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 in⁵ 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

A fuzzy set  defined on a non empty set X is the family A = { (x, μ _A (x)/x ∈ X}, where μ _A : X → I is the membership function. In classical fuzzy set theory the set  is usually defined as the interval [0,1] such that

It takes any intermediate value between 0 and 1 represents the degree in which x ∈ A. The set I could be discrete set of the form I  = {0,1,...k} where μ _A (x) < μ (x') indicates that the degree of membership of  to is lower than the degree of membership of  x to A is lower than the degree of membership of x'.

Definition

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

Definition

A fuzzy graph Ì‚G = (V, σ, μ) is called a complete fuzzy graph if μ (uv) = σ (u) ∧ σ (v)for all u, v ∈ V and u, v ∈ E . We denote this complete fuzzy graph by Ì‚Gk.

Definition

Two vertices u and v in Ì‚G are called adjacent if (1/2) [σ (u) ∧ σ (v)] ≤  μ (uv).

Definition

The edge uv of Ì‚G is called strong if  u and v are adjacent. Otherwise it is called weak.

Definition

A family ⌈ = { _{ϒ1, ϒ2,....,ϒk} } of fuzzy sets on V is called a k-fuzzy coloring of Äœ = (V, σ, μ) if

a) ∧⌈ = σ

b) _{ϒi ^∧ϒj} = 0

c) For every strong edge of uv of Äœ,  _{ϒi  (u)^∧ϒj (v)} = 0 for 1 ≤ i ≤ k.
The above definition of k-fuzzy coloring was defined by the authors Eslahchi and Onagh [1] on fuzzy set of vertices.

Definition

The least value of k for which Ghas a fuzzy coloring, denoted by x^f (G), is called the fuzzy chromatic number of G.

Definition

For a k-fuzzy coloring ⌈= { _{ϒ1, ϒ2,....,ϒk} } of a fuzzy graph ofG, ⌈ chromatic fuzzy sum of G denoted by ∑_⌈(G) is defined as

Where C_i = suppγi and θ_i (x) = {max{σ (x) + μ (xy) / y ∈ C_i}.

Definition

The chromatic fuzzy sum of Gdenoted by ∑(G) is defined as follows
∑(G) = min {∑_Γ (G)/Γ is fuzzy colouring}.
The number of fuzzy coloring of G is finite and so there exist a fuzzy Γ₀ which is called minimum fuzzy coloring of G such that ∑(G) = ∑r₀ (G).

Theorem

Let G be a fuzzy graph and Γ0 = { _{ϒ1, ϒ2,....,ϒk} } is minimum fuzzy sum coloring of G. Then

Theorem

For a fuzzy graph

where h (σ) is height of  σ and |V| is cardinality of V.

Remarks

• Let Äœ = (V, σ, μ) be a connected fuzzy graph with strong edges. Then the lower bound for ∑(G) isw   , where w = max {σ(x) + μ(xy) > o,x ∈ V  is weak edge of G}.
• The fuzzy chromatic sum lies between 

Results and Discussions

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.
Task, {(1, 2), (1, 4), (1, 5), (2, 4),  (4, 8)}conflict together with 0.4 hrs.
Task {(1, 3), (2, 8), (3, 4), (4, 5), (5, 7)}conflict together with 0.3 hrs.
Now, we define the fuzzy graph for above problem>
Let Äœ = (V, σ, μ)  where V is the set of all task, σ(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 x and y. Finding the minimum value of job completion time for this problem is equivalent to the chromatic sum of Äœ.

The fuzzy graph Äœ = (V, σ, μ) corresponding to our example is defined as follows:
Let V = {1, 2, 3, 4, 5, 6, 7, 8},

The fuzzy graph for above problem is

Strong edges: (1, 2), (1,3), (1, 4), (1, 5), (2, 4), (3, 4), (4, 5), (4, 7), (4, 8), (5, 7), (6, 7).

Weak edges: (1, 6), (1, 7), (1, 8), (2, 3), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 6), (5, 6), (5, 8), (6, 8), (7,

Let  Γ1 = { _{ϒ1, ϒ2,....,ϒ8} } be a family of fuzzy set defined on V where

From the above table, we can see that Γ1 satisfied all the properties of k-

Therefore G has 8-Coloring and x^f (G)=8. For this 8-Coloring, Γ1 chromatic number can be calculated as follows:

The  Γ1 chromatic fuzzy sum of G

Let Γ2 = { _{ϒ1, ϒ2,....,ϒ5} } be a family of fuzzy set defined on V where

Again from the above table, we can see that Γ2 satisfied all the properties of k- fuzzy coloring.

Therefore G has 5-Coloring and x^f  =5. For this 5-Coloring, chromatic number can be calculated as follows:

The Γ2 chromatic fuzzy sum of G

Let Γ3 = { _{ϒ1, ϒ2, ϒ3} } be a family of fuzzy set defined on V where

Again from the above table, we can see that Γ3  satisfied all the properties of k- fuzzy coloring.

Therefore G has 3-Coloring and x^f (G) =5. For this 3-Coloring, Γ3 chromatic number can be calculated as follows:

The Γ3 chromatic fuzzy sum of G

Therefore the fuzzy chromatic sum of G is

Calculation for w:

Now,

Conclusion

The fuzzy chromatic number lies between 30 and 1.9592. In our problem ∑(G) = 5.5

Therefore the minimum time of task completion of our problem is 5.5hrs.

References

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