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flying hours per month and chronological age inthe data for the period 1990 to 1995. The correlationin the data between monthly flying hours andage is r = 0.07. The intensity of use appears almostconstant across age, indicating that aircraft are notbeing used less as they age. With constant use perunit time, the accumulated hours of use are almosta scalar multiple of chronological age. So, calendartime was used in the model for predicting the numberof failures; that is, the time between blockrepairs and the time since the last repair will bemeasured in calendar time rather than accumulatedhours of use.Regression ModelThe formulation of a change model for the numberof failures creates a framework suitable for observationand statistical analysis. Based on the model inequation (15), consider the regression modelNt ( 1 , t 2 ) = β q 1 X 1 + β q 2 X 2 + β q 3 X 3+ βq 4 X 4 + β q 5 X 5 + ε , (16)whereN(t 1 ,t 2 ) = the number of failures between ages t 1and t 2 ,X 1 = the indicator for the first τ -repair in theinterval,X 2 = the indicator for the kth δ -repair in theinterval, k ≥ 1 ,X 3 = the difference between the squared number of2 2repairs, k 2 – k 1 ,X 4 = the difference in residual times, r 2 – r 1 ,X 5 = the difference in squared residual times,2 2r 2 – r 1 , andε = the random error.In the regression model, assume that the unscheduledlandings and item failures from degradationare directly related to the number of block repairsand the time since the last block repair. There arealso other factors such as repair skill level, maintenancephilosophy, and operational environmentinvolved in unscheduled landings (Phelan 2003). Wewill assume that these other factors are associatedwith the operator. As well, the aircraft model is afactor in failure rates. So, the coefficients in theregression model depend on the aircraft model andthe aircraft operator. This is reflected in the regressionmodel with a superscript q on the coefficients.The coefficients in the regression model are counterpartsof the coefficients in the failure model, andappropriate tests characterize the role of degradationand repair on failures for a particular modeland operator combination. Table 2 displays the relevantresearch hypotheses.TABLE 2 Research HypothesesRegression modelhypothesisqH: β 2 –qH: β 3 > 0qH: β 4 > 0qH: β 5 > 0β 1q> 0InterpretationBlock repair is incomplete; not togood-as-new.The repair fraction of good-asnewis decreasing.The failure rate increases with thetime since block repair.There is an accelerated failurerate with increasing time sincerepair.A comparison of the coefficients for differentmodel and operator combinations would reveal differencesin model degeneration rates and/or differencesin operator maintenance practices. To includecomparisons, an expanded regression equation isdefined. Consider the indicator variables:U = 1 for model M 1 and 0 otherwiseV 1 = 1 for operator O 1 and 0 otherwiseV 2 = 1 for operator O 2 and 0 otherwise.The regression equation for defining model andoperator effects is5∑Nt ( 1 , t 2 ) = β j X j + λ ij V i X jj = 12∑i = 1 j = 1+ γ j UX 3 + j+ ε . (17)j = 1An equivalent formulation, which reveals theeffect on coefficients, isTo simplify notation, consider the vectors2∑3∑3⎛2Nt ( 1 , t 2 ) = ⎜β j + λ ij V i ⎟+∑j = 1⎝∑i = 12∑ β 3 + j γj = 1⎞Xj( + jU)X 3 j +⎠+ ε. (18)8 JOURNAL OF TRANSPORTATION AND STATISTICS V8, N1 2005
β⎛⎜⎜⎜= ⎜⎜⎜⎜⎝β 1β 2β 3β 4β 5⎞ ⎛λ 11 ⎞ ⎛λ 21 ⎞ ⎛0⎞⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟ ⎜λ 12 ⎟ ⎜λ 22 ⎟ ⎜0⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎟;λ 1 = ⎜λ 13 ⎟;λ 2 = ⎜λ 23 ⎟;γ = ⎜0⎟ .⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎟ ⎜ 0 ⎟ ⎜ 0γ ⎟⎟ ⎜ 1 ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎠ ⎝ 0 ⎠ ⎝ 0 γ ⎟⎠ ⎝ 2⎠Table 3 shows the variations on the regressionequation.TABLE 3 Model Variationsq V 1 V 2 U Parameters1 0002 1003 0104 0015 1016 011Table 4 presents the hypotheses for testing theeffect of differences in models and operators.TABLE 4 Comparison HypothesesHypothesesH: λ 1 ≠ 0H: λ 2 ≠ 0H: γ ≠ 0Fitted ModelInterpretationRepair effect for operator O 1 differsfrom othersRepair effect for operator O 2 differsfrom othersThe failure rate since repair dependson the aircraft modelThe maintenance policies of a carrier as well as theparticular design (components and systems) in anaircraft model are major factors in the operatingcharacteristics of an aircraft. Selecting a single aircraftmodel from a single carrier removes the complicationof varying models and carriers, and thusthe assumption of constant degradation and repairparameters is reasonable. This experimental settingis ideal for focusing on the degradation of the operatingcondition with accumulated use. As such, thedata for operator O 2 were used with degradationmodel (16), because the O 2 fleet consisted only ofB737 aircraft.β 1=ββ 2 = β + λ 1β 3 = β + λ 2β 4 = β + γβ 5 = β + λ 1 + γβ 6 = β + λ 2 + γBecause the number of failures is a counting variable,the error variance is not likely to be constant.Therefore, an iteratively reweighted least squaresestimation method was used, where the weightswere reciprocals of the fitted values (McCullagh andNelder 1989). The effect of weighting was minimal,so the unweighted sums of squares are reported.The results from fitting the degradation model tothe operator O 2 data are given in table 5.TABLE 5 Fitted Failure ModelVariable Parameter Estimate T Pβ 1X 1 –1.54 –6.52 0.000β 2X 2 6.39 5.03 0.000β 4X 4 0.12 6.02 0.000β 5X 5 0.02 10.86 0.000ANOVASource SS df F PRegression 615.79 4 96.22 0.000Error 347.21 217Total 963.00 221Clearly the overall fit of the change model isstrong (F = 96.22). Furthermore, the individualcomponents in the model are highly significant.Maintenance in the observation window and timesince maintenance are important factors in predictingthe number of unscheduled landings that occurin the window. Of particular significance is theacceleration in the number of repairs (increasingfailure rate) as time since repair increasesˆ( β 5 = 0.02, T = 10.86). With reference to Proposition1, the regression provides the following result.Result 1 (degradation): The rate of unscheduledlandings increases with time since the last blockrepair.Furthermore, there is evidence that the blockrepair is not as good-as-new, because the sign onX 1 = I 2 – I 1 is negative and the sign on X 2 = k 2 – k 1 isˆ ˆpositive. A test on the difference β 2 – β 1 = 7.93 ishighly significant ( P < 0.001). However, the data didnot allow for a test of diminishing repair fraction.The aircraft in the operator O 2 fleet are relativelynew and the maximum is k = 1. The regression givesthe analogous result for Proposition 2.MACLEAN, RICHMAN, LARSSON & RICHMAN 9
Page 1 and 2: Volume 8 Number 1, 2005ISSN 1094-88
Page 3 and 4: A PEER-REVIEWED JOURNALJOURNAL OF T
Page 5: JOURNAL OF TRANSPORTATION AND STATI
Page 9 and 10: The Dynamics of Aircraft Degradatio
Page 11 and 12: would show a corresponding increase
Page 13 and 14: FIGURE 2 Aircraft Degradation and R
Page 15: so that- r 2 1 ) . (15)This change
Page 19: and the fleet management practices
Page 22 and 23: of passengers and cargo to the econ
Page 24 and 25: TABLE 3 Cycle Parameters and Final
Page 26 and 27: TABLE 4 Forecasts of Numbers of Pas
Page 28 and 29: FIGURE 4 Slope Components for the Z
Page 30 and 31: 2. LS: this kind of intervention ha
Page 32 and 33: CONCEPTTon-miles is the primary phy
Page 34 and 35: FIGURE 2 Air Freight, Express, and
Page 36 and 37: FIGURE 5 Railroad Ton-MilesTon-mile
Page 38 and 39: Transport Canada. 1990-2002. Transp
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Page 46 and 47: account for premature investments i
Page 48 and 49: zonal definitions. In Supernak et a
Page 50 and 51: TABLE 2 Differences Among the 1972,
Page 52 and 53: FIGURE 1 Research Sequence of Stage
Page 54 and 55: TABLE 3 Regression Models by Metrop
Page 56 and 57: TABLE 5 Tobit Models by Metropolita
Page 58 and 59: Table 9 presents the predicted aver
Page 60 and 61: TABLE 9 Aggregate Estimation Result
Page 62 and 63: TABLE 11 Tobit Estimation of Nontra
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INTRODUCTIONThe Intermodal Surface
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SUPPLEMENTAL STATE-LEVEL DATAIn add
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METHODOLOGYAn exploratory factor an
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Thus, a relatively large p value an
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FIGURE 1 Final Model Specifications
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CONCLUSIONSISTEA and the TEA-21 leg
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Survey Question 2: Do your agency
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Survey Question 11 (continued): Has
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Survey Question 16: Every state has
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Incidents have become one of the ma
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FIGURE 1 Map of the M4 MotorwayA40A
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TABLE 2 Variables in the Fuzzy Logi
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FIGURE 9 Result of the ANN ModelMod
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Garib, A., A.E. Radwan, and H. Al-D
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Fx ( )Working Correlation Matrix in
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that had a lower level of congestio
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TABLE 3 Modeling Results for the MG
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REFERENCESAbdalla, M. 2003. Modelin
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the relevant covariates including t
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BOOK REVIEWSEconomic Impacts of Int
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TABLE 1 Airline Employees by Busine
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JOURNAL OF TRANSPORTATION AND STATI
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U.S. Department of TransportationRe
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