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 solve the transportation problem using Vogel's Approximation Method, we need to find the opportunity cost for each cell.
First, let's calculate the difference between the two lowest transportation costs for each row and column:
For row 1:
Difference between lowest cost (1) and second lowest cost (3): 3 - 1 = 2
Difference between lowest cost (1) and second lowest cost (6): 6 - 1 = 5
Difference between lowest cost (1) and second lowest cost (9): 9 - 6 = 3
Difference between lowest cost (1) and second lowest cost (11): 11 - 9 = 2
For row 2:
Difference between lowest cost (2) and second lowest cost (5): 5 - 2 = 3
Difference between lowest cost (2) and second lowest cost (8): 8 - 5 = 3
Difference between lowest cost (2) and second lowest cost (11): 11 - 8 = 3
Difference between lowest cost (2) and second lowest cost (2): 2 - 2 = 0
For row 3:
Difference between lowest cost (4) and second lowest cost (7): 7 - 4 = 3
Difference between lowest cost (4) and second lowest cost (10): 10 - 7 = 3
Difference between lowest cost (4) and second lowest cost (12): 12 - 10 = 2
Difference between lowest cost (4) and second lowest cost (4): 4 - 4 = 0
Now, let's calculate the difference between the two lowest demands for each column:
For column 1:
Difference between lowest demand (35) and second lowest demand (45): 45 - 35 = 10
Difference between lowest demand (35) and second lowest demand (50): 50 - 35 = 15
Difference between lowest demand (35) and second lowest demand (85): 85 - 50 = 35
For column 2:
Difference between lowest demand (35) and second lowest demand (45): 45 - 35 = 10
Difference between lowest demand (35) and second lowest demand (50): 50 - 35 = 15
Difference between lowest demand (35) and second lowest demand (85): 85 - 50 = 35
For column 3:
Difference between lowest demand (35) and second lowest demand (45): 45 - 35 = 10
Difference between lowest demand (35) and second lowest demand (50): 50 - 35 = 15
Difference between lowest demand (35) and second lowest demand (85): 85 - 50 = 35
For column 4:
Difference between lowest demand (35) and second lowest demand (45): 45 - 35 = 10
Difference between lowest demand (35) and second lowest demand (50): 50 - 35 = 15
Difference between lowest demand (35) and second lowest demand (85): 85 - 50 = 35
Now, let's find the cell with the highest opportunity cost. To do this, we sum the differences for each row and each column, and select the highest value.
For row 1: 2 + 5 + 3 + 2 = 12
For row 2: 3 + 3 + 3 + 0 = 9
For row 3: 3 + 3 + 2 + 0 = 8
For column 1: 10 + 10 + 10 = 30
For column 2: 10 + 10 + 10 = 30
For column 3: 10 + 10 + 10 = 30
For column 4: 15 + 15 + 15 = 45
The highest opportunity cost is 45, which corresponds to column 4. Therefore, we select cell (O1, D4) as the first allocation with a quantity of 45.
Next, we update the supply and demand values accordingly:
Supply:
O1: 70 - 45 = 25
O2: 55
O3: 70
Demand:
D1: 85
D2: 35
D3: 50
D4: 45 - 45 = 0
Now, we recalculate the differences for the updated supply and demand values:
For row 1: 3 - 1 = 2
For row 2: 5 - 2 = 3
For row 3: 7 - 4 = 3
For column 1: 85 - 35 = 50
For column 2: 35 - 35 = 0
For column 3: 50 - 35 = 15
Next, we select the highest opportunity cost, which is 50, corresponding to column 1. We allocate the remaining supply (25) to cell (O3, D1).
Now, we update the supply and demand values:
Supply:
O1: 25
O2: 55
O3: 70 - 25 = 45
Demand:
D1: 85 - 25 = 60
D2: 35 - 35 = 0
D3: 50 - 45 = 5
D4: 45 - 45 = 0
The remaining supply is still greater than the remaining demand, so we can directly allocate the remaining supply (45) to cell (O3, D3).
Now, the transportation table is fully allocated:
Destination
Origins D1 D2 D3 D4 Supply
O1 0 0 25 0 25
O2 0 0 0 0 0
O3 60 0 5 45 110
Supply 60 0 30 45
Finally, let's calculate the total transportation cost:
Total Cost = (Quantity allocated in cell O1, D1 * Cost) + (Quantity allocated in cell O1, D3 * Cost) + (Quantity allocated in cell O3, D1 * Cost) + (Quantity allocated in cell O3, D3 * Cost)
Total Cost = (0 * 6) + (25 * 9) + (60 * 10) + (5 * 4) + (45 * 7)
Total Cost = 0 + 225 + 600 + 20 + 315
Total Cost = 1160