An Application of Lagrange Multiplier for integer Linear Programming in production- Transportation with Flexible Transportation Cost

The Transportation problem is a classic Operations Research problem where the objective is to determine the schedule for transporting goods from source to destination in a way that minimizes the shipping cost while satisfying supply and demand constraints. Although it can be solving as a linear programming problem. Linear programming makes use of the simplex method, an algorithm invented to solve a linear program by progressing from one extreme point of the feasible polyhedron to an adjacent one. The algorithm contains tactics like pricing and pivoting. For a transportation problem, a simplified version of the regular simplex method can be used, known as the transportation simplex method.
In this paper will discuss the functionality of both of these algorithms, and compared their optimized values with non-linear method called the Lagrange Multiplier Method. Lagrange Multiplier is an algorithm that uses different mechanisms to choose the best optimal solutions. This method based on transforming the linear structure transportation problem into the nonlinear structure and solved it directly, by the techniques. The objective of the study was to find out how these algorithms behave in terms of accuracy and speed when a large-scale problem is being solved.