WX93 Production Line — Setup/Inventory Trade-off (EPQ): 23-Ind-A4 Production Management, May 2018, 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.

Neither the number of production days per week nor the annual calendar is stated explicitly; this solution assumes a standard 5-day production week (52 weeks/year, 260 production days/year), consistent with the given hourly production rate and 8-hour production day.

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 (part a) $=17{,}000$ units.

Quantity	Value
Production rate, $p$	$600$/h $\times\,8$ h/day $=4{,}800$ units/day
Demand rate, $d$ (5-day week)	$17{,}000/5=3{,}400$ units/day
Annual demand, $D$	$17{,}000\times52=884{,}000$ 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 once demand jumps to 25,000/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 $t_p\approx88.8$ 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; for part (b), first check whether the new demand rate is even physically achievable on this line before re-optimizing.

Utilization and EPQ batch size (part a). With demand-to-production ratio $u=d/p=3{,}400/4{,}800=0.7083$, the classic EPQ formula (which reduces holding cost by the fraction of the cycle inventory is actually accumulating, $1-u$) gives $$Q^*=\sqrt{\frac{2DS}{H(1-u)}}=\sqrt{\frac{2(884{,}000)(150)}{0.005(1-0.7083)}}=\boxed{Q^*\approx426{,}440\ \text{units}}.$$
Production plan: run length, cycle length, setups/year. Each batch takes $t_p=Q^*/p=426{,}440/4{,}800\approx88.8$ production days to run, after which the line switches to other products for the rest of the cycle $t_c=Q^*/d=426{,}440/3{,}400\approx125.4$ production days ($\approx$25.1 weeks) before WX93 is due again. This gives $D/Q^*\approx\boxed{2.07\ \text{setups per year}}$ — i.e., run a $\approx$426,440-unit batch of WX93 roughly once every 5–6 months, occupying the line for about 17.8 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, $Q^*$	$\approx426{,}440$ units
Production run length per batch, $t_p$	$\approx88.8$ days ($\approx17.8$ 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 25,000/week. At the new demand, $d_2=25{,}000/5=5{,}000$ units/day — but the line's own maximum output at 600 units/hour over an 8-hour, 5-day week is only $600\times8\times5=24{,}000$ units/week. The requested 25,000/week exceeds the line's physical capacity under the standard schedule used in part (a); no batching strategy, however chosen, can make an EPQ plan feasible when $d>p$, since the $(1-d/p)$ term in the EPQ formula itself goes negative. This is a capacity problem, not an inventory-policy problem, and the considerations are: (i) confirm the new demand is genuinely sustained (a temporary popularity surge does not justify a permanent capacity commitment); (ii) add real capacity — overtime, a second shift, or a longer production week (a 6-day week alone would raise the ceiling to $28{,}800$/week, comfortably above 25,000, at the cost of overtime premium and no idle multi-product capacity); (iii) apply SMED (Question 3) to cut the 3-hour setup time, which frees genuine run-time capacity on this shared multi-product line rather than just shrinking $Q^*$; (iv) use the stock already built: if the part-(a) run began at week 0, after four weeks (20 production days) the line has built up $(4{,}800-3{,}400)\times20=28{,}000$ units, enough to cover a 1,000-unit/week shortfall for about 28 weeks while extra capacity is arranged; and (v) coordinate with whichever product is driving WX93's new demand, since a request this close to (and in this case beyond) the line's rated capacity leaves no margin for ordinary demand variability once any extra capacity is added.

Concept & Theory

Topic. Economic Production Quantity (EPQ) — the finite-replenishment-rate extension of EOQ — and the capacity ceiling that limits any batching policy when demand approaches or exceeds the production rate.

Key relations. $Q^*=\sqrt{\dfrac{2DS}{H(1-d/p)}}$; annual holding cost $=H(1-d/p)\dfrac{Q^*}{2}$; annual setup cost $=\dfrac{D}{Q^*}S$; weekly capacity ceiling $=p\times(\text{production days/week})$; as $d\to p$, $(1-d/p)\to0$ and $Q^*\to\infty$, and for $d>p$ the model is undefined — no EPQ batch can ever build ahead of a demand rate the line cannot even match.

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, and the same factor is exactly what signals infeasibility once $d\ge p$.

Common pitfall. Mechanically plugging a demand rate that exceeds the production rate into the EPQ formula (yielding a nonsensical negative-under-the-root result) instead of first checking $d

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)