11.7 Least Cost Method

The Least Cost Method (LCM) is a more cost-conscious approach to generating an initial feasible solution for a balanced transportation model. Unlike the North-West Corner Method, LCM prioritizes allocations based on minimizing transportation costs from the outset.

This method typically results in a lower total transportation cost than the North-West Corner Method, as it incorporates cost efficiency into the allocation process from the beginning.

Determining the Total Transportation Cost Using the Least Cost Method

The model tableau is shown below:

Step 1: Select the Cell with the Lowest Cost

• The cell with the Lowest Cost is (S1, D2) with a cost of $2
• Allocate all 60 units from S1
• Update:
  • S1 supply exhausted
  • D2 remaining demand = [latex]70 - 60 = 10[/latex]

Step 2: Next Lowest Cost

• There is a tie for the next lowest cost at $4, (S2, D1) and (S3, D4)
  • Randomly select (S2, D1)
• Allocate 50 units (the full demand of D1)
• Update:
  • S2 remaining supply = [latex]80 - 50 = 30[/latex]
  • D1 demand fulfilled

Step 3: Next Lowest Cost

• Move to the Next Lowest Cost, (S3, D4), Cost = $4
• Allocate 40 units (total demand of D4)
• Update:
  • S3 remaining supply = [latex]100 - 40 = 60[/latex]
  • D4 demand fulfilled

Step 4: Next Lowest Cost

• There is another tie at $6 (S3, D2) and (S3, D3)
  • Randomly select (S3, D3)
• Allocate 60 units exhausting S3
• Update:
  • S3 supply exhausted
  • D3 remaining demand = [latex]80 - 60 = 20[/latex]

Step 5: Next Lowest Cost

• The Next Lowest Cost is (S2, D3), Cost = $10
• Allocate 20 units (remaining demand from D3)
• Update:
  • S2 remaining supply = [latex]30 - 20 = 10[/latex]
  • D3 demand fulfilled

Step 6: Final Allocation

• The Final Allocation will be to (S2, D2), Cost = $12
• Allocate the remaining 10 units
• Update:
  • S2 exhausted
  • D2 demand fulfilled

After completing the allocation process using the Least Cost Method, all supply capacities have been fully utilized, and all market demands have been satisfied. This confirms that a feasible solution has been reached.

To compute the total transportation cost, 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{S2/D1} \times \small{4}) + (\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 4) + (60 \times 2) + (10 \times 12) + (20 \times 10) + (60 \times 6) + (40 \times 4)\\ \text{Total Cost}&= 200 + 120 + 120 + 200 + 360 + 160 \\ \text{Total Cost}&= 1160 \end{align*}[/latex]

Thus, the initial basic feasible solution obtained using the Least Cost Method results in a total transportation cost of $1160.

This solution is significantly more cost-efficient than the one obtained using the North-West Corner Method ($1760), as it incorporates transportation costs into the allocation process. However, while this method often yields a near-optimal solution, it does not guarantee optimality.

To explore the possibility of further cost reduction, we now turn to Vogel’s Approximation Method (VAM)—a more refined approach that considers both cost and opportunity cost in its allocation strategy.