Decision Analysis — Oil Pipeline Weld Inspection: 23-Ind-A1 Operations Research, Undated paper, Q7

Question 7: Decision Analysis — Oil Pipeline Weld Inspection (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; defect-rate prior: $P(5\%)=0.30$, $P(10\%)=0.50$, $P(20\%)=0.20$; cleanup-team flat cost $130,000 (covers full inspection + repair of all 1000 seams); x-ray inspection of 1 randomly selected completed weld, cost $2,000.

Find. (a) The expected-value-optimal decision (team vs. no team). (b) Whether the value of the extra information from inspecting one weld exceeds its $2,000 cost.

Approach. (a) Compare the expected cost of reworking seams as they occur (no team) against the team's flat fee, using the prior over defect rates. (b) This is a pre-posterior (Bayesian) decision-analysis question: find how a single weld's inspection outcome (defective / good) would update the defect-rate belief, re-decide under each posterior, and compare the resulting expected cost (with the option to act on the sample) against the $2,000 sampling cost — this is the Expected Value of Sample Information (EVSI).

(a) Expected defect rate and expected rework cost with no team: $$E[\text{rate}]=0.05(0.30)+0.10(0.50)+0.20(0.20)=0.015+0.050+0.040=0.105\ (10.5\%)$$ $$E[\text{cost, no team}]=1000\times 0.105\times\$1{,}200=\boxed{\$126{,}000}$$ Compared with the team's flat $130,000: $$\boxed{\$126{,}000\ <\ \$130{,}000\ \Rightarrow\ \text{do NOT hire the cleanup team}}$$ (reworking as-you-go is expected to be $4,000 cheaper).
(b) Marginal probability the sampled weld is defective (law of total probability, using the same prior — the sampled weld's defect chance IS the unknown overall rate): $$P(D)=E[\text{rate}]=0.105,\qquad P(G)=1-0.105=0.895$$
Posterior defect-rate distribution given the sampled weld is Defective (Bayes' rule, $P(\text{rate}\mid D)\propto P(\text{rate})\times\text{rate}$):
Posterior after observing the sampled weld is defective
Rate 5% 10% 20%

Posterior prob. 0.1429 0.4762 0.3810

$$E[\text{rate}\mid D]=0.05(0.1429)+0.10(0.4762)+0.20(0.3810)=0.1310\ (13.10\%)$$ $$E[\text{cost, no team}\mid D]=1000(0.1310)(1200)=\$157{,}142.86\ >\ \$130{,}000$$ — if the sample comes back defective, the optimal follow-up action flips to hire the team, at $130,000.
Posterior given the sampled weld is Good ($P(\text{rate}\mid G)\propto P(\text{rate})\times(1-\text{rate})$):
Posterior after observing the sampled weld is good
Rate 5% 10% 20%

Posterior prob. 0.3184 0.5028 0.1788

$$E[\text{rate}\mid G]=0.05(0.3184)+0.10(0.5028)+0.20(0.1788)=0.1020\ (10.20\%)$$ $$E[\text{cost, no team}\mid G]=1000(0.1020)(1200)=\$122{,}346.37\ <\ \$130{,}000$$ — if the sample comes back good, the optimal action stays no team, at $122,346.37.
Expected cost WITH the option to sample (act optimally after seeing the result): $$E[\text{cost}\mid\text{sample}]=P(D)\min(157{,}142.86,\,130{,}000)+P(G)\min(122{,}346.37,\,130{,}000)$$ $$=0.105(130{,}000)+0.895(122{,}346.37)=13{,}650.00+109{,}500.00=\$123{,}150.00$$
Value of sample information vs. its cost: $$EVSI=\underbrace{\$126{,}000}_{\text{best without sampling}}-\underbrace{\$123{,}150.00}_{\text{expected cost with sampling}}=\$2{,}850.00$$ $$\boxed{EVSI=\$2{,}850.00\ >\ \$2{,}000\text{ inspection cost}\ \Rightarrow\ \text{YES, worth inspecting (net gain \$850)}}$$

Posterior after observing the sampled weld is defective
Rate	5%	10%	20%
Posterior prob.	0.1429	0.4762	0.3810

Posterior after observing the sampled weld is good
Rate	5%	10%	20%
Posterior prob.	0.3184	0.5028	0.1788

Final results — Question 7
Item	Value
Prior expected defect rate	10.5%
(a) Expected cost, no team	$126,000
(a) Decision	Do not hire cleanup team (saves $4,000)
Posterior rate \| defective sample	13.10% → hire team ($130,000)
Posterior rate \| good sample	10.20% → no team ($122,346.37)
Expected cost with sampling	$123,150.00
EVSI	$2,850.00
(b) Decision	Worthwhile (EVSI $2,850 > $2,000 cost)

Topic: Expected-value decision analysis and the value of (imperfect) sample information.

Key relations: $EV=\sum P(\text{state})\times\text{cost}$; Bayes' update $P(\text{state}\mid\text{obs})\propto P(\text{state})P(\text{obs}\mid\text{state})$; $EVSI=EV(\text{no sample})-E_{\text{obs}}[\min_a EV(a\mid\text{obs})]$.

Why this works: A sample is valuable only insofar as it can change the optimal ACTION; here the posterior after a defective observation is large enough to flip the decision from "no team" to "team," while a good observation reinforces the status-quo decision — averaging the resulting optimal costs over both possible sample outcomes gives the true expected cost of "sample, then decide."

Common pitfall: Comparing the inspection cost to the WRONG benchmark — e.g. treating any nonzero information as automatically worth having, instead of computing EVSI, which correctly nets out the $126,000 you would have spent anyway.

Source: Hillier & Lieberman, Introduction to Operations Research (11th ed.), ch. 15–16 (decision analysis, value of information).

Question 7 of 10: Decision Analysis — Oil Pipeline Weld Inspection

Question 7: Decision Analysis — Oil Pipeline Weld Inspection (20 marks)