Exercise set

Maintenance Interval, Condition Monitoring, and Backlog Risk Exercises

Solved maintenance exercises for PM intervals, P-F inspection, condition monitoring, backlog risk, repair readiness and release gates.

Branch: Industrial and Management Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.0.0 · reviewed

These exercises practise maintenance engineering decisions: preventive maintenance workload, reliability-based intervals, P-F inspection frequency, condition-monitoring guards, backlog triage, repair readiness, temporary controls, validation evidence and maintenance release gates.

The goal is not to make maintenance look busy. The goal is to decide whether the asset can remain in service, whether the inspection interval can detect degradation early enough, whether repair resources are ready, and whether deferred work still has controlled risk.

Assume simplified screening models unless an exercise states otherwise. Real maintenance decisions should also check access, permits, isolation, tooling, spare status, skill coverage, vendor support, work-order quality, post-maintenance testing, restart risk and the consequence of each failure mode.

Release Evidence Notes

Maintenance evidence should separate preventive work, condition-based inspection, corrective repair and post-maintenance validation. A completed work order is not release evidence unless it states what was inspected, what was found, what was repaired and what functional test closed the job.

P-F evidence should identify the potential failure signal, detection method, inspection interval, repair lead time and response buffer. A frequent inspection is weak if the method cannot detect the degradation early enough.

Backlog evidence should distinguish low-risk administrative delay from deferred work that protects a critical function. Consequence, likelihood, detectability, temporary control status and due date should be visible.

Condition-monitoring evidence should guard for uncertainty. A trend near a limit should not be accepted using only the central forecast when measurement error, model error and operating change can consume the margin.

Engineering Boundary Notes

This page covers maintenance interval and condition-monitoring decisions. Reliability architecture, redundancy and proof-test claims belong in the reliability availability exercise set. Spare reorder points, stockout probability, repairable pools and shelf-life release belong in the critical spare-parts exercise set.

If the decision is mainly a project schedule, critical path, float or resource-leveling problem, use the operations schedule and queue planning exercise set. If it is a shutdown package with many work fronts, use the maintenance shutdown planning project.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Preventive maintenance interval	1-4	survival target, workload, P-F interval and repair lead time	Set the maximum interval and required response window.
Condition monitoring	5-8, 15	guarded vibration, oil, thermal and trend evidence	Decide whether the asset can wait or needs action.
Backlog and repair readiness	9-14, 17	risk priority, burn-down, queue capacity and temporary controls	Decide whether deferral remains controlled.
Maintenance release	16, 18	evidence completion and hard-gate closure	Release, restrict or hold the asset.

Exercise 1: Preventive Replacement Interval

An exponential failure model has:

\lambda=2.2\times10^{-4}\ \text{h}^{-1}

Find the preventive replacement interval that keeps survival above:

R=0.96

Solution

For an exponential model:

R(t)=e^{-\lambda t}

Solve for time:

t=\dfrac{-\ln(R)}{\lambda}

Substitute:

t=\dfrac{-\ln(0.96)}{2.2\times10^{-4}}=185.6\ \text{h}

Engineering Comment

The interval is credible only if the constant-rate model matches the failure mode. Wear-out or degradation signals may require a Weibull or condition-based rule.

Plausibility Check

The MTBF is about $4545$ hours, so a $96\%$ survival interval should be much shorter than MTBF.

Exercise 2: Monthly PM Workload

A site has:

N=84

assets. Each asset requires preventive maintenance every:

I=6\ \text{months}

Estimate monthly PM events.

Solution

Monthly events are:

M=\dfrac{N}{I}

Thus:

M=\dfrac{84}{6}=14\ \text{events/month}

Engineering Comment

The workload must be converted into hours, skill categories, access windows and outage constraints before staffing is approved.

Plausibility Check

Over six months all eighty-four assets are visited once, so one sixth per month is fourteen.

Exercise 3: P-F Inspection Frequency

A bearing degradation mode has estimated P-F interval:

T_{PF}=15\ \text{weeks}

The maintenance rule requires at least three inspection opportunities inside the interval. Find the maximum inspection interval.

Solution

Maximum interval is:

T_i=\dfrac{T_{PF}}{3}

So:

T_i=\dfrac{15}{3}=5\ \text{weeks}

Engineering Comment

The interval is useful only if the inspection method can see the potential failure near the start of the P-F window and if repair can be completed before functional failure.

Plausibility Check

Three opportunities across fifteen weeks means one opportunity every five weeks.

Exercise 4: P-F Repair Readiness Buffer

A defect has:

T_{PF}=15\ \text{weeks}

The inspection interval is:

T_i=5\ \text{weeks}

Repair lead time is:

T_r=7\ \text{weeks}

Estimate the worst-case response buffer after detection.

Solution

Worst-case detection delay is one full inspection interval:

T_d=T_i=5\ \text{weeks}

Remaining time after repair is:

B=T_{PF}-T_d-T_r=15-5-7=3\ \text{weeks}

Engineering Comment

The positive buffer means the policy can work on paper, but a three-week margin may disappear if access, spares or permits are late.

Plausibility Check

The total detection plus repair time is twelve weeks, leaving three weeks inside the fifteen-week P-F interval.

Exercise 5: Guarded Vibration Trend

A vibration alarm threshold is:

V_{lim}=7.0\ \text{mm/s}

The predicted value at the next planned shutdown is:

V_p=6.3\ \text{mm/s}

Prediction uncertainty is:

u=0.4\ \text{mm/s}

Use a one-sided guard of $2u$ .

Solution

Guarded prediction is:

V_g=V_p+2u=6.3+2(0.4)=7.1\ \text{mm/s}

Since:

7.1>7.0

the shutdown cannot wait under this guarded rule.

Engineering Comment

When a forecast is close to a limit, uncertainty should drive the action. The response may be earlier inspection, reduced load, speed restriction or planned replacement.

Plausibility Check

The unguarded margin is $0.7\ \text{mm/s}$ , while the guard is $0.8\ \text{mm/s}$ , so the guarded value fails by a small amount.

Exercise 6: Oil Analysis Z-Score Trigger

An oil particle-count indicator has baseline:

\mu=18

and standard deviation:

\sigma=4

The latest result is:

x=29

Compute the z-score.

Solution

Use:

z=\dfrac{x-\mu}{\sigma}

Substitute:

z=\dfrac{29-18}{4}=2.75

Engineering Comment

A high z-score should trigger review of contamination source, sampling method, filter condition and matching operating regime before declaring a true degradation trend.

Plausibility Check

The result is eleven units above baseline, a little less than three standard deviations.

Exercise 7: Thermal Trend Margin

A motor bearing temperature limit is:

T_{lim}=85^\circ\text{C}

The current temperature is:

T_0=72^\circ\text{C}

The trend is:

r=1.5^\circ\text{C/week}

Estimate weeks until the limit.

Solution

Temperature margin is:

\Delta T=T_{lim}-T_0=85-72=13^\circ\text{C}

Time to limit is:

t=\dfrac{\Delta T}{r}=\dfrac{13}{1.5}=8.7\ \text{weeks}

Engineering Comment

The estimate assumes the trend continues linearly. Load, ambient temperature, lubrication and sensor placement should be checked before scheduling.

Plausibility Check

At about one and a half degrees per week, thirteen degrees of margin lasts just under nine weeks.

Exercise 8: Condition-Monitoring Alarm Margin

A condition index has release limit:

I_{lim}=100

The current guarded index is:

I_g=92

The rule requires at least:

M_{min}=12

points of margin. Check the release margin.

Solution

Margin is:

M=I_{lim}-I_g=100-92=8

Since:

8<12

the release margin fails.

Engineering Comment

Being below an alarm limit is not always enough. Some assets need guard margin because inspection delay and operating variability can consume the remaining headroom.

Plausibility Check

The index is only eight points below the limit, so it cannot satisfy a twelve-point rule.

Exercise 9: Backlog Risk Priority

A deferred maintenance item has:

C=9,\quad L=4,\quad D=3

where $C$ is consequence, $L$ is deferral likelihood and $D$ is detectability weakness. A temporary control reduces likelihood to $2$ . Compute priority before and after.

Solution

Initial priority is:

P_1=CLD=9(4)(3)=108

After the control:

P_2=9(2)(3)=54

Engineering Comment

The score halves, but high consequence remains. The backlog item still needs an owner, due date and proof that the temporary control is active.

Plausibility Check

Only likelihood changes from $4$ to $2$ , so the score should be half the original.

Exercise 10: Backlog Burn-Down Time

A maintenance backlog contains:

B=126\ \text{work h}

Available weekly capacity after routine work is:

C=18\ \text{work h/week}

Estimate the burn-down time.

Solution

Burn-down time is:

t=\dfrac{B}{C}

So:

t=\dfrac{126}{18}=7\ \text{weeks}

Engineering Comment

This assumes no new backlog arrives and all hours are qualified for the work. Critical backlog should not be hidden inside an average burn-down target.

Plausibility Check

Eighteen hours per week removes $126$ hours in exactly seven weeks.

Exercise 11: Maintenance Intake Utilization

A maintenance desk receives:

\lambda=9\ \text{jobs/day}

and can complete:

\mu=12\ \text{jobs/day}

Estimate utilization.

Solution

Utilization is:

\rho=\dfrac{\lambda}{\mu}

Therefore:

\rho=\dfrac{9}{12}=0.75

Engineering Comment

Seventy-five percent average utilization can still create delay if arrivals cluster or jobs have mixed skill requirements. Priority work needs a reserve, not just average capacity.

Plausibility Check

Nine jobs against twelve jobs of capacity uses three quarters of the desk.

Exercise 12: MTTR Reduction from Maintenance Readiness

An asset has:

MTBF=1200\ \text{h}

Current repair time is:

MTTR_1=12\ \text{h}

Better access and tooling reduce repair time to:

MTTR_2=5\ \text{h}

Compute the availability gain.

Solution

Before:

A_1=\dfrac{1200}{1200+12}=0.9901

After:

A_2=\dfrac{1200}{1200+5}=0.9959

Gain:

\Delta A=0.9959-0.9901=0.0058

or $0.58$ percentage points.

Engineering Comment

Maintenance readiness can improve availability without changing failure frequency. The evidence should include access, tooling, procedures, trained labor and restart checks.

Plausibility Check

Both repair times are small compared with MTBF, so the gain is less than one percentage point.

Exercise 13: PM Compliance Rate

A month requires:

N_p=48

preventive maintenance jobs. Completed on time:

N_c=43

The release rule requires $95\%$ on-time completion. Check compliance.

Solution

Compliance is:

C=\dfrac{N_c}{N_p}=\dfrac{43}{48}=0.896

Since:

89.6\%<95\%

the PM compliance gate fails.

Engineering Comment

PM compliance should be reviewed by criticality. Missing a low-risk inspection and missing a safety-critical test should not have equal release weight.

Plausibility Check

Five misses out of forty-eight is more than ten percent, so the result below ninety percent is expected.

Exercise 14: Deferred Work Exposure

A temporary operating restriction is allowed for:

T_{max}=21\ \text{days}

The backlog item has already been deferred:

T_d=16\ \text{days}

The planned repair will start in:

T_s=6\ \text{days}

Check whether the temporary restriction expires before repair starts.

Solution

Total deferral before repair start is:

T_{tot}=T_d+T_s=16+6=22\ \text{days}

Since:

22>21

the restriction expires before repair starts.

Engineering Comment

Temporary controls need active expiry tracking. A risk that was acceptable for a short deferral can become unacceptable if the repair slips.

Plausibility Check

The remaining allowance is only five days, but the repair starts in six days.

Exercise 15: Guarded Condition Acceptance

A wear measurement has acceptance limit:

w_{lim}=2.0\ \text{mm}

Measured wear is:

w_m=1.72\ \text{mm}

Measurement uncertainty is:

u=0.12\ \text{mm}

Use a two-uncertainty guard.

Solution

Guarded wear is:

w_g=w_m+2u=1.72+2(0.12)=1.96\ \text{mm}

Since:

1.96<2.0

the measurement passes with:

M=2.0-1.96=0.04\ \text{mm}

Engineering Comment

The pass is very narrow. Repeatability, probe access and trend rate should be checked before extending the inspection interval.

Plausibility Check

The unguarded margin is $0.28\ \text{mm}$ and the guard consumes $0.24\ \text{mm}$ , leaving a small margin.

Exercise 16: Work-Order Evidence Completion

A maintenance release package requires:

N=22

evidence items. Completed and reviewed:

N_c=20

The rule requires at least $95\%$ completion and no missing hard-gate item. Compute completion.

Solution

Completion is:

C=\dfrac{20}{22}=0.909

Since:

90.9\%<95\%

the package fails even before checking hard gates.

Engineering Comment

Maintenance records should show measurements, findings, corrective action, functional test and reviewer approval. A checkbox alone is weak evidence.

Plausibility Check

Two missing items out of twenty-two is about nine percent missing, so completion near $91\%$ is expected.

Exercise 17: Temporary Control Effectiveness

A backlog risk score is:

P_0=120

A temporary control is estimated to reduce likelihood by $40\%$ . Compute the residual score.

Solution

Residual likelihood factor is:

1-0.40=0.60

Residual score:

P_r=120(0.60)=72

Engineering Comment

Temporary controls should be verified in the field. A paper control does not reduce risk unless it is active, understood and monitored.

Plausibility Check

A forty percent reduction leaves sixty percent of the score, so $72$ is expected.

Exercise 18: Maintenance Release Gate

A maintenance release package has:

Gate	Requirement	Current result
P-F response buffer	at least $2$ weeks	$3$ weeks
guarded vibration	below limit	fail
PM compliance	at least $95\%$	$89.6\%$
work-order evidence	at least $95\%$	$90.9\%$

Decide whether to release.

Solution

The P-F buffer passes:

3\geq2

But guarded vibration, PM compliance and evidence completion fail. The package is not releasable.

Engineering Comment

Maintenance release is a hard-gate decision. A passing buffer does not compensate for failed condition-monitoring and evidence gates.

Plausibility Check

Several hard gates fail, so the only defensible decision is hold, restrict or repair.

Validation Package Checklist

A strong maintenance interval and backlog solution should check:

whether the failure mode and maintenance objective are explicit;
whether PM intervals match the failure model or degradation signal;
whether P-F inspection frequency includes detection delay and repair lead time;
whether condition-monitoring trends are guarded for uncertainty;
whether backlog priority includes consequence, likelihood and detectability;
whether temporary controls have owners, expiry dates and field evidence;
whether repair readiness includes access, permits, tools, spares and skill coverage;
whether work orders include measurements and functional validation;
whether failed hard gates block release instead of being averaged away.

Common Release Mistakes

Common mistakes include setting PM intervals by habit, using P-F intervals without repair lead time, accepting an unguarded trend near a limit, treating backlog count as backlog risk, ignoring temporary-control expiry, assuming average maintenance capacity covers priority work, reporting MTTR without access and restart time, closing work orders without functional evidence, and releasing an asset because most gates passed while a condition or evidence gate failed.

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