We want to calculate the optimum element of transportation from the transaction above.

Calculate the difference between the two lowest costs for each row and column.

• Row Penalties: $S_1: (2-1)=1$, $S_2: (3-2)=1$, $S_3: (7-5)=2$
• Column Penalties: $D_1: (5-4)=1$, $D_2: (8-1)=7$, $D_3: (3-2)=1$, $D_4: (6-2)=4$

The highest penalty is 7 (Column $D_2$). The minimum cost in $D_2$ is $C_{12}=1$.
Allocate $\min(S_1, D_2) = \min(1600, 1000) = 1000$ units to cell $(S_1, D_2)$.
Update: $S_1 = 1600 - 1000 = 600$, $D_2 = 1000 - 1000 = 0$. Column $D_2$ is satisfied.

• Row Penalties: $S_1: (4-2)=2$, $S_2: (3-2)=1$, $S_3: (7-5)=2$
• Column Penalties: $D_1: (5-4)=1$, $D_3: (3-2)=1$, $D_4: (6-2)=4$

The highest penalty is 4 (Column $D_4$). The minimum cost in $D_4$ is $C_{24}=2$.
Allocate $\min(S_2, D_4) = \min(1200, 900) = 900$ units to cell $(S_2, D_4)$.
Update: $S_2 = 1200 - 900 = 300$, $D_4 = 900 - 900 = 0$. Column $D_4$ is satisfied.

• Row Penalties: $S_1: (4-2)=2$, $S_2: (8-3)=5$, $S_3: (8-5)=3$
• Column Penalties: $D_1: (5-4)=1$, $D_3: (3-2)=1$

The highest penalty is 5 (Row $S_2$). The minimum cost in $S_2$ is $C_{23}=3$.
Allocate $\min(S_2, D_3) = \min(300, 1000) = 300$ units to cell $(S_2, D_3)$.
Update: $S_2 = 300 - 300 = 0$, $D_3 = 1000 - 300 = 700$. Row $S_2$ is satisfied.

• Row Penalties: $S_1: (4-2)=2$, $S_3: (8-5)=3$
• Column Penalties: $D_1: (5-4)=1$, $D_3: (8-2)=6$

The highest penalty is 6 (Column $D_3$). The minimum cost in $D_3$ is $C_{13}=2$.
Allocate $\min(S_1, D_3) = \min(600, 700) = 600$ units to cell $(S_1, D_3)$.
Update: $S_1 = 600 - 600 = 0$, $D_3 = 700 - 600 = 100$. Row $S_1$ is satisfied.

Only row $S_3$ and columns $D_1$, $D_3$ remain.
Allocate $\min(S_3, D_1) = \min(1400, 1300) = 1300$ units to cell $(S_3, D_1)$.
Update: $S_3 = 1400 - 1300 = 100$, $D_1 = 1300 - 1300 = 0$.
Allocate $\min(S_3, D_3) = \min(100, 100) = 100$ units to cell $(S_3, D_3)$.
Update: $S_3 = 100 - 100 = 0$, $D_3 = 100 - 100 = 0$.
All supply and demand are met.

The allocations are:

• $X_{12} = 1000$ units at cost $C_{12}=1$
• $X_{24} = 900$ units at cost $C_{24}=2$
• $X_{23} = 300$ units at cost $C_{23}=3$
• $X_{13} = 600$ units at cost $C_{13}=2$
• $X_{31} = 1300$ units at cost $C_{31}=5$
• $X_{33} = 100$ units at cost $C_{33}=8$

Total Cost = $(1000 \times 1) + (900 \times 2) + (300 \times 3) + (600 \times 2) + (1300 \times 5) + (100 \times 8)$
Total Cost = $1000 + 1800 + 900 + 1200 + 6500 + 800$
Total Cost = 12200