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An optimal dynamic interval preventive maintenance scheduling for series systems
Author's Accepted Manuscript
An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems Gao Yicong, Yixiong Feng, Zixian Zhang, Jianrong Tan
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S0951-8320(15)00098-8 http://dx.doi.org/10.1016/j.ress.2015.03.032 RESS5280
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Reliability Engineering and System Safety
Received date: 15 November 2014 Revised date: 20 March 2015 Accepted date: 29 March 2015 Cite this article as: Gao Yicong, Yixiong Feng, Zixian Zhang, Jianrong Tan, An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j. ress.2015.03.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems Gao Yicong1, Yixiong Feng1, *, Zixian Zhang2, Jianrong Tan1 1.
State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China 2.
Department of Mechanical Science and Engineering, Tokyo Institute of Technology, Tokyo, Japan
Corresponding author: Yixiong Feng Mailing Address: State Key Lab of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China Telephones: +86-13819161642 (Mobile) E-mail: [email protected]
Abstract: This paper studies preventive maintenance (PM) with dynamic interval for a multi-component system. Instead of equal interval, the time of PM period in the proposed dynamic interval model is not a fixed constant, which varies from interval-down to interval-up. It is helpful to reduce the outage loss on frequent repair parts and avoid lack of maintenance of the equipment by controlling the equipment maintenance frequency, when compared to a periodic PM scheme. According to the definition of dynamic interval, the reliability of system is analyzed from the failure mechanisms of its components and the different effects of non-periodic PM actions on the reliability of the components. Following the proposed model of reliability, a novel framework for solving the non-periodical PM schedule with dynamic interval based on the multi-objective genetic algorithm is proposed. The framework denotes the strategies include updating strategy, deleting strategy, inserting strategy and moving forward strategy, which is set to correct the invalid population individuals of the algorithm. The values of the dynamic interval and the selections of PM action for the components on every PM stage are determined by achieving a certain level of system availability with the minimum total PM-related cost. Finally, a typical rotary table system of NC machine tool is used as an example to describe the proposed method. Keywords: dynamic interval; non-periodic preventive maintenance; optimization; machine tools technology 1. Introduction Preventive maintenance (PM) is defined as a set of activities aimed at improving the overall reliability and availability of a system. While planning the PM schedule according to the defined activities, the time between two PM actions is an important aspect of actions adopted and it would affect the maintenance economics. In term of PM interval, PM can be categorized into two kinds, periodic PM (fixed interval) and non-periodic PM (dynamic interval) [1, 2]. In the former approach, the planning horizon is segmented into discrete equal intervals. For each PM stage, a decision must be made to perform one of the three PM actions (inspection, repair and replacement) on each component by maximizing system benefit in PM. Instead of searching for equally spaced PM actions, non-periodic PM approach search for the flexible interval in which PM interventions should be performed[3]. During the last few decades, numerous papers have been published on periodic PM modeling and optimization. Cassady and Kutanoglu et al.[4] developed an integrated mathematical model for a single-machine problem with periodic PM interval. They defined the total weighted expected completion time as the objective function. Their model allowed multiple maintenance activities and explicitly captures the risk of not performing maintenance. Bartholomew-Biggs et al.[5] proposed a new PM formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. The formulation involved the global minimization of a non-smooth performance function. Lim and Park[6] proposed a periodic PM policy, which keeping the pattern of hazard rate unchanged by PM. They evaluated the expected cost rate per unit time based on computing the expected number of failures depending on the hazard rate of the underlying life distribution of the system. Liao et al.[7] developed a reliability-centered sequential PM model for monitored repairable deteriorating system. They supposed that system’s reliability could be monitored continuously and perfectly, whenever it reaches the threshold, the imperfect repair must be performed to restore the system. Wang and Tsai[8] established a bi-objective imperfect PM model of a series-parallel system. They developed a unit-cost cumulative reliability expectation measure to evaluate the extent to which maintaining each individual component benefits the total maintenance cost and system reliability over the operational lifetime. El-Ferik and Ben-Daya[9] developed a hybrid age-based model for imperfect PM involving maintainable and non-maintainable failure modes. They determined the number of PM actions and the length of PM intervals that minimize the total long-term expected cost per unit time. Ebrahimipour et al.[10] developed a multi-objective PM scheduling model in a multiple production line. They defined reliability of production lines, costs of
maintaining, failure and downtime of system as multiple objectives and different thresholds for available manpower, spare part inventory and periods under maintenance was applied. Levitin and Lisnianski et al.[11] presented an optimization model for PM scheduling in multi-state series-parallel systems. They considered the cost of unsupplied demand due to failures of components as an important part of the PM activities cost. Schutz et al.[12] proposed and modeled periodic and sequential PM policies for a system. The objective of periodic PM policy was to determine the optimal number of PM to achieve and the objective of sequential PM policy was to determine the optimal number of PM intervals and the duration of these different intervals. Wang and Lin[13] proposed an improved particle swam optimization of periodic PM. The optimal maintenance periods for all components in the system were determined according to the importance of components on system reliability, to minimize the periodic PM cost for a series–parallel system. Tam et al.[14] analyzed the affection of reliability, budget and breakdown outages cost to the calculation of optimal maintenance intervals. Three models were proposed to calculate optimal maintenance intervals for multi-component system in a factory subjected to minimum required reliability, maximum allowable budget and minimum total cost. Alardhi et al.[15] presented an optimization method for scheduling the PM tasks in separate and linked cogeneration plants and satisfying the maintenance and production constraints. Shirmohammadi et al.[16] presented an optimization method for scheduling the PM of a system that is subject to random failures, and investigated the decision rule for PM. They defined the time between preventive replacements and cut-off age as decision variables to determine the optimal maintenance policy. Bartholomew et al.[17] proposed a model in which each action of the PM reduces the equipment’s effective age. The optimization process involved minimizing a performance function that allows for the costs of the minimal repairs and eventual system replacement as well as for the costs of the PM during equipment is operating lifetime. Harrou et al.[18] formulated the model of imperfect maintenance optimization for series-parallel transmission system structure. They improved the availability of transmission system through selecting the optimal sequence of intervals to perform PM actions. Moghaddam et al.[19] presented mathematical models and a solution approach to determine the optimal PM schedules for a repairable and maintainable series system with equally-sized periods. Lin and Wang[20] identified important components and determined their maintenance priorities of a series-parallel system. The optimal maintenance periods of these important components were determined to minimize total maintenance cost given the allowable worst reliability of a repairable system. According to Tsai et al.[21], the interval and the type of PM action are the main decision variables of non-periodic PM. These make the optimization problem of non-periodic PM more complex. However, the interval of non-periodic PM is more flexible and the schedules of non-periodic PM can be better adjusted, reducing the outage loss on frequent repair parts through the interval (PM frequency) control[22]. In view of the potential important applications of non-periodic PM, some improvement ideas have been proposed. Wang[23] summarizes, classifies, and compares various existing maintenance policies for both single-unit and multi-unit systems. Relationships among different maintenance policies are also addressed. He found that non-periodic approach provided gains in the overall system average availability. Percy and Kobbacy[24] investigated two principal types of general models for determining the optimal maintenance intervals. The first model considered fixed PM intervals, while the second one was adaptable, allowing variable PM intervals. But they did not given clear comparison to help decide between these two approaches. Lapa et al.[25] defined inspections as special PM actions and the PM was said to be non-periodic. They developed a surveillance test optimization methodology based on genetic algorithms. Later, Lapa et al.[26] presented a methodology for PM policy evaluation based upon a cost-reliability model. They used flexible intervals between maintenance interventions. However, the proposed method defined maintenance action as PM action and replacement action was neglected. The method selected components to undergo maintenance at different time, while avoiding this for components in parallel. Lin and Wang[27] proposed a non-periodic PM policy method for series-parallel systems, which is based on failure limit policy. Firstly, they identified the parallel sub-system required to be maintained. Secondly, they identified the maintained parallel sub-system by the measure of unit-cost extended life. But it
was a time-consuming task for searching the optimum values of the parameter and the replacement action was ignored in the method. The detail of the solving method was not mentioned. Fitouhi and Nourelfath[28] proposed an integrates noncyclical preventive maintenance method with tactical production planning in multi-state systems. They suggested noncyclical preventive replacements of components, and minimal repair on failed components. However, only preventive replacements were considered in their method for the sake of simplicity. These studies paid attention on periodic PM modeling and optimization. However, it is found that there is a few works in on non-periodic PM scheduling and for simple. Therefore, we expand upon the previous work of Lapa et al.[25,26]. A comprehensive model for the non-periodic PM scheduling problem was developed and a solving framework based on the multi-objective genetic algorithm (MOGA) was proposed, which integrated a population correction procedure. The main contribution of this research is to define a general model for PM scheduling problem, in which PM actions are performed under dynamic intervals, and then develop a solving framework to determine optimal non-periodic PM schedules. Some new issues that are aspects of real world problem of non-periodic PM scheduling are considered in the general model. The time-based schedule of non-periodic PM actions that maximize the system availability and minimize the total cost of those actions is developed. The following of the paper is organized as follows: In section 2, the notations are used throughout this article are given. The expressions for the reliability function under the non-periodical PM actions are developed in section 3 and PM dynamic interval model are explained in Section 4. A framework based on MOGA for obtaining the optimum solution is discussed in Section 5. To carry out the performance of the result obtained, an illustrative example is analyzed in Section 6 and finally, some concluding remarks and future work are given in Section 7. 2. Notation
m: Number of component.
n: Number of intervals.
nt : Numbers of non-periodic PM up to time t.
T: Length of the planning horizon.
i: Index of components, i = 1, …, m.
j: Index of intervals, j = 1, …, n.
R0(t): Instant reliability of the system at t.
Ri(t): Instant reliability of component i at t.
tpj: The length of the interval of period j.
tptemp : PM interval adjustment integer.
min tpsystem : the minimum allowed PM interval of system.
max tpsystem : the maximum allowed PM interval of system.
min min t j ∈ [ j ⋅ tpsystem , ( j + 1) ⋅ tpsystem − 1] if j ≤ n -1 tj: The time that PM actions are performed, min if j = n t j ∈ [n ⋅ tpsystem , T ]
ρi : Repair efficiency factor of component i. 0 < ρi < 1 , efficient repair; ρi = 1 , replacement (optimal
repair), as good as new; ρ = 0 , inspection (useless repair), as bad as old.
λi : Scale (characteristic life) parameters of component i.
βi : Shape parameters of component i.
eai−, j : Effective age of component i at the start of period j.
eai+, j : Effective age of component i at the end of period j.
TMj : Total repair time during the period j.
TRj : Total replacement time during the period j.
TIj : Total inspection time during the period j.
tMi : Repair time of component i.
tRi : Replacement time of component i.
tIi : Inspection time of component i.
CFj : Total PM cost associated with the possible failures during the period j.
CMj : Total repair actions cost during the period j.
CRj : Total replacement actions cost during the period j.
CIj : Total inspection actions cost during the period j.
cFi : Repair cost of possible failure cost of component i.
cMi : Repair cost of component i.
cRi : Replacement cost of component i.
cIi : Inspection cost of component i.
1 if component i at period j is inspected . yIij = 0 otherwise
1 if component i at period j is maintained . yMij = 0 otherwise
1 if component i at period j is replaced . yRij = 0 otherwise
Suppose there is a new repairable series system of m components, denoted as P = { p1 , , pi , , pm } .For the sake of simplicity, only series systems of components are considered in this paper. In order to model the effects of various PM actions in multi-component systems, three typical PM actions are considered as follows[2]. Inspection: The inspection action includes lubricating, cleaning, checking and adjusting. There is no effect will be on the rate of occurrence of failure of component and the component remains in a state of “as-bad-as-old”. Repair: The repair action restores the degraded strength partly. It effectively reduces state of the components in terms of the rate of occurrence of failure, but the effect of repair is not to return the system to the state of “as-good-as-new”. Replacement: The replacement action makes the components replace with new ones. It returns the state of the component as good as new, which means that the rate of occurrence of failure of component is recovered to its initial condition. 3. Reliability under non-periodical preventive maintenance Modeling the effect of PM actions is the first step of scheduling the PM program. In order to calculate evaluate the PM efficiency, several models with that type of assumption have already been proposed. The Arithmetic Reduction of Age model (ARA) is one of the most famous, in which each repair action reduces the age of the system is the basis of Kijima’s virtual age models[29]. Therefore, we use the approach proposed by Doyen et al.[30] to evaluate the reliability of a component undergoing a PM schedule. The failure intensity of pi is defined as:
nt −1 hi (t ) = hi t − ρi ∑ (1 − ρi ) j tpnt − j j =1
(1)
It is also assumed that component failure corresponds to the well known Non-Homogeneous Poisson Process and the rate of occurrence of failure can be expressed by: hi (t ) = λi ⋅ βi ⋅ t βi −1 λ > 0, β > 0 , for i = 1, , m
(2)
where λi and βi are the scale (characteristic life) and the shape parameters of pi for the Weibull distribution respectively. The Non-Homogeneous Poisson Process is similar to the Homogeneous Poisson Process with the exception that the rate of occurrence of failure is a function of time. For simplicity, we assume that the PM actions occur at the end of the period. The reliability of pi during period j is expressed as eai+, j Ri (t ) = exp − ∫ − hi (t )dt (3) eai , j eai+, j = eai−, j + (t j − t j −1 ) = eai−, j + tp j j − k + + − eai , j +1 = eai , j − ρi ⋅ eai , j = (1 − ρi ) ⋅ eai , j + (t j − t j −1 ) = ∑ (1 − ρi ) ⋅ (t j − k − t j − k −1 ) k =0
(4)
The reliability of system is expressed as m
m
R0 (t ) = ∏ ( Ri (t ) ) = ∏ exp −(λi− βi ⋅ eai+, j i =1 i =1
(
)
βi
(
− λi− βi ⋅ eai−, j
)
βi
)
(5)
4. Preventive maintenance dynamic interval model PM planning with dynamic interval is basically enacted as a bi-level decision-making process. The dynamic interval decision is regarded as a leader and PM action decisions act as the followers. We use a leader-follower joint method[31,32] to modeling the bi-level decision-making process. Fig.1 illustrates the PM planning model with dynamic interval. The non-periodical PM schedule under consideration is over the period [0, T], whose planning horizon is divided into n discrete intervals. The frequencies of PM action are determined by the reliability characteristic of components. The dynamic interval decision can be denoted as
… …
TPset=[tp1,tp2, ,tpj, ,tpn], and ∃tph ∈ TPset , tph ≠ tpk ( k = 1, 2, , n and k ≠ h ) . For interval tpj, the PM actions are adjusted according to the economic object and the reliability constraint, which are denoted as Φset=[Φ1,Φ2, ,Φj, ,Φn] and Φj=[ΩM,j,ΩR,j,ΩI,j], where ΩM,j is the components set of repair at period j, ΩR,j is the components set of replacement at period j, ΩI,j is the components set of inspection at period j, and ΩM,j∩ΩR,j,∩ΩI,j=Ø.
… …
We assume that the maximum allowed PM interval of pi is tpimax and the minimum allowed PM interval of pi is tpimin . Therefore, the maximum allowed PM interval of system is defined as: max tpsystem = min{tpimax i ∈ (1, 2, , m)}
(6)
In the same way, the minimum allowed PM interval of system is defined as: min tpsystem = min{tpimin i ∈ (1, 2, , m)}
(7)
Considering the dynamic intervals whose length is tp j , the number of intervals is calculated as:
min T − tpsystem +1 n = max k ∈ ≤ k min tpsystem
(8)
In this case, we donated a bi-level decision making parameter as shown in Fig.2. The leader parameter
X=[x1,…,xj,…xn] is introduced to describe dynamic interval vector TPset, the derivation of xj should satisfy such constraints as
∑ j =1 x j = T n
(9)
Similarly, the PM actions can be characterized by the follower parameter Y=[Y1,…,Yj,…Yn] as shown in Fig.2, such that ∀ Φj ∈ Φset ( 1 ≤ j ≤ n ), Y j = YIj , YMj , YRj where YIj=( yIj1,…, yIji,…, yIjm), YMj=( yMj1,…, yMji,…, yMjm) and
∈
YRj=( yRj1,…, yRji,…, yRjm), 1 ≤ i ≤ m . When Yj Yset, there occurs a constraint,
{∑
m i =1
( yIji + yMji + yRji ) = m ∀j} .
5. Preventive maintenance planning with dynamic interval PM planning with dynamic interval is a complex decision-making process, which joint PM intervals and PM actions optimization and it is a NP-hard problem. MOGA are powerful optimization techniques, inspired by the principles of natural selection and species evolution[33-36]. However, it is challenging that the individual generated by initialization, crossover or mutation operations may not satisfy the PM actions schedule feasible constraints. Due to the special characteristic of PM planning problem, we develop a framework for solving the non-periodical PM schedule with dynamic interval based on the MOGA. Updating strategy, deleting strategy, inserting strategy and moving forward strategy are denoted in the framework, which are used to correct the individuals against PM actions schedule feasible constraints. 5.1. MOGA encoding In this problem, the chromosome indicates the value of each interval and PM actions that are performed in all system components. Consistent with the bi-level decision making of PM planning optimization, two kinds of chromosomes are composed: the upper-level chromosome (ULC) and the lower-level chromosome (LLC). According to the PM dynamic interval model, a ULC represents solution X that can be divided into a group of LLCs and a group of LLCs represent solution Y. Fig.3 shows an example of MOGA encoding (n=6,m=10,T=36 month). As shown in Fig.3, the ULC represents the planning horizon 36 month to be divided into six periods, which are expressed as TPset=[tp1,tp2, tp3,tp4, tp5,tp6]=[8,4,6,6,5,7]. It means that the first PM action realizes at the 8-th month and the second realizes at the 12-th month. Based on the ULC, LLCI (Lower-level chromosome for inspection schedule), LLCM (Lower-level chromosome for repair schedule) and LLCR (Lower-level chromosome for repair schedule) can be generated. The component indexed by “1” in the LLCIj reveals that the component is performed inspection action at period j. In the same way, the component indexed by “1” in the LLCMj means that it is selected to undergo implementing repair action at period j. The component is executed replacement action when the component is indexed by “1” in the LLCRj.
5.2 Initial population MOGA involves two initialization stages, i.e., the upper- and lower-level initializations. The upper-level initialization takes place in the following two steps: min max Step1: Generate randomly a number of ULC whose elements are between tpsystem and tpsystem . According to (8)
in section 3, the length of each ULC is set as n. Step2: After initiating the ULC, the lower-level initialization begins. Since the ULC is decomposed into a group of LLCs, the number of LLCs should equal triple the number of intervals in ULC. Fig.3 presents an example with three LLCs. Each group of LLCs represents an initial solution of PM planning. Redundant PM actions are generated. 5.3 lower-level chromosome corrections To obtain feasible solutions, each chromosome must satisfy certain PM feasible constraints during initialization, crossover and mutation. PM feasible constraints are related to the restrictions as follows: (1) The minimum level of system reliability Rmin is achieved over the planning horizon. min max and less then tpsystem .Therefore, PM actions can (2) The interval between two PM actions is more than tpsystem min not been performed at t ∈ (t j , t j + tpsystem ]. min max (3) Similarly to constraint (2), PM actions must be performed once in period at t ∈ (t j + tpsystem , t j + tpsystem ]. According to those constraints, we propose four type strategies for chromosome correction: updating strategy, deleting strategy, inserting strategy and moving forward strategy. 5.3.1 Updating strategy
The updating strategy is used to change the PM actions of the j-th LLCs chromosome in order to make R0 (t ) ≥ Rmin ( t ∈ (t j , t j +1 ] ). The procedures of update strategy are elaborated below:
Step1: The system reliability R0 (t ) where t is between tj and tj+1 is calculated. If ∀t ∈ (t j , t j +1 ] , R0 (t ) ≥ Rmin , then turn to Step6. Step2: The reliability of components that are performed inspection action is calculated. Step3: The component which has minimum reliability is selected. Step4: The PM level of the selected component is updated. The element of the selected component in LLCI is set ‘0’and the element in LLCM is set ‘1’. Fig.4 illustrates the process of updating strategy. Step5: Repeat Step1-4. Step6: End.
5.3.2 Deleting strategy Deleting strategy can deal with those redundant PM actions that are generated by crossover and mutation operations of MOGA. The procedures of delete strategy are elaborated below: Step1: Set Atemp is initialized as NULL. min Step2: If ∀t ∈ (t j , t j + tpsystem ] , yIjk+ yMjk+ yRjk=0, then turn to Step5;
Step3: The redundant PM actions are saved to set Atemp and marked. Step4: The redundant PM actions are deleted. The elements of the selected PM actions in LLCI are set ‘1’, the element in LLCM and LLCR are set ‘0’, as shown in Fig.5. Step5: End.
5.3.3 Inserting strategy According to PM feasible constraint (3), PM actions must be performed once in period at min max t ∈ (t j + tpsystem , t j + tpsystem ] . However, this feasible constraint might be not satisfied because of the crossover and max min mutation operations of MOGA. If ∀j ∈ (0, n), x ' j > tpsystem and x ' j ≥ 2tpsystem as shown in Fig.6, PM actions need
to be inserted at t ∈ (t j −1 , t j −1 + x ' j ) , where x ' j is the element x j of ULC after crossover and mutation operations. The procedures of insert strategy are elaborated below: Step1: PM interval adjustment integer tptemp is initialized as 0. min Step2: The system reliability R0 (t ) at t j − tpsystem − tptemp is calculated.
Step3: if R0 (t ) < Rmin , then tptemp = tptemp + 1 and turn to Step2. min Step4: New PM actions are inserted at t ' j , where t ' j = t j − tpsystem − tptemp . Turn to updating strategy, as shown
in Fig.7. Step5: End.
5.3.4 Moving forward strategy Moving forward strategy is another strategy that used to deal with the problem that the ULC may not satisfy the feasible constraint after crossover and mutation operations. We assume that the leader parameter X is changed max min < x ' j < 2tpsystem and PM actions are inserted at to X ' by crossover and mutation operations. If ∀j ∈ (0, n), tpsystem min max t ∈ (t j −1 , x ' j ) and t j −1 + tpsystem < t < t j −1 + tpsystem based on PM feasible constraint (3), then the interval between t min min and t ' is tp ' = t ' j − t = (t j −1 + x ' j ) − t = x ' j − (t − t j −1 ) ⇒ tp ' < x ' j − tpsystem ⇒ tp ' < tpsystem . It conflicts with PM
feasible constraint (2). Therefore, the PM actions at t ' j need to be moved forward, as shown in Fig.8. The procedures of insert strategy are elaborated below: Step1: PM interval adjustment integer tptemp is initialized as 0. max Step2: The PM actions at tj are moved forward to t j −1 + tpsystem − tptemp . The system reliability R0 (t ) at max t j −1 + tpsystem − tptemp is calculated.
Step3: If R0 (t ) < Rmin , then tptemp = tptemp + 1 and turn to Step2.
Step4: The PM actions at tj+1 are deleted, as shown in Fig.9. Step5: End.
6. Example In order to progress example analysis, several assumptions are drawn as follows: 1) The time-dependent failure distributions and the degraded behaviors of the components can be acquired. 2) The improvement factors, the PM times and the related costs in PM can be identified. 3) The PM actions are performed at the end of period. 4) The downtime is defined as the sum of PM operating time. It is not charged if any component is repaired or replaced. A typical mechatronic system (rotary table system of NC machine tool) is used as an example to explain the procedure of PM scheduling. Rotary table system can be represented by 10 main components: (1) grating sensing, (2) electric motor control, (3) rotary guide, (4) torque motor, (5) spindle, (6) brake, (7) cooling system, (8) coupling, (9) bearing, (10) tool. In normal operation its function is to complete the up-going and
down-going of the elevator, as well as to stop the elevator cab to the given floor. A representative data set based on the practical data of rotary table system of NC machine tool is shown in Table 1.
The system availability depends on both reliability and maintainability. The expression to describe the operational availability is by the mean up-time (MUT) and the down-time (MDT) [37]. Considering non-periodical PM problem, the system availability is defined as: MUT F = A MUT+MDT n MUT = tp j − TMj + TRj + TIj j =1 n m m m = tp j − tMi yMij − tRi yRij − tIi yIij j =1 i =1 i =1 i =1 n MDT = TMj + TRj + TIj j =1 n m m m = tMi yMij + tRi yRij + tIi yIij j =1 i =1 i =1 i =1
∑(
(
∑
∑
))
∑
∑(
)
∑∑
∑
∑
(10)
∑
We know that if the system carries a high rate of occurrence of failure through a period, then we are at risk of experiencing high number, and hence cost, of failures. Conversely, a low rate of occurrence of failure in period j should yield a low cost of failure. Hence regardless of any repair or replacement actions (which are assumed to occur at the end of the period) in period j, there is still a cost associated with the possible failures that can occur during the period. Further, the total PM cost of system at any stage is defined as n
FC =
∑(CFj + CMj + CPj + CIj ) j =1 m m m m −β + β − β cfi λi i (eai, j ) i − (eai, j ) i + cMi yMij + cRi yRij + cIi (1− yMij − yRij ) j =1 i =1 i =1 i =1 i =1 n
=
∑∑
=
∑∑{cfiλi−β (eai+, j )β − (eai−, j )β + cmi yMij + cri yRij } + ∑cIi 1− ∏(1− yMij − yRij )
n
) ∑
(
m
∑
n
i
j =1 i =1
∑
i
i
(11)
m
i =1
j =1
The supposed related parameters the PM costs in the example are listed in Table 1. Here, the expected life of system is set to T=10800 day. The initial reliability of the components are all set to Ri(0)=0.999. The minimum reliability for judging whether preventive maintained or not are set to and Rmin=0.85. The particular implementation used in the case study below is based on the Strength Pareto Evolutionary Algorithm 2+ (SPEA2+). A detail of SPEA2+ with a conventional method of Pareto surface generation for a complex engineering design problem is given in [38, 39]. A laptop computer (Intel/Core 2, 1.67 GHz and 2 GB RAM) with MATLAB (R2008 a2) software is utilized to solve the model. The parameters of SPEA2+ are as follows: Number of generations: 300. Population size: 100. Archive size: 50. The crossover rate: 0.8. The mutation rate: 0.03. According to the given parameters in Table 1, an optimal non-periodic PM schedule solution is presented in Table 2. It provides a tradeoff between the system availability and the total PM cost.
Moreover, The fixed interval method of ref.[40] is used to compare with our dynamic interval method. The optimal periodic PM schedule which fixed interval (tpfixed=150 days) set is calculated. The periodic PM schedule and the corresponding availability and cost information are listed in Table 3. The value of total cost for the optimal non-periodic PM schedule solution in Table2 is $7939.1. Comparing it with Table 3, we find that the availability is lower and the total cost is higher than the results in Table 2. The total cost for performing the PM actions has been reduced by 7.44%. The results reveal that the proposed dynamic interval PM policy is more advantage than fixed interval considered. The reliability changing of components in the example are shown in Fig.10, respectively. Observing the reliability of components, as shown in Fig.10, one can see that the reliability curve of the dynamic-interval optimal schedule is smoother than the fixed-interval schedule and PM actions can be better adjusted. The change of reliability of the components could be estimated by the model and then the information can be used to initiate additional PM activities. It makes the system under the proposed PM policy safer than fixed interval considered.
Furthermore, cost-reliability evaluates contribution of extended system reliability per unit-cost in PM schedule and the Cost-reliability of period j can be expressed as: CR(t j ) =
R0+ (t j ) − R0− (t j ) (CFj + CMj + CPj + CIj )
(12)
where R0− (t j ) is the system reliability before PM and R0+ (t j ) is the system reliability after PM.
Cost-reliability of each stage for two kinds of PM policy is calculated as show in Table 4. It can be observed that the average cost-reliability of dynamic interval PM policy is higher than the average cost-reliability of fixed interval PM policy. Maintenance mangers could use the dynamic interval PM policy to make the system safer (higher reliability) under the given budget. Therefore, the dynamic interval PM policy is more effective and the reliability is heightened more than fixed interval PM policy under using the same money.
7. Conclusion This paper studies PM with dynamic interval for a multi-component system. The effects of non-periodic PM actions to reliability are formulated based on the improvements of the survival and failed parts for constructing the reliability model of system following PM. A novel framework for solving the non-periodical PM schedule with dynamic interval based on the MOGA is proposed. By applying the proposed dynamic interval model, it is possible to find the optimal non-periodic PM schedule which achieves a higher level of system availability with less total PM-related cost. The studied results show that considering dynamic interval turns the solution space much more flexible and schedules can be better adjusted, providing gains in the overall system average availability, when compared to those obtained by an optimized periodic PM schedule. A continuation of this work intends to investigate more realistic situations, where the costs contemplate other kinds of impacts obtained by more elaborated models. Another future work is the investigation of a sensitivity analysis on the cost associated with repair and replacement activities and reliability characteristic parameters.
References 1. Martorell S, Sánchez A, Carlos S, et al. Comparing effectiveness and efficiency in technical specifications and maintenance optimization. Reliability Engineering & System Safety, 2002, 77(3): 281-289. 2. Ahmad R, Kamaruddin S. An overview of time-based and condition-based maintenance in industrial application. Computers & Industrial Engineering, 2012, 63(1): 135-149. 3. Tsai Y T, Wang K S, Tsai L C. A study of availability-centered preventive maintenance for multi-component systems. Reliability Engineering & System Safety, 2004, 84(3): 261-270. 4. Cassady C R, Kutanoglu E. Integrating preventive maintenance planning and production scheduling for a single machine. Reliability, IEEE Transactions on, 2005, 54(2): 304-309. 5. Bartholomew-Biggs M, Christianson B, Zuo M. Optimizing preventive maintenance models. Computational Optimization and Applications, 2006, 35(2): 261-279. 6. Lim J H, Park D H. Optimal periodic preventive maintenance schedules with improvement factors depending on number of preventive maintenances. Asia-Pacific Journal of Operational Research, 2007, 24(01): 111-124. 7. Liao W, Pan E, Xi L. Preventive maintenance scheduling for repairable system with deterioration. Journal of Intelligent Manufacturing, 2010, 21(6): 875-884. 8. Wang C H, Tsai S W. Optimizing bi-objective imperfect preventive maintenance model for series-parallel system using established hybrid genetic algorithm. Journal of Intelligent Manufacturing, 2012: 1-14. 9. El-Ferik S, Ben-Daya M. Age-based hybrid model for imperfect preventive maintenance. IIE Transactions, 2006, 38(4): 365-375. 10. Ebrahimipour V, Najjarbashi A, Sheikhalishahi M. Multi-objective modeling for preventive maintenance scheduling in a multiple production line. Journal of Intelligent Manufacturing, 2013: 1-12. 11. Levitin G, Lisnianski A. Short communication optimal replacement scheduling in multi state series–parallel systems. Quality and Reliability Engineering International, 2000, 16(2): 157-162. 12. Schutz J, Rezg N, Léger J B. Periodic and sequential preventive maintenance policies over a finite planning horizon with a dynamic failure law. Journal of Intelligent Manufacturing, 2011, 22(4): 523-532. 13. Wang C H, Lin T W. Improved particle swarm optimization to minimize periodic preventive maintenance cost for series-parallel systems. Expert Systems with Applications, 2011, 38(7): 8963-8969. 14. Tam A S B, Chan W M, Price J W H. Optimal maintenance intervals for a multi-component system. Production Planning and Control, 2006, 17(8): 769-779. 15. Alardhi M, Hannam R G, Labib A W. Preventive maintenance scheduling for multi-cogeneration plants with production constraints. Journal of Quality in Maintenance Engineering, 2007, 13(3): 276-292. 16. Shirmohammadi A H, Zhang Z G, Love E. A computational model for determining the optimal preventive maintenance policy with random breakdowns and imperfect repairs. Reliability, IEEE Transactions on, 2007, 56(2): 332-339. 17. Bartholomew-Biggs M, Christianson B, Zuo M. Optimizing preventive maintenance models. Computational Optimization and Applications, 2006, 35(2): 261-279. 18. Harrou F, Tassadit A, Bouyeddou B, et al. Efficient Optimization Algorithm for Preventive-Maintenance in Transmission Systems. Journal of Modelling & Simulation of Systems, 2010, 1(1): 59–67. 19. Moghaddam K S, Usher J S. Preventive maintenance and replacement scheduling for repairable and maintainable systems using dynamic programming. Computers & Industrial Engineering, 2011, 60(4): 654-665. 20. Lin T W, Wang C H. A hybrid genetic algorithm to minimize the periodic preventive maintenance cost in a series-parallel system. Journal of Intelligent manufacturing, 2012, 23(4): 1225-1236. 21. Tsai Y T, Wang K S, Tsai L C. A study of availability-centered preventive maintenance for multi-component systems. Reliability Engineering & System Safety, 2004, 84(3): 261-270.
22. Nakagawa T. Periodic and sequential preventive maintenance policies. Journal of Applied Probability, 1986, 26(4): 536-542. 23. Wang H. A survey of maintenance policies of deteriorating systems. European journal of operational research, 2002, 139(3): 469-489. 24. Percy D F, Kobbacy K A H. Determining economical maintenance intervals. International Journal of Production Economics, 2000, 67(1): 87-94. 25. Lapa C M F, Pereira C M N A, Frutuoso e Melo P F. Surveillance test policy optimization through genetic algorithms using non-periodic intervention frequencies and considering seasonal constraints. Reliability Engineering & System Safety, 2003, 81(1): 103-109. 26. Lapa C M F, Pereira C M N A, de Barros M P. A model for preventive maintenance planning by genetic algorithms based in cost and reliability. Reliability Engineering & System Safety, 2006, 91(2): 233-240. 27. Lin T W, Wang C H. A new approach to minimize non-periodic preventive maintenance cost using importance measures of components. Journal of Scientific and Industrial Research, 2010, 69: 667-671. 28. Fitouhi, M. C., & Nourelfath, M. Integrating noncyclical preventive maintenance scheduling and production planning for a single machine. International Journal of Production Economics, 2012, 136(2): 344-351. 29. Kijima M. Some results for repairable systems with general repair. Journal of Applied Probability, 1989, 26(1): 89-102. 30. Doyen L, Gaudoin O. Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliability Engineering & System Safety, 2004, 84(1): 45-56. 31. Hansen P, Jaumard B, Savard G. New branch-and-bound rules for linear bilevel programming. Siam Journal on Scientific and Statistical Computing, 1992, 13(5):1194-1217. 32. Du, G., Wang, J.Z. Product family hierarchical associated design and its hierarchical optimization. In: Proceeding of IEEE International Conference on Industrial Engineering and Engineering Management, Hong Kong, China; 2009. p. 1642-1645. 33. Wang, Y., & Handschin, E. A new genetic algorithm for preventive unit maintenance scheduling of power systems. International Journal of Electrical Power and Energy Systems, 2000, 22(5): 343–348. 34. Allaoui, H., & Artiba, A. Integrating simulation and optimization to schedule a hybrid flow shop with maintenance constraints. Computers and Industrial Engineering, 2000, 47(4): 431–450. 35. Limbourg, P., & Kochs, H. D. Preventive maintenance scheduling by variable dimension evolutionary algorithms. International Journal of Pressure Vessels and Piping, 2006, 83(4): 262–269 36. Suresh, K., Kumarappan, N. Combined genetic algorithm and simulated annealing for preventive unit maintenance scheduling in power system. In 2006 IEEE power engineering society general meeting, PES, 18–22 June 2006, Montreal, Quebec, Canada. 37. Ebeling CE. In introduction to reliability and maintainability engineering. New York: McGraw-Hill; 1997. 38. Abido M A. Multiobjective evolutionary algorithms for electric power dispatch problem. Evolutionary Computation, IEEE Transactions on, 2006, 10(3): 315-329. 39. Qiu, J, Wang, X, & Dai, G. Improving the indoor localization accuracy for CPS by reorganizing the fingerprint signatures. International Journal of Distributed Sensor Networks, 2014. 40. Chung S H, Lau H C W, Ho G T S, et al. Optimization of system reliability in multi-factory production networks by maintenance approach. Expert Systems with Applications, 2009, 36(6): 10188-10196.
Acknowledgments This work was supported by the National Natural Science Foundation of China (No.51205347, 51322506), Zhejiang Provincial Natural Science Foundation of China (No. LR14E050003), the Fundamental Research Funds for the Central Universities, Innovation Foundation of the State Key Laboratory of Fluid Power Transmission and Control, and Zhejiang University K.P.Chao's High Technology Development Foundation. Sincere appreciation is extended to the reviewers of this paper for their helpful comments.
Table 1 Parameters for components in the example Component
PM Parameters λi (1/day) βi
ρi
cFi ($)
cMi ($)
cRi ($)
tMi (hour)
tRi (hour)
tIi (hour)
5
p1
50
2.30
0.75
195
47
180
11.25
22.5
p2
46
2.15
0.62
245
37
205
8.75
17.5
5
p3
96
1.70
0.48
215
53
210
12.5
25
5
p4
53
2.20
0.67
250
58
255
13.75
27.5
5
p5
89
1.85
0.52
210
46
215
12
24
5
p6
53
2.05
0.58
235
34
215
8
16
5
p7
47
2.15
0.55
265
62
250
16.25
32.5
5
p8
69
1.98
0.50
205
41
185
10.5
21
5
p9
151
1.75
0.68
220
34
220
7.5
15
5
p10
49
2.10
0.65
275
39
240
9.5
19
5
Table 2 An optimal non-periodic PM schedule of the rotary table system in the example Stage
Time
Interval
(day)
(day)
Component p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
1
150
150
I
I
I
I
M
I
M
M
I
M
2
210
60
M
M
M
M
I
R
R
M
M
M
3
330
120
M
R
M
M
M
R
R
R
I
R
4
510
180
R
M
R
M
R
R
R
R
R
R
5
600
90
I
M
I
M
I
I
I
I
I
I
6
720
120
R
M
M
M
M
R
R
M
M
R
7
900
180
R
R
M
M
M
M
R
R
M
R
‘I’ is defined as inspection, ‘M’ is defined as repair and ‘R’ is defined as replacement
Total cost
System
FC ($)
availability FA
7939.1
96.35%
Table 3 The optimal periodic PM schedule (tpfixed =150 days) of the rotary table system in the example Stage
Time
Interval
(day)
(day)
Component p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
1
150
150
M
M
I
M
I
M
M
M
I
M
2
300
150
R
M
R
M
R
R
R
R
M
M
3
450
150
R
R
R
R
R
R
R
M
I
R
4
600
150
M
M
M
M
I
M
M
M
R
M
5
750
150
M
R
M
R
R
R
R
R
I
R
6
900
150
R
M
R
M
R
R
M
R
M
R
7
1050
150
I
I
I
I
I
I
I
I
I
I
Total cost
System
FC ($)
availability FA
8577.5
96.17%
Table 4 The cost-reliability and average cost-reliability of the rotary table system under two kinds of PM policy Stage
Dynamic interval PM policy Reliability
Fixed interval PM policy Cost-reliability
Before PM
After PM
1
0.936
0.959
2
0.911
0.987
3
0.912
0.988
Reliability
Cost-reliability
Before PM
After PM
3.379E-05
0.936
0.979
7.607E-05
9.066E-05
0.86
0.994
9.607E-05
5.606E-05
0.906
0.992
4.640E-05
4
0.87
0.997
7.187E-05
0.919
0.978
8.769E-05
5
0.965
0.979
1.909E-05
0.853
0.993
8.192E-05
6
0.884
0.988
9.157E-05
0.917
0.995
5.542E-05
0.861
0.986
0.91
0.91
0.000E+00
7
Average Cost-reliability
1.003E-04 6.619E-05
Average Cost-reliability
6.337E-05
Fig.1. PM planning model with dynamic interval Fig.2. Formalization of leader-follower bi-level decision-making process Fig.3. Example of MOGA genotype Fig.4. Process of updating strategy Fig.5. Process of deleting strategy Fig.6. Process of PM scheduling of inserting strategy Fig.7. Process of inserting strategy Fig.8. Process of PM scheduling of moving forward strategy Fig.9. Process of moving forward strategy Fig.10.The reliability changing of the components in the example
min min [ tpsystem , 2 ⋅ tpsystem − 1]
min min [2 ⋅ tpsystem ,3 ⋅ tpsystem − 1]
min min [ j ⋅ tpsystem , ( j + 1) ⋅ tpsystem − 1]
min min [n ⋅ tpsystem − 2, n ⋅ tpsystem − 1]
min [n ⋅ tpsystem ,T ]
Fig.1. PM planning model with dynamic interval
X=[x1
xn]
xj
Y=[Y1
tp1 tp2 tp3
... tpj-1 tpj tpj+1 ... tpn-1 tpn
Φ1 Φ2 Φ3
... Φj-1 Φj Φj+1 ... Φn-1 Φn
Yn]
Yj
Yj=[YIj YMj YRj]
YIj=(yIj1,
0
1
1
...
yIji,
1
1
, yIjm)
0
...
YMj=(yMj1,
0
1
1
0
0
...
yMji,
0
0
, yMjm)
0
...
YRj=(yRj1,
1
0
0
Fig.2. Formalization of leader-follower bi-level decision-making process
0
0
...
yRji,
0
0
, yRjm)
1
...
0
0
Fig.3. Example of MOGA genotype
Fig.4. Process of updating strategy
Fig.5. Process of deleting strategy
Fig.6. Process of PM scheduling of inserting strategy
Fig.7. Process of inserting strategy
Fig.8. Process of PM scheduling of moving forward strategy
Fig.9. Process of moving forward strategy
1
(1) Component 1
(2) Component 2
(3) Component 3
(4) Component 4
(5) Component 5
(6) Component 6
(7) Component 7
(8) Component 8
2
(9) Component 9
(10) Component 10
Fig.10. The reliability changing of the components in the example
3
A non-periodic preventive maintenance scheduling model is proposed. A framework for solving the non-periodical PM schedule problem is developed. The interval of non-periodic PM is flexible and schedule can be better adjusted. Dynamic interval leads to more efficient solutions than fixed interval does.
4
An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems Gao Yicong, Yixiong Feng, Zixian Zhang, Jianrong Tan
www.elsevier.com/locate/ress
PII: DOI: Reference:
S0951-8320(15)00098-8 http://dx.doi.org/10.1016/j.ress.2015.03.032 RESS5280
To appear in:
Reliability Engineering and System Safety
Received date: 15 November 2014 Revised date: 20 March 2015 Accepted date: 29 March 2015 Cite this article as: Gao Yicong, Yixiong Feng, Zixian Zhang, Jianrong Tan, An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j. ress.2015.03.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An optimal Dynamic Interval Preventive Maintenance Scheduling for Series Systems Gao Yicong1, Yixiong Feng1, *, Zixian Zhang2, Jianrong Tan1 1.
State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, China 2.
Department of Mechanical Science and Engineering, Tokyo Institute of Technology, Tokyo, Japan
Corresponding author: Yixiong Feng Mailing Address: State Key Lab of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China Telephones: +86-13819161642 (Mobile) E-mail: [email protected]
Abstract: This paper studies preventive maintenance (PM) with dynamic interval for a multi-component system. Instead of equal interval, the time of PM period in the proposed dynamic interval model is not a fixed constant, which varies from interval-down to interval-up. It is helpful to reduce the outage loss on frequent repair parts and avoid lack of maintenance of the equipment by controlling the equipment maintenance frequency, when compared to a periodic PM scheme. According to the definition of dynamic interval, the reliability of system is analyzed from the failure mechanisms of its components and the different effects of non-periodic PM actions on the reliability of the components. Following the proposed model of reliability, a novel framework for solving the non-periodical PM schedule with dynamic interval based on the multi-objective genetic algorithm is proposed. The framework denotes the strategies include updating strategy, deleting strategy, inserting strategy and moving forward strategy, which is set to correct the invalid population individuals of the algorithm. The values of the dynamic interval and the selections of PM action for the components on every PM stage are determined by achieving a certain level of system availability with the minimum total PM-related cost. Finally, a typical rotary table system of NC machine tool is used as an example to describe the proposed method. Keywords: dynamic interval; non-periodic preventive maintenance; optimization; machine tools technology 1. Introduction Preventive maintenance (PM) is defined as a set of activities aimed at improving the overall reliability and availability of a system. While planning the PM schedule according to the defined activities, the time between two PM actions is an important aspect of actions adopted and it would affect the maintenance economics. In term of PM interval, PM can be categorized into two kinds, periodic PM (fixed interval) and non-periodic PM (dynamic interval) [1, 2]. In the former approach, the planning horizon is segmented into discrete equal intervals. For each PM stage, a decision must be made to perform one of the three PM actions (inspection, repair and replacement) on each component by maximizing system benefit in PM. Instead of searching for equally spaced PM actions, non-periodic PM approach search for the flexible interval in which PM interventions should be performed[3]. During the last few decades, numerous papers have been published on periodic PM modeling and optimization. Cassady and Kutanoglu et al.[4] developed an integrated mathematical model for a single-machine problem with periodic PM interval. They defined the total weighted expected completion time as the objective function. Their model allowed multiple maintenance activities and explicitly captures the risk of not performing maintenance. Bartholomew-Biggs et al.[5] proposed a new PM formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. The formulation involved the global minimization of a non-smooth performance function. Lim and Park[6] proposed a periodic PM policy, which keeping the pattern of hazard rate unchanged by PM. They evaluated the expected cost rate per unit time based on computing the expected number of failures depending on the hazard rate of the underlying life distribution of the system. Liao et al.[7] developed a reliability-centered sequential PM model for monitored repairable deteriorating system. They supposed that system’s reliability could be monitored continuously and perfectly, whenever it reaches the threshold, the imperfect repair must be performed to restore the system. Wang and Tsai[8] established a bi-objective imperfect PM model of a series-parallel system. They developed a unit-cost cumulative reliability expectation measure to evaluate the extent to which maintaining each individual component benefits the total maintenance cost and system reliability over the operational lifetime. El-Ferik and Ben-Daya[9] developed a hybrid age-based model for imperfect PM involving maintainable and non-maintainable failure modes. They determined the number of PM actions and the length of PM intervals that minimize the total long-term expected cost per unit time. Ebrahimipour et al.[10] developed a multi-objective PM scheduling model in a multiple production line. They defined reliability of production lines, costs of
maintaining, failure and downtime of system as multiple objectives and different thresholds for available manpower, spare part inventory and periods under maintenance was applied. Levitin and Lisnianski et al.[11] presented an optimization model for PM scheduling in multi-state series-parallel systems. They considered the cost of unsupplied demand due to failures of components as an important part of the PM activities cost. Schutz et al.[12] proposed and modeled periodic and sequential PM policies for a system. The objective of periodic PM policy was to determine the optimal number of PM to achieve and the objective of sequential PM policy was to determine the optimal number of PM intervals and the duration of these different intervals. Wang and Lin[13] proposed an improved particle swam optimization of periodic PM. The optimal maintenance periods for all components in the system were determined according to the importance of components on system reliability, to minimize the periodic PM cost for a series–parallel system. Tam et al.[14] analyzed the affection of reliability, budget and breakdown outages cost to the calculation of optimal maintenance intervals. Three models were proposed to calculate optimal maintenance intervals for multi-component system in a factory subjected to minimum required reliability, maximum allowable budget and minimum total cost. Alardhi et al.[15] presented an optimization method for scheduling the PM tasks in separate and linked cogeneration plants and satisfying the maintenance and production constraints. Shirmohammadi et al.[16] presented an optimization method for scheduling the PM of a system that is subject to random failures, and investigated the decision rule for PM. They defined the time between preventive replacements and cut-off age as decision variables to determine the optimal maintenance policy. Bartholomew et al.[17] proposed a model in which each action of the PM reduces the equipment’s effective age. The optimization process involved minimizing a performance function that allows for the costs of the minimal repairs and eventual system replacement as well as for the costs of the PM during equipment is operating lifetime. Harrou et al.[18] formulated the model of imperfect maintenance optimization for series-parallel transmission system structure. They improved the availability of transmission system through selecting the optimal sequence of intervals to perform PM actions. Moghaddam et al.[19] presented mathematical models and a solution approach to determine the optimal PM schedules for a repairable and maintainable series system with equally-sized periods. Lin and Wang[20] identified important components and determined their maintenance priorities of a series-parallel system. The optimal maintenance periods of these important components were determined to minimize total maintenance cost given the allowable worst reliability of a repairable system. According to Tsai et al.[21], the interval and the type of PM action are the main decision variables of non-periodic PM. These make the optimization problem of non-periodic PM more complex. However, the interval of non-periodic PM is more flexible and the schedules of non-periodic PM can be better adjusted, reducing the outage loss on frequent repair parts through the interval (PM frequency) control[22]. In view of the potential important applications of non-periodic PM, some improvement ideas have been proposed. Wang[23] summarizes, classifies, and compares various existing maintenance policies for both single-unit and multi-unit systems. Relationships among different maintenance policies are also addressed. He found that non-periodic approach provided gains in the overall system average availability. Percy and Kobbacy[24] investigated two principal types of general models for determining the optimal maintenance intervals. The first model considered fixed PM intervals, while the second one was adaptable, allowing variable PM intervals. But they did not given clear comparison to help decide between these two approaches. Lapa et al.[25] defined inspections as special PM actions and the PM was said to be non-periodic. They developed a surveillance test optimization methodology based on genetic algorithms. Later, Lapa et al.[26] presented a methodology for PM policy evaluation based upon a cost-reliability model. They used flexible intervals between maintenance interventions. However, the proposed method defined maintenance action as PM action and replacement action was neglected. The method selected components to undergo maintenance at different time, while avoiding this for components in parallel. Lin and Wang[27] proposed a non-periodic PM policy method for series-parallel systems, which is based on failure limit policy. Firstly, they identified the parallel sub-system required to be maintained. Secondly, they identified the maintained parallel sub-system by the measure of unit-cost extended life. But it
was a time-consuming task for searching the optimum values of the parameter and the replacement action was ignored in the method. The detail of the solving method was not mentioned. Fitouhi and Nourelfath[28] proposed an integrates noncyclical preventive maintenance method with tactical production planning in multi-state systems. They suggested noncyclical preventive replacements of components, and minimal repair on failed components. However, only preventive replacements were considered in their method for the sake of simplicity. These studies paid attention on periodic PM modeling and optimization. However, it is found that there is a few works in on non-periodic PM scheduling and for simple. Therefore, we expand upon the previous work of Lapa et al.[25,26]. A comprehensive model for the non-periodic PM scheduling problem was developed and a solving framework based on the multi-objective genetic algorithm (MOGA) was proposed, which integrated a population correction procedure. The main contribution of this research is to define a general model for PM scheduling problem, in which PM actions are performed under dynamic intervals, and then develop a solving framework to determine optimal non-periodic PM schedules. Some new issues that are aspects of real world problem of non-periodic PM scheduling are considered in the general model. The time-based schedule of non-periodic PM actions that maximize the system availability and minimize the total cost of those actions is developed. The following of the paper is organized as follows: In section 2, the notations are used throughout this article are given. The expressions for the reliability function under the non-periodical PM actions are developed in section 3 and PM dynamic interval model are explained in Section 4. A framework based on MOGA for obtaining the optimum solution is discussed in Section 5. To carry out the performance of the result obtained, an illustrative example is analyzed in Section 6 and finally, some concluding remarks and future work are given in Section 7. 2. Notation
m: Number of component.
n: Number of intervals.
nt : Numbers of non-periodic PM up to time t.
T: Length of the planning horizon.
i: Index of components, i = 1, …, m.
j: Index of intervals, j = 1, …, n.
R0(t): Instant reliability of the system at t.
Ri(t): Instant reliability of component i at t.
tpj: The length of the interval of period j.
tptemp : PM interval adjustment integer.
min tpsystem : the minimum allowed PM interval of system.
max tpsystem : the maximum allowed PM interval of system.
min min t j ∈ [ j ⋅ tpsystem , ( j + 1) ⋅ tpsystem − 1] if j ≤ n -1 tj: The time that PM actions are performed, min if j = n t j ∈ [n ⋅ tpsystem , T ]
ρi : Repair efficiency factor of component i. 0 < ρi < 1 , efficient repair; ρi = 1 , replacement (optimal
repair), as good as new; ρ = 0 , inspection (useless repair), as bad as old.
λi : Scale (characteristic life) parameters of component i.
βi : Shape parameters of component i.
eai−, j : Effective age of component i at the start of period j.
eai+, j : Effective age of component i at the end of period j.
TMj : Total repair time during the period j.
TRj : Total replacement time during the period j.
TIj : Total inspection time during the period j.
tMi : Repair time of component i.
tRi : Replacement time of component i.
tIi : Inspection time of component i.
CFj : Total PM cost associated with the possible failures during the period j.
CMj : Total repair actions cost during the period j.
CRj : Total replacement actions cost during the period j.
CIj : Total inspection actions cost during the period j.
cFi : Repair cost of possible failure cost of component i.
cMi : Repair cost of component i.
cRi : Replacement cost of component i.
cIi : Inspection cost of component i.
1 if component i at period j is inspected . yIij = 0 otherwise
1 if component i at period j is maintained . yMij = 0 otherwise
1 if component i at period j is replaced . yRij = 0 otherwise
Suppose there is a new repairable series system of m components, denoted as P = { p1 , , pi , , pm } .For the sake of simplicity, only series systems of components are considered in this paper. In order to model the effects of various PM actions in multi-component systems, three typical PM actions are considered as follows[2]. Inspection: The inspection action includes lubricating, cleaning, checking and adjusting. There is no effect will be on the rate of occurrence of failure of component and the component remains in a state of “as-bad-as-old”. Repair: The repair action restores the degraded strength partly. It effectively reduces state of the components in terms of the rate of occurrence of failure, but the effect of repair is not to return the system to the state of “as-good-as-new”. Replacement: The replacement action makes the components replace with new ones. It returns the state of the component as good as new, which means that the rate of occurrence of failure of component is recovered to its initial condition. 3. Reliability under non-periodical preventive maintenance Modeling the effect of PM actions is the first step of scheduling the PM program. In order to calculate evaluate the PM efficiency, several models with that type of assumption have already been proposed. The Arithmetic Reduction of Age model (ARA) is one of the most famous, in which each repair action reduces the age of the system is the basis of Kijima’s virtual age models[29]. Therefore, we use the approach proposed by Doyen et al.[30] to evaluate the reliability of a component undergoing a PM schedule. The failure intensity of pi is defined as:
nt −1 hi (t ) = hi t − ρi ∑ (1 − ρi ) j tpnt − j j =1
(1)
It is also assumed that component failure corresponds to the well known Non-Homogeneous Poisson Process and the rate of occurrence of failure can be expressed by: hi (t ) = λi ⋅ βi ⋅ t βi −1 λ > 0, β > 0 , for i = 1, , m
(2)
where λi and βi are the scale (characteristic life) and the shape parameters of pi for the Weibull distribution respectively. The Non-Homogeneous Poisson Process is similar to the Homogeneous Poisson Process with the exception that the rate of occurrence of failure is a function of time. For simplicity, we assume that the PM actions occur at the end of the period. The reliability of pi during period j is expressed as eai+, j Ri (t ) = exp − ∫ − hi (t )dt (3) eai , j eai+, j = eai−, j + (t j − t j −1 ) = eai−, j + tp j j − k + + − eai , j +1 = eai , j − ρi ⋅ eai , j = (1 − ρi ) ⋅ eai , j + (t j − t j −1 ) = ∑ (1 − ρi ) ⋅ (t j − k − t j − k −1 ) k =0
(4)
The reliability of system is expressed as m
m
R0 (t ) = ∏ ( Ri (t ) ) = ∏ exp −(λi− βi ⋅ eai+, j i =1 i =1
(
)
βi
(
− λi− βi ⋅ eai−, j
)
βi
)
(5)
4. Preventive maintenance dynamic interval model PM planning with dynamic interval is basically enacted as a bi-level decision-making process. The dynamic interval decision is regarded as a leader and PM action decisions act as the followers. We use a leader-follower joint method[31,32] to modeling the bi-level decision-making process. Fig.1 illustrates the PM planning model with dynamic interval. The non-periodical PM schedule under consideration is over the period [0, T], whose planning horizon is divided into n discrete intervals. The frequencies of PM action are determined by the reliability characteristic of components. The dynamic interval decision can be denoted as
… …
TPset=[tp1,tp2, ,tpj, ,tpn], and ∃tph ∈ TPset , tph ≠ tpk ( k = 1, 2, , n and k ≠ h ) . For interval tpj, the PM actions are adjusted according to the economic object and the reliability constraint, which are denoted as Φset=[Φ1,Φ2, ,Φj, ,Φn] and Φj=[ΩM,j,ΩR,j,ΩI,j], where ΩM,j is the components set of repair at period j, ΩR,j is the components set of replacement at period j, ΩI,j is the components set of inspection at period j, and ΩM,j∩ΩR,j,∩ΩI,j=Ø.
… …
We assume that the maximum allowed PM interval of pi is tpimax and the minimum allowed PM interval of pi is tpimin . Therefore, the maximum allowed PM interval of system is defined as: max tpsystem = min{tpimax i ∈ (1, 2, , m)}
(6)
In the same way, the minimum allowed PM interval of system is defined as: min tpsystem = min{tpimin i ∈ (1, 2, , m)}
(7)
Considering the dynamic intervals whose length is tp j , the number of intervals is calculated as:
min T − tpsystem +1 n = max k ∈ ≤ k min tpsystem
(8)
In this case, we donated a bi-level decision making parameter as shown in Fig.2. The leader parameter
X=[x1,…,xj,…xn] is introduced to describe dynamic interval vector TPset, the derivation of xj should satisfy such constraints as
∑ j =1 x j = T n
(9)
Similarly, the PM actions can be characterized by the follower parameter Y=[Y1,…,Yj,…Yn] as shown in Fig.2, such that ∀ Φj ∈ Φset ( 1 ≤ j ≤ n ), Y j = YIj , YMj , YRj where YIj=( yIj1,…, yIji,…, yIjm), YMj=( yMj1,…, yMji,…, yMjm) and
∈
YRj=( yRj1,…, yRji,…, yRjm), 1 ≤ i ≤ m . When Yj Yset, there occurs a constraint,
{∑
m i =1
( yIji + yMji + yRji ) = m ∀j} .
5. Preventive maintenance planning with dynamic interval PM planning with dynamic interval is a complex decision-making process, which joint PM intervals and PM actions optimization and it is a NP-hard problem. MOGA are powerful optimization techniques, inspired by the principles of natural selection and species evolution[33-36]. However, it is challenging that the individual generated by initialization, crossover or mutation operations may not satisfy the PM actions schedule feasible constraints. Due to the special characteristic of PM planning problem, we develop a framework for solving the non-periodical PM schedule with dynamic interval based on the MOGA. Updating strategy, deleting strategy, inserting strategy and moving forward strategy are denoted in the framework, which are used to correct the individuals against PM actions schedule feasible constraints. 5.1. MOGA encoding In this problem, the chromosome indicates the value of each interval and PM actions that are performed in all system components. Consistent with the bi-level decision making of PM planning optimization, two kinds of chromosomes are composed: the upper-level chromosome (ULC) and the lower-level chromosome (LLC). According to the PM dynamic interval model, a ULC represents solution X that can be divided into a group of LLCs and a group of LLCs represent solution Y. Fig.3 shows an example of MOGA encoding (n=6,m=10,T=36 month). As shown in Fig.3, the ULC represents the planning horizon 36 month to be divided into six periods, which are expressed as TPset=[tp1,tp2, tp3,tp4, tp5,tp6]=[8,4,6,6,5,7]. It means that the first PM action realizes at the 8-th month and the second realizes at the 12-th month. Based on the ULC, LLCI (Lower-level chromosome for inspection schedule), LLCM (Lower-level chromosome for repair schedule) and LLCR (Lower-level chromosome for repair schedule) can be generated. The component indexed by “1” in the LLCIj reveals that the component is performed inspection action at period j. In the same way, the component indexed by “1” in the LLCMj means that it is selected to undergo implementing repair action at period j. The component is executed replacement action when the component is indexed by “1” in the LLCRj.
5.2 Initial population MOGA involves two initialization stages, i.e., the upper- and lower-level initializations. The upper-level initialization takes place in the following two steps: min max Step1: Generate randomly a number of ULC whose elements are between tpsystem and tpsystem . According to (8)
in section 3, the length of each ULC is set as n. Step2: After initiating the ULC, the lower-level initialization begins. Since the ULC is decomposed into a group of LLCs, the number of LLCs should equal triple the number of intervals in ULC. Fig.3 presents an example with three LLCs. Each group of LLCs represents an initial solution of PM planning. Redundant PM actions are generated. 5.3 lower-level chromosome corrections To obtain feasible solutions, each chromosome must satisfy certain PM feasible constraints during initialization, crossover and mutation. PM feasible constraints are related to the restrictions as follows: (1) The minimum level of system reliability Rmin is achieved over the planning horizon. min max and less then tpsystem .Therefore, PM actions can (2) The interval between two PM actions is more than tpsystem min not been performed at t ∈ (t j , t j + tpsystem ]. min max (3) Similarly to constraint (2), PM actions must be performed once in period at t ∈ (t j + tpsystem , t j + tpsystem ]. According to those constraints, we propose four type strategies for chromosome correction: updating strategy, deleting strategy, inserting strategy and moving forward strategy. 5.3.1 Updating strategy
The updating strategy is used to change the PM actions of the j-th LLCs chromosome in order to make R0 (t ) ≥ Rmin ( t ∈ (t j , t j +1 ] ). The procedures of update strategy are elaborated below:
Step1: The system reliability R0 (t ) where t is between tj and tj+1 is calculated. If ∀t ∈ (t j , t j +1 ] , R0 (t ) ≥ Rmin , then turn to Step6. Step2: The reliability of components that are performed inspection action is calculated. Step3: The component which has minimum reliability is selected. Step4: The PM level of the selected component is updated. The element of the selected component in LLCI is set ‘0’and the element in LLCM is set ‘1’. Fig.4 illustrates the process of updating strategy. Step5: Repeat Step1-4. Step6: End.
5.3.2 Deleting strategy Deleting strategy can deal with those redundant PM actions that are generated by crossover and mutation operations of MOGA. The procedures of delete strategy are elaborated below: Step1: Set Atemp is initialized as NULL. min Step2: If ∀t ∈ (t j , t j + tpsystem ] , yIjk+ yMjk+ yRjk=0, then turn to Step5;
Step3: The redundant PM actions are saved to set Atemp and marked. Step4: The redundant PM actions are deleted. The elements of the selected PM actions in LLCI are set ‘1’, the element in LLCM and LLCR are set ‘0’, as shown in Fig.5. Step5: End.
5.3.3 Inserting strategy According to PM feasible constraint (3), PM actions must be performed once in period at min max t ∈ (t j + tpsystem , t j + tpsystem ] . However, this feasible constraint might be not satisfied because of the crossover and max min mutation operations of MOGA. If ∀j ∈ (0, n), x ' j > tpsystem and x ' j ≥ 2tpsystem as shown in Fig.6, PM actions need
to be inserted at t ∈ (t j −1 , t j −1 + x ' j ) , where x ' j is the element x j of ULC after crossover and mutation operations. The procedures of insert strategy are elaborated below: Step1: PM interval adjustment integer tptemp is initialized as 0. min Step2: The system reliability R0 (t ) at t j − tpsystem − tptemp is calculated.
Step3: if R0 (t ) < Rmin , then tptemp = tptemp + 1 and turn to Step2. min Step4: New PM actions are inserted at t ' j , where t ' j = t j − tpsystem − tptemp . Turn to updating strategy, as shown
in Fig.7. Step5: End.
5.3.4 Moving forward strategy Moving forward strategy is another strategy that used to deal with the problem that the ULC may not satisfy the feasible constraint after crossover and mutation operations. We assume that the leader parameter X is changed max min < x ' j < 2tpsystem and PM actions are inserted at to X ' by crossover and mutation operations. If ∀j ∈ (0, n), tpsystem min max t ∈ (t j −1 , x ' j ) and t j −1 + tpsystem < t < t j −1 + tpsystem based on PM feasible constraint (3), then the interval between t min min and t ' is tp ' = t ' j − t = (t j −1 + x ' j ) − t = x ' j − (t − t j −1 ) ⇒ tp ' < x ' j − tpsystem ⇒ tp ' < tpsystem . It conflicts with PM
feasible constraint (2). Therefore, the PM actions at t ' j need to be moved forward, as shown in Fig.8. The procedures of insert strategy are elaborated below: Step1: PM interval adjustment integer tptemp is initialized as 0. max Step2: The PM actions at tj are moved forward to t j −1 + tpsystem − tptemp . The system reliability R0 (t ) at max t j −1 + tpsystem − tptemp is calculated.
Step3: If R0 (t ) < Rmin , then tptemp = tptemp + 1 and turn to Step2.
Step4: The PM actions at tj+1 are deleted, as shown in Fig.9. Step5: End.
6. Example In order to progress example analysis, several assumptions are drawn as follows: 1) The time-dependent failure distributions and the degraded behaviors of the components can be acquired. 2) The improvement factors, the PM times and the related costs in PM can be identified. 3) The PM actions are performed at the end of period. 4) The downtime is defined as the sum of PM operating time. It is not charged if any component is repaired or replaced. A typical mechatronic system (rotary table system of NC machine tool) is used as an example to explain the procedure of PM scheduling. Rotary table system can be represented by 10 main components: (1) grating sensing, (2) electric motor control, (3) rotary guide, (4) torque motor, (5) spindle, (6) brake, (7) cooling system, (8) coupling, (9) bearing, (10) tool. In normal operation its function is to complete the up-going and
down-going of the elevator, as well as to stop the elevator cab to the given floor. A representative data set based on the practical data of rotary table system of NC machine tool is shown in Table 1.
The system availability depends on both reliability and maintainability. The expression to describe the operational availability is by the mean up-time (MUT) and the down-time (MDT) [37]. Considering non-periodical PM problem, the system availability is defined as: MUT F = A MUT+MDT n MUT = tp j − TMj + TRj + TIj j =1 n m m m = tp j − tMi yMij − tRi yRij − tIi yIij j =1 i =1 i =1 i =1 n MDT = TMj + TRj + TIj j =1 n m m m = tMi yMij + tRi yRij + tIi yIij j =1 i =1 i =1 i =1
∑(
(
∑
∑
))
∑
∑(
)
∑∑
∑
∑
(10)
∑
We know that if the system carries a high rate of occurrence of failure through a period, then we are at risk of experiencing high number, and hence cost, of failures. Conversely, a low rate of occurrence of failure in period j should yield a low cost of failure. Hence regardless of any repair or replacement actions (which are assumed to occur at the end of the period) in period j, there is still a cost associated with the possible failures that can occur during the period. Further, the total PM cost of system at any stage is defined as n
FC =
∑(CFj + CMj + CPj + CIj ) j =1 m m m m −β + β − β cfi λi i (eai, j ) i − (eai, j ) i + cMi yMij + cRi yRij + cIi (1− yMij − yRij ) j =1 i =1 i =1 i =1 i =1 n
=
∑∑
=
∑∑{cfiλi−β (eai+, j )β − (eai−, j )β + cmi yMij + cri yRij } + ∑cIi 1− ∏(1− yMij − yRij )
n
) ∑
(
m
∑
n
i
j =1 i =1
∑
i
i
(11)
m
i =1
j =1
The supposed related parameters the PM costs in the example are listed in Table 1. Here, the expected life of system is set to T=10800 day. The initial reliability of the components are all set to Ri(0)=0.999. The minimum reliability for judging whether preventive maintained or not are set to and Rmin=0.85. The particular implementation used in the case study below is based on the Strength Pareto Evolutionary Algorithm 2+ (SPEA2+). A detail of SPEA2+ with a conventional method of Pareto surface generation for a complex engineering design problem is given in [38, 39]. A laptop computer (Intel/Core 2, 1.67 GHz and 2 GB RAM) with MATLAB (R2008 a2) software is utilized to solve the model. The parameters of SPEA2+ are as follows: Number of generations: 300. Population size: 100. Archive size: 50. The crossover rate: 0.8. The mutation rate: 0.03. According to the given parameters in Table 1, an optimal non-periodic PM schedule solution is presented in Table 2. It provides a tradeoff between the system availability and the total PM cost.
Moreover, The fixed interval method of ref.[40] is used to compare with our dynamic interval method. The optimal periodic PM schedule which fixed interval (tpfixed=150 days) set is calculated. The periodic PM schedule and the corresponding availability and cost information are listed in Table 3. The value of total cost for the optimal non-periodic PM schedule solution in Table2 is $7939.1. Comparing it with Table 3, we find that the availability is lower and the total cost is higher than the results in Table 2. The total cost for performing the PM actions has been reduced by 7.44%. The results reveal that the proposed dynamic interval PM policy is more advantage than fixed interval considered. The reliability changing of components in the example are shown in Fig.10, respectively. Observing the reliability of components, as shown in Fig.10, one can see that the reliability curve of the dynamic-interval optimal schedule is smoother than the fixed-interval schedule and PM actions can be better adjusted. The change of reliability of the components could be estimated by the model and then the information can be used to initiate additional PM activities. It makes the system under the proposed PM policy safer than fixed interval considered.
Furthermore, cost-reliability evaluates contribution of extended system reliability per unit-cost in PM schedule and the Cost-reliability of period j can be expressed as: CR(t j ) =
R0+ (t j ) − R0− (t j ) (CFj + CMj + CPj + CIj )
(12)
where R0− (t j ) is the system reliability before PM and R0+ (t j ) is the system reliability after PM.
Cost-reliability of each stage for two kinds of PM policy is calculated as show in Table 4. It can be observed that the average cost-reliability of dynamic interval PM policy is higher than the average cost-reliability of fixed interval PM policy. Maintenance mangers could use the dynamic interval PM policy to make the system safer (higher reliability) under the given budget. Therefore, the dynamic interval PM policy is more effective and the reliability is heightened more than fixed interval PM policy under using the same money.
7. Conclusion This paper studies PM with dynamic interval for a multi-component system. The effects of non-periodic PM actions to reliability are formulated based on the improvements of the survival and failed parts for constructing the reliability model of system following PM. A novel framework for solving the non-periodical PM schedule with dynamic interval based on the MOGA is proposed. By applying the proposed dynamic interval model, it is possible to find the optimal non-periodic PM schedule which achieves a higher level of system availability with less total PM-related cost. The studied results show that considering dynamic interval turns the solution space much more flexible and schedules can be better adjusted, providing gains in the overall system average availability, when compared to those obtained by an optimized periodic PM schedule. A continuation of this work intends to investigate more realistic situations, where the costs contemplate other kinds of impacts obtained by more elaborated models. Another future work is the investigation of a sensitivity analysis on the cost associated with repair and replacement activities and reliability characteristic parameters.
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Acknowledgments This work was supported by the National Natural Science Foundation of China (No.51205347, 51322506), Zhejiang Provincial Natural Science Foundation of China (No. LR14E050003), the Fundamental Research Funds for the Central Universities, Innovation Foundation of the State Key Laboratory of Fluid Power Transmission and Control, and Zhejiang University K.P.Chao's High Technology Development Foundation. Sincere appreciation is extended to the reviewers of this paper for their helpful comments.
Table 1 Parameters for components in the example Component
PM Parameters λi (1/day) βi
ρi
cFi ($)
cMi ($)
cRi ($)
tMi (hour)
tRi (hour)
tIi (hour)
5
p1
50
2.30
0.75
195
47
180
11.25
22.5
p2
46
2.15
0.62
245
37
205
8.75
17.5
5
p3
96
1.70
0.48
215
53
210
12.5
25
5
p4
53
2.20
0.67
250
58
255
13.75
27.5
5
p5
89
1.85
0.52
210
46
215
12
24
5
p6
53
2.05
0.58
235
34
215
8
16
5
p7
47
2.15
0.55
265
62
250
16.25
32.5
5
p8
69
1.98
0.50
205
41
185
10.5
21
5
p9
151
1.75
0.68
220
34
220
7.5
15
5
p10
49
2.10
0.65
275
39
240
9.5
19
5
Table 2 An optimal non-periodic PM schedule of the rotary table system in the example Stage
Time
Interval
(day)
(day)
Component p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
1
150
150
I
I
I
I
M
I
M
M
I
M
2
210
60
M
M
M
M
I
R
R
M
M
M
3
330
120
M
R
M
M
M
R
R
R
I
R
4
510
180
R
M
R
M
R
R
R
R
R
R
5
600
90
I
M
I
M
I
I
I
I
I
I
6
720
120
R
M
M
M
M
R
R
M
M
R
7
900
180
R
R
M
M
M
M
R
R
M
R
‘I’ is defined as inspection, ‘M’ is defined as repair and ‘R’ is defined as replacement
Total cost
System
FC ($)
availability FA
7939.1
96.35%
Table 3 The optimal periodic PM schedule (tpfixed =150 days) of the rotary table system in the example Stage
Time
Interval
(day)
(day)
Component p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
1
150
150
M
M
I
M
I
M
M
M
I
M
2
300
150
R
M
R
M
R
R
R
R
M
M
3
450
150
R
R
R
R
R
R
R
M
I
R
4
600
150
M
M
M
M
I
M
M
M
R
M
5
750
150
M
R
M
R
R
R
R
R
I
R
6
900
150
R
M
R
M
R
R
M
R
M
R
7
1050
150
I
I
I
I
I
I
I
I
I
I
Total cost
System
FC ($)
availability FA
8577.5
96.17%
Table 4 The cost-reliability and average cost-reliability of the rotary table system under two kinds of PM policy Stage
Dynamic interval PM policy Reliability
Fixed interval PM policy Cost-reliability
Before PM
After PM
1
0.936
0.959
2
0.911
0.987
3
0.912
0.988
Reliability
Cost-reliability
Before PM
After PM
3.379E-05
0.936
0.979
7.607E-05
9.066E-05
0.86
0.994
9.607E-05
5.606E-05
0.906
0.992
4.640E-05
4
0.87
0.997
7.187E-05
0.919
0.978
8.769E-05
5
0.965
0.979
1.909E-05
0.853
0.993
8.192E-05
6
0.884
0.988
9.157E-05
0.917
0.995
5.542E-05
0.861
0.986
0.91
0.91
0.000E+00
7
Average Cost-reliability
1.003E-04 6.619E-05
Average Cost-reliability
6.337E-05
Fig.1. PM planning model with dynamic interval Fig.2. Formalization of leader-follower bi-level decision-making process Fig.3. Example of MOGA genotype Fig.4. Process of updating strategy Fig.5. Process of deleting strategy Fig.6. Process of PM scheduling of inserting strategy Fig.7. Process of inserting strategy Fig.8. Process of PM scheduling of moving forward strategy Fig.9. Process of moving forward strategy Fig.10.The reliability changing of the components in the example
min min [ tpsystem , 2 ⋅ tpsystem − 1]
min min [2 ⋅ tpsystem ,3 ⋅ tpsystem − 1]
min min [ j ⋅ tpsystem , ( j + 1) ⋅ tpsystem − 1]
min min [n ⋅ tpsystem − 2, n ⋅ tpsystem − 1]
min [n ⋅ tpsystem ,T ]
Fig.1. PM planning model with dynamic interval
X=[x1
xn]
xj
Y=[Y1
tp1 tp2 tp3
... tpj-1 tpj tpj+1 ... tpn-1 tpn
Φ1 Φ2 Φ3
... Φj-1 Φj Φj+1 ... Φn-1 Φn
Yn]
Yj
Yj=[YIj YMj YRj]
YIj=(yIj1,
0
1
1
...
yIji,
1
1
, yIjm)
0
...
YMj=(yMj1,
0
1
1
0
0
...
yMji,
0
0
, yMjm)
0
...
YRj=(yRj1,
1
0
0
Fig.2. Formalization of leader-follower bi-level decision-making process
0
0
...
yRji,
0
0
, yRjm)
1
...
0
0
Fig.3. Example of MOGA genotype
Fig.4. Process of updating strategy
Fig.5. Process of deleting strategy
Fig.6. Process of PM scheduling of inserting strategy
Fig.7. Process of inserting strategy
Fig.8. Process of PM scheduling of moving forward strategy
Fig.9. Process of moving forward strategy
1
(1) Component 1
(2) Component 2
(3) Component 3
(4) Component 4
(5) Component 5
(6) Component 6
(7) Component 7
(8) Component 8
2
(9) Component 9
(10) Component 10
Fig.10. The reliability changing of the components in the example
3
A non-periodic preventive maintenance scheduling model is proposed. A framework for solving the non-periodical PM schedule problem is developed. The interval of non-periodic PM is flexible and schedule can be better adjusted. Dynamic interval leads to more efficient solutions than fixed interval does.
4