AgeReplacementPolicy#

class relife.policy.AgeReplacementPolicy(lifetime_model, cf, cp, discounting_rate=0.0, a0=None, ar=None, ar1=None)[source]#

Age replacement policy.

Behind the scene, a renewal reward stochastic process is used where assets are replaced at a fixed age \(a_r\) with costs \(c_p\) or upon failure with costs \(c_f\) if earlier [1].

Parameters:

lifetime_modelany lifetime distribution or frozen lifetime model: A lifetime model representing the durations between events.
cffloat or 1darray: Costs of failures
cpfloat or 1darray: Costs of preventive replacements
discounting_ratefloat, default is 0.: The discounting rate value used in the exponential discounting function
a0float or 1darray, optional: Current ages of the assets. If it is given, left truncations of a0 will be take into account for the first cycle.
arfloat or 1darray, optional: Ages of preventive replacements, by default None. If not given, one must call optimize to set ar values and access to the rest of the object interface.
ar1float, 2D array, optional: Ages of the first preventive replacements, by default None. If not given, one must call optimize to set ar values and access to the rest of the object interface.

Attributes:

a0float or 1darray, optional: Current ages of the assets. If it is given, left truncations of a0 will be take into account for the first cycle.
ar: Ages of the preventive replacements
first_cycle_tr: Time before the first replacement
cf: Cost of failures
cp: Cost of preventive replacements

References

[1]

Mazzuchi, T. A., Van Noortwijk, J. M., & Kallen, M. J. (2007). Maintenance optimization. Encyclopedia of Statistics in Quality and Reliability, 1000-1008.

Methods

`annual_number_of_replacements`	The expected number of annual replacements.
`asymptotic_expected_equivalent_annual_cost`	Calculate the asymptotic expected equivalent annual cost.
`asymptotic_expected_total_cost`	Calculate the asymptotic expected total cost.
`expected_equivalent_annual_cost`	Calculate the expected equivalent annual cost over a given timeline.
`expected_nb_replacements`	The expected number of replacements.
`expected_total_cost`	Calculate the expected total cost over a given timeline.
`generate_lifetime_data`	Generate lifetime data
`optimize`	Computes the optimal age(s) of replacement and updates the internal `ar` value(s) and,t optionally `ar1`.
`sample`	Renewal data sampling.

annual_number_of_replacements(nb_years, upon_failure=False, total=True)[source]#

The expected number of annual replacements.

Parameters:

nb_yearsint: The number of years on which the annual number of replacements are projected
upon_failurebool, default is False: If True, it also returns the annual number of replacements due to unexpected failures
totalbool, default is True: If True, the given numbers of replacements are the sum of all replacements without distinction between assets

property ar#

Ages of the preventive replacements

Returns:

ndarray

asymptotic_expected_equivalent_annual_cost()[source]#

Calculate the asymptotic expected equivalent annual cost.

It takes into account discounting_rate attribute value.

The asymptotic expected total cost is:

\[\lim_{t\to\infty} \text{EEAC}(t)\]

where \(\text{EEAC}(t)\) is the expected equivalent annual cost at \(t\). See expected_equivalent_annual_cost() for more details.

Returns:

np.ndarray: The asymptotic expected equivalent annual cost.

asymptotic_expected_total_cost()[source]#

Calculate the asymptotic expected total cost.

It takes into account discounting_rate attribute value.

The asymptotic expected total cost is:

\[\lim_{t\to\infty} z(t)\]

where \(z(t)\) is the expected total cost at \(t\). See expected_total_cost() for more details.

Returns:

np.ndarray: The asymptotic expected total cost.

property cf#

Cost of failures

Returns:

ndarray

property cp#

Cost of preventive replacements

Returns:

ndarray

expected_equivalent_annual_cost(tf, nb_steps)[source]#

Calculate the expected equivalent annual cost over a given timeline.

It takes into account discounting_rate attribute value.

The expected equivalent annual cost \(\text{EEAC}(t)\) is given by:

\[\text{EEAC}(t) = \dfrac{\delta z(t)}{1 - e^{-\delta t}}\]

where :

\(t\) is the time
\(z(t)\) is the expected_total_cost at \(t\). See expected_total_cost() for more details.`.
\(\delta\) is the discounting rate.

Parameters:

timeline: np.ndarray: Values of the timeline over which the expected equivalent annual cost is to be calculated.

Returns:

np.ndarray: The expected equivalent annual cost.

expected_nb_replacements(tf, nb_steps)#

The expected number of replacements.

It is computed by solving the renewal equation:

\[m(t) = F_1(t) + \int_0^t m(t-x) \mathrm{d}F(x)\]

where:

\(m\) is the renewal function,
\(F\) is the cumulative distribution function of the underlying lifetime model,
\(F_1\) is the cumulative distribution function of the underlying lifetime model for the fist renewal in the case of a delayed renewal process.

Parameters:

tffloat: Time horizon. The expected number of replacements will be computed up until this calendar time.
nb_stepsint: The number of steps used to compute the expected number of replacements

Returns:

tuple of two ndarrays: A tuple containing the timeline used to compute the expected number of replacements and its corresponding values at each step of the timeline.

expected_total_cost(tf, nb_steps)[source]#

Calculate the expected total cost over a given timeline.

It takes into account discounting_rate attribute value.

The expected total cost \(z(t)\) is computed by solving the renewal equation and is given by:

\[z(t) = \mathbb{E}(Z_t) = \int_{0}^{\infty}\mathbb{E}(Z_t~|~X_1 = x)dF(x)\]

where :

\(t\) is the time
\(X_i \sim F\) are \(n\) random variable lifetimes, i.i.d., of cumulative distribution \(F\).
\(Z_t\) is the random variable reward at each time \(t\).
\(\delta\) is the discounting rate.

Parameters:

timeline: np.ndarray: Values of the timeline over which the expected total cost is to be calculated.

Returns:

np.ndarray: The expected total cost.

property first_cycle_tr#

Time before the first replacement

Returns:

ndarray

generate_lifetime_data(size, tf, t0=0.0, seed=None)#

Generate lifetime data

This function will generate lifetime data that can be used to fit a lifetime model.

Parameters:

sizeint: The size of the desired sample.
tffloat: Time at the end of the observation.
t0float, default 0: Time at the beginning of the observation.
seedint, optional: Random seed, by default None.

Returns:

A dict of time, event, entry and args (covariates)

optimize()[source]#

Computes the optimal age(s) of replacement and updates the internal ar value(s) and,t optionally ar1.

Returns:

Self: Same instance with optimized ar (optionnaly ar1).

sample(size, tf, t0=0.0, seed=None)#

Renewal data sampling.

This function will sample data and encapsulate them in an object.

Parameters:

sizeint: The size of the desired sample.
tffloat: Time at the end of the observation.
t0float, default 0: Time at the beginning of the observation.
sizeint or tuple of 2 int: Size of the sample
seedint, optional: Random seed, by default None.

AgeReplacementPolicy#

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