Economic Order Quantity for Cemented-Carbide Inserts: 22-Mec-B4 Integrated Manufacturing Systems, Undated paper, Q5

Question 5: Economic Order Quantity for Cemented-Carbide Inserts (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Symbol	Value
Monthly usage rate	$U$	1,100 inserts per month
Annual demand	$D$	$12 \times 1{,}100 = 13{,}200$ inserts per year
Unit cost at order quantities over 500	$C$	$4.36 per insert
Ordering cost per batch	$S$	$60 per order
Holding charge, fraction of item cost per year	$i$	0.25 per year

Find. The economic order quantity for the insert — and, because the unit price is conditional on ordering more than 500, a check that the answer is consistent with the price actually used.

Total annual cost against order quantity for the insert. The ordering and carrying components cross exactly at the economic order quantity, and the total curve is very flat around it, so a convenient box quantity near 1,200 costs essentially the same as the exact optimum.

Approach. Convert the monthly usage to an annual demand, convert the percentage holding charge into dollars per insert per year using the unit price, apply the square-root EOQ formula, then verify that the resulting quantity exceeds the 500-unit price break so that the price used is the price that applies.

Put demand on an annual basis. The holding charge is quoted per year, so demand must be too: $$D = 12 \times 1{,}100 = 13{,}200\ \text{inserts per year.}$$ Mixing a monthly demand with an annual holding rate is the single commonest way to lose this question, and it produces an answer low by a factor of $\sqrt{12}$.
Convert the percentage holding charge into dollars. The shop charges 25 % per year of the value of the item held, so $$H = i\,C = 0.25 \times 4.36 = \boxed{H = \textrm{CAD } 1.09 \text{ per insert per year.}}$$
Apply the economic order quantity formula. Balancing the annual ordering cost $DS/Q$ against the annual carrying cost $HQ/2$ gives $$Q^{*} = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2(13{,}200)(60)}{1.09}} = \sqrt{1{,}453{,}211} = \boxed{Q^{*} = 1{,}205\ \text{inserts per order.}}$$
Check the price break. The quoted price of $4.36 applies to quantities over 500, and $Q^{*} = 1{,}205 > 500$, so the price used to build $H$ is the price that will actually be paid and no price-break comparison against a higher unit cost is needed. Had the optimum fallen below 500 the calculation would have had to be repeated at the higher small-quantity price and the two totals compared.
Report the ordering pattern and verify with the cost balance. The shop places $$n = \frac{D}{Q^{*}} = \frac{13{,}200}{1{,}205} = 10.95 \approx 11 \ \text{orders per year,}$$ one about every 1.1 months or 4.7 weeks. At that quantity $$\frac{D}{Q^{*}}S = 656.99, \qquad H\frac{Q^{*}}{2} = 656.99,$$ equal as the optimum requires, so the relevant inventory cost is $$\boxed{TC = \sqrt{2DSH} = \textrm{CAD } 1{,}313.99 \text{ per year,}}$$ against a purchase cost of $13{,}200 \times 4.36 =$ $57,552, giving an all-in annual cost of $58,865.99.
Note how flat the optimum is. Ordering a round 1,200 inserts rather than 1,205 costs $$TC(1{,}200) = \frac{13{,}200}{1{,}200}(60) + 1.09\frac{1{,}200}{2} = 660.00 + 654.00 = 1{,}314.00,$$ about one cent a year more. The recommendation to the shop is therefore a convenient standard box quantity of 1,200 rather than the arithmetically exact figure — the total-cost curve near its minimum is so flat that practicality should win.

Final results.

Quantity	Result
Annual demand	13,200 inserts per year
Holding cost	$1.09 per insert per year
Economic order quantity	1,205 inserts (recommend a 1,200 box quantity)
Orders per year	10.95, say 11 — about every 4.7 weeks
Annual ordering / carrying cost	$656.99 each
Relevant annual inventory cost	$1,313.99 per year
All-in annual cost including purchase	$58,865.99 per year

Topic. The basic economic order quantity with a holding charge expressed as a percentage of item value, and the quantity-discount consistency check.

Key relations. $D = 12U$; $H = iC$; $Q^{*}=\sqrt{2DS/H}$; $n = D/Q^{*}$; $TC_{\text{relevant}} = \sqrt{2DSH}$, with the ordering and carrying components equal at the optimum; total cost including purchase is $DC + \sqrt{2DSH}$.

Why this works. Writing the holding charge as a fraction of item value is how industry actually states it, because the cost of holding stock is dominated by the cost of the capital tied up in it, which is proportional to value; obsolescence, insurance, shrinkage and storage are then folded into the same percentage. Substituting H = iC into the square-root formula shows that the optimum quantity varies inversely with the square root of the unit price, so expensive items are ordered in small quantities and cheap ones in large — the quantitative justification for ABC inventory classification. The square root also explains why the answer is robust: the total cost is the sum of a decreasing and an increasing term of equal size at the optimum, so a proportional error of x in Q costs only about x-squared over two in total cost, and a 20 % error in the order quantity costs under 2 % of the relevant cost.

Common pitfall. Two dominate. Using the monthly usage with an annual holding rate, which understates the order quantity by a factor of the square root of twelve; and treating the 25 % directly as a dollar figure instead of multiplying it by the item cost, which makes H = 0.25 and inflates the answer by a factor of about two. A third, subtler one is forgetting to confirm that the computed quantity is on the right side of the stated price break.

Source. Nahmias & Olsen, Production and Operations Analysis, 7th ed., ch. 4.5 (the basic EOQ model) and 4.7 (quantity discounts); Chase & Jacobs, Operations and Supply Chain Management, 16th ed., ch. 20; Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed., ch. 26 (inventory control in the machine shop).

Question 5 of 7: Economic Order Quantity for Cemented-Carbide Inserts

Question 5: Economic Order Quantity for Cemented-Carbide Inserts (20 marks)