Sunday, May 2, 2010

Transportation Problem and MODI Method of Solving Transportation Problem From MB0032 Assignment

The question is – “What do you understand by the transportation problem? What is the basic assumption behind the transportation problem? Describe the MODI method of solving transportation problem.” It has been taken from MB0032 assignment of SMU MBA. It is the question of Operation Research book of Sikkim Manipal University. I am back with this question after the - Matrix Minimum Method, Integer Programming Problem and Penalty Cost Method or Big-M Method for Solving LPP.

Transportation Problem:

Here we study an important class of linear programs called the transportation model. This model studies the minimization of the cost of transporting a commodity from a number of sources to several destinations. The supply at each source and the demand at each destination are known.

The objective is to develop an integral transportation schedule that meets all demands from the inventory at a minimum total transportation cost.

Basic Assumption behind Transportation Problem:

Let us consider a T.P involving m-origins and n-destinations. Since the sum of origin capacities equals the sum of destination requirements, a feasible solution always exists. Any feasible solution satisfying m+n-1 of the m + n constraints is a redundant one and hence can be deleted. This also means that a feasible solution to a T.P can have at the most only m + n – 1 strictly positive components, otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T. P. in such a manner that the rim requirements are satisfied. This can be achieved either by inspection or by following some simple rules. We begin by imagining that the transportation table is blank i.e. initially all Xij = 0. The simplest procedures for initial allocation discussed in the following section.

MODI Method of Solving Transportation Problem:

The first approximation to (2) is always integral and therefore always a feasible solution. Rather than determining a first approximation by a direct application of the simplex method it is more efficient to work with the table given below called the transportation table. The transportation algorithm is the simplex specialized to the format of table it involves:

a) Finding an integral basic feasible solution
b) Testing the solution for optimality
c) Improving the solution, when it is not optimal
d) Repeating steps (1) and (2) until the optimal solution is obtained

The solution of T.P. is obtained in two stages. In the first stage we find basic feasible solution by any one of the following methods a) North-west corner rale b) Matrix Minima method or least cost method c) Vogel’s approximation method. In the second stage we test the B.Fs for its optimality either by MODI metod or by stepping stone method.

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