A Preventive Maintenance Model Subject to Sequential Inspection for a Three-Stage Failure Process

doi:10.23940/ijpe.19.01.p8.7687

Abstract

Abstract:

For a system subject to gradual degradation, it can be in one of three different states: normal, minor defective, or severe defective stages, which can overall be referred to as a three-stage failure process. For this system, periodic inspection may not be the most ideal policy.Fewer inspections will lead to lower costs if the system is in the normal stage, but if the system is in the defective stage, frequent inspections are recommended to prevent failure. Therefore, it may be more economical to take the sequential time${{T}_{j}}(j=0,1,2,\cdots )$as the inspection interval. In view of this, a preventive maintenance (PM) model subject to sequential inspection for a three-stage failure process is proposed. Two-level sequential inspections, postponed maintenance and opportunistic maintenance (OM), are introduced into the PM model. The minor inspection is taken at the successive time${{S}_{j}}(j=0,1,2,\cdots )$, where${{S}_{j}}=\sum\limits_{k=0}^{j}{{{T}_{k}}}$, ${{T}_{0}}=0$, and${{S}_{0}}=0.$ Minor inspection is an imperfect inspection that can identify the minor defective stage with a certain probability but can reveal other two stages completely. The nth major inspection is taken to substitute for the A_n ^{th$(n=0,1,2,\cdots )$} minor inspection, where ${{A}_{n}}=\sum\limits_{k=0}^{n}{{{N}_{k}}}$,${{N}_{0}}=0$, and${{A}_{0}}=0.$ Major inspection is a perfect inspection that can distinguish the state of the system perfectly. Once the severe defective stage is identified, the inspection is stopped and the maintenance action is postponed to the next OM if the time to the next OM is less than a threshold level; otherwise, the system is maintained immediately. A numerical example is given to demonstrate the proposed model by comparing with other models and analysing the influence of the parameters on the expected cost.

Key words: preventive maintenance, three-stage failure process, sequential inspection, postponed maintenance, opportunistic maintenance

Xiaoxiao Cao, Chao Guo, Huasheng Xiong, Duo Li, and Xiaojin Huang. A Preventive Maintenance Model Subject to Sequential Inspection for a Three-Stage Failure Process [J]. Int J Performability Eng, 2019, 15(1): 76-87.

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Figures/Tables 9

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${{X}_{m}}$	Random variables representing the durations of the m^th stage of the system $(m=1,2,3)$
${{f}_{{{X}_{m}}}}\left( x \right)$	Probability density function (pdf) of ${{X}_{m}}$
${{T}_{k}}$	Interval from (k-1)^thminor inspection to k^th minor inspection
${{N}_{n}}$	There are ${{N}_{n}}$minor inspection intervals between (n-1)^thmajor inspection and n^thmajor inspection, ${{N}_{0}}=0$
$T$	Minor inspection intervals sequence $T=\left\{ {{T}_{1}},{{T}_{2}},\cdots ,{{T}_{k}},\cdots \right\}$
$N$	Major inspection intervals sequence $N=\left\{ {{N}_{1}},{{N}_{2}},\cdots ,{{N}_{n}},\cdots \right\}$
${{S}_{j}}$	Minor inspection is implemented at successive time${{S}_{j}}$,${{S}_{j}}=\sum\limits_{k=0}^{j}{{{T}_{k}}}$,${{T}_{0}}=0$,${{S}_{0}}=0$
${{A}_{n}}$	The n^th major inspection happens at the time whenA_n^thminor inspection should be taken, ${{A}_{n}}=\sum\nolimits_{k=0}^{n}{{{N}_{k}}}$
${{T}_{OM}}$	The OM interval
$\tau $	Random time to the next OM
$t$	The threshold deciding whether to wait for OM
$\alpha $	Probability of minor inspection detecting minor defective stage
${{T}_{cr}}$	Random time when the corrective replacement happens for the failed system
${{T}_{ir}}$	Random time when the inspection replacement is taken for the defective system
${{T}_{or}}$	Random time when the opportunistic replacement is implemented for the defective system
${{c}_{mi}}$	Cost of a minor inspection
${{c}_{ma}}$	Cost of a major inspection
${{c}_{cr}}$	Cost of a corrective replacement
${{c}_{ir}}$	Cost of an inspection replacement
${{c}_{or}}$	Cost of an opportunistic replacement

${{\lambda }_{1}}$	15.83	${{k}_{1}}$	1.47
${{\lambda }_{2}}$	9.61	${{k}_{2}}$	1.95
${{\lambda }_{3}}$	6.02	${{k}_{3}}$	2.81
$\alpha $	0.4	${{c}_{mi}}$	8
${{c}_{ma}}$	20	${{c}_{cr}}$	3000
${{c}_{ir}}$	400	${{c}_{or}}$	200
${{T}_{om}}$	200

	$t$	${{N}_{1}}(N)$	${{N}_{d}}$	${{T}_{q}}$	${{N}^{0}}$	${{T}^{0}}$
Proposed model	2	10	1	0.8	3	2
Model1	2	3	--	--	--	--
Model2	2	--	--	0.8	--	2
Model3	0	10	1	0.8	3	2

${{T}_{OM}}$	$t$	$T_{1}^{*}$	${{c}^{*}}$	${{T}_{OM}}$	$t$	$T_{1}^{*}$	${{c}^{*}}$
200	0	5	21.6327	150	2	5	21.5905
	1		21.5926	100		5	21.5744
	2		21.5767	50		5	21.5149
	3		21.6317	40		5	21.4559
	4		21.8111	30		5	21.4215
	5		22.0624	20		5	21.3141.
	10		24.3215	10		4	20.8560