Warranty cost analysis with an alternating geometric process

Arnold, Richard; Chukova, Stefanka; Hayakawa, Yu; Marshall, Sarah

doi:10.1177/1748006x18820379

Cited by 5 publications

(17 citation statements)

References 14 publications

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“…A slight modification of the models introduced above yields an alternating Geometric Process (AGP) model . Let

{false{{normalX}_{normalk} false}}_{normalk = 1}^{\infty}

be a sequence of non‐negative random variables.…”

Section: Analytic Results and Special Casesmentioning

confidence: 99%

“…This gives the same inverse behaviour between operating and repair times that we expect: if

a > 1

with ever decreasing operating times, then

b = a^{- β}

leads to increasing repair times (assuming

β > 0

). Note that this is a special case of the AGP, with both baseline distributions being exponential …”

Section: Analytic Results and Special Casesmentioning

confidence: 99%

“…Repair costs are dominated by labour costs, which these are proportional to repair time, so that repair cost is a suitable proxy for repair time. Following methods similar to Arnold et al, we have amalgamated claims from the same vehicle occurring within 3 days of each other and converted by a linear transformation the repair cost variable into a repair duration between 0.01 and 25 days. Where the repair cost was missing (19.7% of cases), we imputed it by hot deck imputation, randomly selecting from the repair costs of claims for cars of similar ages.…”

Section: Examplesmentioning

confidence: 99%

“…A slight modification of the models introduced above yields an alternating Geometric Process (AGP) model. 11 Let {X k } ∞ k=1 be a sequence of non-negative random variables. If they are independent and the distribution function of X k is given by F k (x) = F 1 (a k−1 x) for k = 1, 2, … for some constant a > 0, then the sequence is called a geometric process.…”

Section: Case 3: Connection To the Alternating Geometric Processmentioning

confidence: 99%

Section: Example 2: Car Warranty Claimsmentioning

confidence: 99%

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Nonzero repair times dependent on the failure hazard

Arnold

Chukova

Hayakawa

et al. 2019

Quality & Reliability Eng

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It is common in the literature on the reliability and maintenance of repairable systems to model the repair times as instantaneous. However, this is an unreasonable assumption for some complex systems, especially those requiring a high level of reliability, and such systems may spend a significant proportion of their lifetimes under maintenance and repair. We model the ageing of such a system with alternating stochastic processes. Operational times are generated at random and may have an increasing failure rate. Repair times are generated from a random process where the repair time is related to the hazard rate at failure. This yields lengthened repair times at late stages in a system subject to an increasing failure hazard rate but also accommodates long repair times at young ages in systems with a bathtub‐shaped hazard rate function. We derive analytic results for a set of special cases of the model, show how simulation and inference can be carried out, and apply our method to real data from a large car manufacturer.

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“…A slight modification of the models introduced above yields an alternating Geometric Process (AGP) model . Let

{false{{normalX}_{normalk} false}}_{normalk = 1}^{\infty}

be a sequence of non‐negative random variables.…”

Section: Analytic Results and Special Casesmentioning

confidence: 99%

“…This gives the same inverse behaviour between operating and repair times that we expect: if

a > 1

with ever decreasing operating times, then

b = a^{- β}

leads to increasing repair times (assuming

β > 0

). Note that this is a special case of the AGP, with both baseline distributions being exponential …”

Section: Analytic Results and Special Casesmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

Section: Case 3: Connection To the Alternating Geometric Processmentioning

confidence: 99%

Section: Example 2: Car Warranty Claimsmentioning

confidence: 99%

See 3 more Smart Citations

Nonzero repair times dependent on the failure hazard

Arnold

Chukova

Hayakawa

et al. 2019

Quality & Reliability Eng

Self Cite

View full text Add to dashboard Cite

show abstract

Mean and variance of an alternating geometric process: An application in warranty cost analysis

Arnold

Chukova

Hayakawa

et al. 2021

Quality & Reliability Eng

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An alternating geometric process can be used to model the operational and repair times of an ageing system. In applications such as warranty cost analysis, the mean of an alternating geometric process (i.e. the expected number of events by a given time) and the variance are of interest. In this paper, two new approaches are proposed for computing the mean and variance functions of two counting processes related to the alternating geometric process, namely the number of cycles up to time t$t$ and the number of failures up to time t$t$. In warranty cost analysis, these approaches can be used to compute the expected number of claims and the expected cost over the warranty period. The usefulness of the proposed approaches in warranty cost analysis is demonstrated for a non‐renewing free‐repair warranty policy. The new approaches offer benefits over simulation in terms of computational time and accuracy.

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