BETTER PERFORMANCE OF SIMPLEX METHOD ON TRANSPORT NETWORK WITH NON – INTEGER EDGE CAPACITIES.

Operation Research problems (OR) have been solved by operation research technique efficiently for a long time. Graph theory techniques are competent with OR techniques in some area like transport problem. Identification of a better technique to solve the OR problem is very important. In this paper, the better performance of Big M method on transport network with non – integer edge capacities is reported. The redundant constraints removal in Big M method improve its performance

A Transport Network (TN) is shown in Figure1. The numbers written beside the edge are the edge capacities. The capacity C ij of an edge (i,j) can be thought of as the maximal amount of some commodity that can be transported form station i to j along the edge (i,j) per unit of time in a steady state. The objective of the method is to find the maximal amount of commodity flow from a given vertex s to another specified vertex t via the entire transport network.
Algorithm for finding maximal flow:-In a given transport network G, a flow is an assignment of a non negative number f ij to every directed edge (i,j) such that the following conditions are satisfied.
(i) For every directed edge (i,j) in G, ij ij c f  (ii) There is a specified vertex s in G, called the source, for which where the summation are taken over all vertices in G and W is the value of the flow. (iii) There is another specified vertex t in G, called the sink, for which In a given Transport network G, the value of the flow W from source to sink is less than or equal to the capacity of any cut separating s from t.
Theorem 2 (Max-flow -min cut theorem) In the given Transport Network G, the maximum value of a flow from s to t is equal to the minimum value of the capacities of all the cuts in G that separates s form t.
In this Transport Network of figure 1, there are eight cuts that separate s form t. These cuts (Identified by vertex set P) and their capacities are given below. C(P, ̅ ) is defined to be the sum of the capacities of those edges directed from the vertices in set P to the vertices in ̅ .

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LPP on Transport Network:-Transport network problem can be solved by big M-method. First the transport network problem is formulated, and then solved using Big-M-Method.

Formation of LPP:-
subject to and After Changing to standard form  where "A' is the incidence matrix of the digraph obtained by adding an edge from "t" to "s' in the transport network. Also note that the edges in f, c and "A' must appear in the same order.
In the LPP some of the constraints are of equal type. So big M method is used to solve the LPP [2].

Algorithm for 'Big M -method'
Step 1 Express the LPP in the standard form by introducing slack and / or surplus variables if any.

Step 2
Introduce the non negative artificial variable R 1, R 2 ,……..to the left hand side of all the constraints of or =type.
A very large penalty is assigned (-M for maximization problems and +M for minimization Problems) as the coefficients of the artificial variable in the objective function.
Step 3 Solve the modified linear problem by Simplex Method.  Fist table of big Method  Table 2 The result is shown above in Table 2 and Table 3. Table 2 is the initial  Step 1: Find the optimal objective function value to the problem P i , i=1…..m by using the simplex method .let z i be the optimal objective function value of the problem P i

Result and discussions:-Graph Theory Technique
Step 2: Check whether z i b i , The i th constraint .