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2.5.6.2 Algorithm: Warranty Analysis

Contents

Transform Warranty Data

Arbitrary-censored data is expected as input for the warranty analysis tool. If raw data in Nevada structure is available, the following algorithm is used to transform it into arbitrary-censored data.

Number of failures in the time interval \((i-1, i)\): \(f_i=\sum_{j=1}^{k-i+1}f_{j, i+j-1}\)
Number of right censored units at time \(i\): \(s_i=n_{k-i+1}-\sum_{j=k-i+1}^{k}f_{k-i+1,j}\)

where

  • \(k\): total number of shipments.
  • \(f_{i,j}\): number of items shiped at time \(i\) and failed at time \(j\), where \(i = 1,2,...,k\); \(j = i+1,i+2,...,k\)
  • \(n_i\): number of items shiped at time \(i\)

The input data are now 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 arbitrary-censored dataset is first fitted with a selected 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.

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.
  • 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 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 \(\mathcal{L}_{i}( \beta, \eta \,;tl_i, tr_i) = 1-R(min(L,tl_i))\)

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
    \[x_{U}= \frac{1}{2}\chi^{2}_{2(s+1),\alpha}\]
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:
    \[ PNF(\Delta) = \sum_{i=1}^{m} n_i(1-\frac{R(tl_i + \Delta)}{R(tl_i)}) + \sum_{i=1}^{q}d_j(1-R(\Delta+1-i))\]

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:
    \[ PNF(\Delta) = \sum_{i=1}^{m, tl_i<L} n_i(1-\frac{R(tl_i + \min(\Delta, L-tl_i)}{R(tl_i)})+\sum_{i=1}^{q}d_j(1-R(\min(\Delta+1-i,L))\]
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