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A periodicity metric for assessing maintenance strategies
CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
Contents lists available at ScienceDirect
CIRP Journal of Manufacturing Science and Technology journal homepage: www.elsevier.com/locate/cirpj
A periodicity metric for assessing maintenance strategies K.T. Meselhy, W.H. ElMaraghy, H.A. ElMaraghy * Intelligent Manufacturing Systems Centre, University of Windsor, 204 Odette Building, 401 Sunset Avenue, Windsor, Ontario, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Maintenance Periodicity Complexity Manufacturing systems
Maintenance schemes in manufacturing systems are devised to reset the machines functionality in an economical fashion and keep it within acceptable levels. The notion of periodic functional resetting to reduce complexity has been utilized in this research to introduce and formulate a novel metric to evaluate and quantify function resetting due to a given maintenance policy. The two main parameters are the rate and extent of function resetting. The developed periodicity metric can be used as a criterion for comparing different maintenance policy alternatives and as a tool for predicting system performance under a given maintenance policy. An industrial example is used to illustrate the application of the new metric. The proposed method and metric for assessing maintenance strategies in the functional domain are sufficiently general and can also be applied in other applications that involve system resetting. ß 2010 CIRP.
1. Introduction
1.2. Literature review
1.1. Problem description
Maintenance policies and programs in manufacturing systems must be designed to ensure predetermined machine reliability, availability, efficiency and capability [2]. The current research is focused on the management of maintenance actions at the policy level rather than its technical details at the machines level, and proposes a general strategic approach and high level metric, which are not restricted to certain types of machines, systems or industry. The maintenance actions are normally classified as shown in Fig. 1 [3]: Corrective maintenance includes all actions performed as a result of a failure to restore an item to a specified working condition, while preventive maintenance (PM) includes all actions performed on operating equipment to restore them to a better condition [4]. A maintenance strategy is a structured combination of these two maintenance actions [2], which describes the events (e.g. failure, passing of time, certain machine condition, etc.) and the type of action they trigger (i.e. inspection, repair, maintenance or replacement). The literature contains numerous suggested maintenance policies/strategies [5], which can be categorized as follows: Age-dependent PM polices. The PM actions (minimal, imperfect or perfect) are triggered by the age of the component such as (T, n) policy [6], where T stands for the time between perfect preventive maintenance and n stands for the number of failures between perfect maintenance. Periodic PM policies. The PM are pre-planned at fixed time intervals [7,8]. Sequential PM policies. PM is carried out at age-dependent decreasing time intervals [8,9].
Manufacturing systems are planned, controlled and maintained with the objective to supply products with a predetermined quality level and maximize the utilization of available production capacity. The system features and capabilities are designed a priori. Initially, a manufacturing system performs as designed. As time passes, the machines age and un-planned failures occur, causing the system performance to drift away from its initial state. Therefore, the function of a manufacturing system must be periodically restored to the desired level; this is practically achieved by maintenance operations. This periodic resetting is performed to ensure that the machines are kept in an acceptable condition throughout their useful life. Therefore, introducing periodicity into manufacturing systems to re-initialize their functional state can be considered among the main outcomes of the maintenance action. Nevertheless, there is a lack of reported literature that assesses the maintenance from the periodicity perspective. Furthermore, metrics are needed to evaluate the effectiveness of maintenance strategies and support decisions regarding designing a new maintenance policy or re-designing an existing one. Such metrics should be simple to use to facilitate their application in today’s changeable and reconfigurable manufacturing environment [1].
* Corresponding author. Tel.: +1 519 253 3000x3439; fax: +1 519 973 7053. E-mail addresses: [email protected], [email protected] (H.A. ElMaraghy). 1755-5817/$ – see front matter ß 2010 CIRP. doi:10.1016/j.cirpj.2010.06.004
[(Fig._2)TD$IG]
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136
Nomenclature and units PM PML PMF RL pr
preventive maintenance preventive maintenance level preventive maintenance frequency repair level periodicity
Repair number counting and reference time policies. The maintenance action depends on the number of previous failures and the item’s age at the time of maintenance [5]. Failure limit policies. The PM is carried out when the unit failure rate or reliability indices reach a predetermined level. The intervening failures are corrected by repairs [10]. Reliable evaluation methods are needed to compare the effectiveness of these diverse maintenance policies/strategies. Different criteria have been used in the literature to assess maintenance strategies, such as cost [11], availability [12], reliability [13], and quality [14]. This brief overview of existing maintenance evaluation methods and criteria highlights the need for a new criterion to evaluate the main role of maintenance strategies in defining the required and sufficient frequency and the extent of the maintenance actions. 2. Maintenance strategies and periodic system complexity The main task of maintenance is to periodically reset the manufacturing system, either by repairing failures or by carrying out preventive maintenance. This resetting can be quantitatively described in terms of functional parameters, such as the production rate or available capacity and/or physical parameters, such as the power efficiency of machine tools. The notion of periodic resetting in the functional domain has been defined by Suh [15] as a mechanism to reduce complexity and to restore the desired state of operation. Suh [16] explained that introducing functional periodicity transforms the combinatorial complexity into periodic complexity, which serves to ensure long-term stability for engineered and natural systems. In this context, complexity is defined as a measure of the uncertainty in satisfying the system functional requirements and is measured by the information content defined as follows [16]: Isys ¼
m X log2 P i
(1)
i¼1
where Isys: information content of the system; m: number of functional requirements, FRs; Pi probability that a function requirement, FRi, is satisfied. The complexity is generally categorized by Suh [17] as shown in Fig. 2.
[(Fig._1)TD$IG]
Fig. 1. Classification of maintenance actions.
Fig. 2. Complexity categorization.
The ‘periodicity’ causes the deteriorating functions to exhibit a cyclic behavior that periodically restores their desired characteristics. Therefore, periodicity re-initializes the system functionality to a ‘‘like new’’ state, which assures a high degree of functional certainty. Hence, introducing periodicity reduces, if not eliminates, uncertainty and consequently decreases the complexity associated with combinatorial complexity. Lee [18] introduced many examples of periodicity in systems found in different fields including manufacturing systems. The deterioration of manufacturing system performance is characterized by a time moving system range and may be considered as a time dependent complexity. Hence, carrying out maintenance actions would serve to reset the system performance characteristics. It is clear from this discussion that periodicity is important for the long-term sustainable system functionality. However, how often a system should be reset, to what extent the system should be reset, what is a desirable level of periodicity and at what cost, remain un-answered questions. Another important consideration is quantifying the level of periodicity that would result from adopting a certain maintenance regime and comparing it with the level of periodicity required to achieve the desired functionality goals. Therefore, a metric to quantify the amount of required periodicity is needed to help design new and effective maintenance strategies and evaluate existing ones. 3. Maintenance modeling In order to develop a periodicity metric for evaluating the maintenance strategies, it is necessary to establish a model to define the maintenance strategies in a parametric form. The maintenance policies are currently described in a textual nonmathematical form. Therefore, there is a need to develop a mathematical methodology for modeling maintenance strategies. In this research, the main focus is placed on the time-based maintenance strategies because they are the most commonly used in industry. The developed maintenance policies, reported in the literature, and the maintenance strategies applied in industry, indicate that there are two sub-strategies for any time-based maintenance policy: - The failure repair sub-strategy describes when to repair failures and the level of repair. - The preventive maintenance sub-strategy describes the number of PM classes and the frequency and the level of each class (an example of a system under multi-class maintenance policy is a manufacturing system subject to daily, weekly, mid-annual, and annual preventive maintenances). Therefore, the maintenance strategy can be fully determined by defining the five criteria shown in Table 1. The first two criteria determine the failure repair strategy and the last three determine the PM strategy. Since the failure is normally repaired when it happens (assuming that a perfect failure detection system is in
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
place), the first parameter – when to repair a failure – is excluded from the proposed model. The repair/PM level is represented by a continuous real variable in range [0,1], where 0 means restoring the machine to its state just before failure and 1 means restoring it to the original as a new state [19]. 4. Periodicity modeling Periodicity is a result of a resetting plan. Each resetting action re-initializes system functionality, according to a certain pattern (plan). A system may mean a single machine or a whole manufacturing system. First, the case of a single resetting plan is introduced and the formula for the resulting periodicity is developed. The periodicity resulting from multiple resetting plans is then investigated. 4.1. Periodicity of single resetting plan A resetting plan has two essential dimensions that completely define it: - Frequency of resetting. - Extent of resetting, which expresses the level of re-initialization.
These two aspects are illustrated in Fig. 3, where different resetting levels are shown with a time between resetting, T. Fig. 3(a) represents the reference resetting policy with time between resetting T and full resetting. Fig. 3(b) shows the effect of decreasing the resetting frequency by increasing the time between resetting to 2T while maintaining a full resetting level. The resetting policy represented by Fig. 3(b) has less periodicity than the one represented by Fig. 3(a), which leads to a noticeable increase in the average complexity level. The effect of the resetting level is demonstrated in Fig. 3(c) where different resetting levels are shown while keeping the time between resetting at T. The resetting level is shown in the figure by the ratio a/L where L is the
137
Table 1 Maintenance strategy parameters. Maintenance action
Defining criterion
Maintenance policy parameter
Failure repair
When to repair a failure? Repair level
Failure repair time RL
PM
Number of PM classes Frequency of carrying out each PM class Level of each PM class
N PMF1, . . ., PMFN PML1, . . ., PMLN
total system complexity before the resetting process and a is the amount of complexity actually decreased by the resetting process. Therefore, the resetting policy represented by Fig. 3(c) has less periodicity than the one represented by Fig. 3(a), which causes an increase in the average complexity level. The resetting frequency is the number of resets per unit of time, which is expressed as 1/T. It assumes real values in the range [0,1] where 0 means no resetting at all and 1 means system resetting at infinitesimal time intervals. The resetting extent quantifies the amount of resetting and it can be expressed by the following relationship: resetting extent ¼
amount of resetting amount of full re-initialization
(2)
The amount of resetting for any functional parameter (such as production rate and availability) is the amount of change of the parameter value due to the resetting. Furthermore, to ensure that the resetting extent measure is generic and dimensionless, it would be expressed in terms of the uncertainty of fulfilling the functional requirement, which represents the complexity [16]. Therefore, assuming that the complexity related to a defined functional requirement, such as availability [20] of a new system, is zero (i.e. designed system certainly fulfills the specified functional requirement), then the resetting extent is expressed as:
[(Fig._3)TD$IG]
resetting extent
complexity before resetting complexity after resetting complexity before resetting complexity after resetting ¼1 complexity before resetting
¼
(3)
For simplicity, it will be assumed that the complexity increases linearly with a rate y, which will be called complication rate; however, other relationship functions may be used. The complexity change in the presence of a resetting plan, where the time between resetting is T and the resetting extent, is x is shown in Fig. 4. When there is no periodicity (pr = 0, combinatorial complexity case), the complexity continues to increase without resetting as [(Fig._4)TD$IG]
Fig. 3. Effect of periodicity aspects on complexity.
Fig. 4. Complexity versus time with resetting policy (T, x).
[(Fig._5)TD$IG]
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138
represented by the line of zero periodicity (dashed line). The other theoretical extreme is when the system is being fully reset at infinitesimal time periods, such that its complexity is always zero (pr = 1). This case is represented by the line of full periodicity. Therefore, as the periodicity increases, Area B increases and Area A decreases. The periodicity, therefore, can be expressed as: pr ¼ limt ! 1
Area B ðArea AÞ t
(4)
where t is the time elapsed since the system was new. Area A can be expressed by the following relationship: A¼
1 X yT 2 T x0 C i1 þ 2 i¼1
(5)
where Ci complexity at time iT where i represents the number of resets; x0 = (1 x). The complexity at any time iT is described by the following relationship: C i ¼ x0 C i1 þ yT
(6)
where T time between resetting; y complication rate. The complication rate is a new term introduced in this research to express the increase of complexity per unit time. It is a property of each system that depends on the rate of functionality deterioration. It is represented in Fig. 4 by the slope of the complexity line. From Eq. (5), the area A can be calculated as follows: ! yT 2 yT 2 yT 2 A ¼0þ þ x 0 yT 2 þ þ::::::::::: þ x0 ðx0 yT þ yTÞT þ 2 2 2 1 1 1 0 þ yT 2 þx0 þ yT 2 þx0 þ x 2 ¼ yT 2 2 2 2 1 0 0 þx0 þ x 2 þ x 3 þ ::::::: þ yT 2 2 ! n i X 1 X 0 yT 2 þ x k1 ¼ limn ! 1 2 k¼1 i¼1 From Eqs. (4), (6) and (7), the periodicity relationship can be formulated as follows: 2 2
pr
¼ limn ! 1 ¼ limn ! 1 ¼ limn ! 1
ð1=2Þyn T Pn
Pn
Pi
2
0 k1
i¼1 yT ðð1=2Þ þ k¼1 x Pi 0 k1 2 y T ðð1=2Þ þ x Þ i¼1 k¼1
T
Pn
i¼1
Pn
ð1=2Þn
yT 2 ðð1=2Þ þ n
Pi
Pi
k¼1
x0 k1 Þ
1 nT
!
Þ
1 nT
i¼1 ðð1=2Þ
4.2. Multiple reseting maintenance plans Most real systems are maintained using multiple resetting plans. These plans can be independent or dependent. In this research, independent resetting plans are considered. The independency condition makes it possible to express the total periodicity as a summation of all the periodicity elements. Therefore: pr ¼
n n X X pri ¼ i¼1
i¼1
1 T i ðð2=xi Þ 1Þ
(10)
5. Periodicity-based maintenance policy evaluation The periodicity due to maintenance programs, introduced by two independent sources; the failure repair and the pre-planned PM, will be evaluated. 5.1. Failure repair periodicity
¼ limn ! 1
pr
corresponding time between resetting at a certain periodicity level. For any maintenance policy, periodicity metric physically means the equivalent number of full resets (perfect maintenances) per unit time. This interpretation can be explained by assuming x = 1 (full resetting), then pr = 1/T, which is the number of resets per unit time. Following this interpretation, it can be concluded that 1/pr for a maintenance policy represents the equivalent time between full resetting. Therefore, the developed periodicity metric can be used as a common basis for comparing different maintenance strategies.
0 k1
þ k¼1 x Þ n P P 2Tððn=2Þ þ ni¼1 ik¼1 x0 k1 Þ 1 ¼ limn ! 1 P P 2Tðð1=2Þ þ ð1=nÞ ni¼1 ik¼1 x0 k1 Þ 1 ¼ 2Tðð1=2Þ þ limn ! 1 ðððn 1Þ=nÞx0 0 0 þððn 2Þ=nÞx 2 þ ::::::::: þ ð1=nÞx n1 ÞÞ P k Given the mathematical formula limk ! 1 1 k¼0 x ¼ 1=ð1 xÞ; jxj 1, then, with approximation, the periodicity relationship can be expressed as follows: 2T
Fig. 5. Resetting extent versus time between resetting at different periodicity levels.
1 1 ¼ 2Tðð1=2Þ þ ð1=ð1 x0 ÞÞ 1Þ Tðð2=1 x0 Þ 1Þ 1 ¼ Tðð2=xÞ 1Þ ¼
(9)
The periodicity relationship is plotted in Fig. 5 where each curve represents the relation between resetting extent and the
The machine periodicity due to failure repair is calculated assuming that the machine has a known failure rate l. It is assumed that the machine is repaired (i.e. functionality reset) as soon as it fails, which is the common practice in industry. Therefore, the resetting rate is the same as the failure rate l. Using Eq. (3), it can be stated that: complexity after resetting ¼ ð1 resetting extentÞ complexity before reresetting (11) The resetting extent, in the maintenance context, is represented by the repair/maintenance level (more information about the different imperfect maintenance modeling approaches can be found in Pham and Wang [19]). Therefore, the failure repair periodicity can be expressed by the following equation: prrepair ¼
EðlÞ ð2=RLÞ 1
(12)
[(Fig._6)TD$IG]
[(Fig._7)TD$IG]
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139
Fig. 7. Virtual age and time relationship in case of imperfect resetting.
Fig. 6. Virtual age concept [19].
where E(l) is the expected value of the failure rate. 5.2. Preventive maintenance periodicity Preventive maintenance is the main source of periodicity when the maintenance strategy calls for minimal repair of failures. The machine is reset with each PM, therefore, the resetting rate is the same value as the PM frequency (PMF) and the periodicity extent is expressed by the level of PM (PML). Therefore, the periodicity resulting from the PM of n classes can be described by the following relationship: prPM ¼
n X i¼1
PMFi ð2=PMLi Þ 1
(13)
Pham and Wang [19] defined virtual age as the accumulated age, such that after repair (maintenance), the lifetime of a system will be reduced to a fraction x of the one immediately preceding maintenance where x represents the repair (maintenance) level. This concept is depicted in Fig. 6 [19]. Consider a maintained system with time between maintenance T and maintenance level x, at t = 0, V = 0, where V increases till it reaches T whenever, the system is reset by level x, so, given the virtual age definition, V reaches a value of (1 x) T. Then V increases with time till it reaches (1 x) T + T at time 2T and so on. Therefore, at steady state, the V will range between P P limn ! 1 ðT ni¼1 ð1 xÞi Þ and limn ! 1 ðT ni¼0 ð1 xÞi Þ as explained in Fig. 7. Using mathematical approximation: lim ðT
n!1
lim ðT
n!1
5.3. Total maintenance policy periodicity From Eqs. (12) and (13), the total system periodicity resulting from a given maintenance policy can be expressed as: pr ¼
n X 2 PMFi þ ð2=RLÞ 1 i¼1 ð2=PMLi Þ 1
(14)
Therefore, given the maintenance policy parameters, RL, PMLi, and PMFi and the machine failure rate l, the maintenance policy periodicity can be calculated. This calculated periodicity (pr) is a measure of the relative ability of the maintenance strategy to reset the machine’s functionality. 6. Application of periodicity metric to predict maintained system performance The new developed periodicity metric can be utilized to predict the long-run future performance of maintained systems. The virtual age (V) concept will be utilized to develop this relationship.
[(Fig._8)TD$IG]
i¼n X i¼0 i¼n X
ð1 xÞi Þ T
1
x
1 ð1 xÞi Þ T 1
i¼0
x
(15) (16)
Therefore, at the steady state the V will range from T((1/x) 1) and T(1/x). Therefore, the average V would be: V average
T
ðð1=xÞ 1Þ þ ð1=xÞ 2 1 1 ¼T x 2 1 2 pr
(17)
This Eq. (17) explains the relationship between the system steady state average virtual age and the applied maintenance plan periodicity. Therefore, in the cases where there is a defined relationship between the system performance and its age – similar to the case described by Finkelstein [21] – where a degrading system where the quality of the system performance is decreasing in some suitable sense, e.g. increasing tool wear, the system performance can be predicted given the machine/system age (or the virtual age in the case of maintained systems).
Fig. 8. Virtual age model framework for expecting performance of maintained systems.
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
140 Table 2 Original maintenance plan parameters. Repair level
RL
0
Number of PM classes PM frequency PM level
N PMFi PMLi
4 1.0 between shifts 0.0
0.1 weekly 0.05
0.004167 semi-annually 0.5
0.002083 annually 0.95
Table 3 Suggested maintenance plan parameters. Repair level
RL
0
Number of PM classes PM frequency PM level
N PMFi PMLi
4 0.5 daily 0
Hence, if the maintenance plan is known, the periodicity can be calculated using Eq. (14), and the average virtual age can be calculated using Eq. (17). Hence, the future long-term performance can be estimated, which represents a considerable enhancement in maintenance decisions. The framework of this procedure is illustrated in Fig. 8.
7. Illustrating example In the following example, the maintenance policy adopted at an auto-manufacturer assembly plant is used to illustrate the application of the proposed new periodicity-based maintenance strategies assessment metric. The plant maintenance policy can be described as follows: when a machine fails, it is instantaneously minimally repaired to quickly restore the production. The PM policy comprises four classes: between shifts, weekly, semiannual, and annual. Each PM class has associated courses of action for each machine. The exact determination of maintenance level for each maintenance class requires historical data. Nevertheless, based on the courses of action in each maintenance class, the maintenance levels are estimated to be approximately 0, 0.05, 0.5, and 0.95, respectively. The plant operates two shifts per day, 5 days per week. One shift is considered to be the time unit. Using the proposed maintenance modeling approach, the plant maintenance strategy can by fully described by the following parameters (Table 2): This maintenance strategy applies to every resource/machine in the plant. One of these resources is a frame-welding robot, which experiences random failures with an average of one failure/week. The amount of functional periodicity introduced by implementing this maintenance strategy is calculated, using Eq. (14), to be 0.00583. This periodicity measures the relative ability of the maintenance strategy to re-initialize the robot functionality. The company is considering an alternative maintenance strategy, which is described by the parameters in Table 3. Using Eq. (14), the periodicity of the new maintenance strategy is calculated to be 0.00675. A comparison between the two maintenance policies is explained in Table 4.
0.025 monthly 0.2
0.0083 quarterly 0.4
0.002083 annually 0.95
The comparison shows that the proposed maintenance policy introduces more periodicity into the system than the one currently applied, hence, the proposed maintenance policy is more capable of reducing system complexity, which leads to more stability and more certainty that it will satisfy its functional requirements. Furthermore, the comparison quantitatively shows that the proposed maintenance policy results in less virtual age (in the long-run) compared with the currently applied policy. Therefore, system performance (corresponding to the low virtual age) is expected to be better than the performance resulting from the currently applied policy. It is important to note that this conclusion is based only on considering the periodicity/complexity. However, in real life cases, there may be other criteria to be included as well in selecting a maintenance policy. 8. Discussion and conclusions Maintenance in manufacturing systems introduces periodicity, which is required to ensure the system functional stability throughout its life. A novel general metric for quantifying this periodicity has been developed and a formula for calculating it is derived. A new term called complication rate has been introduced to measure the system’s functional deterioration. The proposed periodicity metric can be used to quantitatively compare the resetting ability of different maintenance policies, which combined with other performance metrics like cost, availability and product quality, can vastly enhance decision making in selecting appropriate maintenance strategies. It has been shown that the calculation of periodicity introduced by a maintenance strategy, using the proposed model and formulation, is quite simple and makes it practically applicable in industry. Although the application of the developed periodicity metric has been discussed in the context of the engineering/ manufacturing maintenance field, the method is general enough and can be applied in any application that involves system resetting, such as natural and political systems.
References Table 4 Comparison between maintenance policies.
Periodicity (pr) Average long-run virtual age Equivalent time between full resettings
Policy 1
Policy 2
0.00583 85.76 shift 171.53 shift
0.00675 74 shift 148.15 shift
[1] Wiendahl, H.P., et al, 2007, Changeable Manufacturing – Classification, Design and Operation, CIRP Annals – Manufacturing Technology, 56/2: 783–809. [2] Dekker, R., 1996, Applications of Maintenance Optimization Models: A Review and Analysis, Reliability Engineering and System Safety, 51/3: 229–240. [3] Aurich, J.C., Siener, M., Wagenknecht, C., 2006, Quality Oriented Productive Maintenance Within the Life Cycle of a Manufacturing System, 13th CIRP International Conference on Life Cycle Engineering, . [4] Dhillon, B.S., 1999, Engineering Maintainability, Gulf Professional Publishing.
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141 [5] Wang, H., 2002, A Survey of Maintenance Policies of Deteriorating Systems, European Journal of Operational Research, 139/3: 469–489. [6] Sheu, S.-H., Griffith, W.S., Nakagawa, T., 1995, Extended Optimal Replacement Model with Random Minimal Repair Costs, European Journal of Operational Research, 85/3: 636–649. [7] Xiao-Gao, L., Makis, V., Jardine, A.K.S., 1995, A Replacement Model with Overhauls and Repairs, Naval Research Logistics, 42/7: 1063–1079. [8] Nakagawa, T., 1986, Periodic and Sequential Preventive Maintenance Policies, Journal of Applied Probability, 23/2: 536–542. [9] Nakagawa, T., 1988, Sequential Imperfect Preventive Maintenance Policies, IEEE Transactions on Reliability, 37/3: 295–298. [10] Cassady, C.R., et al, 2000, Combining Preventive Maintenance and Statistical Process Control: A Preliminary Investigation, IIE Transactions, 32/6: 471– 478. [11] Moore, W.J., Starr, A.G., 2006, An Intelligent Maintenance System for Continuous Cost-based Prioritisation of Maintenance Activities, Computers in Industry, 57/6: 595–606. [12] Wang, H., Pham, H., 2006, Availability and Maintenance of Series Systems Subject to Imperfect Repair and Correlated Failure and Repair, European Journal of Operational Research, 174/3: 1706–1722.
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[13] Demers, J.-M., 2005, Reliability Centred Maintenance, CIM Bulletin, 98/1086: 50. [14] Ben-Daya, M., 1999, Integrated Production Maintenance and Quality Model for Imperfect Processes, IIE Transactions (Institute of Industrial Engineers), 31/6: 491–501. [15] Suh, N.P., 2004, On Functional Periodicity as the Basis for Long-term Stability of Engineered and Natural Systems and its Relationship to Physical Laws, Research in Engineering Design, 15/1: 72–75. [16] Suh, N.P., 2005, Complexity in Engineering, CIRP Annals – Manufacturing Technology, 54/2: 581–598. [17] Suh, N.P., 2005, Complexity, Theory and Applications, MIT, Oxford University Press. [18] Lee, T., 2003, Complexity Theory in Axiomatic Design, Mechanical Engineering, MIT. p. 182. [19] Pham, H., Wang, H., 1996, Imperfect Maintenance, European Journal of Operational Research, 94/3: 425–438. [20] ElMaraghy, H.A., Kuzgunkaya, O., Urbanic, R.J., 2005, Manufacturing Systems Configuration Complexity, CIRP Annals – Manufacturing Technology, 54/1: 445–450. [21] Finkelstein, M., 2009, Virtual age of Non-repairable Objects, Reliability Engineering and System Safety, 94/2: 666–669.
Contents lists available at ScienceDirect
CIRP Journal of Manufacturing Science and Technology journal homepage: www.elsevier.com/locate/cirpj
A periodicity metric for assessing maintenance strategies K.T. Meselhy, W.H. ElMaraghy, H.A. ElMaraghy * Intelligent Manufacturing Systems Centre, University of Windsor, 204 Odette Building, 401 Sunset Avenue, Windsor, Ontario, Canada
A R T I C L E I N F O
A B S T R A C T
Keywords: Maintenance Periodicity Complexity Manufacturing systems
Maintenance schemes in manufacturing systems are devised to reset the machines functionality in an economical fashion and keep it within acceptable levels. The notion of periodic functional resetting to reduce complexity has been utilized in this research to introduce and formulate a novel metric to evaluate and quantify function resetting due to a given maintenance policy. The two main parameters are the rate and extent of function resetting. The developed periodicity metric can be used as a criterion for comparing different maintenance policy alternatives and as a tool for predicting system performance under a given maintenance policy. An industrial example is used to illustrate the application of the new metric. The proposed method and metric for assessing maintenance strategies in the functional domain are sufficiently general and can also be applied in other applications that involve system resetting. ß 2010 CIRP.
1. Introduction
1.2. Literature review
1.1. Problem description
Maintenance policies and programs in manufacturing systems must be designed to ensure predetermined machine reliability, availability, efficiency and capability [2]. The current research is focused on the management of maintenance actions at the policy level rather than its technical details at the machines level, and proposes a general strategic approach and high level metric, which are not restricted to certain types of machines, systems or industry. The maintenance actions are normally classified as shown in Fig. 1 [3]: Corrective maintenance includes all actions performed as a result of a failure to restore an item to a specified working condition, while preventive maintenance (PM) includes all actions performed on operating equipment to restore them to a better condition [4]. A maintenance strategy is a structured combination of these two maintenance actions [2], which describes the events (e.g. failure, passing of time, certain machine condition, etc.) and the type of action they trigger (i.e. inspection, repair, maintenance or replacement). The literature contains numerous suggested maintenance policies/strategies [5], which can be categorized as follows: Age-dependent PM polices. The PM actions (minimal, imperfect or perfect) are triggered by the age of the component such as (T, n) policy [6], where T stands for the time between perfect preventive maintenance and n stands for the number of failures between perfect maintenance. Periodic PM policies. The PM are pre-planned at fixed time intervals [7,8]. Sequential PM policies. PM is carried out at age-dependent decreasing time intervals [8,9].
Manufacturing systems are planned, controlled and maintained with the objective to supply products with a predetermined quality level and maximize the utilization of available production capacity. The system features and capabilities are designed a priori. Initially, a manufacturing system performs as designed. As time passes, the machines age and un-planned failures occur, causing the system performance to drift away from its initial state. Therefore, the function of a manufacturing system must be periodically restored to the desired level; this is practically achieved by maintenance operations. This periodic resetting is performed to ensure that the machines are kept in an acceptable condition throughout their useful life. Therefore, introducing periodicity into manufacturing systems to re-initialize their functional state can be considered among the main outcomes of the maintenance action. Nevertheless, there is a lack of reported literature that assesses the maintenance from the periodicity perspective. Furthermore, metrics are needed to evaluate the effectiveness of maintenance strategies and support decisions regarding designing a new maintenance policy or re-designing an existing one. Such metrics should be simple to use to facilitate their application in today’s changeable and reconfigurable manufacturing environment [1].
* Corresponding author. Tel.: +1 519 253 3000x3439; fax: +1 519 973 7053. E-mail addresses: [email protected], [email protected] (H.A. ElMaraghy). 1755-5817/$ – see front matter ß 2010 CIRP. doi:10.1016/j.cirpj.2010.06.004
[(Fig._2)TD$IG]
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
136
Nomenclature and units PM PML PMF RL pr
preventive maintenance preventive maintenance level preventive maintenance frequency repair level periodicity
Repair number counting and reference time policies. The maintenance action depends on the number of previous failures and the item’s age at the time of maintenance [5]. Failure limit policies. The PM is carried out when the unit failure rate or reliability indices reach a predetermined level. The intervening failures are corrected by repairs [10]. Reliable evaluation methods are needed to compare the effectiveness of these diverse maintenance policies/strategies. Different criteria have been used in the literature to assess maintenance strategies, such as cost [11], availability [12], reliability [13], and quality [14]. This brief overview of existing maintenance evaluation methods and criteria highlights the need for a new criterion to evaluate the main role of maintenance strategies in defining the required and sufficient frequency and the extent of the maintenance actions. 2. Maintenance strategies and periodic system complexity The main task of maintenance is to periodically reset the manufacturing system, either by repairing failures or by carrying out preventive maintenance. This resetting can be quantitatively described in terms of functional parameters, such as the production rate or available capacity and/or physical parameters, such as the power efficiency of machine tools. The notion of periodic resetting in the functional domain has been defined by Suh [15] as a mechanism to reduce complexity and to restore the desired state of operation. Suh [16] explained that introducing functional periodicity transforms the combinatorial complexity into periodic complexity, which serves to ensure long-term stability for engineered and natural systems. In this context, complexity is defined as a measure of the uncertainty in satisfying the system functional requirements and is measured by the information content defined as follows [16]: Isys ¼
m X log2 P i
(1)
i¼1
where Isys: information content of the system; m: number of functional requirements, FRs; Pi probability that a function requirement, FRi, is satisfied. The complexity is generally categorized by Suh [17] as shown in Fig. 2.
[(Fig._1)TD$IG]
Fig. 1. Classification of maintenance actions.
Fig. 2. Complexity categorization.
The ‘periodicity’ causes the deteriorating functions to exhibit a cyclic behavior that periodically restores their desired characteristics. Therefore, periodicity re-initializes the system functionality to a ‘‘like new’’ state, which assures a high degree of functional certainty. Hence, introducing periodicity reduces, if not eliminates, uncertainty and consequently decreases the complexity associated with combinatorial complexity. Lee [18] introduced many examples of periodicity in systems found in different fields including manufacturing systems. The deterioration of manufacturing system performance is characterized by a time moving system range and may be considered as a time dependent complexity. Hence, carrying out maintenance actions would serve to reset the system performance characteristics. It is clear from this discussion that periodicity is important for the long-term sustainable system functionality. However, how often a system should be reset, to what extent the system should be reset, what is a desirable level of periodicity and at what cost, remain un-answered questions. Another important consideration is quantifying the level of periodicity that would result from adopting a certain maintenance regime and comparing it with the level of periodicity required to achieve the desired functionality goals. Therefore, a metric to quantify the amount of required periodicity is needed to help design new and effective maintenance strategies and evaluate existing ones. 3. Maintenance modeling In order to develop a periodicity metric for evaluating the maintenance strategies, it is necessary to establish a model to define the maintenance strategies in a parametric form. The maintenance policies are currently described in a textual nonmathematical form. Therefore, there is a need to develop a mathematical methodology for modeling maintenance strategies. In this research, the main focus is placed on the time-based maintenance strategies because they are the most commonly used in industry. The developed maintenance policies, reported in the literature, and the maintenance strategies applied in industry, indicate that there are two sub-strategies for any time-based maintenance policy: - The failure repair sub-strategy describes when to repair failures and the level of repair. - The preventive maintenance sub-strategy describes the number of PM classes and the frequency and the level of each class (an example of a system under multi-class maintenance policy is a manufacturing system subject to daily, weekly, mid-annual, and annual preventive maintenances). Therefore, the maintenance strategy can be fully determined by defining the five criteria shown in Table 1. The first two criteria determine the failure repair strategy and the last three determine the PM strategy. Since the failure is normally repaired when it happens (assuming that a perfect failure detection system is in
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
place), the first parameter – when to repair a failure – is excluded from the proposed model. The repair/PM level is represented by a continuous real variable in range [0,1], where 0 means restoring the machine to its state just before failure and 1 means restoring it to the original as a new state [19]. 4. Periodicity modeling Periodicity is a result of a resetting plan. Each resetting action re-initializes system functionality, according to a certain pattern (plan). A system may mean a single machine or a whole manufacturing system. First, the case of a single resetting plan is introduced and the formula for the resulting periodicity is developed. The periodicity resulting from multiple resetting plans is then investigated. 4.1. Periodicity of single resetting plan A resetting plan has two essential dimensions that completely define it: - Frequency of resetting. - Extent of resetting, which expresses the level of re-initialization.
These two aspects are illustrated in Fig. 3, where different resetting levels are shown with a time between resetting, T. Fig. 3(a) represents the reference resetting policy with time between resetting T and full resetting. Fig. 3(b) shows the effect of decreasing the resetting frequency by increasing the time between resetting to 2T while maintaining a full resetting level. The resetting policy represented by Fig. 3(b) has less periodicity than the one represented by Fig. 3(a), which leads to a noticeable increase in the average complexity level. The effect of the resetting level is demonstrated in Fig. 3(c) where different resetting levels are shown while keeping the time between resetting at T. The resetting level is shown in the figure by the ratio a/L where L is the
137
Table 1 Maintenance strategy parameters. Maintenance action
Defining criterion
Maintenance policy parameter
Failure repair
When to repair a failure? Repair level
Failure repair time RL
PM
Number of PM classes Frequency of carrying out each PM class Level of each PM class
N PMF1, . . ., PMFN PML1, . . ., PMLN
total system complexity before the resetting process and a is the amount of complexity actually decreased by the resetting process. Therefore, the resetting policy represented by Fig. 3(c) has less periodicity than the one represented by Fig. 3(a), which causes an increase in the average complexity level. The resetting frequency is the number of resets per unit of time, which is expressed as 1/T. It assumes real values in the range [0,1] where 0 means no resetting at all and 1 means system resetting at infinitesimal time intervals. The resetting extent quantifies the amount of resetting and it can be expressed by the following relationship: resetting extent ¼
amount of resetting amount of full re-initialization
(2)
The amount of resetting for any functional parameter (such as production rate and availability) is the amount of change of the parameter value due to the resetting. Furthermore, to ensure that the resetting extent measure is generic and dimensionless, it would be expressed in terms of the uncertainty of fulfilling the functional requirement, which represents the complexity [16]. Therefore, assuming that the complexity related to a defined functional requirement, such as availability [20] of a new system, is zero (i.e. designed system certainly fulfills the specified functional requirement), then the resetting extent is expressed as:
[(Fig._3)TD$IG]
resetting extent
complexity before resetting complexity after resetting complexity before resetting complexity after resetting ¼1 complexity before resetting
¼
(3)
For simplicity, it will be assumed that the complexity increases linearly with a rate y, which will be called complication rate; however, other relationship functions may be used. The complexity change in the presence of a resetting plan, where the time between resetting is T and the resetting extent, is x is shown in Fig. 4. When there is no periodicity (pr = 0, combinatorial complexity case), the complexity continues to increase without resetting as [(Fig._4)TD$IG]
Fig. 3. Effect of periodicity aspects on complexity.
Fig. 4. Complexity versus time with resetting policy (T, x).
[(Fig._5)TD$IG]
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138
represented by the line of zero periodicity (dashed line). The other theoretical extreme is when the system is being fully reset at infinitesimal time periods, such that its complexity is always zero (pr = 1). This case is represented by the line of full periodicity. Therefore, as the periodicity increases, Area B increases and Area A decreases. The periodicity, therefore, can be expressed as: pr ¼ limt ! 1
Area B ðArea AÞ t
(4)
where t is the time elapsed since the system was new. Area A can be expressed by the following relationship: A¼
1 X yT 2 T x0 C i1 þ 2 i¼1
(5)
where Ci complexity at time iT where i represents the number of resets; x0 = (1 x). The complexity at any time iT is described by the following relationship: C i ¼ x0 C i1 þ yT
(6)
where T time between resetting; y complication rate. The complication rate is a new term introduced in this research to express the increase of complexity per unit time. It is a property of each system that depends on the rate of functionality deterioration. It is represented in Fig. 4 by the slope of the complexity line. From Eq. (5), the area A can be calculated as follows: ! yT 2 yT 2 yT 2 A ¼0þ þ x 0 yT 2 þ þ::::::::::: þ x0 ðx0 yT þ yTÞT þ 2 2 2 1 1 1 0 þ yT 2 þx0 þ yT 2 þx0 þ x 2 ¼ yT 2 2 2 2 1 0 0 þx0 þ x 2 þ x 3 þ ::::::: þ yT 2 2 ! n i X 1 X 0 yT 2 þ x k1 ¼ limn ! 1 2 k¼1 i¼1 From Eqs. (4), (6) and (7), the periodicity relationship can be formulated as follows: 2 2
pr
¼ limn ! 1 ¼ limn ! 1 ¼ limn ! 1
ð1=2Þyn T Pn
Pn
Pi
2
0 k1
i¼1 yT ðð1=2Þ þ k¼1 x Pi 0 k1 2 y T ðð1=2Þ þ x Þ i¼1 k¼1
T
Pn
i¼1
Pn
ð1=2Þn
yT 2 ðð1=2Þ þ n
Pi
Pi
k¼1
x0 k1 Þ
1 nT
!
Þ
1 nT
i¼1 ðð1=2Þ
4.2. Multiple reseting maintenance plans Most real systems are maintained using multiple resetting plans. These plans can be independent or dependent. In this research, independent resetting plans are considered. The independency condition makes it possible to express the total periodicity as a summation of all the periodicity elements. Therefore: pr ¼
n n X X pri ¼ i¼1
i¼1
1 T i ðð2=xi Þ 1Þ
(10)
5. Periodicity-based maintenance policy evaluation The periodicity due to maintenance programs, introduced by two independent sources; the failure repair and the pre-planned PM, will be evaluated. 5.1. Failure repair periodicity
¼ limn ! 1
pr
corresponding time between resetting at a certain periodicity level. For any maintenance policy, periodicity metric physically means the equivalent number of full resets (perfect maintenances) per unit time. This interpretation can be explained by assuming x = 1 (full resetting), then pr = 1/T, which is the number of resets per unit time. Following this interpretation, it can be concluded that 1/pr for a maintenance policy represents the equivalent time between full resetting. Therefore, the developed periodicity metric can be used as a common basis for comparing different maintenance strategies.
0 k1
þ k¼1 x Þ n P P 2Tððn=2Þ þ ni¼1 ik¼1 x0 k1 Þ 1 ¼ limn ! 1 P P 2Tðð1=2Þ þ ð1=nÞ ni¼1 ik¼1 x0 k1 Þ 1 ¼ 2Tðð1=2Þ þ limn ! 1 ðððn 1Þ=nÞx0 0 0 þððn 2Þ=nÞx 2 þ ::::::::: þ ð1=nÞx n1 ÞÞ P k Given the mathematical formula limk ! 1 1 k¼0 x ¼ 1=ð1 xÞ; jxj 1, then, with approximation, the periodicity relationship can be expressed as follows: 2T
Fig. 5. Resetting extent versus time between resetting at different periodicity levels.
1 1 ¼ 2Tðð1=2Þ þ ð1=ð1 x0 ÞÞ 1Þ Tðð2=1 x0 Þ 1Þ 1 ¼ Tðð2=xÞ 1Þ ¼
(9)
The periodicity relationship is plotted in Fig. 5 where each curve represents the relation between resetting extent and the
The machine periodicity due to failure repair is calculated assuming that the machine has a known failure rate l. It is assumed that the machine is repaired (i.e. functionality reset) as soon as it fails, which is the common practice in industry. Therefore, the resetting rate is the same as the failure rate l. Using Eq. (3), it can be stated that: complexity after resetting ¼ ð1 resetting extentÞ complexity before reresetting (11) The resetting extent, in the maintenance context, is represented by the repair/maintenance level (more information about the different imperfect maintenance modeling approaches can be found in Pham and Wang [19]). Therefore, the failure repair periodicity can be expressed by the following equation: prrepair ¼
EðlÞ ð2=RLÞ 1
(12)
[(Fig._6)TD$IG]
[(Fig._7)TD$IG]
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139
Fig. 7. Virtual age and time relationship in case of imperfect resetting.
Fig. 6. Virtual age concept [19].
where E(l) is the expected value of the failure rate. 5.2. Preventive maintenance periodicity Preventive maintenance is the main source of periodicity when the maintenance strategy calls for minimal repair of failures. The machine is reset with each PM, therefore, the resetting rate is the same value as the PM frequency (PMF) and the periodicity extent is expressed by the level of PM (PML). Therefore, the periodicity resulting from the PM of n classes can be described by the following relationship: prPM ¼
n X i¼1
PMFi ð2=PMLi Þ 1
(13)
Pham and Wang [19] defined virtual age as the accumulated age, such that after repair (maintenance), the lifetime of a system will be reduced to a fraction x of the one immediately preceding maintenance where x represents the repair (maintenance) level. This concept is depicted in Fig. 6 [19]. Consider a maintained system with time between maintenance T and maintenance level x, at t = 0, V = 0, where V increases till it reaches T whenever, the system is reset by level x, so, given the virtual age definition, V reaches a value of (1 x) T. Then V increases with time till it reaches (1 x) T + T at time 2T and so on. Therefore, at steady state, the V will range between P P limn ! 1 ðT ni¼1 ð1 xÞi Þ and limn ! 1 ðT ni¼0 ð1 xÞi Þ as explained in Fig. 7. Using mathematical approximation: lim ðT
n!1
lim ðT
n!1
5.3. Total maintenance policy periodicity From Eqs. (12) and (13), the total system periodicity resulting from a given maintenance policy can be expressed as: pr ¼
n X 2 PMFi þ ð2=RLÞ 1 i¼1 ð2=PMLi Þ 1
(14)
Therefore, given the maintenance policy parameters, RL, PMLi, and PMFi and the machine failure rate l, the maintenance policy periodicity can be calculated. This calculated periodicity (pr) is a measure of the relative ability of the maintenance strategy to reset the machine’s functionality. 6. Application of periodicity metric to predict maintained system performance The new developed periodicity metric can be utilized to predict the long-run future performance of maintained systems. The virtual age (V) concept will be utilized to develop this relationship.
[(Fig._8)TD$IG]
i¼n X i¼0 i¼n X
ð1 xÞi Þ T
1
x
1 ð1 xÞi Þ T 1
i¼0
x
(15) (16)
Therefore, at the steady state the V will range from T((1/x) 1) and T(1/x). Therefore, the average V would be: V average
T
ðð1=xÞ 1Þ þ ð1=xÞ 2 1 1 ¼T x 2 1 2 pr
(17)
This Eq. (17) explains the relationship between the system steady state average virtual age and the applied maintenance plan periodicity. Therefore, in the cases where there is a defined relationship between the system performance and its age – similar to the case described by Finkelstein [21] – where a degrading system where the quality of the system performance is decreasing in some suitable sense, e.g. increasing tool wear, the system performance can be predicted given the machine/system age (or the virtual age in the case of maintained systems).
Fig. 8. Virtual age model framework for expecting performance of maintained systems.
K.T. Meselhy et al. / CIRP Journal of Manufacturing Science and Technology 3 (2010) 135–141
140 Table 2 Original maintenance plan parameters. Repair level
RL
0
Number of PM classes PM frequency PM level
N PMFi PMLi
4 1.0 between shifts 0.0
0.1 weekly 0.05
0.004167 semi-annually 0.5
0.002083 annually 0.95
Table 3 Suggested maintenance plan parameters. Repair level
RL
0
Number of PM classes PM frequency PM level
N PMFi PMLi
4 0.5 daily 0
Hence, if the maintenance plan is known, the periodicity can be calculated using Eq. (14), and the average virtual age can be calculated using Eq. (17). Hence, the future long-term performance can be estimated, which represents a considerable enhancement in maintenance decisions. The framework of this procedure is illustrated in Fig. 8.
7. Illustrating example In the following example, the maintenance policy adopted at an auto-manufacturer assembly plant is used to illustrate the application of the proposed new periodicity-based maintenance strategies assessment metric. The plant maintenance policy can be described as follows: when a machine fails, it is instantaneously minimally repaired to quickly restore the production. The PM policy comprises four classes: between shifts, weekly, semiannual, and annual. Each PM class has associated courses of action for each machine. The exact determination of maintenance level for each maintenance class requires historical data. Nevertheless, based on the courses of action in each maintenance class, the maintenance levels are estimated to be approximately 0, 0.05, 0.5, and 0.95, respectively. The plant operates two shifts per day, 5 days per week. One shift is considered to be the time unit. Using the proposed maintenance modeling approach, the plant maintenance strategy can by fully described by the following parameters (Table 2): This maintenance strategy applies to every resource/machine in the plant. One of these resources is a frame-welding robot, which experiences random failures with an average of one failure/week. The amount of functional periodicity introduced by implementing this maintenance strategy is calculated, using Eq. (14), to be 0.00583. This periodicity measures the relative ability of the maintenance strategy to re-initialize the robot functionality. The company is considering an alternative maintenance strategy, which is described by the parameters in Table 3. Using Eq. (14), the periodicity of the new maintenance strategy is calculated to be 0.00675. A comparison between the two maintenance policies is explained in Table 4.
0.025 monthly 0.2
0.0083 quarterly 0.4
0.002083 annually 0.95
The comparison shows that the proposed maintenance policy introduces more periodicity into the system than the one currently applied, hence, the proposed maintenance policy is more capable of reducing system complexity, which leads to more stability and more certainty that it will satisfy its functional requirements. Furthermore, the comparison quantitatively shows that the proposed maintenance policy results in less virtual age (in the long-run) compared with the currently applied policy. Therefore, system performance (corresponding to the low virtual age) is expected to be better than the performance resulting from the currently applied policy. It is important to note that this conclusion is based only on considering the periodicity/complexity. However, in real life cases, there may be other criteria to be included as well in selecting a maintenance policy. 8. Discussion and conclusions Maintenance in manufacturing systems introduces periodicity, which is required to ensure the system functional stability throughout its life. A novel general metric for quantifying this periodicity has been developed and a formula for calculating it is derived. A new term called complication rate has been introduced to measure the system’s functional deterioration. The proposed periodicity metric can be used to quantitatively compare the resetting ability of different maintenance policies, which combined with other performance metrics like cost, availability and product quality, can vastly enhance decision making in selecting appropriate maintenance strategies. It has been shown that the calculation of periodicity introduced by a maintenance strategy, using the proposed model and formulation, is quite simple and makes it practically applicable in industry. Although the application of the developed periodicity metric has been discussed in the context of the engineering/ manufacturing maintenance field, the method is general enough and can be applied in any application that involves system resetting, such as natural and political systems.
References Table 4 Comparison between maintenance policies.
Periodicity (pr) Average long-run virtual age Equivalent time between full resettings
Policy 1
Policy 2
0.00583 85.76 shift 171.53 shift
0.00675 74 shift 148.15 shift
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