Modelling spare parts dynamics by multi-state Monte Carlo simulation Marzio Marseguerra, Enrico Zio, Luca Podofillini Department of Nuclear Engineering, Polytechnic of Milan Via Ponzio 34/3, 20133 Milan, Italy
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Abstract In this paper, we resort to Monte Carlo simulation for achieving a realistic modelling of the dynamics of a system supported by spare parts. The spare dynamics, namely the start-up of a spare unit upon failure of an on line unit, is caught by means of the multi-state correlation method previously introduced by some of the present authors to describe stand-by and load-share dependencies. The Monte Carlo simulation model is embedded within a genetic algorithm multiobjective search for the maximization of the system revenues and the minimization of the total spares volume.
1. Introduction The determination of the optimal spare parts allocation is a concern of great interest in many industrial applications, not only for the relevance of the spare parts management costs with respect to the whole system life cycle cost, but also in view of the possible effects that the presence of spare parts may have on the safety of risky plants such as the nuclear ones. Several works have addressed the problem of determining the optimal spare parts inventory. In the literature, gradient methods, dynamic programming, integer programming, mixed integer and nonlinear programming are the main tools suggested (Dinesh 1998, Dinesh 1997, Messinger 1970, Burton 1971). Unfortunately, such optimization techniques typically entail the use of simplified plants or systems models whose predictions may be of questionable realism and reliability. For the modern complex industrial plants and systems with significant safety implications such as, for example the nuclear power or chemical plants, a realistic model is in order. In these cases the plant behavior is typically described by multivariate, nonlinear relations which can hardly be put in an explicit analytical form: this poses severe limitations to the applicability of the above optimization methods. In this work we use multi-state Monte Carlo (MC) simulation for modelling the availability and reliability of a plant with supporting spare parts available on site. In the MC modelling, we allow the components to stochastically move among multiple states (operational, failed, in start-up, in restoration, in storage) and handle the components’ state dynamics by means of the correlation method originally introduced by some of the authors to describe stand-by and load-share dependencies (Marseguerra & Zio 1993). By means of this method we are able to catch the recycling process characterising the dynamics of a system supported by spare parts. The Monte Carlo simulation model has been implemented within a genetic algorithm multiobjective search aiming at finding the optimal numbers of spares which maximizes the net profit of the plant while minimizing the total volume of the spare parts. The paper is organized as follows. In Section 2, we describe the MC approach for modelling the availability/reliability of a plant with spares available on site. In Section 3, the MC approach is tested with reference to a simplified system for which analytical results can be obtained. The multiobjective optimisation by genetic algorithms is also presented in this Section. Conclusions will close the paper. All the results reported have been obtained with a code especially developed at our Department.
2. Modelling systems supported by spare parts via Monte Carlo We consider a system made up of Nc operative components and Nsp spares. The modes of operation of a generic component are described by means of four states: 1-operating; 2-spare; 3-in start-up; 4-under repair.
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In our model, when a component fails, it is replaced by a spare part. The failed component is sent for repair to a specialized facility and, after being repaired, it is shipped back to the plant depot where it serves as a spare. Thus, for the generic component, three probability density functions (PDFs) are involved in the process: f(t), the failure time distribution of the component, g(t) the replace time distribution, and h(t) the distribution of the recycling time, i.e. the time required for the failed unit to be shipped to the repair facility, repaired and returned to the local storage. For simplicity, and without loss of generality, in our simulations all distributions are assumed exponential, with corresponding rates λ, η, α. This assumption can be easily removed within the Monte Carlo simulation scheme. We start the simulation from the nominal configuration: Nc components in state 1 and Nsp in state 2. The components’ transition rates from state 1 to state 2 and 3 are zero by definition of the system logic. Moreover, we assume that a spare part cannot fail and that it cannot be set into operation before being started up: this means that the rates of the transitions from state 2 to state 4 and 1 are zero. The components’ transition rates from state 2 into state 3, are initially set to very small values as compared to those of the rates governing the transitions (failures) from state 1 into state 4 (typically λ = 10-3 h-1 , in our applications). By doing so, the probability that a spare unrealistically starts operating spuriously is negligible so that in this situation the only possible transition is the failure of one of the operating components. During the simulation, when the failure of an operating component is sampled (say, component A), a correlation is activated such that the transition probability of spare unit B, of the same type of component A, from state 2 to state 3 is multiplied by a very large coefficient so that this transition is sampled next with certainty and instantaneously (Marseguerra & Zio 1993). At this time we have component A in state 4, sent off to repair, and its substitute B in the process of being started up (state 3). The replacement of A with the spare B is completed, with a stochastic delay, when the transition of B from state 3 to state 1 is sampled. Figure 1 and Figure 2 sketch the components’ states and recycling process dynamics, respectively. Evidently, the replacement of component A occurs only if a spare of that type is available: if a spare part were not available, due to the fact that all components of this kind are already operating, being started up or under repair, we update two counters: the first one registers the time interval during which the system is unavailable if lack of spares makes it so; the second one records that a spare of that type is needed in the position of the failed component. As for the repair of the failed component A, the process is dependent on the availability of a repair facility. If no facility for that kind of repair is available, because all are currently working on other units, component A has to wait before its repair process is started. To complete the spare dynamics modeling, when a repaired component (say, component C) is transferred into state 2, a new spare is now available. In this case, we check if a spare of the same type of component C is needed somewhere in the system. If the spare is required (this occurs if there were no spare parts of the same type of C in state 2 before the arrival of C) a correlation of the type illustrated above is activated so as to transfer instantaneously component C from state 2 to state 3, from which it will transfer into the operative state 1 at its characteristic replacement rate. under repair
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3. Numerical applications As validation of the Monte Carlo code, a simple case study is considered for which an analytical solution can be readily found. The analytical model is based on a Markov approach, which, for systems with few components, allows to describe the recycling process in Figure 2, upon which the system dynamics rely. The system under modelling is made up of two identical, repairable units in parallel logic, supported by two spare units. The four components have exponential failure and repair times with rates λ = 5 ⋅ 10 −3 h −1 , α = 1 ⋅ 10 −3 h −1 , respectively, whereas the time needed for a replacement is assumed negligible. The mission time is TM = 1000 h. The analytical time-dependent unavailability obtained by a Markov approach (solid line) is shown in Figure 3 to be in agreement with the Monte Carlo results obtained with 104 trials (circles). 1
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Figure 3: Time-dependent unavailability of a parallel system of two identical, repairable units supported by two spare units with negligible replacement time: analytical (solid line) and Monte Carlo (circles).
We now consider a system made up of four different components (Nc = 4) in series. All components are assumed to have exponentially distributed failure times with rates λ1 = 5 ⋅ 10 −4 h −1 , λ 2 = 1 ⋅ 10 −3 h −1 , λ3 = 5 ⋅ 10 −3 h −1 , λ 4 = 1 ⋅ 10 −2 h −1 . The replace time is negligible and no repair facilities are available. The problem is that of determining the optimal spare parts allocation in terms of how many spares have to be kept in store for each kind of component. The objective functions to be maximized are the net profit achievable by the system during the mission time, TM = 1000 h, and the inverse of the total volume of stored spares. The detailed definition of the objective functions and the data pertaining to the system under analysis can be found in (Marseguerra, Zio, Podofillini, 2001). The optimization was performed by means of a genetic algorithm which embeds the Monte Carlo simulation model of the spare dynamics previously illustrated. For details on the functioning of the genetic algorithms, the reader may consult (Goldberg, 1989). The multiobjective optimization is driven by the search for the set of dominant solution according to a Pareto optimality concept (Giuggioli, Marseguerra, Zio, 2001). The possible number of spare components of each node ranges from 0 to 15, so that there are 65536 possible allocation strategies. In the genetic algorithm framework, the alternative spare parts allocation solutions were coded into chromosomes made up of four genes, each representing the number of spares of one of the four components’ types. Each gene contains four bits to code the 16 possible spare allocations alternatives of a component type. Under the above simplifying model assumptions, the spare parts dynamics can be evaluated analytically, thus allowing the validation of the optimization procedure by crudely spanning all the alternative allocation strategies. In Figure 4 we report the values of the objective functions in correspondence of all the Pareto dominant solutions found by the genetic algorithm which have been confirmed, by crude and
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analytical evaluation of the objective functions of all the possible alternatives in the search space (Figure 5), to be optimal with respect to either the net profit or the total spares volume. The Pareto optimal set provides the decision maker with the whole spectrum of performances with respect to the objectives, and he or she must ultimately select the preferred one according to some subjective preference values. Thus, the closure of the problem must still rely on techniques of decision making such as utility theory, multi-attribute value theory or fuzzy decision making, to name a few. 4
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Figure 4: Multiobjective Pareto optimal solution set
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4. Conclusions Analytical models of systems supported by spares often yield unsatisfactory results when the spare parts dynamics is realistically complex. In this paper, we exploit a multi-state Monte Carlo simulation technique for modelling the availability of a system when spare parts are present on site. The components’ dynamics is tackled by means of the correlation method originally introduced by some of the authors to describe stand-by and loadshare dependencies. For validation purposes, the Monte Carlo modelling of the spare parts recycling process has been tested against analytical results. The agreement thereby obtained turned out to be satisfactory. The simulation model developed has been successfully embedded within a GA multiobjective search of the optimal number of spares which maximizes the net profit of a system and minimizes the total volume of the spare parts. References Burton B. M., Howard G. T. (1971) Optimal design for system reliability and maintainability. IEEE Transactions on Reliability, 20, 56-60. Dinesh Kumar U., Knezevic J. (1998) Availability based spare optimization using renewal process. Reliability Engineering and System Safety 59 217-223. Dinesh Kumar U., Knezevic J. (1997) Spares optimization models for series and parallel structures. Journal of Quality in Maintenance Engineering, , Vol. 3. No3, pp. 177, 188 Giuggioli Busacca P., Marseguerra M, Zio E. (2001) Multiobjective optimization by genetic algorithm: application to safety systems. Reliability Engineering and System Safety 72 pp. 59-74 Goldberg DE. (1989) Genetic Algorithms in Search, Optimization, and Machine Learning. AddisonWesley Publishing Company Marseguerra M. Zio E. Podofillini L. (2001) Multiobjective Spare Part Allocation By Means Of Genetic Algorithms And Monte Carlo Simulation. Submitted to IEEE Transactions on Reliability. Marseguerra, M. and Zio, E., (1993) Nonlinear Monte Carlo reliability analysis with biasing towards top event, Reliab. Engng. and Sys. Safety, 40, pp. 31-42.
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