomena. This mathematics is developed
by informatics domain, for their purpose. It consists of all kind of discrete phenomena found in
informatics, including logique, algorythm, network anlysis and others. In case network
optimization, the content is the same as in operation research (Munir 2012)
Linear Algebra
Linear algebra deals with all aspects of linear equation. The most important part to be noted are :
the simultaneous linear equations, the linear inequalities, the vector presentation of linear equation.
The simultaneous liniear equations are deeply used in structural engineering. Threre are three type
of simultaneous linear equations calculation method : elimination type, heuristics type and matrix
type (Arifin 2001; Assakkaf 2001).
Matrix Theory
It’s all about matrix. Matrix can be of any form : column matrix, row matrix, nxm, nxn, etc. Matrix
operation : multiplication, addition, substraction, invers, determinant, eigenvalue, eigenvector, etc.
Matrix utilisation for solving simultaneous linear equation (Brand & Sherlock 1970).
Set Theory
Set Theory is a branch of mathematics dealing with the problem and characteristics of set (Weiss
2008). This will be useful for matrix operation with a nature of a set theory operation, like
intersection, union, etc.
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Max-Plus Algebra
Max-plus algebra was developed by control system domain in France for discrete event system
calculation. It’s about special algebra mathematic operation for constants, variable and matrix. It
has been started to be used to solve transportation problem for network schedulling (Baccelli et al
1992; Schutter & Boom 2008; Vries, Schutter & Moor 1998).
Matrix Calculation in Transportation Science
Trip Distribution Calculation
Considered often as a second step of transport modelling, the trip distribution is to calculate OriginDestination matrix, based on trip generation data and certain impedance model. Since the objective
is to produce OD matrix, the whole calculation is done by using matrix calculation technics (Ofyar
2008; Ortuzar & Willumsen 1994). The basic form of OD matrix must be very considered in
defining the special matrix technic to be developed.
Geography of Transport Calculation Method
Certain matrix calculation method for calculating network quality has been developed by
geography of transport. It mus be noted that this network quality has been derived from graph
theory principal. Therefore, certain quality components is not quite right. Certain calculation is still
limited (Taaffe et al 1996).
Summary
The literature review gave the following important conclusions. The appropriate network model is
simple node-link network model. The network analysis are calculated based on list of network data,
and not based on matrix data. The matrix theory has its form of matrix which is different from the
defined special matrix form. Therefore, the calculation method is still to be developed based on
network data presentation in a matrix form and matrix operation for calculation. The matrix must
be in concordance with the matrix used in trip distibution model calculation.
SPECIAL MATRIX TECHNIC DEVELOPMENT
Basic Reason
In almost of all cases, network analysis deal with a big network consisting of a lot of data. To
calculate a big data manually without using a computer help is nonsense. We know two kind of
computer softwares : spread sheet and programming language. The popular spreadsheet type
software is Microsoft Excell. The advantage of spreadsheet type software can only be used in a
maximum, only if the network data are structured in a matrix form, thus the calculations are also in
a form of matrix calculation.
In the existing mathematics dealing with network analysis, graph theory, network theory, operation
researches and discrete mathematics, practically all of the time the network data are presented in a
list form. Therefore Special Matrix Form and Special Matrix Calculation need to be developed.
This development dealt with : problem, network model, network data, matrix convention and
matrix calculation.
First Objective
The first objective of this special matrix development is to identify the existence of un-connected
node and miss-conneccted node. These are thus :
 Node with no connection at all.
 Node with one-way connection : to or from.
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Network Model
Network model is the means to represent the road network. Traffic flow is not yet considered here.
The link length calculation is based on a straight line length. Thus the road segment model is a
straight line. Therefore, the network model must be able to model : road segment, intersection and
curve segments in several straight lines. This model are presented in the following Figure 1 below.
Figure 1 Road Network Model
Basic Network Data
Basic data on road network are road segments data and nodes data. For this first objective, the main
node data are the location coordinate. While the road segment data consists mainly of data on the
segment existence. These data will vary in function of problem considered.
Basic Matrix Conventions
To fulfill the objective of developing this special matrix, a general matrix convention has been
defined as follows :
 Matrix is for representing road network model data : node and link data.
 Basic matrix is an n x n matrix.
 Nodes data are written in the diagonal cells, thus Node Matrix is an diagonal matrix.
 Links data are written in the non-diagonal cells, thus in Link Matrix diagonal cells are
always empty.
 The matrices can be simetrical, while the whole links are two way links.
Basic Matrix Notations
All notation must be conventioned systematically, wether for constant, variable, matrix and
mathematic operation. Basic notations is taken as follow :
 Notation is an abbreviation of general lexic in English.
 All mathematics operation use the existing Standard Mathematical Notation, but if there
is a new one, it wil be given a notation of a Greek alphabet or others.
 All Constants are given a capital alphabet : C.
 All Variables are given a small case alphabet : v.
 All Sets are given general notation as follow : s.S
 All Matrices are given the following notations : m.M, me.M, mi.M
 Mathematic Operators needed are :
o If
: 
o Member
: 
o Node Data Import
: |
o Link Data Import
: ||
o Matrix Expansion
: χ
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Basic Matrix Forms
To fulfill the research objective, three types of matrix are needed : Basic Matrix, Expanded Matrix
and Identification Matrix. Those three are presented in Table 1 below. Those three are related each
other, as presented in formula written below.
Table 1 BM - basic matrix, EM - expanded matrix and IM - identification matrix.
BM 1
2
3
4
EM 1
2
3
4
SR
IM
1
1
1
2
2
2
3
3
3
4
4
4
SC
SC
1
2
3
4
SR IR
IC
Expanded Matrix
Expansion Matrix is always an expansion of a basic matrix, to contain : row, column and matrix
summation. The summation formula is presented below.
me.E = m.B + SRi + SCj + SM
SRi =  mIj
SCj =  miJ
SM =  mij or  SRj or  SCi
Identification Matrix
Identification Matrix, most of all is an expansion of an Expanded Matrix, to contain certain
identification, based on Expanded Matrix Value, for certain characteritics of the network.
mi.I = me.E + IRi + ICj + SI
IRi =
ICj =
SIR
SIC
C1 for IRi = mathematical condition, else C2
C1 for ICj = mathematical condition, else C2
=  IRi
=  ICj
Basic Interpretation of Basic Matrix and Expanded Matrix
Data written on Basic Matrix and Expanded Matrix can be interpreted as follow. In general mij
indicate links connecting node i to j. The data can represent various informations, such as the
existence, length, speed and others. For the Expanded Matrix, while the Basic Matrix indicate the
link existence, then SRi indicate number of links connected from node i and SCj indicate number of
links connected toward node j.
Basic Mathematical Formulae
To acomplish the research objective, to identify un-connected node and miss-connected node,
several mathematical formula are needed : basic matrix development – either node matrix and link
matrix, expanded matrix development, identification matrix development. Thus different formula
are presented below.

Basic Matrix Development – Node Matrix
m.X = | t.X[i,a.c,a.d]
m.Xci,ci = di
m.Xi,j
= 0, untuk i  j
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015

Basic Matrix Development – Link Matrix
m.X = || t.X[i,a.c1,a.c2,a.d]
m.Xc1i,c2i
= di
m.Xi,i
= 0

Expanded Matrix Development
me.X = χ m.X

Identification Matrix Development – Link Existence
mi.X = χ me.X
IRi = C1 for IRi = C, else C2
ICj = C1 for ICj = C, else C2
Identifying Un-connected Node and Miss-connected Node
Un-connected Node : Isolated Node
Un-connected Node, can be said as Isolated Node, is a node onwhich there is no link connected at
all. In geoghraphy of transport it is called as a node with a degree value equal to zero. An example
of this case is presented by Figure 1 and Tabel 1 below.
Figure 2 Network with Un-connected Node
Table 1 Expanded Matrix for Example Network
me.LE
1
2
3
4
5
SC
1
0
1
0
1
0
2
2
1
0
1
1
0
3
3
0
1
0
0
0
1
4
1
1
0
0
0
2
5
0
0
0
0
0
0
SR
2
3
1
2
0
Node 5 is an Isolated Node with SR5 = SC5 = 0. The existence of an Isolated Node (Un-connected
Node) as above then mathematically can be formulated below.
n = isolated,  SCn = SRn = 0, where SCn & SRn  me.LE
Miss-connected Node : One One-Way Node Connection
Miss-connected Node is a node with only one way connection, either to or from. It can be of one
connection or more than one connections. This paper discuss a One One-Way Connected Node
only. An example is given below.
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Figure 3 Network with Miss-connected Node
Table 2 Expanded Matrix for Sample Network
me.LE
1
2
3
4
5
SC
1
0
1
0
1
0
2
2
1
0
1
1
0
3
3
0
1
0
0
0
1
4
1
1
0
0
0
2
5
0
0
1
0
0
1
SR
2
3
2
2
0
Node 5 is a Miss-connected Node, with SR5 = 0 and SC5 = 1. The existence of a Miss-Conneceted
Node – One One-Way then mathematically can be formulated below.
n = miss-connected,  SRn = 0 & SCn ≠ 0 or SRn ≠ 0 & SCn = 0, where SRn & SCn  me.LE
CALCULATION TRIAL
Trial Case
The method developed above has to be tried. A Network with 5 nodes and 9 links, 5 two-way links
and 4 one-way links was taken as a Trial Case. The problem is to find the un-connected node and
the miss-connected nodes. The Trial Case is presented in Figure 4 and Table 3 below.
Figure 6 Trial Network
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Table 3 Basic Matrix – Links Existence
m.LE
1
2
3
4
5
6
7
8
9
1
0
1
0
1
0
0
0
0
0
2
1
0
1
1
0
0
0
0
0
3
0
1
0
0
0
0
0
0
0
4
1
1
0
0
0
1
0
0
0
5
0
0
0
0
0
0
0
0
0
6
0
0
0
1
0
0
0
1
1
7
0
0
0
0
0
1
0
0
0
8
0
0
0
0
0
0
0
0
0
9
0
0
1
0
0
0
0
0
0
Trial Case Calculation
An Expanded Matrix and an Identification Matrix are needed to find or to calculate the existence of
the un-connected node and the miss-connceted node. The Expanded Matrix is calculated as
mentioned above. The Expanded Matrix is presented in Table 4.
me.LE = χ m.LE
Table 5 Expanded Matrix – Links Existence
me.N
1
2
3
4
5
6
7
8
9
SCi
1
0
1
0
1
0
0
0
0
0
2
2
1
0
1
1
0
0
0
0
0
3
3
0
1
0
0
0
0
0
0
0
1
4
1
1
0
0
0
1
0
0
0
3
5
0
0
0
0
0
0
0
0
0
0
6
0
0
0
1
0
0
0
1
1
3
7
0
0
0
0
0
1
0
0
0
1
8
0
0
0
0
0
0
0
0
0
0
9
0
0
1
0
0
0
0
0
0
1
SRj
2
3
2
3
0
2
0
1
1
14
Due to the Miss-Connected Node formulated above, the Indentification Matrix is developed mainly
to identify the existence of SRn = 0 and SCn = 0. Therefore the matrix development formula is
written as follow.
mi.LE = χ me.LE
IRj = 1  SRj = 0, else = 0
ICi = 1  SCi = 0, else = 0
Table 6 Identification Matrix – Links Existence
mi.LE
1
2
3
4
5
6
7
8
9
1
0
1
0
1
0
0
0
0
0
2
1
0
1
1
0
0
0
0
0
3
0
1
0
0
0
0
0
0
0
4
1
1
0
0
0
1
0
0
0
5
0
0
0
0
0
0
0
0
0
6
0
0
0
1
0
0
0
1
1
7
0
0
0
0
0
1
0
0
0
8
0
0
0
0
0
0
0
0
0
9
0
0
1
0
0
0
0
0
0
SRj
IRj
2
3
2
3
0
2
0
1
1
0
0
0
0
1
0
1
0
0
SCi
ICi
2
3
1
3
0
3
1
0
1
14
0
0
0
0
1
0
0
1
0
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Calculation Result
The Identification Matrix gave us three nodes which are conform to the characteritics of unconnected node and miss-connected node. Thus three node are analysed below.
 Nodes 5 : since IR5 = 0 & IC5 = 0 →
Node 5 = un-connected node.
 Nodes 7 : since IR7 = 0 & IC7 = 1 →
Node 7 = miss-connected node.
 Nodes 8 : since IR8 = 0 & IC8 = 1 →
Node 8 = miss-connected node.
CONCLUSIONS
The research objective has been successfully fulfilled. The most important thing is that a network
analysis calculation method which can be executed by using spread-sheet type program has been
developed, the advantage of the spread-sheet can be utilized in maximum by creating special
matrix and special matrix calculation. More than this, this special method can be programmed also
by using a programming language. The principal research result are mentioned as follow :
 Special matrix form has been developed.
 Matrix special characteristics has been mentioned.
 Several calculation method has been developed :
o Expansion matrix
o Identification matrix
It can be understood easily that the utilisation of this Special Matrix Technic can be expanded for
any kind of transportation mode network. For the reason of practicality and easiness, it is better to
give a name to this method. The researcher think of the following name :
 Special Matrix Technics for Transportation Network Analysis
 Matrix for Transportation Network Analysis.
ACKNOWLEDGEMENT
This paper is a small part of writter’s dissertation, the very basic part. Since this is new and useful, it is very
important to be socialized and to be criticized. The other parts will be published step by step.
REFERENCES
Arifin, Achmad. (2001). Aljabar Linier. Edisi Kedua. Penerbit ITB, Bandung.
Assakkaf, Ibrahim A (2001). “Chapter 5f. Simultaneous Linear Equations”. ENCE 2003 –
Computation Methods in Civil Engineering II. Department of Civil and Enviromental
Engineering. University of Maryland, College Park. www.assakkaf.com/courses. Accessed
11 April 2015.
Baccelli, F., Cohen, G., Olsder, G.J. & Quadrat, J-P. (1992). Synchronization and Linearity – An
Algebra for Discrete Event Systems. John Wiley & Son. New York.
Binning, J.C. & Crabtree, M.R. (2002). Transyt 11 User Guide. TRL Software Bureau.
Crowthorne.
Bondy, J.A. & Murty, U.S.R. (1982). Graph Theory with Applications. Fifth Printing. NorthHolland. New York.
Brand, T. & Sherlock, A. (1970). Matrices : Pure and Applied. Contemporary Mathematics Series.
Edward Arnold. London.
Brandes, U. & Erlebach, T. (1998). Network Analysis – Methodological Foundations. Springer,
Berlin.
Goldberg, A.W. (2000). “Basic Shortest Path Algorithms”. DIKU Summer School on Shortest
Paths. Microsoft Research. Silicon Valey.
Hillier, F.S. & Lieberman, G.J. (1990). Introduction to Operation Research. Fifth Edition, HoldenDay Inc.. San Francisco.
The 18th FSTPT International Symposium, Unila, Bandar Lampung, August 28, 2015
Chevalier, A. & Hirsch, G. (1980). Méthodes Quantitatives pour le Management : Finances,
Marketing, Production. Entreprise Moderne d’Édition. Paris.
Daganzo, Carlos (1994). “The Cell Transmision Model : Network Traffic”. Working Papers.
Institute of Transportation Studies, University of California. Berkeley.
Dimyati, T.T. & Dimyati A. (1994). Operation Research – Model-Model Pengambilan Keputusan.
Sinar Baru Algesindo. Bandung.
INRO (1999). EMME/2 User’s Manual – Software Release 9. INRO Consultant Inc. Montreal.
MathWorks (2005). Learning Matlab 7 – Matlab & Simulink Student Version. The MathWorks,
Inc. Natick, Massachusetts.
Munir, Rinaldi (2012). Matematika Diskrit. Edisi Kelima. Penerbit Informatika. Bandung.
Ortuzar, J.D. & Willumsen, L.G. (1994). Modelling Transport. Second Edition. John Wiley &
Sons. New York.
Pressman, R.S. (2002). Rekayasa Perangkat Lunak. Terjemahan dari ”Software Engineering, A
Practitioner’s Approach”. Penerbit ANDI. Yogyakarta.
Schutter, B de. de & Boom, T van den. de (2008). “Max-plus Algebra and Max-Plus Linear
Discrete Event Systems : An Intoduction”. Proceeding of 9th International Workshop on
Discrete Event Systems (WODES ’08). Goteborg.
Suprayitno, Hitapriya (2008). “Metoda Enumerasi Total bagi Perhitungan Lintasan Terpendek
Antara Dua Titik”. Seminar Nasional Teknik Sipil IV, Suirabaya, 13 Februari 2008. Jurusan
Teknik Sipil. Institut Teknologi Sepuluh November (ITS), Surabaya.
Suprayitno, Hitapriya (2009). “A Special Matrix Method to Identify a Tree Network”; ISSTEC
2009, 24-25 January 2009 – International Seminar on Science and Technology. Universitas
Islam Indonesia (UII), Yogyakarta.
Suprayitno, Hitapriya (2010). “Identifikasi Graf Pohon Berbasis Aljabar Maks-Plus – Pada Graf
Tak Berarah Tak Berbobot”; Seminar Nasional Teknik Sipil VII, Januari 2010; Jurusan
Teknik Sipil, Intitut Teknologi Sepuluh Nopember (ITS); Surabaya.
Suprayitno, H., Mochtar, I.B. & Wicaksono, A (2014). “Assessment to A Max-Plus Algebra Power
Operation on UnWeighted Transportation Network Model of Its Behavior, Connotation, and
Utilization”. Prosiding Seminar Nasional X – 2014. Jurusan Teknik Sipil. Institut Teknologi
epuluh Nopember.
Suprayitno, H., Mochtar, I.B. & Wicaksono, A (2014a). “A Special Matrix Power Operation
Development for Simultaneous Calculation of All Network’s Shortest Path”. JATIT, 10 April
2014, Vol 62 no 1 2014. Journal of Applied and Theoritical Information Technology.
Suprayitno, Hitapriya (2014). Metoda Penilaian Kualitas Jaringan Jalan Utama – di Wilayah
Kabupaten. Disertasi RC 09-3399. Jurusan Teknik Sipil. Fakultas Teknik Sipil dan
Perencanaan. Institut Teknologi Sepuluh November. Surabaya.
Taaffe, J.E.; Gauthier, H.L. & O’Kelly, M.E. (1996). Geography of Transportation. Second
Edition. Prentice Hall. New Jersey.
Taha, Hamdy A. (1992). Operation Research – An Introduction. Fifth Edition. Maxwell Macmillan
International. New York.
Tamin, O.Z. (2008). Perencanaan, Pemodelan dan Rekayasa Transportasi : Teori, Contoh Soal
dan Aplikasi. Penerbit ITB. Bandung.
UAG (1995). TRANPLAN – Transportation Planning Software – User Manual. The Urban Analisis
Group and RUST PPK. Danville.
Vries, R. de; Schutter, B. de & Moor, B. de (1998). “On Max-Algebraic Models for Transportation
Networks”. Proceeding of 4th International Workshop on Discrete Event Systems (WODES
’98). Cagliari.
Weiss, A.R. (2008). Introduction to Set Theory. University of Toronto, Toronto.