North-West Corner Rule

North-west corner rule is one of the easiest methods to find a feasible solution to a transportation problem. Before getting into detail about the North-west corner rule, let’s recall what a transportation problem is.

Transportation problem:

It is a special type of Linear Programming Problem (LPP) in which goods are transported from one set of sources to another set of destinations based on the supply and demand of the origins and destination, respectively, such that the total cost of transportation is minimized.

There are two types of transportation problems. They are

Balanced transportation problems: Both supply and demand are equal.

Unbalanced transportation problem: Supply and demand are not equal. In this case, either a dummy column or row is added based on the necessity to make it a balanced problem.

Below are the methods to find the initial basic feasible solution for a transportation problem:

1. North West Corner Cell Method (or) North West Corner Rule

2. Least Call Cell Method

3. Vogel’s Approximation Method (VAM)

In this article, you will learn one of the most commonly used methods to find an initial feasible solution to a transportation problem called the North-west corner rule.

North West Corner Rule of Transportation Problem

The North West corner rule is a technique for calculating an initial feasible solution for a transportation problem. In this method, we must select basic variables from the upper left cell, i.e., the North-west corner cell.

North West Corner Rule Steps

Go through the steps given below to understand how to find a feasible solution for a transportation problem.

Step 1: Select the upper-left cell, i.e., the north-west corner cell of the transportation matrix and assign the minimum value of supply or demand, i.e., min(supply, demand).

Step 2: Subtract the above minimum value from Oi and Di of the corresponding row and column. Here, we may get three possibilities, as given below.

If the supply is equal to 0, strike that row and move down to the next cell.

If the demand equals 0, strike that column and move right to the next cell.

If supply and demand are 0, then strike both row and column and move diagonally to the next cell.

Step 3: Repeat these steps until all the supply and demand values are 0.

North West Corner Method Solved Example

Question:

Get an initial basic feasible solution to the given transportation problem using the North-west corner rule.

Solution:

For the given transportation problem, total supply = 950 and total demand = 950.

Thus, the given problem is the balanced transportation problem.

Now, we can proceed with the North-west corner rule to find the initial feasible solution.

Step 1: Consider the upper-left corner cell, which has the value 11. The minimum value of the corresponding cell’s supply and demand is 200.

Step 2: The difference between the corresponding cell’s supply and demand from the minimum value obtained in the previous step is:

Supply = 250 – 200 = 50

Demand = 200 – 200 = 0

As demand is 0, we need to allocate 200 to that cell and strike the corresponding column and then move right to the next cell, i.e., the cell with the value 13.

Step 3: For the cell with value 13, the minimum of supply and demand is min(50, 225) = 50.

Step 4: The difference between the corresponding cell’s supply and demand from the minimum value obtained in the previous step is:

Supply = 50 – 50 = 0

Demand = 225 – 50 = 175

As the supply is 0, we need to allocate 50 to that cell and strike the corresponding column and then move down to the next cell, i.e., the cell with the value 18.

Step 5: For the cell with value 18, the minimum of supply and demand is min(300, 175) = 175.

Step 6: The difference between the corresponding cell’s supply and demand from the minimum value obtained in the previous step is:

Supply = 300 – 175 = 125

Demand = 175 – 175 = 0

As demand is 0, we need to allocate 175 to that cell and strike the corresponding column and then move right to the next cell, i.e., the cell with the value 14.

Step 7: For the cell with value 14, the minimum of supply and demand is min(125, 275) = 125.

Step 8: The difference between the corresponding cell’s supply and demand from the minimum value obtained in the previous step is:

Supply = 125 – 125 = 0

Demand = 275 – 125 = 150

As the supply is 0, we need to allocate 125 to that cell and strike the corresponding column and then move down to the next cell, i.e., the cell with the value 13.

Step 9: For the cell with value 13, the minimum of supply and demand is min(400, 150) = 150.

Step 10: The difference between the corresponding cell’s supply and demand from the minimum value obtained in the previous step is:

Supply = 400 – 150 = 250

Demand = 150 – 150 = 0

As demand is 0, we need to allocate 125 to that cell and then move right to the next cell, i.e., the cell with the value 10. Here, we don’t get any further cells to strike off.

Also, we can see that the corresponding supply and demand for the left-out cell with the value 10 are equal. Now allocate the supply or demand value to that cell. Therefore, we can get 0’s for all supplies and demands.

Now, we should calculate the total minimum cost using the allocated values and the corresponding cell values.

Here, the transportation path is:

O1 → D1, O1 ​​→ D2, O2 → D2, O2 → D3, O3 → D3, O3 → D4

Therefore, the total cost = (200 × 11) + (50 × 13) + (175 × 18) + (125 × 14) + (150 × 13) + (250 × 10)

= 2200 + 650 + 3150 + 1750 + 1950 + 2500

= Rs. 12,200

North West Corner Rule Problems

1. Find the initial basic feasible solution of the following transportation problem.
   
2. Find the initial feasible solution of the given transportation problem so that the total cost of transportation is minimum.
   
3. Obtain the initial basic feasible solution for the below transportation problem.