Birnbaum’s component importance (1969)

Which component should be improved first? Birnbaum (1969) defined a component’s importance as the rate at which system reliability changes with its reliability, \(I_i = \partial R_s/\partial R_i\). In a series system that is the product of all the other reliabilities (birnbaum_importance_series()), so the weakest link is the most important. The example checks the derivative numerically and shows that improving the most important component buys the most.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.reliability import birnbaum_importance_series, series_reliability

Importance of each stage of a pumping train

names = ["intake screen", "pump", "motor", "control valve", "flow meter"]
R = np.array([0.99, 0.90, 0.97, 0.95, 0.995])
importance = np.array([birnbaum_importance_series(R, i) for i in range(R.size)])
eps = 1e-6
numerical = []
for i in range(R.size):
    up, down = R.copy(), R.copy()
    up[i] += eps
    down[i] -= eps
    numerical.append((series_reliability(*up) - series_reliability(*down)) / (2 * eps))
for name, I, d in zip(names, importance, numerical):
    print(f"{name:14s} I = {I:.4f} (finite difference {d:.4f})")

intake screen  I = 0.8252 (finite difference 0.8252)
pump           I = 0.9077 (finite difference 0.9077)
motor          I = 0.8422 (finite difference 0.8422)
control valve  I = 0.8599 (finite difference 0.8599)
flow meter     I = 0.8211 (finite difference 0.8211)

Spend a fixed improvement on each component in turn

Remove 0.02 of unreliability from one component at a time.

gain = []
for i in range(R.size):
    better = R.copy()
    better[i] = min(1.0, better[i] + 0.02)
    gain.append(series_reliability(*better) - series_reliability(*R))
fig, ax = plt.subplots(figsize=(6.5, 3.5))
ax.bar(names, importance, alpha=0.6, label="Birnbaum importance")
ax.plot(names, np.array(gain) / 0.02, "ko", label="gain per unit improvement")
ax.set(ylabel="dR_system / dR_i", title=f"System reliability {series_reliability(*R):.4f}")
ax.legend()
fig.tight_layout()

assert np.allclose(importance, numerical)
assert names[int(np.argmax(importance))] == "pump" == names[int(np.argmax(gain))]

plt.show()