2.5.3.2 Algorithm: Warranty Analysis
The data of the app are start and end times in the form \((tl_1, tr_1), (tl_1, tr_2),...,(tl_k, tr_k)\) and each interval \((tl_i, tr_i)\) contains \(n_i\) failures (if \(tr_i < \infty\)) or \(n_i\) suspensions (if \(tri = \infty\)), \(i = 1, 2,..., k \). This arbitrarily censored dataset is first fitted with a weibull distribution using either the MLE(maximum likelihood estimation) or LS (least squares) method to obtain the parameters \(\beta\) and \(\eta\). The results shown below are then computed based on the fitted model.
Contents
Summary of the warranty claims
- Total number of units: \(\sum_{i=1}^k n_i\)
- Observed number of failures: \(\sum_{i=1}^l n_i\)
where
- \(k\): total times.
- \(l\): the number of distinct interval censored times.
Expected Number of Failures
- Reliability Function:
- \(R(t) = 1 - F(t;\beta,\eta) \) where \(F(t;\beta,\eta)\) is the CDF of weibull distribution.
- Reliability Function:
- Expected number of failures without known warranty length (L):
- \[ENF = \sum n_i\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i ) \]
- if \(tl_i < tr_i\), then \(\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i) = R(tl_i)-R(tr_i)\)
- if \(tl_i = tr_i\) or \(tr_i = \infty\), then \(\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i) = 1-R(tl_i)\)
- Expected number of failures without known warranty length (L):
- Expected number of gailures with known warranty length (L):
- \[ENF = \sum n_i\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i ) \]
- if \(tl_i < tr_i\), then \(\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i) = R(tl_i)-R(min(L,tr_i))\)
- if \(tl_i = tr_i\) or \(tr_i = \infty\), then \(\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i) = 1-R(min(L,tl_i))\)
- Expected number of gailures with known warranty length (L):
Confidence intervals for the expected number of failures
- Two-sided \(100(1-\alpha)\%\) confidence interval
- \[x_{L} = \frac{1}{2}\chi^2_{2s, 1-\alpha/2}\]
- \[x_{U} = \frac{1}{2}\chi^2_{2(s+1), \alpha/2}\]
- One-sided \(100(1-\alpha)\%\) lower confidence bound
- \[x_{L} = \frac{1}{2}\chi^2_{2s, 1-\alpha}\]
- One-sided \(100(1-\alpha)\%\) upper confidence bound
- \[x_{U}= \frac{1}{2}\chi^{2}_{2(s+1),\alpha}\]
- Two-sided \(100(1-\alpha)\%\) confidence interval
- where
- \(s\) is the predicted number of future failures \(PNF\).
Number of units at risk for future time periods
- Without known warranty length (L): \(\sum_{i=1}^m n_i\)
- With known warranty length (L): \(\sum_{i=1}^{m, tl_i < L} n_i\)
- \(m\): the number of distinct right censored times.
Predicted number of future failures
Predicted Number of failures without known warranty length (L):
- If production quantity for each future period \(d_i\) is not provided:
- \[ PNF(\Delta) = \sum_{i=1}^{m} n_i(1-\frac{R(tl_i + \Delta)}{R(tl_i)})\]
- If production quantity for each future period \(d_i\) is provided:
- \[ PNF(\Delta) = \sum_{i=1}^{m} n_i(1-\frac{R(tl_i + \Delta)}{R(tl_i)}) + \sum_{i=1}^{q}d_j(1-R(\Delta+1-i))\]
- If production quantity for each future period \(d_i\) is not provided:
Predicted Number of failures without known warranty length (L):
- If production quantity for each future period \(d_i\) is not provided:
- \[ PNF(\Delta) = \sum_{i=1}^{m, tl_i<L} n_i(1-\frac{R(tl_i + \min(\Delta, L-tl_i)}{R(tl_i)})\]
- If production quantity for each future period \(d_i\) is provided:
- \[ PNF(\Delta) = \sum_{i=1}^{m, tl_i<L} n_i(1-\frac{R(tl_i + \min(\Delta, L-tl_i)}{R(tl_i)})+\sum_{i=1}^{q}d_j(1-R(\min(\Delta+1-i,L))\]
- If production quantity for each future period \(d_i\) is not provided:
- where
- \(d_i\) is the production quantities \(d_1, d_2,...,d_r\) for future periods \(1, 2,...,r\)
- \[q = \min(r, \Delta) \]
Predicted cost of future failures
- \[ PCF = PNF \times C\]
- where
- \(C\) is the provided cost per failure