North West Corner Rule

The North West corner rule is a method for computing a basic feasible solution of a transportation problem, where the basic variables are selected from the North – West corner ( i.e., top left corner ).

The standard instructions for a transportation model are paraphrased below. Please read them carefully.

Steps in North West Corner Rule

1. Select the upper left-hand corner cell of the transportation table and allocate as many units as possible equal to the minimum between available supply and demand, i.e., min(s1, d1).
2. Adjust the supply and demand numbers in the respective rows and columns.
3. If the demand for the first cell is satisfied, then move horizontally to the next cell in the second column.
4. If the supply for the first row is exhausted, then move down to the first cell in the second row.
5. If for any cell, supply equals demand, then the next allocation can be made in cell either in the next row or column.
6. Continue the process until all supply and demand values are exhausted.

Example 1: North West Corner Rule - Transportation Problem

The Amulya Milk Company has three plants located throughout a state with production capacity 50, 75 and 25 gallons. Each day the firm must furnish its four retail shops R₁, R₂, R₃, & R₄ with at least 20, 20 , 50, and 60 gallons respectively. The transportation costs (in Rs.) are given below.

The economic problem is to distribute the available product to different retail shops in such a way so that the total transportation cost is minimum

Solution.

Starting from the North west corner, we allocate min (50, 20) to P₁R₁, i.e., 20 units to cell P₁R₁. The demand for the first column is satisfied. The allocation is shown in the following table.

Table 1

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Now we move horizontally to the second column in the first row and allocate 20 units to cell P₁R₂. The demand for the second column is also satisfied.

Table 2

Proceeding in this way, we observe that P₁R₃ = 10, P₂R₃ = 40, P₂R₄ = 35, P₃R₄ = 25. The resulting feasible solution is shown in the following table.

Final Table

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Here, number of retail shops(n) = 4, and
Number of plants (m) = 3

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

Initial basic feasible solution

The total transportation cost is calculated by multiplying each x_ij in an occupied cell with the corresponding c_ij and adding as follows:

20 X 3 + 20 X 5 + 10 X 7 + 40 X 8 + 35 X 2 + 25 X 2 = 670