International Journal of Automation and Computing
6(4), November 2009, 364-378 DOI: 10.1007/s11633-009-0364-4
On Problems of Multicomponent System Maintenance Modelling Tomasz Nowakowski
Sylwia Werbi´ nka∗
Department of Logistics and Transportation Systems, Wroclaw University of Technology, Wroclaw, Poland
Abstract: We present an overview of some recent developments in the area of mathematical modeling of maintenance decisions for multi-unit systems. The emphasis is on three main groups of multicomponent maintenance optimization models: the block replacement models, group maintenance models, and opportunistic maintenance models. Moreover, an example of a two-unit system maintenance process is provided in order to compare various maintenance policies. Keywords:
1
Multicomponent systems, corrective and preventive maintenance, maintenance free operating periods (MFOP) concepts.
Introduction
The prime maintenance objective is to ensure that a system performs its intended functions. Thus, generally maintenance can be divided into two main types: corrective maintenance and preventive maintenance. Corrective maintenance (CM) is any maintenance action that occurs when a system has been already failed; so, there is no possibility to optimize its performance with respect to a given economic or reliability criteria. In the situation, when it is necessary to avoid system failures during operation, especially when such an event is costly or/and dangerous, it is important to perform planned maintenance actions. Preventive maintenance (PM), according to MIL-STD-721B, means all actions performed in an attempt to retain an item in a specified condition by providing systematic inspection, detection, and prevention of incipient failures. The interest in development and investigation of maintenance problems has been extensively discussed in the literature since the early 1960s. Thus, there are many possible ways to classify the literature related to maintenance optimization models based on the following[1−4] : 1) information availability, 2) single-unit versus multi-unit systems, 3) time-dependent/action relationship, 4) model types, 5) optimization criterion, 6) methods of solution, 7) planning time horizon (finite/infinite), 8) maintenance effect (perfect, minimal, imperfect). The excellent basic review in the area of maintenance models for proper scheduling maintenance actions was prepared by Pierskalla and Voelker[2] , where authors investigated discrete time vs. continuous time maintenance models, which was later updated by Valdez-Flores and Feldman[5] . For other surveys see [1, 4, 6–11]. A review of the current literature in maintenance modeling problems indicates that the existing models can be classified under four main categories[12] : Manuscript received July 24, 2008; revised February 13, 2009 *Corresponding author. E-mail address:
[email protected]
1) inspection maintenance, 2) preventive maintenance for single-unit and multi-unit systems, 3) condition-based maintenance, 4) maintenance information management. The first category consists of those studies that consider the problem of optimally scheduling of inspections for systems that deteriorate or age. The second category of research work, which is particularly relevant to this paper, includes studies that examine the problems of execution of maintenance actions for singleunit or multi-unit systems. If the deterioration of a system or a control parameter, strongly correlated with the state of the system, can be directly measured, and if the system is subject to failure only if it deteriorates beyond a given threshold level, it is more appropriate to base the maintenance decision on the actual deterioration state of the system. This leads to the choice of the third category – condition-based maintenance policy. The last category takes into account all those studies that discuss the proper organization of maintenance information management processes necessary to effective performance of a system. In the investigated literature, most of the PM models consist in optimizing the execution of maintenance processes of single-unit systems and consider a single decision variable. The well-known maintenance models for singleunit systems are the age and the block replacement models. The basic references in this area are [3, 4]. Aven et al.[13] and Frostig[14] give a comparison of those maintenance policies for stochastically failing equipment. However, for a complex multi-component system, it may not be advisable to replace the entire system just because of the failure of one component. In fact, the system comes back into operation after repair or replacement of the failed component by an operative one. As a result, there is an increasing interest in analyzing the maintenance models for multi-unit systems. One reason for that is that the problem of proper maintenance scheduling is much more difficult to analyze. On the one hand, interactions between components complicate
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
the maintenance modeling and optimization process. On the other hand, the components dependencies also offer the opportunity to group maintenance actions, which may save costs of system performance. The other reason is that there is now more awareness of the importance of developing and using more realistic models. Maintenance and replacement models for singleunit systems are usually too simple when compared to the complex systems where the applications occur. Moreover, improvements in analytical techniques and the availability of fast computers have allowed more complex systems to be investigated. The maintenance optimization surveys by Cho and Parlar[1] , Nicolai and Dekker[7] , Thomas[5] , Wang[4] , review the growing body of literature on multi-unit systems. More comprehensive discussion in maintenance from application point of view can be found in [12, 15]. The definition of multi-component maintenance models is given by Cho and Parlar[1] as: multi component maintenance models are concerned with optimal maintenance policies for a system consisting of several units of machines or many pieces of equipment, which may or may not depend on each other. Interactions between components can be classified into three different types (see e.g., [5]): economic dependence, structural dependence, and stochastic dependence. In this paper, an economic dependence implies that an opportunity for a group replacement of several components costs less than separate replacements of the individual components. Stochastic dependence, also named as failure or probabilistic dependence, occurs if the condition of components influences the lifetime distribution of other components. Finally, structural dependence means that components structurally form a part, so that maintenance of a failed component implies maintenance of working components. If all units in a system are economically or stochastically independent of one another, maintenance policies for single-unit models can be applied to the multi-unit maintenance problems analysis. However, if there can be defined components interactions, then the optimal maintenance policy is not the one of considering each subsystem separately, and maintenance decisions will not be independent. Following the introduction, this paper is focused on multi-unit systems. Moreover, authors present the review of maintenance modeling devoted to the case of binary states. Consequently, the paper is organized as follows. In Section 2, we present an overview of the most often applied multicomponent maintenance models. We do not aim to give a list of all papers that have appeared. Instead, we want to investigate the main ways of the multicomponent maintenance models development, presented in the recent literature. Moreover, in Section 2.5, we mention about the problem of multi-state deteriorating system maintenance modeling, investigating the main research directions in this area. Later, there is an example provided of a two-unit system to compare the obtained results for various maintenance policies, and a briefly summary.
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Multicomponent maintenance models
The general classification scheme for multicomponent maintenance models is presented in Fig. 1. Nicolai and Dekker[7] surveyed and summarized the research in the area of multicomponent maintenance optimization models. Their classification scheme mostly bases on the type of dependency between system components. Thus, the focus of this overview is on the other defined group of multicomponent maintenance models. As it has been mentioned, the two standard maintenance policies, the age and the block replacement models, are generally used for optimization of maintenance action performance of single-unit systems or such multi-unit systems, where neither economic dependence, failure dependence, nor structural dependence exists. Fig. 1 shows Classification scheme of maintenance optimization models for multiunit systems.
Fig. 1 Classification scheme of maintenance optimization models for multi-unit systems[1,4,7]
The most common maintenance policy is the block replacement policy. Under such a policy, an operating unit is preventively replaced by new ones at times kT (k = 1, 2, 3, · · · ) independently on the age and state of the system. Such a policy is rather wasteful, since sometimes almost new systems are replaced. To overcome this undesirable feature, various modifications have been developed. The age replacement policy can also be used. Under this policy, a unit is always replaced at its deterioration age T or failure, which ever occurs first, where T is a constant. If there exists any dependence between components (economic/stochastic/structural) to optimize maintenance decisions, we may use one of the three groups of maintenance policies: 1) group maintenance policies, 2) opportunity-based replacement policies, 3) cannibalization maintenance policies. First, the group maintenance policies may be used. Under such a policy, a group of items is replaced at the same time to take advantage of economies of scale. The main problems connected with analyzing the group maintenance policies for multicomponent systems have been defined as: 1) definition of categories of units that should be replaced when failure occurs,
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2) cost reduction by including redundant parts into system design, 3) maintenance scheduling for systems of independently operating machines. Opportunity-based replacement models base on the rule, that replacement is performed at the time when an opportunity arrives, like scheduled downtime, planned shutdown of the machines, or failure of a system in close proximity to the item of interest. The analyzed problems are likely the same either for opportunity replacement models or for group replacement models. In the situation, when one machine is inoperative due to, e.g., lack of components and, at the same time, one or more other machines are inoperative due to the lack of different components, maintenance personnel may “cannibalize” operative components from one or more machines to repair the other(s). This practice is common in systems that are composed of sufficiently identical component parts (see, e.g., [16]). Thus, cannibalization is performed to increase system availability when spare parts or repair facilities are not available. As a result, the main problems that are investigated in this area mostly regard to inventory planning problem, spares allocation problem, or supply cannibalization issue. Following these considerations, in the next sections, we examine various types of maintenance policies for multi-unit systems, which are the most commonly used.
2.1
Block replacement models
Taking into account the basic assumptions of simple block replacement policies, each unit is replaced at failure. Moreover, all units in the system are replaced at periodic intervals regardless of their individual age. The maintenance problem is usually aimed at finding the optimal cycle length T in order, either to minimize total maintenance and operational costs or to maximize system availability. The classification of block replacement policies dependent on the types of considered maintenance problems of deteriorating systems is presented in Fig. 2.
Fig. 2
Block replacement policies for deteriorating systems
Scarf and Deara[17] investigated various block replacement policies under general type of costs structure for twounit system in series. Taking into account the following assumptions, 1) perfect repair policy, 2) failures detected immediately, 3) negligible replacement times of components and system, there are proposed two simple maintenance policies. First, independent block replacement policy, which assumes that the replacement of system components is performed on system failure and at fixed intervals Ti (i= 1, 2). The total long-run cost per unit time for defined policy over [0, T1 ] is given by T2 1 [1 + H(T2 − t)] u(t)dt+ c12 w1 H(T2 ) + T 0 T12 12 2 cw1 H(T1 − T2 )+ cw2 + 0 T1 −T2 [1 + Hy (T1 − T2 − y)]uy (t)dt dFy (t) + c12 w2 0
C(T1 ) =
(1) where c12 w1 is cost of failure replacement of both components, H(t) is system renewal function, Hy (t) is system renewal function as a function of time y, Ti is time between PM actions of unit i, u(t) is supportive function dependent on maintenance costs and distribution function of time to system renewal[17] , and c2w2 is cost of preventive replacement of component 2. The second interesting maintenance policy is combined block replacement policy. Under this policy, replacement of both components simultaneously is performed at fixed intervals or on failure of the system, whether units failed or not. When system satisfies the same assumptions, the analytical function of the long-run cost per unit time for this policy is given as T 1 H(T ) + [1 + H(T − t)] u1 (t)dt + 2 · c12 w1 T 0 c2w2 + c12 w2 (2) where c12 w2 is cost of preventive replacement of both components, and u1 (t) is supportive function dependent on maintenance costs and distribution function of time to system renewal[17] . The solutions of presented replacement policies were obtained with the use of simulation process, by approximating the time to failure distribution of the system by a Gamma distribution. The problems of periodic replacement of failed components, named as modified block policies, were extended by other researchers. Lai and Yuan[18] considered a parallel redundant system that consists of n identical components and fails when all components have failed. In the presented model, there are taken into account two types of failures: 1) independent failures of one component in a system, 2) failures of many components of the system at the same time, not necessary independent. The second type of failure event may be synchronized and is named as a common cause shock failure. This kind of event is classified depending on its effect into two kinds: 1) Non-lethal. When each component is assumed to fail C(T ) =
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
independently with probability p; the number of failed components is then a random variable, 2) Lethal. When every components in the system fails. The defined maintenance actions performed in a system include replacement when system fails (lethal shock) or at scheduled times kT (k = 1, 2, 3, · · · ), and minimal repair in case of non-lethal shock occurrence. Taking into account the following assumptions: 1) independent failure process of any component, 2) continuous monitoring and detection of component failures, 3) negligible replacement and minimal repair times, the long-run expected cost per unit time is obtained as C(T ) = n−1
cmr
T
k=1
0
T λk (u)F (u)du + c0 0 λn (u)F (u)du + cw2 (3) T F (u)du 0
where cw2 is cost of periodic replacement of a system, c0 is additional penalty cost, cmr is cost of minimal repair of a component, F (t) is common survival function of a system, F (t) = 1−F (t), λn (t) is intensity function of the unplanned replacement process, and λk (t) is intensity function of the independent failures of components process. Most studies on the periodic replacement policy focus on the expected maintenance and operational costs function. However, there are also studies, which also take into account the reliability criteria. One of those is a simple block replacement model for series component system developed by Duarte et al.[19] The problem is to determine the optimum frequency to perform preventive maintenance in equipment in order to ensure its availability. For the simplified assumptions: 1) constant repair rate of components, 2) increasing hazard rate of the components, 3) perfect repair policy, the objective function (defined as a cost function per unit time) is evaluated by n i
ai · ciw1 cw2 · ai · Ai + (4) C(A1 , A2 , · · · , An ) = 2μ(1 − Ai ) μ(1 − Ai ) i=1 subject to
n
Ai A
(5)
i=1
where ciw2 is cost of preventive maintenance of component i, ciw1 is cost of corrective maintenance of component i, ai is linear coefficient of hazard rate of component i, Ai is component iavailability, and μ is constant repair rate of components. The algorithm calculates the interval of time between PM actions for each component, minimizing the costs in a certain period of time when times to failure are increasingly and repair times are non-negligible. However, presented models are aimed at optimization of the cycle length T between preventive maintenance actions performance. There is also a number of research works that deal with the problem of cyclically scheduling maintenance activities assuming a fixed cycle length. Grigoriev et al.[20]
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formulated the maintenance scheduling problem to maintain a set of machines for a given determined T . The study presents the completely deterministic approach to decide for each period t ∈ T , which machine to service (if any) such that total servicing costs and operating costs are minimized. The solution is obtained with the use of branch and price algorithm. The block replacement policy is usually the basis for group maintenance policies presented in the next section.
2.2
Group maintenance models
Maintenance activity carried out on technical systems usually regards to the group of components. A group maintenance is performed either when a fixed time interval is expired or when a fixed number of units are failed, whichever comes first. There are four main classes of group replacement policies (see Fig. 3). A T -age policy, which assumes that system replacement is performed after every T units of time. An m-failure policy calls for replacing the system at the time of the m-th failure. The last policy, named (m, T )policy, combines features of both of the described classes. Under such a policy, system replacement is performed at the time of the m-th failure or at time T , whichever occurs first. The T -policy refers to the assumptions of the block replacement.
Fig. 3
Group replacement policies classification
Group replacement policies are very popular due to the ease with which they can be implemented in a real production setting. As systems become more complicated and require new technologies or methodologies, various modifications of basic group maintenance model have been developed to solve maintenance problems. Types of group replacement models are presented in Fig. 4. A number of replacement models have been proposed for two-unit systems. Scarf and Deara[17] considered group maintenance policies for two-unit system with failure dependence. In their paper, the failure dependence is assumed that whenever one component fails, it can induce the failure of the other components. The simple replacement policy assumes that the system is replaced when either both components fail or at fixed intervals, whichever occur first. Taking into account the assumptions presented in Section 2.1, where it described block replacement policies investigated, the long-run cost per unit time can be estimated as T 1 12 [1 + H(T − t)] u1 (t)dt + c12 cw1 H(T ) + C(T ) = w2 . T 0 (6) More recently, Zequeira and Berenguer[21] introduced a simple group replacement policy with periodic testing and inspections for two-unit standby parallel system with failure
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Fig. 4
Static models for deteriorating systems
dependence. Under such a policy, components failures occur randomly but are detected only by periodic testing or inspections. If the component is found failed during the inspection, then the corrective maintenance of the whole system is performed. Besides periodic inspections, preventive maintenance actions are scheduled for the system at a fixed time T since the end of the last maintenance action (corrective or preventive). Taking into account the following assumptions: 1) maintenance actions render both components to “as good as new” condition, 2) maintenance actions have constant durations, 3) lifetimes of both components after system maintenance action are s-independent up to the failure of a component, 4) inspections or tests are reliable, the expected costs per unit time due to testing, maintenance, and accident consequences are given by C(T ) =
1 [cins Nins (T, M ) + cw1 (1 − Pop (M T ))+ Tc cw2 Pop (M T ) + κcdw E(Tdw )]
(7)
where cins is mean cost of testing or inspection, cw1 is mean cost of CM or replacement of the system when at least one components has failed, cw2 is mean cost of renewal of the system when neither component has failed, cdw is mean cost of downtime in a cycle, Nins (T, M ) is expected number of components tests or inspections in a cycle, Pop (t) is probability that neither component is failed at time t, Tdw is random variable of downtime during a cycle, κ is frequency of true demands calling for the system to start up or function, and E(Tdw ) mean downtime in a cycle. The solution of presented model is provided for a simple case with numerical examples. In another study, Lai and Chen[22] presented an economic periodic replacement model for a two-unit system
with failure dependence and minimal repair. In the presented model, whenever unit 1 fails, it causes an increase in the failure rate of unit 2 by a certain degree. Moreover, each unit 2 failure induces unit 1 into instantaneous failure. As a result, there is a system failure occurrence. Investigated maintenance policy bases on system replacement at age T or at failure, whichever occurs first. Before the complete replacement of the two-unit system, each unit 1 failure is assumed to be reconditioned by minimal repairs. For simplified assumptions: 1) continuously monitoring of the system, 2) negligible repair and replacement times, the long-run expected cost per unit time in the steady state are evaluated by T cw2 F 2 (t) + cw1 F 2 (t) + 0 F 2 (y)cmr (y)r1 (y)dy C(T ) = T F 2 (z)dz 0 (8) where r1 is failure rate of unit 1, F2 (t) is lifetime distribution function of unit 2 and F 2 (t) = 1 − F2 (t), and cmr (y) is cost of minimal repair of a unit (dependent on the occurrence time of its failure). Possible extension of the presented maintenance models for a two-unit system is to develop multi-unit systems. Haurie and L Ecuyer[23] considered a simple group replacement model for a multicomponent system having identical elements. The system is comprised of n elements working independently under the same conditions. During the operational cycle, if the element fails it has to be replaced by a new one. Simultaneously, there are performed preventive actions, when a repairman can replace any number of working elements. The maintenance problem is solved by using the dynamic programming equation in the framework of the theory of optimal control of jump processes. In another study, Yasui et al.[24] summarized the main basic
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group replacement policies for an n-unit parallel redundant system. The first investigated replacement policy assumes that the system is replaced when all components fail or at the determined time T1 . The expected cost rate is derived from nc1 + cw2 F (T1 )n C(T1 ) = T1 (9) [1 − F (t)n ] dt 0 where c1 is acquisition cost of one unit, and F (t) is probability density function (pdf) of time to failure of a component. Next, suppose that system is replaced only at scheduled moments kT (k = 1, 2, 3, · · · ) if the total number of unit failures exceeds N2 until this moment. Expected cost ratio may be obtained as C(N ) = 2 ∞ N −1 1 i (k+1)T (n [F (t) − F (kT )]n−i dt + cdw i )[F (kT )] kT i=0 k=0
−1 ∞ N −1 1 n i n−i nc1 } × T (i )[F (kT )] [F (kT )] k=0 i=0
(10) where cdw is downtime cost per unit per unit of time. To modify this policy, assume that the replacement of the system is performed at system failure or at periodic times kT (k = 1, 2, 3, · · ·) if the total number of unit failures exceeds N1 until moment of preventive replacement. Expected cost ratio is given as C(N ) = 1 ∞ N −1 1 i n−i cw2 (n + i )[F (kT )] [F ((k + 1)T ) − F (kT )] k=0 i=0 N −1 ∞ 1 n nc1 } × (i )[F (kT )]i [F (kT )]n−i · i=0 k=0 n−i −1 (k+1)T F (t) . 1− 1− dt kT F (kT ) (11) The expected cost ratio for the presented group replacement policies is also obtained for a “k-out-of-n” system. Later, Popova and Wilson[25] presented a comparison of closed-form results for expected cost function and variance per unit time, derived for the three major classes of group replacement policy (T -age, m-failure and (m, T )). There is a system investigated comprising of n independent components working in parallel structure. Failure time of the components has a phase distribution. The assumptions taken into account are 1) variability of costs from cycle to cycle, 2) maintenance actions render system components to “as good as new” condition, 3) negligible repair time. Expected cost per unit time for T -age policy equals to T 1 cw2 +ncs +(cw1 −cs )nF (T )+ncdw F (t)dt C(T ) = T 0 (12) where cs is cost of a functioning component maintenance. Expected cost per unit time for m-failure policy is given by C(T ) =
1 [cw2 + mcw1 + (n − m)cs + cdw E(Tdw ) ] (13) Tc
where T c is mean of the cycle length. Finally, the expected cost per unit time for (m, T ) policy can by estimated as 1 [cw2 + ncs + (cw1 − cs )E(N ) + cdw E(Tdw ) ] Tc (14) where N is total number of components that fail during a cycle. However, in order to adapt the presented basic group replacement policies for practical use, the minimal repair of failed components in a system before the scheduled preventive maintenance action performance is introduced. Sandve and Aven[26] considered the optimal replacement problem of a multicomponent system, where components are minimally repaired at failures. Taking into account the following assumptions: 1) random and independent components failures, 2) repair times of component i are independent and identically distributed, 3) maintenance actions restores system to “as good as new” condition, two main replacement policies under the cost constraint are investigated. First, the standard T -policy is analyzed. The expression for the long-run average cost per unit time is given by T 1 E [a(t)] dt (15) cw2 + C(T ) = T 0 C(T ) =
where a(t) is the expected cost per unit of time at tdue to minimal repairs and downtime costs[26] . An extension of the presented policy is the (T, S)-policy. Under such a policy, the system is replaced at time S or at the first failure after time T , whichever comes first. Now, the average cost per unit time has the following form: cw2 + E
C(T ) =
T+
S
min{YT ,S}
0
a(t)dt
−1 P (YT > t)dt
· (16)
T
where YT is time to the first component failure after T . Unfortunately, the computation of E[a(t)] is possible only to obtain with an approximation under some simplified assumptions[26] . Another possible extension of the described replacement policy is to propose a two-phase maintenance policy for a group of identical repairable units with two types of component failure (minor/catastrophic), presented in Sheu and Jhang[27] . Under such a policy, there is a defined time-interval (0, T ] as the first phase, and the time interval (T, T + W ] as the second phase. Individual units have two types of failures. Type I failures (minor) are removed by minimal repairs (in both phases), whereas type II failures (catastrophic) are removed by replacements in the first phase, or are left idle in the second phase. The group maintenance in a system is performed at time T + W or upon the k-th idle, whichever comes first. At an inspection, all idle units are replaced with new units and all functioning units are overhauled so that they become as good as new.
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The long-run average cost per unit time for a generalized group maintenance policy is given by Cop + Cmr1 + Cdw1 + Con + Cw1 + Cins1 T + E(b) (17) where Cop is expected total operational costs over the time interval (0, T ], Cmr1 is expected total minimal repair costs over the time interval (T, T + τ ], Cdw1 is expected total downtime costs over the time interval (T, T + τ ], Con is expected total overhaul costs at the time T + τ , Cw1 is expected total replacement costs at the time T + τ , Cins1 is expected total inspections costs at the time T + τ , and b is minimum of the order statistics[27] . Most of the multicomponent maintenance models are aimed at optimizing the expected maintenance and operational cost per unit time. In the article, Chelbi et al.[28] proposed a preventive maintenance strategy for a serial system consisting of ncomponents, which are not necessarily identical. For the following assumptions: 1) random and stochastically independent components failures, 2) instantaneous detection of failures, 3) availability at any given time of resources required to undertake the replacements, there are two replacement policies investigated under reliability constraint. The first proposed strategy is defined as follows: a preventive replacement of a system is undertaken at the same moments kT (k = 1, 2, 3, · · · ). When any component fails between consecutive PM actions, it is replaced by a new one. Taking into account, that the maintenance of every components in a system is made separately, the PM of every component i is undertaken at moments kT i . The stationary availability of every component i would be given by C(T, W, k) =
Tw1i Hi (Ti ) + Tw2i Ai (Ti ) = 1 − Ti
(18)
where Hi (t) is component i renewal function, Tw1i is average time of corrective replacement of component i, and Tw2i is average time of preventive replacement of component i. The penalty function that allows measuring the difference between group PM or separately replacement of every component for any given period T ∗ follows: Dp (T ) =
n
i=1
(Ai (T ∗ ) − Ai (T )) −
(n − 1)TF T
(19)
where TF is fixed average time duration of PM operation. For the situation, when there is a considered minimal repair performance instead of replacement of failed components, the stationary availability of the system is given by n n T Tw1i 0 ri (t)dt + TF + Tiv i=1 i=1 (20) A(T ) = T where ri (t) is failure rate of component i, and Tiv is variable average time duration of a preventive replacement relating to the component i.
The presented classes of maintenance models are based on the assumption, that the failure distribution of the system is known with certainty. However, the failure distribution of a system is usually unknown or known with uncertain parameters in practice (see Fig. 5). In this case, the Bayesian group replacement policies are proposed. In particular, Popova[29] presented the optimal structure of Bayesian group replacement policies for a parallel system of n items with exponential failure times and random failure parameter. Each time a replacement decision is made, all n items are replaced.
Fig. 5
Types of group replacement models
From the renewal theory, the expected cost per unit time equals to C(T ) =
cw2 − csal E(Nsal ) + cdw E(Tdw ) T
(21)
where csal is salvage cost of a component, and Nsal is number of working items salvaged. For the defined cost constraint, there is a discussion presented about optimality results for group replacement policies for system withn machines. Taking a further step, Sheu et al.[30] proposed an adaptive preventive maintenance model with minimal repair for repairable system and developed a Bayesian technique to derive optimal maintenance policy. In the discussed model, a planned maintenance is carried out as soon as T time units have elapsed since the last maintenance action. When the system fails before age T , it is either correctively maintained (or replaced after (N 1) maintenances) or minimally repaired, depending on the random repair cost at failure. At the N -th maintenance, the system is replaced rather than maintained. There are also defined two types of system failures defined: minor failure, when the minimal repair is performed, and catastrophic failure, where corrective maintenance takes place. For the following assumptions: 1) random repair cost for system failures, 2) increasing failure rate of a system, 3) negligible times of any maintenance actions, 4) infinite time span, the objective is to determine the optimal maintenance plan that minimizes expected cost per unit of time, which is
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
given by the following equation: C(N, T ) = cdw
1 + pII
δ(cw2 +cwN ) 0
(N − 1)cw2 + cwN } ×
xhc (x)dx
N T i=1
0
N
Fi,pII (T ) +
i=1
−1
F i,pII (z)dz
(22) where cwN is planned replacement cost at the N -th maintenance, hc (t) is density function of random repair cost cr , pII is probability of catastrophic failure occurrence, and Fi,pII (t) is survival distribution of the time between the (i − 1)-th and i-th unplanned maintenance. The maintenance of deteriorating systems is also frequently modeled using Markov decision theory. Gurler and Kaya[31] considered a multicomponent system where the lifetime of each component is described by several stages, which are further classified as good, doubtful, PM due, and down. The system is composed of n identical and independently operating components that are connected in series. The maintenance policy assumes that system is replaced when a component enters a PM due or down state, and the number of components in doubtful states at that moment is at least N . The maintenance time is assumed to be negligible. The proposed maintenance policy is described by a multi dimensional Markov process. For the model under study, there is an expression derived for the long-run average cost per unit time given by C(K, N ) = cw1 (E(N1 ) + (1 − p)E(N2 )) + cw2 pE(N2 ) + cw3 E(Tr ) + P (Tr = 0)E(Tr1 )
(23)
where cw3 is system replacement cost, N1 is number of transition from node i to itself before a system replacement performance, N2 is number of transition from node i to (i-1) before a system replacement performance, p is probability of preventive replacement performance, Tr is time to system replacement, Tr1 is time to system replacement given that there are N +1 doubtful components at t= 0, and P (T2 = 0) is probability of system termination when there are more than N doubtful components. Optimization of the maintenance parameters is carried out with the use of numerical methods. Other model of group maintenance policy based on marginal cost considerations, formulated as a Markov decision chains, is given by Dekker and Roelvink[15] . The group age replacement problem is considered as a replacement decision which is based on sufficient information about the history of the process, being a vector containing all component ages. Although, lots of studies can be found that investigate the group maintenance of multicomponent systems as a Markov decision process, the state space in such problems grows exponentially with the number of components. Therefore, the Markov decision modeling is not tractable for systems with more than three components. There are also a limited number of problems that can be solved based on assumptions of Markov decision theory. Taking into account the planning aspect, the group main-
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tenance models can be classified as stationary or dynamic. In stationary models, there is assumed a long-term stable situation when the rules for maintenance do not change over the planning horizon. The presented models in this overview mostly regard to this type. However, stationary models cannot incorporate dynamically changing information during operational process performance, such as a varying deterioration of components or unexpected opportunities. To take such short-term circumstances into account, there are dynamic models proposed which can adapt the long-term plan according to information becoming available on the short term. This yields a dynamic grouping policy. Wildeman et al.[32] described a rolling-horizon approach that takes a long-term tentative plan as a basis for a subsequent adaptation according to information that becomes available on the short term. In this paper, there is a multicomponent system considered with n components. On each component i, a PM activity i can be carried out. The approach presented in this work enables interactive planning, taking into account opportunities and a varying use of components during operational processes performance. There is proposed a dynamicprogramming algorithm. During the performance processes of a multi-unit system, some maintenance opportunities may occur due to, e.g., breakdowns of a unit in a series configuration. In most cases, opportunities cannot be predicted in advance, and because of their random occurrence, there can be used opportunistic maintenance models to make effective maintenance planning.
2.3
Opportunistic maintenance models
In a system with several stochastically failing parts, it may be advantageous to follow an opportunistic maintenance policy, in which the maintenance action to be taken on a given part at a given time depends on the rest of the system. Types of group maintenance policies considered in the paper are presented in Fig. 6.
Fig. 6
Types of opportunistic maintenance models
One of the earliest treatments of the opportunistic replacement policy is the study of Radner and Jorgensen[33] . There is an opportunistic replacement of a single uninspected part considered in the presence of several monitored parts. The policy has the (ni , N ) structure such that an uninspected part is replaced alone on its failure or the
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arrival of its preventive replacement age N , and replaced opportunistically with a failed part i, if its age has reached a critical age ni . The presented approach is continued by van der Duyn Schouten and Vanneste[34] , where authors investigated the maintenance problem of a two-component series system, taking into account the possibility of (n, N )-strategies application. In another study, Epstein and Wilamowsky[35] presented the deterministic approach to investigate an opportunistic replacement problem of a two-component system. The considered problem is defined as finding the optimal maintenance plan when the exact time of both failure and maintenance opportunities are known at the outset. More recently, Fard and Zheng[36] discussed an opportunistic failure rate replacement policy for a non-repairable multi-unit system. The considered maintenance policy assumes that a unit is replaced when it fails or when its failure rate reaches a given limit L. When a failed unit is replaced or its hazard rate exceeds limit L, all operating units with their failure rates falling in (L − u, L) are also replaced. Taking into account the following simplified assumptions: 1) increasing in cycle time hazard rate of units, 2) negligible replacement time of units, 3) infinite planning horizon, 4) s-independent failure events, from the renewal theory, the expected system cost rate in the steady state is given: n1
1 [(cw2 + cw5 )P (fi ) + cw2 P (pi ) + T ci i=1 (cw2 + cw5 )P (ai )] (24) where cw5 is fixed replacement cost, fi , pi , ai are event of corrective, preventive, or active replacement of type i unit, n1 is number of types of units in a system, and ni is number of type i units. The presented model has been modified by Zheng and Fard[37] . In this paper, there is a repairable multi-unit system considered operating under the same replacement policy with one exception. When a unit fails with the hazard rate in (0,L − u), then it is minimally repaired, with known repair rate, instead of replacing it. For the same assumptions presented above, the expected system cost rate is evaluated as C(L, u) =
ni
C(L, u) = n1 rmr1 1 )H(Di ) + cw2 P (pi)+ (cmr1 + ni T μ ci i=1 cw5 )P (ai) (cw1 + cw5 )P (f i) + (cw2 + 1 + mi
(25)
where cmr1 is fixed repair cost, rmr1 is repair cost rate, μ is repair rate, mi is mean value of number of type i units which are actively replaced at the end of cycle j given an active replacement on unit l of type i at the end of cycle j, hi (t) is hazard rate of a type i unit, and Di is h−1 i (L − u). Taking a further step, Jhang and Sheu[38] investigated a multi-unit system that has two types of failures. Type I failures (minor failures) are removed by minimal repairs, whereas type II failures (catastrophic failures) are removed
by replacements. Both types of the failures are age dependent. The system is replaced at type II failure or at the opportunity after age T , whichever occurs first. Taking into account the following assumptions: 1) cost of minimal repair depends on the random and deterministic part, 2) instantaneous detection and reparation of failures, and 3) infinite horizon planning, the total expected long-run cost per unit time is given as ∞ C(T ) = 0 [cwT + (cwII − cwT )Fp (T + ω)+ T u2 (z)F p (z)pI (z)r(z)dz gw (ω)dω × 0 −1 T F (T − z)g (z)dz p p 0
(26)
where cwT is cost of replacement at the opportunity after T , cwII is cost of replacement at type II failure, Fp (t) is survival distribution of the time between successive type II failures, u2 (z) is expectation with respect to random repair costs, pI (t) is probability of type I failure occurrence, r(t) is system failure rate, and gw (t) is pdf of time between successive opportunities. In another study, Pham and Wang[39] discussed the opportunistic maintenance of a k-out-of-n: G system with imperfect PM and allowable partial failure occurrence. The following maintenance policy was designed: each failure of a system component in the time interval (0, τ ) is immediately removed by a minimal repair. Components that fail in the time interval (τ, T ) can be lying idle. The system is replaced when the total operating time reaches T or with CM and PM actions when there is exactly m components idle, whichever occurs first. That is, if m components fail in the time interval (τ, T ), CM combined with PM is undertaken; if less than m components fail in the time interval (τ, T ), then PM is carried out at time T . For the following supplementary assumptions: 1) s-independent failure events, 2) negligible time of minimal repair, 3) age-dependent and number of minimal repairsdependent minimal repair costs, 4) increasing failure rate of every component, 5) perfect preventive maintenance, the limiting average system availability for an infinite planning horizon are obtained: T −τ F m (t)dt × A(τ, T ) = τ + 0 T −τ F m (t)dt + τ + Tw2 + Fm (T − τ )(Tf − Tw2 )}−1 0 (27) where Fm (t) is cumulative distribution function (cdf) of time between maintenance, Tf is time to perform CM together with PM, and Tw2 is time to perform PM alone the long-run. The expected system maintenance cost per unit time is τ C(τ, T ) = n 0φ(y)rc (y)dy+Fm (T − τ )(cf − cw2 )+ T −τ F m (t)dt + Fm (T − τ )(Tf − Tw2 )+ cw2 } × 0 τ + Tw2 }−1
(28) where φ(t) is expectation with respect to minimal repair costs and number of performed minimal repairs, rc (t) is
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
failure rate of a component, and cf is cost of CM combined with PM. When taking into account, that PM actions are imperfect, the expressions for expected system maintenance cost per unit time and limiting system stationary availability also have been developed. However, a key conclusion from the literature on multicomponent maintenance models is that the optimal maintenance policies are difficult to compute. In addition, because of their complex form, it is very difficult to use them in practice. For this reason, some other methods have been developed, which give the opportunity to obtain models designed to yield practical and easy to implement policies. For example, Hopp and Kuo[40] developed the three heuristics and a lower bound for a system with all non-safety-critical components. First, hierarchical approach for scheduling replacement epochs for n components is defined. Later, the obtained results are compared to the other heuristics: sequential approach and base interval approach. In another study, Haque et al.[41] applied genetic algorithm with fuzzy logic controller to get a near optimal decision for opportunistic replacement of a multi-unit system.
2.4
Maintenance modeling using MFOP concepts
Maintenance and reliability problems of technical systems have received considerable attention by many researchers and practitioners. The main models that address maintenance strategy have been presented above. The presented literature review should be also extended on the reliability measure that considers reliability requirements to be based on operational requirements. Moreover, the investigated measure depends on the planned and unplanned maintenance action performance. Reliability can be measured and investigated in a different ways. Generally, overall multi-system performance depends on individual system availability, which is quantifiable by the measures time to failure (TTFi ) or time to restoration (TTRi ) (see, e.g., [42]). Example of such a real system can be found in [43]. However, the most common reliability measure is maintenance time between failures (MTBF), which has become a way of defining the acceptability of failures by counting the number of failures per hour[44,45] . However, the presented measure has limitations, e.g., [44]: 1) MTBF assumes a constant failure rate,
Fig. 7
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2) MTBF can only be accurately calculated after a considerable period of time, 3) MTBF relies on probabilistic. As a result, the high costs driven by the consequences of the systems unreliability have been the starting point for searching new measures to describe the reliability of a system — especially in a field of aircraft maintenance. The new reliability measure, having been investigated since mid 1990s, is maintenance free operating periods (MFOP). This measure is defined as a period of operation during which the equipment must be able to carry out all its assigned missions without any maintenance action and without the operators being restricted in any way due to system faults or limitations[45,46] . The main classification scheme is presented in Fig. 7, and the comparison of both the measures, MTBF and MFOP, is given in [44]. The first MFOP mathematical models are developed in [45]. In the presented paper, MFOP is defined as a period during which the equipment shall operate without failure and without the need for any maintenance (however, minor, planned maintenance is permissible). The probability of not having any unscheduled maintenance for a period of tmf life units is given as maintenance-free operation period survivability (MFOPS). Authors developed two mathematical models to predict MFOPS: 1) MFOP prediction is the mission reliability approach. 2) MFOP prediction is the alternating renewal theory. Let us consider a system with n components connected in series. Let us also assume that 1) system time to failure and time to repair follows arbitrary distributions, 2) time to failure distributions of various items of a system are independent. The probability that the system will survive the i-th cycle of MFOP, given that it survives (i − 1) cycles is given as n Rk (i · tmf ) (29) M F OP S(tmf , i) = Rk ([i − 1] · tmf ) k=1
where Rk (tmf ) is reliability of the k-th component for (the first) tmf life units. In the second model, there is used alternating renewal theory approach. MFOPS is found during a stated period of T along with the maintenance recovery period. Let us consider a repairable system, assuming that 1) time to failure distribution of an item follows arbitrary distribution with density function f (t),
MFOP modelling concepts
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2) maintenance recovery time of the item follows arbitrary distribution with density function g(t), 3) the item can be in one of two states {1,0}, where “1” is up state and “0” is down state. The probability which is the system will operate for at least tmf life units before it fails during T hours of operation is given by T f (μ |tmf ) P0 (T − μ) dμ (30) P1 (T ) = R (tmf ) + 0
and
P0 (T ) =
T 0
g (ν) P1 (T − ν) dν
(31)
where f (μ |tmf ) is probability that system fails at time μ. There are also some numerical solutions for various failure distributions provided in the paper. Another way of MFOP concepts model is presented in [47]. Authors describe the ultra reliable aircraft model (URAM): an aircraft reliability and maintenance discrete event simulation model that is designed to investigate MFOP concepts. There are also identified key factors determining MFOP achievement. The extension of MFOP model is given by Todinov[48−50] . The author proposed the new reliability measure: minimum failure-free operating period (MFFOP), which can be defined as a combination of specified minimum intervals before random variables in a finite time interval, whose existence is guaranteed with a minimum probability PM F F OP [49] . Assuming that a system is composed of a non-repairable component, and taking into account the following assumptions: 1) “as good as new” replacement of a component, 2) “critical repair” to be a critical event that leads to a system halt or degeneration of the required, function below a minimum acceptable level and to require an immediate intervention for repair, 3) random failures following a homogeneous Poisson process, minimum probability PM F F OP is given by k r
ks (λa)k exp(−λa) × 1− PM F F OP = (32) k! a
of standby components, subject to factors such as maintenance capability, or system availability maximization, cost of standby items, or operational cost minimization. Many of maintenance models which investigate such problems are presented below. However, real systems and its components can function in degraded states. Systems that can exist in more than two states are called multi-state systems. For a stochastically failure system in which deterioration takes place only from state to state (state-dependent deterioration), replacement policies and inspection policies using Markov processes were investigated. An interesting example of such a model is presented in [51], where an optimal inspection time interval for a periodically inspected 4-state Markov system is determined. However, under the Markovian formulation, the deterioration of a system is indicated only by the changes of states. Thus, there is an assumption made that performance of a system within each state does not age. In practice, this assumption may be invalid. To overcome this simplification, more general system configurations, in which random shocks could occur or deterioration could take place within a state (state-agedeterioration) were examined with the use of semi-Markov processes. State-age-dependent replacement policies for a multistate deteriorating system is considered, e.g., in [52]. The optimization criterion in this model is to minimize the expected long-run cost rate. Some practical application frameworks for multi-state systems maintenance modeling also have been developed. For example, see [53], where a flow transmission water pipe system is analyzed. In that paper, a statistical model for condition-based maintenance policy of multi-state systems has been developed by combining the universal generating function and Markov chain analysis theories. Another interesting problem in the area of multi-state system maintenance modeling is selective maintenance optimization for multi-state systems (see, e.g., [54]), where the problem is what maintenance activities should be performed during the limited amount of time. To solve the integer, non-linear programming problem is used the shortest path method.
k=0
where (λa) exp(−λa)/k! is probability of exactly k failures in the finite time interval a, and p(S |k) = (1 − ks/a)k is conditional probability that given k random failures, before each failure there will be a failure-free gap of length at least s. Author also provide some application examples, and presents simulation algorithms for evaluating the probability of MFFOP existence. k
2.5
Multi-state system maintenance modeling
The plethora of existing studies in reliability and maintenance modeling has been devoted to the case of binary states. That is, a system and its components could, at any point in time, presume only two operational states: function or failure. For this issue, maintenance policy problem is usually connected with determination of optimal number
3
An example
According to the replacement policies, described in the presented overview, a system is maintained to different operation levels with various time intervals. Moreover, every maintenance strategy takes into consideration various input and output parameters (e.g., cost constraints) or model assumptions (e.g., non-/negligible maintenance time, random/constant repair cost, different dependencies between components) in order to obtain the optimal maintenance policy. Consequently, the issue of this section is to study a described below system performance under the different maintenance policies. The presented comparison is limited only to those policies that can be evaluated analytically. We consider a system that is composed of two identical components (denoted as units 1 and 2), whose failures are random and stochastically independent. This kind of a system with no redundancy will often be better to model
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
as one component or to model each component separately. Despite this, the various maintenance policies, described in Section 2, can be compared on the basis of economic cost and system reliability. The problem of proper maintenance schedule planning for a two-unit system can be considered for the following main cases: 1) identical/various probability distributions of time to failure of a given components, 2) identical/various failure rates of components, 3) minimal repair performance instead of replacement of failed components, 4) failure dependence occurrence. First, to obtain the maintenance scheduling plan for the presented system, the simple group replacement model is proposed by Scarf and Deara[17] . Under such a policy, failed components are replaced immediately. The system itself is only renewed when both components are replaced simultaneously at failure or at scheduled moment kT (k = 1, 2, 3, · · ·) (see Fig. 8).
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According to the presented model assumptions, to evaluate the long-run cost per unit time, there must be defined cost of minimal repair of failed unit 1 denoted by cm (t). Suppose that cm (t) is a constant equal to 60 per failure, for the exponential time to failure of both components the expected cost per unit time given by (8) are easily obtainable. Fig. 10 shows plots of the expected cost per unit time as a function of T for both age policies: T -age policy and periodic replacement with minimal repair. Moreover, Fig. 11 illustrates the long-run cost per unit time as a function of T for various failure rates of system components.
Fig. 9 The expected cost per unit time as a function of T for considered group replacement model Fig. 8
System maintenance plan
The exponential distribution F (t) = 1 − exp(−λt) is used to model the lifetimes of both components. The parameters are λ1 = λ2 = 0.001, and λ1 = 0.1 and λ2 = 0.001. Considered costs are as follows: 1) cost of failure replacement of both components c12 w1 = 1500, 2) cost of corrective replacement of component 2 c2w1 = 1000, 3) cost of preventive replacement of both components c12 w2 = 100. According to the defined model assumptions, the longrun cost per unit time described by formula (6), when p = 1 and q = 0 is given by C(T ) = 2 c12 w1 H1 (T ) + cw1
T 0
Fig. 10 The expected cost per unit time as a function of T for T -age policy and periodic replacement with minimal repair
[1 + H1 (T − t)] F 1 (t)dH2 (t) + c12 w2 T (33)
Fig. 9 presents the long-run cost per unit time as a function of time interval T for various values of components failure rates. In the second approach, the T -age policy proposed by Popova[29] is investigated. To obtain the long-run cost per unit time for this model, according to (13), there is a need to estimate the downtime cost of cdw per unit time until the failed component is replaced and the cost of either replacing or repairing a functioning component to as good as new, denoted by cs . Thus, the long-run cost per unit time is evaluated for the same simplified assumptions, when cdw is equal to 250 and cs equals 70. Taking a further step, a more difficult model, defined by Lai and Chen[22] , can be considered.
Fig. 11 The expected cost per unit time as a function of T for periodic replacement with minimal repair
To complete the analysis, the expected cost per unit time as a function of failure rate λ for the investigated replacement policies (see Figs. 12–14) is also presented.
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Presented examples are given to illustrate the methods, and let us make the following conclusions: 1) It can be seen that the cost C(T ) can vary for a different level of parameter λi . When the failure intensity for both components is the same, the lower long-run cost per unit time as a function of T can be obtained for age replacement policy with minimal repairs. However, taking into account the failure dependence between components the decision which policy should be chosen is less obvious. 2) It is also clear, that for small values of parameter λ, there will be more failures in the system. Thus, the choice between the policies in terms of system performance should be confirmed also by the reliability analysis. Moreover, the implementation of replacement policies requires some level of maintenance reporting and control to gain successful results. The principal advantage of the block policies is that component age does not need to be monitored. Thus, the requirements of the maintenance management becomes simpler. The incorporation of minimal repair performance instead of failed component replacement requires the definition and evaluation of any costs which are involved in this maintenance action. This demonstrates once again, that the decision whether to consider independent or grouped block replacement policies or an age policy depends on the relative mean times to failure of the two components working in a system. A possible extension of this work is to model the system performance with the use of other probability distributions of components lifetimes, and to propose a simulation model that let us obtain the optimal policy parameters.
Fig. 12 The expected cost per unit time as a function of λ when T = 2000 time units
Fig. 13 The expected cost per unit time as a function of λ1 when T = 2000 time units and λ2 = 0.001
Fig. 14 The expected cost per unit time as a function of λ2 when T = 2000 time units and λ1 = 0.001
4
Conclusions
In this paper, we have reviewed the literature on the most commonly used optimal multicomponent maintenance models. This let us draw following conclusions: 1) The main mathematical methods used for analyzing maintenance scheduling problems include applied probability theory, renewal reward processes, and Markov decision theory. However, there are a lot of maintenance problems, where the functional relationship between the system s input and output parameters cannot be described analytically. Thus, in practice, various maintenance models have been developed, which apply linear and nonlinear programming, dynamic programming, simulation processes, and heuristic approaches. These were only mentioned in the presented overview. 2) Most maintenance models, to obtain the optimal maintenance parameters, take into account only the cost constraint. However, maintenance actions are aimed at improving system dependability. For complex systems, where various types of components have different maintenance cost and different reliability importance in the system, it is more appropriate to analyze the optimal maintenance policy under cost and reliability constraints simultaneously. 3) Many maintenance models consider the grouping of maintenance activities on a long-term basis with an infinitive horizon. In practice, planning horizons are usually finite for a number of reasons: information is only available over the short term, a modification of the system changes the maintenance problem completely, and some events are unpredictable. 4) In most existing literature on maintenance theory, the maintenance time is assumed to be negligible. This assumption makes, e.g., availability modeling impossible or unrealistic. Obtained results are not traceable to practical situations. 5) Most of the maintenance models for complex system are based on the following assumptions: infinite system planning horizon, steady-state conditions, perfect repair policy, etc. The models resulting from these assumptions are often an oversimplified version of the real-world system behavior. 6) Most maintenance models assume that whenever a system component is to be replaced, a new component is immediately available. This implies that either the components
T. Nowakowski and S. Werbi´ nka / On Problems of Multicomponent System Maintenance Modelling
are highly standardized so that they can be immediately delivered from suppliers, or that they are so inexpensive, that there can be stored large amount of spares as a protection against system failures. Taking into account the “real life situations”, the number of spare parts is usually limited, and the procurement lead time is non-negligible. This implies, that the maintenance policy and spare provisioning policy must be closely coupled, because separate treatment of them will not result in the system optimal maintenance policy. 7) Maintenance modelling development during the last decades has taken into account the application of imperfect PM, system performance under uncertainty (e.g., lack of information, unknown distribution functions of components), different type of system failures occurrence, dynamic grouping, inspection maintenance, etc. However, the more extended model is considered with various maintenance parameters and more complex system behavior, to obtain the the robust optimal solution. From the theoretical point of view, much of the maintenance work is of mathematical interest only exploring the modelling methods. That is one of the reasons, why the application of maintenance models has been rather limited in practice. The difficulty with application of maintenance models lies in making the models simple enough to be both tractable and accessible to practitioners. 8) Moreover, maintenance and replacement decisions base on the information, e.g., failure data of the equipment under consideration, maintenance performance times, and type and number of necessary support resources. Sufficient data rarely exist for estimating parameters in a complex model, and if data do exist, they are often unreliable. This makes the application of mathematical models to support maintenance and replacement decisions less obvious.
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Tomasz Nowakowski is a professor of Wroclaw University of Technology, Wroclaw, Poland. He is a head of Division of Logistics and Transportation Systems at Mechanical Engineering Faculty. He is also a president of Polish Logistic Association, member of teams of Polish Academy of Sciences, vice-president of Polish Maintenance Society. He is editor of Terotechnology section of quarterly Scientific Problems of Machines Operation and Maintenance. His research interests include reliability, maintainability, safety of technical (generally transportation and logistic) systems, computer aided operation, knowledge based (expert) systems, uncertainty of operation, and maintenance data.
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Sylwia Werbi´ nka received her M. Sc. in mechanical engineering (specialty: logistics) from Wroclaw University of Technology, Poland in 2004. She is currently a Ph. D. candidate in Division of Logistics and Transportation Systems at Wroclaw University of Technology. Her research interests include logistic support systems modelling, systems reliability, and maintenance processes design-
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