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Fuzzy optimization scheduling using coloring fuzzy graph
Safar Mohammad noori
1
Department of Mathematic, Shahrood university of technology
Shahrood, Republic Islamic of Iran
Mohammad.noori@gmail.com
Abstract
The ( d , f )  extended coloring of a fuzzy graph was defined
by the authors in S. Munez, M.TeresaOrtuna, J. Ramirez J.
Yanez (2006). In this paper is defined a type of coloring that give
the chromatic fuzzy sum for ( d , f )  extended coloring of a
fuzzy graph and is expressed a model for this coloring using
integer linear programming. In this paper one of the applications
of this coloring in exam scheduling problem is studied that the
exam primary days have higher graduation and the exam last
days have less graduation. For this graduation express a
membership function. This coloring, for this membership
function present optimal answer, that sum of graduated assigned
colors of fuzzy graph is maximum value over all proper the
( d , f )  extended coloring of the fuzzy graph.
Keywords: chromatic fuzzy sum, graph theory, optimization,
exam scheduling problem, fuzzy set.
1. Introduction
Graph coloring is one of the most studied problems of
combinatorial optimization .Many problems of practical
interest can be modeled as coloring problems. The general
form of this application involves forming a graph with
nodes representing items of interest. The basic graph
coloring problem is to group items in as few groups as
possible, subject to the constraint that no incompatible
items end up in the same group.
Definition1.1. Let G  (V , E ) be a graph with a vertex
set V and an edge set E . A proper coloring of the vertices
of the graph G is assigning different colors to adjacent
vertices, a coloring function a mapping c : V  N
identifying c (i ) as the color of node i V , in such a way
that two adjacent nodes cannot share the same color, i.e.,
c(i )  c( j ) if {i, j}  E . The chromatic number
 (G ) is the smallest number of colors in a proper
coloring of G .
Definition1.3 A proper coloring c of a graph G is called a
best coloring of whenever
 G   c (v )
[1].
vV
It is known that computing the chromatic sum of an
arbitrary graph is an NP-complete problem [1]. The
vertex-strength of the graph G , denoted by s (G), is the
smallest integer s such that
 G is attained using colors
{1, 2, . . . , s}.
Fuzzy-set theory, introduced by Zadeh [2], is a
mathematical tool to handle uncertainties like vagueness,
ambiguity, and imprecision in linguistic variables, see also
[3, 4]. The first definition of fuzzy graph was proposed by
Kaufmann [5], from the fuzzy relations introduced by
Zadeh [2]. Although Rosenfeld [6] introduced another
elaborated definition, including fuzzy nodes and fuzzy
edges, in this paper we deal with graphs with crisp nodes
and fuzzy edges, see, for example, [7,8].See also [9] for an
overview on some optimization problems in fuzzy graphs.
We will use the classical definition of fuzzy set a defined
on a non empty set X as the family:
{( x,  A~ ( x)) / x  X }
membership function and
Where
 A~ ( x) : X  I is
 A~ ( x)
the
reflects the ambiguity
of the assertion x belongs to A . In classical fuzzy-set
theory the set I is usually defined as the interval [0, 1], in
such a way that
belong to A ,
 A~ ( x) =
0 indicates that
 A~ ( x) = 1 indicates that x
x
does not
strictly belongs
~
to A , and any intermediate graduation represents the
~
degree in which x could belong to A . In general, the set
I can be any ordered set, not necessarily numerical, for
instance, I = {null, low, medium, high, total}.
Definition 1.2 Let G  (V , E ) be a graph with vertex set
V
and edge E .chromatic sum of graph
the minimum sum
G,
 G ,to be
 c(v) taken over all proper coloring
1. 2 the ( d , f )  extended coloring function of a fuzzy
graph
vV
c
of
graph
G
using
natural
c : V  N , c(v)  c( w), (v, w)  E .
number
i.
e
In order to model certain types of problems as coloring
Problems, it is necessary to define dissimilarity or distance
measure on the color set. It can be done in the following
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IJCSMS International Journal of Computer Science & Management Studies, Vol. 11, Issue 01, May 2011
ISSN (Online): 2231 –5268
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way [10]. Let S be the available color set. Let d be the
dissimilarity measure defined by
d : S  S  [0, )
With the following properties:
1  d (r , s)  0
r , s  S
2  d (r , s )  0  r  s r , s  S
Definition 2.1Let
vertex set V and fuzzy edges E , Extended Minimum sum
~
This dissimilarity measure d can take into account the
incompatibility degree in the sense that the more
incompatible two nodes are, the more distant their
associated colors. In this way, an extended coloring
function is introduced. Given I , the image of the
~
~
membership function of the fuzzy graph G  (V , E ) let
f : I  [0, ) Be a non-negative, none decreasing (with
f (  )  f (  )
 ,    I s.t
  
(1)
The above two concepts of dissimilarity measure and scale
function allow the following definition.
~
~
Definition1.3. Given a fuzzy graph G  (V , E ) , color set
S,
a dissimilarity measure d defined on
S
~
~
G  (V , E ) is a fuzzy graph with
~
3  d (r , s )  d ( s, r ) r , s  S
respect to the order  ) and real scale function, i.e.
competition on resources, that is, two vertices are adjacent
if the corresponding processors cannot run their jobs
simultaneously. The objective is to minimize the average
response time, or equivalently to minimize the sum of the
job completion times. In MSC problem, conflict between
vertices is crisp, and then vertices and edges are crisp. For
condition that conflict between vertices is fuzzy, using
definition of 1-3, extending
MSC problem for fuzzy
graph we will have bellow definitions:
of coloring fuzzy graph G (EMSC) is the smallest sum of
colors over all proper the ( d , f )  extended coloring of a
fuzzy graph with natural numbers that can be dictated with

E
G.
~
Definition 2.2The vertex-strength of the fuzzy graph G ,
~
s , such that
E G is attained using colors {1,2,3,..., s} for short as
~
~
s .Taking Graph G  (V , Eij ) with vertices
denote by s (G ) , is the smallest integer
{v1 , v2 , v3 ,..., vn }
~
{ ij  Eij /{i, j}  E}
and
~  f (  ) , d (r , s ) 
That w
ij
ij
fuzzy
edges
r  s then modeling of
EMSC coloring problem is brought in section of 2-2.
and a scale
~
function f , a ( d , f )  extended coloring function of G ,
2.2 . A linear programming model for EMSC problem
denoted as C d , f , for short as c , is a mapping
c : V  S With the property:
d (c (i )  c ( j ))  f ( ij )
i , j  V i  j
2. Chromatic fuzzy sum the ( d , f )  extended
coloring of fuzzy graph
assigned colors of all the vertices of G is minimized. The
MSC problem has a natural Application in scheduling
theory. One of the applications is the problem of resource
allocation with constraints imposed by conflicting resource
requirements.
In a common representation of the
distributed resource allocation problem [11, 12], the
constraints are given by a conflict graph G, in which the
vertices represent processors, and the edges indicate
There are not many known polynomial algorithms for
computing chromatic sum for some families of graphs. Not
surprisingly, there is an efficient linear algorithm for tree
[1]. There is also a linear algorithm to compute chromatic
sum for unicycle graphs and a cubic algorithm for planar
graph [13].
For fuzzy graph haven’t expressed any algorithm yet.
Expressed model in (2) hold in all of arbitrary fuzzy and
weighted graph.
n
min
 c (v )
i 1
s.t
i
(2)
~
c (vi )  c (v j )  w
ij
i  1...n  1, j  i  1...n;
c(vi ), c(v j )  {1,..., s}
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c(vi ), c(v j )  {1,..., s} Are assigned colors to vertex,
~  f (  ) in crisp graph define following bellow
and w
ij
ij
wij


t
4
3. Maximum sum of graduated assigned
colors of fuzzy graph
Following definition 1.1 that colors is defined with natural
number, in normal case this number is crisp and has equal
graduation, in cases uncertainties like vagueness,
ambiguity, and imprecision use from fuzzy set theory and
is graduated number.
~
Definition (SGF): let G  (V , E ) be a fuzzy graph then
via coloring of fuzzy graph with colors of
{ c(vi ) / i  1,..., n} , SGF is sum of graduated assigned
colors to vertices of graph, so that color set of A is
graduated via membership function(3):



A
That
i
~
A  {( c(vi ),  A~ (c(vi ))
n

 c (vi )  (3)
i 1

i  {1...n}}
~
A Is a fuzzy set in interval [0, 1] that dictate grade of
each assigned color to vertex.
In other words if fuzzy graph of
then have:
c(vi )  {1,2,3,...}
be allocated colors to the vertices
c(vi )  {1,..., s} be
allocated
{vi i  1,2,3,..., n} and s is vertex strength, then have:
n
This model is a model of integer linear programming that
can be solved with soft ware. This coloring model can
solve EMSC coloring problem.
 ~ (c (vi ))  1   (c (v )  1)
If
colors to the vertices via coloring of EMSC that
~  f (  ) is defined in (1) and in
And in fuzzy graph w
ij
ij
~
Proof:
via coloring of C d , f and
1 {i , j }E
0 O .W
this paper scale function of f be defined by
I
n
l
m
h
f
0
1
2
3
3
~
~
G  (V , E ) is colored
n
~
SGF (G )    A~ (c(vi )
n
n
n
 c(v )   c(v )  1  c(v )  1  c(v )
i 1
n
i
n
i 1
i
 c(vi )  n
i 1
i 1
i
i 1
i
n
 c (v ) 
i 1
i
n
n




1   n  c(vi )   1   n  c(vi )  
 i 1

 i 1

n
n


 ( c(vi )  n)  c(vi )  
i 1
 i 1

n
n


 (  c (vi )  n )  c (vi ) 
i 1
 i 1

n
n


 n   ( c (vi )  n)  c(vi )  
i 1
 i 1

n
n


n   (  c (v i )  n )  c (vi )  
i 1
 i 1

n






1

(
c
(
v
)

1
)
c
(
v
)



1
1


i

1




n




1   (c(v2 )  1)  c(v2 )  
 
i 1



n



1   (c(v3 )  1)  c(v3 )    ... 
i 1



n





1

(
c
(
v
)

1
)
c(vn )  


n

 
i 1



n
 

1   (c(v1 )  1)  c(v1 )   
i 1

 
n




1

(
c
(
v
)

1
)
c(v2 )  


2


i 1



i 1
3.1. Theorem: Let G  (V ,  ) be a fuzzy graph, via
coloring of EMSC, amount of SGF is Maximum over
another proper coloring.
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


1   (c(v3 )  1)





1   (c(vn )  1)



i{1,..., n}
~
A
(c(vi )) 

c(v3 )  

  ... 
i 1

n

c(vn )  


i 1


i{1,..., n}
n
~
A
(c (vi )) (3)
n
In the coloring of EMSC,
 c(v ) is
i 1
According to (3),
i
minimum, then
 ~ (c (vi )) is maximum

i{1,..., n} A
 ~ (c (vi ))

i{1,..., n} A
over all proper the (d, f)-extended coloring of the fuzzy
graph.
Application of This theorem is considered in next
subsection 3.2 for scheduling problem with fuzzy time that
all of times don’t have equal graduation.
3.2. Application of EMSC problem in exam scheduling
problem
Following Bullnheimer [14], the examination scheduling
problem consists of assigning a number of exams to a
number of potential time periods or slots within the
examination period, taking into account that students
cannot take more than one exam at the same time, as well
as several other constraints. A simplified version of this
problem can be modeled as a classical coloring problem.
Each exam is a node of the graph, and the edges link those
incompatible exams in the sense that there is at least one
student sharing both of them. Each color is identified with
one slot and all equally colored nodes are assigned to the
associated slot.
If incompatibility of exams is characterized by distance
between examination days and graduation of days of exam
is equal so accompanying coloring of C
can be
amount
and an Upper bound for
time has lowest graduation so therefore coloring EMSC
give us value maximum of

i{1,..., n}
~
A
(c (v i )) .
Therefore via coloring of ESMC obtain that sum of times
for exam that days are valued with membership function
(3) is value maximum over another coloring.
Bring two examples to compare the coloring of ESMC and
other coloring for amount of SGF.
Example 3.1There are five exams {A, B, C, D, E} and,
taking into account their different characteristics
(difficulty, the implied specialties or profiles, etc.), the
incompatibility between them is valued following way: n
if they are compatible, and l, m and h if the incompatibility
degree is, respectively, low, medium and high. This
problem can be modeled by means of the fuzzy graph
~
G  (V ,  ) where
V = {A, B, C, D, E}
m
A
a
B
t
h
m
t
C
l
E
D
h
Fig1. Fuzzy graph
~
G
for Example 3.2
d, f
achieved optimal time period of exam days. Now if the
graduation the examination days not equal and this
graduation be characterized by membership function (3),
considering the colors as exam days and vertices as
examinations, according to theorem (3.1) Coloring of
EMSC Gives maximum amount of fuzzy time sum of the
examination this means that Sum of the days that allocated
for exams will be minimal And amount of SGF is
maximum. In this issue membership function (3) means
that the primal days have more graduation as compared to
last days.
Conditions imposed by this membership function (3) so
that the first time has the highest graduation and the final
 m

m   t h

m n
t n

t
h
l
n
m
n
l
h
t

n
n

h
 
This fuzzy graph is depicted in Fig.1.
Via coloring of EMSC have:
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 G  16 , s  6 
  (c(v ))  4.3125
i{1,..., 5}
e
~
A
m

n
l 

- 
 m t

m - h
  t h 
m n l

c( A)  6 c(B)  4 c(C)  1 c(D)  4 c(E)  1 
E
5
i

Then of EMSC
 A~ (have:
c(vi ))  4.3125
Then via coloring
i{1,..., 5}
c(v1 )  3, c(v
2 )  1 , c(v 3 )  7 , c(v 4 )  1
Via coloring of C d , f using exact algorithm in [10] will
have:
c( A)  1 c(B)  5 c(C)  6 c(D)  3 c(E)  6
 6
then

i{1,..., 5}
~
A
(c(vi ))  4.142857
Taking the above example the result that if valuation of
exam days is graduated according to (3) then the best
schedule this is that exam A be in first day and exam B be
in fifth day of exam and exam C,E be in third day of exam
and exam D be in sixth day of exam.
Coloring of EMSC schedule the five exams with these
constrains and gives sum maximum of graduated assigned
colors to vertex and coloring of

 se (G)  7 ,
E

. Then

iv
G  12 
i{1,..., 5}
~
A
~
A
 A~ (c(vi ))  3.33333
(c(vi ))  4
.142857
iv
(c(vi ))  3.3076
And via coloring C d , f
and getting chromatic number have:
c(v1 )  1 , c(v 2 )  3 , c(v 3 )  6 , c(v 4 )  3
 d , f  6
, iV c(v i )  13 
C d , f give minimum the
overall exam period. For above information, minimum of
slot is six and optimum schedule for both coloring is the
following:
~
SGF (G ) via
~
coloring of ESMC is bigger than amount of SGF (G ) via
coloring of C d , f .
In this example s 

and amount of
In order to comparing amount of SGF via coloring of
EMSC with colors of c (v ) and coloring of C d , f with
colors of c (v ) , That C d , f acquires via exact algorithm
in[10](that give chromatic number), three different fuzzy
graph types with 6,7,8 nodes and for each arbitrary three
graph, have been considered. In Table 1 is depicted

difference between
iv
In this example, chromatic number from coloring of
Cd , f
is equal with vertex strength s from coloring of ESMC.
In below example vertex strength is bigger than chromatic
number i.e. s   .
~
Example3.2. let G  (V ,  ) be a fuzzy graph where V=
{A, B, C, D}, I = {n, l, m, h, t};
~
A
(c (vi )) ,   A~ (c(vi )) ,
c(vi ), iv c(vi )
that in some coloring despite   s , have
iv c(vi )  iv c(vi ) . And difference and
and relation between
, s ,
improvement between

iv

iv
~
A

iv
iv
~
A
(c (vi )) ,
(c(vi )) via coloring of EMSC as compared to
coloring of C d , f
respectively.
depicted in figure 2, figure 3,
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Fig 3 Difference of
~
SGF (G ) via EMSC and C d , f
~
Obtaining amount of SGF (G ) for coloring of ESMC is
used from model of coloring (2).for all nine fuzzy graph
~
amount of SGF (G ) is bigger or equal than this amount
for coloring of C d , f f, this difference between this two
coloring represented in fig3 that for all nine graph
~
SGF (G ) bigger or equal than other coloring.
4. Conclusions
In this paper, extending MSC problem for fuzzy graph is
expressed the concept EMSC problem. And is expressed
application of this coloring in exam scheduling problem
that examination time is graduated with membership
function (3). This means that sum of graduated allocating
days for exams, is maximum value over all proper the
( d , f )  extended coloring of the fuzzy graph.
~
Fig 2 Comprising of SGF (G ) via EMSC and
Cd , f
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