Exercise set
Engineering Decision Analysis, Risk, and Trade-Off Exercises
Solved decision-analysis exercises for expected value, risk, schedule delay, bottleneck value, Pareto trade-offs, pilots, regret and gates.
These exercises practise decision analysis for engineering choices under uncertainty, risk, scenarios, schedule exposure, bottlenecks, pilot tests, trade-off tables, weighted scoring, regret, sensitivity and approval gates. The focus is the decision: what action should be taken when technical outcomes, costs and benefits are uncertain.
The goal is not only to compute an expected value. A decision can have positive expected value and still be unacceptable because downside consequence is severe, a dominated option remains in the shortlist, a pilot does not change the decision, a schedule delay lies on the critical path, or an approval gate depends on an unvalidated assumption.
Assume simplified decision models unless an exercise states otherwise. Real decisions should also check safety constraints, regulatory obligations, stakeholder weights, common-cause risks, ethical limits, model uncertainty, data quality, implementation capability and the cost of being wrong.
Release Evidence Notes
Decision-analysis evidence should start with alternatives, states of the world, probabilities, consequences, constraints and decision authority. The model should state what is being optimized and what is not allowed to be traded away.
Expected-value evidence should preserve downside information. A single average can hide rare but unacceptable losses, schedule slips, safety exposure, customer impacts or regulatory violations. Report threshold cases and sensitivity where the recommendation can reverse.
Trade-off evidence should identify dominated options before detailed scoring. Weighted scoring should show criteria, weights, normalization and minimum hard gates. If a hard gate fails, a high weighted score should not release the option.
Information-value evidence should show whether testing, pilots or staged commitments can change the decision enough to justify delay and cost.
Engineering Boundary Notes
The boundary for these exercises is decision quality under uncertainty, not detailed cash-flow accounting. If the decision mainly depends on LCC, NPV, payback, EAC, IRR, tax, depreciation, replacement cost or make-buy break-even, use the companion engineering economics exercise set.
Within the decision boundary, define whether the output is a recommendation, a hold point, a pilot decision, a sensitivity threshold, a risk-control requirement or a rejection.
The release action should match the decision result. A dominated option should be removed; a high-uncertainty option may need a pilot; a severe downside may require a hard constraint; a marginal gate may require staged approval.
Scenario Map
| Scenario | Exercises | Primary check | Engineering decision |
|---|---|---|---|
| Expected value and risk exposure | 1, 2, 3, 7, 12 | Downtime, schedule delay, bottleneck value, risk reduction and downside cap | Fund, reject or constrain an action based on risk-adjusted value. |
| Staged decisions and information | 4, 6, 10, 11 | Staged expansion, pilot value, decision tree and value of information | Test, stage, cancel or proceed. |
| Trade-off ranking | 5, 8, 9, 13, 14 | Pareto dominance, weighted scoring, threshold sensitivity, regret and normalized score | Remove dominated alternatives and explain ranking. |
| Validation and release gates | 15, 16, 17, 18 | Post-audit realization, approval sensitivity, percentile risk and hard gates | Release, hold or revise the recommendation. |
Exercise 1: Expected Annual Downtime Cost
A failure mode occurs on average:
Each event causes:
Downtime cost is:
Estimate expected annual downtime cost.
Solution
Expected annual downtime hours:
Expected annual cost:
Engineering Comment
Expected cost is a planning estimate, not a guarantee. If one event can create safety, regulatory or customer-critical damage, the decision needs consequence limits beyond expected value.
Plausibility Check
A few events per year at several hours each creates about fifteen lost hours. At nearly USD 5000 per hour, cost near USD 70,000 is plausible.
Exercise 2: Schedule Delay Cost on the Critical Path
A commissioning activity is on the critical path. Delay cost is:
The activity has 30\% probability of a two-day delay and 10\% probability of a five-day delay. Otherwise it has no delay. Estimate expected delay cost.
Solution
Expected delay is:
Expected delay cost:
Engineering Comment
The cost matters because the activity is on the critical path. If it had float, the same task delay might not create project delay until the float was consumed.
Plausibility Check
Most scenarios have zero or small delay, but the five-day tail lifts the expected delay above one day. Multiplying by USD 18,000 gives just under USD 20,000.
Exercise 3: Capacity Value at the Real Bottleneck
A process improvement adds:
only if applied at the bottleneck. Contribution margin is:
If there is 80\% probability that the proposed station is the true bottleneck, estimate expected weekly value.
Solution
Value if the station is the bottleneck:
Expected value:
Engineering Comment
The decision depends on bottleneck evidence. A local improvement at a non-bottleneck station may create no system value and can increase WIP.
Plausibility Check
Expected value is 80\% of the full bottleneck value, so it should be slightly above USD 3300 per week.
Exercise 4: Expected Value of a Staged Expansion Option
A modular expansion costs USD 60,000 more now than a fixed design. If demand is high, the option is worth USD 240,000. If demand is medium, it is worth USD 90,000. If demand is low, it has no value. Probabilities are:
| Scenario | Probability |
|---|---|
| high demand | 0.25 |
| medium demand | 0.45 |
| low demand | 0.30 |
Find expected net value of the option.
Solution
Expected future value:
Net option value:
Engineering Comment
The staged option has positive expected value, but the model should still check whether high demand can be recognized early enough to use the option.
Plausibility Check
The high-demand scenario alone offsets the option cost in expectation. Medium demand adds more value, so positive net value is expected.
Exercise 5: Pareto Dominance in a Trade-Off Table
Three alternatives are compared. Lower cost is better, higher reliability is better, and shorter implementation time is better.
| Alternative | Cost | Reliability | Time |
|---|---|---|---|
| A | 100 | 0.94 | 10 |
| B | 105 | 0.96 | 9 |
| C | 120 | 0.93 | 12 |
Identify any dominated alternative.
Solution
Compare C with A:
- C costs more than A: 120>100.
- C has lower reliability than A: 0.93<0.94.
- C takes longer than A: 12>10.
C is worse than A on all criteria, so C is dominated.
B is not dominated by A because B has higher reliability and shorter time, despite higher cost.
Engineering Comment
Dominated options should be removed before weighted scoring. Keeping them adds noise and can distort stakeholder discussion.
Plausibility Check
C is visibly worse than A in every column, so it should not remain in the shortlist.
Exercise 6: Expected Value of a Pilot Test
Immediate rollout has 60\% probability of USD 250,000 net benefit and 40\% probability of USD 120,000 net loss. A pilot costs USD 30,000. If the pilot is used, there is 70\% probability of a pass signal; after a pass, rollout expected value is USD 220,000. After a fail signal, rollout is cancelled.
Compare expected value of immediate rollout with pilot strategy.
Solution
Immediate rollout:
Pilot strategy:
Pilot strategy is higher by:
Engineering Comment
The pilot has positive value because it filters out bad rollout conditions. The decision should still check whether the pilot is representative and whether delay has hidden cost.
Plausibility Check
Immediate rollout already has positive expected value, so the pilot must add enough information to overcome its cost. The computed advantage is modest, which is credible.
Exercise 7: Expected Loss Reduction and Break-Even Control Effectiveness
A control costs USD 40,000 per year. Baseline expected loss is USD 110,000 per year. If installed, the control reduces loss by fraction e. Find break-even effectiveness.
Solution
Annual benefit is:
Break-even occurs when:
The control must reduce expected loss by at least 36.4\%.
Engineering Comment
The control should not be justified by expected loss alone if the baseline includes safety or compliance consequences. Effectiveness also needs verification evidence.
Plausibility Check
The control cost is a little over one third of baseline expected loss, so break-even effectiveness near one third is expected.
Exercise 8: Weighted Decision Matrix
Two alternatives are scored from 1 to 5. Weights sum to 1.
| Criterion | Weight | Option A | Option B |
|---|---|---|---|
| performance | 0.40 | 4 | 5 |
| maintainability | 0.25 | 5 | 3 |
| schedule | 0.20 | 3 | 4 |
| cost score | 0.15 | 4 | 3 |
Compute weighted scores.
Solution
Option A:
Option B:
Option A is slightly higher.
Engineering Comment
The ranking is close. A small change in weights or scores could reverse it, so hard gates and sensitivity review matter.
Plausibility Check
B wins performance and schedule, but A wins maintainability and cost. Close scores are expected.
Exercise 9: Weight Sensitivity Threshold
In Exercise 8, suppose only the performance weight changes. Option B has one-point performance advantage over A. Option A has total advantage of 0.45 from all non-performance criteria at the current normalized basis. Find performance weight above which B overtakes A.
Solution
B overtakes A when performance advantage exceeds non-performance disadvantage:
Therefore:
If performance weight rises above 45\%, B becomes preferred.
Engineering Comment
This threshold tells stakeholders what belief must change to reverse the decision. It is more useful than arguing about a single weighted score.
Plausibility Check
Current performance weight is 40\%, where A barely wins. A threshold slightly above current weight is consistent with the close score.
Exercise 10: Decision Tree with Rework Chance
A new process can be released now or held for rework. Release now gives USD 180,000 benefit if successful, but there is 25\% probability of a USD 90,000 failure cost. Holding for rework costs USD 35,000 and reduces failure probability to 8\%, while preserving the same success benefit. Compare expected values.
Solution
Release now:
Rework first:
Rework first is higher by USD 10,900.
Engineering Comment
The hold point has value because it reduces downside probability enough to offset rework cost. The assumption that rework preserves benefit should be validated.
Plausibility Check
Failure cost is large, so reducing failure probability from 25\% to 8\% can justify a moderate rework cost.
Exercise 11: Expected Value of Perfect Information
A decision has two demand states:
| State | Probability | Best value if known |
|---|---|---|
| high | 0.40 | USD 300,000 |
| low | 0.60 | USD 120,000 |
The best decision without perfect information has expected value USD 170,000. Estimate EVPI.
Solution
Expected value with perfect information:
Expected value of perfect information:
Engineering Comment
No test or market study should cost more than the value of the information it can realistically provide. Imperfect information is worth less than EVPI.
Plausibility Check
Perfect information improves the decision but cannot add more than the gap between the current decision and state-specific best choices.
Exercise 12: Risk-Adjusted Expected Value with Downside Cap
An option has expected value USD 85,000 but includes a worst-case loss of USD 260,000. The approval policy allows expected-value selection only if worst-case loss is no worse than USD 200,000. Decide whether the option can be selected by expected value alone.
Solution
Expected value:
Worst-case loss:
The option fails the downside cap. It cannot be selected by expected value alone.
Engineering Comment
Downside constraints are common in safety, compliance, customer and resilience decisions. A positive average does not make an unacceptable tail acceptable.
Plausibility Check
The expected value passes, but the stated hard cap fails. A hard constraint should govern.
Exercise 13: Minimax Regret Choice
Two alternatives are evaluated under high and low demand.
| Alternative | High demand value | Low demand value |
|---|---|---|
| A | 300 | 80 |
| B | 220 | 150 |
Values are in thousand USD. Choose by minimax regret.
Solution
Best value under high demand is 300, so regret:
Best value under low demand is 150, so regret:
Maximum regret:
Minimax regret chooses A.
Engineering Comment
Minimax regret is useful when probabilities are weak but decision makers want to limit the pain of being wrong.
Plausibility Check
A is best in high demand and only moderately poor in low demand. B’s high-demand regret is larger, so A wins.
Exercise 14: Normalized Multi-Criteria Score
Two options are scored on cost and reliability. Cost is lower-is-better from USD 80,000 to USD 120,000. Reliability is higher-is-better from 0.90 to 0.98. Option A costs USD 90,000 and has reliability 0.94. Use equal weights.
Solution
Cost score with 1 best and 0 worst:
Reliability score:
Equal-weight score:
Engineering Comment
Normalization makes unlike criteria comparable, but the chosen bounds strongly affect scores. Bounds should be defensible and shared before ranking.
Plausibility Check
A is closer to best cost than best reliability, so its combined score should be between 0.50 and 0.75.
Exercise 15: Post-Audit Benefit Tracking
A project promised annual benefit:
Measured first-year benefit after normalization is:
The acceptance rule requires at least 85\% realization. Check acceptance.
Solution
Realization is:
The rule requires 85\%, so the post-audit fails.
Engineering Comment
The decision model should be updated. The failure may come from implementation ramp-up, wrong baseline, operating changes, measurement error or overestimated technical effect.
Plausibility Check
Measured benefit is USD 28,000 below promise, which is one fifth of the promised value. Realization at 80\% is expected.
Exercise 16: Approval-Gate Sensitivity
A project’s base NPV is:
The approval gate is:
Annual benefit appears in a five-year annuity with present-worth factor:
How much can annual benefit fall before the gate fails?
Solution
Current margin is:
Equivalent annual benefit margin:
Annual benefit can fall by about USD 3166 before the gate fails.
Engineering Comment
This is a narrow margin if annual savings are uncertain by more than a few thousand dollars. The release should define a measurement plan before approval.
Plausibility Check
Spreading USD 12,000 of present margin across a five-year discounted annuity gives a few thousand dollars per year.
Exercise 17: Percentile Risk from Scenario Samples
Seven simulated net project values in thousand USD are:
Use the second-lowest value as a rough 20th percentile screen. Does the project meet a policy that P20 must be nonnegative?
Solution
The second-lowest value is:
The policy requires:
The project fails the screen.
Engineering Comment
Even if the median is positive, downside percentile risk can block release. More simulation samples would be required for a real percentile estimate.
Plausibility Check
Two of seven scenarios are negative, so a low-percentile screen should likely be negative.
Exercise 18: Decision Release Gate
A decision package has these gates:
| Gate | Requirement | Current result |
|---|---|---|
| expected value | positive | USD 124,000 |
| worst-case loss | no worse than USD 200,000 | USD 180,000 |
| dominated alternatives removed | yes | yes |
| post-audit plan | required | missing |
Decide whether to release.
Solution
Expected value passes:
Worst-case loss passes because:
Dominated alternatives were removed. The post-audit plan is missing, so the decision package fails release.
Engineering Comment
Decision quality includes learning after implementation. Without a post-audit plan, the organization cannot verify whether the assumption that drove the decision was true.
Plausibility Check
The numerical decision gates pass, but a required evidence gate is absent. A hard evidence gate blocks release.
Validation Package Checklist
A strong decision-analysis solution should check:
- whether alternatives, states, probabilities and consequences are explicit;
- whether expected value is reported with downside or threshold information;
- whether critical-path and bottleneck assumptions are verified before valuing delay or capacity;
- whether dominated options are removed before scoring;
- whether weights, normalization and hard gates are documented;
- whether pilot or test value exceeds the cost and delay of testing;
- whether sensitivity thresholds show when the recommendation reverses;
- whether post-audit evidence is planned before implementation.
Common Release Mistakes
Common mistakes include presenting expected value without downside risk, treating a critical-path delay like a noncritical delay, valuing a bottleneck improvement before proving the bottleneck, keeping dominated options in a trade-off table, changing weights after seeing the preferred answer, treating a pilot as useful when it cannot change the decision, using perfect-information value as if a real test were perfect, accepting positive EV despite a failed downside cap, and approving a decision without post-audit evidence.