Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2019 — 17-Ind-A1 Operations Research. Three-hour, open-book exam (any non-communicating calculator permitted); the paper totals 135 marks across 9 questions (all worth 15 marks) and only 100 marks are required, so a candidate would normally answer a subset — all nine are solved below for completeness.
Reference texts: Hillier & Lieberman, Introduction to Operations Research (11th ed., McGraw-Hill) — linear programming & the simplex method (ch. 3–4), duality & sensitivity analysis (ch. 6), integer programming (ch. 12), network optimization & CPM/PERT (ch. 9–10), queueing theory (ch. 17), decision analysis (ch. 15–16), Markov chains (ch. 16), equipment replacement (ch. 11/19). Nahmias, Production and Operations Analysis — single-period (newsvendor) inventory models.
Check: in this steady-state buy-back problem (5,000,000 total, 90%/80%/60% return rates, “four or more uses → cover falls off”), the paper prints 90% (new), 80% (once-used) and 60% (twice-used). Counting uses matters here: a student who buys a twice-used copy is its 3rd user, so when they sell it back it has been used only three times and can still be resold; only the next (4th) user cannot sell it back. The 60% rate therefore creates a fourth sales class (thrice-used), and the balance below includes it.
Given. Total copies sold every fall (new + once-used + twice-used + thrice-used) = 5,000,000; sell-back rates: 90% of new-copy buyers, 80% of once-used-copy buyers, 60% of twice-used-copy buyers; a copy that has been used four times cannot be sold back.
Find. The steady-state number of NEW copies the publisher sells each fall.
Life of one copy through the buy-back cycle. Each arrow is the fraction of that class's buyers who sell the copy back. The buyer of a thrice-used copy is its 4th user, so that copy is discarded (dashed arrow): no fifth sales class exists.
Approach. In steady state each used class sold this fall equals the sell-back rate times the previous class sold, so every class is a fixed multiple of the new-copy sales $N$. Add the classes up, set the total to 5,000,000 and solve for $N$. (Equivalently, this is the steady state of a 4-state Markov chain tracking which class of copy each of the 5 million buyers gets.)
Express each class's sales in terms of $N$ (new copies sold/yr). Once-used: $O=0.9N$. Twice-used: $T=0.8O=0.72N$. A twice-used copy's buyer is its 3rd user; three uses is fewer than four, so 60% of those buyers sell it back as a thrice-used copy: $R=0.6T=0.432N$. The buyer of a thrice-used copy is its 4th user, and the cover then falls off, so the chain stops there.
Total annual sales balance. New + once-used + twice-used + thrice-used must equal the 5,000,000 total:
$$N+0.9N+0.72N+0.432N=5{,}000{,}000 \;\Longrightarrow\; 3.052N=5{,}000{,}000$$
Solve for $N$.
$$N=\frac{5{,}000{,}000}{3.052}$$
$$\boxed{N\approx1{,}638{,}270\text{ new copies sold per year}}$$
(with $O\approx1{,}474{,}443$ once-used, $T\approx1{,}179{,}554$ twice-used and $R\approx707{,}733$ thrice-used copies sold, summing back to 5,000,000). As a Markov-chain check, the steady-state probability that a buyer gets a new copy is $\pi_{\text{new}}=1/3.052=0.3277$, and $0.3277\times5{,}000{,}000\approx1{,}638{,}270$.
