A DIRECT METHOD TO OBTAIN AN OPTIMAL SOLUTION IN THE TRANSPORTATION PROBLEM

A. Seethalakshmy 1 and DR. N. Srinivasan 2 . 1. Research Scholar, Department of mathematics. St. Peter’s university, Avadi, Chennai, India. 2. Professor, Department of mathematics, St. Peter’s university, Avadi, Chennai, India ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

The SS method is a direct method proposed for deriving an optimal solution towards transportation problem. This method solves the problem optimally. Herewith in this research paper, an optimal solution is evinced by row/ column reduction to form a transformed matrix by the systematic allocation of zero by position. Paramount benefit this method produces is lessened the time for calculation, interoperability, and negating degeneracy but not limited to these. Examples herewith make the case for better understanding this method.

Introduction:-
The special type of linear programming problem is transportation problem. In the transportation problem, there are many origins and many destinations in which the commodities are transported from one place to other in a minimum cost. Each source has a certain amount of the supply and each destination has a certain demand. The objective of this model is to determine the minimized total cost while satisfying all the supply and demand restrictions.
The decision variable of a transportation model the i th supply at the source to the j th demand at the destination. Until now, MODI method was used for obtaining/ solving the transportation problem, this method would be futile without obtaining initial feasibility and degeneracy. Whereas this proposed (SS) method would decapitate the initial feasibility and degeneracy steps thereby leaping to the solution in shortened steps.
The chronology of the events to derive in this method delves as below:-Section 2 -Presents the mathematical form of TP. Section 3 -Algorithms formulated and Section 4 -Illustrates numerical examples and finally Section 5 -Summarizes with the conclusion and brief discussion of the results.

Mathematical form of transportation problem:-
The transportation problem can be formulated as an LP problem. Let T ij , i=1…..m, j = 1 ….. n be the number of units transported from source i to destination j.

Sources Destinations
Units of supply Units of demand Algorithm:- Step 1:-For the given data construct a transportation table. If the problem is unbalanced, make it as a balanced one.
Step 2:-Identify the minimum cost element in each row of the transportation table and subtract in their corresponding row.
Step 3:-Similarly, identify the minimum cost element in each column in the resulting table and subtract from their corresponding column Step 4:-Each position of zero is to be discussed Consider the zero of (i , j) th position and find the sum of the value in the row and the column for each zero and allocate the minimum of supply/demand for the position of the zero which is having a maximum value. Delete the corresponding row or column where the supply or demand is exhausted.
Step 5:-Iterate the table from step 2 to step 4, till m + n -1 cell are allocated and all the supply and demand is exhausted.

Conclusion:-
A direct method proposed here solves the transportation problem. This method can be applied to all transportation problems and its kind.SS algorithm contains a systematic procedure, easy to comprehend. Here a maximum value of the sum of row and column of the zero's are given allocation. This (SS) method aids in decision-making process and stands out in comparison to other methods given its quick steps with tangible solutions.