Operations Planning and Reliability Engineering Exercises

These exercises practise operations planning and reliability engineering for schedules, queues, production capacity, equipment effectiveness, risk ranking, availability, redundancy, spare parts, and operational validation. The purpose is not only to calculate a metric. The purpose is to connect planning assumptions to real work systems, failure modes, recovery time, and evidence.

Assume simplified deterministic inputs unless an exercise states otherwise. Real operations should also check variability, calendars, resource constraints, supplier lead time, human workload, maintenance access, data quality, recurrence evidence, and the consequences of missed assumptions.

How to Use These Exercises

For each calculation, define:

1. the system boundary and time basis;
2. whether the metric uses calendar time, operating time, cycles, units, or jobs;
3. the constraint being tested: schedule, capacity, queue, failure mode, spare, or validation criterion;
4. the action that follows if the result fails;
5. the data that must be captured to improve the next estimate.

The common mistake is using operational metrics as labels rather than controls. A useful metric changes a schedule, staffing plan, maintenance interval, spare policy, alarm threshold, or validation decision.

For each result, state whether it supports schedule control, WIP reduction, capacity action, maintenance priority, redundancy claim, spare reorder policy, or validation release. Reliability metrics are useful only when exposure basis, failure definition, and corrective action are explicit.

Exercise 1: Critical Path from a Dependency Network

A commissioning plan has the following activities:

Activity	Duration	Immediate predecessors
A	4 days	none
B	6 days	A
C	5 days	A
D	3 days	B
E	4 days	B, C
F	2 days	D, E

Find the project duration and critical path.

Solution

Candidate path durations are:

The longest path is:

Project duration is:

Engineering Comment

The critical path is a dependency result, not a list of important tasks. If activity B slips by one day, the project slips. If activity C slips by one day, it may still have float. The schedule should be updated when actual progress, access constraints, or failed tests change the logic.

Exercise 2: Little’s Law and Lead Time

A maintenance engineering queue has average work-in-process:

and average completion rate:

Estimate average lead time. Then find the WIP target needed to reach a 5-day average lead time at the same throughput.

Solution

Little’s Law is:

Lead time:

WIP target for 5-day lead time:

Engineering Comment

Reducing lead time at fixed throughput requires lower WIP or less variability. If the queue is fed faster than it can complete work, lead time will grow even if every engineer is busy. High utilization can look efficient while making service worse.

Exercise 3: Takt Time and Effective Capacity

A production cell has available time:

Demand is:

The process cycle time is:

Expected downtime is:

Check whether the cell can meet demand.

Solution

Takt time:

Available run time after downtime:

Effective capacity:

The cell cannot meet 140 units/shift under this downtime assumption.

Engineering Comment

Nominal cycle time is below takt, but effective capacity fails once downtime is included. The practical response could be downtime reduction, parallel capacity, overtime, buffer policy, preventive maintenance, or demand smoothing.

Exercise 4: Overall Equipment Effectiveness

A machine has planned production time:

and downtime:

The ideal cycle time is:

The machine produces 485 total units, of which 462 are good. Calculate availability, performance, quality, and OEE.

Solution

Run time:

Availability:

Performance:

Quality:

OEE:

The OEE is approximately:

Engineering Comment

OEE is useful only if downtime, speed loss, scrap, rework, changeover, and planned time are defined consistently. A higher OEE may not increase system output if the machine is not the bottleneck.

Exercise 5: RPN Before and After a Control Action

A failure mode has rankings:

A corrective action reduces occurrence to 3 and improves detection ranking to 3. Calculate initial and revised RPN.

Solution

Initial RPN:

Revised RPN:

Reduction:

Engineering Comment

The ranking improves, but severity remains high. High-severity failures may still require design change, interlock, redundancy, inspection, or management approval even after RPN is reduced. RPN is a prioritization aid, not a substitute for consequence review.

Exercise 6: MTBF, MTTR, and Availability from Field Data

A fleet records:

with:

and total repair downtime:

Estimate MTBF, MTTR, and steady-state availability.

Solution

Mean time between failures:

Mean time to repair:

Availability:

The screened availability is:

Engineering Comment

The result depends on exposure quality and failure definitions. Calendar time, operating time, starts, cycles, and standby time should not be mixed without a clear data contract. The failure modes should also be separated before corrective action is selected.

Exercise 7: Redundant Availability with a Common Series Element

Two identical standby pumps are arranged so the system works if at least one pump is available. Each pump has availability:

The shared controller required by both pumps has availability:

Estimate system availability assuming independent pump failures and controller in series with the redundant pump set.

Solution

Availability of at least one pump:

System availability with shared controller:

The system availability is approximately:

Engineering Comment

Redundancy helps only if common-cause and shared-support failures are controlled. A shared controller, power supply, suction blockage, maintenance error, or software fault can dominate the availability of redundant hardware.

Exercise 8: Weibull Reliability at Mission Time

A component life model has Weibull shape parameter:

and scale parameter:

Estimate reliability at:

using:

Solution

Compute the exponent:

Reliability:

The estimated reliability at 3000 h is:

Engineering Comment

Because \beta>1, the model implies increasing failure rate with time. The result should be tied to the same operating duty, environment, failure definition, censoring treatment, and confidence bounds used to fit the Weibull model.

Exercise 9: Spare-Part Reorder Point

A spare part has average demand:

Supplier lead time is:

Lead-time demand standard deviation is:

Use service factor:

Estimate safety stock and reorder point.

Solution

Safety stock:

Round to:

Expected demand during lead time:

Reorder point:

Engineering Comment

The reorder point depends on demand variability and lead-time reliability. Critical spares may need a different policy based on downtime consequence, obsolescence, repairability, supplier risk, shelf life, and whether the item is shared across assets.

Exercise 10: Operational Validation Trial

A line trial must demonstrate average throughput of at least:

and no hourly result below:

Five one-hour trial counts are:

Check acceptance against these criteria.

Solution

Average throughput:

Minimum hourly result:

The average meets the 52 units/h requirement, and the minimum result is above 49 units/h. The trial passes these stated criteria.

Engineering Comment

Passing this short trial does not prove full operating readiness. The validation record should state product mix, staffing, downtime, rework, material availability, measurement method, operator workload, maintenance response, and whether the trial represents normal demand variability.

Review Checklist

A strong operations and reliability solution should check:

• whether the boundary, time basis, exposure basis, and failure definition are explicit;
• whether calendar time, operating time, starts, cycles, and standby time are kept separate;
• whether schedule metrics include real dependencies, access constraints, failed tests, and recovery logic;
• whether capacity calculations include downtime, variability, bottlenecks, staffing, material availability, and demand mix;
• whether OEE, MTBF, MTTR, and availability are linked to failure modes rather than averaged into weak signals;
• whether redundancy claims include common-cause failures, shared utilities, controls, software, maintenance error, and access;
• whether spare policies include lead-time risk, criticality, shelf life, obsolescence, and downtime consequence;
• whether validation trials represent normal variability and define what happens if the criterion fails.

The useful result is not only a number; it is a controlled change to a schedule, queue, spare policy, reliability model, maintenance plan, or validation gate.