APEM jowatal Advances in Production Engineering & Management Volume 11 | Number 1 | March 2016 | pp 38-48 http://dx.doi.Org/10.14743/apem2016.1.208 ISSN 1854-6250 Journal home: apem-journal.org Original scientific paper A bi-objective inspection policy optimization model for finite-life repairable systems using a genetic algorithm Ramadan, S.a* department of Mechanical and Industrial Engineering, Applied Science Private University, Shafa Badran, Amman, Jordan A B S T R A C T A R T I C L E I N F O This paper presents a bi-objective optimization model for finding the optimal number and optimal aperiodic times for the inspections of finite-life repairable systems when the availability of the component and the total maintenance cost are under consideration. The model utilizes the delay-time concept under perfect inspection assumption. The defect arrival process is modelled using the nonhomogeneous Poisson process and the failure times are probabilistic. The solution to this problem is NP-hard, therefore, a mutation-based genetic algorithm has been designed to solve the model. The effectiveness of the model was demonstrated using seven illustrative examples and compared to an existing classical periodic inspection model that uses a fixed number of inspections. The results showed that the proposed model did better (in all of the attributes) than the aperiodic model that using a fixed number of inspections. Furthermore, the results showed that the proposed model gave better results than a single-objective aperiodic model. The proposed model is a general model that can be implemented with different rates of occurrence of defects and different delay-time distributions. Also this model can be extended easily to cover complex systems and imperfect inspection cases. © 2016 PEI, University of Maribor. All rights reserved. Keywords: Maintenance Aperiodic inspection Periodic inspection Delay-time Multi-objective optimization Genetic algorithms *Corresponding author: s_ramadan@asu.edu.jo (Ramadan, S.) Article history: Received 14 September 2015 Revised 29 September 2015 Accepted 19 October 2015 1. Introduction As equipment age, the failure and deterioration related maintenance costs and interruptions increase; hence, the need for effective maintenance policies become more obvious. Traditionally, corrective maintenance is the most prevailing maintenance type practiced. It was estimated that 80 % of the industry dollars is spent on maintaining chronic failures of machines, systems, and people. Despite this huge figure, corrective maintenance cannot improve the reliability of the machines/systems as the maintenance action is taken after the failure. In the other hand, it was estimated that eliminating many of those chronic failures by implementing an effective maintenance policies can reduce this percentage between 40 % and 60 % [1]. Preventive maintenance (PM) is one of the most widely used maintenance types that can reduce the cost of maintaining machines and systems due to its ability of discovering hidden failures that may constitute up to 40 % of the failure modes in complex industrial systems [2]. Many inspection models were developed in literatures to optimize the inspection process in order to reduce the number of chronic failures. Earlier inspection models aimed to optimize the number of inspections per unit of time by minimizing the total downtime or maximizing the profit which were expressed as a function of number of inspections [3-7]. These models did not discuss the periodicity of the inspections but rather found the optimum number of inspections per unit of time. More recent inspection models were developed based on delay-time concept introduced by [8] which is very 38 A bi-objective inspection policy optimization model for finite-life repairable systems using a genetic algorithm similar to the Potential Failure interval in reliability centered maintenance developed later [2]. Delay-time concept divides the failure process into two stages: defect initialization stage and failure stage and defines the time elapsed between the defect initialization and the corresponding actual failure as the delay-time. This concept is very important in preventive maintenance PM because it shows that there is a time window (equals to the delay-time) that the maintenance crew can detect and fix the defect before it turns into a chronic failure. This concept inspired many researchers to develop optimization inspection models to reduce the number of chronic failures. The essence of those inspection models is to find the optimal periodicity of the inspections that will reduce the expected number of chronic failures. Christer et al. and Baker, used the delay-time concept in the industrial plant to find the periodicity of the inspections where the value of the delay-time was considered probabilistic [9-13]. Wang and Majid [14] used the concept of delay-time in offshore oil platform plant to optimize the periodicity of the inspections by minimize the system downtime. The work of Dawotola et al. [15] used the concept of delay-time in very long cross-country petroleum pipeline system where the periodicity of the inspections was determined by minimizing the total economic loss of failure while taking the human risk and maintenance budget as constraints. Abdel-Hameed [16] implemented increasing jump Markov process to optimize the periodicity of inspections. Okumura et al. [17, 18] proposed a stochastic-process free method for optimizing the discrete time point inspections for single unit system using stochastic processes. Wang [19] proposed two models one for single component and another one for complex component based on delay-time concept and in [20] the author extended the delay-time concept and instead of assuming that the failures can be detected only by inspections, he assumes that the failures can be revealed by themselves. Based on this extension, he proposed an inspection model for two types of inspections and repairs to determine the optimal constant periodicity of the inspections. Later, Wang et al. [21] extended the work of Wang [20] to multi-component multi-failure mode inspection model. Unfortunately, very little work was devoted to consider the multi-objective optimization of the inspection models under delay-time concept. Under delay-time concept, most of the literatures aimed to optimize the inspection policy based on a single objective namely, minimizing some form of maintenance cost [17, 22-27]. Other objectives are also found in the literatures such as maximizing the availability or the reliability of the system [6, 28, 29]. Few of the studies in the literatures considered both the number of inspections and the timing of these inspections in there models. The majority of them optimized either the number of inspection per unit time [4-7] or considered a constant number of inspections and optimized the times at which the inspections were made [30]. Moreover, a lot of the optimization models in the literatures were solved by a special designed algorithms that can be used only to the corresponding inspection model or algorithms that were time inefficient like enumeration. In this paper a bi-objective inspection optimization model is considered to optimize the number and the timing of inspections utilizing two objectives: maximizing the availability and minimizing the maintenance cost of the system. The model utilized the delay-time concept under perfect inspection assumption. The defect arrival process is modelled using nonhomogeneous Poisson process and the failure times are probabilistic. Genetic algorithm, which is a generic and efficient optimization algorithm, was used to optimize this model. The paper contains the following sections: Section 2 shows the notations and the assumptions of the model. Section 3 presents the model formulation based on delay-time concept. Section 4 presents the details of the genetic algorithm used. Section 5 presents the experimentations and discussion, and finally, section 6 concludes. 2. Assumptions and notations This section lists the assumptions and notations used in this paper. The following assumptions and notations can be explained on the light of Fig. 1 which shows a typical defect-failure -inspection relation under delay-time concept. Advances in Production Engineering & Management 11(1) 2016 39 Ramadan O Initial pMOt of a defect # Failure pouit | Defect identified 31 inspection O O Q à O |0 "i n¿ u^ t-i lUt^u* Time Fig. 1 The relationship between defects, failures, and inspections under the delay-time concept Consider a system with a finite life L, the objective of this proposed model is to find the optimal inspection policy; i.e., the optimal number of inspections n and the optimal timing for the inspections t to achieve the highest possible availability and the lowest possible maintenance cost Cm for the system. The assumptions underlying the proposed model are as follows: • The system is treated as single unit. • One mode of failures (defects) is analyzed and the defects are assumed to be independent. • The defects arise as a nonhomogenous Poisson process with Rate of Occurrence of Defects (ROCOD) A(u) at time u. • A failure happens after the initialization of a defect and the corresponding delay-time h is passed. • The delay-time distribution is independent of the time origin u. • The probability density function for the delay-time h is f(K) with cumulative density function F(h). • Inspections are carried out at t = {t1,t2,t3,^, tn}, hence the decision variables are t and n where t takes discrete values. • Only one type of inspection is considered and thus the inspections are identical. • Inspections are perfect in that all the defects present at the time of inspection will be recognized. • The mean inspection time is dins during which the system is down. • The mean time to rectify a defect is dr during which the system is down. • The mean time to repair a failure is df during which the system is down. • The average inspection cost is cins. • The average rectification cost is cr. • The average repairing cost is df. • E[Nd(ti_1,ti)] represents the expected number of defects in the interval (ti_1,ti). ti) |t(_i] represnts the expected number of failures in the interval (ti_1,ti). • E[Nr(ti-1,ti)] represents the expected number of rectified defects by inspection i at time • As denotes the nonparametric availability of the system during its life L. • Cm denotes the expected maintenance cost of the system during its life L. • Bm denotes the maintenance budget allocated for the system during its life L. • SLAgis the satisfaction level at As. • SLr is the satisfaction level at Cm. cm m 3. Model formulation Consider a nonhomogeneous defect arrivals process with arrival rate given by A(u), then the number of defects in the infinitesimal time S(u) is A(u)S(u). Integrating A(u)5(u) over the interval (ti_1,ti) gives the expected number of defects in that interval. Mathematically, the expected number of defects in the interval (ti_1,ti) is 40 Advances in Production Engineering & Management 11(1) 2016 A bi-objective inspection policy optimization model for finite-life repairable systems using a genetic algorithm E[Nd(ti_1,ti)]= i 1 A(u)du (1) Jti-1 The probability that any of these defects who arose in time u and is in the interval (ti_1,ti) will develop into a failure in the interval (u,u + 5(u)) is A(u)F(u)8(u). Integrating A(u)F(u)S(u) over the interval (ti_1,ti) will give the expected number of failures over that interval. Mathematically, the expected number of failures in the interval (ti_1,ti) is rti E[Nf(tí.1,tí)]= I A(u)F(t[ — Jti-1 u)du (2) Since perfect inspection is assumed, at the ith inspection which is conducted at time tu the expected number of rectifications is simply the difference between the expected number of defects arrived in the interval (ti_1,ti) and the expected number of defects developed into failures, i.e., the expected number of failures, in the same interval. Mathematically the expected number of rectifications in the interval (ti_1,ti) is E[Nr(ti)] = E[Nd(ti_1,ti)] -E[Nf(ti_1,ti)] (3) The nonparametric availability of the system can be seen as the ratio between the uptime and the down time. Mathematically the nonparametric availability As can be given as Uptime As = Uptime + Downtime (4) The uptime of the system is simply the life time of the system, L, minus the downtime of the system during the system's life. This means that the uptime plus the downtime is the L, the life of the system. The system downtime is calculated as the sum of four components, namely: the total expected rectification time corresponding to the n inspections; the total expected correction time corresponding to the n inspections, the total time for the n inspections, and finally, the expected correction time corresponding to the period between the last inspection time tn and the life of the system,!. Mathematically, the expected availability of the system during its life L can be given as As = L- [Tl=1(drE[Nd(ti.í,ti)] +dfE[Nf(ti.i,ti)]) +ndins + dfE[Nf(tn,L)]] (5 The system corrective maintenance cost during its life L, is also the sum of four components namely: the total expected rectification cost corresponding to the n inspections; the total expected correction cost corresponding to the n inspection periods, the total cost for the n inspections, and finally, the expected correction cost corresponding to the period between the last inspection time tn and the life of the system, L. Mathematically, the expected maintenance cost of the system during its life L can be given as n Cm = ^{crE[Nd(ti-1,ti)] + cfE[Nf(ti-1,ti)]) +ncins + cfE[Nf(L, tj] (6) i=i The two objective functions of the proposed model can be expressed as the total satisfaction level TSL about the inspection policy. The total satisfaction level can be calculated as the weighted average of the maintenance cost satisfaction level SLCm and the availability satisfaction level SLAs. To develop the two satisfaction levels, two membership functions were defined: one for As (Fig. 2) and one for Cm (Fig. 3). Advances in Production Engineering & Management 11(1) 2016 41 Ramadan Fig. 2 Membership function for the Fig. 3 Membership function for the C, Using those two membership functions, the SLAs and SLCjn are given by Si, = As SLr =1-^ (7) Bm (8) TSL is TSL = wSLAs + (1- w)SLCm (9) Putting all this together, gives the proposed inspection model as max TSL subject to Cm < Bm (10) — ti-1 ^^ins In this model the decision variables are the number of inspections,n, and the inspection times, t.The objective function of this mode will maximize the total satisfaction level for the inspection policy, (n, t), i.e, find (n, t) corresponding to the highest possible availability (highest SLAs) and lowest possible maintenance cost (highest SLCjn). The constraint >ti^1 + dins dictates that the ith inspection should be at least dins apart from the previous inspection, i.e., the inspection times are discrete. The constraint Cm 0 and exponential delay-time /(t) given by ye~yt, y > 0 are used traditionally in the literatures such as references [30-33]. For such ROCOD and f(t), the expected number of defects, rectifications, and failures are given as follows: E[Nd(ti_1,ti)] = Í ' aeßtdt = ^[e^ Jtí-! P ] (12) rti E[Nf(t¿_1,t¿)]= I A(u)F(t¿ —u)du = ''ti-i rti = 1 ae?u(l - e-Y^-^du Jtí-i -f = l[eßtl_eßtl.1]_ ßu -ae^e'Y^-^du a ~[eßtl -e/?tí"1] -ae~yti ae^e-vtie^du e(ß+Y)ti _e(ß+Y)ti-i (ß + Y) (13) EiNsití)] = E[Nd(ti.1,ti)] -E[Nf(ti.1,ti)] e(ß+Y)ti _e(iS+y)tí-i"l = ae -Yti (ß + Y) Table 1 shows the parameters used in the 7 examples. (14) Table 1 Parameters used in Examples 1-7 Example # À(u) m w Life (year) Bm 1 X(u) = 0.025e(18e"2)u f(h) = 0.0625e"°.°625h w = 0.5 20 $5.0E6 2 X{u) = 0.025 f(h) = 0.0625e-°.°625h w = 0.5 20 $5.0E6 3 Ku) = 0.025e(18e"2)u f(h) =0.1e"olh w = 0.5 20 $5.0E6 4 X{u) = 0.025 f{h) = 0.0625e"°.°625h w = 0.5 10 $5.0E6 5 X(u) = 0.025e(18e"2)u f(h) = 0.0625e-°.°625h w = 0,0 20 $5.0E6 6 À(u) = 0.025e(18e"2)u f(h) = 0.0625e-°0625h w = 1.0 20 $5.0E6 7 A(u) = 0.025e(18e"2)u f(h) = 0.0625e-°.°62Sh w = 0.5 20 $2.5E6 The results for the first example will be discussed in details to show how the model works. The results for the rest of the examples will be listed in Table 2 for comparison. Fig. 5 shows the evolution of the TSL values throughout the generations using the proposed model. The figure shows that the algorithm converged to a value of 0.9400. This convergence happened after 120 generations and stayed for the rest of the generations through the generation number 150. The processing time was 0.57 seconds with population size of 10 chromosomes and 150 generations. 44 Advances in Production Engineering & Management 11(1) 2016 A bi-objective inspection policy optimization model for finite-life repairable systems using a genetic algorithm Generation Number Fig. 5 The evolution of the TSL value throughout the generations using the proposed model The best inspection policy produced by the proposed model for Example 1 consisted of 123 inspections at the following timing (in days): 156 350 537 742 893 1095 1288 1437 1547 1703 1887 1960 2080 2165 2266 2341 2486 2530 2613 2678 2747 2839 2917 3016 3097 3170 3291 3365 3435 3509 3559 3627 3712 3775 3859 3924 3960 4025 4061 4140 4195 4261 4295 4355 4418 4460 4509 4571 4606 4682 4748 4820 4880 4921 4971 5001 5043 5085 5125 5170 5193 5254 5273 5332 5373 5417 5465 5524 5563 5613 5651 5696 5745 5769 5813 5834 5869 5891 5927 5952 5978 6011 6035 6064 6092 6103 6149 6180 6209 6236 6276 6317 6342 6364 6396 6426 6445 6467 6497 6514 6530 6571 6608 6625 6654 6681 6694 6735 6762 6800 6832 6846 6886 6911 6936 6982 6999 7020 7048 7075 7087 7112 7127 } Fig. 6 shows a histogram for the number of inspections in each of the twenty years. The histogram shows that the number of inspections increased with the life of the system. For example in the first 1200 days of the system life, the model suggested 6 inspections while in the last 1200 days of the system life the model suggested to have 51 inspections. This increase in the number of inspections coincides with the fact that the system is aging. As the system ages, the number of defects increases and the delay-time of the defects decreases which force the model to assign more inspections toward the end of the system life. Table 2 shows the results of the 7 examples for the proposed model along with the results for the aperiodic model with fixed number of inspections where the number of inspections was 30 inspections. The table shows that the proposed model is better, in all of the attributes, than the aperiodic model with fixed number of periods except for Example 2 where the number of inspections is equal. Basically, in Example 2, the two models are equivalent. Example 4 shows that the proposed model chose 12 inspections with lower maintenance cost and higher TSL than the aperiodic model with the fixed number of inspections 30. Moreover, the rest of the examples (except Example 2) show that even the number of inspections is higher in the proposed model than the number of inspections in the aperiodic model with fixed number of inspections, both the maintenance cost and the availability is better in the proposed model. These results show that treating the number of inspections as a variable, that need to be optimized in the inspection model, is better than treating it as a constant in the model as this will enhance the maintenance cost and the availability of the system simultaneously. 0.94 0.92 0.88 0.86 0.84 0.82 0 50 100 Advances in Production Engineering & Management 11(1) 2016 45 Ramadan 10 ¡p a 6 4 3000 4000 5000 System life (Days) Fig. 6 A histogram for the number of inspections in the twenty years of life o Comparing the results of Examples 1, 5, and 6 for the proposed model, one can see that the number of inspections chosen by the model is significantly different In Example 5, where the objective of the model was to maximize the availability of the system alone, the number of inspections was significantly higher than the number of inspections in Example 6 where the objective was to minimize the maintenance cost only. Moreover the TSL for example 5 was lower than the TSL in Example 6. The average of TSLs of Example 5 and Example 6 is almost the same as the TSL in Example 1 where the two objectives were considered. Moreover the average number of inspections for the two examples was almost the same as the number of inspections in Example 1 but the average cost of the two examples was higher than the average cost in Example 1. To better understand what happened in Examples 1, 5, and 6 and why it happened. Consider Fig. 7 which shows the relation between the As and Cm. The figure shows that there may be more than one value of As for the same value of Cm. This result can be understood on the light of that different inspection policies may have the same cost but different effect on the availability of the system. For this reason, it is not wise to use maintenance cost as the only objective in the inspection models. On the same taken, using availability as the only objective in the inspection model may result in choosing an expensive inspection policy when we can have the same availability using other inspection policies that have lower costs. This result emphasizes the importance of treating the inspection-policy optimization problem as a multi-objective optimization problem rather than a single objective problem. By comparing the results of Example 1 and the results of Example 3, it is easy to see that the increase in the delay-time rate caused an increase in the number of inspections (to increase the availability of the system) but this increase also increased the maintenance cost, the matter that caused a decrease in the TSL. This result is expected because the increase in the delay-time rate means that the defects will turn into failures faster and thus more inspections are needed to prevent the defects from turning into failures and hence reducing the availability of the system. Table 2 The results of the 7 examples for the proposed model along with the results for the aperiodic model with 30 inspections Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Results for the proposed model As,SLAs 0.9305 0.9903 0.9231 0.9908 N/A 0.9313 0.9295 r 2.53e+05 4.58e+04 2.99e+05 2.298e+04 2.52e+05 2.67e+05 2.54e+05 Sir 0.9495 0.9908 0.9401 0.9954 0.9496 N/A 0.8985 TSL 0.9400 0.9906 0.9316 0.9931 0.9496 0.9313 0.9140 n 120 30 134 12 138 91 127 Results for the aperiodic model with fixed number of inspections (30 inspections) As,SLAs 0.9096 0.9898 0.8882 0.9875 N/A 0.9085 0.9086 r 4.75e+05 4.87e+04 6.48e+05 2.57e+04 4. 80e+05 4.89e+05 4.84e+05 SLr 0.9050 0.9903 0.8703 0.9949 0.9040 N/A 0.8064 TSL 0.9073 0.9900 0.8792 0.9912 0.9040 0.9085 0.8575 n 30 30 30 30 30 30 30 46 Advances in Production Engineering & Management 11(1) 2016 A bi-objective inspection policy optimization model for finite-life repairable systems using a genetic algorithm 0.86r 0.850.84-y 0.830.820.81 -0.80.79-0.78^ "'s* .'""S» 1.1 1.2 1.3 Maintenance Cost ($) 1.4 1.5 1.6 x 106 Fig. 7 The relation between the and C„ The proposed model responded to the increase in the delay-time rate by increasing the number of inspections but this also increased the maintenance cost as well. The model chose the optimal inspections number that compromised between the availability of the system and the maintenance cost of the system 6. Conclusion In this paper an aperiodic inspection model is proposed and solved using mutation-based Genetic algorithm. The proposed inspection model is based on delay-time concept and nonhomogene-ous Poisson process of defect arrivals rather than renewal theory and periodic inspection modelling that are used classically. The proposed model also optimizes the number of inspections and the timing of inspections simultaneously rather than optimizing either the number of inspections or the timing of inspections as in the case of the majority of the available inspection models. Moreover, the proposed model uses two objectives, namely: system availability and maintenance cost, to optimize the inspection policy whereas the available inspection models use only one objective. The results showed that the proposed model is better (in all of the attributes) than the aperiodic model that uses fixed number of inspections. Moreover, the results showed that using two objectives (system availability and maintenance cost) in the inspection models rather than one objective, can improve the quality of the inspection policy in terms of system availability and maintenance cost. The proposed model is a general model that can be implemented with different ROCOD and different delay-time distributions. Also this model can be extended easily to cover complex systems and imperfect inspection cases. Acknowledgement The author is grateful to the Applied Science Private University, Amman, Jordan, for the full financial support granted to this research (Grant No. DRGS-2015). References [1] Dhillon, B.S. (2002). Engineering maintenance: A modern approach, CRC press, Boca Raton, USA, doi: 10.1201/ 9781420031843. [2] Moubray J. (1997). 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