Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Technical Examinations — December 2016 — 98-Ind-A4 Production Management. Three-hour, closed-book exam; Casio or Sharp approved calculators only. Format: seven questions, each worth 20 marks (sub-part weights as tabulated on the front page); only the first five questions appearing in the answer book are marked, so candidates effectively choose 5 of 7. All seven 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) and aggregate planning; Sipper & Bulfin, Production: Planning, Control, and Integration — production-management systems; 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, makespan and tardiness; Hopp & Spearman, Factory Physics (3rd ed.) — variability and production-system inefficiency; Niebel & Freivalds, Methods, Standards, and Work Design — division of labour and work-design history; Liker, The Toyota Way, and the Toyota Production System literature — 5S, Five Whys, and lean root-cause analysis.
25% of unit cost, annualized $\Rightarrow H=0.25(0.002)=\$0.0005$/unit-yr
Find. (a) Order interval, EOQ, and total annual ordering+holding cost at the EOQ; (b) the cost of continuing to order the part-(a) quantity if true demand is 12,000/week; (c) the true optimal cost at 12,000/week, and the difference.
Approach. Apply the classical EOQ model $Q^*=\sqrt{2DS/H}$ to get the order quantity and interval, then evaluate the total ordering+holding cost function $TC(Q)=\frac{D}{Q}S+\frac{Q}{2}H$ at the part-(a) quantity under both the original and the corrected demand, and separately at the true optimum for the corrected demand, to isolate the cost of having used the wrong order quantity.
Economic order quantity and annual cost (part a).
$$Q^*=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(312{,}000)(12)}{0.0005}}=\boxed{Q^*\approx122{,}376\ \text{fasteners}}.$$
Orders per year $=D/Q^*=312{,}000/122{,}376\approx2.55$/yr, i.e. an order interval of $52/2.55\approx\boxed{20.4\ \text{weeks}}$ (about every 143 days, or roughly 5 months). The total annual ordering+holding cost at $Q^*$ is
$$TC(Q^*)=\frac{D}{Q^*}S+\frac{Q^*}{2}H=\frac{312{,}000}{122{,}376}(12)+\frac{122{,}376}{2}(0.0005)=30.59+30.59=\boxed{\$61.19/\text{yr}}.$$
Cost if actual demand is 12,000/week but still ordering $Q^*$ from (a) (part b). True annual demand is now $D_2=12{,}000\times52=624{,}000$/yr, double the forecast, but the order quantity is still the part-(a) value:
$$TC_{\text{wrong }Q}=\frac{D_2}{Q^*}S+\frac{Q^*}{2}H=\frac{624{,}000}{122{,}376}(12)+\frac{122{,}376}{2}(0.0005)=61.19+30.59=\boxed{\$91.78/\text{yr}}.$$
True optimal cost at 12,000/week, and comment (part c). Re-solving the EOQ formula at the corrected demand:
$$Q_2^*=\sqrt{\frac{2D_2S}{H}}=\sqrt{\frac{2(624{,}000)(12)}{0.0005}}\approx173{,}066\ \text{fasteners},\qquad TC(Q_2^*)=\boxed{\$86.53/\text{yr}},$$
ordered about every $52/(624{,}000/173{,}066)\approx14.4$ weeks. Ordering the wrong (part-a) quantity instead of the true optimum costs $91.78-86.53=\boxed{\$5.25/\text{yr}}$ more, a $\boxed{6.1\%}$ penalty. This is a modest penalty despite the demand forecast being wrong by a full 100%, 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
≈122,376 fasteners, every ≈20.4 weeks (2.55 orders/yr)
Total ordering+holding cost at EOQ (part a)
\$61.19/yr
Cost using part-a quantity at true 12,000/wk demand (part b)
\$91.78/yr
True optimal EOQ and cost at 12,000/wk (part c)
≈173,066 fasteners; \$86.53/yr
Penalty for having used the wrong EOQ
\$5.25/yr (6.1% above true optimum)
Check
The EOQ (≈122,000 fasteners, about a 5-month supply) is large because the fastener is so cheap (\$0.002 each, so annualized holding cost is only \$0.0005/unit), and the whole ordering+holding bill is only about \$61/yr. The EOQ formula has no awareness of practical limits such as warehouse space, shelf life or supplier pack sizes; those are not part of the given data. With costs this small, rounding the order to a convenient pack size (for example 120,000) changes the annual cost by only cents.
