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J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special IssueWe assume that each component has a Weibull hazardfunction and a constant repair rate. Components aremaintained preventively at periodic times.Data is presented on Table 1.First we present the nomenclature.θ i , β i – parameters of hazard function.TTR – Mean Time to Repair (corrective maintenance).TTP – Time of one preventive maintenance action.PMC – Preventive maintenance cost.CMC – Corrective maintenance cost.τ - time between two consecutive preventivemaintenance tasks.Table 1. Initial conditionsComponentsθ i β i TTR TTP PMCostCMCost1 4472,136 2 100 10 2000 4000 20002 1873,1716 2 50 40 2500 5000 15003 500,94 2 80 10 1000 2000 250With this preventive maintenance plan the availabilityachieved is about 90,30% and the life cycle cost is122055,79.The target for availability is 90%.The objective function was slightly modified in orderto include the cost of down time.MATLAB was used to optimise the objective function.Table 2 shows the results. With this new preventivemaintenance policy we have a reduction of 5,5% inLife Cycle Cost (LCC) and simultaneously theavailability A achieved (92,70%) is greater than theexisting one (90,30%).Table 2. Results of MatLab optimisationMatLabOptimisationτ 1 1600.2τ 2 1246.8τ 3 170.7535A - % 92.70LCC 115345.22∆ LCC - % -5.5With these results as initial conditions we have appliedthe tool “SOLVER” of Excel and we got a bettersolution (Table 3).Table 3. Results of MatLab + Excel optimisationMatLab + ExcelOptimisation5. Conclusionτ 1 1606.498τ 2 1255.498τ 3 175.4996A - % 93.02LCC 113809.75∆ LCC - % -6.8This paper deals with a maintenance optimisationproblem for a series system. First we have developedτ ian algorithm to determine the optimum frequency toperform preventive maintenance in systems exhibitingWeibull hazard function and constant repair rate, inorder to ensure its availability. Based on this algorithmwe have developed another one to optimisemaintenance management of a series system based onpreventive maintenance over the different systemcomponents. We assume that all components of thesystem still exhibit Weibull hazard function andconstant repair rate and that preventive maintenancewould bring the system to the as good as newcondition. We define a cost function for maintenancetasks (preventive and corrective) for the system. Thealgorithm calculates the interval of time betweenpreventive maintenance actions for each component,minimizing the costs, and in such a way that the totaldowntime, in a certain period of time, does not exceeda predetermined value. The maintenance interval ofeach component depends on factors such as failurerate, repair and maintenance times of each componentin the system. In conclusion, the proposed analyticalmethod is a feasible technique to optimise preventivemaintenance scheduling of each component in a seriessystem.Currently we are developing a software package for theimplementation of the algorithms presented in thispaper.References[1] Andrews, J.D. & Moss, T.R. (1993). Reliability andRisk Assessment, Essex, Longman Scientific &Technical.[2] Dekker, R. (1996). Applications of maintenanceoptimisation models: a review and analysis,Reliability Engineering and System Safety, 51, 229-240.[3] Duarte, J.C., Craveiro, J.T. & Trigo, T. (2006).Optimisation of the preventive maintenance plan ofa series components system, International Journalof Pressure Vessels and Piping 83, 244-248.[4] Duffuaa, S. A. (2000). Mathematical Models inMaintenance Planning and Scheduling. In M. BenDaya et al (eds.), Maintenance, Modelling andOptimisation, Massachusetts, Kluwer AcademicPublishers.[5] Elsayed, E. A. (1996). Reliability Engineering,Massachusetts, Addison Wesley Longman, Inc.[6] Garg, A. & Deshmukh, S.G. (2006). Maintenancemanagement: literature review and directions.Journal of Quality in Maintenance Engineering. 12No. 3, 205–238.[7] Gertsbakh, I. (2000). Reliability theory: withapplications to preventive maintenance, Berlin,Springer.- 38 -
J.Duarte, C.Soares Optimisation of the preventive maintenance plan of a series components system with Weibull hazard functionRTA # 3-4, 2007, December - Special Issue[8] Henley; Ernest J. & Kumamoto, Hiromitsu (1992).Probabilistic risk assessment: reliabilityengineering, design, and analysis, New York, IEEEPress.[9] McCormick, N. J. (1981). Reliability and RiskAnalysis, San Diego, Academic Press Inc.[10] Modarres, M., Kaminskiy, Mark and Krivtsov,Vasily (1999). Reliability and risk analysis, NewYork, Marcel Dekker, Inc.[11] Nakagawa, T. (2005). Maintenance Theory ofReliability, London, Springer-Verlag[12] Valdez-Flores, C. & Feldman, R. M. (1989). ASurvey of Preventive Maintenance Models forStochastically Deteriorating Single-Unit Systems,Naval Research Logistics, 36: 419-446.[13] Vatn, J., Hokstad, P. & Bodsberg, L. (1996). Anoverall model for maintenance optimisation,Reliability Engineering and System Safety, 51, 241-257.- 39 -
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