Question 1 of 8: Newsvendor Model — Christmas Tree Order Quantity
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2016 — 98-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 160 marks across 8 questions (each worth 20) and only 100 marks are required, so a candidate would normally answer 5 — all eight are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming and the simplex method & sensitivity analysis (ch. 3–4/6), network optimization & PERT/CPM (ch. 9–10), integer programming (ch. 12), Markov chains (ch. 16), decision analysis (ch. 15). Nahmias, Production and Operations Analysis — the single-period (newsvendor) inventory model.
Question 1: Newsvendor Model — Christmas Tree Order Quantity (20 marks)
Given. Purchase cost $c$=$$10$/tree; selling price $p$=$$25$/tree; demand $D\sim N(\mu=100,\sigma=30)$; unsold trees have no salvage value (single, one-shot buying decision before the season, classic newsvendor setting).
Find. The order quantity $Q^*$ that maximizes Joe's expected profit.
Approach. This is a single-period (newsvendor) inventory problem: balance the cost of overstocking (a tree bought but not sold) against the cost of understocking (a lost sale), and set the order quantity at the critical fractile of the demand distribution.
Identify the underage and overage costs. Underage cost (profit lost on a sale Joe could not fill) $C_u = p-c = 25-10 = 15$ ($15) per tree short. Overage cost (sunk cost on a tree bought but never sold, salvage $0) $C_o = c = 10$ ($10) per tree over.
Compute the critical ratio (service level).
$$CR=\frac{C_u}{C_u+C_o}=\frac{15}{15+10}=\boxed{0.60}$$
Joe should stock enough trees to cover demand 60% of the time — the point where one more tree's expected marginal profit equals its expected marginal loss.
Convert the critical ratio to a standard-normal z-score and apply it to the demand distribution:
$$z=\Phi^{-1}(0.60)=0.2533,\qquad Q^*=\mu+z\sigma = 100+0.2533(30)=107.6$$
$$\boxed{Q^*\approx 108\text{ trees (rounding up to a whole tree)}}$$
