An Ordering Policy Based on Uncerain Renewal Process wih Applicaion o Aircraf Spare Pars Chunxiao Zhang, Congrong Guo College of Science, Civil Aviaion Universiy of China, Tianjin 300300, China Absrac: Consider an ordering problem of spare pars for a flee consising of muliple new aircrafs. By dealing wih he inerarrival imes as uncerain variables wih a common uncerainy disribuion, an ordering policy of spare pars based on uncerain renewal heory is proposed, and an analyic formula of he opimal ordering quaniy is derived. Finally, a numerical example is provided o illusrae he uiliy of he resul in pracice. Keywords: ordering policy, aircraf spare pars, renewal process, uncerainy heory 1 Inroducion The serious accidens caused by he failure of aircraf pars will no only bring ou exraordinarily heavy damage and bad social repuaion, bu also endanger he lives of passengers and crew. Thus, he airlines usually adop prevenive mainenance policy and impor more han enough aircraf pars for ensuring he replacemen of pars imely a predeermined age or a failure. However, he reserve accouns of spare pars is abou 75 percenage of invenory asses and 25 percenage of liquidiy of an airline, so he high invenory cos of spare pars has been he main componen of operaing coss and he key issue for airlines. In his case, making an opimal ordering policy of spare pars is recognized as one of he mos poenial way o reduce he operaing cos of he airlines. An opimal ordering policy of spare pars is deermined by balancing he risk/cos of exended downime of criical pars agains he holding coss of he invenory. On he one hand, due o exorbian price and ordering coss, he high invenory of aircraf pars is uneconomic. On he oher hand, once he shorage of spares happen, i is difficul o obain he pars from he supplier in a shor ime, so ha he downime of aircraf is prolonged, which will resul in heavy economic losses and inesimable negaive social impac o civil aviaion enerprises. Therefore, he ordering problem of aircraf pars is difficul o handle for an airline. Renewal process is an imporan process in renewal heory and widely used o solve replacemen 1
problem of spare pars. Feller developed renewal heory as a mahemaical discipline [7]. Blackwell [4, 5] deermined he expeced number of renewal evens in a given ime. Cox had imporan conribuion in he sochasic poin processes heory, made an ousanding impac on he renewal heory developmen, and opened new ways o apply sochasic processes in mainenance managemen [6]. In he nex several decades, various renewal processes based on probabiliy heory were well developed [1, 2, 8, 20], and he inerarrival imes in hose processes were considered as iid random variables. Recenly, some researchers sudied fuzzy heory and applied i o renewal process. Employing he concep of fuzzy number defined by Zadeh[24], Badard[3] discussed fuzzy processes by sequence of relaed fuzzy numbers. Kwakernaak[11, 12] firsly proposed fuzzy random variable which became a useful ool o sudy such a phenomena where fuzziness and randomness exis a he same ime. Zhao and Liu [26] considered a fuzzy renewal process which is generaed by a sequence of posiive iid fuzzy variables, and obained a fuzzy elemenary renewal heorem and a fuzzy renewal reward heorem. Hong [9] discussed a renewal process in which he inerarrival imes and rewards are depiced by L-R fuzzy numbers under -norm-based fuzzy operaions. By dealing wih inerarrival imes as exponenially disribued fuzzy random variables, Li e al. [13] considered a fuzzy random homogeneous Poisson process and a fuzzy random compound Poisson process, and esablished several imporan properies of hese wo ypes of processes. Zhao and Tang [27] discussed some properies of fuzzy random renewal processes, and obained a Blackwell s renewal heorem and a Smih s key renewal heorem for fuzzy random inerarrival imes. In his paper, we focus on an ordering problem of one ype of spare par in a flee consising of some new aircrafs. Due o he high price of he pars, i is impracical o conduc an experimen daa for is lifeime. Besides, so far here are no observed daa of pars lifeimes because he fligh ime of hose new ype of aircrafs is so shor ha no failure of spare par has happened. Thus, we have no daa o deermine he probabiliy disribuion of par s lifeime. In his siuaion, we have o invie a number of domain expers o evaluae he lifeimes of pars and heir degree of belief. because people ends o overweigh unlikely evens (Kahneman and Tversky [10]), he degree of belief may have much wider range han ha of he real frequency. If we sill use probabiliy heory o deal wih he kind of problem, some counerinuiive resuls will be obained [18]. Uncerainy heory was founded by Liu [14] in 2007 and refined by Liu [16] in 2010. Nowadays uncerainy heory has become a branch of axiomaic mahemaics o deal wih he degree of belief and has been applied o many fields. The concep of uncerain process was firsly proposed by Liu [15] as a sequence of uncerain variables indexed by ime in 2008. No long laer, an elemenary renewal heorem for deermining he average renewal number was proved by Liu [16] in 2010. Meanwhile, Liu [16] proposed he renewal reward process whose inerarrival imes and rewards were considered o be uncerain variables, and he renewal reward heorem was also proved for deermining he long-run reward rae. In 2012, Yao and Li [22] inroduced he concep of alernaing renewal processand proved an alernaing 2
renewal heorem for deermining he availabiliy rae. Furhermore, a delayed renewal process in which he firs inerarrival ime is allowed o have a differen uncerainy disribuion from he remaining ones was discussed by Zhang e al [25]. Recenly, Yao and Ralescu [23] researched he opimal age replacemen policy when he lifeime of uni follows uncerainy disribuion. Nex we will consider he inerarrival imes of aircraf pars as uncerain variables and find he opimal ordering policy based on uncerain process. The res of his paper is organized as follows: Secion 2 recalls some basic conceps and properies on uncerain renewal process. In Secion 3, an uncerain renewal process is inroduced for beer modeling he age replacemen policy; an applicaion case is presened in Secion 4. Finally, some remarks are made in Secion 5. 2 Preliminaries In his secion, we will inroduce some useful definiions and heorems of uncerainy heory needed hroughou his paper. Le Γ be a nonempy se. A collecion L of Γ is called a σ-algebra if (a) Γ L; (b) if Λ L, hen Λ c L; and (c) if Λ 1, Λ 2, L, hen Λ 1 Λ 2 L. Each elemen Λ in he σ-algebra L is called an even. Uncerain measure is a funcion from L o [0, 1]. In order o presen an axiomaic definiion of uncerain measure, i is necessary o assign o each even Λ a number M{Λ} which indicaes he belief degree ha he even Λ will occur. In order o ensure ha he number M{Λ} has cerain mahemaical properies, Liu[14] proposed he following hree axioms: Axiom 1: (Normaliy Axiom) M{Γ} = 1 for he universal se Γ. Axiom 2: (Dualiy Axiom) M{Λ} + M{Λ c } = 1 for any even Λ. Axiom 3: (Subaddiiviy Axiom) For every counable sequence of evens Λ 1, Λ 2,, we have { } M Λ i M{Λ i }. (1) Definiion 1. (Liu [14]) The se funcion M is called an uncerain measure if i saisfies he normaliy, dualiy, and subaddiiviy axioms. Besides, in order o provide he operaional law, Liu [17] defined he produc uncerain measure on he produc σ-algebre L as follows. Axiom 4: (Produc Axiom) Le (Γ k, L k, M k ) be uncerainy spaces for k = 1, 2, Then he produc uncerain measure M is an uncerain measure saisfying { } M Λ k = M k {Λ k }, (2) k=1 where Λ k are arbirarily chosen evens from L k for k = 1, 2,, respecively. 3
Definiion 2. (Liu [14]) An uncerain variable is a measurable funcion ξ from an uncerainy space (Γ, L, M) o he se of real numbers, i.e., for any Borel se B of real numbers, he se {ξ B} = {γ Γ ξ(γ) B} (3) is an even. For a sequence of uncerain variables ξ 1, ξ 2,, ξ n and a measurable funcion f, Liu [14] proved ha ξ = f(ξ 1, ξ 2,, ξ n ) defined as ξ(γ) = f(ξ 1 (γ), ξ 2 (γ),, ξ n (γ)), γ Γ (4) is also an uncerain variable. A concep of uncerainy disribuion is inroduced o describe an uncerain variable as follows. Definiion 3. (Liu [14]) The uncerainy disribuion Φ of an uncerain variable ξ is defined by Φ(x) = M{ξ x} (5) for any real number x. An uncerain variable ξ is said o be normal if i has a normal uncerainy disribuion Φ(x) = ( ( )) 1 π(e x) 1 + exp, x R (6) 3σ denoed by N(e, σ) where e and σ are real numbers wih σ > 0. In his case, he uncerain variable exp(ξ) has a lognormal uncerainy disribuion Ψ(x) = ( ( )) 1 π(e ln x) 1 + exp, x R + (7) 3σ denoed by LOGN(e, σ). Definiion 4. ( Liu [16]) Le (ξ 1, ξ 2,, ξ n ) be an uncerain vecor. Then he join uncerainy disribuion Φ : R n [0, 1] is defined by Φ(x 1, x 2,, x n ) = M{ξ 1 x 1, ξ 2 x 2,, ξ n x n } (8) for any real numbers x 1, x 2,, x n. Definiion 5. ( Liu [17]) The uncerain variables ξ 1, ξ 2,, ξ m are said o be independen if { m } m M (ξ i B i ) = M{ξ i B i } (9) for any Borel ses B 1, B 2,, B m of real numbers. 4
Peng and Iwamura [21] proved ha a funcion Φ : R [0, 1] is an uncerainy disribuion if and only if i is a monoone increasing funcion excep Φ(x) 0 or Φ(x) 1. The inverse funcion Φ 1 is called he inverse uncerainy disribuion of ξ. Inverse uncerainy disribuion is an imporan ool in he operaion of uncerain variables. Theorem 1. (Liu [15]) Le ξ 1, ξ 2,, ξ n be independen uncerain variables wih regular uncerainy disribuions Φ 1, Φ 2,, Φ n, respecively. If f is a sricly increasing funcion, hen is an uncerain variable wih inverse uncerainy disribuion Ψ 1 (α) = f(φ 1 1 ξ = f(ξ 1, ξ 2,, ξ n ) (10) (α), Φ 1 2 (α),, Φ 1 n (α)) (11) Corollary 1. ( Liu [16]) Le ξ 1, ξ 2,, ξ n be iid uncerain variables. Then ξ 1 + ξ 2 + + ξ n and nξ are idenically disribued uncerain variables. In oher words, if Φ is he common uncerainy disribuion of ξ 1, ξ 2,, ξ n, hen 1 n ξ i has also he uncerainy disribuion Φ(bu no independen of ξ 1, ξ 2,, ξ n ). n An uncerain process (Liu [18]) is essenially a sequence of uncerain variables indexed by ime, and is usually used o model uncerain phenomena ha vary wih ime. And hen, independen incremen process and saionary independen incremen process are furher sudied by Liu [16]. As an imporan ype of uncerain process, a renewal process is an uncerain process in which evens occur coninuously and independenly of one anoher in uncerain imes. The concep of uncerain renewal process was firs given by Liu [18] in 2008. Definiion 6. (Liu [18]) Le ξ 1, ξ 2, be iid posiive uncerain variables. Define S 0 = 0 and S n = ξ 1 + ξ 2 + + ξ n for n 1. Then he uncerain process is called an uncerain renewal process. N = max n 0 {n S n } (12) If ξ 1, ξ 2, denoe he inerarrival imes of successive evens, hen S n can be regarded as he waiing ime unil he occurrence of he n-h even, ha is S n = ξ 1 + ξ 2 + + ξ n. The renewal process N can be regarded as he number of renewals in (0, ], hen he even {N n} is equal o he even {S n } for any ime and ineger n. Furhermore, he even {N n} is equal o {S n+1 > }. Noe ha N is no sample-coninuous, bu each sample pah of N is a righ-coninuous and increasing sep funcion aking only nonnegaive ineger values. Furhermore, he size of each jump of N is always 1. Tha is, N has a mos one renewal a each ime. In paricular, N does no jump a ime 0. Since N akes ineger values, for any x 0, he uncerainy disribuion Υ of N is Υ (x) = M{N x} = M{N x } = 1 M{S x +1 } (13) 5
Theorem 2. (Liu [15]) Le N be a renewal process wih uncerain inerarrival imes ξ 1, ξ 2,. If hose inerarrival imes have a common uncerainy disribuion Φ, hen N has an uncerainy disribuion ( ) Υ (x) = 1 Φ (14) x + 1 where x represens he maximal ineger less han or equal o x. Liu [15] proved ha N / converges in disribuion o 1/ξ 1 which is an uncerain variable. Based on his, he elemenary renewal heorem was proved by Liu [19] as he following. Theorem 3. (Liu [19], Elemenary Renewal Theorem) Le N be a renewal process wih uncerain inerarrival imes ξ 1, ξ 2,. If E[1/ξ 1 ] exiss, hen lim E [ N ] [ ] 1 = E. ξ 1 3 The Ordering Policy Based on Uncerain Renewal Process Consider a sysem conains some of idenical pars ha operae independenly of each oher and subjec o an idenical paern of malfuncion and replacemen. Suppose he daa of hese pars lifeimes are unavailable, herefore we deal wih he lifeimes as uncerain variables, and he ordering policy of spare pars wih uncerainy inerarrival ime is researched based on uncerain renewal process in his secion. If he ordering quaniy is needed o deermine a he beginning of a fixed cycle (0, ], hen he problem we encouner is: how many pars should be provided in order o assure wih confidence level α ha a aircraf will remain in operaion hours? To make our ideas concree, we firs discuss he ordering problem of a spare par. Suppose ha he ordered pars is arrived wihou delay, and each par is replaced immediaely a failure. Le N denoe he number of failures (or renewals if spares are available when needed) before ime, noe ha he number of spares needed is he number of failures. Given α we require he smalles ordering quaniy k such ha Then we can ge he following heorem. M{k N } α. (15) Theorem 4. Le N be a renewal process of a spare par wih inerarrival imes ξ 1, ξ 2,. Assume hose inerarrival imes are independen uncerain variables and have a common uncerainy disribuion Φ. If he uncerain measure ha he ordering quaniy k is larger or equal o he number of renewals N before ime is hold wih confidence level α, hen he opimal ordering quaniy is k = Φ 1 (1 α) 1. (16) where Φ 1 (1 α) 1 is he minimum ineger larger han or equal o ( Φ 1 (1 α) 1). 6
Proof: Noe ha N is he number of renewals in (0, ], and saisfies he inequaliy M{k N } α. (17) From Equaion (14), we immediaely have M{N k} = Υ (k) = 1 Φ( ) α. (18) k + 1 Since Φ is an increasing disribuion funcion, we can ge ( ) k Φ 1 (1 α) 1. (19) where k is an ineger, so k akes he value of Φ 1 (1 α) 1. The heorem is verified. Nex we consider he ordering problem of all his ype of spare pars for a sysem. Theorem 5. Assume ha a sysem conains m iid spare pars. Le N i ()(i = 1, 2,, m) be he renewal process of he i-h spare par wih inerarrival imes ξ ij (i = 1, 2,, m, j = 1, 2,, n i ), where n i denoes he renewal number of he i-h par in (0, ]. Le all of he ξ ij be he iid uncerain inerarrival imes wih a common uncerainy disribuion Φ, K be he ordering quaniy of he enire sysem. Given he confidence level α, hen he minimum K wih uncerainy chance consrain is where m( Proof: Φ 1 (1 α) M {N 1 () + N 2 () + + N m () K} α, (20) K = m( Φ 1 (1 α) 1), (21) 1) is he minimum ineger larger han or equal o m( Φ 1 (1 α) 1). We will find an opimal ordering quaniy K under he chance consrain { m } M N i () K α. (22) From Theorem 2, N i () is a renewal process wih an uncerainy disribuion Υ (K). Le N() = hen Therefore, Thus we can ge N() m Υ (K) N() Υ m N i (), ( ) K. (23) m ( ) ( ) K M{N() K} = Υ = 1 Φ m K m + 1 α. (24) ( ) K m Φ 1 (1 α) 1. (25) Since K akes only ineger values, he minimum value is K = m( Φ 1 (1 α) 1). (26) The heorem is proved. 7
(a) (b) Figure 1: (a) The Laser Gyro in ADIRU of Aircraf (b) The New Type of Aircraf 4 Applicaion In his secion we consider he ordering problem of Laser gyros for an airline s flee. Laser gyro (see Fig.1(a)) is an imporan elemen of he aircrafs, each conains six gyros ha are insalled in he air daa/inerial reference uni (ADIRU) in he former equipmen cabin of he aircraf(see Fig.1(b)). Laser gyros are used for inegraed navigaion and posiioning, real-imely oupuing aircrafs angular velociy, line acceleraion, velociy and so on, offering accurae daa for he mechanical conrol device of he followup anenna as well. Suppose ha a flee is composed of 40 aircrafs of one new and idenical ype (see Fig 1), and he number of gyros in his flee m is 240. Le he seled ordering cycle be 2 years and confidence level α is 0.99. The daa of he lifeime of gyros are unavailable because he aircraf is compleely new, hen he lifeimes are regarded as independen uncerain variables wih a common uncerainy disribuion. In his case, we invie some expers in he field of civil aviaion o evaluae he possible lifeime in order o obain he uncerainy disribuion. According o he expers belief by using he principle of leas squares given by Liu[19], we esimae he unknown parameers of he uncerainy disribuion. Assume ha an uncerainy disribuion o be deermined has a given funcional form Φ(x e, σ) wih unknown parameers e and σ. The principle of leas squares minimizes he sum of he squares of he disance of he exper s experimenal daa o he uncerainy disribuion. This minimizaion can be performed in eiher he verical or horizonal direcion. If he exper s experimenal daa (x 1, α 1 ), (x 2, α 2 ),, (x n, α n ) are obained and he verical direcion is acceped, hen we have min e,σ n (Φ(x i e, σ) α i ) 2. (27) 8
φ(parameer,x) 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 x Figure 2: The fiing curve of he lognormal uncerainy disribuion based on he leas squares The opimal soluion ê and ˆσ of Equaion (26) is called he leas squares esimae of e and σ, and hen he leas squares uncerainy disribuion is Φ(x ê, ˆσ). The expers experimenal daa we obained are calculaed as follows: (1.1, 0.02), (1.8, 0.07), (2.5, 0.20), (3.2, 0.40), (3.9, 0.72), (4.6, 0.89), (5.3, 1.0). Noe ha he measure uni of service ime is en housand hours. The lifeimes of gyros are assumed as iid uncerain lognormal disribuion wih parameers e and σ. Then he leas squares esimae of parameers are obained: ê = 1.2000, ˆσ = 0.3045 and he uncerainy disribuion is Φ(x) = Then he opimal ordering quaniy K of he flee is ( 1 1.2 ln x 1 + exp( )) (28) 0.30 K 1.752 = 240( Φ 1 (1 0.99) 1) = 31. (29) Tha is, minimizing he oal ordering cos and aking ino accoun he high uncerainy measure agains he shorage of spares, 31 gyros can be chosen o order and sock for he flee of he airline every 2 years. 5 Conclusions In his paper, by employing uncerain variables o describe he inerarrival imes, he periodical invenory model of spare pars for a flee was firs proposed. The analyic formula of he ordering quaniy based on uncerain renewal process was found. By a numerical case, we proved he possibiliy of apply he 9
uncerain renewal process model ino pracical mainenance. The fuure work is o find he opimal ordering policy o minimize he cos wih discoun of price and uncerain chance consrains. Acknowledgemens This work is suppored by he Fundamenal Research Funds for he Cenral Universiies under Gran No.ZXH2012K005. The research is also suppored by Uncerainy Theory Laboraory, Deparmen of Mahemaical Sciences, Tsinghua Universiy, Beijing 100084, China. References [1] Allan G., Oleq K., and Josef S., Equivalences in srong limi heorems for renewal couning process, Saisics and Probabiliy Leers, Vol.35, No.4, 381-394, 1997. [2] Alsmeyer G., Superposed and coninuous renewal processes a Markov renewal approach, Sochasic Processes and Their Applicaions, Vol.61, No.2, 311-322, 1996. [3] Badard R., The law of large numbers for fuzzy processes and he esimaion problem, Informaion Science, Vol.28,161-178, 1982. [4] Blackwell D., A renewal heory, Duke Mahemaical Journal, vol.15, 145-150, 1948. [5] Blackwell D., Exension of a renewal heory, PACIFIC Pacific Journal of Mahemaics, Vol.3, 315-320, 1953. [6] Cox D. R., Renewal Theory, London: Mehuen Press, 1962. [7] Feller W., On he inegral equaion of renewal heory, Annals of Mahemaical Saisics, Vol.12, 243-267,1941. [8] Ferreira J.A., Pairs of renewal processes whose superposiion is a renewal process, Sochasic Processes and Their Applicaions, Vol.86, No.2, 217-230, 2000. [9] Hong D.H., Renwal process wih T-relaed fuzzy iner-arrival imes and fuzzy rewards, Informaion Sciences, Vol.176, 2386-2395, 2006. [10] Kahneman D., and Tversky A., Prospec heory: An analysis of decisions under risk, Economerica, Vol.47, No.2, 263-291,1979. [11] Kwakernaak H., Fuzzy random variables-i. Definiions and heorems, Informaion Sciences Vol.15, 1-29, 1978. [12] Kwakernaak H., Fuzzy random variables-ii. Algorihm and examples, Informaion Sciences, Vol.17, 253-278, 1979. [13] Li S., Zhao R., and Tang W., Fuzzy random homogeneous Poisson process and compound Poisson process, Journal of Informaion and Compuer Science, vol.1, 207-224, 2006. [14] Liu B., Uncerainy Theory, 2nd ed., Springer-Verlag, Berlin, 2007. 10
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