Question

Three plants P1, P2, and P3 produce 6, 1, and 9 thousand liters of fruit juice, respectively. The produced fruit juice is transported to three distribution centers D1, D2, and D3 with a requirement of 7, 5, and 4 thousand liters of juice, respectively. The transportation cost (in hundreds of Rupees) from each plant to each distribution center is given in the table. The total transportation cost (in hundreds of Rupees) in the initial basic feasible solution using Vogel’s approximation method is ............. (Answer in integer) 

Hint:

Use Vogel’s Approximation Method to minimize transportation costs by considering the highest penalty differences and allocating according to the least cost.

Answer 1

Correct Answer: 95

 To solve this problem using Vogel's Approximation Method (VAM), follow these steps:

1. Calculate Penalties: For each row and column, compute the penalty by subtracting the smallest transportation cost from the second smallest in that row or column.
2. Select Maximum Penalty: Choose the cell with the maximum penalty. If there is a tie, select the lowest cost in a tied row or column.
3. Allocate Supplies: Allocate as much supply as possible to the selected cell without exceeding demand.
4. Update Table: Adjust the supply and demand, and remove the row or column with zero supply or demand. Repeat the process until all demands are met.

Initial Steps:

	D1	D2	D3	Supply
P1	2	3	11	6
P2	1	0	6	1
P3	5	8	15	9
	7	5	4

Step 1 (Penalties): Calculate penalties for each row and column.

• Row P1: 3−2=1
• Row P2: 6−0=1
• Row P3: 8−5=3
• Column D1: 5−1=4
• Column D2: 8−0=3
• Column D3: 11−6=5

Step 2 (Select Maximum Penalty): Max penalty is in Column D3 (5).

Step 3 (Allocate): Allocate 4 units from P1 to D3 at cost 11.

Current Allocation Cost: 4×11=44

Update Supply and Demand:

• P1: 6−4=2
• D3 Demand: 4−4=0

Continue Process:

• Re-compute penalties, select the next maximum, and allocate:
• Allocate 5 units from P2 to D2 at cost 0.
• Allocate 2 units from P1 to D1 at cost 2.
• Allocate 2 units from P3 to D1 at cost 5.
• Allocate 4 units from P3 to D2 at cost 8.

Total Cost Calculation:

• 4×11 + 5×0 + 2×2 + 2×5 + 4×8 = 44 + 0 + 4 + 10 + 32 = 90

The total transportation cost (in hundreds) is 90, which is within the given range of 95,95.