Question 4 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 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.
Question 4: Decision Analysis with Sample Information — The Don Harnett Story (15 marks)
the numbers below are independently recomputed here and agree with that solved paper.
Given. Prior $P(\text{hit})=0.10$, $P(\text{flop})=0.90$; payoffs: hit $+\$15$M, flop $-\$4$M; Alert's accuracy: $P(\text{predict hit}\mid\text{actual hit})=0.60$, $P(\text{predict flop}\mid\text{actual flop})=0.90$; his fee is $1M.
Find. The best strategy (film / don't film / consult Alert first) and the resulting maximum expected profit, via a decision tree.
Approach. Compute the best expected-profit action (and its value) with no information; then use Bayes' rule to find the posterior hit probability given each of Alert's two possible predictions, the best action's expected value under each posterior, and the resulting expected value of his information; finally compare that (net of his fee) against the no-information decision.
Best action with no information (prior only).$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 hit 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:
$$0.54-1.00=-\$0.46\text{M}$$
$$\boxed{-\$0.46\text{M (pay Alert)} \;<\; \$0\text{ (don't film, no consultation)} \;\Rightarrow\; \text{do NOT pay Alert; best strategy is DON'T FILM, max EMV}=\$0}$$
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.
