Question 4 of 8: WX93 Production Line — Setup/Inventory Trade-off (EPQ)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Examinations — May 2019 — 17-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: eight questions, each worth 20 marks (10/10 sub-part split per the front-page marking scheme); candidates do two questions from Section A and three from Section B, and only the first five questions appearing in the answer book are marked. All eight are solved below for completeness. The paper asks for point-form answers wherever possible; the solutions below use full working for clarity.
Reference texts: Nahmias & Olsen, Production and Operations Analysis (7th ed., Waveland/McGraw-Hill) — forecasting, inventory (EOQ/EPQ) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production scheduling, JIT/kanban and shop-floor implementation gaps; Hillier & Lieberman, Introduction to Operations Research (11th ed.) — LP formulation and project scheduling (CPM/PERT); Pinedo, Scheduling: Theory, Algorithms, and Systems (5th ed.) — parallel-machine scheduling and days-off workforce scheduling; Hopp & Spearman, Factory Physics (3rd ed.) — variability, buffering, and production scheduling; Liker, The Toyota Way, Shingo, A Revolution in Manufacturing: The SMED System, and Shingo, Zero Quality Control: Source Inspection and the Poka-Yoke System — 5S, Five Whys, poka-yoke, SMED and lean root-cause analysis; R.W. Hall, Zero Inventories — the “seven zeros” JIT framework.
Question 4: WX93 Production Line — Setup/Inventory Trade-off (EPQ) (20 marks)
part (b) is re-derived here, because a second shift (not just extra days) closes the capacity gap. 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), 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:
$$\text{Setup cost/yr}=\frac{D}{Q^*}S=2.07\times\$150\approx\$311,\qquad \text{Holding cost/yr}=H(1-u)\frac{Q^*}{2}\approx\$311,$$
$$\boxed{\text{Total annual setup}+\text{inventory cost}\approx\$622/\text{yr}}.$$
Feasibility of the new demand (part b). At the new demand, $d_2=34{,}000/5=6{,}800$ 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, so 34,000/week exceeds the line's rated capacity by over 40%. Even running the line every day of the week — $600\times8\times7=33{,}600$ units/week, a 7-day schedule with no maintenance or changeover downtime at all — still falls $\boxed{400\ \text{units/week short}}$ of 34,000. No batching strategy can make an EPQ plan feasible when $d>p$ under a given schedule, since the $(1-d/p)$ term in the EPQ formula goes negative. But the ceiling is set by line-hours, not by days: the line needs $34{,}000/600\approx56.7$ h/week, versus 40 h on a single 8-h shift, 5 days. A second shift ($600\times16\times5=\boxed{48{,}000\ \text{units/week}}$) closes the gap with margin; so do about 16.7 h/week of overtime or weekend hours.
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
Line capacity for 34,000/week (part b)
24,000 (1 shift, 5 d) and 33,600 (1 shift, 7 d) fall short; 56.7 line-h/week needed; 2 shifts, 5 d = 48,000/week suffices
(b) Considerations for the demand jump to 34,000/week. The gap is a capacity problem first and a batching problem second. The considerations are:
Stock on hand. Four weeks (20 production days) into the first 88.8-day run, WX93 inventory has built up to $(4{,}800-3{,}400)\times20=28{,}000$ units. At the new rate the line still makes 4,800/day but ships 6,800/day, so that stock lasts only $28{,}000/2{,}000=14$ production days (under 3 weeks). Extra capacity must be running before then, or shortages start.
Line-hours. 34,000/week needs about 56.7 line-hours/week. One 8-h shift on 5 days gives 40 h (24,000/week), and even 7 days gives only 56 h (33,600/week). A second shift, or overtime plus weekend hours, is needed. That adds labour cost (a second operator or an overtime premium), and fatigue and maintenance windows must be planned.
Other products on the line. This is a shared multi-product line. If WX93 takes about 57 h/week, almost no time is left for the other products. The added shift must cover them too, or WX93 needs a dedicated line or an outside supplier.
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 in part (a). The EPQ logic still applies with the new $D=34{,}000\times52$ and the extra-shift cost included.
Setup reduction. Apply SMED (Question 3) to the 3-hour setup. This frees run-time on the shared line.
Is the surge real? Confirm the new demand will last before committing to a permanent second shift; temporary overtime or a subcontractor may be cheaper for a short spike. Material supply for WX93 must also scale to the new rate.
