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

Part (b)'s demand jump differs and is genuinely fresh here: 34,000/week, a far more severe jump than May-2018's 25,000/week. 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 34,000/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 considering any batching change.

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 34,000/week. At the new demand, $d_2=34{,}000/5=6{,}800$ units/day. The line's output on the present single 8-hour, 5-day shift is only $600\times8\times5=24{,}000$ units/week, so 34,000/week is more than 40% above what the current schedule can make even if the line ran WX93 and nothing else. Running 8-hour days on all seven days ($600\times8\times7=33{,}600$/week) is still 400 units/week short. The requirement is $34{,}000/600\approx56.7$ line-hours of WX93 per week, before any time for setups or the line's other products. Considerations:

Short-term buffer. If the WX93 run began at week 0, four weeks (20 production days) in the line has built $20\times(4{,}800-3{,}400)=28{,}000$ units of stock (the run lasts about 89 days, so it is still running). At $6{,}800$/day demand against $4{,}800$/day output, that stock covers only $28{,}000/2{,}000=14$ production days, about 3 weeks. Extra capacity is needed within that window, or shipments must be rationed.
Add hours. A second 8-hour shift doubles weekly output to $600\times16\times5=48{,}000$ units, enough for WX93 with about 14,000 units/week of line capacity left for the other products. Weekend or extended-day overtime can bridge the gap until the second shift is staffed. Either way a second operator, at $50/h plus any shift or overtime premium, is needed.
Protect the other products. This is a shared multi-product line. Under the part-(a) plan WX93 already takes about 71% of line time ($d/p=0.708$). At 34,000/week it would need all of a single shift and more, so the other products' schedules must be re-planned, or WX93 moved to a second or dedicated line.
Re-plan the lot size. On two shifts, $p=9{,}600$/day and $d/p=6{,}800/9{,}600=0.708$, the same ratio as part (a). With annual demand $34{,}000\times52=1{,}768{,}000$, the EPQ becomes $Q^*=\sqrt{2(1{,}768{,}000)(150)/(0.005\times0.2917)}\approx603{,}000$ units ($\sqrt2$ times the part-(a) batch), with about $880/yr of setup plus holding cost. Setup and holding cost stay small next to the capacity decision.
Cut setup time. Applying SMED (Question 3) to the 3-hour setup gives back run time on the shared line and makes smaller, more frequent WX93 batches practical.
Confirm the demand and cost. Check that the jump is sustained and not a short popularity spike before hiring a shift or buying a line. Price the extra labour and overtime against the product's margin. Agree delivery quantities with the customer while 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 $d\gt p$ 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 the demand against weekly capacity ($p\times$ production hours); and, in part (b), proposing a bigger batch as the fix when the real constraint is capacity, not batch size. Capacity here is set by the hours worked, so a second shift or overtime closes the gap even though a single 8-hour shift cannot.

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)