WX93 Production Line — Setup/Inventory Trade-off (EPQ): 23-Ind-A4 Production Management, December 2017, Q4

Question 4: WX93 Production Line — Setup/Inventory Trade-off (EPQ) (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 — assumed 5-day production week

Neither the number of production days per week nor the annual calendar (weeks/year) is stated explicitly. This solution assumes a standard 5-day production week (52 weeks/year, 260 production days/year) to convert the given weekly demand into a daily rate consistent with the given hourly production rate and 8-hour production day — a 7-day-week assumption would scale every daily rate by $5/7$ but leave the annual totals (which depend only on annual demand $D=7{,}000\times52$) unchanged.

Given. Multi-product line producing WX93 at $p=600$ units/hour ($=4{,}800$ units/production day); material value $c=\$0.02$/unit; setup time $=3$ h at a $\$50$/h worker wage; annualized holding-cost rate $=25\%$; weekly demand $=7{,}000$ units.

Quantity	Value
Production rate, $p$	$600$/h $\times\,8$ h/day $=4{,}800$ units/day
Demand rate, $d$ (5-day week)	$7{,}000/5=1{,}400$ units/day
Annual demand, $D$	$7{,}000\times52=364{,}000$ units/yr
Setup cost, $S$	$3\text{ h}\times\$50\text{/h}=\$150$/setup
Holding cost, $H$	$0.25\times\$0.02=\$0.005$/unit-yr

Find. (a) A production plan (batch size / cycle) trading off setup and inventory cost, and the resulting total annual setup + holding cost; (b) the considerations needed if demand triples to 20,000/week.

Figure 1 — WX93 EPQ inventory profile (two cycles shown). Inventory ramps up at rate $p-d$ while the line is set up and running WX93 (duration $t_p\approx36.6$ production days), then declines at rate $d$ while the line produces other products, over a repeating cycle $t_c\approx125.4$ production days.

Approach. This is a finite-replenishment-rate (Economic Production Quantity) problem, not a simple instantaneous-delivery EOQ, because WX93 is produced at a finite rate $p$ that exceeds but does not vastly outstrip demand $d$: compute the EPQ batch size that minimizes total annual setup + holding cost, then derive the run length, cycle length, and number of setups/year that make up the production plan.

Utilization and EPQ batch size. With demand-to-production ratio $u=d/p=1{,}400/4{,}800=0.2917$, the classic EPQ formula (which reduces holding cost by the fraction of the cycle inventory is actually accumulating, $1-u$) gives $$Q^*=\sqrt{\frac{2DS}{H(1-u)}}=\sqrt{\frac{2(364{,}000)(150)}{0.005(1-0.2917)}}=\boxed{Q^*\approx175{,}600\ \text{units}}.$$
Production plan: run length, cycle length, setups/year. Each batch takes $t_p=Q^*/p=175{,}600/4{,}800\approx36.6$ production days to run, after which the line switches to other products for the rest of the cycle $t_c=Q^*/d=175{,}600/1{,}400\approx125.4$ production days ($\approx$25.1 weeks, $\approx$5.8 months) before WX93 is due again. This gives $D/Q^*\approx\boxed{2.07\ \text{setups per year}}$ — i.e., run a $\approx$175,600-unit batch of WX93 roughly once every 5–6 months, occupying the line for about 7.3 weeks each time.
Total annual setup and inventory cost. At the optimum, annual setup cost equals annual holding cost: $$\text{Setup cost/yr}=\frac{D}{Q^*}S=2.07\times\$150\approx\$311,\qquad \text{Holding cost/yr}=H(1-u)\frac{Q^*}{2}\approx\$311,$$ $$\boxed{\text{Total annual setup}+\text{inventory cost}\approx\$622/\text{yr}}.$$

Quantity	Value
EPQ batch size, $Q^*$	$\approx175{,}600$ units
Production run length per batch, $t_p$	$\approx36.6$ days ($\approx7.3$ weeks)
Cycle length, $t_c$	$\approx125.4$ days ($\approx25.1$ weeks)
Setups per year	$\approx2.07$
Total annual setup + holding cost	$\approx\$622$/yr

(b) Considerations for the demand jump to 20,000/week. At the new demand, $d_2=20{,}000/5=4{,}000$ units/day, so utilization on WX93 alone rises to $d_2/p=4{,}000/4{,}800=83.3\%$ (up from 29.2%). Re-running the EPQ formula with $D_2=1{,}040{,}000$/yr gives $Q_2^*\approx611{,}900$ units, with a run length of $Q_2^*/p\approx127.5$ production days ($\approx25.5$ weeks) inside a cycle of $Q_2^*/d_2\approx153$ production days ($\approx30.6$ weeks) — i.e., the line would need to run WX93 almost continuously (about 83% of every cycle) just to keep up. Because this is a multi-product line that must also serve other products' setups and runs, this leaves very little calendar time for anything else, so the original plan cannot simply continue at a larger batch size. Key considerations: (i) check whether the demand spike is genuinely sustained or a temporary popularity surge before committing capacity, since the EPQ commits the line to a long, capacity-heavy run either way; (ii) if sustained, add capacity — a second shift/overtime on WX93, or a second (possibly dedicated) line, since 83% utilization leaves almost no slack for changeovers or the line's other products; (iii) apply SMED (Question 3) to cut the 3-hour setup time, which both frees real capacity and lowers $S$, pulling $Q^*$ and the run-length fraction back down; and (iv) coordinate with whichever product is driving WX93's new demand, since a demand this close to the line's rated capacity for a single component leaves no margin for normal demand variability.

Concept & Theory

Topic. Economic Production Quantity (EPQ) — the finite-replenishment-rate extension of EOQ — and the capacity implications of a large demand increase on a shared multi-product line.

Key relations. $Q^*=\sqrt{\dfrac{2DS}{H(1-d/p)}}$; annual holding cost $=H(1-d/p)\dfrac{Q^*}{2}$; annual setup cost $=\dfrac{D}{Q^*}S$; as $d\to p$, $(1-d/p)\to0$ and $Q^*\to\infty$, reflecting that a line running at its own production rate can never build ahead of demand.

Why this works. Unlike classic EOQ (instantaneous delivery), EPQ inventory only accumulates during the fraction of the cycle the line is actually producing the item, at the net rate $p-d$; the $(1-d/p)$ factor correctly discounts the holding cost for this slower build-up, which is why a cheap component with a low holding rate (as here, $\$0.02\times25\%$) yields a very large optimal batch and a long cycle between runs.

Common pitfall. Using the plain EOQ formula ($Q^*=\sqrt{2DS/H}$, no $(1-d/p)$ term) when a finite production rate is given, which understates the true optimal batch size; and, in part (b), treating the new EPQ batch size as an automatic fix without checking whether the resulting utilization leaves any capacity for the line's other products.

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

Question 4 of 8: WX93 Production Line — Setup/Inventory Trade-off (EPQ)

Question 4: WX93 Production Line — Setup/Inventory Trade-off (EPQ) (20 marks)