Question 2 (a)
EOQ by calculus, minimum total cost, and a quantity-discount offer
9 marks
- Time to write
- About 18 minutes
Question
Question 2 (a) · 9 marksA publishing house buys 6000 units of a particular item each year at a unit cost of ₹24. The ordering cost per order is ₹50 and the storage cost equals 10% of the value of average inventory. Using calculus, determine the optimal order quantity and the minimum total cost. Would you accept an offer from a supplier who proposes a discount of 10% on an order size of 1000 units?
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
- Would you accept…?A yes-or-no question: answer it in your first line, then explain why.
- 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 6 min
- 2 Minimum total cost 2 marks 4 min
- 3 Cost under the discount offer 3 marks 5 min
- 4 Decision 1 mark 1 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Short on time? End with step 4's one-line decision, whatever happens: never leave the figures without it.
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 = 6,000 units; unit cost ₹24; ordering cost C₀ = ₹50 an order; storage (carrying) cost 10% of the value of average inventory, so Cₕ = 0.10 × 24 = ₹2.40 a unit a year.
Optimal order quantity by calculus
Annual inventory cost for order size Q: TC(Q) = (D/Q)C₀ + (Q/2)Cₕ = 3,00,000/Q + 1.2Q.
dTC/dQ = −3,00,000/Q² + 1.2 = 0 ⟹ Q² = 2,50,000 ⟹ Q = 500 units. d²TC/dQ² = 6,00,000/Q³ > 0, so this is a minimum (second derivative test); derivatives by the power rule.
| Cost | EOQ = 500 (₹) | Offer: 1,000 at 10% discount (₹) |
|---|---|---|
| Purchase cost | 6,000 × 24 = 1,44,000 | 6,000 × 21.60 = 1,29,600 |
| Ordering cost | 12 orders × 50 = 600 | 6 orders × 50 = 300 |
| Carrying cost | (500/2) × 2.40 = 600 | (1,000/2) × 2.16 = 1,080 |
| Total cost | 1,45,200 | 1,30,980 |
The minimum total cost at the EOQ is ₹1,45,200 a year (₹1,200 of it ordering and carrying).
With a 10% discount the price is ₹21.60 and the carrying cost 10% × 21.60 = ₹2.16 a unit. Ordering 1,000 units at a time costs ₹1,30,980 a year, which is ₹14,220 less than ₹1,45,200. Accept the offer.
The other reading. If the carrying cost stays at ₹2.40 a unit, the offer costs 1,29,600 + 300 + 1,200 = ₹1,31,100, still ₹14,100 below the EOQ cost: the decision is the same.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | EOQ by calculus with second-order condition | 3 |
| 2 | Minimum total cost | 2 |
| 3 | Cost under the discount offer | 3 |
| 4 | Decision | 1 |
| Total | 9 |
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