Maintenance optimization using probabilistic cost-benefit analysis

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1 Maintenance optimization using probabilistic cost-benefit analysis Devarun Ghosh,Sandip Roy Department of Chemical Engineering, Indian Institute of echnology, Bombay , India abstract Keywords: Reliability Maintenance optimization Preventive maintenance (PM) Reliability Centered Preventive Maintenance (RCPM) Benefit-to-cost ratio (BCR) Over the recent decades, plant maintenance strategies have evolved from a corrective to a preventive approach. Also, deterministic models have been increasingly replaced by those based on reliability and risk, which are probabilistic. Approaches to obtaining the optimum maintenance interval have typically involved minimization of the total associated cost. he present work demonstrates an improved technique involving the maximization of reliability-based benefit-to-cost ratio (BCR), i.e., the ratio of potential monetary benefit that can accrue from an optimized preventive maintenance (PM) schedule to the costs incurred in implementing such a schedule. It is shown that the methodology can be used to optimize the PM schedule for process units whose reliability function is either exponential or follows a Weibull distribution. A sensitivity analysis has also been performed to demonstrate the effect of various model parameters on the benefit-to-cost ratio. he proposed approach constitutes an improvement over the cost minimization methodology reported in contemporary literature, and can even be extended to plant shutdown planning. 1. Introduction Maintenance strategies have witnessed a paradigm shift over the recent decades from breakdown maintenance to more sophisticated strategies like condition monitoring and Reliability Centered Preventive Maintenance (RCPM). Plant safety/loss prevention is directly linked to the reliability of its operation. A robust maintenance program is necessary for the process industry as it deals with hazardous substances, often under severe operating conditions. hus, plant managers and engineers today are faced with important preventive maintenance (PM) decisions aimed at integrated loss prevention. Preventive maintenance (PM) can help minimize the probability of losses due to accident situations and unscheduled failure of process units. he growing interest in reliability/risk-based PM and process safety management (PSM) is driven by the need to develop strategies that lead to an optimum safety vs. cost balance. Quantitative approaches which link component deterioration to condition improvement by maintenance can help determine the effect of maintenance on reliability. A large number of papers have been recently published on the subject of optimizing maintenance through the use of mathematical models (Dey, 2004; Khan & Haddara, 2003a, 2003b; Montgomery & Serratella, 2002; Willcocks & Bai, 2000). raditionally, optimal PM intervention schedules have been obtained using models, deterministic or probabilistic, which involves minimization of total costs incurred in relation to maintenance activities. here are other objectives besides economics, which may influence preventive maintenance scheduling. For example, safety is an objective if combinations of equipment failures can cause a hazardous event, and if preventive maintenance can reduce the number of failures. In order to optimally trade-off multiple objectives, a single objective function needs to be constructed. his paper formulates a model to optimize the expected financial gain due to enhanced reliability deriving from PM against the costs incurred due to such an intervention. Cost minimization has been the traditional objective in maintenance planning. Deterministic models (Vintr & Holub, 2003) on preventive maintenance optimization have established minima in costs based on operating cost parameters (repair, maintenance and acquisition). he use of deterministic methods, however, does not provide information about potential risk that results in nonoptimal maintenance planning for process plants (Desjardins, 2002). Probabilistic models, on the other hand, use probability distributions to describe and represent natural variability and uncertainty in parameter, model and scenario (Bedford & Cook, 2001). Probabilistic models of scheduling preventive maintenance also minimize objective functions that reflect repair, replacement and PM costs (Zuo, Christianson, & Bartholomew-Biggs, 2006). he

2 404 preventive maintenance interval is optimized when the increasing rate of corrective maintenance costs (with respect to time) equals the decreasing rate of preventive maintenance costs. Flexible maintenance intervals have been conceptualized by leveraging the step nature of the average change in reliability (with PM) over the service life of the component. Lapa, Pereira, and de Barros (2005) have used genetic algorithms to model flexible maintenance intervals for multi-component systems. he present paper attempts to model PM using cost-benefit analysis (CBA). Our results show that while probabilistic PM cost minimization does not yield a definitive optimum for even the simplest of situation where a process unit has a constant failure rate; the CBA does show that an optimum PM interval can exist. Further, it is shown that units which typically undergo wear and tear during working life pumps, compressors, turbines, etc. and which show a Weibull type failure probability distribution are also amenable to CBA for obtaining optimal PM schedule. Lastly, we show that the benefit-to-cost ratio (BCR) is reasonably high in cases the PM and repair costs are less than 10% of the loss incurred due to an unscheduled breakdown. 2. heory he approach to PM optimization can be extended beyond cost minimization to a cost-benefit analysis. o solve for the optimal maintenance interval, a parameter benefit-to-cost ratio (BCR) may be defined. BCR is the ratio of the financial benefit (from increased reliability of a process unit) due to PM to the costs incurred due to maintenance interventions. Maximization of this ratio identifies the longest maintenance cycle that trades off the benefit from maintenance with the cost incurred to achieve an acceptable level of reliability of a process unit under maintenance. Such an approach can be applied to any process unit. In the present paper, we illustrate it for units with (i) constant failure rate (faults/yr) and (ii) with a linearly increasing failure rate (faults/yr). While the former type of failure rate behavior is used for simple reliability analysis of a variety of process units, the latter type is usually applicable to units which undergo regular wear and tear (pumps, compressors, turbines). he model developed in this work is an extension of that due to Lapa et al. (2005), which itself is a generalized form of the model proposed earlier (Lewis, 1996). Let C m and C r, respectively, be the costs of planned preventive maintenance and unplanned replacement/repair. he total cost, C, referred to the component s operation during the interval from the beginning of its operation and the time it suffers the first maintenance m (1), indicated by the superscript index (0 / 1), is given by: Generalizing for the z number of intervals between maintenance interventions, the last one (z þ 1) between the last maintenance intervention and the end of the service life ( ser ), we get: C 0/ser ¼ Xz C ðj 1Þ/j m j¼1 þc z/ser r R½m ðjþš þc ðj 1Þ/j r 1 R½ mðjþš R½ m ðj 1ÞŠ R½ m ðj 1ÞŠ : ð5þ 1 R½ mðserþš R½ m ðzþš Assuming N maintenance interventions over the entire service life, the total cost incurred due to maintenance for the entire equipment life is: CðÞ ¼ XN C m RðjÞ Rfðj 1Þg þ C RðjÞ r 1 : (6) Rfðj 1Þg Let R(t) be a general reliability function for an equipment and let the equipment be restored to an As Good As New (AGAN) condition after every PM intervention. For the jth maintenance interval, i.e., (j 1) / j, the average difference in the reliability of the equipment with and without PM is given by: DR ðjþ m ðþ¼ R RðtÞdt dt 0 R 0 R j ðj 1Þ R j ðj 1Þ RðtÞdt : (7) dt hus, the average difference in the two reliability functions for the entire life time of the process unit is (Fig. 1): DR m ðþ ¼ 1 N N ¼ ser X N DR ðjþ m (8) ½xŠdenotes greatest integer x: (9) If C inc be the cost incurred due to lost production and other financial losses due to process unit breakdown, the benefit B() derived from periodic PM for the entire equipment life is: BðÞ ¼C inc,dr m ðþ: (10) C 0/1 ¼ C 0/1 m R½ mð1þš þ C 0/1 r ½1 Š: (1) Generalizing this concept to the other intervals, a conditional probability R[t/ m (1)] may be defined; this represents the probability of the system to survive till time t, given that it did not fail until m (1). If t a is the time period until the component s failure, this is given by: R½tj m ð1þš ¼ Pft a > m ð1þþtjt a > m ð1þg (2) R½tj m ð1þš ¼ R½t þ mð1þš : (3) Using t ¼ m (2) m (1) for interval (1 / 2) we get, C 1/2 ¼ Cm 1/2 R½m ð2þš þ Cr 1/2 1 R½ mð2þš : (4) Fig. 1. Reliability vs. time for a single component with and without PM.

3 405 able 1 Model parameters. ype of failure rate Failure rate Equipment ser (yrs) failure PDF a parameter(s) Exponential (case I) Constant l (faults/yr) ¼ Weibull (case II) Linearly increasing a (/yr) ¼ b (/yr 2 ) ¼ 0.02 a Probability Density Function. hus, the benefit-to-cost ratio, BCR {i.e., B()/C()} is given by: BCR ¼ P N C m C inc,dr m ðþ : (11) RðjÞ Rfðj 1Þg þ C RðjÞ r 1 Rfðj 1Þg he optimal PM policy corresponds to the maximum of the function BCR(). For an equipment with a constant failure rate l, the reliability R(t) ¼ exp ( lt). hus, for a service life ser, the following expressions may be obtained using Eqs. (6) and (10). " ( ) ( )# 1 e l BðÞ ¼C inc þ 1 e lser 1 l l ser 1 e l þ 1 e lser 1 l ser 1 e l (12) CðÞ ¼ ser hc m e l þ C r 1 e li : (13) he Weibull distribution is a more generalized failure model and has been widely used for process equipment life data analyses (Herbaty, 1990). he primary advantage of the Weibull analysis is its capability to provide accurate failure analysis and risk predictions with small samples (Abernethy, 1993). he failure rate and reliability functions are, respectively, given by: lðtþ ¼ b t b 1 (14) h h " # t b RðtÞ ¼exp (15) h where, b ¼ shape parameter, h ¼ characteristic life he Weibull distribution is often used in the reliability analysis due to its inherent flexibility it can mimic the behavior of other statistical distributions such as the normal (b ¼ 3.4) and the exponential (b ¼ 1). A decreasing failure rate (b < 1) would correspond to early life failure or infant mortality. his happens as defective items fail early and the failure rate decreases over time as they fall out of the population. A constant failure rate (b ¼ 1) suggests that items are failing from random events. An increasing failure rate (b > 1) suggests wear out parts are more likely to fail as time goes on. However, process units with moving parts can show an increasing failure rate during the useful life period due to wear and tear, and, therefore, require surveillance through PM. For such units one can write the failure rate and reliability functions as: Fig. 2. Variation of total cost as a function of PM interval for case I. lðtþ¼ a þ bt (16) RðtÞ ¼exp at þ bt2 : (17) 2 As may be evident from comparing Eqs. (14) and (16), this constitutes a special case of the Weibull distribution. Using Eqs. (16) and (17), and the general relations as expressed in Eq. (6 11) once can again develop the final expressions for B() and C() as in Eqs. (12) and (13) and, hence for BCR using Eq. (11). his, however, makes it complicated to analytically derive an expression for the BCR. hus, Matlab codes have been used for numerically calculating the BCR in this case. 3. Results and discussions his section presents the results of both cost minimization and BCR maximization models developed in the foregoing section. In each case, we demonstrate the results for both exponential (referred to as case I) and the special case of Weibull failure density functions (linearly increasing failure rate). he latter (referred to as case II) corresponds to Eq. (16). he values of the relevant model parameters used are shown in able 1 while the values of the cost parameters used are enlisted in able 2. hese values have been used for all subsequent calculations. he Weibull parameters have been calculated based on an average failure rate of 0.5 (faults/yr) over a service life of 20 years. able 2 Cost parameters. a Parameter Description Value ($) C inc Incident cost (cost of lost production and safety related 1,000,000 financial losses due to random equipment failure) C m Cost of planned preventive maintenance 10,000 C r Costs of unplanned replacement/repair 25,000 a hese figures do not necessarily reflect actual figures but are used only to illustrate the proposed approach. Fig. 3. Variation of total cost as a function of PM interval for case II.

4 406 Fig. 4. Variation of benefit and costs as a function of PM interval (case I) Cost minimization model results Figs 2 and 3 present plots of the total cost function (Eq. (5)) over the life span of the equipment for both cases I and II. he model parameters used are given in ables 1 and 2. he constant failure rate model does not exhibit a cost minimum with respect to the PM interval,. Also, for case II the behavior of the cost function does not appear to possess a distinctive, strong minimum. It is more or less similar to the cost function in Fig. 2. hese results suggest that the use of cost minimization paradigm for a reliability-based (probabilistic) PM optimization model does not yield practical results which may be used to fix an actual PM schedule for a process unit. It may be noted that deterministic models used traditionally for this purpose, however, suggests the existence of an optimal PM schedule (Vintr & Holub, 2003). Clearly, a better paradigm is needed for optimizing PM schedule within the framework of a reliabilitybased PM model. As we show next, the maximization of the benefit-to-cost ratio (BCR) provides a practicable solution BCR maximization results As evident from Fig. 4, the constant failure rate (case I) model both cost and benefit functions decrease with increasing maintenance intervals; however, the rate of decease for the two functions is not the same. his suggests that the ratio of the two (i.e., BCR) can possess an extremum. Indeed this is displayed in Fig. 5, which shows the BCR over the service life of equipment with constant failure rate (case I). he optimal maintenance interval corresponds to that at which the BCR shows a distinct maximum, i.e., around Fig. 6. BCR using Weibull distribution for reliability (case II) years. his, however, is a value that would appear to be relatively high for an optimal or effective PM interval. As we show later, the optimum PM interval is dependent on the service life. Indeed for equipments with lesser service life the corresponding optimal PM interval is also low, i.e., in the range of 2 5 years. Nevertheless, the relatively high value of optimal PM interval obtained here on the basis of the given service life and failure rate is possibly suggestive of a measure of inaccuracy involved in assuming a constant failure rate. As is shown below, the results are substantially improved by use of a Weibull failure density function instead. A Matlab code plots the BCR vs. the maintenance interval () for the special case of Weibull distribution (case II) for reliability; the results are shown in Fig. 6. he behavior of the BCR is identical with that for case I (constant failure rate); however, the optimal PM is reduced to about 5 years. he similarity of the BCR results of the two cases essentially suggests that the basic theory of cost-benefit analysis provides a consistent approach for optimizing the PM schedule of process units. As is evident from Fig. 6, there is some discrete fluctuation in the BCR value. his results from inclusion of a potential cost of repair for the period between the last maintenance intervention and the end of service life i.e., only the second term in Eq. (5) is used in the last or residual time interval for computing the potential cost of repair. he BCR curve, however, Fig. 5. BCR using constant equipment failure rate model (case I). Fig. 7. Variation of optimal maintenance interval with equipment service life.

5 407 Again, as shown in Fig. 9, when the PM cost is less than 10% of the incident cost, the BCR is high for equipments with low failure rates. However, for components with failure rate 1 (faults/yr), the sensitivity of the BCR to the ratio of PM cost to incident cost is relatively weak. his suggests that equipments with a relatively high failure rate need a cold stand-by in order to make PM effective for such components. 4. Conclusion Fig. 8. Variation of BCR with ratio of repair to incident cost. displays a global maximum. Given the fact that the linearly increasing failure rate is a more realistic model for most process units requiring PM, the results demonstrated by case II here is expected to be more dependable Sensitivity analysis (constant failure rate model) o explore the response of the BCR function (using constant failure rate) to variations in the model parameters, namely equipment service life and relative cost parameters, a sensitivity analysis of the BCR to these parameters has been carried out. he results are presented next. For simplicity, the results are illustrated assuming constant equipment failure rate. hree different component failure rates are assumed the calculations for which the results are presented in Figs. 7 9 below, i.e., have failure 0.1, 0.5 and 1, (fault/yr), respectively. Fig. 7 plots the variation of the optimum maintenance interval as a function of equipment service life using cost parameters shown in able 2. he figure shows that as expected the optimum maintenance interval increases with increasing service life; however, it is not very sensitive to the failure rate. Fig. 8 shows that the BCR is substantial (5), when the repair cost is less than ~ 5% of incident cost. As may also be seen, for a given repair to incident cost ratio, the BCR is (expectedly) greater for equipments with lower failure rates. hus, PM would be a preferred option over breakdown maintenance for components (with a relatively low failure rate), which do not require frequent maintenance interventions. Fig. 9. Benefit-to-cost ratio sensitivity to ratio of maintenance cost to incident cost. Efficient maintenance policies are of fundamental importance in process system engineering because of their impact on safety and economics of plant operation. his paper presents a cost-benefit based modeling approach for the establishment of an optimum reliability-based preventive maintenance schedule. An optimum maintenance plan based on such an analysis is inherently superior to existing models of maintenance optimization because it simultaneously accounts for PM-related cost and the financial benefit accruing from PM. As demonstrated, a maximum in the benefit-tocost ratio (BCR) is obtained for both constant and linearly increasing equipment failure rate. he PM interval at the maximum BCR is the optimal one. A sensitivity analysis based on the constant failure rate model was also undertaken to study the effect of changing the relative cost parameters on the BCR parameter. Equipments with low failure rates have an appreciable BCR (>5) if the repair and maintenance costs are typically less than 5 10% of the incident cost. he modeling approach outlined in this paper is essentially based on unconstrained optimization. As a related piece of work in the field of techno-economic optimization of process plant maintenance, the decision-making framework for an overall maintenance approach has been modeled using a fuzzy Multi-Criteria Decision-Making approach (Ghosh, 2008). However, for further improvement in decision-making, both the reliability of a system (constrained to be above a minimum permissible level) as well as the risk that would result as a consequence of a random equipment failure need (constrained to be below a maximum permissible level) to be considered. References Abernethy, R. B. (1993). he new weibull handbook. North Palm Beach, FL. Bedford,., & Cook, R. (2001). Probabilistic risk analysis: Foundations & methods. Cambridge University Press. 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