Least Cost (Matrix Minima) Method

Least Cost (Matrix Minima) Method

The Least Cost Method (also called the Matrix Minima Method) improves on NWC by allocating to the cheapest cells first, so the starting solution is usually closer to optimal.

Rule: Find the cell with the lowest unit cost in the whole table (breaking a tie in favour of the cell that allows the larger allocation). Allocate the maximum possible there. Cross out the row or column whose supply/demand is now exhausted. Repeat on the remaining table until all supply and demand is used up.

Worked example — same cost matrix as the North-West Corner example

Step-by-step allocation:

1. The lowest cost in the table is 4 at $(O_3,D_1)$: allocate $\min(20,25)=20$. $O_3$ is exhausted; $D_1$ has 5 units left.
2. Among the remaining cells, the lowest cost is 6 at $(O_1,D_1)$: allocate $\min(30,5)=5$. $D_1$ is exhausted; $O_1$ has 25 units left.
3. The next lowest remaining cost is 8 at $(O_1,D_2)$: allocate $\min(25,35)=25$. $O_1$ is exhausted; $D_2$ has 10 units left.
4. Only the $O_2$ row remains, against $D_2$ (10 left) and $D_3$ (40). The lower of its two costs is 11 at $(O_2,D_2)$ (tied with $(O_2,D_3)$, resolved here in favour of clearing the smaller remaining column first): allocate $\min(50,10)=10$. $D_2$ is exhausted; $O_2$ has 40 units left.
5. Final cell $(O_2,D_3)$: allocate $\min(40,40)=40$.

Allocation table (LCM)