Worked Examples
Real-world calculations for all 20 reliability and quality tools
Core Reliability Metrics
MTTR
Open CalculatorProduction Line — 5 breakdowns in a month
Given
- • Breakdown 1: 2.5 h
- • Breakdown 2: 1.8 h
- • Breakdown 3: 4.2 h
- • Breakdown 4: 3.1 h
- • Breakdown 5: 2.4 h
Calculation
MTTR = 2.8 hours
Slightly above the 2.0 h manufacturing benchmark. Review Breakdown 3 (4.2 h) for improvement.
IT Server — 12 incidents over 6 months
Given
- • 8 incidents × 0.75 h = 6 h
- • 3 incidents × 2 h = 6 h
- • 1 major incident × 8 h = 8 h
Calculation
MTTR = 1.67 hours
Within the 1–4 h IT benchmark. The single major incident raises the average — resolve its root cause.
MTBF
Open CalculatorFleet of 10 delivery trucks — 1 year operation
Given
- • 10 trucks × 8 h/day × 250 days = 20,000 h
- • 25 failures across fleet
Calculation
MTBF = 800 hours
Schedule preventive maintenance every ~640 h (80% of MTBF) to catch failures before they occur.
Industrial pump operating 24/7 for 6 months
Given
- • Operating time = (6×30×24) − 48 h downtime = 4,272 h
- • 3 major failures
Calculation
MTBF = 1,424 hours
Good reliability for industrial pumps. Consider condition monitoring to detect degradation between PM intervals.
MTTF
Open Calculator100 LED bulbs tested — 20 fail over testing period
Given
- • Total component-hours = 3,000,000 h
- • 20 failures observed
Calculation
MTTF = 150,000 hours
Suitable for applications requiring ≥ 50,000 h lamp life. Replace batch proactively at 80% of MTTF.
Availability
Open CalculatorWeb server — one calendar month (744 h)
Given
- • Total planned time: 744 h
- • Planned maintenance: 4 h
- • Unplanned downtime: 6 h
Calculation
Availability = 98.66%
Below the 99.9% target. 10 h downtime/month = 120 h/year. Focus on eliminating the 6 h unplanned outages.
Using MTBF/MTTR formula
Given
- • MTBF = 500 h
- • MTTR = 5 h
Calculation
Availability = 99.01%
Reducing MTTR from 5 h to 2 h would improve availability to 99.60%.
Injection moulding machine — 8 h shift
Given
- • Planned: 480 min
- • Breakdowns: 30 min
- • Changeovers: 20 min
- • Ideal cycle: 2 min, Actual: 200 units
- • Defects: 10 units
Calculation
OEE = 79.2%
Good for a typical plant. Availability is the weakest factor — target the 50 min of stoppage time first.
Downtime Cost
Open CalculatorAutomotive assembly line — 3-hour unplanned stoppage
Given
- • Lost revenue: $8,000/h
- • Overtime labour: $2,000 total
- • Customer penalty: $5,000
- • Repair parts: $800
Calculation
Total Downtime Cost = $31,800
A $5,000 predictive maintenance programme that prevents one such event per year delivers 6× ROI.
Spare Parts
Open CalculatorCritical pump seal — stockroom optimisation
Given
- • Mean annual demand (λ) = 4 seals/yr
- • Lead time: 3 weeks
- • Target service level: 95%
Calculation
Stock 1 seal to achieve 95% SL
For 99% SL, stock 2 seals. For a $15 seal, the cost difference is trivial vs hours of production downtime.
Advanced Reliability Analysis
Reliability R(t)
Open CalculatorAircraft component — 500-hour mission
Given
- • MTBF = 2,000 h
- • Mission time t = 500 h
- • λ = 1 / 2,000 = 0.0005 /h
Calculation
R(500) = 77.88%
Below the 90% threshold for aviation-critical components. Consider redundancy or shorter inspection interval.
System Reliability
Open CalculatorParallel pump system (2-of-2 redundancy)
Given
- • Pump A: R = 0.90
- • Pump B: R = 0.90
Calculation
System R = 99%
Adding one redundant pump improves system reliability from 90% to 99%. Worth the investment for critical services.
3-component series: pump → valve → filter
Given
- • R_pump = 0.95
- • R_valve = 0.98
- • R_filter = 0.99
Calculation
System R = 92.1%
The pump is the weakest link. A 1% reliability improvement in the pump yields more than improvements in other components.
Failure Rate
Open CalculatorElectronic controller — fleet data
Given
- • 200 units tested for 5,000 h
- • 8 failures observed
- • Total unit-hours = 200 × 5,000 = 1,000,000 h
Calculation
λ = 8,000 FIT | MTBF = 125,000 h
Good for industrial electronics. Automotive targets are typically < 100 FIT — more work needed for that application.
Weibull Analysis
Open CalculatorBearing wear-out failure analysis
Given
- • β = 2.8 (wear-out mode)
- • η = 8,000 h (characteristic life)
Calculation
R(6,000 h) = 63.5%
Only 63.5% survival at 6,000 h. Set replacement interval at 4,000–5,000 h for 85%+ reliability.
FMEA
Open CalculatorAutomotive fuel injector failure mode analysis
Given
- • Failure Mode: Clogged injector
- • Severity (S) = 8 — engine stall
- • Occurrence (O) = 4 — occasional
- • Detection (D) = 6 — not easily caught in test
Calculation
RPN = 192 (High Priority)
Improve detection with better test coverage. Reducing D from 6 to 3 drops RPN to 96 — below the action threshold.
Gage R&R
Open CalculatorCaliper measurement system validation
Given
- • 2 operators, 10 parts, 2 replicates each
- • σ_repeatability = 0.025 mm
- • σ_reproducibility = 0.015 mm
- • σ_Part = 0.100 mm
Calculation
%GRR = 28% (Marginal)
Repeatability dominates. Investigate fixture consistency. Retrain operators and recalibrate caliper before accepting this MSA.
Quality & Statistical Tools
Process Capability
Open CalculatorChemical reactor temperature control
Given
- • USL = 205°C, LSL = 195°C
- • Process mean μ = 201°C
- • Process std dev σ = 1.2°C
Calculation
Cp = 1.39 | Cpk = 1.11
Process is capable (Cpk > 1.0) but off-center. Shift mean from 201°C toward 200°C to raise Cpk to ≥ 1.33.
DPMO & Sigma Level
Open CalculatorInvoice processing — accounts payable team
Given
- • 1,000 invoices processed
- • 3 opportunities per invoice (amount, vendor, date)
- • 36 defects found in audit
Calculation
DPMO = 12,000 → ~3.8 Sigma
Below the 4σ (6,210 DPMO) target. Top defects: incorrect amounts (55%) and wrong vendor codes (30%) — use Pareto to prioritize.
Sample Size
Open CalculatorIncoming inspection — bolt tensile strength
Given
- • Confidence level: 95% (Z = 1.96)
- • Process std dev σ = 12 kN
- • Acceptable margin of error E = 2 kN
Calculation
Required sample n = 139 bolts
If sampling 139 is too costly, widen margin of error to 3 kN → n = 62. Use control charts to reduce required future sample sizes.
Control Chart
Open CalculatorAutomotive shaft diameter — X̄-R chart (n=5)
Given
- • Grand mean X̄̄ = 49.98 mm
- • Average range R̄ = 0.06 mm
- • Constants: A₂=0.577, D₃=0, D₄=2.114
Calculation
UCL=50.015, LCL=49.945 mm
All points in control. Cpk = 1.67 — this is a world-class process. Continue monitoring at current frequency.
Pareto Chart
Open CalculatorCustomer complaint analysis — 6 months
Given
- • Late delivery: 245 (42%)
- • Product defects: 156 (27%)
- • Poor service: 89 (15%)
- • Wrong product: 58 (10%)
- • Other: 35 (6%)
Calculation
Late delivery + defects = 69% of all complaints
Fix the delivery process and product quality. Addressing just these two issues eliminates 69% of customer complaints.
Fishbone Diagram
Open CalculatorHigh scrap rate on CNC machined parts (target 2%, actual 8%)
Given
- • Man: New operators (3 hired last month)
- • Machine: Worn cutting tools, vibration
- • Material: Hardness variation in new supplier batch
- • Method: Inadequate SOP for new operators
Calculation
Primary cause: worn cutting tools + inadequate SOP
Implement daily tool wear inspection and launch a 2-week operator training programme on updated SOPs.
Histogram
Open CalculatorFill volume analysis — 50 bottles sampled
Given
- • Target: 500 mL
- • Sample mean x̄ = 501.4 mL
- • Sample std dev σ = 3.2 mL
- • USL = 510, LSL = 490 mL
Calculation
Cp = 1.04 — marginally capable
Right-skewed histogram: mean is above target. Reduce fill head valve opening slightly to center the distribution.
Scatter Diagram
Open CalculatorCure temperature vs. bond strength (n=30 samples)
Given
- • Temperature range: 150–180°C
- • Bond strength: 12–28 MPa
- • Pearson r calculated from data
Calculation
r = 0.87 — Strong positive correlation
Higher cure temperature → stronger bonds. Raise cure temperature to 175°C as the optimal operating point identified from the regression line.
Check Sheet
Open CalculatorDefect tally on finished assemblies — one shift
Given
- • Missing screw: 14
- • Wrong label: 8
- • Scratch/cosmetic: 22
- • Wrong colour: 3
- • Other: 2
Calculation
Scratch (44.9%) | Missing screw (28.6%) are dominant
Transfer these two categories to a Pareto chart. Together they account for 73.5% of defects — focus quality action here.
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