Formula sheet

Operations and Reliability Formula Sheet

Operations and reliability formulas for critical path timing, Little's Law, capacity, yield, inventory, RPN, MTBF, availability, Weibull models, and validation.

Branch: Industrial and Management Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects common first-pass relationships for operations planning, project control, capacity analysis, risk ranking, reliability engineering, and validation review. Use it for screening and consistency checks. Detailed decisions still require clear assumptions, data quality review, confidence bounds, and context-specific acceptance criteria.

State the system boundary, time basis, failure definition, resource assumptions, and data source before using any metric.

Reliability data normalization

Failure rate from observed failures and exposure:

\displaystyle \hat{\lambda}=\frac{N_f}{T_{exposure}}

Mean time between failures from exposure-normalized data:

\displaystyle MTBF=\frac{T_{exposure}}{N_f}

Mean time to repair:

\displaystyle MTTR=\frac{T_{downtime}}{N_f}

Repair rate:

\displaystyle \mu=\frac{1}{MTTR}

Do not mix calendar time, operating time, cycles, starts, or distance without stating the exposure basis. A failure count without exposure is not a reliability metric.

Critical path timing

Activity duration:

D=EF-ES

where $ES$ is earliest start and $EF$ is earliest finish.

Forward pass:

ES_i=\max(EF_{predecessors})

EF_i=ES_i+D_i

Backward pass:

LF_i=\min(LS_{successors})

LS_i=LF_i-D_i

Total float:

TF_i=LS_i-ES_i=LF_i-EF_i

Activities with zero or near-zero total float are critical under the current schedule logic.

Project duration and path length

Path duration:

D_{path}=\sum_i D_i

Critical path duration:

D_{project}=\max(D_{path})

Schedule variance from baseline:

SV=t_{planned}-t_{actual}

The critical path is valid only if the dependency network, calendars, constraints, and resource assumptions are valid.

Little’s Law

Little’s Law:

L=\lambda W

where $L$ is average number of items in the system, $\lambda$ is throughput or arrival rate, and $W$ is average time in the system.

Lead time from work-in-process:

\displaystyle W=\frac{L}{\lambda}

Throughput from WIP and lead time:

\displaystyle \lambda=\frac{L}{W}

Little’s Law applies to stable systems using consistent definitions of system boundary, throughput, and time.

Utilization

Single-server utilization:

\displaystyle \rho=\frac{\lambda}{\mu}

For $c$ parallel identical servers:

\displaystyle \rho=\frac{\lambda}{c\mu}

where $\lambda$ is arrival rate and $\mu$ is service rate per server.

Stable operation requires:

\rho<1

High utilization can cause large waiting times when variability is present.

Capacity and takt checks

Cycle time:

\displaystyle CT=\frac{\text{available production time}}{\text{units produced}}

Takt time:

\displaystyle TT=\frac{\text{available production time}}{\text{customer demand}}

Capacity:

\displaystyle C=\frac{\text{available time}}{CT}

A process can meet demand only if effective capacity exceeds demand after downtime, yield loss, setup, rework, and variability are included.

Effective capacity and OEE

Operating availability:

\displaystyle A_o=\frac{\text{run time}}{\text{planned production time}}

Performance factor:

\displaystyle P_o=\frac{\text{ideal cycle time}\times\text{total count}}{\text{run time}}

Quality factor:

\displaystyle Q_o=\frac{\text{good count}}{\text{total count}}

Overall equipment effectiveness:

OEE=A_oP_oQ_o

Effective capacity:

C_{eff}=C\cdot OEE

Use the same boundary for planned time, downtime, speed loss, scrap, rework, and changeover before comparing OEE across lines or sites.

Yield and rework

First-pass yield:

\displaystyle FPY=\frac{\text{units passing without rework}}{\text{units entering process}}

Rolled throughput yield for sequential steps:

RTY=\prod_i FPY_i

Scrap rate:

\displaystyle S=\frac{\text{scrapped units}}{\text{units entering process}}

Rework can hide process weakness because final yield may look acceptable while time, cost, and capacity are consumed.

Inventory and reorder screening

Reorder point:

ROP=dL+SS

where $d$ is average demand rate, $L$ is replenishment lead time, and $SS$ is safety stock.

Safety stock for normal lead-time demand:

SS=z\sigma_L

Economic order quantity:

\displaystyle EOQ=\sqrt{\frac{2DS}{H}}

where $D$ is annual demand, $S$ is ordering cost per order, and $H$ is annual holding cost per unit.

Inventory turns:

\displaystyle Turns=\frac{\text{annual usage or cost of goods sold}}{\text{average inventory}}

Inventory formulas should be checked against demand variability, supplier reliability, batch constraints, shelf life, obsolescence, and service-level targets.

Risk Priority Number

Traditional FMEA risk priority number:

RPN=SOD

where $S$ is severity, $O$ is occurrence, and $D$ is detection ranking.

Residual RPN after action:

RPN_{res}=S_{res}O_{res}D_{res}

RPN uses ordinal rankings. It should guide discussion and prioritization, not be treated as a physical measure of absolute risk.

Reliability

Reliability function:

R(t)=P(T>t)

For constant failure rate:

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

Failure probability by time $t$ :

F(t)=1-R(t)

Mean time between failures for a constant failure rate:

\displaystyle MTBF=\frac{1}{\lambda}

The exponential model is not appropriate for wear-out, fatigue, corrosion, infant mortality, or mixed failure populations without justification.

Availability

For a repairable system with mean time between failures and mean time to repair:

\displaystyle A=\frac{MTBF}{MTBF+MTTR}

Downtime fraction:

\displaystyle 1-A=\frac{MTTR}{MTBF+MTTR}

Expected downtime over operating time $T$ :

D_T=(1-A)T

Availability depends on diagnostics, access, spare parts, repair skill, logistics, and restoration quality.

Mini example: availability

For:

MTBF=500\ \text{h}

and:

MTTR=5\ \text{h}

availability is:

\displaystyle A=\frac{500}{500+5}=0.990

Expected downtime in 2000 operating hours is:

D_T=(1-0.990)(2000)=20\ \text{h}

For three required independent assets in series, each with availability 0.990:

A_{system}=0.990^3=0.970

This screen assumes independence, stable repair process, and representative MTBF and MTTR data.

Series and parallel reliability

Series system reliability for independent components:

R_{series}=\prod_i R_i

Two independent parallel paths:

R_{parallel}=1-(1-R_1)(1-R_2)

For identical independent parallel components:

R_{parallel}=1-(1-R)^n

These formulas assume independence. Common-cause failures, shared power, shared environment, shared software, and maintenance errors can invalidate simple redundancy claims.

Weibull reliability

Weibull reliability:

R(t)=e^{-(t/\eta)^\beta}

Failure distribution:

F(t)=1-e^{-(t/\eta)^\beta}

Failure rate:

\displaystyle \lambda(t)=\frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}

Interpretation:

\beta<1 \quad \text{decreasing failure rate}

\beta=1 \quad \text{constant failure rate}

\beta>1 \quad \text{increasing failure rate}

Weibull models require appropriate life data, censoring treatment, and confidence bounds.

Validation metrics

Mean error:

\displaystyle ME=\frac{1}{N}\sum_i(x_i-r_i)

Mean absolute error:

\displaystyle MAE=\frac{1}{N}\sum_i|x_i-r_i|

Root-mean-square error:

\displaystyle RMSE=\sqrt{\frac{1}{N}\sum_i(x_i-r_i)^2}

Pass rate:

\displaystyle P_{pass}=\frac{\text{number passing}}{\text{number tested}}

Validation metrics must be matched to intended use, risk, reference uncertainty, and acceptance criteria.

Confidence and uncertainty

Sample mean:

\displaystyle \bar{x}=\frac{1}{N}\sum_i x_i

Sample standard deviation:

\displaystyle s=\sqrt{\frac{1}{N-1}\sum_i(x_i-\bar{x})^2}

Standard error of the mean:

\displaystyle SE=\frac{s}{\sqrt{N}}

Combined independent uncertainty:

u_c=\sqrt{\sum_i u_i^2}

Do not treat systematic bias, missing data, or model-form error as random variation without justification.

Monte Carlo schedule or risk simulation

For uncertain inputs:

y=f(x_1,x_2,\ldots,x_n)

Sample each trial:

y_j=f(x_{1,j},x_{2,j},\ldots,x_{n,j})

Estimate probability of meeting a target:

\displaystyle P(y\leq y_{target})\approx \frac{\text{successful trials}}{\text{total trials}}

Monte Carlo results depend on input distributions, correlations, truncation limits, and model logic.

Pareto and improvement priority

Cumulative fraction for ranked causes:

\displaystyle C_k=\frac{\sum_{i=1}^{k} x_i}{\sum_{i=1}^{n} x_i}

Benefit-cost ratio:

\displaystyle BCR=\frac{\text{expected benefit}}{\text{implementation cost}}

Expected value:

EV=\sum_i p_i v_i

Use these as decision aids. Safety, compliance, severity, strategic value, and uncertainty may override simple economic ranking.

Validation record

For operations and reliability calculations, record:

boundary, time basis, exposure unit, and failure definition;
data source, censoring rules, missing-data treatment, and confidence bounds;
whether failures are independent, repairable, recurring, or common-cause;
assumptions for demand, downtime, setup, rework, repair time, and supplier lead time;
acceptance criteria for schedule, capacity, availability, reliability, or validation error;
owner and action when a metric crosses its trigger threshold.

The metric is useful only when it changes a planning, maintenance, quality, or design decision.

Practical checklist

Use these formulas with a short review checklist:

Define boundary, time basis, resource assumptions, and data source.
Check schedule logic before trusting float or critical path.
Separate average demand, peak demand, variability, downtime, and rework.
Treat RPN as prioritization, not proof of safety.
Match reliability model to the failure mechanism and mission time.
Include confidence bounds and uncertainty where data are sparse.
Validate controls with evidence tied to the actual failure mode.

The calculations support engineering judgement. They do not replace clear scope, field data, operating feedback, or cross-functional review.

REF

Disciplines