Use / VAN to calculate the optimum cost element of transportation from the transaction above.

Question

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To calculate the optimum cost element of transportation using the Vogel's Approximation Method (VAM), we first need to set up the transportation table.

From the image, we can identify the following:

Costs ( $C_{ij}$ ):
- Row 1: 4, 1, 2, 6
- Row 2: 8, 8, 3, 2
- Row 3: 5, 9, 8, 7
Demand ( $D_j$ ):
- $D_1 = 1300$
- $D_2 = 1000$
- $D_3 = 1000$
- $D_4 = 900$
- Total Demand = $1300 + 1000 + 1000 + 900 = 4200$
Supply ( $S_i$ ):
- $S_2 = 1400$
- $S_3 = 1400$
- The supply for $S_1$ is missing. Since the total demand is 4200, we assume the total supply must also be 4200 for a balanced problem.
- Therefore, $S_1 = 4200 - (1400 + 1400) = 4200 - 2800 = 1400$ .

The balanced transportation table is:

\begin{array}{|c|c|c|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_2 (1000) & D_3 (1000) & D_4 (900) & \text{Supply} (S_i) \\ \hline S_1 & 4 & 1 & 2 & 6 & 1400 \\ S_2 & 8 & 8 & 3 & 2 & 1400 \\ S_3 & 5 & 9 & 8 & 7 & 1400 \\ \hline \text{Demand} (D_j) & 1300 & 1000 & 1000 & 900 & 4200 \\ \hline \end{array}

We will now apply the VAM steps.

Calculate initial penalties.
Calculate the difference between the two lowest costs for each row and column.
$\begin{array}{|c|c|c|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_2 (1000) & D_3 (1000) & D_4 (900) & \text{Supply} (S_i) & \text{Row Penalty} \\ \hline S_1 & 4 & 1 & 2 & 6 & 1400 & 2-1=1 \\ S_2 & 8 & 8 & 3 & 2 & 1400 & 3-2=1 \\ S_3 & 5 & 9 & 8 & 7 & 1400 & 7-5=2 \\ \hline \text{Demand} (D_j) & 1300 & 1000 & 1000 & 900 & & \\ \hline \text{Col Penalty} & 5-4=1 & 8-1=7 & 3-2=1 & 6-2=4 & & \\ \hline \end{array}$
Make the first allocation.
The highest penalty is 7 (Column $D_2$ ). In $D_2$ , the minimum cost is $C_{12}=1$ .
Allocate $\min(S_1, D_2) = \min(1400, 1000) = 1000$ units to cell $(S_1, D_2)$ .
Update supply and demand: $S_1 = 1400 - 1000 = 400$ , $D_2 = 1000 - 1000 = 0$ . Column $D_2$ is satisfied.
Recalculate penalties and make the second allocation.
$\begin{array}{|c|c|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_3 (1000) & D_4 (900) & \text{Supply} (S_i) & \text{Row Penalty} \\ \hline S_1 & 4 & 2 & 6 & 400 & 4-2=2 \\ S_2 & 8 & 3 & 2 & 1400 & 3-2=1 \\ S_3 & 5 & 8 & 7 & 1400 & 7-5=2 \\ \hline \text{Demand} (D_j) & 1300 & 1000 & 900 & & \\ \hline \text{Col Penalty} & 5-4=1 & 3-2=1 & 6-2=4 & & \\ \hline \end{array}$
The highest penalty is 4 (Column $D_4$ ). In $D_4$ , the minimum cost is $C_{24}=2$ .
Allocate $\min(S_2, D_4) = \min(1400, 900) = 900$ units to cell $(S_2, D_4)$ .
Update supply and demand: $S_2 = 1400 - 900 = 500$ , $D_4 = 900 - 900 = 0$ . Column $D_4$ is satisfied.
Recalculate penalties and make the third allocation.
$\begin{array}{|c|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_3 (1000) & \text{Supply} (S_i) & \text{Row Penalty} \\ \hline S_1 & 4 & 2 & 400 & 4-2=2 \\ S_2 & 8 & 3 & 500 & 8-3=5 \\ S_3 & 5 & 8 & 1400 & 8-5=3 \\ \hline \text{Demand} (D_j) & 1300 & 1000 & & \\ \hline \text{Col Penalty} & 5-4=1 & 3-2=1 & & \\ \hline \end{array}$
The highest penalty is 5 (Row $S_2$ ). In $S_2$ , the minimum cost is $C_{23}=3$ .
Allocate $\min(S_2, D_3) = \min(500, 1000) = 500$ units to cell $(S_2, D_3)$ .
Update supply and demand: $S_2 = 500 - 500 = 0$ , $D_3 = 1000 - 500 = 500$ . Row $S_2$ is satisfied.
Recalculate penalties and make the fourth allocation.
$\begin{array}{|c|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_3 (500) & \text{Supply} (S_i) & \text{Row Penalty} \\ \hline S_1 & 4 & 2 & 400 & 4-2=2 \\ S_3 & 5 & 8 & 1400 & 8-5=3 \\ \hline \text{Demand} (D_j) & 1300 & 500 & & \\ \hline \text{Col Penalty} & 5-4=1 & 8-2=6 & & \\ \hline \end{array}$
The highest penalty is 6 (Column $D_3$ ). In $D_3$ , the minimum cost is $C_{13}=2$ .
Allocate $\min(S_1, D_3) = \min(400, 500) = 400$ units to cell $(S_1, D_3)$ .
Update supply and demand: $S_1 = 400 - 400 = 0$ , $D_3 = 500 - 400 = 100$ . Row $S_1$ is satisfied.
Make the final allocations.
Only row $S_3$ and columns $D_1$ , $D_3$ remain.
$\begin{array}{|c|c|c|c|} \hline \text{From/To} & D_1 (1300) & D_3 (100) & \text{Supply} (S_i) \\ \hline S_3 & 5 & 8 & 1400 \\ \hline \text{Demand} (D_j) & 1300 & 100 & \\ \hline \end{array}$
Allocate remaining demand to $S_3$ .
Allocate $\min(S_3, D_1) = \min(1400, 1300) = 1300$ units to cell $(S_3, D_1)$ .
Update supply and demand: $S_3 = 1400 - 1300 = 100$ , $D_1 = 1300 - 1300 = 0$ .
Allocate $\min(S_3, D_3) = \min(100, 100) = 100$ units to cell $(S_3, D_3)$ .
Update supply and demand: $S_3 = 100 - 100 = 0$ , $D_3 = 100 - 100 = 0$ .
All supply and demand are met.
Calculate the total transportation cost.
The allocations are:
- $X_{12} = 1000$ units at cost $C_{12}=1$
- $X_{24} = 900$ units at cost $C_{24}=2$
- $X_{23} = 500$ units at cost $C_{23}=3$
- $X_{13} = 400$ 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 = $(1000 \times 1) + (900 \times 2) + (500 \times 3) + (400 \times 2) + (1300 \times 5) + (100 \times 8)$
Total Cost = $1000 + 1800 + 1500 + 800 + 6500 + 800$
Total Cost = 12400

Use / VAN to calculate the optimum cost element of transportation from the transaction above.

Answer

Step-by-step solution

Calculate initial penalties.

Make the first allocation.

Recalculate penalties and make the second allocation.

Recalculate penalties and make the third allocation.

Recalculate penalties and make the fourth allocation.

Make the final allocations.

Calculate the total transportation cost.

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