# Matrix Minimum Method

Matrix minimum (Least cost) method is a method for computing a basic feasible solution of a transportation problem, where the basic variables are chosen according to the unit cost of transportation. This method is very useful because it reduces the computation and the time required to determine the optimal solution.

The following steps summarize the approach.

## Steps in Matrix Minimum Method for Transportation Problem

1. Identify the box having minimum unit transportation cost (cij).
2. If the minimum cost is not unique, then you are at liberty to choose any cell.
3. Choose the value of the corresponding xij as much as possible subject to the capacity and requirement constraints.
4. Repeat steps 1-3 until all restrictions are satisfied.

Example-1, Example-2

#### Example 1

Consider the transportation problem presented in the following table:

Factory Retail Shop Supply
1 2 3 4
1 3 5 7 6 50
2 2 5 8 2 75
3 3 6 9 2 25
Demand 20 20 50 60

Solution.

We observe that c21 =2, which is the minimum transportation cost. So x21 = 20. The demand for the first column is satisfied. The allocation is shown in the following table.

Table 1

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Factory Retail Shop Supply
1 2 3 4
1 3 5 7 6 50
2 5 8 2 75  55
3 3 6 9 2 25
Demand 20 20 50 60

Now we observe that c24 =2, which is the minimum transportation cost, so x24 = 55. The supply for the second row is exhausted.

Table 2

Factory Retail Shop Supply
1 2 3 4
1 3 5 7 6 50
2 5 8 75
3 3 6 9 2 25
Demand 20 20 50 60 5

Proceeding in this way, we observe that x34 = 5, x12 = 20, x13 = 30, x33 = 20. The resulting feasible solution is shown in the following table.

Final Table

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Factory Retail Shop Supply
1 2 3 4
1 3 6 50
2 5 8 75
3 3 6 25
Demand 20 20 50 60

Number of basic variables = m + n –1 = 3 + 4 – 1 = 6.

#### Initial basic feasible solution

The total transportation cost associated with this solution is calculated as given below:
20 X 2 + 20 X 5 + 30 X 7 + 55 X 2 + 20 X 9 + 5 X 2 = 650.