Question 9 of 9: Decision Analysis with Sample Information — The Don Harnett Story
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).
Question 9: Decision Analysis with Sample Information — The Don Harnett Story (20 marks)
Find. Whether paying Alert for his opinion increases expected profit relative to deciding on the prior alone.
Approach. Compute the best expected-profit action (and its value) with no information; then use Bayes' rule to find the posterior success probability given each of Alert's two possible predictions, the best action's expected value under each posterior, and the resulting expected value of his information; finally compare that (net of his fee) against the no-information decision.
Best action with no information (prior only).$E[\text{film}]=0.10(15)+0.90(-4)=1.5-3.6=-2.1$M, versus $E[\text{don't film}]=0$:
$$\boxed{\text{best no-information action: DON'T FILM, EMV}=\$0}$$
Bayes' rule — probability of each Alert prediction. With $P(\text{predict flop}\mid\text{hit})=0.40$ and $P(\text{predict hit}\mid\text{flop})=0.10$:
$$P(\text{predict hit})=0.60(0.10)+0.10(0.90)=0.06+0.09=0.15$$
$$P(\text{predict flop})=0.40(0.10)+0.90(0.90)=0.04+0.81=0.85$$
Posterior success probability given each prediction (Bayes' rule):
$$P(\text{hit}\mid\text{predict hit})=\frac{0.06}{0.15}=0.40,\qquad P(\text{hit}\mid\text{predict flop})=\frac{0.04}{0.85}=0.0471$$
Best action's expected value under each prediction. Given a "predict hit" report: $E[\text{film}]=0.40(15)+0.60(-4)=6-2.4=3.6$M $>0$, so film. Given a "predict flop" report: $E[\text{film}]=0.0471(15)+0.9529(-4)=0.706-3.812=-3.106$M $<0$, so don't film (value 0):
$$\text{best value}\mid\text{predict hit}=\$3.6\text{M},\qquad \text{best value}\mid\text{predict flop}=\$0$$
Expected value of Alert's information, gross of his fee.
$$\text{EVSI}_{\text{gross}}=P(\text{predict hit})(3.6)+P(\text{predict flop})(0)=0.15(3.6)=\$0.54\text{M}$$
Net value of paying Alert, and the decision. Subtracting the $1M fee:
$$0.54-1.00=-\$0.46\text{M}$$
$$\boxed{-\$0.46\text{M (pay Alert)} \;<\; \$0\text{ (don't film, no consultation)} \;\Rightarrow\; \text{do NOT pay Alert}}$$
The information Alert provides is genuinely useful (it would raise expected profit by $0.54M if free), but at $1M his fee costs almost twice what that information is worth, so the best overall policy is simply not to film, without consulting him.
