LEONARD MACLEAN ^{1,
*}
ALEX RICHMAN ^{2}
STIG
LARSSON ^{3}
VINCENT
RICHMAN ^{4}
This paper looks at the predictability of system failures of aging aircraft. We present a stochastic, dynamic model for the trajectory of the operating condition with use. With failure defined as the operating condition below a critical level, the dynamics of the number of failures with accumulated use is developed. The important factors in the prediction of mechanical failures are the number of previous repairs and the time since last repair. Those factors are related to repair procedures, with the time of repair and the extent of repair (fraction of good-as-new) being variables under the control of the operator. The methodology is then applied to data on non-accident mechanical failures affecting safety that result in unscheduled landings.
KEYWORDS: Aircraft failures, aircraft degradation and repair, airline schedule reliability.
An aircraft is a complex machine composed of many interrelated parts, components, and systems. Electrical and mechanical systems are designed with an expected life length, where length refers to time units (hours) of use. As the aircraft and systems age and their use accumulates, they gradually degenerate until they are no longer able to perform the functions for which they were designed; that is, the system is in a failed state.
A nonfunctional part, component, or system can be upgraded through replacement or repair, in which case the condition of the aircraft is restored to some degree. Maintenance can be based on the condition; that is, items are repaired when they fail. However, failure during operation can have serious consequences, so detection of items with a high probability of failure through periodic inspection becomes a major component of maintenance.
The failure rate (the probability of failure at a point in time) for a degenerating system increases with use and age. Figure 1 depicts alternative patterns of failure rates for an aircraft that undergoes periodic maintenance (a similar figure appears in Lincoln 2000). In case A, the aircraft has an increasing failure rate with age and reaches an acceptability threshold, at which point the aircraft would need to be replaced. The failure rate declines with periodic maintenance, but the improvement through maintenance diminishes over time. The threshold is not reached in case B, likely because of increased effort and cost put into maintaining the aircraft.
The cost of maintenance required to keep aircraft airworthy (below the threshold) is a major concern of operators. Although replacement time was set by manufacturers at 20 years for many aircraft models, this life length was extended by operators. An assumption has been made that aircraft operating condition can be kept at an acceptable level beyond the intended life through maintenance, but costs are high.
In 1997, 46% of U.S. commercial aircraft were over 17 years of age and 28% were over 20 years. In 2001, 31% of the U.S. commercial fleet were over 15 years of age, and those aircraft accounted for 66% of the total cost of maintenance per block hour.^{1} Although aging (the degeneration in operating condition with accumulated use) inevitably occurs, it is modified by a number of factors: quantity and quality of repair work; intensity of use; deferral of the schedule for planned maintenance; and the environment (Alfred et al. 1997). It should be noted, however, unscheduled maintenance accounts for up to 60% of the overall maintenance workload (Phelan 2003).
In addition to the possibility of maintaining an airworthy operating condition, other reasons exist for not retiring an aircraft from the fleet: the high cost of new aircraft; the increase in demand requiring an expanded fleet; and an earlier shortage of production capacity for new planes (Friend 1992). These and other factors result in a large number of aircraft in use beyond their planned retirement. A claim could be made that commercial aircraft are being strained to perform well beyond their intended operating life. Of course, if this is true we would expect that either the rate of aircraft failure would show a corresponding increase or the operating hours per aircraft would decrease because planes would be out of service for repairs more frequently.
In the United States, Service Difficulty Reports (SDRs) contain records of the safety problems an aircraft experiences during operation. This database, maintained by the Federal Aviation Administration, is considered a potential source of important information on aircraft failures (Sampath 2000). A study comparing failure rates by carrier identified significant factors that explain the differences in the rate of SDRs across carriers (Kanafani et al. 1993). Because accumulated aircraft use (age) was not included in that study, degradation with age and differences over time in the safety of individual aircraft were not considered.
With age and accumulated use, the many interrelated parts and components in an aircraft can be assumed to degrade. The operating condition or airworthiness of the aircraft is based on the status of individual parts, components, and systems, with the items that are most degenerated being the main determinants. A certain level of degeneration implies failure; that is, the item is no longer operational. As well, failure of certain components or combinations of components may render the aircraft not airworthy, which means the aircraft is in a failed state. To address the failure of operating systems, airline management undertakes a program of maintenance, with scheduled preventive maintenance and unscheduled repair/replacement of failed parts, components, and systems. This section presents a conceptual model for the degradation and repair of aircraft. The model provides a foundation for hypotheses about operations that can be tested with field data.
To characterize the degradation process, consider that the operating condition of an aircraft is captured by an unobserved health status index. The value of the index is derived from the condition of the various parts, components, and systems in the aircraft. Let t be the age of an aircraft, defined by the accumulated hours of use, and let
Y(t) = the health status of an aircraft at age t.
The status is a dynamic stochastic process, with the change in status at any age being a random variable. Assume that the average condition declines with age, but at any point variation in the status, based on environmental factors and operating characteristics, can occur. The dynamics of degradation at a point in time can be represented by a stochastic differential equation as
d Y (t ) = μ _{t} d t + σ _{t} d Z _{t} (1)
where
μ_{t} < 0 is the degradation
rate,
σ_{t} > 0 is a scaling factor,
and
dZ_{t} is an independent random process.
For example, if the random process is white noise, the stochastic differential equation defines a Wiener process, and the distribution of the health status at a given age is Gaussian (Aven and Jensen 1998). So, with starting state y_{0} and constant parameters μ and σ, the distribution of status after t time periods is Normal,
Y (t) ∝ N (y _{0} + μ t , t σ ^{2}) (2)
In this degradation framework, at any age (hours of use) the possibility exists that the status of a item during operation will drop below the critical level for functionality and the component reaches a failure state. Degradation and failure of components lower the value of the health status index Y. In particular, failure of parts and components included on a minimum equipment list (MEL) indicates the aircraft remains airworthy. Beyond the MEL, moderate mechanical failures that occur during aircraft operation would render the aircraft not airworthy. Assume that the critical health status level y* defines airworthiness. Then an aircraft failure occurs when Y (t ) < y*.
Based on the stochastic model, the many parts, components, and systems have a probability of failure during operation and, therefore, the aircraft has a probability of failure. For an airworthy aircraft, the important variable is the time to failure. Let T be the length of life (hours of use before failure) of an aircraft, with the probability distribution F (t ) = Pr (T ≤ t ), and the corresponding density f (t ). Then
is the failure rate at time t (Aven and Jensen 1998). The failure time distribution is determined by the failure rate because
In the example, where the state dynamics are defined by a Wiener process, the failure time is inverse Gaussian with density
. (3)
Failure during operation may precipitate unscheduled maintenance, particularly when items beyond the MEL fail and consequently the aircraft is not airworthy. The repair/replacement of failed items is called on-condition repair. On-condition repair brings the system back to the operating status expected of the system given its age, that is, the same status as just prior to failure. With these minimal repairs (Block et al. 1985), the aircraft failure rate is unchanged since other parts, components, and systems are still in the degraded state attained just before repair. Typically, moderate mechanical failures result in such minimal repair.
In addition to unscheduled maintenance, the whole system is subject to time-based or block repair, where items are inspected and replaced/refurbished before failure. This scheduled preventive maintenance improves the operating condition to a status greater than expected for its age and correspondingly reduces the system failure rate (Brown and Proschan 1983). To incorporate repair into the degradation model, the age variable is partitioned into intervals based on the block repair times. Assume that the first scheduled block repair is at age (hours of use) τ, and subsequent block repairs are at regular intervals of δ hours of use, where δ ≤ τ. Then age t can be written as
t = I τ + k δ + r (4)
where I = 1 if t ≥ τ, I = 0 if t < τ
if t ≥ τ, k = 0 if t < τ and
r = t - I τ - k δ.
The notation]x[defines the greatest integer less than x. Equation (4) gives the age in terms of (I + k) = the number of repairs, and r = the use since the last block repair. The intervention with a block repair improves the health status of an aircraft above the level expected for its age. Let the improvement level from a block repair at age t be up to the line
y (t) = α + β t, (5)
where α > y* and β ≤ 0.
The repair line is theoretical and the important parameter is β, which describes the repair policy to return the aircraft to a fraction of good-as-new at scheduled times. If β = 0, then repair always brings the plane to the same health status regardless of age.
To simplify the presentation, repair policies that are equivalent in terms of the total repair effort will be considered. Let
L = the expected length of the operating life of an aircraft.
Assume that all feasible repair policies have the same total repair over the expected life of the aircraft. That is,
,
for some constant α*. With this condition, the repair policy is determined by β, which also determines the distribution of repair over the aircraft's lifetime.
The partition of age at block repairs generates renewal cycles for the degradation process, with the first cycle starting at the initial status y_{0}, and subsequent cycles beginning at the status defined by the repair line:
y _{j} (τ, δ, β) = α + β t _{j} , (6)
where t _{j} = τ + (j - 1) δ, j = 1, ., k. The repair policy is defined by: τ- the hours of use until the first block repair; δ- the hours of use between subsequent block repairs; and β- the repair fraction. Using the definitions of t_{j} and y_{j}, the increase in the health status at each block repair from β can be calculated. The policy determines the starting state and length of renewal phases or cycles for the degeneration process. Figure 2 illustrates the cycles of degradation and repair for an aircraft.
In each renewal phase of the degradation model, there is a chance that the aircraft fails; that is, its status drops below the critical level y*. Let T_{j} = time to critical condition y* in cycle j starting from y_{j-1}, j = 1,2,... . The failure time distribution for T_{j} is written as F_{j}(s | τ, δ, β), with density f_{j}(s | τ, δ, β), where the repair policy (τ, δ, β) determines the starting status. Given the failure time distribution, the failure rate in the jth cycle is
. (7)
Consider
N(t) = the number of aircraft failures to age t for repair policy (τ, δ, β). (8)
Because
the expected number of failures is
. (9)
In the failure rate for each renewal phase, the probability distribution for the time to failure has the same form, but the starting state in each phase declines if β < 0. With t_{j} = τ + (j - 1)δ, and the starting state in phase j + 1 as
,
define
ψ (x, y _{j}) = -ln (1 - F _{j + 1} (x)). (10)
Thus, ψ(x,y_{j}) is the expected number of failures between times 0 and x in phase j + 1, with failure time distribution F_{j+1} and starting state y_{j}. Then
. (11)
The degradation process and repair policy are determined by parameter values, and those policies determine the properties of the expected number of failures over time. Let ψ′_{x} and ψ′_{y} denote first derivatives of ψ with respect to x and y, respectively. Thus, ψ′_{x} is the change in expected failures with use (degradation) within a phase, and ψ′_{y} is the change with respect to the phase starting state, determined by the block repair policy. The following general results establish the expectations for mechanical failures when the degeneration model applies.
Proposition 1 (degeneration): If the health status of an aircraft degenerates with use, then between block repairs, the failure rate with use increases, as does the expected number of failures in a fixed-width use interval.
In the dynamic model, degeneration follows from μ < 0. With degradation, ψ′_{x} = λ > 0, and ψ′′_{x} > 0, which implies an increasing failure rate between repairs.
Proposition 2 (imperfect repair): If the block repair is imperfect, then the failure rate with use since the last repair is nondecreasing with the number of previous block repairs, and the expected number of failures in a fixed (use since last repair) interval is nondecreasing with the number of repairs. If the repair fraction decreases over time, then the expected number of failures is increasing.
In the model, ψ′_{y} > 0. If α < y_{0}, β = 0, then the expected number of failures in a fixed interval is constant after the initial block repair. If β < 0, then the starting state y decreases, with increasing failure rates in successive phases between block repairs.
Proposition 3 (repair interval): If the imperfect repair fraction is decreasing over time, then the expected number of failures in a fixed-use interval increases/decreases as the block repair interval increases/decreases.
Block repair increases the health status above that expected for accumulated use, so more block repairs (shorter times between block repairs) raise the expected value of y and decrease the failure rate and number of failures.
The link between the latent state model for degradation/repair and the model for the number of failures shows how the operating practices of airlines can manifest themselves in mechanical failures, safety problems, and unscheduled maintenance. Historical data on failures and maintenance will have that complex relationship embedded. The information on failures and block repairs is available, but the degradation rate and extent of repair (fraction of good-as-new) are unknown. However, from Proposition 1, the time since the last block repair reflects degradation, and from Proposition 2, the extent of the repair is directly related to the number of block repairs. The transformation of equation (11) for the expected number of failures to an expression in terms of the number of block repairs and the time since the last block repair is achieved by a series approximation to the function for E(N(t)).
From the model, the average level of repair is α*. Consider the first order approximation to ψ(δ, y) around (δ, α*):
ψ (δ, y _{j}) ≈ C_{0} (τ, δ, α*) + C_{1} (τ, δ, α*) j. (12)
In the last (incomplete) phase, a second order approximation to the number of failures around (0,α*) is reasonable, assuming the failure rate is increasing monotonically with use. Then
ψ (r, y_{k + 1}) ≈ D_{0} (τ, δ, α*) + D_{1} (τ, δ, α*) k
+ D_{2} (τ, δ, α*) r
+ D_{3} (τ, δ, α*) r ^{2}. (13)
The coefficients in the approximating functions are defined by derivatives of ψ, evaluated at (τ, δ, α*). Substituting the approximations in equations (12) and (13) into equation (11), the expected number of failures has the form
E (N (t)) ≈ B _{0} + B _{1}I + B _{2}k + B _{3}k ^{2}
+ B _{4}r + B _{3}r ^{2}. (14)
(Note that .)
A representation of the number of failures is shown in figure 3.
Thus, B_{1} is the expected number of failures in the first phase. {B_{2}, B_{3}} capture the expected number of failures in subsequent phases, and {B_{4}, B_{5}} capture the expected number in the last (incomplete) phase. The coefficients in the expected number function that relate to the propositions are B_{3} and B_{5}. If the block repair is imperfect, then B_{3} > 0. Figure 3 shows this effect with the failure function starting above the origin at block repair times. An accelerated failure rate between block repairs implies B_{5} > 0, which is shown with a steeper slope in successive phases between repairs.
The approximating equation for the expected number of failures is in a very suitable form for analysis. Consider an observation window (interval) (t_{1},t_{2}), where t_{2} - t_{1} < δ. With t_{1} = I_{1}τ + k_{1}δ + r_{1}, and t_{2} = I_{2}τ + k_{2}δ + r_{2}, the number of failures in the interval (t_{1},t_{2}) is approximately
η = E (N (t _{1}, t _{2})) = E (N (t _{2})) - E (N (t _{1})),
so that
η = B _{1} (I _{2} - I _{1}) + B _{2} (k _{2} - k _{1}) + B _{3 }(k ^{2}_{2} - k ^{2}_{1})
+ B _{4} (r _{2} - r _{1}) + B _{5} (r ^{2}_{2} - r ^{2}_{1}). (15)
This change model relates the number of failures in an observation window to the degeneration and the block repairs in the window. In equation (15), (I_{2} - I_{1}) = 1 if the first repair is in the interval, and zero otherwise; (k_{2} - k_{1}) = 1 if a later repair occurs in the window, and zero otherwise.
We used data from AlgoPlus (2004) to test the failure model on operating failures and AvSoft (2004) on aircraft use. For the purposes of this study, an operating failure is defined as an unscheduled landing due to mechanical problems affecting safety. Thus, an unscheduled landing is a record of an operating condition at or below a critical or intervention level. In figure 2, the unscheduled landings (failures) occur when the health status drops to the critical level, where airworthiness fails. It is also possible that components fail and the event does not lead to an unscheduled emergency landing. As mentioned earlier, a minimum equipment list details which components may fail without the need for an unscheduled landing. In terms of the degradation/repair model, the critical condition line is below the condition for failures on the MEL, so that reaching the critical line implies unsafe operation and a need to interrupt the flight of an aircraft.
The record of unscheduled landings over time provides an information base for analyzing the degeneration in the operating condition of an aircraft. The AlgoPlus data contain detailed records on all unscheduled landings as reported in the Service Difficulty Reports for all commercial aircraft in the United States. The AvSoft data maintain records on departures and flying hours for all commercial aircraft in North America. Both datasets have the serial number, chronological age, model, and carrier/operator for each aircraft.
An observation window from 1990 to 1995 inclusive was chosen, and all aircraft operating during that time for three operators and two models were selected for this study. For each aircraft, the following information was recorded: 1) model; 2) operator; 3) age on December 30, 1995; 4) use (block hours, cycles) by month from January 1990 to December 1995; 5) dates out of service for at least one month between 1990 and 1995; and 6) number of unscheduled landings between 1990 and 1995. We interpreted the out-of-service period in the observation window as a time when scheduled repair was undertaken. The identification of these periods is within a record of otherwise continuous use. Outside the observation window, the block repair (preventive maintenance) cycle was set at 10 years for the first block repair and 8 years for subsequent block repairs. This is based on the recommendations for D-check cycles.^{2} Of course, in practice the time of block repairs would be variable across aircraft and using a fixed value (outside the window) could reduce the power of the fitted models.
Table 1 presents a brief description of the aircraft in the dataset. For the aircraft in the study group, table 1 shows substantial differences across models and operators in the age of aircraft as of December 1995 and the number of unscheduled landings between January 1990 and December 1995.
The definition of age in the degradation of aircraft refers to hours of use rather than chronological age. However, an aircraft operator might make little distinction between airworthy aircraft of varying ages when making decisions on use. To consider this point, we looked at the relationship between flying hours per month and chronological age in the data for the period 1990 to 1995. The correlation in the data between monthly flying hours and age is r = 0.07. The intensity of use appears almost constant across age, indicating that aircraft are not being used less as they age. With constant use per unit time, the accumulated hours of use are almost a scalar multiple of chronological age. So, calendar time was used in the model for predicting the number of failures; that is, the time between block repairs and the time since the last repair will be measured in calendar time rather than accumulated hours of use.
The formulation of a change model for the number of failures creates a framework suitable for observation and statistical analysis. Based on the model in equation (15), consider the regression model
, (16)
where
N(t_{1},t_{2}) = the number of failures between ages t_{1} and t_{2},
X_{1} = the indicator for the first τ-repair in the interval,
X_{2} = the indicator for the kth δ-repair in the interval, k ≥ 1,
X_{3} = the difference between the squared number of repairs, k ^{2}_{2} - k ^{2}_{1} ,
X_{4} = the difference in residual times, r_{2} - r_{1, }
X_{5} = the difference in squared residual times, (r ^{2}_{2} - r ^{2}_{1}), and
ε = the random error.
In the regression model, assume that the unscheduled landings and item failures from degradation are directly related to the number of block repairs and the time since the last block repair. There are also other factors such as repair skill level, maintenance philosophy, and operational environment involved in unscheduled landings (Phelan 2003). We will assume that these other factors are associated with the operator. As well, the aircraft model is a factor in failure rates. So, the coefficients in the regression model depend on the aircraft model and the aircraft operator. This is reflected in the regression model with a superscript q on the coefficients.
The coefficients in the regression model are counterparts of the coefficients in the failure model, and appropriate tests characterize the role of degradation and repair on failures for a particular model and operator combination. Table 2 displays the relevant research hypotheses.
A comparison of the coefficients for different model and operator combinations would reveal differences in model degeneration rates and/or differences in operator maintenance practices. To include comparisons, an expanded regression equation is defined. Consider the indicator variables:
U = 1 for model M_{1} and 0 otherwise
V_{1} = 1 for operator O_{1} and 0 otherwise
V_{2} = 1 for operator O_{2} and 0 otherwise.
The regression equation for defining model and operator effects is
. (17)
An equivalent formulation, which reveals the effect on coefficients, is
. (18)
To simplify notation, consider the vectors
.
Table 3 shows the variations on the regression equation.
Table 4 presents the hypotheses for testing the effect of differences in models and operators.
The maintenance policies of a carrier as well as the particular design (components and systems) in an aircraft model are major factors in the operating characteristics of an aircraft. Selecting a single aircraft model from a single carrier removes the complication of varying models and carriers, and thus the assumption of constant degradation and repair parameters is reasonable. This experimental setting is ideal for focusing on the degradation of the operating condition with accumulated use. As such, the data for operator O_{2} were used with degradation model (16), because the O_{2} fleet consisted only of B737 aircraft.
Because the number of failures is a counting variable, the error variance is not likely to be constant. Therefore, an iteratively reweighted least squares estimation method was used, where the weights were reciprocals of the fitted values (McCullagh and Nelder 1989). The effect of weighting was minimal, so the unweighted sums of squares are reported. The results from fitting the degradation model to the operator O_{2} data are given in table 5.
Clearly the overall fit of the change model is strong (F = 96.22). Furthermore, the individual components in the model are highly significant. Maintenance in the observation window and time since maintenance are important factors in predicting the number of unscheduled landings that occur in the window. Of particular significance is the acceleration in the number of repairs (increasing failure rate) as time since repair increases . With reference to Proposition 1, the regression provides the following result.
Result 1 (degradation): The rate of unscheduled landings increases with time since the last block repair.
Furthermore, there is evidence that the block repair is not as good-as-new, because the sign on X_{1} = I_{2} - I_{1} is negative and the sign on X_{2} = k_{2} - k_{1} is positive. A test on the difference is highly significant (P < 0.001). However, the data did not allow for a test of diminishing repair fraction. The aircraft in the operator O_{2} fleet are relatively new and the maximum is k = 1. The regression gives the analogous result for Proposition 2.
Result 2 (imperfect repair): The rate of unscheduled landings decreases after a block repair, with the decrease greater for the initial repair than for subsequent repairs.
To consider the issue of differential effects for the aircraft model and operator, the additional terms with indicator variables were included. Table 6 presents the regression results. The additional sum of squares for models and operators indicated in table 6 are considered after including other effects. That is, the outcome (number of unscheduled landings) was adjusted for the model effect when considering the operator effect, and it was adjusted for the operator effect when considering the model effect. In both cases, the effects are statistically significant (i.e., there is a differential effect for operators and models). In the context of the regression equation, the effect of maintenance on unscheduled landings was not the same for the operators in the study. As well, the effect of the time since maintenance was not the same for the models selected.
Result 3: The relationship between the rate of unscheduled landings and the time since the last block repair and the number of block repairs depends on the aircraft model and operator.
Using the indicator variables, it is possible to write out the fitted equation for each (operator and model) type. The estimates for equation parameters are given in table 7. The equation for operator O_{2}(q = 3) is slightly different from the equation using only O_{2} data, owing to the greater variation in using multiple operators and models. However, it is a good reference for understanding the changes in the equation with the operator/model variations. The biggest operator effect is the difference of O_{3}(q = 2,5) from the others on the estimate . For aircraft models, the estimate of is most affected.
The operating condition of aging aircraft has been a hotly discussed topic for more than a decade. The Federal Aviation Administration's position is that the operation of older aircraft is an economic decision and not a safety issue; that is, aircraft can be repaired to a safe operating condition and the cost of those repairs is the issue.
This study considers the trajectory of the health status of an individual aircraft, with an emphasis on episodes where flights are interrupted because of mechanical failures affecting safety. In the context of a model for mechanical failure, two experiments were carried out. In the first experiment, a single model and carrier were analyzed for the potential impact of aircraft age (accumulated use) and repair on schedule reliability. The study assumes that all the selected aircraft are equivalent except for age, and the fleet management practices of the carrier remain consistent over time. In this setting, the variability in the failure rate (unscheduled landings) can be partly attributed to the aging of the aircraft and incomplete repair during preventive maintenance. The second experiment involved multiple operators and aircraft models. With the same failure model, the differential effect of operational practices and aircraft design can be studied.
The following can be concluded from the results of this study.
The clear relationship between unscheduled landings and degradation/repair in the regression model has implications for the maintenance policies of operators. The operator has control over the repair intervals-(τ,δ) and the repair effort β-the fraction of good-as-new. The dependence of the regression coefficients on the maintenance parameters (τ,δ,β) is implied in this paper, but that connection can be made more explicit by using the actual derivatives in the series approximations. In that way, changes in the values of the maintenance parameters would translate into changes in the rate of unscheduled landings. Therefore, an operator could explore the outcome (in terms of unscheduled landings) of changes in the repair parameters, for example, the block repair interval.
The purpose of our research was to establish the feasibility of predicting unscheduled landings from data on use and maintenance. An earlier study (Nowlan and Heap 1978) found that 89% of aviation mechanical malfunctions were unpredicted using operating limits or undertaking repeated checks of equipment. The results of this work indicate that important problems in the operation of aircraft can be studied with existing field data. Use of these results in the management of an airline would require additional study, but a step in that direction has been taken here.
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1. A block hour refers to flying time in hours, including takeoff and landing.
2. A D-check refers to the major maintenance and overhaul programs in which the aircraft is completely stripped down and inspected, with many parts and components replaced or refurbished.
^{1} Corresponding Author: L. Maclean, School of Business Administration, Dalhousie University, Halifax, Canada B3H 3J5. E-mail:L.C.MacLean@dal.ca
^{2} A. Richman, AlgoPlus Consulting Ltd., Halifax, Canada B3H 1H6. E-mail:arichman@algoplusaviation.com
^{3} S. Larsson, School of Business Administration, Dalhousie University, Halifax, Canada B3H 3J5. E-mail:S.O.Larsson@dal.ca
^{3} V. Richman, Sonoma State University, Rohnert Park, CA 94928-3609. E-mail:Vincent.richman@sonoma.edu