CC BY
7
PREDICTIVE MAINTENANCE SCHEME FOR PHASED
MISSION SYSTEMS
1Preeti Wanti srivastava and 2satya rani
•
Department of Operational Resear ch University of Delhi, Delhi-7, India 1 pr eetisriv asta va.saxena@gmail.com
Abstract
In both industrial and military fields, many systems are phase mission systems (PMSs) which execute mission composed of different phases in sequence. The structure, failure behaviour, and working condition of such a system may change from phase to phase. Maintenance actions comprising corrective and preventive maintenance schemes studied in the literature are aimed at retaining the maintained system in a proper condition and improving its availability and extending its life. The present paper deals with finding optimal periodic inspection time using multi-objective criteria comprising objectives of minimizing expected maintenance cost incurred due to predictive, breakdown and periodic maintenance of a PMS , and maximizing its expected residual lifetime. The predictive maintenance is condition-based preventive maintenance that anticipates system failures in order to plan timely interventions on the system and hence improve its performance. The dependency is modelled using Gumbel-Haugaard copula.
An aircraft flight PMS comprising Taxiing phase, Take-Off phase, Cruising phase and Landing phase has been used to illustrate the method developed.
Keywords: phased mission system, reliability , Gumbel-Haugaar d copula, predictiv e maintenance, periodic maintenance, mean residual life, cost optimization, cumulativ e exposur e model
1. Introduction
The reliability of a supply chain depends on the reliability of all the equipment involved including transportation vehicles, sophisticated machines and computer -based information systems in network of suppliers , manufactur ers and distributors whose sole aim is to provide goods and services in a timely manner . The reliability of such equipment in turn depends on their design, maintenance and subsequent repairs. Reliability engineering is therefore part and parcel of operations management.
In real life, systems such as coal transportation systems [1][2], air crafts [3], avionic parts of airborne weapon systems [4], machining line [5], and nuclear plants are required to execute missions sequentially . Such systems called phased mission systems (PMSs) are subject to multiple, consecutiv e, non-overlapping operation phases. Failur es of these systems during the mission may cause great economic losses to enter prises, serious security threats to personnel, or extensiv e damage to the environment. Some maintenance activities need to be undertaken during the mission break to reduce the probability of system failur e of a PMS in the succeeding mission.
Unlike a non-repairable PMS in a repairable PMS, the state of the system depends not only on failur e characteristics of its components but also on maintenance conducted during the mission.
Further , the system reliability depends on its age and the maintenance policy applied. It usually decreases as components deteriorate. Performing proper maintenance actions is necessar y to keep the reliability of a system at a desir ed level. Maintenance is classifie into two main categories: corrective maintenance (CM) and preventive maintenance (PM). Corrective maintenance
is generally performed after the system breakdown. Preventive maintenance corresponds to the scheduled actions which are perfor med while the system is still operational. It aims at keeping the system in available state by improving the condition of its components. Usually, preventive maintenance is more advantageous as it may prevent catastrophic losses due to unpr edicted failures [6][7][8][9][10][11][12][13]. The PM actions are usually performed at predeter mined points in time to keep the reliability of the system at a desir ed level.
Predictiv e maintenance (PdM) also known as condition-based maintenance is meant to minimize unscheduled equipment failures, lost production, and maintenance costs. It involves the use of information such as maintenance logs and sensor data to predict maintenance needs in advance. PdM plays a very important role in the airline industr y by helping in reducing delays and costs, while impr oving and maintaining air craft operational reliability .
The aim of this paper is to deter mine optimal periodic inspection time using multi-objectiv e criteria of minimizing the expected maintenance cost due to predictiv e, breakdo wn and periodic maintenances, of the PMS, and maximizing its mean residual lifetime. The decision variable is the length of the periodic interval, t, subject to the constraints that the reliability of each phase does not exceed the pre-specifie values.
The paper is organized as follows: Section 2 is a brief literatur e review. The model of Predictiv e maintenance cost is explained in Section 3. In Section 4 the phased mission system is explained. Traditional maintenance models involving periodic and breakdown maintenances, and integrated models involving predictiv e maintenance besides periodic and breakdo wn maintenances are discussed in Section 5. The concept of Remaining Useful Life (RUL) is highlighted in Section 6, and multi-objectiv e optimization problem is formulated in Section 7. In Section 8, the proposed method is explained using an aircraft fligh PMS. The concluding remarks have been made in the last section.
2. LITERATURE REVIEW
The maintenance models used in the literatur e predict problems that can help timely replacement or repair of an equipment before it fails for a single system. The resear chers have used knowledge about degradation state of the equipment for prediction purpose [14] out-of-contr ol condition using statistical process control [15][16][17][18][19][20] and on-line sensors [21][22] for prediction purpose for a single system. Maintenance at system-le vel of a PMS without considering predictiv e maintenance has been studied by [23]. The present paper deals with maintenance of a PMS taking into account predictiv e, periodic and breakdo wn maintenances along with its mean residual lifetime. It is assumed that the components are dependent within a phase, and all the phases involved are dependent. The dependency is modelled using Gumbel-Hougaar d copula.
3. Predictive Maintenance Mqdel
Defin fpMS (t) as the density function that specifie the probability of failur e of a PMS at time t and g(s | t) as the conditional density function that specifie the probability that the signal of a potential failure is received at time s given that the actual failure would have occurred at time t. The conditional density, g(s | t), define the capability (i.e., accuracy and precision) of the prediction system.
The choice of the distribution form for the prediction signal, conditional on the equipment failure, is based on the concept of "P-F curves" for prediction systems [24] as well as diagnosis of the sensor equipment by the concer ned technician(s).
Thus
g (s I t)
k (1 - p) sk—1 t-k
P
0 < s < t s > t
G (s | t)
(1 - m?r 1
0<s<t s>t
wher e s is the tim e of the signal, t is the time of failur e if no replacement is made, k is the prediction precision, (1 — ft) is the prediction accuracy and k > 1,0 < ft < 1, are respectiv ely, the conditional density and distribution function used for the purpose. This form of the conditional distribution function characterizes the featur es of typical signal and failur e times seen in industr y [22]).
In this paper , the objectiv e is to minimize the expected maintenance cost of a PMS per unit time. The maintenance costs include costs of periodic and predictiv e replacements and that of failur es. It is assumed that the PMS will go for maintenance after completing the mission and restored to "as good as new" condition, therefore, using renewal reward process the expected maintenance cost per period is:
E[Predictive Maintenance cost + Breakdown cost + Periodic maintenance cost]
E[Time until maintenance]
(See for reference [25]).
4. Phased Mission System (PMS)
A phased mission system (PMS) is define as a system comprising multiple, consecutiv e, and non-overlapping phases. During each phase, a PMS needs to complete a specifie task without failur e. In these phases, the system maybe subject to different working conditions and environmental stresses, as well as different performance requirements. For example, in a twin-engine airplane with two phases, namely, taxiing phase and take-of f phase, one engine is requir ed in the former phase, and both the engines are necessar y in the latter phase. In contrast to the other phases of the fligh profile the engines are more prone to failur e during the take-off period due to enormous pressure they undergo during this period [26][27]. So in different phases, the system configurations and the components, failure rates and even failure criteria could be vastly different.
Let Tmn denote lifetime of component m of phase n with reliability Hmn (t). Let Fm1 (t), Fm2 (t), ... and Fmn (t) be the reliability of phase 1, phase 2, ... and phase n, respectiv ely. Then, reliability of PMS is:
~PMS
(t)
Fm1 (t) ,0 < t < T1 Fm2 (t) , T1 < t < T2
(1)
Fmn (t) , Tn—1 < t < Tn,
where (Tn—1, Tn) represents time-duration of functioning of phase n of the phased mission system n = 1,2,3,4,...,n, T0 = 0.
Since considering phase n has m dependent components and reliability of phase n dnoted by Fmn (t) so dependency is modelled using Gumbel-Haugaar d copula [28] gives,
Fmn (t) = C (H1n (t) , H2n (t) ,..., Hmn (t)) . (2)
And , reliability of PMS is:
FpMS (t) = C (Fm1 (T1) , Fm2 (T2) , Fm3 (T3 ) ,. . ., Fmn (Tn )) . (3)
The cumulativ e exposur e model [29] is used in equation (2), to obtain the reliability of phase n at Tn. We obtain,
Fmn (Tn) = C (H1n ( Tn — Tn 1 + l 1n ) , H2n (Tn — Tn—1 + l2n ) ,. . ., Hmn ( Tn — Tn 1 + l mn )) (4)
lmn , wher e m denotes the components and n denotes the phase of the system, m = 1,.., m, & n = 1,.., n, is deter mined in such a way that [30])
Hmn (lmn( — Hmn-1 (Tn-1 Tn-2 + lmn-1 ) , and l1n-1 — 0,
wher e C (H1n (t), H2n (t), H3n (t),..., Hmn (t)) is the m-dimensional Gumbel-Hougaar d. Thus,
C(Hm (t), H2n (t), H3n (t),..., Hmn (t))
exp
- ((-log ( H1n (t)))9 + (-log (H2n (t)))9 + (-log ( H3n (t)))9 + ■■■ + (-log ( Hmn (t)))']
e V 9'
5. Maintenance Model
The present section focuses on the traditional periodic maintenance model (TM) and integrated model.(IM)
5.1. Traditional Model
In TM no predictiv e maintenance is used, periodic maintenance is conducted if there has been no failure prior to time T, and breakdown maintenance is conducted if the equipment fails prior to time T.
For the TM, the decision variable is the periodic interval T and the objective function value is as follows:
Ctm (T)
E[ CBP] E[ Ct1 ],
(5)
wher e
E[CBp ] — Mb
fPMS (t) dt
+ M
fPMS (t)dt
is sum of expectation of breakdo wn maintenance costs and periodic maintenance cost, and
E [ Ct1 ]
tfPMS (t)dt
+ T
fPMS (t)dt
is mean time between failure (replacement).
5.2. Integrated Model (IM)
The second model utilizes both predictive and periodic maintenance and is referred to as the Integrated Model. For IM, the decision variable is the periodic interval, T, and the objective function is:
Cim (T)
E[ CPdBP]
E[ CT2 ] ,
(6)
wher e,
E[CPdBP] — Mpd
r T r œ
(1 - ß) fPMS(t)dt +/ G(T | t)fpMS(t)dt J0 JT
+ Mb
ß IT fPMS (t)dt
J0
+ Mp
[1 - G(T | t)] fPMS(t)dt
œ
T
œ
œ
T
is sum of expectation of predictive maintenance cost, breakdown maintenance costs and periodic maintenance cost, Mpd is predictive maintenance cost, Mb is breakdown maintenance and Mp is periodic maintenance, and
E [ Ct2 ]
fT f t r œ <■ T
sg (s | t) /PMS (t)dsdt + / / sg (s | t) /PMS(t)dsdt
10 j0
+
T0
M t/PMS (t)dt
0
+
¡> œ
TJT [1 - G (T | t)] /pms(t)dt
is sum of expected time betw een replacement with signal and without signal.
6. Remaining Useful Life (RUL)
RUL is the residual life time of a system used to perfor m its functional capabilities befor e failur e. It is a key metric and critical for predicting the failure of a machine in the production line, and is used by engineers to decide whether to do maintenance or delay it due to production requirements [31].
Let Tpms be the time to failur e of the phased mission system, and suppose the phased mission system has survived until time t. Then the "conditional" random variable
XPMS = TPMS - t(TPMS > ^
i.e., the remaining time to failur e, is called "RUL " of the phased mission system. The conditional reliability function
Fpms (t) = Ppms (Xpms > x) = P (Tpms - t > x\Tpms - t), x > 0,
incorporates all the information relevant for prediction and future planning. The mean residual life (MRL) used as a point estimate of RUL or a prediction interval for RUL is define as:
Fpms (t) = epms[Xpms] = E [[Tpms — tlTPMS > t]. Then, ^pms (0) = Vpms = E[T] and
œ
Fpms (t)= I Fpms (x) dx = irFPMS (X dX. (7)
Jo Fpms (t)
7. Optimization Problems
Amongst various approaches used to solve a multi-objectiv e optimization problem, one of the commonly used approach is to combine the objectives involved into one single composite objective so that the traditional mathematical programming method can be used for the propose.
In this paper , the weighted sum multi-objectiv e optimization problem is used to minimize the expected maintenance cost per unit time and maximize mean residual lifetime function for the PMS subject to the constraints that the reliability of the each phase does not exceed the pre-specifie values, Rj, i = 1,2,..., n.
Let T1 be the periodic inspection time for the traditional model and T2 be that for the Integrated Model.
The optimization problem is formulated as:
subject to, T1 > Tn,
7.1. Optimizing CTM
Min Z1 = w1CTM (T1) + w2(-^PMs (T1))
1 > Fmi (T1) > Ri,i = 1,2, ..., nCTM > 0.
7.2. Optimizing C1M
Min Z2 = w1CiM (T2) + w2(-^pms (T2))
subject to, T2 > Tn,
1 > Fmi (T2) > Ri, i = 1,2, ..., nCiM > 0. Mathematica 11.0 has been used for solve the optimization problem.
8. N umerical Illustrations
In this section aircraft fligh PMS used for the illustrativ e purpose, Figure 1(a)-(d), shows reliability block diagrams for the four-phase aircraft flight comprises The firs phase is taxiing in which the navigation system, one out of the four engines and all three landing gears are needed, the second phase is take-of f wher e in all four engines, the navigation system and all three landing gears are needed, the third phase is cruising in which the navigation system and three of the four engines are required. Finally, the fourth phase islanding comprising the navigation system, two of the four engines and all three landing gears.
8.1. Reliability of Aircraft Flight PMS system
Let T1denote lifetimes of navigation with reliability H1n (t). T2, T3, T4 and T5 denote lifetimes of the four engines E1, E2, E3 and E4 with reliabilities H2n (t), H3n (t), H4n (t) and H5n (t), respectiv ely, and T^, T7 and T8 denote lifetimes of landing gear 1 (G1), landing gear 2 (G2) and landing gear 3 (G3) with reliabilities H(,n (t), H7n (t)and H8n (t), respectiv ely. Let Fp1 (t), Fp2 (t), Fp3 (t), and Fp4 (t) be the reliability of subsystems in phase 1, phase 2, phase 3 and phase 4, respectiv ely. Then, reliability of 4-PMS is:
~PMS
(t)
Fp1 (t) ,0 < t < T1 Fp2 (t), T1 < t < T2 Fp3 (t), T2 < t < T3 Fp4 (t), T3 < t < T4.
(8)
PHASE-1(Taxiing Phase)
Let H11 (t) be life distribution of navigation, H21 (t), H31 (t), H41 (t) & H51 (t) be life distribution of components E1, ¿2, E3 & E 4, respectiv ely further let Hg1 (t), H71 (t) & H81 (t) be life distribution of components 'G', 'G2 & 'G3, respectiv ely. Reliability of navigation,
Fn (t) = p [T1 > t].
Reliability of engines,
F21 (t) =
p [T2 > t, T3 < t, T4 < t, T5 < t]+ p [T2 < t, T3 > t, T4 < t, T5 < t]+ p [T2 < t, T3 < t, T4 > t, T5 < t
p [T2 < t, T3 < t, T4 < t, T5 > t}+ p [T2 > t, T3 > t, T4 < t, T5 < t}+ p [T2 > t, T3 < t, T4 > t, T5 < t
p [T2 > t, T3 < t, T4 < t, T5 > t]+ p [T2 < t, T3 > t, T4 > t, T5 < t]+ p [t2 < t, T3 > t, T4 < t, T5 > t
p [T2 < t, T3 < t, T4 > t, T5 > t]+ p [T2 > t, T3 > t, T4 > t, T5 < t]+ p [t2 > t, T3 > t, T4 < t, T5 > t
p [T2 > t, T3 < t, T4 > t, T5 > t]+ p [T2 < t, T3 > t, T4 > t, T5 > t]+ p [T2 > t, T3 > t, T4 > t, T5 > t
Preeti Wanti Srivastava and Satya Rani RT&A, No 1 (77)
PREDICTIVE MAINTENANCE SCHEME FOR PMSS Volume 19, March 2024
Figure 1(d): Landing Phase Figure 1: 1(a)-(d) Reliability Block Diagrams for the four- phase aircraft flight[32]
Preeti Wanti Srivastava and Satya Rani CT&A^ No 1 (77)
PREDICTIVE MAINTENANCE SCHEME FOR PMSS Volume 19, March 2024
Reliability of landing gear ,
F31 (t) = p [T6 > t, T7 > t, T > t].
Thus, Reliability of phase-1,
Fpi (t) = F11 (t) .F21 (t) .F31 (t). (9)
PHASE-2 (Take-Off Phase)
Let H12 (t) be life distribution of navigation, H22 (t), H32 (t), H42 (t) & H52 (t) be life distribution of components E1, ¿2, E 3 & E 4, respectiv ely further let Hg2 (t), H72 (t) & H82 (t) be life distribution of components 'G', 'G2 & 'G3, respectiv ely. Reliability of navigation,
F12 (t) = p [T1 > t].
Reliability of engines,
F22 (t) = p [T2 > t, T3 > t, T4 > t, T5 > t].
Reliability of landing gear ,
F32 (t) = p [T6 > t, T7 > t, T8 > t].
Thus, Reliability of phase-2,
Fp2 (t)= F12 (t) .F22 (t) .F32 (t). (10)
PHASE-3 (Cruising Phase)
Let H13 (t) be life distribution of navigation, H23 (t), H33 (t), H43 (t) & H53 (t) be life distribution of components E1, ¿2, E 3 & E 4, respectiv ely further let Hg3 (t), H32 (t) & H83 (t) be life distribution of components 'G', 'G2 & 'G3, respectiv ely. Reliability of navigation,
F13 (t) = p [T1 > t].
Reliability of engines,
F23 (t) = p [T2 > t, T3 > t, T4 > t, T5 < t]+ p [T2 > t, T3 > t, T4 < t, T5 > t] + p [T2 > t, T3 < t, T4 > t, T5 > t]+ p [T2 < t, T3 > t, T4 > t, T5 > t]+ p [T2 > t, T3 > t, T4 > t, T5 > t].
Thus, Reliability of phase-3,
Fp3 (t)= F13 (t) .F23 (t). (11)
PHASE-4 (Landing Phase)
Let H14(t) be life distribution of navigation, H24(t), H34(t), H44(t) & H54(t) be life distribution of components E1, E2, E 3 & E 4, respectiv ely further let Hg4 (t), H74 (t) & H84 (t) be life distribution of components 'G', 'G2 & 'G3, respectiv ely. Reliability of navigation,
F14 (t) = p [T1 > t].
Reliability of engines,
F24 (t) = p [T2 > t, T3 > t, T4 < t, T5 < t]+ p [T2 > t, T3 < t, T4 > t, T5 < t] + p [T2 > t, T3 < t, T4 < t, T5 > t]+ p [T2 < t, T3 > t, T4 > t, T5 < t]+ p [T2 < t, T3 > t, T4 < p [T2 < t, T3 < t, T4 > t, T5 > t]+ p [T2 > t, T3 > t, T4 > t, T5 < t]+ p [T2 > t, T3 > t, T4 < t p [T2 > t, T3 < t, T4 > t, T5 > t]+ p [T2 < t, T3 > t, T4 > t, T5 > t]+ p [T2 > t, T3 > t, T4 > t
t, T5 > t] + , T5 > t] + , T5 > t].
Reliability of landing gear,
F34 (t) = p [T6 > t, T7 > t, T8 > t].
Thus, Reliability of phase-2,
Fp4 (t) = F14 (t) .F24 (t) .F34 (t). (12)
Reliability of Aircraft fligh PMS system using equation (3), we have
F (t) = C (Fpi (t), Fp2 (t), Fp3 (t), Fp4 (t)) (13)
equations(9), (10), (11) and (12) give reliability of the four phases in PMS.
After using p [AB] + p [ABc] = p[A] and Gumbel-Hougaar d copula equation (2) in above equations we get,
F11 (t) = Hn (t),
F21_(t) = CJH21 (t), 1,1,1) + C (1,H31W ,1,1)+ C (M,H41 (t), 1)_+ C (), 1,1,H51 (t)) _
- C (H2±(t), H31 (t) ,1,1) - C (H2±(t), 1, H41 (t) ,1) - C (H21 (t), M, H51 (t)) - C (), 1, H41 (t), H51 (t))
- C (1, H31 (ty, 1, H51 (ty - C ) 1, H31_(f), H41 (t), 0 + CJH21 )t), H(1 (t), H41 (t)_, 1) _
+ C (H21 (t), H31 (t), 1.H51 (t))_+ C (H21 (t), 1, H41 (t), H51 (t)) + C (1, H31 (t), H41 (t), H51 (t))
- C (H21 (t), H31 (t), H41 (t), H51 (t)),
F31 (t) = C (H61 (t), H71 (t), H81 (t)), F12 (t) = H12 (t),
F22 (t) = C (H22 (t), H32 (t), H42 (t), H52 (t)), F32 (t) = C (H62 (t) , H72 (t) , H82 (t)) , F13 (t) = H13 (t),
F23 (0 = C (H23_(f) , H33^(t) , H43 (t),l)_+ C (% (t), H33 (t) ,1, H53 (())___
+ C(H 23 (t) , 1, H43 (t) , H53 (t)) + C (X H33 (t) , H43 (t) , H53 (t)) — 3C (H23 (t) , H33 (t) , H43 (t) , H53 (t)) ,
F14 (t) = H14 (t) ,
F24 (t) = c (H24 (t), H34 (t), 1,1) + C (H24 (t), 1, H44 (t), 1)_+ C (% (t), 1,1, H54 (t)) + C (M, H44 (t), H54 (t)} + C (1, H34 (t)^1, H54 (0) + C (1.H34 (t), H44 (t)_, 1) _ _
- 2C (H2±(t), H34(t), H4A(t) ,1) - 2C (H24 (t), H34 (t), 1,H54 (t))_- 2C (H24 (t), 1, H44 (t), H54 (t))
- 2C (1, H34 (t), H44 (t), H54 (t)) + 3C (H24 (t), H34 (t), H44 (t), H54 (t)),
F34 (t) = p [T6 > t, T7 > t, T8 > t] = C (H 64 (t) , H74 (t) , H84 (t)) .
The cumulativ e exposur e model is used in above equations, to obtain the reliability of subsystems in phase 1, phase 2, phase 3 and phase 4 at T1, T2, T3 and t4, respectiv ely. Thus,
F11 (T1 ) = H11 (T1) ,
F21 (T1) = C (H21 (T1), 1,1,1) + C (1, H31 (T1) ,1,1) + C (1,1, (T1), 1) + C(1,1,1, ~H 51 (T1))
- C (H21 (t), H31 (T1) ,1,1) - C (H21 (T1), 1, H41 (T1) ,1) - C (H21 (T1), M, H51 (T1))
- C (1, 1, H41 (T1) , H51 (T1)) - C () H31 (T1 ) ,1, H5i_(T1)) - C (1, H31 (T1) , H41_(T1) ,1) _ _
+ C (H2±(T1) , H31 (T1 ) , H4±(T1) ,1) + C (H21 (T1 ), H31 (T1) ,1,H51 (T1 )]_+ C (fi21 (T1) ,1, H41 (T1 ) , H51 (T1)) + C (1, H31 (T1), H41 (T1), H51 (T1)) - C (H21 (T1), H31 (T1), H41 (T1), H51 (T1)),
F31 (T1) = C (H61 (T1), H71 (T1), H81 (T1)),
F12 (T2 ) = H12 (T2 - T1 + ¡12 ) ,
F22 (T2 ) = C (H22 (T2 - T1 + ¡22 ) , H32 (T2 - T1 + ¡32 ) , H42 (T2 - T + ¡42 ) , H52 (T2 - T1 + ¡52 )) ,
F32 (T2 ) = C (H62 (T2 - T1 + ¡62 ) , H72 (T2 - T1 + ¡72 ) , Hg2 T - T1 + ¡82 )) ,
F13 (T3 ) = H13 (T3 - T2 + ¡13 ) ,
F23 (T3) = C (H23 (T3 - ¡23 ) , H33 (T3 - T2 + ¡33 ) , H43 (T3 - T2+ ¡43 ) , 1) + C (H23 (T3 - T2 + ¡23 ) , H33(T3 - T2 + ¡33 ) , 1, H_53 (T3 - T2 + ¡53 )) + C (H23(T3 - T2 + ¡23 ) , 1, H43 (T3 - T2 + ¡43 ) , H53 (T3 - T2 + ¡53 )) + C (1, H33 (T3 - T2 + ¡33 ) ,_H43 (T3 - T2 + ¡43 ) ,H53 (T3 - T2 + ¡53
- 3C (H23 (T3 - T2 + ¡23 ) , H33 (T3 - T2 + ¡33 ) , H43 (T3 - T2 + ¡43 ) , H53 (T3 - T2 + ¡53 )) ,
F14 (T4) = H14 (T4 - T3 + ¡14) ,
F24 (T4) = C (H24 (T4 - T3 + ¡24 ) , H34 (T4 - T3 + ¡34 ) , 1, 1) + C (H24 (T4 - T3 + ¡24 ) , 1, H44 (T4 - T3 + ¡44 ) , 1) + C (H24 (T4 - T3 + ¡24 ) , 1, 1, H54 (T4 - T3 + ¡54 )) + C (1, 1, H44 (T4 - T3 + ¡44 ) , H54 (T4 - T3 + ¡14 ))
+ C (1, H34 (t) , 1, H54 (t)) + C (1, H34 (t) , H44 (t) ,1) - 2C (H24 (t) , H34 (t) , H44 (t) ,1)
- 2C (H24 (t), H34 (t), 1, H54 (t)) - 2C (H24 (t), 1, H44 (t), H54 (t)) - 2C (1, H34 (t), H44 (t), H54 (t))
+ 3C (H24 (t), H34 (t), H44 (t), H54 (t))
F34 (T4) = p [T6 > t, T7 > t, T8 > t] = C (H64 (t) , H74 (t) , H84 (t)) .
It is assumed that a component's life distribution in a phase is Weibull with reliability function: Hmn (t) = exp [-(t/ amn)Y] , t > 0; amn > 0; Y > 0; n = 1,2,3,4, m = 1,2,3,4,5,6,7,8.
To illustrate the above model, assume that each of the phase- Taxiing and Take-Off has duration of 15 minutes, cruising phase has duration of 130 minutes and landing phase has duration of 20 minutes. Components of the aircraft follow weibull distribution with 7 = 1.8 with amn = 1000 hours for navigation system, amn = 950 hours for engines and amn = 925 hours for the landing gear. The value of Mp=10000, Mpd=Mp, Mb = 5.500 * Mp, ft = 0.260, k = 2.00 [22].
Tables 1.1- 1.4 are obtained using these data for both the optimization problems formulated in Section 7, with R = 0.995, i = 1,2,..., 4
Table 1.1: Values of Multi-objective functions and T (in minutes) for 9 = 1.0 with different weights
w1 w2 Ctm (T1) RUL1 Cim (T2) RUL2 T1 T2
1/2 1/2 22309.9 123841 7091.89 123841 4033.45 4033.45
1/3 2/3 22309.9 123841 7729.81 124301 4033.45 3894.98
1/4 3/4 22309.9 123841 8005.13 124416 4033.45 3839.89
2/3 1/3 22309.9 123841 7091.89 123841 4033.45 4033.45
3/4 1/4 22309.9 123841 7091.89 123841 4033.45 4033.45
Table 1.2: Values of Multi-objective functions and T (in minutes) for 9 = 1.182 with different weights
w1 w2 Ctm (T1) RUL1 Cim (T2) RUL2 T1 T2
1/2 1/2 12647.6 121420 4352.71 123822 4832.62 3917
1/3 2/3 15018.8 123150 5135.21 124046 4085.97 3817.32
1/4 3/4 16155.5 123620 5250.38 124094 3977.96 3781.49
2/3 1/3 12082.4 120810 4360.13 123125 4426.73 4091.04
3/4 1/4 12082.4 120810 3998.01 122226 4426.73 4242.69
Table 1.3: Values of Multi-objective functions and T (in minutes) for 9 = 2.182 with different weights
w1 w2 Ctm (T1) RUL1 Cim (T2) RUL2 T1 T2
1/2 1/2 4207.26 121771 2847.66 123458 4207.26 3818.37
1/3 2/3 11723.3 122956 3755.75 123547 3986.95 3755.75
1/4 3/4 12465.1 123263 3012.93 123566 3899.44 3733.48
2/3 1/3 8159.79 118946 2652.79 123170 4546.32 3929
3/4 1/4 6953.79 115971 2499.36 122789 4818.2 4025.64
Table 1.4: Values of Multi-objective functions and T (in minutes) for 9 = 3.182 with different weights
w1 w2 Ctm (T1) RUL1 Cim (T2) RUL2 T1 T2
1/2 1/2 9877.32 121727 2649.01 123346 3814.62 4192.79
1/3 2/3 11438.3 122870 2769.67 123436 3976.54 3751.94
1/4 3/4 12150.9 123165 2814.46 123165 3890.79 3729.71
2/3 1/3 7992.92 118990 2453.59 123057 4526.04 3925.37
3/4 1/4 6821.02 116098 2299.76 122675 4793.23 4022.06
Table 1.1- Table 1.4 gives optimal cost and optimal residual useful life for the two models using different weight combinations. It is observed that the integrated model in almost all the cases yields lower cost and higher RUL with smaller periodic inspection time. Table 1.1 shows that for IM the minimum cost is obtained when w1 = 1/4 and w2 = 3/4 and the optimal periodic insepection time is T2 = 3839.89 implying that four-phase aircraft fligh needs to be send for maintenance after every 21 cycles. Similar interpretation holds for data depicted in Table- 1.2 to Table 1.4
9. Conclusion
In this paper predictiv e maintenance frame work is proposed for a phased missio n system. The multi-objectiv e problem is used wher ein weighted sum of expected maintenance cost and mean residual life function of the PMS is minimized subject to the constraints that the reliability of each phase doesn't exceed the pre-specifie values. The decision variable is the length of the periodic inter val. The optimal solution obtained using IM model is compar ed with traditional model (TM). For illustrativ e purpose aircraft fligh PMS composed of four phases, namely; taxiing, take-of f, cruising, and landing is used with dependency betw een components of each phase modelled using Gumbel-Haugaar d copula. The cumulativ e exposur e model is used to deter mine the reliability of the PMS. It is found that the integrated model yields lower cost and higher RUL with smaller periodic inspection time. Thus, the use of predictive tools with
periodic maintenance reduces overall equipment maintenance costs with higher mean residual life.
ACKNOWLEDGEMENT
This resear ch work is financiall supported by University of Delhi, Delhi-7, INDIA.
DECLARATION OF CONFLICT INTEREST
The authors have declar ed that no conflic inter ests exist.
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