EOQ for Plastic Fasteners: 23-Ind-A4 Production Management, May 2017, Q4

Question 4: EOQ for Plastic Fasteners (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.

Check — scenario numbers

This question uses the “plastic fasteners” EOQ scenario with a unit cost of $0.02 each, and the part-(b) “actual demand” scenario is 15,000/week.

Given. A single EOQ item with a forecast usage rate, unit cost, order cost, and an annualized holding-cost rate.

Quantity	Value
Forecast usage rate	6,000 fasteners/week (52 operating weeks/yr) $\Rightarrow D=312{,}000$/yr
Unit cost $c$	$0.02 each
Order cost $S$	$12/order
Holding-cost rate	25% annualized $\Rightarrow H=0.25c=\$0.005$/unit/yr

Find. (a) The order interval, order quantity, and total annual ordering+holding cost at the economic order quantity; (b) using part (a)'s quantity at the true (higher) demand of 15,000/week, the resulting total cost, and its penalty relative to re-optimizing at that demand.

Approach. Apply the classical EOQ model to the forecast demand for part (a); for part (b), hold the order quantity fixed at part (a)'s $Q^{*}$ but evaluate the total-cost function at the true demand, and compare against the total cost of re-optimizing $Q$ at that same true demand.

Economic order quantity and annual cost (part a). $$Q^{*}=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(312{,}000)(12)}{0.005}}=\boxed{Q^{*}\approx38{,}698.8\ \text{fasteners}}.$$ Orders per year $=D/Q^{*}=312{,}000/38{,}698.8\approx8.06$/yr, i.e. an order interval of $52/8.06\approx\boxed{6.45\ \text{weeks}}$ (order roughly every 6–7 weeks). The total annual ordering+holding cost at $Q^{*}$ is $$TC(Q^{*})=\frac{D}{Q^{*}}S+\frac{Q^{*}}{2}H=\frac{312{,}000}{38{,}698.8}(12)+\frac{38{,}698.8}{2}(0.005)=\boxed{\$193.49/\text{yr}}.$$
Cost of using part-a's quantity at the true 15,000/week demand (part b). The true annual demand is $D_2=15{,}000\times52=780{,}000$/yr. Holding $Q$ fixed at the part-(a) $Q^{*}=38{,}698.8$ but evaluating at $D_2$: $$TC_{\text{wrong }Q}=\frac{D_2}{Q^{*}}S+\frac{Q^{*}}{2}H=\frac{780{,}000}{38{,}698.8}(12)+\frac{38{,}698.8}{2}(0.005)=\boxed{\$338.61/\text{yr}}.$$
True optimal cost at 15,000/week, and comparison. Re-solving the EOQ formula at the corrected demand: $$Q_2^{*}=\sqrt{\frac{2D_2S}{H}}=\sqrt{\frac{2(780{,}000)(12)}{0.005}}\approx61{,}188.2\ \text{fasteners},\qquad TC(Q_2^{*})=\boxed{\$305.94/\text{yr}}.$$ Ordering the wrong (part-a) quantity instead of the true optimum costs $338.61-305.94=\boxed{\$32.67/\text{yr}}$ more, a $\boxed{10.7\%}$ penalty. This is a modest penalty despite the demand forecast being wrong by a full 150%, because the $TC(Q)$ curve is shallow near its minimum — the EOQ model is forgiving of moderate-to-large forecast error precisely because ordering and holding costs trade off against each other symmetrically around the optimum.

Quantity	Value
EOQ (part a), demand as forecast	38,698.8 fasteners, every ≈6.45 weeks
Total ordering+holding cost at EOQ (part a)	$193.49/yr
Cost using part-a quantity at true 15,000/wk demand (part b)	$338.61/yr
True optimal EOQ and cost at 15,000/wk (part b)	61,188.2 fasteners; $305.94/yr
Penalty for having used the wrong EOQ	$32.67/yr (10.7% above true optimum)

Check: fastener quantities are treated as continuous for the EOQ formula (standard practice for a high-volume, low-cost item); rounding to a whole fastener count changes the boxed totals by well under 0.01%.

Concept & Theory

Topic. The classical economic order quantity (EOQ) model, and the cost of ordering a quantity sized for the wrong demand forecast.

Key relations. $Q^{*}=\sqrt{2DS/H}$; total annual cost $TC(Q)=\dfrac{D}{Q}S+\dfrac{Q}{2}H$; order interval $=52\,Q^{*}/D$ weeks.

Why this works. $TC(Q)$ trades a falling ordering-cost term against a rising holding-cost term as $Q$ grows; setting $dTC/dQ=0$ shows the two terms are exactly equal at the optimum, which is the standard EOQ derivation. Because $TC(Q)$ is flat (quadratic, not steep) near its minimum, a moderately wrong $Q$ — even one sized for a demand forecast that was very wrong — produces only a modest cost penalty rather than a proportionally large one.

Common pitfall. Forgetting to convert the 25% holding rate into an absolute annual holding cost per unit ($H=0.25c$, not 25 or 0.25 in isolation); and assuming a large forecast error must produce a large cost penalty, when in fact the EOQ total-cost curve's shallow minimum means the opposite is often true.

Source. Nahmias & Olsen, Production and Operations Analysis, Ch. 4 (deterministic inventory models, EOQ).

Question 4 of 8: EOQ for Plastic Fasteners

Question 4: EOQ for Plastic Fasteners (20 marks)