Task Scheduling Problem Using Fuzzy Graph

The concept of obtaining fuzzy sum of fuzzy colorings problem has a natural application in scheduling theory. The problem of scheduling N jobs on a single machine and obtain the minimum value of the job completion times is equivalent to finding the fuzzy chromatic sum of the fuzzy graph modeled for this problem. The aim of this paper is to solve task scheduling problems using fuzzy graph. keywords: Fuzzy Graph, k-fuzzy coloring, Chromatic fuzzy sum of Graph, -chromatic sum of Graph.

keywords: Fuzzy Graph, k-fuzzy coloring, Chromatic fuzzy sum of Graph, -chromatic sum of Graph.

inTRodUcTion
The field of mathematics plays vital role in various field.One of the important areas in Mathematics is graph theory, which is used in several models.The origin of graph theory started with Konigsberg bridge problem, in 1735.It was a long standing problem until solved by Leonhard Euler, by means of graph.The colouring problem consists of determining the chromatic number of a graph and an associated colouring function.Let G be a simple graph with n vertices.A colouring of the vertices of G is a mapping f: V (G) '! N, such that adjacent vertices are assigned different colours.The chromatic sum of a graph introduced in 5 is defined as the smallest possible total over all vertices that can occur among all colourings of G. Senthilraj S. 10  generalize these concepts to fuzzy graphs.He define fuzzy graphs with fuzzy vertex set and fuzzy edge set and generalize the concept of the chromatic joins and chromatic sum of a graph to fuzzy graphs and define the fuzzy chromatic sum of fuzzy graph.Author consider the problem of scheduling N jobs on a single machine and obtain the minimum value of the job completion times which is equivalent to finding the fuzzy chromatic sum of the fuzzy graph modeled for this problem by considering the example of scheduling 6 jobs on a single machine.
In this paper we generalize the above said result by considering the case of scheduling 8 tasks on a single machine and obtain a minimum value of the task completion time.

Preliminaries and results definition 2.1:[11]
A fuzzy set defined on a non empty set X is the family , where is the membership function.In classical fuzzy set theory the set is usually defined as the interval such that It takes any intermediate value between 0 and 1 represents the degree in which .The set I could be discrete set of the form where indicates that the degree of membership of to is lower than the degree of membership of x′ .

definition 2.2
Let be a finite nonempty set.The triple Ĝ = (V,σ,μ) is called a fuzzy graph on V where s and μ are fuzzy sets on V and E, respectively, such that μ(uv)≤ σ(u)Ʌ σ(v) for all u,v∈V and uv ∈ E. For fuzzy graph Ĝ = (V,σ,μ) the elements V and E are called set of vertices and set of edges of G respectively.

definition 2.3
A fuzzy graph Ĝ = (V,σ,μ) is called a complete fuzzy graph if μ(uv) = σ(u)Ʌ σ(v) for all u,v∈V and uv ∈ E. We denote this complete fuzzy graph by Ĝk.

definition 2.4
Two vertices u and The edge uv of Ĝ is called strong if u and v are adjacent.Otherwise it is called weak.

definition 2.6
A family Γ = {γ_1,γ_2,…..,γ_k} of fuzzy sets on The above definition of k-fuzzy coloring was defined by the authors Eslahchi and Onagh 1 on fuzzy set of vertices.

definition 2.7:[10]
The least value of k for which has a fuzzy coloring, denoted by x f (G), is called the fuzzy chromatic number of G.
The number of fuzzy coloring of is finite and so there exist a fuzzy which is called minimum fuzzy coloring of G such that Ʃ(G)=Ʃ Г0 (G) Let G be a fuzzy graph and Г 0 ={γ 1 ,γ 2 ,…..,γ k } is minimum fuzzy sum coloring of G.Then For a fuzzy graph Ĝ = (V,σ,μ),Ʃ(G)≤3/4[(x f (G)+1)h(σ)|V|,, where h(s) is height of s and |V| is cardinality of v.

2.1
Let Ĝ = (V,σ,μ) be a connected fuzzy gra p h w i t h s t r o n g e d g e s.T h e n t h e lower bound for Ʃ(G)isw√8e is, where w=max{σ(x)+μ(xy)>o,x∈V,(x,y) is weak edge of G}.

ReSUlTS and diScUSSionS
Result 3.1: Find a minimum value of the task completion time for scheduling 8 tasks on a single machine.
Assume that at any time the machine is capable to perform any number of tasks and these tasks are independent or conflicts between them are less than one.Consider the time consuming for task 1and 4 is 0.4hrs, for tasks 3 and 6 is 0.3hrs, for tasks 2 and 5 is 1hrs.for tasks 7 and 8 is 0.2hrs.Also, Task {(2, 5), (5,6), (6, 7)}conflict together with 0.1 hrs.
Let Ĝ = (V,σ,μ) where is the set of all task, s(x) is the amount of consuming time of machine for each x ∈ V and μ(x,y) is the measure of the conflict between the task and.Finding the minimum value of job compwletion time for this problem is equivalent to the chromatic sum of Ĝ.