# Optimizing Spare Parts Inventory for Time-Varying Task

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2 2. Model analyss Equpment wll generate spare parts demand when the tems fal, and the supply tme and mantenance tme wll both affect the operatonal readness. n order to optmze the nventory, the frst step s to forecast the demand rate on the bass of the relablty parameters and usage actvtes. As the relablty parameters s determned n the desgn phase, and use frequency s related to the task requrement, so the demand rate s fluctuatng wth the dynamcal task. Regardng wth the stuaton mentoned above, ths paper proposes a new calculaton model of spare part demand both consderng the dynamcal task and dynamcal spare part program. 2. Effects of dynamcal task to the demand rate Demand rate s the number of spares generatng from the faled tems to the nventory as the equpment perform task, and the formula s: UR() t λ ()= t QPA Nsys MTBF Where MTBF s mean tme between falure of LRU, UR() t s the daly usage of equpment, QAP s the nstallaton number of LRU n the system, Nsys s the number of system. Accordng to formula (), when the task ntensty s dynamcally changng, the usage of equpment s tmevaryng as well. Assumng the ntal demand for spares s 0, the demand s mutually ndependent n each dsjont tme perod. λ () t s ntensty functon and n a small tme nterval t,t+ t the probablty of Δε spare demand s ε () t of whch ε () t s a tny varable related to λ () t, that s, 0 when Δ Δt 0. Then accordng the defnton of non-homogeneous Posson Dstrbuton, the demand of spares follows the non-homogeneous Posson Dstrbuton, and the parameter s: t () Q =Λ( t ) Λ( t ) 2 (2) t 0 Where Λ () t = λ ( s) ds. Λ () t s a mean functon and represents the average falure number wthn the tme perod (0,t). When the task requrement changes wth tme and ts lfetme follows the Exponental dstrbuton, the falure number could approxmate the Posson dstrbuton.the demand of spares s a nonhomogeneous Posson process wth the ntensty of λ () t, and the repar tme s Exponental dstrbuton. As the tems n reparng are unavalable, the expected number of reparng spares can be consdered as the expected number of falure spares. Assumng the repar tme s Exponental dstrbuton, whose average value s / μ = TAT, the reparng spares s: t ξ() t = λ() s ds t TAT (3) Where TAT s the repar tme of LRU. Thus accordng to the above concluson, the tme-varyng demand dstrbuton s: k ( ξ ( t)) Λ () t Pr( D ( t) = k) = e (4) k! Where D () t s the demand number of LRU at tme t. 2.2 Tme-varyng EBO Spare backorder s the sum of lack of all parts per unt tme, the mathematc functon of EBO s: n EBO = ( k s) Pr( D = k) (5) = k= s + Where s s the stock of LRU. 638

3 When the task changes wth tme, spare backorder wll change as well. How to calculate the tme-varyng backorder s the key ssue of spare optmzaton n the non-statonary. Accordng to the expected formula of dscrete stochastc varable, EBO(s,t) of LRU s EBO(,) s t = Pr( D() t = s() t + ) + 2Pr( D () t = s () t + 2) + 3Pr( D () t = s () t + 3) + (6) = ( k s ( t))pr( D ( t) = k) k= s () t + 3. Optmal model and algorthm 3. Optmal model The backorder number of spare parts s a sgnfcant ndcator of that f the spare parts nventory can meet the ablty of spare parts requrements. EBO s the demand number that can not satsfy the expectatons of the supply of spare parts n some tmes, whch s an mportant ndcator of the spare parts guarantee. As long as we can not meet the needs, these wll contnue untl one of the supplements or fault parts has repared. The smaller the backorder number, the hgher combat readness rate of spare parts s. EBO can be seen as optmzaton objectves, the cost of spare parts s a constrant, and we consder the cost of the purchase of spare parts n ths paper, C s the cost of LRU, and sc = s the total cost of spare parts. The mantenance cost s lmted wthn a certan range, then we buld a constraned formula of mantenance costs, C m s maxmum cost, and the constrant condton s task strength, we create a optmzaton model s as follows: mn EBO ( s, t E( D ( t))) = = st.. cs C m FHP() t = 24 UR() t Nsys 0 t T ts, N, (7) c s the unt prce of LRU, FHP() t s daly workng hours, T s the total msson tme. The types of tems are so many and the confguraton s dfferent from each other, so the combnatons are large and t s necessary to apply convex optmal method for the reason that t can guarantee the global optmzaton and effcency. Frstly analyze the convexty of EBO() s Defnte h(s) a functon wth the dscrete varable s. When frst-order dfferental Δ hs () = hs ( + ) hs () 0 and second-order dfference Δ 2 hs () = hs ( + 2) 2( hs+ ) + hs () 0, h(s) s convex functon. To prove that EBO s a convex functon n any probablty dstrbuton and any varable s, the defnton of EBO s ntroduced to convex functon. Δ EBO() s = EBO( s + ) EBO() s = Pr{ D = s + 2} + 2Pr{ D = s + 3} + Pr{ D = s + } 2Pr{ D = s + 2} 3Pr{ D = s + 3} = Pr{ D = s + } Pr{ D = s + 2} Pr{ D = s + 3} 0 (8) 639

4 EBO() s Pr{ D s 3} 2Pr{ D s 4} 2 Δ = = + + = + + 2Pr{ D = s + 2} 4Pr{ D = s + 3} 6Pr{ D = s + 4} + Pr{ D = s + } + 2Pr{ D = s + 2} + 3Pr{ D = s + 3} + = Pr{ D = s + } 0 (9) t satsfes the requrement of convex functon, so EBO s a convex functon. Gven c s the prce of spare parts, then the expected margnal value { EBO( s ) EBO( s)}/ c s not ncreasng functon. As the sum of convex functon s stll convex functon, the nventory curve wth the object of the sum of EBO EBO-C s a optmal curve. 3.2 Optmal Algorthm Due to the wde varety of spares, the nventory confguraton s dfferent n dfferent task. n order to quckly and effcently calculate the optmal nventory t needs a vald algorthm. Convex optmal algorthm precsely meet these requrements. Here are convex optmzaton algorthm specfc steps: Step : ntalze spare parts nventory = S = 0, S 0 (0) Step2: push forward the tme to t = t' +Δ t, where t 'ndcates the last tme, Δ t ndcates the tme nterval and t represents the current moment * Step 3: calculate EBO( S0,..., S,..., SM; t ) and EBO ( t) = EBO( S0,..., S +,..., SM; t) of LRU at tme t Step4: compare each knd of spare part s [ EBO( sk) EBO( sk + )]/ ck, and fnd the knd of spare pare k that make the formula maxmum. Step5: ncrease the nventory of spare parts k, and add the cost of spare parts k to the total cost. * set EBO ( S,..., S +,..., S ; t) = EBO ( t) 0 M * Step 6: f EBO( S0,..., S,..., SM; t) EBO ( t) then return to step 2, otherwse stop. Accordng to the above steps the optmal nventory and smallest EBO can be acqured, where EBO* s the smallest value at that tme. 4. Case analyss Make arplane as example, and optmze the ntal orderng program of the group of arcraft. Usage scenaro determnes the group of arcraft operated wth tme-varyng task, and each arplane conssts of 0 types of LRU. n Fgure there are the daly usage rate λ () t under the whole task cycle. n Table there are the parameters related to the ntal spare parts, ncludng relablty and mantenance desgn parameters and cost of spares. The optmal purpose s to obtan the confguraton that makes EBO smallest under the stuaton of tme-varyng task and cost constrant. Fgure : The daly usage of arplane 640

5 Table : The nput parameters of LRU tem Falure rate(year) Prce(thousand) Repar tme(day) QPA Cost(mllon) LRU LRU LRU LRU LRU LRU LRU LRU LRU LRU Accordng to formula ()-(3) the average reparng number of LRU can be calculated. The reparng number of LRU n the whole lfe cycle s shown n Table 2. Fgure 2: The reparng number of LRU From the above fgure, the reparng number of each LRU s changng wth tme n the whole task cycle, whch results n the optmal nventory per tme nterval. So n order to get the proper nventory, t s necessary to optmze t makng EBO( s, t E( D( t))) largest per 30 days, and adopt the confguraton at that tme as the global optmal nventory schedule. = Fgure 3: The changes of EBO per month under the constrants of cost Fgure 3 shows the optmal curve of EBO. The tme nterval s 30 days, and the cost constrant s 2 mllon dollars. From the fluctuaton of task ntensty, EBO changes sgnfcantly as well, and there even s a gap of 4-5 tmes n some months. So t s necessary to optmze the nventory n each nterval to guarantee the best system effcacy under the cost constrant. 64

6 Table 2: Optmal nventory of LRU per month nventory Jan Feb Mar Apr May Jun LRU LRU LRU LRU LRU LRU LRU LRU LRU LRU Fgure 2 shows the optmal nventory per month. 5. Concluson Ths paper analyzes the tme-varyng varaton of non-statonary spares demand based on the dynamcal task, sets up functon relatonshp between dynamcal spare parts requrements and tme-varyng task, consderng the dynamcal task nfluence to spare parts requrements, apples the non-homogeneous Posson dstrbuton to propose a calculaton model of backorder wth non-statonary Posson process. The convexty of object functon s analyzed and the tme-varyng convex optmal method calculatng the nventory s proposed as well as the typcal case. Ths method can gve out the optmal nventory accordng to the demand of each LRU, so there s a broad applcaton prospects on the tradeoff decsons and optmal schedules of spares nventory. n future wore the followng areas wll be carred out: ) further consder to ntroduce costs constrants of spare parts procurement batch costs, storage costs, as well as spare parts transportaton; 2) apply smulaton method to further verfy the valdty and accuracy of the optmzaton algorthm. 3) based on the dynamcal task calculate the optmal nventory n each nterval when t approaches 0 to control the nventory n real tme. Acknowledgement Ths research s supported by the Natonal Natural Scence Foundaton of Chna (Grant No.60432, No , No.77008). References Eduardo S.B., Eduardo U., 2009, A faclty locaton and nstallaton of resources model for level of repar analyss, European Journal of Operatonal Research, 92, , DO: 0.06/j.ejor Kennedy W.J., Patterson J.W., Lawrence D.F., 2002, An overvew of recent lterature on spare parts nventores, nt. J. Producton Economcs, 76, Lau H.C., Song H.W., 2004, Two-echelon reparable tem nventory system wth lmted repar capacty under nonstatonary demands, Proc. 35th Meetng of the Decson Scence nsttute, Boston, USA. Lau H.C., Song H.W., See C.T., Cheng S.Y., 2006, Evaluaton of tme-varyng avalablty n mult-echelon spare parts systems wth passvaton, European Journal of Operatonal Research, 70, 9 05, DO: 0.06/j.ejor Sherbrooke C.C., 968, METRC:a mult-echelon technque for recoverable tem control, Operatons Research, 6, Tongdan J., Yu T. 202, Optmzng relablty and servce parts logstcs for a tme-varyng nstalled base, European Journal of Operatonal Research, 28, 52 62, DO: 0.06/j.ejor Wenbn W., 202, A stochastc model for jont spare parts nventory and planned mantenance optmzaton, European Journal of Operatonal Research, 26, 27 39, DO: 0.06/ j.ejor

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