Research Paper

A quantitative model for aviation safety risk assessment Huan-Jyh Shyur * Department of Information Management, Tamkang University, 151 Ying-Chuan Road, Tamsui, Taipei, Taiwan Received 2 August 2006; received in revised form 14 June 2007; accepted 14 June 2007 Available online 21 June 2007 Abstract The objective of this research is to develop an analytic method that uses data on both accident and safety indicators to quantify the aviation risk which are caused by human errors. A speciﬁed proportional hazard model considering the base- line hazard function as a quadratic spline function has investigated and demonstrated its applicability in aviation risk assessment. The use of the proposed model allows investigation of non-linear eﬀects of aviation safety factors and ﬂexible assessment of aviation risk. A subset of data gathered from the Fight Safety Management Information System (FSMIS) developed by the oﬃce of the Taiwan Civil Aeronautics Administration (CAA) was applied to accomplish this study. The results demonstrate that the proposed model is a more promising approach with the potential of becoming very useful in practice and leading to further generalization of aviation risk analysis.

Keywords:Risk assessment; Aviation safety; Human error; Proportional hazard model 1. Introduction As the worldwide air transportation traﬃc volume grows rapidly, safety in aviation becomes a burning problem over many countries today. Aviation accident may result in human injury or even death. It inﬂuences the reputation and the economy of the air transportation industry of a country. According to the analysis of Mineata (1997), when today’s accident rate is applied to the traﬃc forecast for 2015, the result would be the crashing of an airliner somewhere in the world almost every week.Braithwaite, Caves, and Faulkner (1998) stated that in order to achieve safety and reduce accident rate, we must quantify risk and balance it with appropriate safety measures.

In order to ensure the public safety and maintain a safe aviation environment, developing an analytic method that moves beyond the essential identiﬁcation of risk factors to assess the safety performance and dis- cover the potential hazards of airlines is indispensable.McFadden and Towell (1999)mentioned, while appre- ciating the value of accident investigations in identifying the cause and initiating corrective actions to prevent future errors, that a fundamental shift in the emphasis to ‘‘proactive safety’’ would be necessary. To achieve 0360-8352/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.cie.2007.06.032 *Tel.: +88 6226215656 2881.

E-mail address:[email protected] Available online at www.sciencedirect.com Computers & Industrial Engineering 54 (2008) 34–44 www.elsevier.com/locate/dsw ‘‘proactive safety’’, an idea risk assessment tool should be developed enabling an analyst to examine a wide variety of accidents quickly, systematically, and probabilistically and assisting a risk manager in priority set- ting and policy decision making. However, only few attempts have been made so far at how to analyze the aviation risk systematically and quantitatively.

Risk assessment is a structured science-based process to estimate the likelihood and severity of risk with attendant uncertainty (Coleman & Marks, 1999). The most obvious approach to study aviation risk focused on analyzing the accident data. For example,Janic (2000) and Lee (2006)treated the pattern of accidents as a Possion process to assess the probability of future events by using a sample of global accident records. This approach neglects the ordinary safety performance of the airlines, which may inﬂuence the aviation safety environment directly. Civil aviation is a complex mosaic of many varied, yet interrelated human, technical, environmental, and organizational factors that aﬀects safety and sys- tem performance. Aviation accidents result from multiple contributing factors.Logan (1999)mentioned that operational safety data such as aircraft reliability, ﬂight data records, employee safety reports, enforcement information, inspector investigations or oversight information were also essential to aviation risk analysis.

The Airline Safety Assessment System, currently under development by the Taiwan Civil Aeronautics Administration (CAA), will contain indicators of air carrier safety performance that can identify potential problem areas for inspectors. The objective of this research is to develop an analytic method that uses data on both accident and safety performance to quantify the aviation risk. Our approach takes into account the more complex relationships among relevant aviation risk contributing factors. In this study, risk involves a measure of probability of the occurrence of a hazardous event caused by human error. A speciﬁed propor- tional hazard model considering the baseline hazard function as a quadratic spline function has investigated and demonstrated its applicability in aviation risk assessment. A subset of data gathered from the Fight Safety Management Information System (FSMIS) developed by the oﬃce of the Taiwan Civil Aeronautics Admin- istration (CAA) was remodeled to accomplish this study. The results demonstrate that the proposed model is a more promising approach with the potential of becoming very useful in practice and leading to further gen- eralization of aviation risk analysis.

Statistics indicate that more than 70% of aviation accidents are related to human errors and 56% of world- wide hull lose accidents are caused by ﬂight crew errors (McFadden, 1993; Boeing Commercial Airplane Group, 2005). It has also been claimed that all accidents have some forms of human error attached to their causes (Braithwaite et al., 1998). Estimation of the human error related risk in a given time interval that a particular airline would be expected to have, upon adjusting for the airline’s corresponding safety performance indicators, could help to identify situations in need of heightened level of surveillance by the safety inspectors.

2. Research methodology There are two proposed approaches for assessing the aviation risk and safety: The ﬁrst one is looking at the number of accidents and fatalities continuously for oﬀering an indicator on the improvement of the sector’s safety. The second approach statistically models the occurrence of air accidents over time by assuming the accident events following Poisson process (Janic, 2000). Such a process is based on the following assumptions:

•An event can occur at random and at any time.

•The numbers of events, which occur in non-overlapping intervals are independent.

•The probability of an event occurring for a small intervalDtis proportional toDtand can be estimated by kDtwhere,kis the hazard or failure rate.

According to the assumptions, the time interval between any two consecutive events will follow an expo- nential distribution, which is a fundamental model in parametric survival analysis. The probability of the occurrence of at least one accident in timetcan be written as PðT6tÞ¼1 PðT>tÞ¼1 e kt ;ð1Þ H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–4435 whereTis the random variable representing the time between any two consecutive events andkis a constant.

If there are safety related factors upon which accident inter arrival time may depend, it becomes of interest to consider generalizations of the model to take account of the dependent information. The above model ignores the possible inﬂuence of safety factors to event inter arrival time.

Regression models for survival analysis have been extensively studied in the past 30 years. They allow the hazard rate to be a function of the observed explanatory variables (or covariates). Generally, regression mod- els can be generally categorized in two classes. The ﬁrst one is called the parametric statistical model, which assumes the shapes of time to event distributions are known.McFadden (2003)used logistic regression model to predict pilot-error accident and incident rates on an airline-by-airline basis. However, when the survival time data involve complex distributional shapes that are not well-known or when the number of observations is small making it diﬃcult to test, the second model type of survival analysis – semi-parametric or non-para- metric statistical model appears to be an attractive method to the parametric ones. The model is ‘‘distribution- free’’ since no assumptions need to be made about the shapes of time to event distributions. For example, Cox’s proportional hazards (PH) model (Cox, 1972) is one of the most famous semi-parametric or non-para- metric statistical models for time-to-event data with explanatory variables. It is widely applied in the medical ﬁeld. Recently, the model is also gaining acceptance in many sectors, including reliability engineering, trans- portation, and ﬁnance.

A log linear PH model is expressed as kðtjzÞ¼k 0ðtÞ e zb ð2Þ wherek(tjz) is the hazard rate at timetand covariate vectorz,k 0(t) which is the modiﬁed multiplicatively by covariates is referred to as the baseline hazard function, andbis the regression coeﬃcients vector. A PH model is a class of models with the property that diﬀerent individuals have hazard functions that are proportional to each other. That is, the ratiok(tjz 1)/k(tjz 2) of the hazard functions for two individuals with diﬀerent covariate vectorsz 1andz 2does not vary with time. In other words,k(tjz 1) is directly proportional tok(tjz 2).

There are two unknown components in Eq.(2): the vector of regression coeﬃcients and the baseline hazard functionk 0(t).Cox (1972)uses an attractive approach, in which a likelihood function that does not depend uponk 0(t) is obtained for. This function is referred to as a partial likelihood function and is expressed as LðbÞY n i¼1 exp bzi P l2Sðt iÞexp bzl ð3Þ where,nis the number of observed failure times andS(t i) is the risk set at timet i. This function can be max- imized to give an estimate ofbin the absence of any knowledge onk 0(t). The motivation for the likelihood function is that givenS(t) and given that a failure occurs att, the probability that the componenti(i 2S(t)) fails is kðtjz iÞ P l2SðtÞ kðtjz lÞ¼ k0ðtÞ exp bzi P l2SðtÞ k0ðtÞ exp bzl¼ exp bzi P l2SðtÞ exp bzl:ð4Þ Gill (1984)gave a discussion on how martingale approach could be used to give a ﬁrm mathematical basis to Cox proportional hazard model.

If we assumek 0(t)=k 0, (2) will reduce to an exponential regression model. It is a special case of the pro- portional hazard model where the base line hazard is speciﬁed by a single parameter. The conditional density function oftgivenzis fðt;zÞ¼k 0ezbe k0ezbt;ð5Þ and the conditional probability of the occurrence of at least one accident in timetfor covariatezis given by PðT6tÞ¼1 e Rt 0kðsjzÞds ¼1 e k0ezbt ð6Þ The maximum likelihood theory is used to evaluate the unknown parameters of the above models. Considern independent observations distributed according to (2). Lett ibe the observed event inter arrival time with the 36H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–44 corresponding covariate vectorz i=(z 1i,z 2i,...,z si) for theith observation. The natural logarithm of the like- lihood function for this model is given by lnLðk 0;bÞ¼lnY i k0ezibe k0ez ibti !

¼n lnðk 0ÞþX i zib k 0 X i ezibti ð7Þ This function can be maximized to give an estimate ofk 0andbby setting the ﬁrst derivative of lnL(k 0,b), with respect tok 0andb, equal to zero and by solving the resulting equations. Here, olnLðk 0;bÞ ok 0 ¼ n k 0 X i ezibti¼0ð8Þ olnLðk 0;bÞ ob j ¼X z jið1 e zibtiÞ¼0ð9Þ The standard likelihood approach outlined above does not adequately take the advantage of the particular structure of this model. However, for the speciﬁed proportional hazard model, it can determine all the un- knowns at once.

The Weibull distribution can be generated to the regression situation essentially in the same way, when a nonlinear expression for the baseline hazard rate function is used. The hazard rate function under this condi- tion is kðtjzÞ¼hcðtÞ c 1 ezb;ð10Þ where bothhandcare positive and are referred to as the scale parameter and the shape parameter of the dis- tribution, respectively. In our study, the baseline hazard functionk 0(t) is speciﬁed by a quadratic spline func- tion to estimate the unknown underlying distribution. We use the Heaviside function, where,U +=UifU=0 andU +=0 ifU< 0 to create a spline function. The formula is given by k 0ðtÞ¼X 2 n¼0 cntnþX l m¼1 hmðt s mÞ2 þ ð11Þ wherelis the number of knots,s mis the location of knots,h mis the added linear eﬀect following knots, andc n is the coeﬃcient of the underlying base polynomial. Splines are presented as a non-parametric function esti- mating technique (Wegmen & Wright, 1983). A spline function of degree m is a piecewisem-degree polynomial with pieces joining at deﬁned points, which are called ‘‘knots’’. A detailed discussion of spline functions is gi- ven in our previous paper (Shyur, Elsayed, & Luxhoj, 1999). To estimate the parameters of the spline function and the coeﬃcients of the covariates, a general likelihood function is used. Since the baseline hazard rates are always non-negative, we must ensure that the results of the estimation will satisfy the constraints. The details of the proposed model will be provided at a later point in this paper.

3. Data description A subset of data gathered from the Flight Safety Management Information System (FSMIS) developed by the Taiwan Civil Aeronautics Administration (CAA) oﬃce was remodeled to accomplish this study. The FSMIS is an analytical tool intended to support CAA inspection activities, which contains data related to the surveillance records of air operators, maintenance facilities, and manufacture of aircraft parts and acci- dents/incidents investigation reports.

Any member of a set of human actions that exceeds some limit of acceptability will cause human error (Latorella & Prabhu, 2000). The main goal of this research is to develop a model to provide relative risk prob- ability inference and trend analysis among diﬀerent kinds of human errors, which may cause any major avi- ation events. Here, a major event is deﬁned as a ﬂight event, which may lead any person to suﬀer death or serious injury, or the aircraft to receive substantial damage. The risk of this kind of event is much more essen- tial to be managed. For this reason, the analysis focuses only on the major event, but not all kinds of accidents and incidents. We analyze the aviation safety risk using a sample of 61 major accident records for the period H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–4437 from January 2003 to December 2004. The provided database contains a general cause category for each acci- dent/incident. The aviation accidents/incidents that were coded as (1) improper maintenance, (2) operator deﬁciency, (3) crew induced, (4) operation and maintenance, (5) inadequate maintenance, and (6) crew, ground crew, and ground handle system were analyzed in this paper. All the events are related to human error.

According to the accident/incident records, the time between every two consecutive events for each cause cat- egory and airline can be simply calculated.

Based on the FSMIS database, numerous safety performance indicators for signaling the potential problem areas considered for inspection are currently deﬁned by CAA (Shyur, 2006). These indicators assist in diag- nosing an airline’s ‘‘proﬁle’’ compared with others in the same peer class and provide insights as to whether an airline is more or less likely to undertake unsafe practices. The airline safety performances inﬂuence the whole aviation safety environment directly and assist in diagnosing the proﬁle of an airline. So the safety per- formances are explored as the eﬀecting factors or explanatory factors of the aviation risk. Three integrated major corresponding performance indicators are considered in this study – airworthiness surveillance (AS), operations surveillance (OS), and frequency of general events (FE). Surveillance is one of the most signiﬁcant duties of the CAA oﬃce in its larger responsibility of assuring air transportation safety. According to the safety report published by FAA (Federal Aviation Administration, 1997), the information on factors that could aﬀect airline safety practices can be found in the inspection and surveillance reports on air carrier operations.

The CAA monitors the airworthiness-related activities performed within an air carrier using various sur- veillance techniques. The airworthiness surveillance activities conducted are based on the risks associated with the scope and depth of airworthiness authority assigned to the airline, and are performed against the CAA approved airworthiness process manual of that carrier. The CAA is also responsible for monitoring all phases of air carrier operations including: training programs and records; base and station facilities; airports and route systems. One of the limitations of the FSMIS surveillance database is that it does not use a quantitative way to measure the surveillance result.

The recorded surveillance report in FSMIS contains a result category for each check item, for instance,S represents that the check item has satisﬁed the certain requirements andFmeans there are some ﬁndings in this item, etc. For quantifying the surveillance result, each code was assigned a weighted score according to its order of severity. A ratio scale approach, the Analytic Hierarch Process (AHP), was conducted to make the decision. AHP was proposed by Saaty as a method of solving socio-economic decision making problems and has found its widest applications in multi-criteria decision making (Satty, 1980). Using the weighted scores, the surveillance indicators were measured by the percentage of unfavorable surveillance records asso- ciated with a given smoothed time period. The exact calculation formula is Unfavorable rate¼ PN i¼1ðw i niÞ W Nð12Þ where W: predeﬁned maximum weighted score, N: total number of inspections during timeT, w i: weighted score for the resultiof individual inspection item,iis the result code, n i: resultiof individual inspection item during timeT.

Table 1contains a sample of the summary of input data from FSMIS for air carrier A. For data security, only 10 records are shown.

4. Application of the methodology Our approach takes into account the more complex relationships among relevant aviation risk factors.

Using the models presented, risk has been assessed as the probability of occurrence of a speciﬁc type of human error related aviation accident. The potential human error related risk could be identiﬁed and monitored timely. The results can provide better references to the civil aviation communities to manage the aviation- safety risk, thus corrective action can be taken to reduce the occurrence of aviation accidents. 38H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–44 4.1. Model development Considernindependent categorykevents distributed according to (2). LetT k irefers to the time between the ith and (i 1)th events. The covariatesðz 1 i;z2 i;z3 iÞrepresent the unfavorable rates of AS, OS, and FE measured in theith time interval. Since the time interval is not a constant, the event frequency is normalized by thou- sands of ﬂight landings. The likelihood for this model is given by L k¼Y n i¼1 fðT k i;ziÞ¼Y n i¼1 k0ðT k iÞezi bkexp e zibkZ Tk i 0 k0ðtÞdt !

ð13Þ where,z i b k¼z 1 ibk 1þz 2 ibk 2þz 3 ibk 3, fori=1–n. The log-likelihood function provides more ﬂexibility in the parameter estimation for the spline function,k 0(t), in the extreme tails and in estimating the coeﬃcients of the covariates. The natural logarithm of the likelihood function used in this paper is l¼lnL¼X n i¼1 ðzi b kþlnk 0ðT k iÞ e zi bkZ Tk i 0 k0ðtÞdtÞ:ð14Þ Because the baseline hazard rates are always nonnegative, we must make sure that the resulting estimate will satisfy this constraint. To approximate the maximum value of the log-likelihood, the Generalized Reduced Gradient (GRG) algorithm (Lasdon, Waren, Jain, & Ratner, 1978) has been applied to obtain the optimized solution. The GRG solves a sequence of reduced problems by a gradient method to prevent a more complex searching problem. The algorithm has been shown to be eﬃcient and reliable when solving small to moderate nonlinear programming problems. To estimate the spline function,Etezadi-Amoli and Ciampi (1987)suggest starting with zero knots and constant hazard. The number of knots is increased, adding one knot at a time, until no improvement in the ﬁt is obtained.

To estimate the baseline spline functions,k 0(t), models with a diﬀerent number of knots are created. Results show that one knot spline functions can provide good approximations of the hazard functions for all analyzed data sets. The created models related to air carrier A are shown inTable 2. Three covariates are introduced to 2 sets of accident/incident data for air carrier ‘‘A’’ since this carrier contains only two types of major events in the analyzed time period. The standard errors (SE) of the estimates of the proposed model coeﬃcients are pro- vided in the parentheses. We use the information matrix only to determine standard errors for regression parameters. In this case, we found that the log-likelihood is quite insensitive to the changes in the position of the knots. The situation will be reﬂected by ‘‘ridges’’ in the likelihood surface and a nearly singular Hessian matrix (Ciampi & Etezadi-Amoli, 1984). Therefore, we kept the position of the knots ﬁxed when calculating the information matrix ofb 1,b 2, andb 3. Table 1 Sample FSMIS ‘‘Organized’’ data Air carrier A Major event causeImproper maintenance Event no. Time elapsed before next event (days)Airworthiness performanceOperation performanceFrequency of general accidents per thousand landings 1 196 0.01556 0.00203 2.116995 2 10 0.01613 0.00358 1.770303 3 68 0.01284 0.0054 2.971014 4 78 0.00756 0.00117 1.461988 5 311 0.01735 0.00245 2.705822 6 3 0.00649 0 0.605938 7 14 0.00629 0.00399 0.807918 8 28 0.01055 0.0012 0.420698 9 29 0.02442 0.00374 0.386453 10 20 0.01144 0.00118 0.420062H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–4439 According to the results, it appears that the estimated hazards increase as the unfavorable rate of airwor- thiness surveillance and operation surveillance increase, with coeﬃcient estimates ofb 1= 0.0044 and b 2= 0.0014 for the improper maintenance induced event, andb 1= 4.4477 andb 2= 0.0096 for the crew induced event. In this case, it can also be noted that the estimated eﬀect of airworthiness performance is stron- ger than the operation performance sinceb 1>b 2. Moreover, it can assess the probability of an occurrence of any kind of major event by providing the values of air carrier performances (covariates). The corresponding mathematical function of risk assessment is Risk¼PðT6tÞ¼1 e Rt 0kðsjzÞds ¼1 e ezbRt 0k0ðsÞds ð15Þ Fig. 1illustrates the probabilities of the occurrence of at least one major event within the time periodtfor an air carrier ‘‘A’’ when the same safety performances existing in the current will continue to exist in the future. In this case, we set AS = 0.01144, OS = 0.00118, and FE = 0.420062. According to the risk assessment function, the probabilities rise over time until the next event. For example, the probability of at least one major event caused by improper maintenance is about 0.372 by 10 days, 0.654 by 30 days, and 0.752 by 60 days. The prob- abilistic risk inferred by the models can be further used to facilitate the organization to identify relative haz- ardous human error. For example, asFig. 1indicates, the extent of risk can be compared in the order:

improper maintenance > crew induced > others. Therefore, the inspection plan or prevention action should be taken to strengthen the routine monitoring for the two types of human error risk especially for the impro- per maintenance problem in order to prevent the occurrence of the corresponding events. Table 2 Results of proposed risk assessment model for air carrier ‘‘A’’ Parameters Major event cause Improper maintenance induced Crew induced c 0 0.049868021 0.0067334 c 1 0.000260578 0.0000094 c 2 0.000092883*0 h 1 0.000102162*0 s 1 5.38 15.35 b 1(SE) 0.00440092(0.0028) 20.703049(0.76) b 2(SE) 0.00139579(0.00035) 0.000013(0.0000087) b 3(SE) 0.00973966(0.00065)*0 Log-likelihood 48.54 29.43 0 0.2 0.4 0.6 0.81 The time until the main event (days) Probability of one main event within the time t Improper Maintenance Crew Induced 0 102030405060708090 Fig. 1. Risk assessment for air carrier A. 40H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–44 In our developed model, the covariates are measured as unfavorable inspection rate and the frequency of general event, so the risk increases if the performance of the related eﬀecting operations is not right. On the contrary, if there is improvement in performance, then the aviation risk can be reduced. Brieﬂy, the results of the relationship of the covariates and the human error induced risk can direct the inspectors to focus on more related or important aviation operations inspection in order to control the hazardous risk in the future.

4.2. Comparing models Although we wish to summarize our risk assessment with a single model, there are usually other choices to be made.Kullback and Leibler (1951)addressed such issues and developed a measure, the Kullback–Leibler information, to represent the information loss when approximating reality.Akaike (1974)proposed using Kullback–Leibler information for model selection. He established a relationship between the maximum like- lihood and the Kullback–Leibler information. In essence, he developed an information criterion to estimate the Kullback–Leibler information, Akaike’s information criterion (AIC), which is deﬁned as AIC¼ 2ðlog likelihoodÞþ2kð16Þ where,kis the number of estimated parameters included in the model. For a given data set, the log-likelihood of the model reﬂects the overall ﬁt of the model. The AIC penalizes for the addition of parameters, and thus selects a model that ﬁts well but with a minimum number of parameters. The model with the lowest AIC being the best model among all models speciﬁed for the data at hand, when it is compared to the AIC of a series of models speciﬁed a priori.

In this paper, two semi-parametric methods for estimating the function form of aviation safety risk assess- ment have been investigated to compare with the proposed approach. One is the exponential regression and the other is the Weibull regression model. Using the identical data sets, the maximum likelihood approach was also conducted to evaluate the unknown parameters for the alternative models.Table 3shows the estimation results. The AIC statistics are presented inTable 4.

In general, the proposed models tend to provide better ﬁt in both the given data sets with the advantage of being ‘‘distribution-free’’. Comparing all the alternative models including the Poisson process technique, a graphical analysis of the risk estimation curves for each of the models under two diﬀerent levels of safety per- formance for air carrier ‘‘A’’ was performed. The risk assessment models consider safety performance indica- tors as the contributing factors of aviation events.Figs. 2–5show that the proposed model better represents the eﬀects of safety performances on aviation risk no matter what type of human error related risk has been Table 3 Results of semi-parametric models for air carrier ‘‘A’’ Parameters Improper maintenance induced Crew induced Exponential regression Weibull regression Exponential regression Weibull regression Scale parameter 0.01321 0.033877 0.007153 0.009374 Shape parameter N/A 0.805614 N/A 0.941930 b 1 0 0 23.442 24.40633 b 2 0.00024 0.131236 0.013409 0.452157 b 3 0 0 0 0 Log-likelihood 53.26 52.79 33.46 33.44 Table 4 Akaike’s information criterion of the hazard regression models AIC Proposed model Exponential regression Weibull regression Improper maintenance induced 113.08 114.52 115.58 Crew induced 74.86 74.92 76.88H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–4441 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 0 10 203040 506070 8090 Time Risk Proposed model Poisson process Poisson process Exponential Regression Weibull Regression Fig. 2. Risk assessment for improper maintenance induced event with low AS and OS (AS = 0.01144, OS = 0.00118, and FE = 0.420062). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 0 102030405060708090 Time Risk Proposed model Poisson process Exponential Regression Weibull Regression Fig. 3. Risk assessment for improper maintenance induced event with high AS and OS (AS = 0.5, OS = 0.5, and FE = 0.420062). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 0 10203040 5060708090 Time Risk Proposed model Poisson process Exponential Regression Weibull Regression Fig. 4. Risk assessment for crew induced event with low AS and OS (AS = 0.01144, OS = 0.00118, and FE = 0.420062). 42H.-J. Shyur / Computers & Industrial Engineering 54 (2008) 34–44 studied. Similar results were also appeared when we studied the accident and safety data of the other ﬁve air carriers.

5. Conclusions In the past, only accident or fatality data were investigated and used to measure the risk or/and safety level of airlines. This is just a reactive way to manage the aviation risk. However, commercial aviation is a complex mosaic of many varied, yet interrelated human, technical, environmental, and organizational factors that aﬀect safety and system performance. The possible inﬂuencing factors should be included while assessing risk.

The application of the hazard regression models to the analysis of aviation risk has not been previously reported in the research literature.

This paper proposed a new quantitative methodology for the assessment of risk in civil aviation. The spline function is used to present the baseline hazard function. Our proposed approach allows ﬁnding fundamental cause of human error related accidents through the analysis of operational safety data. The modiﬁed propor- tional hazards model takes into account the relationships among relevant aviation risk factors. Furthermore, the dependence of the aviation risk on operational performance of airlines can also be measured. Finally, the results of the case study demonstrate that the proposed model is a more promising regression model with the potential of becoming very useful in practice.

Acknowledgements The author would like to thank the Taiwan CAA for their assistance in problem formulation and data collection.

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