This question comes from Operation Research assignment of MB0032 for SMU MBA. The question is “Describe the Matrix Minimum method of finding the initial basic feasible solution in the transportation problem.” From the SMU MBA MB002 assignment I already have shared about classification of Operations Research models and Penalty Cost Method or Big-M Method for Solving LPP.

Matrix Minimum Method:

Step 1: Determine the smallest cost in the cost matrix of the transportation table. Let it be Cij, Allocate Xij = min (aj, bj) in the cell (i, j).

Step 2: If Xij = aj cross off the ith row of the transportation table and decrease bj by ai go to step 3.

If xij = bj cross off the ith column of the transportation table and decrease ai by bj go to step 3.

If Xij = ai = bj crosss off either the ith row or the ith column but not both.

Step 3: Repeat steps 1 and 2 for the resulting reduced transportation table until all the rim requirements are satisfied whenever the minimum cost is not unique make an arbitrary choice among the minima.

The Initial Basic Feasible Solution:

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 component, 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 = o. The simplest procedures for initial allocation discussed in the following section.

Matrix Minimum Method:

Step 1: Determine the smallest cost in the cost matrix of the transportation table. Let it be Cij, Allocate Xij = min (aj, bj) in the cell (i, j).

Step 2: If Xij = aj cross off the ith row of the transportation table and decrease bj by ai go to step 3.

If xij = bj cross off the ith column of the transportation table and decrease ai by bj go to step 3.

If Xij = ai = bj crosss off either the ith row or the ith column but not both.

Step 3: Repeat steps 1 and 2 for the resulting reduced transportation table until all the rim requirements are satisfied whenever the minimum cost is not unique make an arbitrary choice among the minima.

The Initial Basic Feasible Solution:

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 component, 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 = o. The simplest procedures for initial allocation discussed in the following section.

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