Optimal preventive maintenance and replacement schedules with ...

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METHODOLOGY AND THEORY

Optimal preventive maintenance and replacement schedules with variable improvement factor

Optimal preventive maintenance 271

Kamran S. Moghaddam and John S. Usher Department of Industrial Engineering, University of Louisville, Louisville, Kentucky, USA Abstract Purpose – This paper seeks to develop and present a new mathematical formulation to determine the optimal preventive maintenance and replacement schedule of a system. Design/methodology/approach – The paper divides the maintenance-planning horizon into discrete and equally-sized intervals and in each period decide on one of three possible actions: maintain the system, replace the system, or do nothing. Each decision carries a specific cost and affects the failure pattern of the system. The paper models the cases of minimizing total cost subject to a constraint on system reliability, and maximizing the system reliability subject to a budgetary constraint on total cost. The paper presents a new mathematical function to model an improvement factor based on the ratio of maintenance and repair costs, and show how it outperforms fixed improvement factor models by analyzing the effectiveness in terms of cost and reliability of the system. Findings – Optimal decisions in each period over a planning horizon are sought such that the objectives and the requirements of the system can be achieved. Practical implications – The developed mathematical models for this improvement factor can be used in theoretical and practical situations. Originality/value – The presented models are effective decision tools that find the optimal solution of the preventive maintenance and replacement scheduling problem. Keywords Preventive maintenance, Optimization techniques, Manufacturing systems Paper type Research paper

1. Introduction Preventive maintenance and replacement describes a wide range of activities designed and performed in order to improve the overall reliability and availability of a system. All types of systems found in manufacturing, health care, communications, and transportation industries have prescribed maintenance plans designed by reliability engineers to reduce the risk of system failure. Preventive maintenance and replacement activities generally consist of inspection, cleaning, lubrication, adjustment, alignment, and/or replacement of systems that wear-out. Regardless of the specific system, preventive maintenance and replacement activities can be categorized in one of two ways, maintenance or replacement. An example of system maintenance would be checking and, if needed, correcting the air pressure in the tires of an automobile. Note that this activity changes the aging characteristics of the tires and, if done correctly, ultimately decreases their rate of occurrence of failure. An example of replacement would be replacing the tires with new ones.

Journal of Quality in Maintenance Engineering Vol. 16 No. 3, 2010 pp. 271-287 q Emerald Group Publishing Limited 1355-2511 DOI 10.1108/13552511011072916

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It is clear that preventive maintenance and replacement strategies involve a basic trade-off between the conflicting desires to minimize of the total costs of maintenance and replacement activities and maximization of the overall reliability of the system. System that are maintained and/or replaced often will experience high reliability (low cost of failures), but with high costs of maintenance and replacement. Designers of preventive maintenance and replacement policies must weigh these individual costs along with cost of system failure in order to minimize the overall cost of system operation. They may also be interested in maximizing the system reliability, subject to some sort of budget limitations. In this research, mathematical models for planning the preventive maintenance and replacement schedules for a repairable system are presented. In each period, (which could be defined as an hour, a day, a week, a month, etc.) we assume that one of three distinct actions can be performed: (1) Do nothing. In this case, no action is to be performed. This is often addressed to as leaving the system in a state of “bad-as-old”, where it continues to age normally. (2) Maintenance. In this case, the system is maintained or repaired, which places it into a state somewhere between “good-as-new” and “bad-as-old”. In this research, we assume that the maintenance action reduces the effective age of the system by a stated percentage of its actual age. This is only applicable to systems subject to wear out, that experience an increasing ROCOF. As such, the reduction in effective age places the system at a lower point on the increasing ROCOF function. (3) Replacement. In this case, the system is replaced by a new one, immediately placing it in a state of “good-as-new”, i.e. its age is effectively returned to time zero. The preventive maintenance and replacement scheduling problem can be defined as designing a sequence of actions (1), (2) or (3) for the system for each period in the planning horizon such that overall costs are minimized and/or reliability is maximized. The organization of the paper is as follows. In section 2, we briefly review the current literature. We describe the failure rate, and characteristics of maintenance and replacement activities used in our models in section 3. In section 4, we develop and present optimization models that include a new mathematical model for defining the improvement factor. We discuss how to find optimal preventive maintenance and replacement policies. In section 5, we present a numerical example and demonstrate computational results. Conclusions and recommendations for further study are presented in section 6. 2. Literature review Nakagawa (1988) presents models that utilize an improvement factor approach to maintenance planning. His work has been referenced by many researchers. He develops two analytical models in order to find optimal preventive maintenance schedules based on an assumption of increasing failure rate over time. The first model, called a preventive maintenance hazard rate model, calculates the average failure cost of minimal repairs along with costs of preventive maintenance and replacement actions under the assumption that preventive maintenance actions reduce the next effective age to zero. He also assumes the failure rate is increased by increasing the frequency of

preventive maintenance actions. Furthermore, this model assumes that maintenance activities take place at fixed intervals between each predetermined replacement. The second model, called an age reduction preventive maintenance model, considers the average failure cost of minimal repairs as well as costs of preventive maintenance and replacement actions by assuming that the effective age of a system is reduced by an improvement factor after performing minimal repairs. In order to find the optimal schedule, both models are optimized by calculus methods. He applies the models in a numerical example and describes that based on obtained computational results the second model is more practical than the first model. Jayabalan and Chaudhuri (1992) propose another often-cited work on age reduction and improvement factors models. They develop an optimization model and a branching algorithm that minimizes the total cost of preventive maintenance and replacement activities. They assume a constant improvement factor and define a required failure rate. In addition, they assume a zero failure cost and do not consider time value of money for future costs. Their algorithm determines an optimal schedule of maintenance actions before each replacement action in order to minimize the total cost in a finite planning horizon. They utilize FORTRAN programming environment to implement the algorithm and prove the effectiveness of the algorithm via several numerical examples. Dedopoulos and Smeers (1998) develop a nonlinear optimization model to find the best preventive maintenance schedule by considering the degree of age reduction as the variable in the model. The researchers assume a constant improvement factor but a variable amount of age reduction, which depends on the schedule of preventive maintenance actions. They define amount of age reduction, time and duration of preventive maintenance activities as the decision variables and consider fixed cost and variable cost for maintenance actions. They present the variable cost as a function of the amount of age reduction and duration of action and the effective age of the system. Moreover, they present the failure rate in each period as a recursive function of age reduction from a previous period and consider the net profit as the objective function in the model. They implement the model in GAMS programming environment and use GAMS/MINOS optimization software. Finally, the effectiveness of the model is shown via three numerical examples. Martorell et al. (1999) present an age-dependent preventive maintenance model based on the surveillance parameters, improvement factor, and environmental and operational conditions of the equipment in a nuclear power plant. They consider risk and cost as the main criteria of the model based on the age of the system and perform a sensitivity analysis to show the effect of the parameters on the preventive maintenance policies. They express that the results obtained from their model are different from those resulted from the models that do not consider the improvement factor parameter and working conditions. Lin et al. (2001) combine the models that were developed by Nakagawa (1988) and present hybrid models in which the effects of each preventive maintenance action are considered in two ways; one for its immediate effects and the other one for the lasting effects when the equipment is put to use again. The authors construct two models that reflect the concept of maintainable and non-maintainable failure modes. In the first model, they assume that preventive maintenance and replacement time are independent decision variables and consider the mean cost rate as the objective function that should be minimized. In the second model, they


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assume that preventive maintenance activities are performed whenever the failure rate of the system exceeds the certain level and same as the first model, the mean cost rate is considered as the objective function. Finally, they present numerical examples to show the application of the developed models and mention that for a system with failure rate corresponds to Weibull distribution function, optimal schedules can be achieved analytically, but for the general case, it cannot be solved by analytic methods. Cheng and Chen (2003) consider the improvement factor as a variable of total number of preventive maintenance actions performed over the planning horizon, and the cost ratio of preventive maintenance to replacement actions. They assume different types of restoration effect based on the cost ratio of maintenance and replacement actions and propose three different models. They consider the total number of preventive maintenance actions as the decision variable and develop an objective function to minimize the total cost of the system. By using a numerical analysis method, they mention that the proposed improvement factor model provides a variety of options to evaluate the restoration effect of a deteriorating system. Xi et al. (2005) develop a sequential preventive maintenance optimization model over a finite planning horizon. They define a recursive hybrid failure rate based on the improvement factor concept and increasing failure rate in order to estimate the systems reliability in each period of planning horizon. In addition, they consider the total cost of preventive maintenance activities and assume that the mean cost in each period is a function of required reliability and the improvement factor parameter. Finally, they utilize a simulation approach to optimize the model and mention that the computational results can be used in a maintenance decision support system for job shop scheduling problems. Jaturonnatee et al. (2006) develop an analytical model in order to find the optimal preventive maintenance schedule of leased equipment by minimizing the total cost function. They define maintenance actions as preventive and corrective, each with associated costs, and then consider the concept of reduction in failure intensity function along with penalty costs due to violation of leased contact issues. They present a numerical example for a system with Weibull failure rate, solve the model analytically, and examine the effect of penalty terms on the optimal preventive maintenance policies. Bartholomew-Biggs et al. (2006) present several preventive maintenance scheduling models that consider the effect of imperfect maintenance on effective age of the system. The researchers develop optimization models that minimize the total cost of preventive maintenance and replacement activities. In this study, they assume a known failure rate to express the expected failures as a function of age and consider age reduction in the effective age, based on the concept of an improvement factor. They develop a new mathematical programming formulation to achieve optimal maintenance schedule and utilize automatic differentiation as numerical approach, instead of an analytical approach, to compute the gradients and Hessians in the optimization procedure, which is a global minimization of a non-smooth performance function. Finally, the effectiveness of the proposed model and algorithm is shown in several numerical examples. El-Ferik and Ben-Daya (2006) present an age-based hybrid model for imperfect preventive maintenance scheduling. The authors review different policies and propose a new sequential age-based analytical model that assumes imperfect preventive

maintenance activities reduce the effective age of the system but increase the failure rate. They present mathematical formulations to determine the adjustment factors for both failure rate and the age reduction coefficient. They construct an optimization model based on their analytical models, consider the total cost as the objective function, and solve the optimization model via a new heuristic algorithm in a numerical example. One of the recent works on methods for estimating age reduction factor is presented by Che-Hua (2007). In this research, he considers an optimal preventive maintenance for a deteriorating one-component system via minimizing the expected cost over a finite planning horizon. He develops a mathematical model for estimating improvement factor to measure the restoration of system under the minimal repair. The proposed improvement factor is a function of effective age of the system, the number of preventive maintenance actions, and the cost ratio of maintenance action to the replacement action. Finally, he obtains an optimal preventive maintenance schedule for a case study with the Weibull hazard function by applying a particle swarm optimization method. Cheng et al. (2007) present a paper about models to estimate the degradation rate of the age reduction factor. They present two optimization models, which minimize the cost subject to required reliability. The first model has a periodic preventive maintenance time interval for every replacement and the second one contains the maintenance schedule where the time interval between the final maintenance and replacement is not constant. Lim and Park (2007) present three analytical preventive maintenance models that consider the expected cost rate per unit time as the objective function. In this research, they assume that each preventive maintenance activity reduces the starting effective age but does not change the failure rate and consider the improvement factor as the function of number of preventive maintenance activities. They also assume that the failure function corresponds to a Weibull distribution function and develop a mathematical formulation for three different situations; preventive maintenance period is known, number of preventive maintenance is known, and number and period of preventive maintenance is unknown. They obtain optimal preventive maintenance and replacement schedules by taking an analytical approach and apply them to a numerical example to show an application of their models. By reviewing the literature, we find that most researchers assume a constant improvement factor and develop optimization models to determine the optimal schedule of preventive maintenance actions; see Jayabalan and Chaudhuri (1992), Martorell et al. (1999) and Martorell et al. (1999). Some assume a constant improvement factor with a variable amount of age reduction, which depends on when maintenance actions are performed; see Dedopoulos and Smeers (1998). Nakagawa (1988) assumes a variable improvement factor as a function of time intervals before any replacement and presents equations (1) and (2) for hazard rate improvement factor and effective age improvement factor which are also used by Lim and Park (2007): ak ¼

2k þ 1 fork ¼ 1; :::; n kþ1

ð1Þ

bk ¼

k fork ¼ 1; :::; n kþ1

ð2Þ


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Lin et al. (2001) consider equations (3) and (4) for the same purpose, which are also used by El-Ferik and Ben-Daya (2006) and Bartholomew-Biggs et al. (2006):

276

ak ¼

6k þ 1 fork ¼ 1; :::; n 5k þ 1

ð3Þ

bk ¼

k fork ¼ 1; :::; n 2k þ 1

ð4Þ

In this paper, we introduce a more practical formulation to model the improvement factor in preventive maintenance and replacement scheduling problems. The proposed function assumes the variable improvement factor and it is based on ratio of maintenance and replacement costs and the effective age of the system. 3. Effects of maintenance, replacement, and do nothing actions Notation: . T : Length of the planning horizon. . J : Number of intervals. . l : Characteristic life (scale) parameter of system. . b : Shape parameter of system. . a : Improvement factor of system. . F : Unexpected failure cost of system. . M : Maintenance cost of system. . R : Replacement cost of system. . RR : Required reliability of system. . GB : Given budget for system. . X j : Effective age of system at the start of period j. . X 0j : Effective age of system at the end of period j. ( 1 if component at period j is maintained mj ¼ 0 otherwise ( rj ¼

1 if component at period j is replaced 0 otherwise

Suppose we have a new repairable and maintainable system, which is subject to deterioration. The system is assumed to have an increasing rate of occurrence of failure (ROCOF), vðtÞ, where t denotes actual time, ðt . 0Þ. We assume that the ROCOF corresponds to the well-known Non-Homogeneous Poisson Process (NHPP) and takes on the following form: vðtÞ ¼ l · b · t b21

ð5Þ

l and b are the characteristic life (scale) and the shape parameters of the system respectively. For more on this well-known stochastic process see Ascher and Feingold (1984). We seek to find a schedule of future maintenance and replacement actions for the system over the planning period [0, T ], which is segmented into J discrete intervals, each of length T/J. At the end of period j, the system is either, maintained, replaced, or no action is performed. We assume that maintenance or replacement activities in period j reduce the “effective age” and thus the failure rate of the system. For simplicity we also assume that these activities are instantaneous, i.e. the time required to replace or maintain is negligible, relative to the size of the interval, and thus is zero. However, we do consider a cost associated with each repair or maintenance action. 3.1 Maintenance We define effective ages of a system at the start and end of each period denoted by X j , and X 0j respectively and presented an equation to relate them to each other by the length of each period T/J as follows:

X 0j ¼ X j þ

T forj ¼ 1; :::; T J

ð6Þ

In addition, we assume the initial age of the system is equal to zero. We also assume that the maintenance activity occurs at the end of the each period and effectively reduces the age of the system at the start of the next period based on an “improvement factor” (aka “age reduction factor”). This kind of maintenance does not change the failure characteristics of the system but reduces its effective age, and is sometimes referred to as “minimal repair”:

X jþ1 ¼ a · X 0j for i ¼ 1; :::; N ; j ¼ 1; :::; Tandð0 # a # 1Þ

ð7Þ

Note that when a ¼ 0, the effect of maintenance is to return the system to a state of “good-as-new” and it corresponds to replacement of the system. When a ¼ 1, maintenance has no effect, and the system remains in a state of “bad-as-old” which corresponds to “do nothing”. Without lose of generality, we can always assume that 0 # a # 1. The maintenance action at the end of period j results in an instantaneous drop in the ROCOF of the system, as shown in Figure 1. Thus at the end of period j, the system ROCOF is vðX 0j Þ. At the start of period j þ 1 we find that the ROCOF drops to vðX 0j Þ. Suppose a system is maintained during its service life without any replacement. We can calculate its effective age at the start and end of each period as a function of length of each period, number of maintenance actions, and amount of improvement factor based on the following equations:


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8 > > > > > > > > > > > > > >
> > > > > .. > > > . > > > > > : X j ¼ a · X 0j ¼ TJ a j21 þ a j22 þ a j23 þ · · · þ a ; X 0j ¼ X j þ TJ ¼ TJ a j21 þ a j22 þ a j23 þ · · · þ a þ 1

T Xj ¼ J

X 0j

T ¼ J

j21 X

ð8Þ

!

a

r

ð9Þ

r¼1

j21 X

r

a þ1

! ð10Þ

r¼1

Now if we assume an unlimited service life for a system with a large number of maintenance actions, we can find the lower-bound for its effective age by taking a limit of the effective ages when number of maintenance actions goes to infinity (note that 0 , a , 1): !! j21 T X r T a lim X j ¼ lim a ¼ ð11Þ j!1 j!1 J J 12a r¼1

lim

j!1

X 0j

T ¼ lim j!1 J

j21 X r a þ1 r¼1

!!

T ¼ J

1 12a

ð12Þ

The equations (11) and (12) provide a useful perspective to figure out how maintenance actions affect the effective ages of a system over a long-term planning horizon. For example, suppose the length of each period is equal to one month, the planning horizon is long enough and the system is maintained every month with an improvement factor equal to 0.8, which means that each maintenance action reduces the effective age by 20 percent. Under these assumptions, a lower-bound for the starting and ending effective

Figure 1. Effect of period-j maintenance on system ROCOF

ages in any given month would be 4 and 5 months respectively. That is, the best one could ever do over the long run, by maintaining a system every period, would be to have a minimum effective age of 4 months at the start of any future period. These values can be therefore be considered as the minimum for starting and ending effective ages of the system over the long run. In addition, if a maintenance is performed on the system in period j, a maintenance cost constant M is incurred at the end of the period. 3.2 Replacement If the system is replaced at the end of period j with a new identical one then, it is returned to a state of “good-as-new” and the ROCOF of system instantaneously drops from vðX 0j Þ to vð0Þ as shown in Figure 2. We also assume a replacement cost for the system, denoted R: X jþ1 ¼ 0 for j ¼ 1; . . .; T


ð13Þ

3.3 Do nothing If no action is performed in period j, then no effect will be on the ROCOF of the system, and we find that: X 0j ¼ X j þ

T for j ¼ 1; :::; T J

ð14Þ

X jþ1 ¼ X 0j for j ¼ 1; :::; T

ð15Þ

vðX jþ1 Þ ¼ vðX 0j Þfor j ¼ 1; :::; T

ð16Þ

When we plan the future schedule of operation for the system, we must consider the inevitable costs caused by unexpected system failures. Based on the assumptions, at the start of period j ¼ 1 however, we cannot know when such failures will be observed. However, we know that if the system carries a high ROCOF through a period, then the system is at risk of experiencing high number, and hence the cost, of failures. On the other hand, a low ROCOF in period j should yield a low cost of failure. In order to take into account this unplanned cost, we propose the computation of the expected number

Figure 2. Effect of period-j replacement on system ROCOF

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of failures in each period. Here we compute the expected number of failures of system in period j, as: Z X0 j vðtÞdtfor j ¼ 1; :::; T ð17Þ E Nj ¼ Xj

280

Under the NHPP failure rate, we can find the expected number of system failures in period j to be: Z X0 b b j for j ¼ 1; :::; T ð18Þ l · b · t b21 dt ¼ l X 0j 2 X j E Nj ¼ Xj

We assume that the cost of each failure is F (in units of $/failure event). Hence regardless of any maintenance 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 and unexpected failures that can occur during the period. 4. Optimization models Using the results of section 3, we now present models to minimize total cost subject to the constraint that some minimum level of system reliability over the planning horizon is achieved. From our definitions of each type of cost, we can write the total cost function as follows: T X b b TotalCost ¼ þ M · mj þ R · r j ð19Þ F ·l X 0j 2 X j j¼1

This function computes total cost as a simple sum of system costs in each period based on any maintenance or repair cost, and the cost of the expected number of failures. It is certainly possible to compute a more accurate economic measure of present value of these future costs by using a suitable interest rate, or even including the effects of inflation, by adding an inflation rate into the calculation of future costs. While these may make the model more accurate, we have avoided those minor refinements for the sake of notational simplicity. One may also be interested in determining the system reliability (probability of operating without failure survival over the planning horizon). Based on the assumption on a NHPP, we can define the system reliability in period j (the probability of surviving to the end of period j given survival to the start of period j) as follows: hR X 0 i h b b i 2 X j vðtÞdt 2 l X 0j 2 X j j Rj ¼ e ¼e forj ¼ 1; :::; T ð20Þ Therefore, the probability of the system surviving the entire planning horizon is: h b b i T Y 2 l X 0j 2 X j ð21Þ e Reliability ¼ j¼1

Now we must define the improvement factor because it affects the effective ages of the system, which in turn affects the cost and reliability defined in the two previous

objective functions. Here we will propose a new improvement factor model as a function of maintenance and replacement costs, and effective age of the system at the end of previous period: ! X 0j21 R2M 0 · aj ¼ fðR; M ; X j21 Þ ¼ forj ¼ 1; :::; T ð22Þ X 0j21 þ 1 R The first term is the constant coefficient based on the ratio of difference of replacement and maintenance costs, which is always between zero and one. It is designed so that if a costly maintenance action is performed on a system, the effective age improves more than when inexpensive maintenance is performed. That is, more expensive maintenance results in a greater amount of age reduction. For example, overhauling an engine results in more age reduction that does changing the oil. Note that if maintenance cost is equal to the replacement cost, the numerator of the fraction becomes zero, and maintenance action will coincide with replacement action. On the other hand, if the maintenance cost is equal to zero, the ratio becomes one and it means that maintenance does not affect the effective age and it can be considered as do nothing. The second term is a ratio is the effective age at the end of previous period, which is always less than one. The minimum value is obtained whenever the system is replaced at the previous period. It can be seen that the ratio increases by increasing the effective age and the amount of age reduction decreases as the system ages over the planning horizon. We present the following nonlinear mixed-integer programming model that minimizes the total cost subject a required reliability of the system: MinTotalCost ¼

T X

b b þ M · mj þ R · r j F ·l X 0j 2 X j

j¼1

s:t: : X1 ¼ 0 X j ¼ ð1 2 mj21 Þð1 2 r j21 ÞX 0j21 þ mj21 ðaj · X 0j21 Þj ¼ 2; :::; T X 0j ¼ X j þ TJ j ¼ 1; :::; T

ð23Þ

mj þ r j # 1j ¼ 1; :::; T h b i T b Y 2 l X 0j 2ð X j Þ $ RR e j¼1

mj ; r j ¼ 0or1j ¼ 1; :::; T X j ; X 0j $ 0j ¼ 1; :::; T The first constraint addresses the initial age of the system at the beginning of the planning horizon. The second and third sets calculate the effective age of the system based on preventive maintenance activities recursively. The fourth set prevents the


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occurrence of simultaneous maintenance and replacement actions on the system. The main constraint of the model ensures that reliability of the system is greater or equal than the required reliability. Finally, the last two sets of constraints restrict the decision variables to be binary and positive. Alternatively, we can modify the formulation and introduce a budgetary constraint, GB. The objective of this model is to maximize the reliability, through our choice of maintenance and replace decisions, such that we do not exceed the budgeted total cost. This model can be formulated as: h b b i T Y 2 l X 0j 2 X j Max Re liability ¼ e j¼1

s:t: : X1 ¼ 0 X j ¼ ð1 2 mj21 Þð1 2 r j21 ÞX 0j21 þ mi;j21 ðaj · X 0j21 Þj ¼ 2; :::; T X 0j ¼ X j þ TJ j ¼ 1; :::; T T X

ð24Þ

mj þ r j # 1j ¼ 1; :::; T b b þ M · mj þ R · r j # GB F ·l X 0j 2 X j

j¼1

mj ; r j ¼ 0or1j ¼ 1; :::; T X j ; X 0j $ 0j ¼ 1; :::; T

5. Computational results In order to show the effectiveness of the proposed mathematical models, we consider a system with l ¼ 0:00025 and b ¼ 2:20 as the characteristic life (scale) and the shape parameters and consider failure, maintenance, and replacement costs equal to $2,500, $300, $1,500 respectively. In addition, we assume RR ¼ 92 percent as the required reliability for model 1, GB ¼ $6000 as the given budget for model 2, and 36 months as the planning horizon. In addition, we define three types of improvement factor function as follows: R2M ð25Þ a1;j ¼ f1 ðR; M Þ ¼ R

a2;j ¼ f2 ðX 0j21 Þ ¼

a3;j ¼

f3 ðR; M ; X 0j21 Þ

¼

X 0j21 X 0j21 þ 1

R2M R

·

! ; forj ¼ 1; :::; T

X 0j21 X 0j21 þ 1

ð26Þ

! ; forj ¼ 1; :::; T

ð27Þ

The first function (25) calculates the improvement factor of the system based on the ratio of difference of replacement and maintenance costs, which is constant over the planning horizon. The second function (26) is a simple version of the original formulation that uses only the ratio of effective age at the end of previous period. The last function (27) is the original proposed model. We employ the functions in our optimization models (23) and (24) and solve them using LINGO (www.lindo.com) software. The optimal objective function value for both models with different improvement factor functions is presented in Table I. As we can see, by applying a variable improvement factor, equations (26) and (27), we can obtain lower optimal value in model 1, minimizing total cost subject to reliability constraint, and higher optimal value in model 2, maximizing overall reliability subject to budgetary constraint, than considering constant improvement factor function; equation (25). It is clear that variable improvement factor functions have an advantage over constant improvement factor in terms of objective function value in optimal solution. Tables II and III illustrate the optimal schedules based on different improvement factor functions in both models. As can be seen, by using a variable improvement factor, the optimal schedules contain more maintenance activities than replacements activities. Especially, by applying the third function the optimal schedule consists of only maintenance actions. We also plot the variation of improvement factor functions over the planning horizon in Figures 3 and 4. It can be seen that the constant coefficient smoothes the second function and reduces its variability. We can state that equation (27), which combines maintenance and replacement costs as a constant coefficient along with effective age of the system as an independent variable can model the improvement factor variations very well and result the better optimal solution than the second function. Finally, we conclude that the proposed improvement factor model has advantage over the constant improvement factor and the variable improvement factor function that uses just concept of effective age without considering maintenance and replacement costs.


6. Conclusion In this paper, we reviewed current improvement factor models applied in maintenance scheduling optimization models. We developed mathematical equations to determine a lower-bound for effective age of a maintainable and repairable system over a long-term planning horizon. We presented two nonlinear mixed-integer optimization models to find the optimal preventive maintenance and replacement scheduling for the system. These models seek to minimize the total cost subject to achieving some minimal reliability and maximize the reliability subject to a budgetary constraint. We also introduced a variable improvement factor model and analyzed it using computational

Improvement factor

Total cost

Function 1 Function 2 Function 3

$8,002.54 $7,707.74 $6,506.86

Model 1 Reliability (%) 92 92 92

Model 2 Reliability (%) 89.45 89.66 91.17

Budget $6,000 $6,000 $6,000

Table I. Optimal objective function values

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Table II. Optimal maintenance and replacement schedules in model 1

Month/function 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Function 1

Function 2

Function 3

– – – – – R – – – R – – – – M – – R – – – – – – R – – – R – – – – – – –

– – – – – R – M – – – – – R – – – – R – M M M – – – – R – M – – – – – –

– – M – M M M M – – – – – – M M M – M M M – – – – – R – – – M – – – – –

results of the optimization models. We showed the advantages of it over the constant models. The developed models in this paper can be applied in a wide variety of industries such as semiconductor manufacturing, transportation, material handling, and power generation. Maintenance engineers and managers can use these models to design maintenance plans in order to meet systems requirements and objectives. These models can also be used to quickly generate new preventive maintenance and replacement plans even after unexpected failures occur in the system. In such a situation, the original schedule should be updated and the new optimal schedule will be used over the remaining of the planning horizon. They can also be applied into condition-based simulation models as a real-time optimization procedure to refine and update maintenance plans during the simulation run. Future work in this area is

Month/function 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Function 1

Function 2

Function 3

– – – – M – – – R – – – – – – R – – – – M M – – – R – – – – M – – – – –

– M – – – – – – R – M M M – – – – – – R – – – – – – – R – – – – – – – –

– – – M – M – M M M – M M M – M – M M – M M – M M – M M M M – – – – – –


Table III. Optimal maintenance and replacement schedules in model 2

Figure 3. Variation of improvement factor functions in model 1

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286 Figure 4. Variation of improvement factor functions in model 2

needed to investigate the effects of other constraints on system availability and demand for limited maintenance resources. It is also recommended to use of more complex stochastic programming techniques for developing optimal preventive maintenance plans that better account for the stochastic nature of the problem. References Ascher, H.E. and Feingold, H. (1984), Repairable Systems Reliability: Modeling, Inference, Misconceptions, and Their Causes, Marcel Dekker, New York, NY. Bartholomew-Biggs, M., Christianson, B. and Zuo, M. (2006), “Optimizing preventive maintenance models”, Computational Optimization and Applications, Vol. 35 No. 2, pp. 261-79. Che-Hua, L. (2007), “Preventive maintenance scheduling via particle swarm optimization method”, International Multi Conference of Engineers and Computer Scientists, 21-23 March, Kowloon, China, Vol. 1, pp. 121-6. Cheng, C.Y. and Chen, M.C. (2003), “The periodic preventive maintenance policy for deteriorating systems by using improvement factor model”, International Journal Applied Science and Engineering, Vol. 1 No. 2, pp. 114-22. Cheng, C.Y., Chen, M. and Guo, R. (2007), “The optimal periodic preventive maintenance policy with degradation rate reduction under reliability limit”, 2007 IEEE International Conference on Industrial Engineering and Engineering Management, 2-4 December, Singapore, pp. 649-53. Dedopoulos, I.T. and Smeers, Y. (1998), “Age reduction approach for finite horizon optimization of preventive maintenance for single units subject to random failures”, Computers and Industrial Engineering, Vol. 34 No. 3, pp. 643-54. El-Ferik, S. and Ben-Daya, M. (2006), “Age-based hybrid model for imperfect preventive maintenance”, IIE Transactions, Vol. 38 No. 4, pp. 365-75. Jaturonnatee, J., Murthy, D.N.P. and Boondiskulchok, R. (2006), “Optimal preventive maintenance of leased equipment with corrective minimal repairs”, European Journal of Operational Research, Vol. 174 No. 1, pp. 201-15. Jayabalan, V. and Chaudhuri, D. (1992), “Cost optimization of maintenance scheduling for a system with assured reliability”, IEEE Transactions on Reliability, Vol. 41 No. 1, pp. 21-5. Lim, J.H. and Park, D.H. (2007), “Optimal periodic preventive maintenance schedules with improvement factors depending on number of preventive maintenances”, Asia-Pacific Journal of Operational Research, Vol. 24 No. 1, pp. 111-24.

Lin, D., Zuo, M.J. and Yam, R.C.M. (2001), “Sequential imperfect preventive maintenance models with two categories of failure modes”, Naval Research Logistics, Vol. 48 No. 2, pp. 172-83. Malik, M.A.K. (1979), “Reliable preventive maintenance scheduling”, AIIE Transactions, Vol. 11 No. 3, pp. 221-8. Martorell, S., Sanchez, A. and Serradell, V. (1999), “Age-dependent reliability model considering effects of maintenance and working conditions”, Reliability Engineering and System Safety, Vol. 64 No. 1, pp. 19-31. Nakagawa, T. (1988), “Sequential imperfect preventive maintenance policies”, IEEE Transactions on Reliability, Vol. 37 No. 3, pp. 295-8. Usher, J.S., Kamal, A.H. and Hashmi, S.W. (1998), “Cost-optimal preventive maintenance and replacement scheduling”, IIE Transactions, Vol. 30 No. 12, pp. 1121-8. Xi, L.F., Zhou, X.J. and Lee, J. (2005), “Research on sequential preventive maintenance policy in finite time horizon”, Computer Integrated Manufacturing Systems, Vol. 11 No. 10, pp. 1465-8. About the authors Kamran S. Moghaddam is a Post-Doctoral Associate in the Department of Industrial Engineering at the University of Louisville. He received his BS degree in Applied Mathematics from the University of Tehran, his MS degree in Industrial Engineering from Tehran Polytechnic and his PhD degree in Industrial Engineering from the University of Louisville in 2001, 2003, and 2010 respectively. He was an Industrial Engineer, Transportation Systems Analyst, and later a Project Control Manager at the Tehran Comprehensive Transportation and Traffic Studies Company (TCTTS) during 2002-2006. His research interests are systems reliability optimization, healthcare operations management, supply chain management, and applied operations research. Kamran S. Moghaddam is the corresponding author and can be contacted at: [email protected] John S. Usher is a Professor and Chair of the Department of Industrial Engineering at the University of Louisville. He received his BS and MEng degrees in Industrial Engineering from the University of Louisville in 1980 and 1981, respectively. He received his PhD in Industrial Engineering from North Carolina State University in 1987. While completing this degree he was employed by IBM in Research Triangle Park, North Carolina for two years as a Reliability Engineer. He has conducted funded research in the areas of systems reliability, quality control, and facilities design and analysis for organizations such as the National Science Foundation, US Navy, IBM, AT&T Bell Laboratories, General Electric, Quaker and KFC. He is Past Director of the Facilities Planning and Design Division of IIE and a Past President of the College Industry Council for Material Handling Education (CIC-MHE). He is also member of IIE, ASQ, and IEEE, and is a Registered Professional Engineer in the State of Kentucky.

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