Application of Graph Theory in Operations Research

Volume 2, Issue 5, May 2017 International Journal of Innovative Science and Research Technology
ISSN No: - 2456 - 2165
Application of Graph theory in Operations Research

Sanjay Kumar Bisen
Faculty Mathematics
Govt. P.G. College, Datia (M.P.)
(Affiliated to Jiwaji University Gwalior) India
ABSTRACT:- One of the common themes in operation A. Job scheduling

research is the modeling approach, many accurate model
of operations research. Problems turn out to be intractable Is the process of allocating system resources to many
when subjected to standard technique this research paper different tasks by an operating system, The system handles
show that how graph theory and networks may be prioritized job queues that are awaiting CPU time and it
profitably used to model certain discrete operations should determine which job to be taken from which queue and
research problem from a different view-point effective the amount of time to be allocated for the job.
algorithms.
B. Time table scheduling
Keyword:- Graph, Direct graph, Graph networks, Simple
graphs. Consider a college with M teachers T1, T2, T3,Tn
and n classes C1, C2, C3,.Cn given that teacher T1 is
I. INTRODUCTION required to teach class Cj for Pij periods schedule a complete
time table in the minimum possible numbers of periods. The
Graph purpose in operating system. Processes are problem can be formulated as graph theory model in which a
represented in graph theory. One of the common themes in bipartite graph G = V, E is devised with V = V1 U V2
operation research is the modeling approach. Unfortunately where V1 = T1,T2,. And V2 = C1,C2,..Cn
many accurate models of operations research problems. Turn and points T1 and Cj are joined by Pi lines.
out to be intractable when subjected to standard techniques.
However certain discrete problems can be profitably analyzed The timetabling problem is equivalent to coloring the
using graph theoretic models. points in V1 so that no two adjacent points have the same
colour with as few colours as possible each Colour
This paper introduces useful concepts from graph theory representing a distinct period.
and shows how they may be used to look at certain operations
research problems from the viewpoint of the graph theory. C. One more timetabling problem
Many of the Indian industries making use of operations The purpose of this paper is to explain, Now that
research activity are cloth mills, Indian Railway, Indian Graph theory has become a systematic tool. How it can be
Airline and Defense organizations and other corporation in used to yield new insights in to operations research models.
India While making use of the techniques of operations One simple use of graphs in problem solving in any field
research. A mathematical model of the problem in formulated including operations research is the following. It is often
.this model is actually a simplified representation of the convenient to depict the relationship between elements of a
problems in graph theory. system by means of a graph. Thus one often sees a
transportation, assignment, or PERT problem represented by a
Graph theoretic models part 1 and 2 discuss paper of graph. Such representations are often an aid in describing a
certain operations research .problems based on graph. Directed problem and are useful as such. This paper is concerned not
graphs respectively in this paper. just with way of representing an operations research problem
in terms of graph theory but more with using the results of
II. APPLICATION OF GRAPH THEORY IN graph theory to actually solve the problem. Not every
OPERATIONS RESEARCH conceivable graph theory applications are documented rather.
Some of the major applications are presented to give an
Graph theory is a very natural and powerful tool in overall flavor to this area.
combinatorial operations research. Some important operations
research problems that can be solved using graphs. A D. Graph theoretic Algorithms
networks called transport network where a graph is used to
model the transportation of commodity from one place to In this paper we adopt the following specialized
anothers the objective is to maximize the flow or minimize definition of an algorithm. An algorithm for a problem is a
the cost within the prescribed flow scientific procedure which is guaranteed to converge to an
optimal solution of the problem. This part discusses effective
IJISRT17MY45 www.ijisrt.com 162

ISSN No: - 2456 - 2165
graph theory algorithms for certain operations research In order to establish the differences, we must consider the
problems. order of the vertices for each graph. The following
E. Chinese postmans problem Vertex Order

A 3
In 1962, A chiness mathematician called Kuan Mei- B 3
ko was interested in a postman delivering mail to a number of C 3
streets. Such that the total distance walked by the postman was D 3
as short possible. How could the postman ensure that the Graph 01
distance walked was minimum.
Vertex Order
Following example: - A postman has to start at A, A 3
walk along all 13 streets and return to A. The numbers on each B 4
edge represent the Length, in meters, of each street. The
C 4
problem is to find a train that uses all the edges of a graph
D 3
with minimum Length
E 2
Graph 02
Vertex Order
A 4
B 4
C 4
D 4
E 2
F 2
Graph 03
When the order of all the vertices is even the graph is

Fig. 01. Chinese postmans problem Traversable. When there are two odd vertices we can draw the
graph but the start and end vertices are different. When there
We will return to solving this actual problem later, but initially are four odd vertices the graph cant be drawn without
we will look at drawing various graphs. The Chinese postman repeating an edge.
is traversable graphs given below.
Chinese postman algorithm :-
An algorithm for finding an optimal Chinese postman route is.

Step 1 :- List all odd vertices.
Step 2 :- List all possible paring of odd vertices.
Step 3 :- For each pairing find the edges that connect the
vertices with the minimum weight.
Step 4 :- Find the pairing such that the sum of the weights is
minimized.
Step 5 :- On the original graph add the edges that have been
found in step 4.
Step 6 :- The length of an optimal Chinese postman route is
the sum of all the edges added to the total found in step 4.
From these Graph we find; Step 7 :- A route corresponding to this minimum weight can
It is impossible to draw graph 1 without either taking then be easily found.
the pen off the paper or re- tracing an edge. Now we apply the algorithm to the original problem
We can draw graph 2, but only by starting at either A in fig. 01.as;
or D-in each case the path will end at the other vertex Step 01 the odd vertices are A and H.
of D or A. Step 02 there is only one way of pairing these odd vertices
Graph 3 can be drawn regardless of the starting namely AH.
position and you will always return to the start Step 03 the shortest way of joining A to H is using the path
vertex. AB, BF, FH a total length of 160.
Step 04 these edges on to the original network in this fig

ISSN No: - 2456 - 2165
Step 05 the length of the optimal Chinese postman route is the REFERENCES
sum of all the edges in the original network. Which is 840 mtr
Plus the answer found in step 4, which is 160 mtr., Hence the [1]. Apple J. M. Plant layout and materials Handling 2nd
length of the optimal Chinese postman route is 100 mtr. Ronald press New York (1963).
Step 06 one possible route corresponding to this length is [2]. Bondy J.A. and U.S.R. Murty, graph theory with
ADCGHCABDFBEFHFBA, but many other possible routes applications MacMillan London.
of the sum minimum length can be found. [3]. Application of graph theory of computer science an
overview by S. G. Shrinivas, S. Vetrivel and Dr.
III. DIGRAPH METHODS IN OPERATIONS N.M. Elango. (International journal of engineering
RESEARCH science and technology vol. 2 (9) 2010.4610-4621.
[4]. Application of graph theory in communication
A directed graph or digraph for short is a graph in which networks by suman Deswal and Anita Singhrova.
each line has been given a direction. The lines have arrows International journal of Application of Application or
attached to them to indicate their direction many real world innovation in Engineering and Management Volum
systems which operations research .Analysts study can be 1, issu 2 And October 2012.
modeled as binary relation between the elements of the system [5]. ORE, O, theory of graphs American mathematical
.Since this is what a digraph is .digraph theory can often be society providence R.I. 1962.
applied to analyses operations research. [6]. Graph theory with application to engineering and
computer science Narshing Deo.
A. Game theory [7]. Graph theory A survey of its use in orations research
by L.R.FOULDS University of Canterbury
When I first skimmed the theory of games and economic Christchurch, N. Z jan (1982).
behavior in 1948 I did not really understand it, but I sensed [8]. Operations research by S.D. Sharma MEERUT
that this was the way to go in the study of multi person publishers India.
conscious strategic behavior. I had heard a little about [9]. Operations research by R.K . Gupta Krishna
operations research casually in 1944 in two applications. One Prakashan MEERUT India.
concerning how to aim an anti-aircraft gun to take account of [10]. Game theory and operations research 1-2002 Martin
the planes motions during the time it took to reach it after Shubik (http:IItrace.tennessee.edu/utk-harlanabout).
firing and I had a vague idea that one could try to analyze the
best ways for convoy defense by using some form of
mathematics.
The game theory methods to operations research

economics, political science, and management science. My
main purpose is to cover the evolution of the relationship
between game theory and operation research rather than to
review all of operation research. I limit my broader comments
on operation research with a few pan glossies remarks. It is
my belief that operations research has been so successful that
it may have put itself out of business. At least in its easy to-
recognize sense It has succeeded to the extent that it is taught
in a move of less routine and watered- down manner in every
business school linear programming, queuing studies, and
elementary competitive models.
IV. CONCLUSIONS
It must now be clear to the author that graph theory and

its offshoots, the theory of digraphs and networks, have
blossomed not only as a branch of mathematics but also as
systematic tools in operations research it has been shown that
graph theoretic variety of operations research .this paper is
designed to benefit the students of computer science to gain
depth knowledge operations research.

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Volume 2, Issue 5, May 2017 International Journal of Innovative Science and Research Technology

ISSN No: - 2456 - 2165

Application of Graph theory in Operations Research

ABSTRACT:- One of the common themes in operation A. Job scheduling

IJISRT17MY45 www.ijisrt.com 162

E. Chinese postmans problem Vertex Order

When the order of all the vertices is even the graph is

An algorithm for finding an optimal Chinese postman route is.

IJISRT17MY45 www.ijisrt.com 163

The game theory methods to operations research

It must now be clear to the author that graph theory and

IJISRT17MY45 www.ijisrt.com 164

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