AGE-BASED REPLACEMENT WITH IMPERFECT REPAIR AND RISK AS LOW AS REASONABLY PRACTICABLE

Araújo, Joyce; Lopes, Rodrigo; Scarf, Philip

doi:10.1590/0101-7438.2022.042nspe1.00263588

Joyce Araújo ^**Corresponding author

RANDOM - Research Group on Risk and Decision Analysis in Operations and Maintenance, Management Engineering Department, Federal University of Pernambuco, Rodovia BR-104, Km 59, s/n, Nova Caruaru, 55002-970 Caruaru, PE, Brazil - E-mail: joyce.danielle.araujo@gmail.com

http://orcid.org/0000-0002-8015-0011

Rodrigo Lopes

RANDOM - Research Group on Risk and Decision Analysis in Operations and Maintenance, Management Engineering Department, Federal University of Pernambuco, Rodovia BR-104, Km 59, s/n, Nova Caruaru, 55002-970 Caruaru, PE, Brazil - E-mail: r.s.lopes@random.org.br

http://orcid.org/0000-0001-5339-3301

Philip Scarf

RANDOM - Research Group on Risk and Decision Analysis in Operations and Maintenance UFPE, Cardiff Business School, Cardiff University, Cardiff, CF10 3AT, UK - E-mail: ScarfP@cardiff.ac.uk

http://orcid.org/0000-0001-5623-906X

*Corresponding author

C ^′ _r	C ^′ _p	C ^′ _m	v	p _A	PLL _A	β	θ	p ₁	r ₀	r ₁	u
1	2	0.25	100	0.1	0.01	2	1	0.9	10⁻⁵	10⁻³	1

Changes (%)		p = 0, 8				p = 0.9				p = 1
		Fixed VPF		Crescent VPF		Fixed VPF		Crescent VPF		Fixed VPF		Crescent VPF
		T _f ^∗	B(T _f ^∗ )	T _c ^∗	B(T _c ^∗ )	T _f ^∗	B(T _f ^∗ )	T _c ^∗	B(T _c ^∗ )	T _f ^∗	B(T _f ^∗ )	T _c ^∗	B(T _c ^∗ )
C ^′ _m	-50	1.10	2.073	1.05	2.286	1.05	2.174	0.95	2.390	1.05	2.270	0.95	2.479
	-25	1.10	2.085	1.05	2.297	1.05	2.179	0.95	2.395	1.05	2.270	0.95	2.479
	+25	1.10	2.108	1.05	2.320	1.05	2.190	0.95	2.406	1.05	2.270	0.95	2.479
	+50	1.10	2.120	1.05	2.331	1.05	2.196	0.95	2.411	1.05	2.270	0.95	2.479
C ^′ _p	-50	2.00	1.205	2.00	1.440	2.00	1.224	2.00	1.416	2.00	1.245	2.00	1.398
	-25	1.55	1.686	2.00	1.930	1.50	1.741	2.00	1.940	1.50	1.795	2.00	1.954
	+25	0.90	2.445	0.80	2.631	0.85	2.560	0.80	2.745	0.85	2.669	0.75	2.852
	+50	0.80	2.751	0.70	2.908	0.75	2.888	0.70	3.045	0.70	3.019	0.65	3.174
C ^′ _r	-50	0.60	1.686	0.55	1.784	0.60	1.762	0.55	1.865	0.55	1.838	0.50	1.935
	-25	0.85	1.942	0.70	2.110	0.80	2.028	0.70	2.196	0.80	2.111	0.70	2.280
	+25	1.45	2.177	2.00	2.430	1.40	2.263	2.00	2.472	1.40	2.346	2.00	2.516
	+50	2.00	2.205	2.00	2.440	2.00	2.288	2.00	2.480	1.95	2.368	2.00	2.521
PLL _A	-50	1.15	2.049	1.05	2.104	1.10	2.139	1.05	2.195	1.05	2.226	1.00	2.280
	-25	1.10	2.073	1.05	2.193	1.10	2.162	1.00	2.286	1.05	2.248	1.00	2.368
	+25	1.10	2.120	1.05	2.450	1.05	2.207	0.85	2.534	1.00	2.291	0.90	2.613
	+50	1.05	2.143	0.75	2.592	1.05	2.229	0.75	2.681	1.00	2.312	0.80	2.763
p _A	-50	1.15	2.049	1.05	2.104	1.10	2.139	1.05	2.195	1.05	2.226	1.00	2.280
	-25	1.10	2.073	1.05	2.193	1.10	2.162	1.00	2.286	1.05	2.248	1.00	2.368
	+25	1.10	2.120	1.05	2.450	1.05	2.207	0.85	2.534	1.00	2.291	0.90	2.613
	+50	1.05	2.143	0.75	2.592	1.05	2.229	0.75	2.681	1.00	2.312	0.80	2.763
v	-50	1.15	2.049	1.05	2.158	1.10	2.139	1.05	2.251	1.05	2.226	1.00	2.334
	-25	1.10	2.073	1.05	2.233	1.10	2.162	1.00	2.328	1.05	2.248	0.95	2.407
	+25	1.10	2.120	1.05	2.383	1.05	2.207	0.90	2.472	1.00	2.291	0.90	2.549
	+50	1.05	2.143	0.85	2.458	1.05	2.229	0.85	2.541	1.00	2.312	0.85	2.619
θ	-50	0.55	4.193	0.35	4.986	0.55	4.370	0.30	5.310	0.50	4.540	0.25	5.730
	-25	0.85	2.796	2.00	3.149	0.80	2.913	2.00	3.195	0.75	3.027	2.00	3.255
	+25	1.40	1.677	1.10	1.817	1.35	1.748	1.05	1.888	1.30	1.816	1.05	1.956
	+50	1.65	1.398	1.35	1.509	1.60	1.456	1.35	1.568	1.55	1.513	1.30	1.624
β	-25	2.00	2.080	2.00	2.251	2.00	2.201	2.00	2.351	2.00	2.320	2.00	2.449
	+25	0.90	2.016	0.75	2.179	0.85	2.083	0.75	2.242	0.85	2.146	0.75	2.303
	+50	0.80	1.925	0.75	2.040	0.80	1.976	0.70	2.090	0.80	2.027	0.70	2.137

		p = 0.8		p = 0.9		p = 1
		T ^∗	B(T ^∗ )	T ^∗	B(T ^∗ )	T ^∗	B(T ^∗ )
Infinite-Horizon	Without ALARP	1.20	2.001	1.15	2.093	1.10	2.182
	Fixed VPF	1.10	2.097	1.05	2.185	1.05	2.270
	Increasing VPF	1.05	2.308	0.95	2.400	0.95	2.479
One-replacement-cycle	Without ALARP	0.80	2.966	0.75	3.126	0.70	3.278
	Fixed VPF	0.75	3.084	0.70	3.244	0.65	3.398
	Increasing VPF	0.70	3.244	0.60	3.482	0.55	3.643

β	π	VPF	p₂ = 0.9		p₂ = 0.95		p₂ = 1
			T ^∗	B(T ^∗ )	T ^∗	B(T ^∗ )	T ^∗	B(T ^∗ )
β = 2	π1 = 0.5 e π2 = 0.5	Fixed	1.10	2.139	1.10	2.159	1.10	2.178
	π1 = 0.5 e π2 = 0.5	Increasing	1.05	2.356	0.95	2.376	0.95	2.393
	π1 = 0.4 e π2 = 0.6	Fixed	1.10	2.148	1.10	2.173	1.05	2.196
	π1 = 0.4 e π2 = 0.6	Increasing	1.05	2.366	0.95	2.388	0.95	2.409
β = 2.5	π1 = 0.5 e π2 = 0.5	Fixed	0.90	2.049	0.90	2.064	0.90	2.080
	π1 = 0.5 e π2 = 0.5	Increasing	0.75	2.210	0.75	2.225	0.75	2.239
	π1 = 0.4 e π2 = 0.6	Fixed	0.9	2.056	0.90	2.075	0.85	2.093
	π1 = 0.4 e π2 = 0.6	Increasing	0.75	2.217	0.75	2.234	0.75	2.251
β = 3	π1 = 0.5 e π2 = 0.5	Fixed	0.80	1.950	0.80	1.962	0.80	1.974
	π1 = 0.5 e π2 = 0.5	Increasing	0.70	2.065	0.70	2.077	0.70	2.088
	π1 = 0.4 e π2 = 0.6	Fixed	0.80	1.955	0.80	1.970	0.80	1.984
	π1 = 0.4 e π2 = 0.6	Increasing	0.70	2.070	0.70	2.084	0.70	2.097

B (T) = \frac{[C'_{m} + p_{A} P L L_{A} d (R (T)) v] \int_{0}^{T} \bar{H} (t) r_{G} (t) d t + [C'_{p} + p_{A} P L L_{A} d (R (T)) v] H (T) + C'_{r} \bar{H} (T)}{\int_{0}^{T} \bar{H} (t) d t}

E [W (T)] = C_{p} \int_{0}^{T} \frac{1}{t} d H (t) + \frac{C_{r} \bar{H} (T)}{T} + \frac{C_{m} \int_{0}^{T} r_{G} (t) d t}{T} - C_{m} \int_{0}^{T} [\frac{H (t)}{t} r_{G} (t) - \frac{1}{t} \int_{0}^{t} r_{G} (z) d z] d t

E [W (T)] = (C'_{p} + p_{A} P L L_{A} d (R (T)) v) \int_{0}^{T} \frac{1}{t} d H (t) + \frac{C_{r} \bar{H} (T)}{T} + \frac{(C'_{m} + p_{A} P L L_{A} d (R (T)) v) \int_{0}^{T} r_{G} (t) d t}{T} - (C'_{m} + p_{A} P L L_{A} d (R (T)) v) \int_{0}^{T} [\frac{H (t)}{t} r_{G} (t) - \frac{1}{t} \int_{0}^{t} r_{G} (z) d z] d t

E [W (T)] = (C'_{p} + p_{A} P L L_{A} v) \int_{0}^{T} \frac{1}{t} d H (t) + \frac{C_{r} \bar{H} (T)}{T} + \frac{(C'_{m} + p_{A} P L L_{A} v) \int_{0}^{T} r_{G} (t) d t}{T} - (C'_{m} + p_{A} P L L_{A} v) \int_{0}^{T} [\frac{H (t)}{t} r_{G} (t) - \frac{1}{t} \int_{0}^{t} r_{G} (z) d z] d t

r_{H} (t) = \frac{β}{θ} t^{β - 1} (\frac{p_{1} π_{1} e^{- p_{1} {(\frac{t}{θ})}^{β}} + p_{2} π_{2} e^{- p_{2} {(\frac{t}{θ})}^{β}}}{π_{1} e^{- p_{1} {(\frac{t}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{t}{θ})}^{β}}})

r_{G} (t) = \frac{β}{θ} t^{β - 1} (\frac{(1 - p_{1}) π_{1} e^{- (1 - p_{1}) {(\frac{t}{θ})}^{β}} + (1 - p_{2}) π_{2} e^{- (1 - p_{2}) {(\frac{t}{θ})}^{β}}}{π_{1} e^{- (1 - p_{1}) {(\frac{t}{θ})}^{β}} + π_{2} e^{- (1 - p_{2}) {(\frac{t}{θ})}^{β}}})

R (T) = \{[1 - (π_{1} e^{- p_{1} {(\frac{T}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{T}{θ})}^{β}})] (1 + ⌊\frac{u}{T}⌋) - 1 - (π_{1} e^{- p_{1} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}})\} p_{A} P L L_{A}

h (t) = \frac{β}{θ} t^{β - 1} (p_{1} π_{1} e^{- p_{1} {(\frac{t}{θ})}^{β}} + p_{2} π_{2} e^{- p_{2} {(\frac{t}{θ})}^{β}} + p_{3} π_{3} e^{- p_{3} {(\frac{t}{θ})}^{β}} + p_{4} π_{4} e^{- p_{4} {(\frac{t}{θ})}^{β}})

r_{H} (t) = \frac{β}{θ} t^{β - 1} (\frac{p_{1} π_{1} e^{- p_{1} {(\frac{t}{θ})}^{β}} + p_{2} π_{2} e^{- p_{2} {(\frac{t}{θ})}^{β}} + p_{3} π_{3} e^{- p_{3} {(\frac{t}{θ})}^{β}} + p_{4} π_{4} e^{- p_{4} {(\frac{t}{θ})}^{β}}}{π_{1} e^{- p_{1} {(\frac{t}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{t}{θ})}^{β}} + π_{3} e^{- p_{3} {(\frac{t}{θ})}^{β}} + π_{4} e^{- p_{4} {(\frac{t}{θ})}^{β}}})

r_{G} (t) = \frac{β}{θ} t^{β - 1} (\frac{(1 - p_{1}) π_{1} e^{- (1 - p_{1}) {(\frac{t}{θ})}^{β}} + (1 - p_{2}) π_{2} e^{- (1 - p_{2}) {(\frac{t}{θ})}^{β}} + (1 - p_{3}) π_{3} e^{- (1 - p_{3}) {(\frac{t}{θ})}^{β}} + (1 - p_{4}) π_{4} e^{- (1 - p_{4}) {(\frac{t}{θ})}^{β}}}{π_{1} e^{- (1 - p_{1}) {(\frac{t}{θ})}^{β}} + π_{2} e^{- (1 - p_{2}) {(\frac{t}{θ})}^{β}} + π_{3} e^{- (1 - p_{3}) {(\frac{t}{θ})}^{β}} + π_{4} e^{- (1 - p_{4}) {(\frac{t}{θ})}^{β}}})

R (T) = \{\begin{matrix} [1 - (π_{1} e^{- p_{1} {(\frac{T}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{T}{θ})}^{β}} + π_{3} e^{- p_{3} {(\frac{T}{θ})}^{β}} + π_{4} e^{- p_{4} {(\frac{T}{θ})}^{β}})] (1 + ⌊\frac{u}{T}⌋) \\ - 1 - (π_{1} e^{- p_{1} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}} + π_{2} e^{- p_{2} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}} + π_{3} e^{- p_{3} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}} + π_{4} e^{- p_{4} {(\frac{T - (u - ⌊\frac{u}{T}⌋ T)}{θ})}^{β}}) \end{matrix}\} p_{A} P L L_{A}

[1] *Corresponding author

Brasil

Brasil

AGE-BASED REPLACEMENT WITH IMPERFECT REPAIR AND RISK AS LOW AS REASONABLY PRACTICABLE

ABSTRACT