Question 2 OR (a)
EOQ by calculus, cost at EOQ, order frequency and the break-even quantity discount
9 marks
- Time to write
- About 18 minutes
Question
Question 2 OR (a) · 9 marksTo maintain the stock of one of its components, a factory follows an economic order quantity system. The annual demand is 14,400 units, the cost of placing an order is ₹64, and the component cost is ₹80 per unit. The factory has imputed 10% as the carrying rate on inventory.
You are required to determine the following:
The Economic Order Quantity (EOQ) by calculus.
Total cost at the EOQ.
Optimal number of orders.
Optimal interval between orders, taking a year as 360 days.
The discount rate that the supplier should offer so that the purchase of 7,200 units is as attractive as the purchase of EOQ.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- determineFormula first, then the working, then the answer with its unit (₹, %, units).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 EOQ by calculus with second-order condition 3 marks 5 min
- 2 Total cost at EOQ 2 marks 4 min
- 3 Number of orders and interval 2 marks 4 min
- 4 Break-even discount 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Data. Annual demand D = 14,400 units; ordering cost C₀ = ₹64 an order; unit cost u = ₹80; carrying rate 10% a year, so the carrying cost per unit a year is Cₕ = 0.10 × 80 = ₹8.
With order size Q, the annual inventory cost (ordering + carrying) is
TC(Q) = (D/Q)C₀ + (Q/2)Cₕ = 9,21,600/Q + 4Q.
dTC/dQ = −9,21,600/Q² + 4 = 0 ⟹ Q² = 2,30,400 ⟹ Q = 480 units.
d²TC/dQ² = 2 × 9,21,600/Q³ > 0 for Q > 0, so Q = 480 gives the minimum (second derivative test), with the derivatives taken by the power rule. (Equivalently EOQ = √(2DC₀/Cₕ) = √(2 × 14,400 × 64/8) = 480.)
| Cost item | Working | ₹ |
|---|---|---|
| Ordering cost | (14,400/480) × 64 | 1,920 |
| Carrying cost | (480/2) × 8 | 1,920 |
| Inventory cost at EOQ | 3,840 | |
| Purchase cost | 14,400 × 80 | 11,52,000 |
| Total cost at EOQ | 11,55,840 |
At the EOQ the ordering and carrying costs are equal (₹1,920 each).
(iii) Optimal number of orders = D/EOQ = 14,400/480 = 30 orders a year.
(iv) Optimal interval between orders = 360/30 = 12 days.
(v) Discount that makes an order of 7,200 units as attractive as the EOQ
Let the discount be d (as a fraction of the price). The carrying cost is 10% of the value of stock held, so it is taken at the discounted price. With Q = 7,200 (2 orders a year):
TC(7,200) = 14,400 × 80(1 − d) + 2 × 64 + (7,200/2) × 0.10 × 80(1 − d) = 11,52,000(1 − d) + 128 + 28,800(1 − d) = 11,80,800(1 − d) + 128.
Setting this equal to the total cost at the EOQ: 11,80,800(1 − d) + 128 = 11,55,840
1 − d = 11,55,712/11,80,800 = 0.978753 ⟹ d = 2.1247%, about 2.12%.
The supplier must offer a discount of about 2.12% on the price for the 7,200-unit order to be as attractive as ordering the EOQ; any larger discount makes the bulk order cheaper.
The other reading. If the carrying cost is kept at the undiscounted ₹8 a unit, TC(7,200) = 11,52,000(1 − d) + 128 + 28,800, and the break-even discount is d = 2.1778%, about 2.18%.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | EOQ by calculus with second-order condition | 3 |
| 2 | Total cost at EOQ | 2 |
| 3 | Number of orders and interval | 2 |
| 4 | Break-even discount | 2 |
| Total | 9 |
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