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name reliability-engineering description Use this Skill for reliability engineering: Weibull analysis, fault tree analysis, FMEA, reliability block diagrams, and accelerated life testing.
tags ["engineering","reliability","weibull","fmea","fault-tree"] version 1.0.0 authors [{"name":"Rosetta Skills Contributors","github":"@xjtulyc"}] license MIT platforms ["claude-code","codex","gemini-cli","cursor"] dependencies {"python":["numpy>=1.24","scipy>=1.11","matplotlib>=3.7","pandas>=2.0","reliability>=0.8"]} last_updated 2026-03-17 status stable
Reliability Engineering Analysis
One-line summary : Quantify product reliability using Weibull analysis, fault trees, FMEA risk prioritization, and reliability block diagrams with the reliability library and scipy.
When to Use This Skill
When fitting Weibull distributions to failure time data
When computing MTTF, B10/B50 life, and hazard rates
When building fault tree diagrams and computing top-event probability
When performing FMEA and calculating Risk Priority Numbers (RPN)
When modeling series/parallel/k-of-n reliability block diagrams
When designing accelerated life tests (Arrhenius or power-law stress)
Trigger keywords : Weibull, reliability, MTTF, B10 life, fault tree, FMEA, RPN, hazard function, accelerated life test, reliability block diagram, failure rate
Background & Key Concepts
Weibull Distribution
The 2-parameter Weibull PDF:
$$
f(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1} \exp\left[-\left(\frac{t}{\eta}\right)^\beta\right]
$$
$\beta$: shape parameter — $\beta < 1$ decreasing failure rate (infant mortality), $\beta = 1$ constant (exponential), $\beta > 1$ increasing (wear-out)
$\eta$: scale parameter (characteristic life at 63.2% failure)
Reliability function: $R(t) = \exp\left[-\left(t/\eta\right)^\beta\right]$
B-life: $B_p = \eta \left[-\ln(1-p/100)\right]^{1/\beta}$
Fault Tree Analysis (FTA)
Top-down deductive analysis: top event probability computed by propagating basic event probabilities through AND/OR gates.
AND gate: $P = \prod_i P_i$
OR gate: $P = 1 - \prod_i (1-P_i)$
Risk Priority Number (FMEA)
$$
\text{RPN} = \text{Severity} \times \text{Occurrence} \times \text{Detection}
$$
Each scored 1–10; RPNs ≥ 100 typically require corrective action.
Environment Setup
Install Dependencies
pip install numpy>=1.24 scipy>=1.11 matplotlib>=3.7 pandas>=2.0 reliability>=0.8
Verify Installation
import reliability
from reliability.Fitters import Fit_Weibull_2P
import numpy as np
np.random.seed(42
2.0
20
1000
False
False
print
f"Weibull fit: beta={f.beta:.3 f} , eta={f.eta:.1 f} h"
Core Workflow
Step 1: Weibull Parameter Estimation and B-life import numpy as np
import matplotlib.pyplot as plt
from reliability.Fitters import Fit_Weibull_2P
from reliability.Distributions import Weibull_Distribution
from reliability.Utils import colorprint
np.random.seed(42 )
failures = np.array([
342 , 487 , 623 , 741 , 856 , 980 , 1105 , 1230 , 1380 , 1520 ,
1650 , 1790 , 1940 , 2100 , 2280 , 2450 , 2700 , 3010 , 3400 , 4200
])
right_censored = np.array([500 , 750 , 1000 , 1500 , 2000 , 3000 , 5000 ])
f = Fit_Weibull_2P(
failures=failures,
right_censored=right_censored,
show_probability_plot=False ,
print_results=True ,
)
beta_hat = f.beta
eta_hat = f.eta
print (f"\nEstimated parameters: beta={beta_hat:.4 f} , eta={eta_hat:.2 f} h" )
dist = Weibull_Distribution(alpha=eta_hat, beta=beta_hat)
def b_life (dist, p_pct ):
"""Return the age by which p% of units have failed."""
return dist.inverse_SF(1 - p_pct/100 )
for p in [1 , 5 , 10 , 50 ]:
print (f" B{p:2d} life = {b_life(dist, p):8.1 f} h "
f"(R={100 -p} % survive to this age)" )
mttf = dist.mean
sigma = dist.standard_deviation
print (f"\nMTTF = {mttf:.1 f} h, σ = {sigma:.1 f} h" )
t_range = np.linspace(1 , 8000 , 500 )
fig, axes = plt.subplots(1 , 3 , figsize=(15 , 4 ))
R = dist.SF(xvals=t_range)
axes[0 ].plot(t_range, R, 'b-' , linewidth=2 )
axes[0 ].axhline(0.9 , color='red' , linestyle='--' , alpha=0.7 , label='R=0.9' )
axes[0 ].set_xlabel("Time (h)" ); axes[0 ].set_ylabel("Reliability R(t)" )
axes[0 ].set_title("Survival Function" ); axes[0 ].legend()
axes[0 ].grid(True , alpha=0.3 )
F = dist.CDF(xvals=t_range)
axes[1 ].plot(t_range, F * 100 , 'r-' , linewidth=2 )
axes[1 ].set_xlabel("Time (h)" ); axes[1 ].set_ylabel("Failures (%)" )
axes[1 ].set_title("Cumulative Failure Probability" ); axes[1 ].grid(True , alpha=0.3 )
h = dist.HF(xvals=t_range)
axes[2 ].plot(t_range, h * 1000 , 'g-' , linewidth=2 )
axes[2 ].set_xlabel("Time (h)" ); axes[2 ].set_ylabel("Hazard rate (×10⁻³ /h)" )
axes[2 ].set_title("Hazard Function" ); axes[2 ].grid(True , alpha=0.3 )
plt.suptitle(f"Weibull Reliability (β={beta_hat:.2 f} , η={eta_hat:.0 f} h)" , y=1.02 )
plt.tight_layout()
plt.savefig("weibull_reliability.png" , dpi=150 , bbox_inches="tight" )
plt.show()
Step 2: Fault Tree Analysis import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from matplotlib.patches import FancyBboxPatch
basic_events = {
"BE1" : {"name" : "Pump A fails" , "prob" : 0.02 },
"BE2" : {"name" : "Pump B fails" , "prob" : 0.03 },
"BE3" : {"name" : "Control valve fails" , "prob" : 0.01 },
"BE4" : {"name" : "Sensor fails" , "prob" : 0.05 },
"BE5" : {"name" : "Power supply fails" , "prob" : 0.005 },
}
def or_gate (*probs ):
"""Probability that AT LEAST ONE event occurs."""
return 1.0 - np.prod([1 - p for p in probs])
def and_gate (*probs ):
"""Probability that ALL events occur simultaneously."""
return np.prod(list (probs))
p_be1 = basic_events["BE1" ]["prob" ]
p_be2 = basic_events["BE2" ]["prob" ]
p_be3 = basic_events["BE3" ]["prob" ]
p_be4 = basic_events["BE4" ]["prob" ]
p_be5 = basic_events["BE5" ]["prob" ]
p_g1 = or_gate(p_be1, p_be2)
p_g2_sub = or_gate(p_be4, p_be5)
p_g2 = and_gate(p_be3, p_g2_sub)
p_top = or_gate(p_g1, p_g2)
print ("=== Fault Tree Analysis Results ===" )
print (f"G1 (Pump subsystem OR): P = {p_g1:.6 f} ({p_g1*100 :.3 f} %)" )
print (f"G2 (Control subsystem AND): P = {p_g2:.6 f} ({p_g2*100 :.3 f} %)" )
print (f"TOP (System failure OR): P = {p_top:.6 f} ({p_top*100 :.3 f} %)" )
def birnbaum_importance (basic_event_key, p_top_original ):
"""
Compute Birnbaum structural importance:
I_B = P(top | event=1) - P(top | event=0)
"""
results = {}
for state in [0 , 1 ]:
p = {k: v["prob" ] for k, v in basic_events.items()}
p[basic_event_key] = state
_g1 = or_gate(p["BE1" ], p["BE2" ])
_sub = or_gate(p["BE4" ], p["BE5" ])
_g2 = and_gate(p["BE3" ], _sub)
results[state] = or_gate(_g1, _g2)
return results[1 ] - results[0 ]
print ("\n=== Birnbaum Importance Measures ===" )
importances = {}
for key in basic_events:
imp = birnbaum_importance(key, p_top)
importances[key] = imp
print (f" {key} ({basic_events[key]['name' ]:25s} ): I_B = {imp:.6 f} " )
fig, ax = plt.subplots(figsize=(8 , 4 ))
labels = [f"{k} \n{basic_events[k]['name' ]} " for k in importances]
values = list (importances.values())
colors = ['#e74c3c' if v == max (values) else '#3498db' for v in values]
bars = ax.bar(labels, values, color=colors, edgecolor='black' , linewidth=0.8 )
ax.set_ylabel("Birnbaum Importance" )
ax.set_title("Fault Tree: Birnbaum Structural Importance" )
ax.grid(axis='y' , alpha=0.3 )
for bar, val in zip (bars, values):
ax.text(bar.get_x() + bar.get_width()/2 , bar.get_height() + 0.0001 ,
f"{val:.4 f} " , ha='center' , va='bottom' , fontsize=9 )
plt.tight_layout()
plt.savefig("fault_tree_importance.png" , dpi=150 )
plt.show()
print (f"\n[Top-event failure probability per mission: {p_top:.4 %} ]" )
Step 3: FMEA and RPN Analysis import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm as cm
fmea_data = {
"Component" : ["Pump Motor" , "Pump Motor" , "Bearing" , "Seal" ,
"Control PCB" , "Control PCB" , "Power Supply" , "Sensor" ],
"Failure Mode" : ["Winding short" , "Overheating" , "Fatigue crack" , "Leakage" ,
"ADC drift" , "Firmware hang" , "Capacitor fail" , "Bias offset" ],
"Effect" : ["No flow" , "Reduced life" , "Vibration" , "Contamination" ,
"Wrong reading" ,"No control" , "Shutdown" , "False alarm" ],
"S" : [9 , 7 , 8 , 6 , 7 , 9 , 8 , 5 ],
"O" : [3 , 4 , 2 , 5 , 3 , 2 , 4 , 6 ],
"D" : [5 , 6 , 4 , 3 , 4 , 3 , 5 , 4 ],
}
df = pd.DataFrame(fmea_data)
df["RPN" ] = df["S" ] * df["O" ] * df["D" ]
df["Criticality" ] = df["S" ] * df["O" ]
df = df.sort_values("RPN" , ascending=False ).reset_index(drop=True )
df["Rank" ] = range (1 , len (df)+1 )
RPN_THRESHOLD = 100
print ("=== FMEA Results (sorted by RPN) ===" )
print (df[["Rank" ,"Component" ,"Failure Mode" ,"S" ,"O" ,"D" ,"RPN" ]].to_string(index=False ))
print (f"\nFailure modes requiring corrective action (RPN ≥ {RPN_THRESHOLD} ):" )
high_risk = df[df["RPN" ] >= RPN_THRESHOLD]
for _, row in high_risk.iterrows():
print (f" [{row['Rank' ]} ] {row['Component' ]} — {row['Failure Mode' ]} : RPN={row['RPN' ]} " )
fig, (ax1, ax2) = plt.subplots(1 , 2 , figsize=(14 , 5 ))
colors = ['#e74c3c' if r >= RPN_THRESHOLD else '#3498db' for r in df["RPN" ]]
bars = ax1.bar(range (len (df)), df["RPN" ], color=colors, edgecolor='black' , linewidth=0.7 )
cumulative = df["RPN" ].cumsum() / df["RPN" ].sum () * 100
ax1_twin = ax1.twinx()
ax1_twin.plot(range (len (df)), cumulative, 'ko-' , markersize=5 , linewidth=1.5 )
ax1_twin.set_ylabel("Cumulative % of total RPN" )
ax1_twin.axhline(80 , color='gray' , linestyle='--' , alpha=0.5 , label='80%' )
ax1.axhline(RPN_THRESHOLD, color='red' , linestyle='--' , linewidth=1.5 , label=f'Threshold={RPN_THRESHOLD} ' )
ax1.set_xticks(range (len (df)))
ax1.set_xticklabels([f"{r} \n{m[:8 ]} " for r, m in zip (df["Component" ], df["Failure Mode" ])],
fontsize=7 , rotation=30 , ha='right' )
ax1.set_ylabel("RPN" ); ax1.set_title("FMEA Pareto Chart (RPN)" )
ax1.legend(loc='upper right' , fontsize=8 )
sc = ax2.scatter(df["O" ], df["S" ], s=df["D" ]*40 , c=df["RPN" ],
cmap='RdYlGn_r' , vmin=0 , vmax=400 , edgecolors='black' , linewidth=0.7 )
plt.colorbar(sc, ax=ax2, label='RPN' )
for _, row in df.iterrows():
ax2.annotate(f"{row['Component' ][:6 ]} " , (row['O' ], row['S' ]),
textcoords="offset points" , xytext=(4 , 4 ), fontsize=7 )
ax2.set_xlabel("Occurrence (O)" ); ax2.set_ylabel("Severity (S)" )
ax2.set_title("FMEA Criticality Matrix\n(bubble size ∝ Detection)" )
ax2.set_xlim(0 , 11 ); ax2.set_ylim(0 , 11 )
ax2.grid(True , alpha=0.3 )
ax2.fill_between([7 , 11 ], [7 , 7 ], [11 , 11 ], alpha=0.07 , color='red' )
ax2.text(8.5 , 9.5 , "High\nRisk" , ha='center' , color='red' , fontsize=9 )
plt.tight_layout()
plt.savefig("fmea_analysis.png" , dpi=150 )
plt.show()
Advanced Usage
Reliability Block Diagram (Series-Parallel System) import numpy as np
import matplotlib.pyplot as plt
from reliability.Distributions import Weibull_Distribution
def R_subsystem_A (t ):
"""Single Weibull component."""
dist = Weibull_Distribution(alpha=2000 , beta=2.0 )
return dist.SF(xvals=t)
def R_component_B (t ):
"""Individual B component reliability."""
dist = Weibull_Distribution(alpha=3000 , beta=1.5 )
return dist.SF(xvals=t)
def R_subsystem_B_parallel (t ):
"""1-of-2 parallel: R = 1 - (1-R_b)^2."""
R_b = R_component_B(t)
return 1 - (1 - R_b)**2
def R_system (t ):
"""Series combination of A and B_parallel."""
return R_subsystem_A(t) * R_subsystem_B_parallel(t)
t = np.linspace(1 , 6000 , 500 )
fig, ax = plt.subplots(figsize=(10 , 5 ))
ax.plot(t, R_subsystem_A(t), 'b--' , label='Subsystem A (single)' , linewidth=1.5 )
ax.plot(t, R_component_B(t), 'g:' , label='Component B (single)' , linewidth=1.5 )
ax.plot(t, R_subsystem_B_parallel(t),'g--' , label='Subsystem B (1-of-2 parallel)' , linewidth=2 )
ax.plot(t, R_system(t), 'r-' , label='SYSTEM (A series B_parallel)' , linewidth=2.5 )
ax.axhline(0.9 , color='gray' , linestyle='--' , alpha=0.6 , label='R=0.9 target' )
from scipy.optimize import brentq
t_mission = brentq(lambda x: R_system(np.array([x]))[0 ] - 0.90 , 1 , 5999 )
ax.axvline(t_mission, color='orange' , linestyle=':' , linewidth=2 )
ax.text(t_mission + 100 , 0.5 , f"T(R=0.9)={t_mission:.0 f} h" ,
color='orange' , fontsize=10 )
ax.set_xlabel("Time (h)" ); ax.set_ylabel("Reliability R(t)" )
ax.set_title("Reliability Block Diagram — Series-Parallel System" )
ax.legend(loc='upper right' ); ax.grid(True , alpha=0.3 )
plt.tight_layout()
plt.savefig("rbd_system.png" , dpi=150 )
plt.show()
Accelerated Life Testing (Arrhenius Model) import numpy as np
from scipy.optimize import curve_fit
import matplotlib.pyplot as plt
k_B = 8.617e-5
test_temps_K = np.array([398 , 423 , 448 ])
mttf_observed = np.array([5000 , 1800 , 700 ])
def arrhenius_mttf (T, A, Ea ):
return A * np.exp(Ea / (k_B * T))
popt, pcov = curve_fit(arrhenius_mttf, test_temps_K, mttf_observed, p0=[1e-5 , 0.8 ])
A_hat, Ea_hat = popt
print (f"Arrhenius fit: A={A_hat:.4 e} , Ea={Ea_hat:.4 f} eV" )
T_use = 298.0
mttf_use = arrhenius_mttf(T_use, A_hat, Ea_hat)
print (f"Extrapolated MTTF at {T_use-273 :.0 f} °C: {mttf_use:.0 f} h ({mttf_use/8760 :.1 f} years)" )
for T_test in test_temps_K:
AF = mttf_use / arrhenius_mttf(T_test, A_hat, Ea_hat)
print (f" AF ({T_test-273 :.0 f} °C → 25°C) = {AF:.1 f} x" )
T_range = np.linspace(280 , 460 , 200 )
fig, ax = plt.subplots(figsize=(8 , 5 ))
ax.semilogy(1000 /T_range, arrhenius_mttf(T_range, A_hat, Ea_hat),
'b-' , linewidth=2 , label='Arrhenius fit' )
ax.semilogy(1000 /test_temps_K, mttf_observed, 'rs' , markersize=10 ,
label='ALT data' , zorder=5 )
ax.semilogy(1000 /T_use, mttf_use, 'g^' , markersize=12 , label=f'Use condition (25°C): {mttf_use:.0 f} h' )
ax.set_xlabel("1000/T (1/K)" ); ax.set_ylabel("MTTF (h)" )
ax.set_title(f"Arrhenius Plot (Ea = {Ea_hat:.3 f} eV)" )
ax.legend(); ax.grid(True , which='both' , alpha=0.3 )
plt.tight_layout()
plt.savefig("arrhenius_plot.png" , dpi=150 )
plt.show()
Troubleshooting
Error: Fit_Weibull_2P fails with small samples Cause : MLE requires at least ~5 failures. With fewer, use median rank regression.
from reliability.Fitters import Fit_Weibull_2P
f = Fit_Weibull_2P(failures=failures, method='RRX' , show_probability_plot=False )
Error: brentq root not found for B-life Cause : Target reliability outside the distribution range.
from scipy.optimize import brentq
R_0 = dist.SF(xvals=np.array([1e-6 ]))[0 ]
print (f"R at t→0: {R_0:.6 f} " )
Warning: Weibull fit with censored data Use right_censored parameter — never mix censored and failure times in the failures array:
f = Fit_Weibull_2P(
failures=np.array([100 , 200 , 350 ]),
right_censored=np.array([400 , 500 ]),
)
Version Compatibility Package Tested versions Notes reliability 0.8.x API stable since 0.8; check changelog for 0.9 scipy 1.11, 1.12 curve_fit p0 required for Arrheniusnumpy 1.24, 1.26 No known issues
External Resources
Official Documentation
Key Papers
Meeker, W.Q. & Escobar, L.A. (1998). Statistical Methods for Reliability Data . Wiley.
ReVelle, J.B. (2004). Manufacturing Handbook of Best Practices . CRC Press.
Examples
Example 1: Fleet Replacement Interval Optimization import numpy as np
from scipy.optimize import minimize_scalar
from reliability.Distributions import Weibull_Distribution
C_p = 500
C_f = 5000
beta, eta = 2.5 , 8000
dist = Weibull_Distribution(alpha=eta, beta=beta)
def cost_rate (tp ):
"""Expected cost per unit time for preventive replacement interval tp."""
if tp <= 0 :
return np.inf
R_tp = dist.SF(xvals=np.array([tp]))[0 ]
F_tp = 1 - R_tp
t_arr = np.linspace(0 , tp, 1000 )
mean_cycle = np.trapz(dist.SF(xvals=t_arr), t_arr)
return (C_p * R_tp + C_f * F_tp) / mean_cycle
result = minimize_scalar(cost_rate, bounds=(100 , 20000 ), method='bounded' )
t_opt = result.x
print (f"Optimal replacement interval: {t_opt:.0 f} h" )
print (f"Cost rate at optimum: ${cost_rate(t_opt):.4 f} /h" )
print (f"Annual cost (8760 h/yr): ${cost_rate(t_opt)*8760 :.0 f} " )
t_no_pm = minimize_scalar(lambda t: cost_rate(t), bounds=(100000 , 200000 ), method='bounded' ).x
cr_rtf = C_f / dist.mean
print (f"\nRun-to-failure cost rate: ${cr_rtf:.4 f} /h (annual: ${cr_rtf*8760 :.0 f} )" )
print (f"Savings from PM: {(cr_rtf - cost_rate(t_opt))/cr_rtf*100 :.1 f} %" )
Example 2: System Availability with Repair import numpy as np
from reliability.Distributions import Weibull_Distribution
beta, eta = 2.0 , 5000
MTTR = 8.0
dist = Weibull_Distribution(alpha=eta, beta=beta)
MTTF = dist.mean
A_steady = MTTF / (MTTF + MTTR)
unavailability = 1 - A_steady
print (f"MTTF = {MTTF:.1 f} h" )
print (f"MTTR = {MTTR:.1 f} h" )
print (f"Availability A = {A_steady:.6 f} ({A_steady*100 :.4 f} %)" )
print (f"Unavailability = {unavailability:.6 f} ({unavailability*1e6 :.1 f} PPM)" )
print (f"Downtime/year = {unavailability*8760 :.1 f} h/yr" )
target_A = 0.999
MTTR_required = MTTF * (1 - target_A) / target_A
print (f"\nTo achieve A={target_A} : MTTR ≤ {MTTR_required:.2 f} h" )
Last updated: 2026-03-17 | Maintainer: @xjtulyc
Issues: GitHub Issues
