3. a) Find the total transportation cost using Vogel’s Approximation Method when the unit transportation costs, demands and supplies are as given below:
Destination
Origins D1 D2 D3 D4 Supply
O1 6 1 9 3 70
O2 11 5 2 8 55
O3 10 12 4 7 70
Demand 85 35 50 45
5 10
To find the total transportation cost using Vogel's Approximation Method, follow these steps:
Step 1: Calculate the Penalty Cost Matrix
To calculate the penalty cost for each cell, subtract the smallest cost in the row from the second smallest cost in the row, and do the same for each column by subtracting the smallest cost in the column from the second smallest cost in the column.
The penalty cost matrix is as follows:
Destination
Origins D1 D2 D3 D4
O1 5 1 8 2
O2 3 1 0 3
O3 0 2 2 3
Step 2: Identify the Highest Penalty Cost Cell
Find the cell with the highest penalty cost in the matrix. In this case, it is the cell (O1, D3) with a penalty cost of 8.
Step 3: Allocate Units to the Cell with the Highest Penalty Cost
Allocate the maximum number of units possible to the cell with the highest penalty cost. In this case, the maximum supply is 70 and the maximum demand is 50, so allocate 50 units to cell (O1, D3).
Destination
Origins D1 D2 D3 D4 Supply
O1 5 1 0 2 20
O2 3 1 0 3 55
O3 0 2 2 3 70
Demand 85 35 40 45
0 10
Step 4: Reduce Supply and Demand
Update the supply and demand values after allocating units to the highest penalty cost cell.
Destination
Origins D1 D2 D3 D4 Supply
O1 5 1 0 2 20
O2 3 1 0 3 55
O3 0 2 2 3 20
Demand 35 35 40 45
0 10
Step 5: Recalculate the Penalty Cost Matrix
Calculate the penalty cost matrix again based on the updated supply and demand values.
Destination
Origins D1 D2 D3 D4
O1 5 1 0 2
O2 3 1 0 3
O3 0 2 2 3
Step 6: Identify the Highest Penalty Cost Cell
Find the cell with the highest penalty cost in the matrix. In this case, it is the cell (O2, D4) with a penalty cost of 3.
Step 7: Allocate Units to the Cell with the Highest Penalty Cost
Allocate the maximum number of units possible to the cell with the highest penalty cost. In this case, the maximum supply is 55 and the maximum demand is 45, so allocate 45 units to cell (O2, D4).
Destination
Origins D1 D2 D3 D4 Supply
O1 5 1 0 2 20
O2 3 1 0 0 10
O3 0 2 2 3 20
Demand 35 35 40 0
0 0
Step 8: Reduce Supply and Demand
Update the supply and demand values after allocating units to the highest penalty cost cell.
Destination
Origins D1 D2 D3 D4 Supply
O1 5 1 0 2 20
O2 3 1 0 0 0
O3 0 2 2 3 20
Demand 35 35 40 0
0 0
Step 9: Calculate the Transportation Cost
To calculate the total transportation cost, multiply the number of units in each cell by their respective transportation cost per unit and sum them up.
Total transportation cost = (20 * $6) + (1 * $1) + (2 * $4) + (2 * $8) + (2 * $12) + (2 * $4) + (2 * $7) = $442
Therefore, the total transportation cost using Vogel's Approximation Method is $442.