Inventory Decision Model of Single-echelon and Two-indenture Repairable Spares

Transcription

1 Te nd nternatonal Conference on Computer Applcaton and System Modelng (0 nventory Decson Model of Sngle-ecelon and Two-ndenture Reparable Spares Lu Cenyu Naval Aeronautcal and Astronautcal Unversty Qngdao Branc Qngdao 6604, Cna Zou Bn Naval Aeronautcal and Astronautcal Unversty Qngdao Branc Qngdao 6604, Cna Guo Feng Naval Aeronautcal and Astronautcal Unversty Qngdao Branc Qngdao 6604, Cna Zang Suqn Naval Aeronautcal and Astronautcal Unversty Qngdao Branc Qngdao 6604, Cna Abstract-Accordng to te tree condtons tat te varanceto-mean rato of te spares demand s larger tan, equal to or less tan, put forward te tree demand dstrbutons suc as te Negatve bnomal dstrbuton, Posson dstrbuton and Bnomal dstrbuton. For te problem tat te two-level materals demand follows te Posson dstrbuton, use te Negatve bnomal dstrbuton to mprove te forecast accuracy. n te gven total securty funds as constrant condtons, troug total sortfall mnmum nstead of supply avalablty maxmum to smplfy te objectve functon, establs te spares nventory decson model, and optmzed by te margnal analyss and comple algortm. Te example proves tat te model as good accuracy and practcalty. Keywords-nventory decson, Reparable spares, Supply avalablty, Two ndentures, Varance-to-mean rato. NTRODUCTON wen predctng te stock of spares n te past, generally assume tat te demand of te reparable spares follow te Posson dstrbuton, but actually purcasng next spare, not only demand s random, but te average may cange, so not all spares are subject to te Posson dstrbuton; n addton, exstng spares nventory decson models do not take nto account te nfluence of te needs of mult-ndenture to dstrbuton of demand, wc would generate more predcton errors. Spares nventory decson requres forecastng average demand and some safety stock, terefore, to determne te dstrbuton of demand means not only to use te mean of spares demand, but also to consder te varance-to-mean rato, wc as often been overlooked n te past decson researc about te spares nventory. Most of spares demand n te sort term follow te Posson dstrbuton wt mean as a constant [-3]; However, wt te observaton growng te dfference average rato as an ncreasng trend, te unque model n lne wt tese observatons s te ncremental non-stable posson process, ten te Posson dstrbuton can be extended to te Negatve bnomal dstrbuton wt ts varance over ts mean, use negatve bnomal to establs te nventory decson model. n addton, wen te mult-ndenture demand of te spares takes place, even f te demand of te frst ndenture spares follows te Posson dstrbuton, but because te sortage of parts brngs about some of ts ndrect for t, wc causes te varance-to-mean rato of te ppelne (product of average demand mean and average repar tme more tan, ten te probablty dstrbuton meetng te needs of te frst ndenture spares s no longer te Posson dstrbuton, but te Negatve bnomal dstrbuton, te predcton accuracy tan te Posson dstrbuton. Tese are te two dstrbutons of random demand, n fact, te demand process of some spares s not stocastc, but caused te loss of spares, suc as turbne blades, landng gear, batteres, etc., wc are subject to te Bnomal dstrbuton wt ts varance less tan ts mean[4]. dentfyng demand dstrbuton s only te frst step of nventory decson, te second step s to determne te assessment ndcators of nventory decson [], and establs te nventory decson model to make te assessment on te supply system performance: t s necessary to evaluate te correspondng error of te temzed parts, but also assess te total ntegraton error of all te varous parts. Te correspondng error standard of te temzed parts s ts sortfall, but te total error standard s not a smple summaton of sortfall. As te sortfall can be converted to te supply avalablty of arcraft, and supply avalablty ndex can predct te arport's arcraft avalablty next year, so t can be used as te evaluaton ndex of spares nventory decson. From wc te optmzaton objectve of nventory decsons can be drawn: wtn te gven total nvestment cost of spares, look for te gest supply avalablty[6-9], or lmt te supply avalablty standard, for te mnmal cost, tese two goals s bot based on seekng te mnmal total sortfall, so ter solutons are smlar, te model establses te nventory decson model only by te frst target.. ASSUMPTONS ( Te number of te falures of all te arcraft parts s ndependent of eac oter. ( Te requrements of te frst ndenture parts (Lnereplaceable Unt, LRU occurs n te base, f currently tere are stocks ten send out one, f not, once sortage of LRU appens. Bot cases, no matter wat knd, are at te base reparng faulty LRU, wle n te repar process fnds ts own second ndenture spares (Sop-replaceable Unt, SRU s faulty, replace tem f te SRU falure parts n stock to complete te repar, or once sortage of SRU occurs. Bot Publsed by Atlants Press, Pars, France. te autors 044

2 Te nd nternatonal Conference on Computer Applcaton and System Modelng (0 cases, no matter wat knd, are at te base reparng SRU. Eac LRU falure only because of a fault n SRU, SRU repar wll not be delayed for ts parts are not n place. Ts paper manly studys te two-ndenture demand problems tat te frst ndenture spares follow te Posson dstrbuton. (3 Fault parts of a spares n te base can be repared wtn a certan tme, and are expressed by means of te probablty dstrbuton of T, ts demand meets te nventory balance equaton s OH + D - BO[0], were, s s te necessary nventory amount, OH s te currently avalable nventory amount n te base, D s te number of te reparng spares, BO s te spares sortage. (4 No cannbalzatons. ( n strct accordance wt te prncple turn over te old and lead te new to provde te spares for te outfeld, no stuaton of LRU arrears.. THREE-REQUREMENT DSTRBUTON OF SPARES A. Negatve bnomal dstrbuton Wen varance-to-mean rato s greater tan, te demand of spares follows te Negatve bnomal dstrbuton, namely r + x - x r Pr ( x p ( x - p x x 0,,,... ( r ( p Were, r>0,0<p<. Troug te mean μ p and varance-to-mean rato V p of te Negatve bnomal dstrbuton, educe tat r μ, p ( V V Tus, parameters r and p can be calculated by mean and observaton, and generate a negatve bnomal probablty dstrbuton. For te second-ndenture demand, te metod to estmate te parameters of ts Negatve bnomal dstrbuton s: Suppose subscrpt as te ndex of LRU spares occurng two-ndenture demand, te subscrpt j s te ndex of ts own SRU spares. Because LRU, wc occurs two-ndenture damand, follows te Posson dstrbuton, so t's te demand of SRU also follows te Posson dstrbuton, namely m m Te actual demand of LRU follows te Negatve bnomal dstrbuton, use te mean E(X and varance Var(X of te ppelne to estmate te parameters r and p and te number of spares sortage EBO(s 0 E(X, Var(X, te mean and varance formulas of te ppelne are (3 E ( X m T + Var ( X m T + EBO( s VBO( s m T m T (4 ( Were, EBO (s, VBO (s are te expectaton, varance of te number of spares sortage, namely EBO( s VBO( s E[ B x s+ x s+ ( x spr{d x} ( X s] [EBO( s] ( x-s Pr{ X x} [EBO( s] (6 (7 B. Posson dstrbuton Wen varance-to-mean rato equal to, te spares demand follows te Posson dstrbuton. Accordng to Palm teorem [], assume tat te demand of any spares s subject to Posson process wt te average annual demand m, and eac faled unt repar tme are ndependent, and obey te same dstrbuton wt te average repar tme T, te steady-state probablty dstrbuton of number of reparng spares follows te Posson dstrbuton wt mean mt, namely x mt ( mt e Pr ( x x 0,,,3,... (8 x! C. Bnomal dstrbuton Wen varance-to-mean rato less tan, te spares demand follows te Bnomal dstrbuton, namely n x n x Pr( x p (- p x 0,,,..., n (9 x Were, n> 0,0 <p <. Troug te mean μ np and varance-to-mean rato V p of te Bnomal dstrbuton, so μ n V, p V (0 Tus, parameters n and p can be calculated by mean and observaton, and generate a bnomal probablty dstrbuton. V. NVENTORY DECSON MODEL AND SOLUTON A. nventory Decson Model Supply avalablty s te assessment ndcators of spares nventory decson, t expresses te expectaton value A of percentage of number of te grounded arcrafts not due to any backorder, namely Publsed by Atlants Press, Pars, France. te autors 04

3 Te nd nternatonal Conference on Computer Applcaton and System Modelng (0 A 00 EBO ( s { } NZ Z ( Were, s te number of tems of spares, Z s te number of spare nstalled n one plane, N s te number of arcraft fleet, s s te stock of spare. Spares mentoned ere, ncludng LRU and SRU, so ere te subscrpt s te ndexes of all te spares. On condton tat tere s no cannbalzaton, to searc for te sortfall mnmum s essentally equvalent to searc for te supply avalablty maxmum. Terefore, use te total sortfall mnmum as te objectve functon, te total securty funds as constrant condton to establs nventory decson optmzaton model, namely mn z EBO ( s ( cs C Were, c s te unt prce of spare n te nventory, C s total costs of te spares of te equpment system (nteger value, ts unt s ten tousand dollars. B. Model soluton Model soluton steps are: Frst, calculate varance-tomean rato to determne wat dstrbuton te demand of spares follows, and calculate ts parameters. Ten, accordng to te nventory decson model, calculate te optmal nventory allocaton, sortfall and supply avalablty of spares under te condtons of gven stock funds and te total sortfall mnmum. Te soluton key s to solve te nventory decson model, t uses funds as resources, te total sortfall as te objectve functon, wc s a typcal problem of optmal allocaton of resources; At te same tme, te sortfall functon s a convex functon, meetng te requrements to searc for te optmal allocaton by te margnal analyss[-4]. V. EXAMPLE ANALYSS Tere s 4 arcrafts n a base, t as tree tems of frstndenture spares LRU, LRU, LRU3, LRU as two tems of second-ndenture spares SRU, SRU, ter probablty causng LRU falure s 0.. Te past ten years, from 000 to 009, of demand statstcal data of te fve spares s sown n Table. Stocks of tree LRU spares n late 009 s respectvely (,3,4, stocks of SRU, SRU s 0, no reparng spares, te budget of te fve spares s 0 mllon dollars n te 00, next, predct te stock confguraton n 00. Te sortage and costs of all te spares n dfferent nventory confguratons are sown n Table. Table. sows, te total cost s 90 mllon dollars, fve spares nventory confguraton s (0,3,7,,, te supply avalablty s equal to 9.783%. Analyss: ( Te sortage of te latter four spares are small, ter backorder are unlkely to occur. Te sortage of te frst spare s.30, wc s prone to occur. So, n te work of te next year, t s necessary to partcularly look at te frst spares of te securty to prevent sortage. But, n fact, te sortage of spares s always nevtable, and spares support personnel can mnmze te sortage of spares to make supply avalablty to mantan a reasonable level. n practce, te supply of spares ncludes normal and abnormal supply, normal supply s to provde te outfeld wt spares by base wareouse, wc can be predcted by ts model, abnormal supply s to supply troug te emergency order, allocaton, borrowng, etc, wc s greater flexblty and s essental way to reduce te sortage of spares to furter enance te supply level of spares. As te supply avalablty of te normal supply s ger tan 9~98%, te stock captal would ncrease dramatcally, so n order to mantan reasonable nventory level, meanwle, to make te supply avalablty of spares to reac 9~98%, generally set normal supply avalablty at 90~9%, and te abnormal supply avalablty at between % to 6% []. n ts example, f all spares are suppled by te base wareouse, to make te supply avalablty to reac %, tere need to nvest 6 mllon dollars, compared wt 00 budget ncreases 30%. f combnng wt abnormal supply mode, you can aceve te same even ger supply level, but t can greatly reduce te budget so tat more money concentrated n te spares supply command sector, spares s easer to rase. ( Te metod to verfy te accuracy of te model predcton s: Calculate te actual supply avalablty of ts year by te actual sortfall n 00, and compare te supply avalablty of normal supply wt te supply avalablty worked out by ts model to determne f te model's predcton s accurate. Wen te base wareouse suppled outfeld wt te frst spare n 00, sortfalls occurred for twce, wc s a normal supply, te supply avalablty at ts tme was %; However, te spares supply sector solved once of tem troug an emergency allocaton of spares, so te actual sortfall s, ts tme te supply avalablty reaced %, te extra avalablty % s accomplsed by way of abnormal supply mode. Clearly, te relatve error of te 00 supply avalablty value forecasted by ts model and actual value was only 0.943%, altoug slgtly ger tan normal formulated range 90 ~ 9%, but stll wtn te allowable range. Terefore, ts model as good predctve accuracy. (3 Excludng te two-ndenture demand, te LRU follows te Posson dstrbuton, oter condtons are constant, te supply avalablty at ts tme s %, compared wt te actual value, te error s greater tan te model predcton. n addton, f fve spares, regardless of ter varance-to-mean rato s greater tan, equal to or less tan, are calculated by te Posson dstrbuton, supply avalablty may be up to %, wc s muc ger compared wt te model predctons and actual values, devaton s very sgnfcant, ts s caused by te Posson dstrbuton calculatons even te varance not equal to, so Publsed by Atlants Press, Pars, France. te autors 046

4 Te nd nternatonal Conference on Computer Applcaton and System Modelng (0 te key to get more accurate results s to predct accordng to te actually subject probablty dstrbuton. V. SUMMARES ( Te Negatve bnomal dstrbuton and Bnomal dstrbuton s te metod to establs te demand forecastng model for te varance ger and lower tan mean, but reparable spares not only obey te Posson dstrbuton wt te varance equal to te mean, but also be subject to te two dstrbuton, as to wat dstrbuton s adopted to predct can be decded by varance-to-mean rato. f only adopt te Posson dstrbuton or not consder multndenture demand, ter predcton results are sometmes large dfferent from te actual, compared to tem, predcton of ts model s more accurate. ( Te model s te same wt te nventory decson of all crtcal spares wt a greater mpact on system performance, as te system effcency generated by tese spares reflects te basc supply level, tus te results of ter nventory decsons can be used as te fundamental bass of spares management decson next year. Ts s a systematc approac to optmze te overall nventory of spares and aceve te ger system effcency. REFERENCES [] HE Ya-qun; TAN Xue-feng; N Fu-lu. Demand Analyss of Arcraft Reparable Spares Based on Avalablty[]. Systems Engneerng and Electroncs, 004,6(6: [] CHEN an-ua. Researc on Plannng and nventory Management of Reparable Spares Parts n Cnese Arlnes[D]. Bejng: Bejng aotong Unversty 009. [3] WANG Kun. Management of Arcraft Equpment Purcase and Stock[D]. Nanjng: Nanjng Unversty of Aeronautcs and Astronautcs, 00. [4] DU un-gang; HE Ya-qun. On te ndexes for te Apprasal of Reparable tems Precse Support n te USAF[]. ournal of Xuzou Ar Force College, 007,8(4: [] LU S-a; ZHENG n-zong; Mng. Arcraft Materal Securty ndex Analyss Based on ARNC Model[]. Wareouse management and tecnology 007(4:9-3. [6] ZHANG Heng; HUA Xng-la; XU Sao-mu. nventory Decson Smulaton Optmzaton Model of Reparable Spares System[]. Systems engneerng and electroncs 009 3(6:0-4. [7] ZHANG Ru-cang; ZHAO Song-zeng. Consumptve Materals Spares to Determne te Order Model[]. Mltary Operatons Researc: and Systems Engneerng, 004,8(4:40-4. [8] FU Hong-yong; ZHAO Yu. Te Analyss of Spare Cost and Operatng Avalablty for Plane-group[]. Avaton mantenance and engneerng, 004(3:-3. [9] LU Yuan; CHEN Yun-xang. Optmzaton Researc on Avaton Spares Reserves Based on Avalablty and Cost Requrement[]. ournal ofar force engneerng unversty(natural scence edton, 009,0(6:-8. [0] Crag C. Serbrooke. Optmal nventory Modelng of Systems: MultEcelon Tecnques, Second Edton[M]. He Buje, Translate. Bejng: Publsng House of Electroncs ndustry 008. [] U Dnes Kumar. Relablty, Mantenance and Logstc Support A Lfe Cycle Approac[M]. LU Qngua, SONG Nngze, Translate. Beng: Publsng House of Electroncs ndustry, 00. [] FU Xng-fang; L -jun. A Stock Model and ts Algortm for Restorng Ar Materel Beneat te Sngle-level Provdng Condton[]. Operatons Researc and Management Scence, 003(:9-9. [3] FU Xng-fang; L -jun; L Zong-z. A Stock Strategy Model for Restorng Ar Materel Based on te Two-level Provdng Condton[]. Systems Engneerng-Teory & Practce, 00,30(6: [4] QU Hong-cun; ANG Bo-song. Study on te Aeronautcal Materal System Relablty-based Optmzaton Model of Spares[]. Avaton manufacturng tecnology, 004(9:79-8. [] ZHAO Su-fang. Study on Arcraft Spare Prognostcatng Metod based on RCM[D]. Nanjng: Nanjng Unversty of Aeronautcs and Astronautcs, 00. TABLE. STATSTCAL DATA OF ALL THE SPARES DEMAND ndex of spares Z c / j T m V mt / / Publsed by Atlants Press, Pars, France. te autors 047

5 Te nd nternatonal Conference on Computer Applcaton and System Modelng (0 TABLE. SHORTAGE, THE SPARES N DFFERENT CONFGURATONS LRU (s LRU (s LRU3 (s 3 SRU (s 4 SRU (s total sortage (sortage of eac base total costs (4.097,0.6,0.4007, , (3.3947,0.6,0.4007, , (3.3947,0.6,0.4007, , (3.3947,0.6,0.4007, (.670,0.6,0.4007, (.670,0.6,0.808, (.0474,0.6,0.808, (.0474,0.8,0.808, (.30,0.8,0.808, (.30,0.060,0.808, (.30,0.060,0.808, , (.30,0.060,0.808, , (.30,0.060,0.808, , (.30,0.060,0.808, , (.30,0.060,0.808, , (.30,0.060,0.808, , Publsed by Atlants Press, Pars, France. te autors 048