On Solving Transportation Problem – Linear Path Approach

: Aim of this study is to determine the optimal cost for the given rectangular (or) square grid corresponding transportation problem along with balanced and unbalanced manner. We proposed the algorithmic way to provide cost from supply to the corresponding demand of the transportation problem in graph theoretical way to obtain minimum cost than the existed method in Operation Research. The proposed research brings out anoptimal basic feasible solution derived through graph theoretical method. It provides more than hundred percentage matching with so called existed method’s optimal basic feasible solution.


Introduction
Organizations need to fix lot of problems and make fruitful decisions in order to manage proper administration.Operation Research is one of the domains that help as a rescue to solve complications by the application of a set of analytical methods [1]- [4].These analytical methods involve an advanced range of mathematical models to get an optimal solution of the given task.
Visual representations like graph help a better understanding of data.Graphs are normally presented by points structured both directed and undirected ways so as to capture the image for analysis in different scientific and real time problems [5].
Expansion of any business that relies on transportation can be successful or failure [8]- [10] based on economic management of cost.Mobility of products from one end to another end may lead to some issues.Issues that are related to monetary reduction and economizing can be sort out by unique kind of Linear Programming problem [6]& [7].

Theorem:
There exists a path of length ( − 1) as an equivalent graph of the corresponding transportation pay off matrix where  is a number of vertices of  is connected.

Proof:
We know that for the TPM having solution satisfied degeneracy condition (i.e)  +  − 1 Where m is number of rows in TPM, n is number of columns in TPM,  is number of vertices of  (as a solution of connected graph).Hence the proof.

Cor 1:
Every TPM will have an unique weighted graphical representation.

Research Article
Cor 2: There exist several OBFS for a TPM in its corresponding graphical representation with different starting point and also consists of the same point.
Step 2: List out all possible paths from certain starting point which covers maximum number of other points once.
Step 3: Shade the cost which is weight of the corresponding two vertices of the paths.
Step 4: Choose the least element which occur in supplyor demand and allotthat cost to the minimum value corresponding to the row and column of the shaded cell if possible.
Step 5: Repeat Step 4 until degeneracy condition is satisfied.
Step 6: Compute the cost value based on Step 5.

Example : TPM :
Table : 1 Graphical Representation of the given TPM: Table : 2 Allotment table for the path P1: Step 1: Shade the cells with the edge weight of the corresponding path.

Table: 3
Step 2: Choose the least cost from supply& demand and allot the possible cost to the minimum element corresponding to the row and column of the shaded cell if possible.

Table: 4
Step 3: Repeat Step 2 until the degeneracy condition satisfied for all shaded cells in Step 1 if possible.(which is explained in step 3.1 to step 3.

Research Article
Allotment table for the path P1:

Table : 9
Note: Yellow shaded indicates the allotment from path P1; Red shaded indicates the new adjusted allotment with respect to supply & demand.