stock level

ASSESSMENT OF POISSON MODEL FOR AIRLINERS’ REPAIRAB LE SPARE PARTS DEMAND FORECASTING

Carlos Federico Gayer

Facultad de Ciencias de la Administración Universidad del Salvador, Argentina

Received: August 30, 2010; Accepted: October 15, 2010

ABSTRACT

Flight delays and cancellations have extreme impact on airlines competitiveness.

Historically, operators have adopted different strategies avoiding critical spare-parts shortages,

reducing AOG situations. Then, in order to evaluate OEM’s method based on Palm’s theorem

for recommending a repairable component operative inventory for an airliners fleet, and its

efficacy estimating parts demand during a certain period, the number of removals for the

cockpit’s audio control panels from a group of A320 family aircrafts was analyzed. As a result

of the demand variance, the method shown not to be appropriate for all type of components and

repair processes.

Keywords: Spare-Parts; Inventory Theory; Palm’s Theorem.

JOURNAL OF THE BRAZILIAN AIR TRANSPORTATION RESEARCH SOCIETY • VOLUME 6 • ISSUE 1 • 2010 26

1. INTRODUCTION

Increasing airlines need to control their operative costs in order to be more competitive, attractive for the investors and especially be able to survive for long time in a complex political, economic and financial context (Doganis, 2006), with changing game rules and sensitive to the globalization effects, forced the operators to work on the reduction of the departure times, the reliability of their flight equipment, and the cash flow optimization. This is traduced especially, due to the high cost of the components, in optimal levels of spare parts inventories that assure the smaller proportion of immobilized capital, without jeopardizing operational efficiency (Chong et al., 2008).

An industry widely spread practice (Kaul et al., 2008) for determining repairable components recommended number of units for replacement (stock), satisfying a required service level, is based on a statistical process with Poisson-type distribution, in which the mean time between unscheduled removals (MTBUR) and the time that passes since the component is sent to repair and it is replaced by a repaired unit, or turn-around time (TAT), are computed.

In order to verify its efficacy, the historical evolution during seven years for the removals of one repairable item was analyzed (demand) within a defined fleet, and its

number per trimester was compared with the expected removals resulting from the method application for every corresponding period. The number of aircrafts in the fleet (FS) and the average flight hours (FH) per period were independent variables.

1.1. Audio Control Panel

The component whose removals have been analyzed was an ATA (Air Transport Association) chapter 23 item named Audio Control Panel -with part number ACP2788AB04- installed in Airbus A320 family airliners. Each aircraft has three units in the cockpit: two in the pedestal, for the captain and first officer (Figure 1), and another one available for the third occupant.

These units are used to control the transmission and reception of audio signals from different intercommunication, radio communication and radio navigation devices installed in the aircraft, through a collector ARINC 429.

They are financially classified as rotable modules (repairables). Each unit price is USD 6.500 and its MEL dispatch inoperative equipment list essentiality is ESS2 (Go-If). It is important to notice that among the studied population, the main cause of removal was the accidental breakage -or mishandling- of its switches, and the fault of its consumable parts.

Figure 1 - Audio Control Panel ACP2788AB04

ASSESSMENT OF POISSON MODEL FOR AIRLINERS’ REPAIRABLE SPARE PARTS DEMAND FORECASTING 27

1.2. Research Population

Data corresponds to a group of Airbus A320 family aircrafts (models A318, A319 and A320) that were part of the same Latin-American airline at the time of the last statistical record. Nevertheless, history is associated to each one of the aircrafts,

although they were in service with different operators for a certain period.

In Table 1, the serial numbers (MSN) for each aircraft model of the fleet were detailed. The average age at the last recorded period was 3,5 years. At the same time, 10,0 and 0,2 years have passed since the first flight of the oldest and youngest aircraft, respectively.

Table 1 - Fleet Composition.

Model MSN

A320-232 990 A320-233 1304, 1332, 1351, 1355, 1491, 1512,

1526, 1548, 1568, 1626, 1854, 1858, 1877, 1903, 3280, 3319, 3535, 3556

A319-131 2096 A319-132 2089, 2295, 2304, 2321, 2572, 2585,

2845, 2858, 2864, 2872, 2886, 2887, 2892, 2894, 3663, 3671, 3770, 3772, 3779

A318-121 3001, 3030, 3062, 3214, 3216, 3371, 3390, 3438, 3469, 3509, 3585, 3602, 2606, 3635, 3642

2. DEFINITION OF THE PROBLEM

For airlines, on-time flight departure is crucial. As noted by Cohen et al. (2006), technical delays are associated with the riskiest situations. To reduce them, airlines usually allocate a specific number of critical components at some strategic airports. These components are selected based on the recommendations made by the Original Equipment Manufacturer (OEM) through a document known as Minimum Equipment List (MEL).

According to MEL, the dispatch inoperative equipment list is created and components are classified into three categories depending on their quantity per aircraft (QPA) and their essentiality for the operation: ESS1, ESS2 and ESS3. If a failure

is detected for a component ESS1, the affected aircraft is not authorized for take-off (No-Go). For ESS2 component failure, the aircraft could take-off under certain conditions (Go-If), and its repair can be deferred for a period of time indicated by MEL (from 1 to 10 days). However, an accumulation of ESS2 failures may lead to ESS1 situation. When a component ESS3 faults, its repair or replacement can be deferred for a longer period (120 days). A No-Go situation leads to an AOG (aircraft on ground) condition where the airplane is inoperative until the failure is repaired.

The terms "failure" and "demand" are used indistinctly in this report. It is assumed when demand exists, that one spare unit is required to replace a failed component.


Repairable parts supply cycle is a specific process, different to that developed for other components such as consumable parts (Ijioui et al., 2008). It includes a number of events of the supply chain designed to ensure that replenishment occurs, beginning a new cycle. It is characterized by the dual role of end users (they must return the faulty unit to start the repair cycle).

Replenishment consists of three processes: recovery of the failed unit from the base, its transport to the repair center, and resupply the user with a functional unit that can be the same serial already repaired, or another one (Muckstadt et al. 2002).

As noted by Kilpi et al. (2004), there are four factors that affect the availability cost of a component: its reliability, its repair process TAT, the required service level, and the number of protected units. The concept of service level (SL) is used to indicate the probability that a spare unit is available when demand is created. Given an inventory level s, Muckstadt (2005) defined service level to the fraction of demand that can be satisfied immediately with the available spare units in the warehouse. As s increases, the service level increases. When calculating the service level, it does not take into account the time it takes to meet the originally unsatisfied demands (shortages).

For an inventory system with stationary demand and inventory policy, the long-term service level can be calculated as the percent ratio between the units available when an order was placed and the total demand (Katok et al., 2008). In a finite horizon, the level attained is a random variable. In the case of repairable components it is usually defined as inventory policy (s-1,s) based on an optimal level s, for which replacement is ordered immediately after a unit consumption.

To recognize that spare parts consumption forecast for unscheduled maintenance events is random in nature (MacDonnell et al., 2007), involves the intensive use of condition-based inspection

and fault analysis to determine the requirements. Better demand forecasts may be developed taking into account causal factors such as aircraft and components aging (Cohen et al., 2006). Traditional MRP techniques can be applied only to scheduled maintenance (A-, B-, C-and D-Check).

2.1. Palm’s theorem

Within the framework of queuing theory, the model was published at Sweden by Conny Palm in 1938 for calculating the traffic through telephone networks. Applied to inventory theory, its importance relies on its ability to estimate the steady state probability distribution for the number of units under repair, by using the probability distribution of the demand process and the repair time distribution average.

It basically states that if demand for an item is a process with mean µ [ea/time], which is constant (VTMR=1), and the repair time for each defective unit is independently and identically distributed according to a distribution with mean T [time], the steady-state probability distribution of the number of units being repaired has a Poisson distribution with mean µT [ea] (Sherbrooke, 2004).

Poisson distribution (Rychlik, 2006) allows knowing repeatability for a particular phenomenon without knowing its causes, assuming that they are independent, and establishing the probability that an accidental event, which causes its occurrence, exists or not. If the relationship between cause and the phenomenon is not accidental, because there is a relationship of dependency between them, this would be evident by increasing the number of intervals of occurrence that were taken as a sample (Poisson, 1837).

Noted Sherbrooke (2004) that it was observed during some tests for U.S. Air Force that the model lost validity when demand varied substantially over time, as the variance (the measure of its dispersion) exceeded the average (VTMR>1), triggering a Poisson process with nonstationary increments. For


these cases, Hodges (1985) proposed a model based on a negative binomial distribution, and recommended to apply Poissonian distribution when Variance-to-Mean-Ratio (VTMR) lies between 0,9 and 1,1.

Assuming the interval between coming units from the repair shop is negative exponentially distributed, and that the failure process is independent from the number of spares available at the warehouse, according to Kilpi et al. (2004) it is not necessary to measure the shape of the distribution. This assumption is violated when a shortage occurs, because the number of operational units is reduced, decreasing the demand for replacement units.

For Adams et al. (1993) the method’s sources of error affecting the demand estimation, are: 1) events random occurrence, 2) a high rate of demand for short periods, 3) insufficient information to determine the real rate of demand, and 4) biased demand rate estimation.

According to Crawford (1981), the steady-state hypothesis (constant mean) clearly results in a demand underestimation. For high-removal rate units, he suggested that demand could be predicted with linear regression techniques.

2.3. Algorithm

Recommended stock of spare units (operational inventory) depends on the components degree of utilization and their rate of removal (due to failure or breakage). In the case of repairable units, demand is limited to that created during replacement time (DTAT) when the inventory is unprotected. From Palm’s theorem (Sherbrooke, 2004) the probability P that the number of unscheduled removals R is less than -or equal to- the number of replacement parts m is a process with Poisson distribution of the form:

∑×=≤ −m m

TATD

m

DemRP TAT

0 !}{

This formula describes the probability

that during TAT the maintenance base will not suffer more removals than the available spares. Demand is caused by unscheduled removals occurring during the replacement cycle. Therefore, it is possible to decrease the stock of spare parts by improving TAT.

Then, the problem of determining the optimal stock of spare parts, given DTAT is reduced to an iterative process where the value of m for which P is greater than -or equal to- the desired service level (SL) is calculated.

Demand forecasting process for a given period n+1, begins with the period n MTBUR calculation. Conceptually, MTBUR represents the average number of hours between unscheduled removals R. It is calculated considering the cumulated number of hours in service provided by all the installed units of the same component in the fleet:

R

FHFSQPAMTBUR

××=

Then, the forecasted demand D [ea] for

the period n is:

MTBUR

FHFSQPAD

××=

Finally, multiplying D by the ratio

between TAT and the period length for which the calculation is performed, DTAT is obtained.

As a decision criterion, the inclusion in the operational inventory of a specific item for a certain period depends on the relationship between its demand for that period, and the minimum number of removals from the previous cycle. If D≥Rmin(n-1) then a number of spare units must be required.

2.3. Reliability

Reliability can be defined as the product's ability to function under certain


conditions during a specific period without exceeding an acceptable level of failure. In other words, it is the probability that no failure occurs (Hnatek, 2003).

It varies by product category, price, quality, user expectations, and the impact of its malfunction. In the field of technical reliability its estimation should be done through testing (Hecht, 2004).

Measuring reliability always involves a time factor. Among the indicators used, MTBF (Mean Time Between Failures) and MTBUR are often mentioned. Technical failures during system operation are considered by MTBF, but also those from accidental breakage are included in MTBUR. The instantaneous rate of failure of a component over time is a law called the "bathtub curve" due to its U-shaped graph (there are distinguished three phases over the part life: infant mortality, random failure, and aging).

According Batchoun et al. (2003), when a failure occurs less than ten times per year it is assumed that removals follow a Poisson process (discrete variable), but when its rate is equal to -or greater than- ten times a year, the number of removals should be assumed to follow a Gaussian distribution (continuous variable).

As an advice, Selivanov said in 1972 that theoretical arguments about the performance of the various elements that compose a machine are insufficient to permit a full appreciation of their real service characteristics, so it is the user who must find the answer to the problem by considering the particular conditions in which the equipment is operated.

3. METHODOLOGY

For this study, historical statistics on the component removals, MTBUR, the number of active aircraft in the fleet (FS) and the average flight hours per period (FH), were collected

consulting quarterly records between 2002 and 2009 from the external repair workshop.

In particular for this analysis, the duration of each period was 90 days so the expected demand during the replenishment period DTAT was calculated as follows:

90

TATDD TAT ×=

With the parameters from Table 2 for

each period n, its corresponding MTBUR, and the values of FS and FH for the period n+1, the expected demand for the next period Dn+1 and its associated DTAT were calculated, and Table 4 prepared with the results.

Table 2 - Parameters.

QPA 3 [ea] TAT 60 [day] SL 98,0 [%]

Thus, the expected demand for period

n+1 was compared to Rn +1 establishing the gap between these numbers.

To estimate the number of replacement units, an iterative process was performed for each period, calculating P for different values of m, and selecting as the recommended stock number that value of m for which P ≥ SL. For example, Table 3 shows the different values P takes for each m between 0 and 2 with n=3 (FH=403 [hour], MTBUR=7000[hour], FS=14[ea]). A TAT of 60 [day] is representative, assuming the unit returned for repair is not replaced with another available in stock. Since P{RTAT≤2}=99,3 is greater than SL, the recommended value of m to support the operation during the next period is 2 [ea].

Table 3: Iteration Process

m P 3 99,9 2 99,3 > 98,0 1 94,3 0 0,0


4. RESULTS

4.1. Analysis of removals

Comparison between quarterly estimated demand Dn, and actual removals Rn for the same period n, showed that 50% of times the expected demand was underestimated, and 14% of times removals were lower than half of expected ones. The ratio between estimated demand and actual removals cumulated during the whole seven years period was 1,06. The mean µ of R was 7,0 [ea] and VTMR=2,4. Table 5 shows the values taken by the frequency distribution of R (mode interval [3;5]) .

While it can be considered that the method was effective to predict the total number of removals, the availability of units

in operation was out of phase with the demand, pressing on the repair cycle. This discrepancy would not exist if MTBUR remains roughly constant for different periods. Periodical addition of aircrafts to the sample undoubtedly was a factor of error for the model application.

Table 5: Distribution of fi(R)

R fi 0-2 0,07 3-5 0,31 6-8 0,28 9-11 0,21 12-14 0,07 15-17 0,07

Table 4: Poisson process results

Period n [90 days]

R [ea]

MTBUR [hour]

FS [ea]

FH [hour]

Dn+1 [ea]

m [ea]

1 1 20871 11 632 1,0 1 2 3 7000 13 538 2,4 2 3 1 16920 14 403 2,6 2 4 4 11082 15 985 2,7 2 5 4 7482 15 665 3,8 3 6 6 4799 15 640 5,2 3 7 4 6215 15 552 7,8 4 8 4 12055 17 945 2,1 2 9 4 6191 17 486 9,1 4

10 11 5124 17 1105 4,1 3 11 10 2125 17 417 9,7 5 12 6 3440 20 344 5,8 3 13 3 6680 20 334 3,1 2 14 9 2290 20 344 9,8 5 15 6 3746 20 375 8,5 4 16 6 5333 22 485 4,2 3 17 13 1719 22 339 12,3 5 18 12 1757 22 319 13,9 6 19 16 1523 26 312 13,7 6 20 4 5198 30 231 7,1 4 21 11 3372 30 412 10,1 5 22 8 4259 31 366 8,4 4 23 7 5140 32 375 7,7 4 24 7 5643 37 356 5,7 3 25 7 4584 38 281 8,0 4 26 16 2278 41 296 24,8 8 27 10 5642 46 409 20,7 7 28 4 29186 51 763 2,1 2 29 9 6914 51 407 6,8 2


As the variance of R exceeded the

average, the demand evolution is not associated to a Poisson-type process, and the first assumption of the method was not satisfied. Then, it can be considered a negative binomial distribution (Figure 2) associated with the same probability function:

( ) rk ppk

rkkRP ×−×

−+== 1

1}{

with k=0, 1, 2,…

Where coefficients r and p are

calculated with: r=int|µ/(VTMR-1) | p=1/VTMR For this particular case r=5 and

p=0,4167. Moreover, in order to determine the

effect of each period n length on the demand estimation, annual periods were also considered. With this data, defined average MTBUR for the first year, the demand for the second year was calculated, repeating the

process for each year. The first four columns of Table 6 show results and actual total removals for each period.

Figure 2: Probability Function P{R=k}

Total forecasted demand (ΣDannual) for

the six years period represented 95% of the removals actually occurred (ΣRannual) during that time. But, 67% of times the demand by year (Dannual) was underestimated, reducing the service level.

Table 6: Annual demand forecast

Year Rannual MTBURannual Dannual MTBURcumulated Dannual

1 18 13968 9 13968 9 2 31 7638 16 10803 11 3 24 4220 23 8609 11 4 45 4512 20 7585 12 5 33 2549 58 6578 22 6 37 4604 53 6249 39

Trying to lessen the effects of varying

MTBUR the same simulation was performed but considering for each year a mean time between removals equal to the cumulative average of previous years. In this case, total estimated demand accounted for 56% of the actual removals during six years, and 83% of times demand was underestimated (columns 5 and 6 of Table 6). Therefore, for short periods of computation over the full term, there is a closer approximation to reality of the total estimated, but if the average time between

failures shows much variation, spares supply timing is strongly out-phased.

To verify the effectiveness of demand forecasting by means of linear regression the relationship between the number of removals and total flight hours cumulated by the units in service within the fleet (considered independent variable) was modeled. The effectiveness achieved in estimating the total number of removals was 100,5%, but 64% of times demand was overestimated (and therefore assets were immobilized


unnecessarily). Its correlation coefficient ρ was 0,2431 (on the other hand, if R is correlated in terms of FH and FS its

coefficients were 0,2952 and 0,3695 respectively).

Figure 3: Relationship between removals and total flight hours

4.2. Operational Inventory estimation

Variations between the values of m for each period indicated little economic viability to adjust inventory levels so regularly. But on the other hand, longer intervals (in terms of annual demand), would lead often to shortage situations.

Also, in those periods for which m≤R, depending on TAT length and the moment when the failure happens (to the beginning or towards the end) it is probable that the replacement units will not return on time from the repair shop to be available to initiate a new cycle (timing). Therefore, actual SL would be less than desired.

In this study, each unit could initiate a replacement cycle up to 2 times per period. That is, m spares would absorb removals up to twice its value. But that will happen, if the removal is deferred after the failure detection until the unit under repair is returned in serviceable condition. Assuming that inventory level s in each period was adjusted perfectly to m (i.e. buying or selling units), 34% of times there had been a shortage, having to make an unbudgeted purchase. Average number of replacement units for all periods was 4[ea]. Adopting a negative binomial distribution, to represent DTAT the following probability function was obtained:

1,53703,06297,01,4

}{ ××

+=≤ k

TAT k

kkDP with k=0, 1, 2,…


Figure 4: DTAT probability function and its distribu tion (TAT 60 days)

Then, for m=15 the probability

P{DTAT≤15}=98,43% exceeded desired SL. Therefore, according to this model the number of recommended spare parts would be 15 units. But, when compared to the number of removals there is a tendency to overestimate the demand.

Moreover, there is a dependency between TAT and the type of distribution that characterizes demand during replenishment time. If DTAT is calculated for different TAT and then VTMR for each case is determined,

there is a proportional relationship between them. Also for a given TAT, VTMR value is 1,0 (in this case for a TAT of 22,5 days). This means that there is a replacement time for which distribution that characterizes the expected demand is Poisson (VTMR=1,0) and can apply Palm's theorem. But, for shorter and longer terms a binomial distribution (VTMR<1,0) and a negative binomial distribution (VTMR>1,0) should be respectively applied for better results.

Figure 5: DTAT probability function and its distribu tion (TAT 25 days)

When DTAT is calculated for TAT of 25

days, the average is equal to 2,14 and VTMR=1,1, yielding the negative binomial probability distribution seen in Figure 5. Then, the recommended m was 6 units for SL>98%, which given the lower TAT could play up to 3 cycles of repair within 90 days, allowing a reasonable coverage of the actual demand.

4.3. Component reliability analysis

Table 7 shows the values of MTBUR relative frequency at intervals of 2.000 hours. According to records, failures often occurred between 4.000 and 6.000 service hours.


Table 7: Distribution of fi(MTBUR )

MTBUR fi

0-2000 0,10 2001-4000 0,21 4001-6000 0,31 6001-8000 0,21 8001-10000 0,00 10001-12000 0,03 12001-14000 0,03 14001-16000 0,00 16001-18000 0,03 18001-20000 0,00 20001-22000 0,03 22001-24000 0,00 24001-26000 0,00 26001-28000 0,00 28001-30000 0,03

4.3.1. Cumulated flight hours effect

In Figure 6, a linear regression was plotted relating MTBUR to a summation for each period of the flight hours contributed by all the installed units. It was noted that MTBUR presented a nearly constant value for any number of hours in service (the difference between maximum and minimum values calculated from regression was 1%), and a

low value of correlation (ρ=0,0223). That is, an increase in the number of flight hours was associated with an increased parts demand, but can be assumed that it was not related to the mean time between removals. 4.3.2. Correspondence between reliability and aircraft utilization

Applying the same procedure for MTBUR based on the average hours flown by each aircraft, the trend in the evolution of mean time between removals was no longer constant. It is noted that a relationship existed between the aircraft utilization and component lifespan (ρ =0,4659). By increasing the average flight hours per aircraft, removals tended to decrease, increasing component reliability. 4.3.3. Fleet size effect

By increasing the number of aircrafts (proportional to the number of parts installed), there was a trend to lowering reliability. The resulting correlation coefficient was ρ =0,3482.

Figure 6: Cumulated flight hours effect on reliability


Figure 7: Aircraft utilization effect on reliability

Figure 8: Fleet size effect on reliability

5. CONCLUSIONS

Method derived from Palm's theorem to estimate the safety stock of a repairable component is based on a Poisson-type probability distribution of demand and a constant mean time between removals. However, if demand dispersion increases (VTMR>1) or if the mean is not stationary, chosen statistical model boundary conditions

are not satisfied. Then, method leads to removals underestimation with a consequent decrease in available stock of spare units, causing a low service level, an increase in deferred items, and the stress of the supply chain, all of these in detriment of the operating costs.

During this analysis, the period used for calculating the MTBUR had strong impact on the estimates. The greater the frequency the


study of reliability was conducted, the closer to the reality was the estimated amount of long-term accumulated demand. However, when mean time between removals presented a high variation from one period to another, or seasonality, predictions for the next period were less accurate than in those cases where reliability was established for longer periods. As the number of aircraft in the fleet has been changed constantly, it was not helpful to use historical accumulated values to calculate reliability (average MTBUR).

Determining demand at the time when units were in the queue for repair, it was observed that when the ratio between TAT and the period length was very small, the information on its variance was lost due to the softening of historical demand dispersion, allowing a dynamic process to be confused with a stationary one, and mistakenly associating removals behavior with a Poisson distribution. To avoid this, it should be considered additional methods for estimating future VTMR.

Model considers an infinite number of available units, meaning that for each removal there is a replacement unit, and estimates with a certain risk level (probability) how many units are being repaired simultaneously for a specified period assuming equally spaced faults. However, this study found that when demand is concentrated in a certain part of the period, the units available (inventory finite) fail to return to the stock on time and there is a shortage.

When demand was forecasted using a linear regression as an alternative method, a better approximation was obtained for high values of the independent variable, but the model demanded more investment in assets than necessary.

Focusing on Audio Control Panel component, given its high removal rate variability, a negative binomial distribution proved to be more representative for its repair process. MTBUR (reliability) showed high dependence on the number of average flight

hours per aircraft (FH) and low dependence on the cumulated time in service, as the removals (R) tended to be related to the number of units installed (FS). That is, taking the number of total flight hours as an independent variable (either flight cycles) was not representative for this process in particular. This behavior can be justified on the main cause of removal indicated by the repair shop (damage during unit handling).

For this component, scheduled removals were not taken into account (for example, due to obsolescence or hard time). However, in the event that any unit at the operational inventory is used as replacement for scheduled removal, it is recommended to include this event in the statistics to avoid triggering a shortage in the medium term.

Based on the simulations it is concluded that Poissonian model is especially suitable for determining initial inventory, when little information from the units in service is available, and it is replaced by industry standard parameters (like MTBF) supplied by OEM. Then, for an already operating fleet, the application should be preferred for those components with very low rotation (slow movers). Different probability distributions of discrete or continuous variable -depending on the number of removals- should be tested for each component and repair process in particular, to find the most suitable parameters to characterize them. In this research, a negative binomial distribution proved to be an appropriate complement to the model that significantly reduces the number of shortages for those components with significant variation in demand and high turnover (fast movers), poorly controlled repair processes (changing TAT), or for those fleets subject to constant size changes.

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