Skip to main content

Reducing degradation and age of items in imperfect repair modeling

Abstract

We develop new models for imperfect repair and the corresponding generalized renewal processes for stochastic description of repairable items that fail when their degradation reaches the specified deterministic or random threshold. The discussion is based on the recently suggested notion of a random virtual age and is applied to monotone processes of degradation with independent increments. Imperfect repair reduces degradation of an item on failure to some intermediate level. However, for the nonhomogeneous processes, the corresponding age reduction, which sets back the ‘clock’ of the process, is also performed. Some relevant stochastic comparisons are obtained. It is shown that the cycles of the corresponding generalized imperfect renewal process are stochastically decreasing/increasing depending on the monotonicity properties of the failure rate that describes the random failure threshold of an item.

This is a preview of subscription content, access via your institution.

References

  • Badia FG, Berrade MD. (2009). Optimum maintenance policy of a periodically inspected system under imperfect repair. Adv Oper Res. Article ID 691203

  • Badıa FG, Berrade MD, Campos CA (2002) Optimal inspection and preventive maintenance of units with revealed and unrevealed failures. Reliab Eng Syst Saf 78:157–163

    Article  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, New York

    MATH  Google Scholar 

  • Belzunce F, Riquelme CM, Mulero J (2015) An Introduction to stochastic orders. Academic Press, Cambridge

    MATH  Google Scholar 

  • Berenguer C, Grall A, Dieulle L (2003) Maintenance policy for a continuously monitored deteriorating system. Probab Eng Inf Sci 17:235–250

    MathSciNet  Article  Google Scholar 

  • Castro IT (2013) An age-based maintenance strategy for a degradation-threshold shock-model for a system subjected to multiple defects. Asia Pac J Oper Res 30:1350016–1350029

    MathSciNet  Article  Google Scholar 

  • Castro IT, Mercier S (2016) Performance measures for a deteriorating system subject to imperfect maintenance and delayed repairs. J Risk Reliab 230:364–377

    Google Scholar 

  • Cha JH, Finkelstein M. (2018). Point Processes for Reliability Analysis. Shocks and Repairable systems. London, Springer.

  • Dijoux Y, Fouladirad M, Nguyen DT (2016) Statistical inference for imperfect maintenance models with missing data. Reliab Eng Syst Saf 154:84–96

    Article  Google Scholar 

  • Doyen L, Gaudoin O (2004) Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliab Eng Syst Saf 84:45–56

    Article  Google Scholar 

  • Etminan J, Kamranfar H, Chahkandi M, Fouladirad M (2022) Analysis of time-to-failure data for a repairable system subject to degradation. J Comput Appl Math. https://doi.org/10.1016/j.cam.2022.114098

    MathSciNet  Article  MATH  Google Scholar 

  • Finkelstein M (2007) On some ageing properties of general repair processes. J Appl Probab 44:506–513

    MathSciNet  Article  Google Scholar 

  • Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, London

    MATH  Google Scholar 

  • Finkelstein M, Cha JH (2013) Stochastic modelling for reliability (shocks, burn-in and heterogeneous populations). Springer, London

    Book  Google Scholar 

  • Finkelstein M, Cha JH (2021) On degradation-based imperfect repair and induced generalized renewal processes. TEST. https://doi.org/10.1007/s11749-021-00765-z

    MathSciNet  Article  MATH  Google Scholar 

  • Kahle W (2019) Imperfect repair in degradation processes: a Kijima-type approach. Appl Stoch Model Bus Ind 35:211–220

    MathSciNet  Article  Google Scholar 

  • Kamranfar H, Fouladirad M, Balakrishnan N (2021) Inference for a gradually deteriorating system with imperfect maintenance. Commun Stat-Simul Comput. https://doi.org/10.1080/03610918.2021.2001528

    Article  Google Scholar 

  • Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26:89–102

    MathSciNet  Article  Google Scholar 

  • Krivtsov VV (2007) Recent advances in theory and applications of stochastic point process models in reliability engineering. Reliab Eng Syst Saf 92:549–551

    Article  Google Scholar 

  • Levitin G, Lisniansky A (2000) Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 67:193–203

    Article  Google Scholar 

  • Meeker WQ, Escobar LA (2014) Statistical methods for reliability data. Wiley, Hoboken

    MATH  Google Scholar 

  • Mercier S, Castro IT (2019) Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system. Eur J Oper Res 273:237–248

    MathSciNet  Article  Google Scholar 

  • Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, Chichester

    MATH  Google Scholar 

  • Nguyen DT, Dijoux Y, Fouladirad M (2017) Analytical properties of an imperfect repair model and application in preventive maintenance scheduling. Eur J Oper Res 256:439–453

    MathSciNet  Article  Google Scholar 

  • Ponchet A, Fouladirad M, Grall A (2011) Maintenance policy on a finite time span for a gradually deteriorating system with imperfect improvements. J Risk Reliab 225:105–116

    Google Scholar 

  • Sadeghi J, Najar M, Mollazadeh M, Yousefi B, Zakeri J (2018) Improvement of railway ballast maintenance approach, incorporating ballast geometry and fouling conditions. J Appl Geophys 151:263–273

    Article  Google Scholar 

  • Salles G, Mercier S, Bordes L (2020) Semiparametric estimate of the efficiency of imperfect maintenance actions for a gamma deteriorating system. J Stat Plan Inference 206:278–297

    MathSciNet  Article  Google Scholar 

  • Shaked M, Shanthikumar J (2007) Stochastic orders. Springer, New York

    Book  Google Scholar 

  • Tanwar M, Rai RN, Bolia N (2014) Imperfect repair modeling using Kijima type generalized renewal process. Reliab Eng Syst Saf 124:24–31

    Article  Google Scholar 

  • van Noortwijk JM (2009) A survey of the application of gamma processes in maintenance. Reliab Eng Syst Saf 94:2–21

    Article  Google Scholar 

  • Wang H, Pham H (2006) Reliability and optimal maintenance. Springer, London

    MATH  Google Scholar 

  • Zhao X, Gaudoin O, Doyen L, Xie M (2019) Optimal inspection and replacement policy based on experimental degradation data with covariates. IISE Trans 5:322–336

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the reviewers for helpful comments and advices. The work of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number: 2019R1A6A1A11051177).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ji Hwan Cha.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

We first need some definitions and preliminary lemmas on multivariate stochastic orders.

Definition A1

Let \({\mathbf{X}}{ = (}X_{1} ,X_{2} ,\ldots,X_{n} )\) and \({\mathbf{Y}}{ = (}Y_{1} ,Y_{2} ,\ldots,Y_{n} )\) be two \(n\)-dimensional random vectors such that

$$ P({\mathbf{X}} \in U) \le P({\mathbf{Y}} \in U),\,{\text{for all upper sets}}\,U \subseteq {\mathbf{R}}^{n} . $$

Then, \({\mathbf{X}}\) is said to be smaller than \({\mathbf{Y}}\) in the usual stochastic order, denoted by \({\mathbf{X}} \le_{st} {\mathbf{Y}}\).

Lemma A1

(Shaked and Shanthikumar 2007) Let \({\mathbf{X}}{ = (}X_{1} ,X_{2} ,\ldots,X_{n} )\) and \({\mathbf{Y}}{ = (}Y_{1} ,Y_{2} ,\ldots,Y_{n} )\) be two \(n\)-dimensional random vectors. If

$$ X_{1} \le_{st} Y_{1} $$

and, for \(i = 2,3,\ldots,n\)

\((X_{i} |X_{1} = x_{1} ,X_{2} = x_{2} ,\ldots,X_{i - 1} = x_{i - 1} ) \le_{st} (Y_{i} |Y_{1} = y_{1} ,Y_{2} = y_{2} ,\ldots,Y_{i - 1} = y_{i - 1} )\) whenever \(x_{j} \le y_{j}\), \(j = 1,2,\ldots,i - 1\), then \({\mathbf{X}} \le_{st} {\mathbf{Y}}\).

Lemma A2

(Shaked and Shanthikumar 2007) Let \({\mathbf{X}}{ = (}X_{1} ,X_{2} ,\ldots,X_{n} )\) and \({\mathbf{Y}}{ = (}Y_{1} ,Y_{2} ,\ldots,Y_{n} )\) be two \(n\) -dimensional random vectors. If \({\mathbf{X}} \le_{st} {\mathbf{Y}}\) and \({\mathbf{g}}\) : \({\mathbf{R}}^{n} \to {\mathbf{R}}^{k}\) is any \(k\)-dimensional increasing (decreasing) function, for any integer \(k\), then the \(k\)-dimensional vectors \({\mathbf{g}}({\mathbf{X}})\) and \({\mathbf{g}}({\mathbf{Y}})\) satisfy \({\mathbf{g}}({\mathbf{X}}) \le_{st} ( \ge_{st} ){\mathbf{g}}({\mathbf{Y}})\) .

Theorem A1

Suppose that the distribution \(G(x)\) is DFR, i.e., \(\lambda_{L} (x)\) is decreasing in \(x\) . Then,

$$ L_{1} \le_{st} L_{2} \le_{st} L_{3} \le_{st} \ldots. $$

Proof

First, we show \(L_{1} <_{st} L_{2}\). Observe that

$$ P(L_{1} > w) = \overline{G}(w) = \exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (x){\rm d}x } \right\}, $$

whereas

$$ P(L_{2} > w|L_{1} = w_{1} ) = \overline{G}(w|qw_{1} ) = \exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qw_{1} + x){\rm d}x}\right\} , $$

and \(P(L_{2} > w) = E[P(L_{2} > w|L_{1} )]\). When \(\lambda_{L} (x)\) is decreasing,

$$ P(L_{1} > w) \le P(L_{2} > w|L_{1} = w_{1} ),\,{\text{for }}\,{\text{all }}\,{\text{realization }}\,{\text{of}}\,w_{1} > 0,\,{\text{for all}}\,w > 0. $$
(A1)

This implies that \(P(L_{1} > w) \le E[P(L_{2} > w|L_{1} )] = P(L_{2} > w)\), for all \(w > 0\).

Observe that

$$ P(L_{3} > w|L_{1} = w_{1} ,L_{2} = w_{2} ) = \overline{G}(w|q(qw_{1} + w_{2} )) =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q(qw_{1} + w_{2} ) + x){\rm d}x}\right\} . $$

Thus, \(P(L_{3} > w|L_{1} = w_{1} ,L_{2} ) =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{2} w_{1} + qL_{2} + x){\rm d}x}\right\} \ge\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qL_{2} + x){\rm d}x }\right\}\), and

$$\begin{aligned}& P(L_{3} > w|L_{1} = w_{1} ) = E_{{(L_{2} |L_{1} = w_{1} )}} [P(L_{3} > w|L_{1} = w_{1} ,L_{2} )] \\ &\qquad\ge E_{{(L_{2} |L_{1} = w_{1} )}} \left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qL_{2} + x){\rm d}x}\right\} \right ].\end{aligned} $$

On the other hand, \(P(L_{2} > w) = E\left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qL_{1} + x){\rm d}x }\right\} \right]\). As \(L_{1} \le_{st} (L_{2} |L_{1} = w_{1} )\), for all realization of \(w_{1} > 0\), due to Lemma 1-(i),(ii),

$$ P(L_{3} > w|L_{1} = w_{1} ) \ge E_{{(L_{2} |L_{1} = w_{1} )}} \left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qL_{2} + x){\rm d}x}\right\} \right] $$

\(\ge E\left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (qL_{1} + x){\rm d}x}\right\} \right] = P(L_{2} > w)\), for all realization of \(w_{1} > 0\), for all \(w > 0\),

which implies that \(P(L_{3} > w) \ge P(L_{2} > w)\), for all \(w > 0\).

In general, for \(n = 3,4,\ldots\),

$$ \begin{aligned} & P(L_{n} > w|L_{1} = w_{1} ,L_{2} = w_{2} ,\ldots,L_{n - 1} = w_{n - 1} ) \\ &\quad = \exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} w_{1} + q^{n - 2} w_{2} + \cdots + qw_{n - 1} + x){\rm d}x}\right\}\end{aligned} $$

and

$$ \begin{aligned} & P(L_{n + 1} > w|L_{1} = w_{1} ,L_{2} = w_{2} ,\ldots,L_{n} = w_{n} ) \\ &\quad =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n} w_{1} + q^{n - 1} w_{2} + \cdots + qw_{n} + x){\rm d}x}\right\} . \end{aligned} $$

Thus,

$$ P(L_{n} > w) = E_{{(L_{1} ,L_{2} \ldots,L_{n - 1} )}} \left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} L_{1} + q^{n - 2} L_{2} + \cdots + qL_{n - 1} + x){\rm d}x}\right\} \right ] $$

and

$$ \begin{aligned} & P(L_{n + 1} > w|L_{1} \\ & \quad = w_{1} ) = E_{{(L_{2} ,L_{3} \ldots,L_{n} |L_{1} = w_{1} )}} [P(L_{n + 1} > w|L_{1} = w_{1} ,L_{2} ,L_{3} ,\ldots,L_{n} )] \\ &\quad \ge E_{{(L_{2} ,L_{3} \ldots,L_{n} |L_{1} = w_{1} )}} \left[\exp\left \{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} L_{2} + \cdots + q^{2} L_{n - 1} + qL_{n} + x){\rm d}x}\right\} \right]. \\ \end{aligned} $$

Now we stochastically compare \((L_{1} ,L_{2} ,\ldots,L_{n - 1} )\) with \((L_{2} ,L_{3} ,\ldots,L_{n} |L_{1} = w_{1} )\). From (A1),

$$ L_{1} \le_{st} (L_{2} |L_{1} = w_{1} ). $$

Also,

$$ \begin{aligned} P(L_{i} > w|L_{1} & = w_{1} ,L_{2} = w_{2} ,\ldots,L_{i - 1} = w_{i - 1} ) \\ & =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{i - 1} w_{1} + q^{i - 2} w_{2} + \cdots + qw_{i - 1} + x){\rm d}x}\right\} , \\ \end{aligned} $$

whereas

$$ \begin{aligned} P(V_{i} & > w|L_{1} = w_{1} ,V_{1} = v_{1} ,V_{2} = v_{2} ,\ldots,V_{i - 1} = v_{i - 1} ) \\ & =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{i} w_{1} + q^{i - 1} v_{1} + \cdots + qv_{i - 1} + x){\rm d}x}\right\} , \\ \end{aligned} $$

where \(V_{j} = L_{j + 1}\), \(j = 1,2,\ldots\). Then, we have

$$ \begin{aligned} &\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{i - 1} w_{1} + q^{i - 2} w_{2} + \cdots + qw_{i - 1} + x){\rm d}x}\right\} \\ &\quad \le\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{i} w_{1} + q^{i - 1} v_{1} + \cdots + qv_{i - 1} + x){\rm d}x}\right\} , \end{aligned} $$

for all \(w > 0\), whenever \(w_{j} \le v_{j}\), \(j = 1,2,\ldots,i - 1\), which implies that

$$\begin{aligned} & (L_{i} |L_{1} = w_{1} ,L_{2} = w_{2} ,\ldots,L_{i - 1} = w_{i - 1} ) \\ &\quad \le_{st} (V_{i} |L_{1} = w_{1} ,V_{1} = v_{1} ,V_{2} = v_{2} ,\ldots,V_{i - 1} = v_{i - 1} ), \end{aligned} $$
(A2)

whenever \(w_{j} \le v_{j}\), \(j = 1,2,\ldots,i - 1\). Inequality (A2) holds for all \(i = 2,\ldots,n - 1\). Thus, by Lemma A1, \((L_{1} ,L_{2} ,\ldots,L_{n - 1} ) \le_{st} (V_{1} ,V_{2} ,\ldots,V_{n - 1} |L_{1} = w_{1} )\) and thus

$$ (L_{1} ,L_{2} ,\ldots,L_{n - 1} ) \le_{st} (L_{2} ,L_{3} ,\ldots,L_{n} |L_{1} = w_{1} ). $$
(A3)

Observe that

$$ h(w_{1} ,w_{2} ,\ldots,w_{n - 1} ) =\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} w_{1} + q^{n - 2} w_{2} + \ldots + qw_{n - 1} + x){\rm d}x}\right\} $$

is an increasing function of \((w_{1} ,w_{2} ,\ldots,w_{n - 1} )\). Then, from (A3) and Lemma A3,

$$ \begin{aligned} P(L_{n} > w) & = E_{{(L_{1} ,L_{2} \ldots,L_{n - 1} )}} \left[ {\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} L_{1} + q^{n - 2} L_{2} + \ldots + qL_{n - 1} + x){\rm d}x}\right\} } \right] \\ & \le E_{{(L_{2} ,L_{3} \ldots,L_{n} |L_{1} = w_{1} )}} \left[\exp \left\{ - \int\limits_{0}^{w} {\lambda_{L} (q^{n - 1} L_{2} + \cdots + q^{2} L_{n - 1} + qL_{n} + x){\rm d}x}\right\} \right] \\ &\le P(L_{n + 1} > w|L_{1} = w_{1} ), \\ \end{aligned} $$

for all realization of \(w_{1} > 0\), for all \(w > 0\). This again implies that

$$ P(L_{n} > w) \le P(L_{n + 1} > w),\,{\text{ for }}\,{\text{all}}\,w > 0. $$

Therefore, we have shown that \(P(L_{n} > w) \le P(L_{n + 1} > w)\), for all \(n = 1,2,\ldots\)

Theorem A2

Suppose that the distribution \(G(x)\) is DFR, i.e., \(\lambda_{L} (x)\) is decreasing in \(x\) . Then,

$$ Z_{1} \le_{st} Z_{2} \le_{st} Z_{3} \le_{st} \ldots. $$

Proof

It is obvious that \(Z_{1} \le_{st} Z_{2}\). In the proof of Theorem A1, we have shown that

$$ (L_{1} ,L_{2} , \ldots ,L_{n} ) \le_{st} (L_{2} ,L_{3} , \ldots ,L_{n + 1} |L_{1} = w_{1} ). $$

Furthermore,

$$ h(w_{1} ,w_{2} ,\ldots,w_{n - 1} ) = q^{n - 1} w_{1} + q^{n - 2} w_{2} + \cdots + q^{{}} w_{n - 1} $$

is an increasing function of \((w_{1} ,w_{2} ,\ldots,w_{n - 1} )\). Then, from Lemma A2,

$$ \begin{aligned}&q^{n-1}L_1+ q^{n-2}L_2+\cdots+qL_{n-1}\le_{st} (q^{n-1}L_2+q^{n-2}L_3+\cdots+qL_n|L_1=w_1 )\\ &\quad\le_{st} (q^{n}w_1+q^{n-1}L_2+q^{n-2}L_3+\cdots+qL_n|L_1=w_1 )\end{aligned}$$

which implies that \(Z_{n} \le_{st} (Z_{n + 1} |L_{1} = w_{1} )\), for all realizations of \(w_{1}\). Therefore, we have

$$ Z_{n} \le_{st} Z_{n + 1} ,\,n = 1,2,\ldots $$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Finkelstein, M., Cha, J.H. Reducing degradation and age of items in imperfect repair modeling. TEST (2022). https://doi.org/10.1007/s11749-022-00813-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11749-022-00813-2

Keywords

  • Imperfect repair
  • Virtual age
  • Renewal processes
  • Remaining lifetime
  • Stochastic comparisons
  • Gamma process

Mathematics Subject Classification

  • 90B25
  • 60K10