Maintenance activities planning and grouping for complex structure systems

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1 Maintenance activities panning and grouping for compex structure systems Hai Canh u, Phuc Do an, Anne Barros, Christophe Berenguer To cite this version: Hai Canh u, Phuc Do an, Anne Barros, Christophe Berenguer. Maintenance activities panning and grouping for compex structure systems. 11th Internationa Probabiistic Safety Assessment and Management Conference & the Annua European Safety and Reiabiity Conference - PSAM 11/ESREL 2012, Jun 2012, Hesinki, Finand. pp.cdrom, <ha > HAL Id: ha https://ha.archives-ouvertes.fr/ha Submitted on 9 May 2012 HAL is a muti-discipinary open access archive for the deposit and dissemination of scientific research documents, whether they are pubished or not. The documents may come from teaching and research institutions in France or abroad, or from pubic or private research centers. L archive ouverte puridiscipinaire HAL, est destinée au dépôt et à a diffusion de documents scientifiques de niveau recherche, pubiés ou non, émanant des étabissements d enseignement et de recherche français ou étrangers, des aboratoires pubics ou privés.

2 Maintenance activities panning and grouping for compex structure systems Hai Canh u a, Phuc Do an b, Anne Barros a, Christophe Bérenguer c a Troyes university of technoogy, Troyes, France b CRAN, Lorraine university, Nancy, France c Grenobe institute of technoogy, Grenobe, France Abstract: This paper presents a dynamic grouping maintenance strategy for compex systems whose structure may ead to both positive and negative economic dependence which impy that combining maintenance activities is cheaper (or more expensive respectivey) than performing maintenance on components separatey. Binary Partice Swarm Optimization (BPSO) agorithm is used to find optima grouping panning which is NP-hard combinatoria probem. The proposed grouping maintenance strategy based on the roing horizon approach can hep to update the maintenance panning by taking into account short-term information which coud be changed with time. A numerica exampe of a 10 components system is finay introduced to iustrate the use and the advantages of the proposed approach in the maintenance optimization framework. Keywords: Grouping maintenance, Economic dependence, Muti-component system, Optimization. 1. INTRODUCTION For muti-component systems, economic dependence which can be either positive or negative have been considered in maintenance optimization. Positive economic dependence impies that costs can be saved when severa components are jointy maintained instead of separatey. Negative economic dependence between components occurs when maintaining components simutaneousy is more expensive than maintaining components individuay. A number of maintenance optimization methods and maintenance strategies have been proposed and deveoped in the iterature, see for exampe [1, 2, 3, 4]. In such papers, they focus mainy on series structures which ead directy to positive economic dependence [5, 6, 7, 8]. Recenty, in [9, 10], both positive and negative economic dependence have been studied by introducing an opportunistic maintenance which attempts to baance the negative aspects and the positive aspects of grouping maintenance. The proposed maintenance modes are however ony appicabe on imited cass of systems with particuar structure. From a practica point of view, with growing up quicky of industry, systems become more and more compex. Hence, the proposed modes may no onger be used. The grouping maintenance probem remains widey open. Moreover, in the grouping maintenance framework, static modes with an infinite panning horizon are usuay used in case of stabe situations [5, 6, 7]. Recenty, dynamic modes have been introduced in order to change the panning rues according to short-term information (e.g. faiures and varying deterioration of components) by using a roing horizon approach [8]. The primary objective of this paper is thus to deveop the roing horizon approach for grouping maintenance of compex structure systems with both positive and negative dependence. For such systems, to find optima grouping panning, dynamic programming (see [8, 3]) is no onger usabe since the combinatoria probem can be formuated as a set partitioning probem, which however can be NP-hard due to the compex structure impact. To this end, the BPSO [11, 12, 13], recognized as a genera search strategy which is simpe for impementation and very efficient for goba search, is used. This paper is organized as foows. Section 2 is devoted to the description of genera assumptions and grouping maintenance probem. The impact of system structure on grouping maintenance is aso anayzed. Section 3 focuses on the deveopment of the roing horizon approach in the context of compex structure systems. A simpe numerica exampe is introduced in Section 4. Finay, the ast section presents the concusions drawn from this work.

3 2. PROBLEM FORMULATION 2.1. Genera assumptions Consider a compex structure coherent system consisting of N independent components in which component state is either operationa or faiure one. For the rate of occurrence of faiures, denoted r i (t), we take a poynomia function with scae parameter λ i > 0, and shape parameter β i > 1: r i (t) = β i λ i ( t λ i ) β i 1 (1) Both corrective and preventive maintenance are considered for each component and we assume aso that preventive and corrective maintenance durations can be negected. According to the system structure, we consider two kinds of components: non-critica component (the system can be sti functioning when the component stops); critica component (a shutdown of the component for whatever raison eads to a shutdown of the system). By definition, maintenance of a critica component may ead to an additiona cost that coud be reying on, for exampe, production quaity oss or/and restart system costs... see [14]. Two kinds of shutdown costs are here considered:(i) panned shutdown cost, denoted Csys, p that is incurred when executing preventive maintenance on a critica component;(ii) unpanned shutdown cost, denoted Csys, u that is incurred when executing corrective maintenance on a critica component. Since the faiure is random and preventive maintenance is panned, an unpanned shutdown is therefore more costy than a panned shutdown, Csys c Csys. p For component i (i = 1,..., N), we assume that after a preventive maintenance (PM) action the component is considered as good as new. The preventive maintenance cost can be divided into three parts: a fixed setup cost, denoted S, represents the common cost for a preventive maintenance activities. The setup cost depends ony on the system. For exampe, the setup cost can be composed the cost of crew traveing and the preparation cost (erecting a scaffoding or opening a machine); a specific cost, denoted c p i, depending on the component; an additiona cost Cp sys is incurred if component i is critica. That mean that this additiona cost depends on the component position in the system structure and the system characteristics. Let π i is an indicator function (π i = 1 if component i is a critica one, and otherwise π i = 0 if component i is a not-critica one). As consequence, if a PM of component i is carried out, we have to pay the foowing cost: C p i = S + cp i + π i C p sys (2) To define the preventive maintenance cyce for each component, a ong-term average cost based approach [15] is used and presented in Section 3. Between two preventive maintenance intervas, if component fais, a minima repair is then immediatey performed to restore the component to an operationa state as bad as od. Let c r i denote the specific repair cost of component i. As in the PM case, when a corrective action is carried out for component i, it requires a corrective maintenance cost, denoted Ci c. Ci c = S + c r i + π i Csys u (3) 2.2. Grouping maintenance and the impact of the system structure In the spirit of grouping maintenance strategies see [8, 1], grouping maintenance can reduce the maintenance cost. The setup cost can be shared when severa maintenance activities are performed simutaneousy since simutaneous maintenance executions on severa (group) components require usuay ony one set-up cost [15, 3]. Note we, however that when a group of components are preventivey performed together, the maintenance cost coud be indirecty penaized with the reduction of components usefu ife if the maintenance dates are advanced and with the increasing of components faiure probabiity which coud impy a system immobiization if the maintenance dates are too ate. Moreover, the maintenance cost of a group may depend on the position of the group s components. We consider two kinds of groups: non-critica group (the system can be sti functioning when a

4 components of the group stop); critica group (the maintenance of the group eads to system stop). As consequence, maintenance on a critica group eads to a panned shutdown cost. Three foowing cases are considered: (i) if the critica group contains at east two critica components, the panned shutdown cost can then be shared; (ii) if the critica group contains one critica component, the panned shutdown cost remains then unchanged; (iii) if the critica group does not contain any critica component, the panned shutdown cost can then be penaized. In order to find the optima groups which coud baance to minimize the system maintenance costs on the scheduing horizon, the roing horizon approach introduced recenty in [8, 15], wi be deveoped by taking into account the system structure, see Section 3. An optimization agorithm, BPSO [11, 12], wi be aso used. 3. EXTENSION OF ROLLING HORIZON APPROACH The deveoped roing horizon approach consists of 5 phases: decomposition, tentative panning, economic profit formuations, grouping optimization using BPSO and roing horizon Phase 1: Decomposition For each individua component, we formuate an infinite horizon maintenance mode to obtain the optima individua preventive maintenance cyce that minimizes the component maintenance costs. Let M i (x) denote the expected deterioration cost incurred in x time units since the atest operation on component i. According to the minima repair poicy, M i (x) can be expressed as the foowing: M i (x) = C r i x From equations (1) and (4), we obtain: M i (x) = C r i ( x λ i ) β i 0 r i (y)dy. (4) If component i is preventivey maintained at x, the expected cost within interva [0, x] is: E i (x) = C p i + M i(x) = C p i + Cr i ( x λ i ) β i (5) By appying the renewa theory when executing the preventive maintenance of component i every x time units amount, the ong-term average cost of component i can be determined as the foowing: ϕ i (x) = E i(x) x = Cp i + Cr i ( x λ i ) β i x Since ϕ i (x) is a convex function, the optima interva ength for the preventive maintenances on component i, denoted x i, can be obtained by setting the derivative of ϕ i(x) to zero: x β i = arg min ϕ i (x) = λ i C p i i x Ci r(β i 1). (7) And the minima ong-term average maintenance cost of component i: (6) ϕ i = ϕ i (x i ) = Cp i β i x i (β i 1) (8) The minima ong-term average maintenance cost of the system assuming that a components are optimay and individuay maintained: N ϕ sys = ϕ i. (9) The optima interva ength x i which represents a nomina PM frequency of component i wi be used to define tentative execution times in the tentative panning phase. i=1

5 3.2. Phase 2: Tentative panning The idea of this phase is to estabish a tentative maintenance panning based on the individua preventive maintenance cyces. Let t begin denote the current date and i j be jth maintenance of component i since t begin and t e i is the operationa time eapsed from the ast preventive maintenance of component i before t begin. Without oss generaity we can set t begin = 0. Based on the individua maintenance rues of phase 1, the current state of component i, the tentative execution time of operation (or activity) i j, denoted t i j, is determined as foows, see [3]: t i 1 = t begin t e i + x i, t i j = t i j 1 + x i if j > 1 (10) In order to evauate the performance of grouping maintenance strategy, a finite panning horizon is usuay defined according to the current date t begin and the ending date t end which guarantees that a components are preventivey maintained at east one time in the horizon interva. It is shown in [3], t end can be determined as foows: t end = max t i 1. i After this phase, the tentative execution times of a individua maintenance activities in the scheduing horizon are identified. In the next phases we try to perform simutaneousy severa maintenance activities by changing their maintenance execution times to minimize tota maintenance cost Phase 3: Economic profit formuations The idea of this phase is to formuate the economic profits when preventive maintenance activities are simutaneousy carried out. To this end, we wi firsty find a cost-effective groups which can hep to identify secondy an optima grouping maintenance panning. Cost-effective group Assume now a group of severa different maintenance operations i j (i, j = 1, 2,...), denoted G k, are simutaneousy performed at time t. The corresponding economic profit of this group can be divided into three parts as foows: Since the execution of a group of m maintenance operations requires ony one set-up cost, the group G k yieds a cost reduction: U G k = (m 1) S (11) The penaty costs due to the changing of components execution maintenance date: H 1 G (t) = h k i (t t i j) = h i ( t i j), (12) i j G k i j G k h i ( t i j) is the penaty cost due to the movement execution date of the maintenance operation i j. Since operation i j is actuay executed at time t = t i j + t i j ( t i j > x i ) instead of t i j. h i ( t i j) = E i (x i + t i j) ( E i (x i )+ t i j.ϕ ) ( i = C r x i i + t ) i j βi C ( r x ) βi ti λ i i C p i j β i i λ i x i (β i 1) (13) The optima execution time of the group G k, denoted t G k, can be found when the H 1 (.) searches G k its minima vaue H 1. That is: G k H 1 G = H 1 k G (t k G k) = min h i (t G k t i j) (14) t i j G k If G k is a critica one, the grouping can conduct to an additiona cost that is expressed as foows: H 2 G k = C G k C notg k, (15) The tota panned shutdown costs when a components of the group are separatey maintained: C notg k = Csys p π i ; (16) i j G k

6 Let π G k is an indicator function (π G k = 1 if G k is a critique group, and π G k = 0 if G k is a not-critique group). The panned shutdown cost when a components of the group are performed together: Thanks to equations (15), (16) and (17), H 2 G k C G k = C p sys π G k (17) can be written as: H 2 G k = C p sys (π G k i j G k π i ). (18) Note we that if the group G k is not critica, a components of the group are then not critica, H 2 G k is hence equa to zero. If G k is a critica group, H 2 can be positive or negative depending on the G k number of critica components of the group (see again Section 2). From Equations (11), (14), and (18), the economic profit of group G k, denoted EP (G k ), can be written as the foowing: EP (G k ) = U G k H 1 G k H 2 G k (19) (i) If EP (G k ) < 0 the grouping maintenance is more expensive than performing maintenance on components separatey, i.e. the grouping impies a negative economic dependence. As consequence, the components of the group shoud not be simutaneousy maintained; (ii) If EP (G k ) 0, the maintenance cost can be saved when components are jointy maintained, i.e. grouping eads to a positive economic dependence. And the group is caed cost-effective. Grouping structure Based on a cost-effective groups, a grouping structure (or partition of a maintenance operations in the scheduing interva), denoted SGM, can be identified. As consequence, the tota economic profit is cacuated as foows: EP S(SGM) = G k SGM EP (G k ) = G k SGM (U G k H 1 G k H 2 G k ). (20) An optima grouping structure when the tota economic profit searches its maxima vaue: SGM optima = arg max EP S(SGM) (21) SGM Note we that the maximum number of groups to be considered is here 2 n 1 for n maintenance activities. The finding of optima grouping structure becomes difficut since the combinatoria probem can be formuated as a set partitioning probem, which however can be NP-hard. The dynamic programming agorithm proposed in [8, 3] can not therefore be used. To sove this probem, BPSO agorithm wi be used and described in next paragraph Phase 4: Grouping optimization using BPSO Binary partice swarm optimization was proposed by Kennedy and Eberhart in 1997, see [13], to sove optimization probems in binary search spaces (0 or 1). This method is oosey modeed on the focking behavior of birds. A soution to a specific probem is presented by a position of a partice. A swarm of fixed number of partices is generated and each partice is initiaized with a random position in the search space. Each partice fies through the search space with a veocity. At each iteration, a new veocity vaue of each partice is cacuated based on its current veocity, the distance from its persona best (the best position of partice so far), and the distance from the goba best (the best position of a partices). Once the veocity of each partice is updated, the partices are then moved to the new positions. The genera structure of BPSO is presented in Fig 1. Coding A coding phase determines how the probem is structured in the agorithm. In our probem, each grouping structure (a soution) is presented by a binary matrix of (n 1) rows and n coumns. Each row represents a group in which entry (k, j) = 1 if maintenance activity j is performed in group k and entry (k, j) = 0 for otherwise. For exampe considering a feasibe soution of 5 preventive

7 Start Coding Initiaize agorithm constants pmax, max, min, r1, r2 Initiaize randomy a swarm of partices and its veocities Evauate the swarm Update the veocity of each partice Update position for a partices No Evauate the swarm Stopping criterion satisfied? Yes Output resuts Stop Figure 1: The genera structure of BPSO maintenance activities contains of 3 groups G 1 = {1, 2}, G 2 = {3, 4}, G 3 = {5}. This soution can be presented by the matrix (22a) X = (22a) = (22b) (22) Create randomy a swarm of partices and its veocities A swarm of s partices is constructed for the BPSO. A sma number of partices in the swarm may resut in oca convergence, on the contrary, a arge number of partices wi increase computationa efforts and may make sow convergence. Therefore, the s is usuay chosen between 60 and 100. The initia position of the th partice is identified by a binary matrix, denoted X 0, whose eements are randomy assigned by 1 or 0. To adjust the position of partices, the initia veocities of a partices are aso randomy generated (1 0,..., s 0 ). eocity vaues are restricted to the imit vaues ( max, min ) to prevent the partice from moving too rapidy from one region in search space to another. We usuay set max = 4, min = 4. The initia veocity of the partice, denoted 0, is represented as matrix (22b). Evauate the swarm The goa of this step is to determine the persona best and goba best in the swarm. To do that, we must firsty evauate the position of each partice at iteration pth according to its fitness function which is defined as foows: { f(x p ) = if EP (G k ) < 0, G k X p EP S(X p ) otherwise The persona best, denoted ibest p, is the best position associated with the best fitness vaue of the partice obtained unti iteration p. ibest p is defined as foows: ibest p = arg max[f(x q )], where q = 1 p q

8 The goba best at iteration p, denoted gbest p, is the best position among a partices in the swarm, which is achieved so far and can be expressed as foows: gbest p = arg max[f(ibest p )], where = 1 s Update the veocity of each partice The objective of this step is to cacuate the new vaue of veocity of each partice to determine the next position of the partice. Based on the current veocity vaue, the persona best and the goba best, the veocity of each partice can be updated as foows: p+1 (k, j) = p (k, j) + r 1 U 1 [ibest p (k, j) Xp (k, j)] + r 2 U 2 [gbest p (k, j) X p (k, j)] U 1, U 2 are uniform random numbers between (0, 1); r 1, r 2 are socia and cognitive parameters which are taken as r 1 = r 2 = 2 consistent with the iterature, see [12]. In BPSO, the veocities of the partices are defined in terms of probabiities that a eement of matrix X p wi change to one. Using this definition a veocity must be restricted within the range [0, 1]. So whenever a veocity vaue is computed, the foowing piece-wise function, whose range is cosed interva [ min, max ], is used to restrict them to minimum and maximum vaue. max if p+1 (k, j) > max h( p+1 (k, j)) = p+1 (k, j) if p+1 (k, j) < max min if p+1 (k, j) < min After appying the piece-wise inear function, the foowing sigmoid function is used to scae the veocities between 0 and 1, which is then used for converting them to the binary vaues. sigmoid( p+1 (k, j)) = 1 + e 1 p+1 (k,j) Update position for a partices Based on new veocities obtained and et U is uniform random numbers between (0, 1), we update the new position for each partice by using the equation beow: { X p+1 1 if U < sigmoid( p+1 (k, j) = (k, j)) 0 otherwise Stopping criterion Stopping criterion is introduced to stop the agorithm process. Herein, imited iterations number (p max ) is used as a stopping criteria Phase 5: Roing horizon The maintenance manager can change the panning when some new information appears or when a panning for a new period and go back to step 4. Due to the previous phase, we have an optima grouping structure within the finite panning horizon [t begin, t end ]. However, with time some new information (ike maintenance resources constraints reying on for exampe management and new technoogy, opportunities,... ) by which this optima grouping structure can be impacted may become avaiabe. To update the maintenance panning, a new optima grouping structure within a new period must be identified. To this end, we simpy go back to phase NUMERICAL EXAMPLES The purpose of this section is to show how the proposed grouping maintenance strategy can be used in preventive maintenance optimization of compex systems. The impact of the system structure on the grouping maintenance panning wi be anayzed. A short comparison with previous grouping approach [8] wi be aso discussed. Consider a 10 components system whose structure is shown in Figure 2. When a component fais, it is immediatey maintained according to a minima-repair poicy. Corrective maintenance restores the component invoved into a state as bad as od. We assume faiure rate of a component i (i = 1,..., 10) is described by a Weibu distribution with scae parameter λ i > 0, and shape parameter β i > 1. Tabe 1 reports the random data for 10 components. For set-up cost and shutdown system costs, we take S = 10 and Csys p = 40, Csys u = 45 respectivey.

9 o o Figure 2: Structure system Tabe 1: Data of 10 components Component i λ i β i c p i c r i t e i Maintenance panning with negigibe system structure Consider now maintenance strategies in which the system structure is not considered. We assume that the system is stopped when performing maintenance on any component, i.e. π i = 1, i = (1,..., 10). Individua maintenance panning We consider an optima maintenance panning in which a components preventive maintenance are separatey performed. To define the optima preventive maintenance cyce of components, the ong-term maintenance cost mode was used. C p i, Cc i, x i, ϕ i and t 1 i are cacuated by substitution of the data input and π i in Equations (2),(3),(7),(6),(10) respectivey. A resuts are shown in tabe 2. Tabe 2: aues of C p i, Cc i, x i, ϕ i and t1 i with negigibe system structure Component i π i C p i Ci r x i ϕ i t 1 i The average maintenance cost of the system when a components are individuay maintained is finay: ϕ SG sys = N ϕ i = (23) i=1 Grouping maintenance panning To estabish an optima grouping maintenance, the proposed grouping maintenance strategy was used. Based on the components individua maintenance execution date, the scheduing horizon is [0, ] (t begin = 0, t end = max t 1 i = ). After 500 iterations, BPSO optimization agorithm provided an optima grouping maintenance panning. The resuts are reported in Tabe 3. The tota cost saving is EP S = which is equivaent to the average saving ϕ sys = EP S t end = = Hence, the average maintenance cost of the system with proposed grouping maintenance panning is: ϕ SG sys = ϕ SG sys ϕ sys = 7.41 (24) The resut shows that grouping maintenance is cheaper than performing maintenance on components separatey. Note we that under the assumptions above, the system can be considered as a series

10 structure for which the roing horizon approach [8] can be used. obtain the same resuts shown in Tabe 3. By appying this approach, we Tabe 3: Optima grouping panning with negigibe system structure Group Group components Group execution date (t G ) Group cost saving 1 1,2,3,4,5,6,7,8,9, Maintenance panning with taking into account the system structure Individua maintenance panning According to the system structure, indicator functions π i (i = 1,..., 10) can be easiy determined. In the same manner, C i p, C r i, x i, ϕ i, t1 i are then cacuated and showed in Tabe 4. The average maintenance cost of the system when a components are individuay maintained is finay: ϕ SG sys = N ϕ i = (25) i=1 The resut shows that the consideration of the system structure in maintenance can save the maintenance cost. If the system structure is taken into account, the individua maintenance panning is even cheaper than an optima grouping maintenance panning in which the system structure is ignored. Tabe 4: aues of individua optimization with taking into account the system structure Component i π i C p i Ci r x i ϕ i t 1 i Grouping maintenance panning In order to estabish a grouping maintenance panning, we used again the proposed grouping approach. According to the components individua maintenance execution date, the panning horizon is [0, 300]. Tabe 5 reports the resuts obtained by using BPSO optimization agorithm with 500 iterations. Tabe 5: Optima grouping panning with taking into account the system structure Group Group components Group execution date (t G ) Group cost saving 1 1,5, ,3,4,6, , The tota maintenance savings cost is EP S = = and the maintenance cost saving par time unit is: ϕ SG sys = ϕ SG sys = (26) 300 From (23), (24), (25) and (26), we obtain an important ranking for the average maintenance cost of the system: ϕ SG sys > ϕsys SG > ϕ SG sys > ϕ SG sys. The resuts assert that grouping can save the maintenance cost and that the system structure can take an important rue and shoud be taken into account in maintenance panning optimization. 5. CONCLUSIONS In this work, the roing horizon approach introduced recenty in [8] is deveoped for grouping maintenance panning of compex systems whose structure can take both positive and negative impact

11 on maintenance grouping. The numerica resuts show that taking into account the system structure into grouping maintenance procedure may reduce significanty maintenance cost. In order to find an optima grouping panning, Binary Partice Swarm Optimization agorithm is proposed to use. This optimization agorithm can hep to sove combinatoria grouping probem which can be NP-hard. Our future research work wi focus on the deveopment of the proposed method for systems which have the stochasticay and structuray dependent between components and aso with the taking into account the duration of maintenance activities. REFERENCES [1] R.P. Nicoai and R. Dekker. Optima maintenance of muti-component systems: a review. Compex system maintenance handbook, pages , [2] D.I. Cho and M. Parar. A survey of maintenance modes for muti-unit systems. European Journa of Operationa Research, 51(1):1 23, [3] P. Do an, K. Bouvard, F. Brissaud, A. Barros, and C. Berenguer. Dynamic grouping maintenance strategy with time imited opportunities. In Avance in Safety, Reiabiity and Risk Management - Proc.ESREL 2011, September 2011, Troyes, France., [4] K. Bouvard, S. Artus, C. Berenguer, and. Cocquempot. Condition-based dynamic maintenance operations panning & grouping. appication to commercia heavy vehices. Reiabiity Engineering & System Safety, [5] R. Dekker. Integrating optimisation, priority setting, panning and combining of maintenance activities. European Journa of Operationa Research, 82(2): , [6] R. Dekker, R.E. Wideman, and R. van Egmond. Joint repacement in an operationa panning phase. European Journa of Operationa Research, 91(1):74 88, [7] R. Laggoune, A. Chateauneuf, and D. Aissani. Preventive maintenance scheduing for a muticomponent system with non-negigibe repacement time. Internationa Journa of Systems Science, 41(7): , [8] R.E. Wideman, R. Dekker, and A. Smit. A dynamic poicy for grouping maintenance activities. European Journa of Operationa Research, 99(3): , [9] H. Pham and H. Wang. Optima (τ, t) opportunistic maintenance of a k-out-of-n: G system with imperfect pm and partia faiure. Nava Research Logistics (NRL), 47(3): , [10] Shey-Huei Sheu and Chung-Ming Kuo. Optimization probems in k-out-of-n systems with minima repair. Reiabiity Engineering and System Safety, 44(1):77 82, [11] J. Kennedy and R. Eberhart. Partice swarm optimization. In Neura Networks, Proceedings., IEEE Internationa Conference on, voume 4, pages IEEE, [12] N. Franken and AP Engebrecht. Investigating binary pso parameter infuence on the knights cover probem. In Evoutionary Computation, The 2005 IEEE Congress on, voume 1, pages IEEE, [13] J. Kennedy and R.C. Eberhart. A discrete binary version of the partice swarm agorithm. In Systems, Man, and Cybernetics, Computationa Cybernetics and Simuation., 1997 IEEE Internationa Conference on, voume 5, pages vo.5, oct [14] WMJ Geraerds. The cost of downtime for maintenance: preiminary considerations. Maint. Manage. Int., 5(1):13 21, [15] R. Dekker, J.R. an Der Meer, R.P. Pasmeijer, R.E. Wideman, and J.J. De Bruin. Maintenance of ight-standards-a case-study. Journa of the Operationa Research Society, pages , 1998.

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