=Paper=
{{Paper
|id=Vol-1839/MIT2016-p17
|storemode=property
|title= Implementation of Weibull’s model for determination of aircraft’s parts reliability and spare parts forecast
|pdfUrl=https://ceur-ws.org/Vol-1839/MIT2016-p17.pdf
|volume=Vol-1839
|authors=Natasa Kontrec,Milena Petrovic,Jelena Vujakovic,Hranislav Milosevic
}}
== Implementation of Weibull’s model for determination of aircraft’s parts reliability and spare parts forecast==
https://ceur-ws.org/Vol-1839/MIT2016-p17.pdf
Mathematical and Information Technologies, MIT-2016 — Information technologies
Implementation of Weibull’s Model for
Determination of Aircraft’s Parts Reliability
and Spare Parts Forecast
Nataša Kontrec, Milena Petrović, Jelena Vujaković, and Hranislav Milošević
University of Priština, Faculty of Natural Sciences and Mathematics,
Kosovska Mitrovica, Serbia
natasa.kontrec@pr.ac.rs
Abstract. Planning of aircraft’s maintenance activities, failure occur-
rences and necessary spare parts are essential for minimizing downtime,
costs and preventing accidents. The aim of this paper is to propose an
approach that supports decision making process in planning of aircraft’s
maintenance activities and required spare parts. Presented mathemati-
cal model is based on Weibull’s model and calculates aircraft’s reliability
characteristics by using data on previous failure times of an aircraft part.
Further, by capitalizing the random nature of failure time, the number
of spare parts and the costs of negative inventory level are determined.
Keywords: aircraft’s spare parts, reliability, forecast, Weibull’s model.
1 Introduction
Optimized maintenance can be used as a key factor in organization’s efficiency
and effectiveness. Maintenance in aviation industry requires replacing of parts to
assure aircraft availability. Aviation companies are often facing aircraft’s down-
time due to spare parts shortage because they simply follow manufacturers’ or
suppliers’ recommendation regarding the required number of spare parts to be
kept on inventory [1]. Furthermore, that leads to unexpected costs of urgent
orders or the passenger accommodation costs in case of flight cancellation, etc.
Adequate spare parts management in the aircraft maintenance system improves
the aircraft availability and reduces downtime. Spare parts forecasting and pro-
visioning is a complex process and there are numerous paper dealing with this
issue [2–6]. In aviation industry some methods described in papers [7–11] found
their application but due to stochastic nature of demand they often failed to
provide accurate results. In recent times, spare parts forecasting with respect to
techno-economical issues (reliability, maintainability, life cycle costs) have been
studied [12–14] but not that extensively in aviation industry. In [15] a method-
ology to forecast the needs for expendable or non-repairable aircraft parts has
been presented. That methodology was based on observing total unit time (Tut)
provided by manufacturer as stochastic process. In the case when parameter
(Tut) is not available, we herewith present a new approach for determination
187
Mathematical and Information Technologies, MIT-2016 — Information technologies
of spare parts requirements. Described approach relies on historical data of pre-
vious failure times of an aircraft part and their stochastic nature. In order to
determine the reliability characteristic of each aircraft part, the Weibull’s model
has been used. The Weibull’s probability density function (PDF) is given by:
𝛽 (︁ 𝑤 )︁𝛽−1 𝑤 𝛽
𝑓 (𝑤) = exp−( 𝜂 ) , 𝑓 (𝑤) ≥ 0, 𝑤 ≥ 0, 𝛽 > 0, 𝜂 > 0, (1)
𝜂 𝜂
where w denotes flight hours, 𝛽 denotes shape parameter or slope, 𝜂 denotes scale
parameter or characteristic life. Based on previous, the cumulative distributive
function (CDF) can be determined as given in eq. (2):
𝑤 𝛽
𝐹 (𝑤) = 1 − exp−( 𝜂 ) . (2)
Further, reliability function of Weibull’s model can be calculated as follows:
𝑤 𝛽
𝑅(𝑤) = exp−( 𝜂 ) . (3)
Also, there is a possibility to calculate the conditional reliability i.e. the reliability
for the additional period of w duration for the parts having already accumulated
W flight hours. It can be calculated as given in eq. (4):
𝑊 +𝑤 𝛽 (︀ )︀
𝑅(𝑊 + 𝑤) exp−( 𝜂 ) 𝛽
− ( 𝑊 𝜂+𝑤 ) −( 𝑊 )
𝛽
𝑅(𝑤|𝑊 ) = = 𝑊 𝛽
= exp 𝜂 . (4)
𝑅(𝑊 ) exp−( 𝜂 )
The mean time to failure (MTTF) of Weibull’s PDF can be determined as in
eq. (5):
(︁ 1 )︁
MTTF = 𝜂 · 𝛤 +1 , (5)
𝛽
where 𝛤 is Gamma function. Failure rate function is given in eq. (6):
𝑓 (𝑤) 𝛽 (︁ 𝑤 )︁𝛽−1
𝜆(𝑤) = = . (6)
𝑅(𝑤) 𝜂 𝜂
In order to calculate reliability characteristic of an aircraft part it is necessary to
estimate the parameters of Weibull’s model. There are several ways to achieve
that, but in the case when we have limited historical data on previous failures, it
is best to perform rank regression on Y [16]. Rank regression on Y is a method
based on the least squares principle, which minimizes the vertical distance be-
tween the data points and the straight line fitted to the data as presented in
Fig. 1. The idea is to bring our function to linear line. In order to achieve that
we are taking natural algorithm of the both sides of the eq. (2).
𝛽
ln[1 − 𝐹 (𝑤)] = ln[𝑒𝑥𝑝(−𝑤/𝜂) ]
ln [− ln[1 − 𝐹 (𝑤)]] = 𝛽 ln(𝑤/𝜂)
ln [− ln[1 − 𝐹 (𝑤)]] = 𝛽 ln 𝑤 − 𝛽 ln 𝜂.
188
Mathematical and Information Technologies, MIT-2016 — Information technologies
Then by setting:
𝑦 = ln[−𝑙𝑛(1 − 𝐹 (𝑤))], 𝑥 = ln 𝑤, 𝑎=𝛽 𝑎𝑛𝑑 𝑏 = −𝛽 ln 𝜂.
the previous equation can be rewritten as 𝑦 = 𝑎𝑥 + 𝑏. Now, assume that we have
sample of failure data set as (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ), ... , (𝑥𝑛 , 𝑦𝑛 ) plotted and 𝑥 values
are predictor variables. According to least square principle, the straight line that
ˆ + ˆ𝑏𝑥, such that:
best fit to these data is 𝑦 = 𝑎
𝑁
∑︁ 𝑁
∑︁
𝑎 + ˆ𝑏𝑥𝑖 − 𝑦𝑖 )2 = 𝑚𝑖𝑛
(ˆ 𝑎 + ˆ𝑏𝑥𝑖 − 𝑦𝑖 )2
(ˆ
𝑖=1 𝑖=1
where 𝑎 ˆ and ˆ𝑏 are the least squares estimates of 𝑎 and 𝑏 and 𝑁 is the number
of failure data. The equations can be minimized by estimates 𝑎 ˆ and ˆ𝑏 as in Eqs.
(7) and (8)
∑︀
𝑁 ∑︀
𝑁
∑︀
𝑁 𝑥𝑖 𝑦𝑖
𝑖=1 𝑖=1
𝑥𝑖 𝑦𝑖 − 𝑁
ˆ𝑏 = 𝑖=1 (︁ 𝑁 )︁2 . (7)
∑︀
∑︀
𝑁 𝑥𝑖
𝑥2𝑖 − 𝑖=1
𝑁
𝑖=1
and
∑︀
𝑁 ∑︀
𝑁
𝑦𝑖 𝑥𝑖
ˆ = 𝑖=1 − ˆ𝑏 𝑖=1 = 𝑦¯ − ˆ𝑏¯
𝑎 𝑥, (8)
𝑁 𝑁
The variable 𝑦¯ is the mean of all the observed values and 𝑥 ¯ is the mean of
all values of the predictor variable at which the observations were taken. Now,
according to the previous, we can easily obtain 𝑦𝑖 and 𝑥𝑖
𝑦𝑖 = ln[−𝑙𝑛(1 − 𝐹 (𝑤𝑖 ))], 𝑥𝑖 = ln(𝑤𝑖 ). (9)
The 𝐹 (𝑤𝑖 ) are values determined from the median ranks, and after we calculate
ˆ and ˆ𝑏, we can easily estimate parameters 𝜂 and 𝛽.
𝑎
2 Numerical analysis
According to the previous formulas we can further perform numerical analysis on
sample of 14 failure-time data for aircraft part number 302634-2 (Igniter plugs
for aircraft Cessna Citation 560XL - provided by Prince Aviation Company,
Serbia). Data are sorted by ascending order and presented in Table 1.
First, it was concluded by using Weibull’s probability plotting that data are
following Weibull’s distribution, as can be seen in Fig. 1. Since the table provide
the sample size less than 15 failed times, rank regression on 𝑌 method, presented
in previous section, has been used for parameter estimation. We applied this
method since it has been considered as more accurate [16]. It has been calculated
189
Mathematical and Information Technologies, MIT-2016 — Information technologies
Table 1. Failure time (flight hours for part no. 302634-2 Igniter Plug)
No. of Failure time
part (flight hours)
1 3258
2 4321
3 5183
4 5223
5 5786
6 5920
7 6004
8 6321
9 6550
10 6893
11 6906
12 7221
13 7305
14 7400
that shape parameter (𝛽) is 4.86 and characteristic life (𝜂) is 6,572.98. According
to the previous conclusions and eq. (3), we further determined reliability function
of the part Igniter plug. Reliability of the part Igniter plug is given in Fig. 2 and
the failure rate is presented in Fig. 3.
According to these figures we can conclude after how many flight hours this
part would most likely stop working.
Fig. 1. Weibull Probability Plot.
190
Mathematical and Information Technologies, MIT-2016 — Information technologies
Fig. 2. Reliability function of the part Igniter plug.
Fig. 3. Failure rate function of the part Igniter plug.
3 Method evaluation
The major contribution of this paper is to determine the number of spare parts
that should be kept on stock in interval [0, 𝑤]. In order to achieve that we are
using an approach presented in paper [15] where the number of part exposed
191
Mathematical and Information Technologies, MIT-2016 — Information technologies
to failure in certain time frame was calculated. These calculation are based
on Rayleigh’s model in the case when only total unit time (usually provided
by parts manufacturer) is available. Similar approach is applied in this paper
but in the case when data of previous failures are available so the reliability
characteristics of the aircraft parts are determined by using the Weibull’s model.
PDF of Weibull’s distributed failure time is given by eq. (1), while the PDF of
Rayleigh’s distribute failure time is:
𝜇 (︁ 𝜇2 )︁
𝑓 (𝜇) = exp − . (10)
𝜎2 2𝜎 2
In the eq. (10), 𝜇 presents Rayleigh’s random variable, while the PDF of Weibull’s
model has been given by eq. (1). According to the above stated equations it can
√ 𝛽
be concluded that 𝜎 = 𝜂/ 2 and 𝜇 = 𝑤 2 .
In order to create relation between these models we are using the following
transformation: (︁ 𝛽 𝛽 𝛽 )︁
˙ = 𝑝𝜇𝜇˙ 𝑤 2 , 𝑤˙ 𝑤 2 −1 |𝐽|,
𝑝𝑤𝑤˙ (𝑤, 𝑤) (11)
2
where |𝐽| presents Jacobian transformation of random variables given by the
following equation: ⃒ 𝑑𝜇 𝑑𝜇 ⃒
⃒ ⃒ 𝛽 2 𝛽−2
|𝐽| = ⃒⃒ 𝑑𝑤 𝑑𝑤˙ ⃒
𝑑𝜇˙ 𝑑𝜇˙ ⃒ = 𝑤 .
𝑑𝑤 𝑑𝑤˙ 4
So, the eq. (11) further transforms into:
𝛽 2 𝛽−2
𝑝𝑤𝑤˙ (𝑤, 𝑤)
˙ = 𝑤 𝑝𝜇𝜇˙ (𝜇, 𝜇).
˙
4
Based on the random nature of failure time of an aircraft part, we are observing
the expected number of variations of Rayleigh’s random variable 𝜇 within an
interval (𝜇, 𝜇 + 𝑑𝜇), for a given slope 𝜇˙ within a specified open neighborhood 𝑑𝜇.
Actually, 𝜇˙ is a gradient of Rayleigh’s random variable, while 𝑤˙ is gradient of
Weibull’s random variable. The number of parts that will be exposed to failure
can be determined as:
∫︁
+∞ ∫︁
+∞
(︁
𝜇 𝜇2 )︁ 1 (︁ 𝜇˙ 2 )︁
𝑛= 𝜇𝑝
˙ 𝜇𝜇˙ (𝜇, 𝜇)𝑑
˙ 𝜇˙ = 𝜇˙ 2 𝑒𝑥𝑝 − 2 √ 𝑒𝑥𝑝 − 2 𝑑𝜇
𝜎 2𝜎 2𝜋𝜎 2 2𝜎
0 0
∫︁
+∞
𝑛= 𝑤𝑝
˙ 𝑤𝑤˙ (𝑤, 𝑤)𝑑
˙ 𝑤.˙
0
According to the previous equations, the number of spare parts exposed to failure
in time w can be finally determined as:
√ 𝛽 (︁ 𝑤𝛽 )︁
4 2𝑤 2
𝑛= 𝑒𝑥𝑝 − 2 .
𝜂 𝜂
192
Mathematical and Information Technologies, MIT-2016 — Information technologies
After we calculated average number of parts that are exposed to failure in inter-
val [0, 𝑤], we can determine the number of parts that should be on inventory. We
are using the approach presented in paper [15] where we observed the expected
amount of time when random variable 𝑤 is below total unit time as quotient of
Rayleighs CDF and 𝑛. Since the characteristic life parameter of Weibull’s distri-
bution 𝜂 is the time at which 63.2% of the units will fail and it is approximately
equal to MTTF [17], in this case, we are assessing the amount of time when 𝑤
is below 𝜂 by dividing CDF function of Weibull’s distributed variable and the
average number of parts to fail in time interval [0, 𝑤] as:
𝐹 (𝑤)
𝑞= . (12)
𝑛
As presented in Fig. 4 for the part Igniter plug it can be concluded at what
time the spare part should be available. In the case that this part is not avail-
7
6
Number of spare parts (q)
5
4
3
2
1
0
0 1000 2000 3000 4000 5000 6000 7000 8000
Flight hours (w)
Fig. 4. Number of spare parts for part Igniter plug.
able when needed, the underage costs appear. The underage costs are difficult to
determine due to their nature. Also, in this paper we are using the well known
Newsvendor method [18] in order to calculate these costs. This method gives
good results when it is necessary to estimate a stochastic variable. The result of
this estimation is a compromise between losses when we decide to order more
spare parts than needed and losses when we order less than required. In both
cases we have costs, either unnecessary inventory costs or costs of urgent or-
ders. Newsvendor method should provide optimal quantity of spare parts. Since
we determined that number in eq. (12), we are using the following formula to
193
Mathematical and Information Technologies, MIT-2016 — Information technologies
calculate the underage costs:
(︁ 𝑐𝑢 )︁
𝑞 = 𝛷−1 ,
𝑐𝑢 + 𝑐𝑜
where 𝛷−1 presents inverse distribution function (complementary error func-
tion), 𝑐𝑢 are underage costs and 𝑐𝑜 are overage costs, which in our case is the
spare part price. Fig. 5 presented the underage costs for aircraft part Ignition
plug. The overage costs for this part are are $1.925,00 and it can be noticed that
the underage costs are growing exponentially in relation to time.
50000
40000
Underage costs (c )
u
30000
20000
10000
0
1000 2000 3000 4000 5000 6000 7000
Flight hours (w)
Fig. 5. Underage cost of the part Igniter plug.
4 Conclusion
This paper presents an approach to determine reliability parameters of each air-
craft part. This has been achieved by using the observed failure times for certain
aircraft part and Weibull’s model. Also, a new methodology for calculation of
parts that are exposed to failure in observed period of time is presented. This ap-
proach was based on random nature of failure or total unit time of each aircraft’s
part. According to the obtained number, we further calculated the quantity of
the aircraft spare parts that should be kept on stock in order to avoid necessary
costs. Also, the Newsvendor model was used in order to assess the potential
underage costs in certain time period. All these calculations aim to support the
decision making process in planning of aircraft maintenance activities and spare
parts needs. As presented in the paper, we evaluated the method for one specific
aircraft part and presented results. Same could be done for any other aircraft
194
Mathematical and Information Technologies, MIT-2016 — Information technologies
part. Also, these analysis could be applied to other industries with no massive
production of spare parts such as weapons industry.
References
1. Ghodrati B. Reliability and operating environment vase spare parts planning, Doc-
torial Thesis, Luea University of Technology, Division of Operation and Mainte-
nance Engineering, 2005
2. Bartman, D., and Beckmann, M. J. Inventory Control: Model and Methods.
Springer Verlag, (1992).
3. Langford, J. W.: Logistics: Principles and Applications. McGraw-Hill, Inc., N.Y.
(1995).
4. Orsburn, D. K.: Spares Management Handbook. McGraw-Hill Inc., (1991).
5. Petrovic, R., and Pavlovic, M. A : New Performance Measure for Multi-level
Spare Parts Inventories. Inventory in Theory and Practice, Elsevier, Editor At-
tilia Chikan, Elsevier, (1986).
6. Petrovic, R. Senborn, A., and Vujosevic, M.: Hierarchical Spare Parts Inventory
Systems. Elsevier, (1986).
7. Croston, J. D.: Forecasting and stock control for intermittent demands, Operational
Research Quarterly, vol. 23, no. 3, pp. 289–303, (1972).
8. Rao, A. V.: A comment on: forecasting and stock control for intermittent demands,
Operational Research Quarterly, vol. 24, no. 4, pp. 639–640, (1973).
9. Syntetos, A. A. and Boylan,J. E.: On the bias of intermittent demand estimates, In-
ternational Journal of Production Economics,vol. 71, no. 1-3, pp. 457–466, (2001).
10. Syntetos, A. A. and Boylan, J. E. :The accuracy of intermittent demand estimates,
International Journal of Forecasting, vol. 21, no. 2, pp. 303–314, (2005).
11. Syntetos, A. A. and Boylan, J. E.: On the stock control performance of intermittent
demand estimators, International Journal of Production Economics, vol. 103, no.
1, pp. 36–47, (2006).
12. Ghodrati, B., Akersten, P.A. and Kumar,U. : Spare parts estimation and risk
assessment conducted at choghart iron oremine: a case study, Journal of Quality
in Maintenance Engineering, vol. 13, no. 4, pp. 353–363, (2007).
13. Barabadi, A., Ghodrati, B., Barabady, J. and Markeset T.: Reliability and spare
parts estimation taking into consideration the operational environmenta case study,
Proceedings of the IEEE International Conference on Industrial Engineering and
Engineering Management (IEEM ’12), Hong Kong, (2012).
14. Ghodrati, B., Banjevic,D. and Jardine A. K. S.: Developing effective spare parts es-
timations results in improved system availability, in Proceedings of the Annual Re-
liability and Maintainability Symposium:The International Symposium on Product
Quality and Integrity (RAMS ’10), pp. 1–6, (2010).
15. Kontrec N., Milovanovic G.V., Panic, S. and Milosevic, H.: A Reliability-Based
Approach to Nonrepairable Spare Part Forecasting in Aircraft Maintenance Sys-
tem. Mathematical Problems in Engineering, 2015. Article ID731437, 7 pages,
doi:10.1155/2015/73143.(2015).
16. Nelson, W. B.: Applied Life Data Analysis, John Wiley & Sons (2004).
17. http://www.barringer1.com/pdf/Chpt1-5th-edition.pdf [Accessed on June 2016]
18. Hill, A. V.:The Newsvendor Problem, Clamshell Beach Press, 2011.
195