Decision Analysis — Pipeline Weld Inspection (EMV and Value of Sample Information): 23-Ind-A1 Operations Research, May 2013, Q6

Question 6: Decision Analysis — Pipeline Weld Inspection (EMV and Value of Sample Information) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. 1000 seams; rework cost $1,200/defective seam; prior on the true defect rate $p$: $P(p{=}0.05)=0.30$, $P(p{=}0.10)=0.50$, $P(p{=}0.20)=0.20$; clean-up team flat fee $130,000; x-ray inspection of one weld costs $2,000.

Defect rate $p$	Prior probability
0.05	0.30
0.10	0.50
0.20	0.20

Find. (a) The EMV-optimal decision (clean-up team vs. repair-as-occur). (b) Whether the $2,000 x-ray inspection is worth its cost, via the expected value of sample information (EVSI).

Approach. (a) Compare the clean-up team's fixed fee against the prior expected rework cost of repairing defects as they occur. (b) Use the single inspected weld as a Bernoulli sample, Bayes-update the defect-rate distribution on "defective" vs. "good", re-optimize the decision under each posterior, and compare the resulting expected cost (including re-optimization) against the no-sampling EMV and the $2,000 sampling cost.

(a) Expected defect rate and repair-as-occur cost. $$E[p]=0.05(0.30)+0.10(0.50)+0.20(0.20)=0.105.$$ Expected seams reworked $=0.105\times1000=105$; expected cost $=105\times\$1{,}200=\boxed{\$126{,}000}$.
(a) Compare to the clean-up team. The flat fee is $\$130{,}000 > \$126{,}000$, so on an EMV basis the company should not hire the clean-up team — repairing welds as they occur is $\$4{,}000$ cheaper in expectation.
(b) Bayesian update from the inspection outcome. $P(\text{defective})=E[p]=0.105$; $P(\text{good})=0.895$. By Bayes' rule, posterior on $p$ given a defective weld observed: $P(p|D)\propto P(p)\cdot p$, giving $P(0.05|D)=0.1429,\ P(0.10|D)=0.4762,\ P(0.20|D)=0.3810$, so $E[p|D]=0.1310$. Given a good weld observed: $P(p|G)\propto P(p)(1-p)$, giving $P(0.05|G)=0.3184,\ P(0.10|G)=0.5028,\ P(0.20|G)=0.1788$, so $E[p|G]=0.1020$.
(b) Re-optimize the decision under each posterior. Repair-as-occur cost given $D$: $0.1310\times1000\times\$1{,}200\approx\$157{,}143$ — worse than the $\$130{,}000$ clean-up fee, so if the sampled weld is defective, hire the clean-up team. Repair-as-occur cost given $G$: $0.1020\times1000\times\$1{,}200\approx\$122{,}346$ — better than $\$130{,}000$, so if the sampled weld is good, repair-as-occur remains the choice.
(b) Expected value of the sampling strategy and EVSI. $$E[\text{cost with inspection}]=P(D)\!\cdot\!\$130{,}000+P(G)\!\cdot\!\$122{,}346 = 0.105(130{,}000)+0.895(122{,}346)\approx\$123{,}150.$$ $$\text{EVSI} = \$126{,}000-\$123{,}150 = \boxed{\$2{,}850},$$ which exceeds the $\$2{,}000$ sampling cost, so the x-ray inspection is worthwhile (expected net benefit $\approx\$850$).

Quantity	Result
(a) EMV, repair-as-occur / clean-up team	$126,000 / $130,000 — do not hire clean-up team
(b) Posterior mean given defective / good weld	0.1310 / 0.1020
(b) Expected cost with inspection	$123,150
(b) EVSI vs. sampling cost	$2,850 > $2,000 — inspection is worthwhile

Concept & Theory

Topic. Expected monetary value (EMV) decision-making under uncertainty, and the expected value of sample information (EVSI) from a single Bernoulli test.

Key relations. $EMV=\sum_s P(s)\,\text{cost}(a,s)$; Bayes' rule $P(s|z)\propto P(s)P(z|s)$; $EVSI=EMV_{\text{no info}}-E_z\big[\min_a EMV(a|z)\big]$; sampling is worthwhile when $EVSI>$ sampling cost.

Why this works. The clean-up team's cost is state-independent (flat fee), so information only ever changes the repair-as-occur estimate — a single weld inspection shifts the posterior mean defect rate enough (0.105→0.131 if defective, or →0.102 if good) to flip the optimal action in the "defective" branch, which is exactly where sample information has value.

Common pitfall. Comparing the $2,000 inspection cost to the wrong baseline — EVSI must be measured against the pre-posterior optimal EMV ($126,000), not against either posterior branch cost in isolation, and must include the probability-weighted re-optimization over both possible outcomes (D and G).

Source. Hillier & Lieberman, Introduction to Operations Research, ch. 16 (Decision Analysis — Bayes' theorem, EMV and the value of information).

Question 6 of 9: Decision Analysis — Pipeline Weld Inspection (EMV and Value of Sample Information)

Question 6: Decision Analysis — Pipeline Weld Inspection (EMV and Value of Sample Information) (20 marks)