Static and Dynamic Maintenance Policies for Repairable Systems under Imperfect Repair Models

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1 Static and Dynamic Maintenance Policies for Repairable Systems under Imperfect Repair Models Gustavo L. Gilardoni (UnB) Joint work with Maria Luiza G. de Toledo (ENCE-IBGE), Enrico Colosimo (UFMG) e Marta Freitas (UFMG) Brasília, March 17, 2016 Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 1 / 24

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3 Repairable systems research group Universidade de Brasília Gustavo L. Gilardoni, Cátia Gonçalves Universidade Federal de Minas Gerais Enrico Colosimo, Marta Freitas Universidade Federal de São Carlos Vera Tomazella, Afrânio Vieira ENCE-IBGE Maria Luiza G. de Toledo Former students Maristela de Oliveira (UFBA), Rodrigo Reis (UFMG), Israel Madureira (UnB), Fernando Oscar (UnB), Rafael Fernandes (UFMG), Cirdêmia Costa (UnB), etc. Current students Hemilhana Pereira (UnB), Rafael Fernandes (UFMG), Marco Pollo (UFSCar) etc. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 3 / 24

4 Porto de Galinhas, PE - 24 a 28 de agosto de 2015 CERTIFICADO A SOBRAPO - Sociedade Brasileira de Pesquisa Operacional, certifica que o trabalho Static and Dynamics Maintenance Policies for Repairable Systems under Imperfect Repair Models de autoria de Maria Luiza Guerra de Toledo, Marta Afonso Freitas, Enrico Colosimo e Gustavo Gilardoni e submetido por Maria Luiza Guerra de Toledo recebeu o prêmio ROBERTO DIÉGUEZ GALVÃO como melhor trabalho apresentado durante o XLVII SBPO - Simpósio Brasileiro de Pesquisa Operacional, realizado em Porto de Galinhas (PE) de 24 a 28 de agosto de Presidente da SOBRAPO Realização Patrocínio Apoio Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 4 / 24

5 Introduction Introduction Designing appropriate maintenance policies helps to maintain equipments running longer and minimize operation costs. Since Barlow and Hunter (1960), much of the literature regarding optimal maintenance of repairable systems has assumed both minimal repair (MR) and perfect maintenance (PM). However, since the MR assumption is too restrictive, attention has been given recently to imperfect repair (IR) models (Kijima, Morimura & Suzuki, 1988; Doyen & Gaudoin, 2004; Toledo et al., 2015). Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 5 / 24

6 Introduction Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 6 / 24

7 Counting Processes and MR Models Counting processes and MR models Let N(t) be the number of failures up to time t. In MR models, N(t) has independent increments and is usually modeled as an NHPP. Therefore, the optimal PM should determine an optimal PM periodicity τ P. Let λ(t), C R and C M be respectively the intensity of the process and the fixed costs of the MR and PM actions. Let C(τ) = C M + C R N(τ) be the total cost between two successive PMs performed τ units of time apart and H(τ) = C(τ)/τ be the corresponding cost per unit of time. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 7 / 24

8 Counting Processes and MR Models Since for an NHPP we have that E[N(t)] = Λ(t) = t λ(u)du, the expected 0 cost per unit of time is E[H(τ)] = [C M + C R Λ(τ)]/τ. Deriving and equating to zero we obtain that the optimal period τ P must be the solution of B(τ) := τλ(τ) Λ(τ) = C M C R, (1) or τ P = B 1 (C M /C R ), where λ(t) is increasing. This is the solution of Barlow & Hunter (1960). Further developments can be found among others in Glasser (1967); Bassin (1973); Gilardoni & Colosimo (2007, 2011) and Gilardoni, Oliveira and Colosimo (2013). This optimal policy is a periodical age-dependent policy. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 8 / 24

9 Counting Processes and IR Models Counting processes and IR models This maintenance policy can be extended for IR models since, from a purely mathematical point of view, an IR model is a general counting process with a random intensity λ(t). Let Φ(t) = E[Λ(t)] and φ(t) = Φ (t), where φ(t) is the rate of occurrence of failures (ROCOF) function, an unconditional version of the intensity λ(t). Now, the expected cost per unit of time becomes E[H(τ)] = [C M + C R Φ(τ)]/τ, and, hence, the optimal solution is τ P = B 1 (C M /C R ), where now B(t) = tφ(t) Φ(t) (compare with the previous PM). However, when one has access to the failure history of the equipment when deciding the maintenance policy, the previous solution is not optimal for IR models. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 9 / 24

10 Counting Processes and IR Models From a mathematical viewpoint, the reason is that, since the resulting process N(t) will not have independent increments, the failure history provides information about the future reliability of the system. Consequently, it should be taken into account when determining the optimal policy. Notwithstanding, most of the literature regarding IR models still focus on the idea of the optimal periodicity (cf. Kijima et al., 1988; Yevkin & Krivtsov, 2013). Hence, the objective is to understand how the information provided by the failure history of the system can be used to determine maintenance policies for IR models with improve the optimal periodical policy. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 10 / 24

11 Counting Processes and IR Models The dynamic approach The dynamic approach If we consider a process where new information is becoming available continuously over time and the corresponding PM policy is being revised accordingly, a PM will occur if and only if λ(t) = φ[b 1 (C M /C R )] (2) For a virtual age model with λ(t) = λ R [V (t)], (2) becomes V (t) = λ 1 R {φ[b 1 (C M /C R )]} := τ VA, (3) meaning that the system will be maintained whenever the virtual age reaches a threshold (i.e., the dyamic policy is periodical in the virtual age scale). Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 11 / 24

12 Counting Processes and IR Models The dynamic approach Implementation of the dynamic policy (2) is straightforward: one needs to compute (an estimate of) φ[b 1 (C M /C R )] and check it against (an estimate of) the intensity of the process. For instance, for ARA 1 model (Doyen & Gaudoin, 2004), we have λ(t) = λ R [t (1 θ)t N(t) ] and hence (2) becomes or, equivalently, λ R [t (1 θ)t N(t) ] = φ[b 1 (C M /C R )] (4) t t N(t) = θt N(t) + λ 1 R {φ[b 1 (C M /C R )]}. (5) In other words, at each failure the operator computes the right hand side of (5). This is the waiting lapse until the next PM provided that no new failure occur before it. On the contrary, if a new failure occurs before the PM is performed, (5) is computed again to obtain a new PM lapse and so on. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 12 / 24

13 Statistical inference for the ARA 1 model Statistical inference for the ARA 1 model Consider K independent, identical repairable systems, where n i failures are observed in the ith system (i = 1, 2,..., K). We denote by T i,j (i = 1, 2,..., K; j = 1, 2,..., n i ) the random variable representing the jth failure of the ith system and by t i,j its observed value. We consider a Power Law Process (PLP) reference intensity λ R (t) = (β/η)(t/η) β 1. The likelihood function is given by K n i L(µ) = λ R [t i,j (1 θ)t i,j 1 ] e Λ R [t i,j (1 θ)t i,j 1 ]+Λ R (θt i,j 1 ) i=1 j=1 e Λ R [t i (1 θ)t i,ni ]+Λ R (θc i ), (6) where c i = t i or c i = t i,ni depending on whether the ith system is time or failure truncated. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 13 / 24

14 Comparison with the periodical policy Comparison with the periodical policy A simulation study compared E[C(τ P )] and E[C(τ D )], using R. For each possible scenario, failure times were generated for N = 100, 000 systems according to an ARA 1 model with a PLP reference intensity. The 120 scenarios were defined by combining β = 1.5, 2.0, 2.5 and 3, θ = 0.1, 0.3, 0.5, 0.7 and 0, 9, C M CR = 1 3, 1 5 and 1 15 and both the periodic and the dynamic policies. The η value remained fixed at 15, 000 units of time. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 14 / 24

15 Comparison with the periodical policy Tabela : Mean cost per unit time point estimates (1) and standard errors (2) for the periodic and dynamic policies C M /C R = 1/3 C M /C R = 1/5 C M /C R = 1/15 β θ periodic Dynamic periodic Dynamic periodic Dynamic (4.41) 2.00 (4.10) 3.50 (10.7) 2.80 (7.69) 7.48 (42.3) 5.96 (29.5) (5.03) 1.80 (3.34) 2.89 (9.61) 2.42 (6.03) 5.18 (30.3) 4.21 (19.6) (4.78) 1.70 (2.63) 2.43 (8.02) 2.06 (4.65) 3.85 (22.9) 3.35 (13.6) (4.29) 1.51 (2.23) 2.12 (6.75) 1.86 (3.63) 3.12 (18.2) 2.70 (10.3) (5.51) 2.10 (4.71) 3.57 (10.9) 3.02 (8.87) 7.62 (43.4) 6.34 (32.6) (5.44) 1.90 (3.94) 2.91 (9.71) 2.46 (6.81) 5.12 (30.6) 4.41 (21.2) (4.88) 1.72 (3.17) 2.44 (8.12) 2.17 (5.24) 3.82 (22.8) 3.38 (15.2) (4.34) 1.57 (2.62) 2.11 (6.69) 1.95 (4.14) 3.10 (17.7) 2.81 (11.3) (5.93) 2.25 (5.36) 3.59 (11.6) 3.17 (9.87) 7.65 (42.2) 6.59 (35.4) (5.78) 2.01 (4.62) 2.91 (9.97) 2.60 (7.80) 5.11 (30.9) 4.52 (23.4) (5.07) 1.79 (3.78) 2.46 (8.17) 2.21 (6.18) 3.86 (23.0) 3.45 (17.1) (4.34) 1.64 (3.09) 2.13 (6.95) 1.94 (4.99) 3.13 (16.6) 2.82 (12.7) (6.51) 2.39 (5.99) 3.65 (12.0) 3.34 (10.7) 7.74 (46.0) 7.04 (39.6) (6.02) 2.11 (5.27) 2.96 (10.2) 2.73 (8.83) 5.26 (32.0) 4.72 (26.3) (5.19) 1.87 (4.40) 2.46 (8.46) 2.30 (6.95) 3.83 (22.2) 3.62 (18.6) (4.47) 1.70 (3.74) 2.14 (7.01) 2.02 (5.77) 3.18 (16.9) 3.00 (13.8) (6.65) 2.53 (6.50) 3.67 (11.8) 3.55 (11.8) 7.63 (44.0) 7.47 (43.1) (6.21) 2.25 (6.01) 2.98 (10.5) 2.90 (9.92) 5.18 (31.6) 5.08 (29.9) (5.25) 1.97 (5.04) 2.49 (8.55) 2.43 (8.06) 3.88 (23.7) 3.74 (21.4) (4.62) 1.78 (4.27) 2.15 (7.24) 2.10 (6.69) 3.13 (18.6) 3.03 (16.1) (1) reported values should be multiplied by 10 4 ; (2) Monte Carlo standard errors, reported values should be multiplied by 10 7 Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 15 / 24

16 Application Application: Engines of off-road trucks Off-road trucks: transport loose materials such as minerals and waste in mining operations. Good performance of this equipment is essential for the financial health of this kind of business. The company wanted to adopt a maintenance policy that favored preventive maintenance, as opposed to repair actions taken after failures. C M /C R = 1/1.23. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 16 / 24

17 Application Data set A sample of 193 engines, 52 being time truncated (PM) 208 failure times (in hours of operation) Number of failures Number of systems Total 193 Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 17 / 24

18 Application systems Expected Number of Failures times time Figura : (a) Failure and PM times (in hours of operation) for each engine; (b) mean cumulative number of failures. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 18 / 24

19 Application Tabela : Point and interval (95% confidence level) estimates for PLP (β, η) and effect of repair (θ) parameters, and values of the maximum of the log-likelihood function (ˆl), AIC e BIC under minimal and imperfect repair models for off-road data. Model: Minimal Repair Imperfect Repair (ARA 1 ) ˆβ 2.125(1.916; 2.357) 2.458(2.185; 2.765) ˆη 16, 715(15, 604; 17, 905) 15, 582(14, 601; 16, 628) ˆθ (0.330; 0.672) ˆl AIC BIC Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 19 / 24

20 Application Mean function hours of operation ROCOF 0e+00 6e hours of operation B(t) hours of operation Figura : Estimated functions for the engine data set: (a) Mean cumulative number of failures, (b) ˆφ(t 0) and (c) ˆB(t), versus t (in hours of operation). Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 20 / 24

21 Application Tabela : Estimations for the off-road engines data set, under different values of C M /C R : Optimal preventive PM period ( ˆB 1 (C M /C R )); Optimal time for preventive PM based on the dynamic approach (ˆτ VA ), and bootstrap (B = 10, 000) 95% confidence intervals (values are in hours) C M /C R ˆB 1 (C M /C R ) 95% CI ˆτ VA 95% CI 1/ , 815 (13, 632; 18, 082) 11, 373 (10, 978; 12, 023) 1/3 9, 207 (8, 608; 10, 173) 8, 141 (7, 572; 8, 647) 1/5 7, 500 (6, 720; 8, 125) 6, 537 (6, 043; 7, 097) 1/10 5, 593 (4, 847; 6, 227) 4, 694 (4, 403; 5, 404) 1/15 4, 621 (4, 017; 5, 386) 4, 031 (3, 656; 4, 596) Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 21 / 24

22 Final remarks Final remarks We proposed a dynamic method to estimate the optimal PM policy for repairable systems. This dynamic policy updates the optimal time continuously as the information provided by the failure history of the system becomes available. A simulation study showed that the expected operating cost per unit of time may be much lower when using the dynamic policy, especially when either (i) the cost ratio CM/CR is large or (ii) the effect of the repair is far from the MR case. The method was applied to a real data set. Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 22 / 24

23 Main references Main references Barlow, R. E. & Hunter, L. C. (1960). Optimum preventive maintenance policies. Operations Research, 8(1): Doyen, L. & Gaudoin, O. (2004). Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliability Engineering & System Safety, 84: Gilardoni, G. L., & Colosimo, E. A. (2007). Optimal maintenance time for repairable systems. Journal of Quality Technology, 39(1): Kijima, M., Morimura, H., & Suzuki, Y. (1988). Periodical replacement problem without assuming minimal repair. European Journal of Operational Research, 37: Toledo, M. L. G.; Freitas, M. A.; Colosimo, E. A. & Gilardoni, G. L. (2015). ARA and ARI imperfect repair models: estimation, goodness-of-fit and reliability prediction. Reliability Engineering & Systems Safety, 140: Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 23 / 24

24 Main references Thank you! Gustavo L. Gilardoni (UnB) Dynamic Maintenance Policies 24 / 24