12/15/2010

ACKNOWLEDGEMENT

First and foremost, we would like to thank to our mentor, Dr. G.N. Patel for his valuable guidance and advice throughout the project. Without his support and guidance, this report would not have been possible.

We would like to extend our sincere regards to the authorities of Birla Institute of Management Technology, Greater Noida (BIMTECH) for providing us with a good environment and facilities to complete this report. Also, we would like to take this opportunity to thank Centre for Business Management (CBM) of BIMTECH for offering this subject, Research Methodology and opportunity to do this project as a part of course curriculum.

Last but not the least we would like to thank our families and friends for their support in completing this project.

TABLE OF CONTENTS

1.Linear Programming ……........…………………………….................................4

2. Transportation Problem ……………………………............................................5

3.Case Study………………………………………..................................................8

4.Other Methods of solving transportation problem..................................................11

LINEAR PROGRAMMING

Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Given a polytope and a real-valued affine function defined on this polytope, a linear programming method will find a point on the polytope where this function has the smallest (or largest) value if such point exists, by searching through the polytope vertices. SIMPLEX METHOD

In mathematical optimization theory, the simplex algorithm or simplex method is a popular algorithm for numerically solving linear programming problems. A linear programming problem is one in which we are to find the maximum or minimum value of a linear expression ax + by + cz + . . .

(called the objective function), subject to a number of linear constraints of the form Ax + By + Cz + . . .≤ N

or

Ax + By + Cz + . . .≥ N.

The largest or smallest value of the objective function is called the optimal value, and a collection of values of x, y, z, . . . that gives the optimal value constitutes an optimal solution. The variables x, y, z, . . . are called the decision variables. In realistic problems, a solution may not be obvious, especially if there are many ingredients each having constraints. A simple procedure is needed to generate an optimal solution no matter how complex the problem is. For this purpose simplex method is used. Transportation problems where cost is to be minimized are one of the problems where Simplex method is used. TRANSPORTATION PROBLEM

The standard scenario where a transportation problem arises is that of sending units of a product across a network of highways that connect a given set of cities. Each city is considered either as a "source," in that units are to be shipped out from there, or as a "sink," in that units are demanded there. Each source has a given supply, each sink has a given demand, and each highway that connects a source-sink pair has a given transportation cost per unit of shipment. This can be visualized in the form of a network, as depicted in figure below.

Given such a network, the problem of interest is to determine an optimal transportation scheme that minimizes the total cost of shipments, subject to supply and demand constraints. Problems with the above structure arise in many applications. For example, the sources could represent warehouses and...