Question 3 of 8: Decision Analysis — Pipeline Weld Inspection
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.
Find. (a) The EMV-optimal action — hire the team or repair-as-occur. (b) Whether the expected value of the one-weld sample information (EVSI) exceeds its $2,000 cost.
Approach. Part (a) compares two flat-cost strategies via expected monetary value (EMV). Part (b) is a pre-posterior (EVSI) analysis: use Bayes' theorem to update the defect-rate distribution on the single inspection's outcome, re-optimize the action under each posterior, then weight by the (prior, marginal) probability of each outcome.
Part (a) — expected defect rate and repair-as-occur cost.
$$E[p]=0.05(0.30)+0.10(0.50)+0.20(0.20)=0.105$$
$$E[\text{defective seams}]=1000(0.105)=105,\qquad \text{Cost}_{\text{repair}}=105(1{,}200)=\boxed{\$126{,}000}$$
Part (a) — compare to the clean-up team's flat cost and pick the cheaper (EMV-optimal) action:
$$\text{Cost}_{\text{team}}=\$130{,}000 \;>\; \text{Cost}_{\text{repair}}=\$126{,}000$$
$$\boxed{\text{Repair seams as they occur; do NOT hire the clean-up team (saves \$4,000 in expectation)}}$$
Part (b) — Bayesian revision on the one-weld sample. The marginal probability the sampled weld is defective equals the prior expected defect rate, $P(\text{def})=0.105$ (from step 1). By Bayes' theorem, $P(p\mid \text{def})=\dfrac{p\cdot P_0(p)}{0.105}$:
Posterior defect-rate distribution given the sampled weld's outcome
$p$
Prior
Posterior | defective
Posterior | OK
0.05
0.30
0.1429
0.3184
0.10
0.50
0.4762
0.5028
0.20
0.20
0.3810
0.1788
Part (b) — re-optimize under each posterior, then weight by outcome probability. Posterior mean defect rate: $E[p\mid\text{def}]=0.1310$, $E[p\mid\text{OK}]=0.1020$. Re-costing repair-as-occur under each: $$1000(0.1310)(1{,}200)=157{,}143$ (defective obs. → team is now cheaper at $130,000) and $$1000(0.1020)(1{,}200)=122{,}346$ (OK obs. → repair-as-occur stays cheaper). Weighting by the marginal outcome probabilities:
$$EV(\text{with sample})=0.105(130{,}000)+0.895(122{,}346)=\boxed{\$123{,}150}$$
Part (b) — compute EVSI and compare to the $2,000 inspection cost.
$$EVSI = EV(\text{no info}) - EV(\text{with sample}) = 126{,}000-123{,}150=\boxed{\$2{,}850}$$
$$\$2{,}850 > \$2{,}000\ \Rightarrow\ \boxed{\text{Yes -- the x-ray inspection is worthwhile (net gain \$850)}}$$
Check: assumes the $2,000 inspection cost is a fixed fee independent of the outcome, and that the single inspected weld is representative (drawn at random from the same population the 1000 seams come from) — both stated directly in the question.
