11.6 North-West Corner Rule (NWCR)

The North-West Corner Method (NWCM) is a straightforward technique used to generate an initial feasible solution for a balanced transportation model. While it does not consider transportation costs during allocation, it provides a valid starting point for further optimization.

This method is named after the north-west (upper-left) cell of the transportation tableau, where the allocation process begins.

Determining the Total Transportation Cost Using the North-West Corner Method

The tableau for this model is shown below:

The steps in solving this problem are:

Step 1: Allocate to (S1, D1)

• Allocate 50 units (the demand of D1).
• Update:
  • S1 remaining supply = [latex]60 - 50 = 10[/latex]
  • D1 demand fulfilled = [latex]50 - 50 = 0[/latex]

Step 2: Allocate to (S1, D2)

• Allocate 10 units (the remaining amount in S1)
• Update:
  • S1 exhausted
  • D2 remaining demand = [latex]70 - 10 = 60[/latex]

Step 3: Allocate to (S2, D2)

• Allocate 60 units (the remaining demand in D2)
• Update:
  • S2 remaining supply = [latex]80 - 60 = 20[/latex]
  • D2 demand fulfilled

Step 4: Allocate to (S2, D3)

• Allocate 20 units (the remaining supply from S2)
• Update:
  • S2 exhausted
  • D3 remaining demand = [latex]80 - 20 = 60[/latex]

Step 5: Allocate to (S3, D3)

• Allocate 60 units (the remaining demand in D3)
• Update:
  • S3 remaining supply = [latex]100 - 60 = 40[/latex]
  • D3 demand fulfilled

Step 6: Allocate to (S3, D4)

• Allocate the remaining 40 units
• Update:
  • S3 exhausted
  • D4 demand fulfilled

After completing the allocation process using the North-West Corner Method, all market demands have been fully satisfied, and the available supply from all farms has been completely utilized. This indicates that a feasible solution has been reached.

To determine the total transportation cost of this initial solution, we multiply the quantity allocated to each cell by its corresponding unit transportation cost and sum the results:

[latex]\begin{align*} \text{Total Cost}&= (\small\text{S1/D1} \times \small{6}) + (\small\text{S1/D2} \times 2) + (\small\text{S2/D2} \times 12)+ (\small\text{S2/D3} \times 10) + (\small\text{S3/D3} \times 6) + (\small\text{S3/D4} \times 4) \\ \text{Total Cost}&= (50 \times 6) + (10 \times 2) + (60 \times 12) + (20 \times 10) + (60 \times 6) + (40 \times 4)\\ \text{Total Cost}&= 300 + 20 + 720 + 200 + 360 + 160 \\ \text{Total Cost}&= 1760 \end{align*}[/latex]

Thus, the initial basic feasible solution obtained using the North-West Corner Method results in a total transportation cost of $1760.

It is important to note that while this method provides a valid starting point, it does not guarantee the optimal solution. Therefore, in the following sections, we will evaluate the total transportation cost using two more refined methods:

• Least Cost Method (LCM)
• Vogel’s Approximation Method (VAM)

These methods incorporate cost considerations during the allocation process and are likely to yield more cost-efficient solutions