Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2017 — 98-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 180 marks across 9 questions (each worth 20) and only 100 marks are required, so a candidate would normally answer 5 — all nine are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming formulation & the simplex method (ch. 3–4), duality & sensitivity analysis (ch. 6), network optimization models (ch. 9), deterministic dynamic programming (ch. 11), integer programming (ch. 12), Markov chains (ch. 16), decision analysis (ch. 15), queueing theory (ch. 17); Nahmias, Production and Operations Analysis (7th ed.) — EOQ with and without planned shortages, the newsvendor (single-period) model (ch. 4–5).
Check: a copy sold as "twice-used" is used for the THIRD time by its buyer; if that buyer sells it back it would need to be resold and used a FOURTH time, which the question rules out ("used four or more times… must be discarded"). So the 60% resale rate on twice-used copies never produces a sellable "thrice-used" class — every twice-used copy exits the market after its one sale, regardless of the 60% figure. This makes the 60% irrelevant to the new-copy count asked for here (it would only matter for a follow-up question about the annual discard rate); the steady-state balance below depends only on the 90% and 80% return rates.
Given. Total copies sold every fall (new + once-used + twice-used) = 5,000,000; return-to-bookstore rates: 90% after a new copy's use, 80% after a once-used copy's use, 60% after a twice-used copy's use (but see the check note); a copy is discarded rather than resold once it would need a fourth use.
Find. The steady-state number of NEW copies the publisher sells each fall.
Generation flow of one printing through the buy-back cycle. The dashed final arrow shows every twice-used copy leaving the sellable market after its sale (no "thrice-used" category exists), independent of the 60% who would try to sell it back.
Approach. Express the once-used and twice-used annual sales as fixed fractions of the new-copy sales using the steady-state return rates, then solve the single equation "new + once-used + twice-used = 5,000,000" for the new-copy count.
Express each generation's sales in terms of $N$ (new copies sold/yr), in steady state. This year's once-used sales equal 90% of new copies sold (any year, since steady state repeats): $O=0.9N$. This year's twice-used sales equal 80% of once-used copies sold: $T=0.8O=0.8(0.9N)=0.72N$. No further generation exists (per the check note).
Total annual sales balance. New + once-used + twice-used must equal the given 5,000,000 total:
$$N+0.9N+0.72N=5{,}000{,}000 \;\Longrightarrow\; 2.62N=5{,}000{,}000$$
Solve for $N$.
$$N=\frac{5{,}000{,}000}{2.62}$$
$$\boxed{N\approx1{,}908{,}397\text{ new copies sold per year}}$$
(with $O\approx1{,}717{,}557$ once-used and $T\approx1{,}374{,}046$ twice-used copies sold, summing back to 5,000,000.)
