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
To find the total transportation cost using Vogel's Approximation Method, we need to follow these steps:
Step 1: Find the penalty costs for each row and column.
Row penalties:
Penalty for row O1 = 9 - 6 = 3
Penalty for row O2 = 11 - 2 = 9
Penalty for row O3 = 12 - 4 = 8
Column penalties:
Penalty for column D1 = 6 - 1 = 5
Penalty for column D2 = 12 - 35 = 23
Penalty for column D3 = 4 - 50 = 46
Penalty for column D4 = 7 - 45 = 38
Step 2: Assign the shipment to the cell with the highest penalty cost.
The highest penalty cost is 46, which corresponds to the cell in row O3 and column D3. We assign the shipment to this cell and subtract the available supply (4) and demand (50) accordingly.
The updated supply and demand table:
Destination
Origins D1 D2 D3 D4 Supply
O1 6 1 5 3 70
O2 11 5 2 8 55
O3 10 12 0 7 70
Demand 85 35 46 45
Step 3: Repeat steps 1 and 2 until all the supply and demand values are zero.
Next, we need to recalculate the penalty costs:
Row penalties:
Penalty for row O1 = 5 - 6 = -1
Penalty for row O2 = 11 - 2 = 9
Penalty for row O3 = 12 - 0 = 12
Column penalties:
Penalty for column D1 = 6 - 1 = 5
Penalty for column D2 = 0 - 35 = -35
Penalty for column D4 = 7 - 45 = 38
The highest penalty cost is 38, which corresponds to the cell in row O3 and column D4. We assign the shipment to this cell and update the supply and demand accordingly.
The updated supply and demand table:
Destination
Origins D1 D2 D3 D4 Supply
O1 6 1 5 3 70
O2 11 5 2 8 55
O3 10 12 0 0 70
Demand 85 35 46 7
Continue the process until all the supply and demand values are zero.
The final supply and demand table:
Destination
Origins D1 D2 D3 D4 Supply
O1 6 1 5 3 70
O2 11 0 2 8 55
O3 10 24 0 0 70
Demand 85 12 46 7
Step 4: Calculate the total transportation cost.
To find the total transportation cost, we multiply the quantity shipped in each cell by its corresponding unit transportation cost and sum them up.
Total transportation cost = (6 * 6) + (1 * 1) + (5 * 10) + (3 * 7) + (2 * 2) + (8 * 8) + (24 * 12) = 36 + 1 + 50 + 21 + 4 + 64 + 288 = 464