Question

A company purchases items in bulk for getting quantity discounts in the item’s price. The price break-up is given in the table. The annual demand for the item is 5000 units. The ordering cost is Rupees 400 per order. The annual inventory carrying cost is 30 percent of the purchase price per unit. The optimal order size (in units) is .......... (Answer in integer) % Table \[ \begin{array}{|c|c|} \hline {Quantity of item (Q in units)} & {Unit price of item (Rupees)}
\hline 0 \leq Q<1200 & 10
1200 \leq Q<2000 & 8
2000 \leq Q & 7
\hline \end{array} \]

Hint:

When dealing with quantity discounts, use the EOQ formula to calculate the optimal order size for each price range. Always select the optimal order size that falls within the constraints of each price range.

Answer 1

Correct Answer: 1995

We use the Economic Order Quantity (EOQ) formula to determine the optimal order size for each price range: The EOQ formula is: \[ EOQ = \sqrt{\frac{2DS}{H}} \] where:
- \( D = 5000 \) (annual demand),
- \( S = 400 \) (ordering cost),
- \( H \) is the holding cost, which is 30% of the unit price.
Case 1: \( 0 \leq Q<1200 \), Unit Price = 10 Rupees \[ H = 0.30 \times 10 = 3 \, {Rupees/unit} \] \[ EOQ = \sqrt{\frac{2 \times 5000 \times 400}{3}} \approx 1154.7 \, {units} \] Since \( Q<1200 \), this EOQ is feasible. Case 2: \( 1200 \leq Q<2000 \), Unit Price = 8 Rupees \[ H = 0.30 \times 8 = 2.4 \, {Rupees/unit} \] \[ EOQ = \sqrt{\frac{2 \times 5000 \times 400}{2.4}} \approx 1291.0 \, {units} \] Since \( 1200 \leq Q<2000 \), this EOQ is feasible. Case 3: \( 2000 \leq Q \), Unit Price = 7 Rupees \[ H = 0.30 \times 7 = 2.1 \, {Rupees/unit} \] \[ EOQ = \sqrt{\frac{2 \times 5000 \times 400}{2.1}} \approx 1380.4 \, {units} \] Since \( Q \geq 2000 \), the optimal order size is 2000 units. Thus, the optimal order size lies between 1995 and 2005 units.