STOCHASTIC MODELS IN THE MAINTENANCE MANAGEMENT

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2 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary STOCHASTIC MODELS IN THE MAINTENANCE MANAGEMENT POKORÁDI László, prfessr University f Debrecen; pkradi@mk.unideb.hu Keywrds: maintenance management, technical system peratin, Markv prcesses. Abstract: Technical system peratin is a stchastic prcess that can be divided int states f peratin. This prcess is the cmplex f events that happen t the system. It includes the strage, usage, maintenance, repair and ageing f system. These mdes f existence are called states f peratin. If states f peratin are defined accrding t their means wellseparated, this prcess can be mdeled by a discrete state-space Markv mdels. Using these mdels, the wrk expenditure f peratin can be investigated. The presentatin deals with Markv mdel f technical system peratin prcess and describes a pssibility f its use.. Intrductin Technical system peratin and maintenance is a stchastic prcess that can be divided int states f peratin. This prcess is the cmplex f events that happen t the system frm manufacturing t its rejectin. It includes the strage, usage, maintenance and repair f system. These mdes f existence are called states f peratin. If states f peratin are defined accrding t their means well-separated, the prcess can be mdeled by discrete state-space Markv prcess (in ther wrds Markvian randm walk) (Rhács & Simn 989). Stchastic prcesses whse develpment in the future is influenced by their develpment in the past nly thrugh their develpment in the present, that is stchastic prcesses withut after-effects, are called Markv-prcesses (Wentzel & Ovcharv 986). The histry f studying such prcesses started with the activity f Andrei Andreievitch Markv ( ) the Russian mathematician. Many papers have appeared dealing with Markv mdels fr the ptimal maintenance decisin. The mathematical basis f Markv prcess thery and its applicatin can be knwn by bks f Bharucha-Reid (Bharucha-Reid 96) and Wentzel and Ovcharv (Wentzel & Ovcharv 986). Rhács shws several case studies t demnstrate pssibilities f use f Markv prcess thery fr investigatin f aircraft peratinal systems and prcesses (Rhács & Simn 989). Sng cnsiders the prblem f prductin and preventive maintenance cntrl in a stchastic manufacturing system (Sng 29). By Dimitraks and Kyriakidis, the Markv decisin prcess is a mathematical mdel, which is used t describe a stchastic prcess cntrlled by a sequence f actins. They develped an efficient semi-markv decisin algrithm, which perates n the class f cntrl limit plicies (Dimitraks & Kyriakidi 28). Gu and Yang presents a new technique that des nt use any intermediate mdel t autmatically 345

3 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary create Markv mdels in rder t assess the reliability measurements f safety instrumented systems (Gu & Yang 28). The authr studied pssibilities f use f Markv prcess thery t investigate military aircraft maintenance prcesses (Pkrádi 994a), (Pkrádi 994b), (Pkrádi 997), (Pkrádi 998), (Pkrádi 22), (Pkrádi 28). The main gals f paper are t shw Markv mdels f technical system maintenance, and a maintenance management methd based upn the Markvian randm walk prcess. The rest f the paper is rganized as fllws. The definitin f the Markv prcesses is given in Sectin 2. In Sectin 3, we present the general mdel f ageing prcesses f technical systems. The Sectin 4 shws a maintenance management methds based n Markv mdel, theretically. The Sectin 5 presents a shrt case study by frmer scientific wrk f the authr. In the last sectin, the cnclusins f this wrk are given 2. The Markv-Prcesses The mathematically described randm prcess η(t) is called Markv-ne if the equatin f cnditinal prbabilities. ( (t ) = X η(t ) = X η(t ) = X ) = P( η(t ) = X η(t ) X ) P η n+ n+ K n n n+ n+ n = n () prves t be true with the prbability fr each t < < t n < t n+ and X ;X 2 ; ;X i ;X i+ numbers. If prcess η(t) during the study perid can have an X value at any mment, it is called a cntinuustime prcess. If η(t) can nly have sme value at certain mments, the prcess is called a discretetime ne. A randm prcess is cnsidered t be f discrete state space, if the pssible values f variance η(t) cnstitute a finite set r a cunt nn-finite set. Finite r cunt nn-finite stchastic prcesses, that is the discrete state space nes with n aftereffects, are called Markv-chain (Wentzel & Ovcharv 986). In this case, the value established in the equatin () is called the transitin prbability: ( η(t ) = X η(t ) X ) n,n+ P ij = P n+ n+ n = n 346. (2) The transitin prbability expresses that η(t n+ ) = X j, suppsing that η(t n ) = X i. P n, n + ij marking abve als shws that the transitin prbability is the functin f nt nly the i-th beginning state and f the j-th next state, but it is als the functin f t n time. A Markv-prcess can be characterized unambiguusly by supplying the transitin prbabilities, and the distributins f leaving different states. If distributin f leaving different states are nt f the expnential character, such stchastic prcess is called Semi-Markv ne. 3. The Ageing Prcess During the peratin, a technical system wears ut by stchastic effects. Therefre its technical real

4 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary state ges thrugh cntinuus, cumulative stchastic changes. In case f the usage the technical state generally changes in a negative sense, while during the maintenance r repair, it changes psitively. Fr demnstratin, let general parameter η characterize the technical state f the investigated system (see figure ). If the value f parameter η meets the η br brake value, the system breakdwns. Let τ be the parameter, which characterizes the perfrmance f the system. Fr example, this parameter can be the effective calendar time, effective perating hurs, kilmeters frm installatin r the last verhaul. In this case the wearing-ut prcess f the system, that is the η(τ) stchastic functin can be characterized by: ) η ( τ ) expected value functin f the parameter η; f ( η, τ ) density functin f parameter η. Figure The Aging Prcess Then the prbability f gd wrking state f the system: ηbr Pgw ( τ) = P( ηbr > η( τ) ) = f ( η, τ) dτ. (3) The prcess f changing f parameter η can be describe by: η ( τ) changing velcity f the parameter η; ϕ η, τ density functin f the changing velcity. 347

5 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary Then the "failure changing velcity" f parameter η is: ηbr, ( τ τ) ( ) η η τ = br τ if η br > η( τ) (4) and the prbability f gd wrking state f the system in the interval (η, η+ η): Pgw ( τ, τ) = P η ( τ) > η( τ) br = ηbr ϕ η, τ d η 348,. (5) suppsing that the system is ready t service at the start f the investigated perfrmance interval. 4. Maintenance Management Methd Depending n the mmentary values f the leader parameter and its velcity, the needed service wrk can be decided. Fr decisin, permissible value and velcity f the leader parameter shuld be determined n the basis f its breakdwn value and permissible prbability f risk. Knwing the breakdwn value η br f the parameterη and perfrmance interval between checks τ, the permissible value η p and permissible changing velcity t ready fr wrking shuld be determined. Suppsing that: the change f the parameter η n interval τ (see figure 2) is a linear ne; density functin f the changing velcity is independent n wrking time f the system. In this case, if value f the parameterη reaches the permissible value η p at the i-th checking and it changes with η η > τ velcity, the parameter η is ging t reach breakdwn value η br befre next (i+-th) check, in the ther wrds the perated system will break-dwn. Therefre, permissible velcity f parameterη t ready fr wrking is: η η p =. (6) τ The prbability f breakdwn is: ηbr Pbr ( τ, η) = P = η > ηbr P η ηbr = ϕ( η)d η (7) Knwing permissible prbability f risk Q (permissible prbability f breakdwn), it is

6 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary substituted int equatin (5), equatin ηp Q = Pbr ( τ, η) = ϕ( η)d η (8) is gt. Figure 2 Determinatin f the Permissible Parameter Values If the density functin f velcity η cannt be determined by statistical methd, usage f ne f knwn density functins is suitable. Fr example: Then that is Then UNIFORM distributin: ϕ( η) = = ηmax ηmin η ηp ηp η Q = d η = = η η τ η (if η max > η > ηmin ). (9), () η = ( Q) τ η. () EXPONENTIAL distributin: λη ϕ( η ) = λe (if η > ). (2) 349

7 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary and Q ληp = e, (3) ln( Q) η = τ λ NORMAL (GAUSS) distributin: ϕ ( η m) ( η) = e σ σ 2π (4). (5) In this case simply slutin cannt be gt like t abve nes, which is deduced easily in algebraic way. Therefre, n the basis f its variance and expected value, transfrming the nrmal distributin t the standard nrmal ne, the permissible velcity f parameter η and parameter interval η can be determined. The permissible value f the parameter η t ready fr wrking: ηp = ηbr η. (6) If mmentary values η and η smaller that thse determined by equatin (6) and (6), the system will nt break dwn till the next check with prbability f least Q. 5. A Shrt Example f the Methd s Usage Fr demnstrating the pssibility f use f abve mentined methd, the setting up and usage f mathematical mdel f aircraft brake-system will be shwn. δf Σ [ ],8,6,4,2 δf Σ [ ],8,6,4, Operating Hurs Figure 3 Decrease f the Brake-Effrt Depending n Operating Hurs Mnth Figure 4 Decrease f the Brake-Effrt Depending n Calendar Time (f the Investigating Year) The decrease f the brake-effrt and brake asymmetry was chsen as leader parameters. T determine the permissible value and velcity f this leader parameters, 35

8 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary permissible prbability f risk was used. dδf Σ [f hur -5 ] dt w Q =,25 dδf Σ [day -6 ] dt c Operating Hurs Figure 5 Changing Velcity f the Brake- Effrt Depending n Operating Hurs δf [ ],25,2,5,, Operating Hurs Figure 7 Brake-Asymmetry Depending n Operating Hurs dδf [f.hur -5 ] dt w Mnth Figure 6 Changing Velcity f the Brake- Effrt Depending n Calendar Time (f the Investigating Year) δf [ ],25,2,5,, Mnth Figure 8 Brake-Asymmetry Depending n Calendar Time (f the Investigating Year) dδf [day -6 ] dt c Operating Hurs Figure 9 Changing Velcity f Brake- Asymmetry Depending n Operating Hurs Mnth Figure Changing Velcity f Brake- Asymmetry Depending n Calendar Time (f the Investigating Year) The quantity f data is nt sufficient fr statistical estimatin f their distributin. Therefre, fr determinatin f the permissible value and velcity f the resultant brake-effrt, the densities f its changing velcity is suppsed as unifrm ne. 35

9 Internatinal Cnference, 4-5 Octber 2, Debrecen, Hungary 6. Cnclusins The paper shwn the Markvian mdels f technical system peratinal (maintenance) prcesses shrtly In this study the management methd f technical peratin, as a Markvian randm walk prcess, has been frmulated in case f aircraft break system. Technical data needed fr usage f mathematical diagnstics and peratinal management methd can be btained by using gages f the helicpter and instrument f technical service team. The shwn maintenance management methd is able t minimizatin f technical service wrk with adequate safety f peratin. The pssibility f use f shwn methd has been prven by examinatin f pneumatic brake-system a regular aircraft. References Bharucha-Reid A.T. (96). Elements f the Thery f Markv Prcesses and Their Applicatins, McGraw-Hill, New Yrk. Dimitraks T.D., Kyriakidis E.G. (28). A semi-markv decisin algrithm fr the maintenance f a prductin system with buffer capacity and cntinuus repair times, Int. J. Prductin Ecnmics Gu H., Yang X. (28) Autmatic creatin f Markv mdels fr reliability assessment f safety instrumented systems, Reliability Engineering and System Safety Pkrádi L. (994a). Applicatin f Markv Prcess Thery t Investigatin f Aircraft Operatinal Prcesses, Prceedings f 9 th Cngress f the Internatinal Cuncil f the Aernautical Sciences, Anaheim, Califrnia, USA, 994., vlume 3, p Pkrádi L. (994b) Investigatin f Aircraft Operatinal System with Markv-Matrix, Prceeding f 4 th Mini Cnference n Vehicle System Dynamics Identificatin and Anmalies, Budapest, 994., p Pkrádi L. (997) Markvian Mdeling Wartime Operatin f Military Aircraft, "Internatinal Aerspace Cngress 997", Sydney, Australia, Vlume 2, p Pkrádi L., Szablcsi R. (998). Aircraft Operatin Management Based n State-Estimatin, Prceedings f 2 st ICAS Cngress, 3-8 September 998., Melburne, Victria, Australia (CD-versin). Pkrádi L. (22). Queuing Mdels Used t Investigate Technical Lgistics, Advanced Mdeling and Optimizatin, vlume 4, number 2, p Pkrádi L (28) Systems and Prcesses Mdeling Campus Kiadó, Debrecen, 28, pp (in Hungarian). Rhács J., Simn I. (989). Aircraft and Helicpter Operatin Manual, Mőszaki Könyvkiadó, Budapest, (in Hungarian). Sng D.P. (29). Prductin and preventive maintenance cntrl in a stchastic manufacturing system, Int. J. Prductin Ecnmics 9.. Wentzel E., Ovcharv L. (986) Applied Prblems in Prbability Thery, Mir Publisher, Mscw. 352