Reliability evaluation for airline network design in response to
#uctuation in passenger demand
Chaug-Ing Hsu
∗, Yuh-Horng Wen
Department of Transportation Technology and Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 30050, ROC
Received 1 October 2000; accepted 22 February 2002
Abstract
This study develops a reliability evaluation method for airline network design. Reliability evaluation in the airline network design phase is considered herein as a post-design task. A series of models for determining #ight frequencies on individual routes, evaluating reliability under normal=abnormal demand #uctuations, and providing a priori adjustment of#ight frequencies are presented. A case study demonstrating the feasibility of applying the proposed models is provided. Study results not only suggest a post evaluation and adjustment method for airline network design in response to future uncertain #uctuation in passenger demand, but also provide ways to improve #exibility in airline #ight frequency decision-making. ? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Reliability evaluation; Airline network design; Passenger demand #uctuations; Flight frequency adjustment
1. Introduction
In the airline network-planning phase, the network design problem is generally stated as one ofdeveloping a design for a single airline’s network and routing policies that sat-is8es origin–destination (OD) passenger demand and mini-mizes total airline=passenger transportation costs [1,2]. The extreme complexity ofdesigning an airline network is due largely to the need for transportation facilities to adhere to passenger demands, and airlines’ #ight schedules must treat demand #uctuations. The extent to which the economic cy-cle in#uences air transportation demand is quite apparent [3,4]. Season=o>-season #uctuations occur and unexpected abnormal #uctuations in#uence future air passenger tra?c. Furthermore, the uncertainty surrounding input parameters complicates airline network design; uncertainty arises from the practice of forecasting only approximate future passen-ger demands between city pairs during network planning phases. Airline network design is a prerequisite for airlines’ ∗Corresponding author. Tel.: +886-3-573-1672; fax:
+886-3-572-0844.
E-mail address: [email protected] (C.-I. Hsu).
medium-run operational planning, such as #ight scheduling and routing. However, airline #ight scheduling and routing policies usually continue and repeat daily over long periods (one or two seasons) to simplify #ight operations and en-hance customer familiarity. Thus, in airline network design phases, planners merely use average estimated OD demand patterns to determine average monthly=weekly=daily #ight frequencies for #ights over one or two seasons, or over in-dividual years covering seasonal peak and o>-peak periods, and then use these frequencies as bases for future opera-tional planning. However, a #exible airline network design that could better respond to future tra?c #uctuations would be more appropriate for operational planning.
Previous studies focused mainly on network modeling and hub-location problems in hub-and-spoke airline networks (e.g., [5–9,2,10–12]). The models proposed in these pa-pers concerned location-allocation p-hub median problems. Other studies on network models for air transportation have addressed the #eet assignment problem (e.g., [13–15]) and crew scheduling problem (e.g., [16–18]). Most ofthis re-search developed deterministic integer programming models and addressed model improvements and algorithms to solve airline #eet assignment and crew scheduling problems.
0305-0483/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0305-0483(02)00027-0
Pertinent literature on airline network design problems focused largely on airline network shape, #ight frequency determination or aircraft choice (e.g., [19–22,1,4]). The au-thors considered airline network design problems involving single airlines from long-run perspectives and constructed problems using mathematical programming. Though such airline network designs can be seen as bases for short-run airlines’ operational planning, the performance results of network designs, apart from short-run demand #uctuations, were generally not evaluated.
Reliability engineering is a well-established area ofengi-neering, and reliability engineering theory has been widely applied to electronic engineering, software reliability, me-chanical reliability, human reliability, power system reli-ability, maintainability engineering, life-cycle costing, etc. [23,24]. However, previous transportation planning models paid little attention to transportation system reliability [25]. Past studies oftransportation network reliability focused largely on assessing the reliability ofroad networks under disaster e>ects or measuring the connectivity and performa-bility in urban transportation networks (e.g., [26–29]). The models proposed in these studies considered unreliability arising from variations in link #ows and variations in link ca-pacities and developed their reliability evaluations, focused on node or link=OD-pair connectivity and travel time ac-ceptability (e.g., [27–29]). However, in the 8eld ofair trans-portation the concept ofreliability engineering is currently applied to airline #eet maintenance planning issues or air tra?c control issues (e.g., [30,31]).
Taking another approach, several studies have used stochastic programming to formulate optimization problems that involves uncertain input parameters (e.g., [32,33]). Chance-constrained (probabilistic constrained) stochastic programming provides a means ofconsidering the con-straints with random parameters [34]. The programming model is well-known and involves 8xing certain reliabil-ity levels for random constraints. This paper presents a two-stage process to tackle the randomness ofinput data for designing an airline network in the airline network-planning phase. Deterministic airline network program is 8rst used to determine average monthly route #ight frequencies, and then the reliability ofthe proposed #ight frequencies is evaluated under #uctuating monthly OD demand, using the chance-constrained formulation. In the present paper, we attempt to devise a reliability evaluation method for assessing how well the results ofan airline network design will work under future potential short-run normal=abnormal tra?c #uctuations; and then propose a 8ne-tuning method for redesigning the network in response to short-run de-mand #uctuations. We evaluate the reliability ofan airline network design by assessing whether its proposed route #ight frequencies can e>ectively maintain cost economies and service quality under the pressure offuture short-run #uctuations in OD passenger demand. Therefore, the reli-ability evaluation in the airline network design problem is de8ned in this paper as post-design evaluation.
In this paper, we de8ne the reliability as the probability that initially proposed #ight frequencies will operate e>ec-tively under future short-run tra?c #uctuations in OD pas-senger demand. The reliability ofproposed #ight frequen-cies for some OD pairs in certain potentially abnormal sit-uations, such as when abrupt #ow shrinkage occurs due to war or economic crisis, or when #ow expansion occurs due to special exhibitions or fairs, are also discussed in this pa-per. We analyze the occurrence probabilities ofvarious ab-normal states and the durations ofthese abab-normal states, and then estimate the reliability of #ight performance for OD pairs that experience these abnormal situations. Further-more, in responding to such potential tra?c #uctuations, we also provide an a priori adjustment method for 8ne-tuning #ight frequencies proposed in the initial airline network de-sign phase. The costs associated with #ight frequency ad-justment, e.g., additional #ight and crew dispatch costs, are also formulated for comparison with expected losses in air-line revenues and=or increases in passenger travel times and costs so as to assess whether it is necessary to perform the adjustments. Instead ofreconstructing the whole network, this dynamic adjustment procedure merely re8nes #ight fre-quencies for some local routes=OD-pairs sustaining severe #uctuations while maintaining overall global network de-sign objectives.
The basic input data when designing an airline network are the estimated passenger tra?c between individual OD pairs. Our earlier work [35,4] forecasted passenger traf-8c, designed airline network con8gurations and proposed #ight frequencies by applying grey theory and multiobjec-tive programming. Grey forecasting models (GM) [36–39] are also applied here to forecast OD demand patterns. The grey model GM(h; N) is de8ned as a linear di>erential equa-tion [39], where h stands for the hth order derivative of grey accumulated generating operation series (AGO-series) ofdependent variables, and N stands for N variables in the model’s di>erential equation. By de8nition, GM(1,1) is a time-series forecasting model and GM(1; N) is a polyfactor system forecasting model [37,39]. Hsu and Wen [35] pre-sented grey time-series models, GM(1,1), for forecasting total passenger tra?c and country-pair passenger tra?c in the Trans-Paci8c market; Hsu and Wen [4] then used these models to forecast route #ows, which were then used as in-put parameters for designing an airline network. The present paper uses the GM(1; N) system forecasting model to con-struct a polyfactor model for OD pair-#ow forecasting. The grey system forecasting model in this paper could be used to examine the e>ects ofsocioeconomic variables on fu-ture passenger demand for di>erent OD pairs in an airline network. Such a tra?c forecasting model incorporates the e>ects ofuncertain socioeconomic variables, thereby fully accounting for the dynamic aspects of demand changes. Ap-pendix A summarizes grey systematic models for forecast-ing OD-pair tra?c.
The rest ofthis paper is organized as follows: Section 2 de-scribes the airline network design problem, and dede-scribes the
framework for evaluating the reliability of proposed #ight frequencies for OD pairs. Section 3 provides a 8ne-tuning method for adjusting the #ight frequencies in response to tra?c #uctuations. A case study is provided in Section 4 to illustrate the application ofthe models. Section 5 presents concluding remarks.
2. Description and formulation of the problems
The airline network design problem in this paper is de-8ned as follows: Given an origin–destination passenger-#ow demand matrix and the capacities and operating costs ofvari-ous types ofaircraft, design an airline network and determine #ight frequencies that satisfy demands and minimize total transportation costs and passenger travel costs [1,2]. Based on the above de8nition, the airline network design problem is typically formulated as a deterministic programming prob-lem that involves a given OD passenger tra?c matrix in the airline network-planning phase (e.g., [20,7,9,2,4]). The pas-senger tra?c #ows between OD pairs are input parameters for the airline network design model. However, the num-ber ofpassengers who travel between an OD pair during a speci8c period #uctuates from time to time; the #uctuations are noted to be monthly, weekly, or even daily. Teodorovic [40] stated that, “it is important to monitor passenger traf-8c from month to month and corresponding changes made in #ight frequencies”. In the present paper, all the quanti-ties apply for 1 month. If the monthly passenger tra?c be-tween OD pairs are considered to be random parameters in airline network design programming, then the random pa-rameters may turn the programming problem into a stochas-tic programming problem. A reliability evaluation approach based on a chance-constrained formulation is designed to tackle stochastic characteristics ofthe input parameter in air-line network design programming. Restated, a deterministic programming problem is formulated subject to constraints that satisfy the expected value of random passenger tra?c on each OD pair (i.e., the average monthly OD #ows over the planning year), to determine the #ight frequency. Next, variations in the random passenger tra?c are analyzed and their e>ects on the reliability ofthe proposed monthly #ight frequency on each OD pair are evaluated. This reliability evaluation method e>ectively uses a chance-constrained for-mulation ofthe problem. Chance-constrained forfor-mulations allow the solution ofbasically deterministic models with penalties for constraints going outside speci8ed limits, and avoid the complexities ofactually modeling what happens when these limits are hit.
2.1. Airline network design programming
The modeling ofthe airline network design programming problem herein, follows the formulation of Hsu and Wen [4]. Consider an airline network G(N; A), where N and A represent the set ofnodes and the set oflinks, respectively,
in graph G. Let R (R ⊆ N) denote the set oforigin cities, and S represent the set ofdestination cities (S ⊆ N), where R ∩ S = ∅. Any given OD pair r–s is connected by a set
ofroutes Prs(r ∈ R; s ∈ S) through the network. An airline
serving international routes typically utilizes several aircraft ofvarious sizes. Accordingly, the main decision variables ofthe airline network modeling are assumed to be monthly #ight frequencies on individual routes served by various types of aircraft in the airline network [4]. The following notation is used.
Nrspq: monthly #ight frequency served by type q aircraft
#ying between OD pair r–s along route p (p ∈ Prs),
Yaq: monthly #ight frequency served by type q aircraft on
link a (a ∈ A),
frsp: monthly number ofpassengers who travel between
OD pair r–s along route p,
fa: monthly number ofpassengers carried through link
a,
Nfrs: expected monthly number ofpassengers who travel
between OD pair r–s in the planning year, r;s
a;p: indicator variable:
r;s a;p=
1; iflink a is part ofpath p between OD pair r–s;
0; otherwise: r;s
a;p;q: indicator variable:
r;s a;p;q=
1; iflink a is part ofroute p served by type q aircraft
between OD pair r–s; 0; otherwise:
The link #ow is the sum ofthe #ows on all routes going through that link and can be expressed as a function of the route #ows. On the other hand, the link #ight frequency is also the sum of#ight frequencies #own on all routes through that link. Those are, respectively,
fa= r;s p r;s a;pfrsp; (1) Yaq= r;s p r;sa;p;qNrspq: (2)
In airline network modeling, two-way OD tra?c #ows are assumed to be symmetric, an assumption commonly made in practice by most airlines when designing their networks. A similar assumption is also made in related papers, such as, Jaillet et al. [2], and Hsu and Wen [4]. Furthermore, the following condition must then be satis8ed such that the sum ofall passengers on individual routes between OD pair r– s equals the total number ofpassengers traveling between
OD pair r–s in a month:p frsp= Nfrs; and Nfrscan be
es-timated from the average monthly forecasted OD #ow over a future planning year. From a planning perspective, the
where nqis the number ofavailable seats on type q aircraft,
lais the speci8ed load factor for link a. The decision-maker
may specify a pro8table load factor la, when designing the
airline network.
Airline operating costs are normally divided into direct operating cost and indirect operating costs. Direct operating costs are all those expenses associated with the type ofoper-ated aircraft, including all #ying costs, all maintenance, and all aircraft depreciation expenses. Indirect operating costs are those expenses related to passengers rather than related
to aircraft. Let CTA
a denote the total airline operating costs
for link a, such as,
CTA a (Yaq) = q CD aq(Yaq) + CaI(Yaq); (3) where CD
aqis the direct operating cost oftype q aircraft for
#ights over link a with stage length da, in US dollars, and CaI
is the total indirect operating cost for link a, in US dollars.
In details, the formulations of cost functions CD
aq(Yaq) and CI a(Yaq) are, respectively, CD aq(Yaq) = (aq+ aqda)Yaq; (4) CI a(Yaq) = ch q nqlaYaq; (5)
where da is the stage length oflink a in miles, aq; aq are
parameters speci8c to type q aircraft, chis the unit handling
cost per passenger in US dollars.
Passenger travel costs are divided into passenger line-haul travel cost and schedule delay cost [41,4]. The total
passen-ger travel cost on link a; CTP
a , is,
CTP
a (Yaq; Nrspq) = CTa(Yaq) + CaS(Nrspq); (6)
where CT
a is the total passenger line-haul travel cost and CaS
is the passenger schedule delay cost for link a. The cost function representing total passenger line-haul travel cost on link a can be expressed as
CaT(Yaq) = ct(r + da+ a)fa
= ct(r + da+ a)
q
nqlaYaq; (7)
where r and are travel-time-function parameters, ctis a unit
time-cost transformation re#ecting the perceived monetary
cost ofline-haul travel time, ais the airport time on link
a. And, the cost function representing passenger schedule delay cost on link a can be expressed as
CaS(Nrspq) = cd r;s p r;sa;p NT qNrspqfrsp ; (8)
where NT is the average airport operating time over a month,
and NT=q Nrspqis the average headway on route p; cdis a
unit time-cost transformation re#ecting the perceived mon-etary cost ofschedule delay time, is a multiplier a>ected by #ight scheduling, and is proved by Teodorovic [20], Teodorovic and Krcmar-Nozic [22] to equal 1=4.
The airline network design programming (P1) can now be formulated as follows. Pl : min Yaq;Nrspq a∈A CTA a (Yaq)CaTP(Yaq; Nrspq) = a q (aq+ aqda)Yaq+ ch q nqlaYaq + ct(r + da+ a) q nqlaYaq + cd r;s p r;sa;p NT q Nrspqfrsp (9a) s:t: q nqlaYaq− r;s p r;sa;pfrsp¿ 0 ∀a ∈ A; (9b) p frsp= Nfrs p ∈ Prs ∀(r; s); (9c) Yaq= r;s p r;sa;p;qNrspq ∀a ∈ A; (9d) a taqYaq6 uqUq ∀q; (9e) Yaq; Nrspq¿ 0 and integer; frsp¿ 0: (9f)
Eq. (9a) is an objective function that minimizes total airline operating costs and total passenger travel costs. Eq. (9b) in-dicates that the transportation capacities o>ered in terms of numbers ofseats for each link must be equal to or greater than the numbers ofpassengers on all routes that include that link. Eq. (9c) indicates that the sum ofthe passengers on any route p between OD pair r–s must equal the total num-ber ofpassengers traveling between the OD pair in a month. Eq. (9d) expresses the relationship between link frequency and route frequency. Eq. (9e) suggests that total aircraft uti-lization must be equal to or less than the maximum possible
utilization, where taqis the block time for type q aircraft on
link a; uqis the maximum possible monthly utilization, and
Uqis the total number oftype q aircraft in the #eet. Finally,
Eq. (9f) constrains variables Yaqand Nrspqto being
nonneg-ative integers, and also constrains frspto being nonnegative.
Our earlier work [4] formulated the airline network programming model as a two-objective nonlinear program-ming problem that minimized total airline operating cost and/or minimized total passenger-travel cost; the trade-o>s between these two objectives were also discussed. The present study focuses on the reliability evaluation of air-line network, and to simplify it, we use single-objective form to formulate airline network modeling. Furthermore, according to the results ofHsu and Wen [4], a com-promise (Pareto-optimal) solution determined from the two-objective programming problem is equivalent to an optimal single-objective programming solution that min-imizes total airline operating cost and total passenger
travel cost. Therefore, single-objective programming im-plies that a compromise solution acceptable to airlines and passengers can be determined. However, nonlinear integer programming problems are extremely di?cult to solve, particularly those with large dimensions. A nonlinear programming (NLP) rounding relaxation method is used to approximate the solution ofinteger programming due to di?culties in obtaining exact solutions to such large combinatorial problems. A similar relaxation approach can be found in several related papers, e.g., Teodorovic and Krcmar-Nozic [22], Teodorovic et al. [1], Hsu and Wen [4].
Let {N∗
rspq} be the optimal solution to the NLP-relaxation
problem. The initial rounded solution to the nonlinear mixed integer programming problem is then de8ned as,
N
Nrspq= Nrspq∗ , ∀r; s; p; q, and NYaq=r;sp r;sa;p;qNNrspq,
∀a; q. NYaq must satisfy constraints (9b) and (9e),
such that q nqlaNYaq − r;s
p r;sa;pfrsp¿ 0; ∀a and
a taqNYaq6 uqUq; ∀q to ensure that NNrspqand NYaqlead to
feasible solutions to the original problem.
2.2. Monthly 5uctuations in OD passenger demand The input data ofairline network design are the forecasted passenger tra?c #ows. We 8rst forecast annual OD-pair tra?c for a future year, then transform this forecast into av-erage monthly tra?c for the future year and use the result as input data for airline network design. However, the un-certainty surrounding socioeconomic variables that a>ect air tra?c #ows exists and the numbers ofavailable data for ob-served tra?c are generally not large due to short accumula-tion times, particularly those concerning city-pairs [42,35,4]. Therefore, in this study, we apply grey systematic models that encompass groups ofdi>erential equations adapted for parameter variance to build OD-pair demand models. Ap-pendix A presents formulations of grey systematic models
used to forecast OD-pair tra?c. Let ˆFrsdenote the annually
forecasted passenger tra?c between OD pair r–s, obtained by grey systematic forecasting models (Eqs. (A.1)–(A.5)).
ˆFrsis then divided by 12 (months) to transform it into
aver-age monthly tra?c, i.e. Nfrs= ˆFrs=12.
The monthly #uctuations in OD passenger demand are further analyzed. Suppose that the monthly OD #ows are random variables. That is, during the planning year, the num-ber ofpassengers who travel between a certain OD pair each month is a random variable. Twelve random variables rep-resent monthly #uctuations in OD passenger #ows between a speci8c OD pair, during the planning year. The following notation is also used.
˜ft
rs: random variable that represents the number
ofpas-sengers who travel between OD pair r–s in month t, ˜ t
rs: random variable that represents the ratios ofrandom
tra?c values in month t to the average monthly
val-ues, such that ˜ t
rs= ˜frst= Nfrs,
t: superscript indicator, where t =1; 2; : : : ; 12, denoting 12 months during the planning year,
I: set of12 months in the planning year, such that I ≡
{1; 2; : : : ; 12},
N t
rs: sample mean of˜ trs,
!( ˜ t
rs): sample standard deviation of˜ trs,
Nft
rs: sample mean of ˜f
t rs,
!( ˜frst): sample standard deviation of ˜frst, and
ft
rs: a random realization of ˜frst.
For a certain OD pair r–s in month t (t ∈ I), a probability space "t
rsof frst is de8ned, where frstrepresents a realization
ofrandom monthly passenger tra?c between OD pair r–s
in month t. A random variable ˜frst, and a probability
dis-tribution of ˜frst are also de8ned. In estimating the
distribu-tion ofrandom variable ˜frst, suppose that the pattern
ofnor-mal monthly demand #uctuations on individual OD pairs is similar over all surveyed years. Historical monthly OD-pair tra?c data was 8rst used during surveyed years as sampling data. For each OD pair r–s, in all surveyed years, the ratios oftra?c values in each month to the average monthly
traf-8c values were obtained. Let ˜ t
rs= ˜frst= Nfrs; ∀t ∈ I, represent
the ratios ofrandom tra?c values in month t ( ˜frst) to the
average monthly tra?c values over the planning year ( Nfrs),
and let ˜ t
rsbe a random variable. That is, ˜ trsrepresents the
potential tra?c #uctuation for OD pair r–s in month t with respect to its mean value over all surveyed years. Twelve
random variables ˜ t
rs, ∀t ∈ I, exist for each OD pair r–s. For
simplicity, these twelve random variables, ˜ t
rs, ∀t ∈ I, are
assumed to be mutually independent. The random variable ˜ t
rsis supposed approximately to follow a normal
distribu-tion with parameters N t
rsand !( ˜ trs). A similar assumption
ofnormal distribution, used to treat #uctuations ofair trans-portation demand, can be found in Swan [19] and Powell [43]. If ˜frst= Nfrs˜ t
rs, then the random variable ˜frst is normally
distributed with parameters, Nfrst and !( ˜frst), where Nfrst= NfrsN t rs
and !( ˜frst) = Nfrs!( N t rs).
Notably, Nfrsequals the average value of Nfrst, ∀t ∈ I, such
that Nfrs=12t=1 Nfrst=12=12t=1 NfrsN t
rs=12, where12t=1 N trs=12=
1 since Nfrs, is the expected value ofmonthly OD tra?c over
the planning year.
2.3. Reliability of proposed 5ight frequencies under OD demand 5uctuations
In reliability engineering, reliability is generally de8ned as the probability that an item will perform its function adequately for the desired period of time when operated according to speci8ed conditions [24]. In this paper it is assumed that proposed monthly #ight frequencies
result-ing from airline network design, NNrspq, associated with
the average monthly tra?c forecasts, Nfrs, are initially
reli-able. In the aforementioned airline network design model, the proposed monthly #ight frequencies must be 8xed in advance, and then the model applied under monthly #uc-tuating demand. The unreliability problem ofthe airline network design phase, arises from the condition that the
proposed monthly #ight frequencies cannot match short-run passenger demand due to seasonal variations and/or abnor-mal variations that fall outside prior acceptable limits. A short-run shrinkage in OD demand results in excess supply and increased costs for airlines due to low OD load factor, although such excess supply enhances high service quality with low delay times and fares for passengers. On the other hand, a short-run expansion in demand causes excessive loading and downgrades service quality through high delay times/costs for passengers, although such excess demand brings cost economies through high or full loads for air-lines. Thus, the results ofairline network design, i.e., the proposed monthly route #ight frequencies produces relia-bility for airlines and passengers only when the tra?c de-mands #uctuate within ranges that allow #ight frequencies to maintain cost economies and/or service levels.
We assume herein that the load factor on monthly #ights between individual OD pairs is the basic criterion for eval-uating the reliability ofproposed monthly #ight frequen-cies between OD pairs over OD demand #uctuations. The monthly OD load factor on #ights between any OD pair r–s
with respect to random OD passenger #ows ˜frst in month
t; lrs( ˜frst), is de8ned as lrs( ˜frst) = ˜f t rs p q nqNNrspq ; (10)
where NNrspqis the initial proposed monthly #ight frequency.
Since NNrspqand #qare 8xed over monthly OD demand
#uc-tuations, lrs( ˜frst) is directly proportional to the realizations
of ˜frst for month t. Let ft
rs represent a random realization
of ˜frst, and a potential value ofOD passenger tra?c under
monthly #uctuations for OD pair r–s, over month t. The
tra?c #uctuations reveal that, if lrs(frst) = 0, it means the
potential OD tra?c is zero, i.e., ft
rs= 0; if lrs(frst) ¿ 1, it
means the potential OD tra?c ft
rsis equal to capacity or
ex-ceeds capacity, it also implies that excess demand will cause schedule delays and increased costs (low service level). We assume that there is a maximally acceptable load factor on
#ights between OD pair r–s; Nlrs, near 100 percent, at which
a minimally acceptable level ofservice can be maintained for passengers, and a minimally acceptable load factor on
#ights between OD pair r–s; lrs, at which a tolerable
mum revenue for airlines is assumed. We specify the
mini-mally acceptable load factor lrsas a break-even load factor
on #ights between OD pair r–s, and suppose that lrs= 55
percent by applying the data proposed in Wells [3]. When the proposed monthly #ight frequencies apply
un-der monthly #uctuating demand, and if ˜frst leads lrs( ˜frst)
to lrs6 lrs( ˜frst) 6 lrs, then the proposed monthly #ight
fre-quencies between the OD pair r–s; NNrspq; ∀p; q, are de8ned
as reliable under #uctuating tra?c ˜frst in month t. If ˜frst leads
lrs( ˜frst) to lrs( ˜frst) ¡ lrsor lrs( ˜frst) ¿ lrs, then the proposed
monthly OD #ight frequencies are de8ned as unreliable un-der tra?c #uctuations in month t. Consequently, the
reli-ability ofthe proposed monthly #ight frequencies between OD pair r–s in month t is then de8ned as the probability that the OD #ows fall between the acceptable limits. The relia-bility for OD pair r–s in month t can be evaluated by using the cumulative distribution functions of normal distribution
since ˜frst follows the normal distribution with parameters Nfrst
and !( ˜frst), that is Rrs( ˜frst) = Pr lrs p q nqNNrspq6 ˜frst 6 lrs p q nqNNrspq = ' lrspq nqNNrspq− Nfrst !( ˜frst) − ' lrspq nqNNrspq− Nfrst !( ˜frst) ; (11)
where Rrs( ˜frst) is the reliability ofthe proposed monthly
#ight frequencies between OD pair r–s in month t, and '(z) is the cumulative distribution function of the standard
nor-mal distribution, that is '(z)= z
−∞(1=
√
2))e−w2=2
dw. This reliability function is equivalent to the chance-constrained formulation. Airline revenue is lost when tra?c #ow falls below a limit and passenger traveling times/costs are in-creased when tra?c #ow is above a limit, because ofthe di>erence between the initial tra?c forecast and the tra?c realization. Section 3 will present a penalty function that represents the expected loss in airline revenue or the in-crease in passenger travel cost. Then, the reliability ofthe proposed monthly #ight frequencies between the OD pair
r–s over the planning year is the average value of Rrs( ˜frst)
over the 12 months ofthe year, given by, NRrs= 12 t=1 1 12Rrs( ˜f t rs); (12)
where NRrs is the reliability ofthe proposed monthly #ight
frequencies between OD pair r–s in the planning year. Consider that some abnormal events (for example, spe-cial festival, political or foreign trade event, international sporting event or political meeting, war, or natural disaster) occur at the origin or destination ofa given OD pair, r–s and cause abnormal OD demand #uctuations during a par-ticular period. An abnormal state is one in which monthly OD passenger tra?c values do not follow the normal tra?c distribution, according to the previously estimated
param-eters, Nfrst and !( ˜frst), due to the occurrence ofan abnormal
event. For an OD pair, r–s, let Krs represent the set ofall
distinct states which occur during the planning year and let Krs≡ {s0; s1; s2; : : : ; sW}, where s1; s2; : : : ; sW, represent
dis-tinct abnormal states; s0represents a normal state (in which,
no abnormal #uctuation occurs), and subscript W gives
probability that state si (i = 0; 1; : : : ; W ) occurs on OD
pair r–s during the planning year, where Pr(si) ¿ 0 and
W
i=0 Pr(si) = 1.
Suppose that, during the planning year, an abnormal state
si; i = 1; 2; : : : ; W , occurs at time t∗i with duration ˜vi, where
t∗
i is the time elapsed from the beginning of year, and
t∗
i; ˜vi∈ R+with 1 month as a unit. The duration of si (i =
1; 2; : : : ; W ); ˜vi, is considered to be a random variable. For
simplicity, ˜vi is supposed to have a 8nite discrete
distribu-tion: {(vj
i; pj); j = 1; : : : ; v} (pj¿ 0; ∀j); i = 1; 2; : : : ; W ,
where vji is a realization of˜vi; pj is its probability, and v
is the number ofrealizations of ˜vi. Let Irsji denote the set
ofmonths belonging to the time interval within which an
abnormal state si (i = 1; 2; : : : ; W ) continues on OD pair r–
s, i.e., Ii
rsj ≡ {t|t∗i 6 t ¡ ti∗+ vji } given state duration
vji; Irsj0 denotes the set ofmonths belonging to the normal
state s0, such that Irsj0 ≡ I − Irsji . Moreover, suppose that the
monthly passenger tra?c between OD pair r–s in an
abnor-mal state, si (i = 1; 2; : : : ; W ), follows another normal
distri-bution with di>erent parametric values. That is, the monthly
OD tra?c #ows associated with abnormal state, si, with the
state duration, vj
i, can now be considered as another random
variable, ˜frsijt ; ∀t ∈ Ii
rsj. Notably, the mean and standard
de-viation ofthe distribution of ˜frsijt is related to the e>ect and
duration ofthe event corresponding to state si.
For example, on a given OD pair r–s during the planning year, a special festival will occur in city r from January 21 to February 5 (for 16 days), and this event will cause ab-normal OD tra?c in January (t = 1) and February (t = 2).
This state is denoted, s1, with t1∗=0:677 (month), v1=0:533
(month). The distributions ofmonthly OD tra?c values in other months are not impacted by this abnormal event and still follow previously estimated normal distributions; these
monthly tra?c values stay in s0. For months January and
February, the distributions ofmonthly OD tra?c values may shift to other normal distributions with di>erent means and standard deviations. Two distinct states exist for passen-ger tra?c between OD pair r–s during the planning year:
{s0 for t = 3; 4; 5; 6; 7; 8; 9; 10; 11; 12}, and {s1 for t = 1; 2}.
Given that an abnormal state, si, with duration vji, has
occurred on OD pair r–s, the conditional reliability of the proposed monthly #ight frequencies associated with ˜ft
rsij; ∀t ∈ Irsji , can be calculated using Eq. (11), as Rrs( ˜frsijt ).
The distribution of ˜frsijt , time interval [t∗
i; ti∗+ vji ), and
Rrs( ˜frsijt ) may all vary with the realization of˜vi; vji, since ˜vi
is a random variable. In this case in which abnormal state,
si, has occurred on OD pair r–s during the planning year,
the conditional reliability ofthe proposed monthly #ight
fre-quencies over the planning year, NRrs|si, can be expressed as
NRrs|si= v j=1 1 12pj t∈Ii rsj Rrs( ˜frsijt ) + t∈I0 rsj Rrs( ˜frst) : (13)
Consequently, the reliability ofthe proposed monthly #ight frequencies between OD pair r–s, under normal/abnormal
#uctuations is E[Rrs] = W i=1 NRrs|siPr(si) + NRrsPr(s0): (14)
3. A Priori adjustment of ight frequencies
Initially proposed monthly #ight frequencies between some OD pairs could be found to have such low reliability that they must be adjusted. Ifwe only consider normal monthly OD demand variations, it is easy to detect the normal tra?c peaks and valleys ofsuch unreliable OD
pairs during one year. Let I0+
rs and Irs0− denote, the set
ofmonths in which such normal tra?c peaks and val-leys, respectively, occur during one planning year, where I0+
rs ∩ Irs0−= ∅. For such OD pairs, #ight frequencies should
be adjusted during months in set I0+
rs or in set Irs0− in the
planning year. Furthermore, the initially proposed monthly #ight frequencies between OD pair r–s ought to be
ad-justed for month t ∈ Ii
rsj in response to abnormal demand
in state si, with duration vji for the planning year. Let
t ≡ {I0+
rs ∪ Irs0−∪ Irsji ; ∀r; s; ∀i} denote the set ofmonths,
in which some OD pairs’ #ight frequencies demonstrate a need for adjustment. Let J be the set ofall OD pairs in the network such that J ≡ {r–s; ∀r ∈ R; ∀s ∈ S}. In a given
month t; t ∈ t, let Jt ≡ { ˙r– ˙s}; Jt ⊆ J, represent unreliable
OD pairs whose #ight frequencies require adjustment in
month t (t ∈ t), and let OD pair Nr– Ns, Nr– Ns∈ J − Jt, denote
reliable OD pairs whose #ight frequencies require no ad-justment in month t; t ∈ t. We then need to adjust the initial
tra?c forecast input for unreliable OD pair ˙r– ˙s∈ Jt, so as
to adjust #ight frequencies on these OD pairs.
Let Nfx˙r˙sdenote the expected monthly tra?c level on OD
pair ˙r– ˙s in a particular month x, and let Nfx˙r˙sijdenote the
ex-pected value ofmonthly tra?c on OD pair ˙r– ˙s associated
with abnormal state si in month x (x ∈ Ii˙r˙sj). Since ˜vi is a
random variable, the expected value, Nfx˙r˙sij, will vary with
di>erent realizations vji. In month t (t ∈ t), let f˙r˙s(t) denote
the adjusted demand on OD ˙r– ˙s; and let f
˙r˙sp(t) denote the
adjusted route tra?c involving OD ˙r– ˙s. The adjusted
de-mand on OD ˙r– ˙s∈ Jt in month t (t ∈ t), which depending
on t, can be given as an average value ofOD tra?c for tra?c-peak or -valley months or for the duration of abnor-mal states, as follows:
f ˙r˙s(t) = average x; x∈I0+˙r˙s { Nfx˙r˙s}; if t ∈ I0+˙r˙s; average x; x∈I0−˙r˙s { Nfx˙r˙s}; if t ∈ I0− ˙r˙s ; v j=1 pj× average x; x∈Ii ˙r˙sj { Nfx˙r˙sij}; if t ∈ Ii ˙r˙sj; (15)
where averagex{ Nfx˙r˙s} is the average value of Nfx˙r˙son OD
pair ˙r– ˙s for all months x belonging to set I0+
averagex;x∈ NIi ˙r˙sj{ Nf
x
˙r˙sij} is the average value of Nfx˙r˙sij on OD
pair ˙r– ˙s for all months x within t∗
i 6 x ¡ ti∗+ vji .
The #ight frequency adjustments on OD pair ˙r– ˙s in any adjustment month t (t ∈ t) can then be determined by solv-ing the followsolv-ing programmsolv-ing model (P2):
P2 min Nt ˙r˙spq a∈A CTA a (Yaq) + CaTP(Yaq; Nt˙r˙spq) (16a) s:t: q nql˙r˙spNt˙r˙spq¿ f˙r˙sp(t); (16b) p f ˙r˙sp(t) = f˙r˙s(t) p ∈ P˙r˙s; ˙r– ˙s∈ Jt; (16c) Yaq= Nr; Ns p NrNs a;p;qNNrspq+ ˙r; ˙s p ˙r; ˙s a;p;qNt˙r˙spq; (16d) a taqYaq6 uqUq ∀q; (16e) Nt˙r˙spq¿ 0 and integer; f˙r˙sp(t) ¿ 0; (16f) where Nt
˙r˙spqare the adjusted monthly #ight frequencies
as-sociated with OD pair ˙r– ˙s in month t; NNrspqare the initially
proposed monthly #ight frequencies associated with OD Nr– Ns
determined using P1, and l˙r˙spis the initially speci8ed load
factor on route p ofOD pair ˙r– ˙s.
We consider the costs associated with #ight frequency adjustment (e.g., extra #ight expenses, additional #ight and crew dispatching costs, and schedule change costs) and com-pare the adjustment costs with the expected loss ofair-line revenue and/or passenger traveling time to determine whether performing the adjustment is justi8ed. The total
ad-justment cost for OD pair ˙r– ˙s in month t (t ∈ t); CADt
˙r˙s , is given as CADt˙r˙s = p q 2t˙r˙spq| NN˙r˙spq− Nt˙r˙spq|; (17) where 2t
˙r˙spqis the adjustment cost per #ight for type q aircraft
on route p associated with OD pair ˙r– ˙s in adjusted month
t (t ∈ t), and NN˙r˙spq is the initially proposed monthly #ight
frequency assigned to OD pair ˙r– ˙s.
Let Nf˙r˙sdenote the initial tra?c forecast input for OD
pair ˙r– ˙s ( ˙r– ˙s∈ Jt). Let l
˙r˙s( Nf˙r˙s) and l˙r˙s(f˙r˙s(t)) represent the
monthly OD load factors on #ights between OD pair ˙r– ˙s
associated with Nf˙r˙sand f
˙r˙s(t), respectively, while l˙r˙s( Nf˙r˙s)
and l˙r˙s(f˙r˙s(t)) are calculated using the de8nition in Eq. (10).
Ifthe tra?c on OD pair ˙r– ˙s is realized to be f
˙r˙s(t) in month
t (t ∈ t), and the initially proposed monthly #ight frequen-cies for OD pair ˙r– ˙s are assessed to be unreliable, there will be losses in airline revenue or increases in passenger trav-eling times/costs owing to the di>erence between the initial tra?c forecast and the tra?c realization. If the OD tra?c
realization in month t (t ∈ t) is f
˙r˙s(t) and f˙r˙s(t) is less than
the initial average monthly forecast OD tra?c, Nf˙r˙s, then
airline revenue is lost because ofthe unsold seats on the #ights between OD pairs ˙r– ˙s in month t. The proportion of unsold seats on OD pairs ˙r– ˙s, in month t can be assessed as
l˙r˙s( Nf˙r˙s) − l˙r˙s(f˙r˙s(t)). The average revenue per passenger
mile #own is assumed roughly to equal the average airline cost per available seat #own. The expected loss in airline
revenue for OD pair ˙r– ˙s in month t when f
˙r˙s(t) 6 Nf˙r˙sis
then, CTA
˙r˙s(l˙r˙s( Nf˙r˙s) − l˙r˙s(f˙r˙s(t))).
However, ifthe OD tra?c realization, f
˙r˙s(t), exceeds the
initial average monthly forecasted OD tra?c, Nf˙r˙s, in month
t (t ∈ t), such that f
˙r˙s(t) ¿ Nf˙r˙s, then passenger traveling
costs will increase because ofexcess passengers who bear line-haul travel costs and endure schedule delays. As with
estimating total passenger travel costs on link a; CTP
a , total
passenger travel costs on OD pair r–s; CTP
rs, can be assessed
by multiplying the average travel cost per passenger (i.e., the average line-haul travel cost plus the average schedule delay cost per passenger) by the total number ofpassengers who #y between OD pair r–s. The expected increase in passenger traveling costs for OD pair ˙r– ˙s in month t, can then be
roughly estimated as CTP
˙r˙s((f˙r˙s(t) − Nf˙r˙s)= Nf˙r˙s). A penalty
function ˆP(f
˙r˙s(t)), is introduced to represent the expected
loss in airline revenue or increase in passenger travel costs
associated with adjusting the expected demand from Nf˙r˙sto
f
˙r˙s(t) for OD pair ˙r– ˙s in month t (t ∈ t). Then, ˆP(f˙r˙s(t))
can be de8ned as ˆP(f ˙r˙s(t)) = CTA ˙r˙s(lrs( Nf˙r˙s) − lrs(f˙r˙s(t))) if f˙r˙s(t) 6 Nf˙r˙s; CTP ˙r˙s f ˙r˙s(t)− Nf˙r˙s N f˙r˙s if f ˙r˙s(t) ¿ Nf˙r˙s: (18)
Finally, by comparing CADt
˙r˙s with ˆP(f˙r˙s(t)), to determine
whether it is necessary to perform adjustments; that is,
deter-mining whether or not CADt
˙r˙s ¡ ˆP(f˙r˙s(t)), #ight frequencies
for the OD pair ˙r– ˙s in month t may or may not be found to require adjustment.
4. Case study
A case study demonstrating application ofthe models proposed here based on available data from China Airlines (CAL) is presented below. The objective ofthe case study is an attempt to design and analyze CAL’s international network for the year 2003. To simplify the study, we se-lected 10 cities (nodes) in eight countries from all cities currently served by CAL, and assumed that there will be 25 wide-body aircraft including 13 Boeing 747-400s (394 seats) and 12 Airbus 300s (268 seats) #ying among those 10 cites in 2003. The nine city-pairs selected are Taipei (TPE)–Hong Kong (HKG), –Tokyo (TYO), –Bangkok (BKK), –Singapore (SIN), –Los Angeles (LAX), –San Francisco (SFO), –New York (NYC), –Frankfurt (FRA), and –Amsterdam (AMS). Tra?c among these selected city-pairs is a major part ofthe tra?c carried by CAL. Base values for the cost-function relevant parameters are given to resolve the problem ofdetermining #ight fre-quencies. However, some ofCAL’s operating cost data are
Table 1
Tra?c forecasts for city-pair passengers carried by CAL in 2003
City-pairs Annual Monthly
forecasts forecasts TPE–HKG∗ 668 252 55 688 TPE–TYO∗ 469 977 39 165 TPE–BKK∗ 186 049 15 504 TPE–SIN∗ 117 612 9 801 TPE–LAX∗ 143 585 11 965 TPE–SFO∗ 71 793 5 983 TPE–NYC∗ 51 281 4 273 TPE–FRA∗ 22 380 1 865 TPE–AMS∗ 108 642 9 054 TYO–NYC∗∗ 49 872 4 156 BKK–FRA∗∗ 18 868 1 572 BKK–AMS∗∗ 39 456 3 288
Source: Historic data,∗Department ofStatistics, M.O.T.C., ROC
[47], and ∗∗Boeing Commercial Airplane Group [48]. ∗7
his-toric data, years 1992–1998, and∗∗10 historic data, years 1984–
1993, respectively, for model forecasting. Note:∗GM(1,3) model:
socio-economic variables include annual per capita GNP and rel-ative exchange rates.∗∗GM(1,1) time series model.
unavailable hence operating cost data reported in Kane [44] were employed to estimate them. Aircraft characteristic data shown in CAL’s #eet facts and those reported in Horonje> and McKelvey [42] were also used to estimate #ight times and airport times. Moreover, the average unit time-cost re-#ecting line-haul travel time and delay time are assumed to be $23:15=h and $30:29=h, respectively, according to slight adjustments to the values oftime obtained by Furichi and Koppelman [45].
Historic data on the airline’s city-pair tra?c is unavail-able so we used annual total 8gures for Taiwan-resident departures and foreign-visitor arrivals (i.e., annual country-pair/city-pair tra?c among the 9 city-pairs), and used annual gross national product per capita and rela-tive exchange rates for the countries as socioeconomic variables for grey systematic forecasting. Eqs. (A.1)– (A.5) in Appendix A were used to forecast the city-pair passenger tra?cs. Forecast values were then adjusted to re#ect CAL’s city-pair tra?c by multiplying the air-line’s market shares for these pairs. Market shares were roughly estimated on the basis ofCAL’s historic data and timetables [4]. Annual and monthly forecast val-ues for each of the 12 city-pairs’ passenger tra?c in the year 2003 are listed in Table 1. The annual values were divided by 12 (months) to obtain average monthly tra?c. The airline can input its own historic city-pair tra?c data to perform grey systematic forecasting (Eqs. (A.1)–(A.5), see Appendix A), ifdata are available. We determined #ight frequencies and routing using the air-line network design model P1 (Eqs. (9a)–(9f)), and the NLP rounding relaxation method to obtain the solution.
Table 2
Initial optimal monthly #ight frequencies and objective function values
Routes Aircraft Monthly
#ight frequencies (two-way) TPE–HKG B747-400 26 A300 239 TPE–TYO B747-400 133 A300 0 TPE–BKK B747-400 53 A300 0 TPE–SIN B747-400 0 A300 49 TPE–LAX B747-400 41 TPE–TYO–LAX B747-400 0 TPE–SFO B747-400 21 TPE–TYO–SFO B747-400 0 TPE–NYC B747-400 0 TPE–TYO–NYC B747-400 20 TPE–FRA B747-400 0 TPE–BKK–FRA B747-400 9 TPE–AMS B747-400 0 TPE–BKK–AMS B747-400 42 Total airline aCTA a = 14 862 876 operating cost ($) Total passenger aCTP a = 32 008 902 travel cost ($) Objective function Z =aCaTA+ CTPa value ($) =46 871 778
The solutions { NNrspq} in this study were obtained
us-ing GINO, a computer-modelus-ing program developed by Liebman et al. [46], based on a generalized reduced gra-dient algorithm. The initial solution results are listed in Table 2.
In this case study, we used historical monthly city-pair tra?c data (arrival/departure tra?c in city-pair markets destined for or originating in Taipei, Taiwan) during 1994 –1999 as sampling data. Table 3 presents estimates ofthe
sample mean Nfrst and sample standard deviation !( ˜frst) of
the sampled data, for normally distributed monthly
traf-8c, ˜frst (∀t ∈ I) on individual OD pairs in 2003. Fig. 1
shows monthly tra?c patterns for individual OD pairs in 2003. For reliability evaluation, we supposed the
mini-mum acceptable load factor to be 55% (i.e., lrs= 0:55),
while maximum acceptable load factors, lrs, were
as-sumed to be 90%, 95% and 100% (i.e., lrs = 0:9, 0.95,
and 1). A numerical experiment for reliability evalua-tion was conducted to observe changes in reliability for
Table 3
Estimated normal distributions ofmonthly tra?c for individual OD pairs OD pairs Normal distributions ofmonthly OD tra?c: ˜frst ∼ N( Nfrst; !( ˜frst))
TPE–HKG Jan. Feb. Mar. Apr. May Jun.
N(50 321; 2457) N(52 984; 3635) N(51 324; 2979) N(58 465; 3618) N(53 348; 5324) N(53 346; 3152)
Jul. Aug. Sep. Oct. Nov. Dec.
N(60 053; 3714) N(63 228; 2808) N(52 577; 2435) N(58 780; 2842) N(50 708; 3540) N(54 366; 4478)
TPE–TYO Jan. Feb. Mar. Apr. May Jun.
N(35 334; 1816) N(39 555; 2050) N(40 793; 4096) N(38 607; 2494) N(36 821; 2250) N(39 703; 1920)
Jul. Aug. Sep. Oct. Nov. Dec.
N(43 489; 4485) N(43 651; 2850) N(36 208; 1943) N(37 926; 3748) N(37 264; 2870) N(35 767; 1855)
TPE–BKK Jan. Feb. Mar. Apr. May Jun.
N(13 052; 1496) N(15 111; 1538) N(14 999; 1271) N(15 895; 1363) N(15 391; 1389) N(16 289; 1530)
Jul. Aug. Sep. Oct. Nov. Dec.
N(17 933; 1398) N(17 953; 826) N(14 527; 611) N(14 434; 647) N(14 083; 1311) N(14 035; 549)
TPE–SIN Jan. Feb. Mar. Apr. May Jun.
N(9435; 733) N(10 492; 532) N(9079; 700) N(9069; 605) N(9250; 530) N(10 670; 604)
Jul. Aug. Sep. Oct. Nov. Dec.
N(11 297; 482) N(11 547; 578) N(9079; 659) N(8687; 722) N(9266; 562) N(10 089; 221)
TPE–LAX Jan. Feb. Mar. Apr. May Jun.
N(11 553; 1344) N(10 041; 816) N(9895; 1291) N(10 282; 945) N(11 781; 1209) N(12 783; 1011)
Jul. Aug. Sep. Oct. Nov. Dec.
N(12 985; 962) N(13 474; 995) N(10 209; 759) N(9549; 687) N(10 335; 1121) N(13 114; 2680)
TPE–SFO Jan. Feb. Mar. Apr. May Jun.
N(5947; 577) N(5381; 331) N(4985; 461) N(5438; 191) N(6090; 268) N(6556; 459)
Jul. Aug. Sep. Oct. Nov. Dec.
N(6969; 585) N(7202; 535) N(5346; 216) N(5004; 278) N(5034; 431) N(5925; 341)
TPE–NYC Jan. Feb. Mar. Apr. May Jun.
N(4012; 1301) N(3843; 1245) N(3444; 900) N(3977; 378) N(4464; 351) N(4869; 605)
Jul. Aug. Sep. Oct. Nov. Dec.
N(4779; 856) N(4967; 566) N(4121; 595) N(3762; 883) N(3990; 819) N(4903; 613)
TPE–FRA Jan. Feb. Mar. Apr. May Jun.
N(1463; 427) N(1825; 380) N(1504; 306) N(1622; 314) N(1727; 558) N(1840; 453)
Jul. Aug. Sep. Oct. Nov. Dec.
N(2232; 503) N(2089; 387) N(1621; 301) N(1588; 342) N(1536; 439) N(1564; 250)
TPE–AMS Jan. Feb. Mar. Apr. May Jun.
N(7552; 874) N(9063; 898) N(9414; 742) N(9498; 796) N(8912; 811) N(9360; 894)
Jul. Aug. Sep. Oct. Nov. Dec.
N(11 056; 817) N(10 852; 482) N(8363; 357) N(8525; 378) N(7865; 766) N(7925; 321) Source: Historic data,∗Department ofStatistics, M.O.T.C., ROC [47], years 1994–1999.
individual OD pairs with respect to variations in the
value of lrs. Table 4 lists the reliabilities ofthe
pro-posed monthly #ight frequencies for individual OD pairs over normal monthly #uctuations in 2003. Table 4 shows that OD pairs TPE–NYC, TPE–FRA, and TPE–AMS have relatively low reliabilities. Comparing the various parameter values in Table 3, shows that the monthly tra?c values on the relatively low reliability OD pairs (TPE–NYC, TPE–FRA, and TPE–AMS) all have rela-tively high coe?cients ofvariation. According to Tables
3 and 4, a lower dispersion ofmonthly OD tra?c val-ues implies higher reliability ofthe proposed monthly OD #ight frequencies. In the test airline network, most OD pairs did not exhibit severe enough monthly tra?c #uctuations to evoke unreliability; the proposed monthly #ight frequencies for these pairs were therefore assessed to be highly reliable. Figs. 1(g)–(i) also identify traf-8c peaks and valleys for those relatively low reliability OD pairs (TPE–NYC, TPE–FRA, and TPE–AMS), as follows:
Fig. 1. Monthly average tra?c patterns in 2003 for OD pairs: (a) TPE–HKG, (b) TPE–TYO, (c) TPE–BKK, (d) TPE–SIN, (e) TPE–LAX, (f) TPE–SFO, (g) TPE–NYC, (h) TPE–FRA, (i) TPE–AMS.
TPE–NYC: Peak tra?c months: June, July, August
and December, e.g., I0+
TPE−NYC =
{6; 7; 8; 12}.
Valley tra?c months: February, March and
October, e.g., I0−
TPE−NYC= {2; 3; 10}.
TPE–FRA: Peak tra?c months: July, and August, e.g., I0+
TPE−FRA= {7; 8}.
Valley tra?c months: January, March, Novem-ber and DecemNovem-ber,
e.g., I0−
TPE−FRA= {1; 3; 11; 12}.
TPE–AMS: Peak tra?c months: July and August, e.g., I0+
TPE−AMS= {7; 8}.
Valley tra?c months: January, November and
December, e.g., I0−
Table 4
Reliability ofproposed monthly #ight frequencies on individual OD pairs for normal monthly #uctuations in 2003
OD pairs Reliability ( NRrs)
Acceptable max. and min. load factors lrs= 0:9, lrs= 0:95, lrs= 1, lrs= 0:55 lrs= 0:55 lrs= 0:55 TPE–HKG 0.9863 0.9991 0.9997 TPE–TYO 0.9674 0.9900 0.9969 TPE–BKK 0.9422 0.9753 0.9830 TPE–SIN 0.9532 0.9891 0.9941 TPE–LAX 0.8902 0.9160 0.9264 TPE–SFO 0.9224 0.9463 0.9580 TPE–NYC 0.7650 0.7939 0.8077 TPE–FRA 0.6016 0.6244 0.6392 TPE–AMS 0.8494 0.8508 0.8509
A hypothetical scenario involving abnormal situations was also considered in this case study. We supposed that a sudden explosion in passenger tra?c occurred on OD pair TPE–TYO owing to tourism promotion during the Lunar New Year holiday period (from January 15 to the middle ofFebruary) in 2003. The data concerning this abnormal
Table 5
Hypothetical data regarding abnormal state on TPE–TYO
Abnormal tra?c distributions Abnormal months State occurrence duration (from Jan. 25)
t∗
1 = 0:806 411= 0:5 (15 days) 421= 0:73 (22 days) 431= 0:93 (28 days)
p1= 0:5 p2= 0:3 p3= 0:2
Jan. N(45 934; 4540) N(45 934; 4540) N(45 934; 4540)
Feb. N(47 466; 5125) N(49 444; 5330) N(51 421; 5535)
Table 6
Reliability ofproposed monthly #ight frequencies on TPE–TYO, considering normal/abnormal tra?c #uctuations in 2003 Acceptable max. Reliability over the planning year, E[Rrs]
and min. load
factors Pr(s1) = 0:6 Pr(s1) = 0:7 Pr(s1) = 0:8 Pr(s1) = 0:9
lrs= 0:9, lrs= 0:55 0.9168 0.9084 0.9000 0.8916
lrs= 0:95, lrs= 0:55 0.9421 0.9426 0.9430 0.9435
lrs= 1, lrs= 0:55 0.9706 0.9706 0.9706 0.9706
Acceptable max. Months Reliability in abnormal months and min. load
factors Pr(s1) = 0:6 Pr(s1) = 0:7 Pr(s1) = 0:8 Pr(s1) = 0:9 lrs= 0:9, lrs= 0:55 Jan. 0.7639 0.7246 0.6853 0.6459 Feb. 0.6296 0.5678 0.5061 0.4444 lrs= 0:95, lrs= 0:55 Jan. 0.8810 0.8611 0.8413 0.8214 Feb. 0.7429 0.7000 0.6572 0.6143 lrs= 1, lrs= 0:55 Jan. 0.9537 0.9460 0.9383 0.9306 Feb. 0.8460 0.8204 0.7947 0.7690
state, including occurrence duration, abnormal tra?c distributions and duration probabilities, were listed in Table 5. The reliabilities ofproposed monthly #ight frequencies on OD pair TPE–TYO, considering nor-mal/abnormal states in year 2003, were calculated using Eq. (14), and the resulting reliability values then are listed in Table 6.
The set ofadjustment months, t, during the test year was determined from the aforementioned reliability evaluation
as, t = {I0+
TPE−NYC ∪ ITPE−NYC0− ∪ ITPE−FRA0+ ∪ ITPE−FRA0− ∪
I0+
TPE−AMS∪ITPE−AMS0− ∪ITPE−TYO1 }={1; 2; 3; 6; 7; 8; 10; 11; 12}.
The subsets ofunreliable OD pairs in month t (t ∈ t); Jt ≡
{ ˙r– ˙s}, were then determined as, respectively: J1= {TPE–
TYO, TPE–FRA, TPE–AMS} in January, J2={TPE–TYO,
TPE–NYC} in February, J3= {TPE–NYC, TPE–FRA} in
March, J6= { TPE–NYC} in June, J7= J8= {TPE–NYC,
TPE–FRA, TPE–AMS} in July and August, J10= {TPE–
NYC} in October, J11 = {TPE–FRA, TPE–AMS} in
November, and J12= {TPE–NYC, TPE–FRA, TPE–AMS}
in December. Furthermore, we used Eq. (15) to calculate the adjusted demand for OD pairs TPE–NYC, TPE–FRA and TPE–AMS, respectively, in normal peak/valley tra?c months and for TPE–TYO during January and February. The adjusted #ight frequencies on OD pairs TPE–NYC,
Table 7
Monthly #ight frequency adjustment in 2003: adjusted #ight frequencies, related adjustment costs, expected penalty values and the results ofadjust/do-nothing judgements
TPE–NPC Monthly #ight frequencies (two-way)
Initial Months
proposed
Routes Aircraft Feb. Mar. Jun. Jul. Aug. Oct. Dec.
TPE–NYC B747-400 0 0 0 0 0 0 0 0
TPE–TYO–NYC B747-400 20 17 17 23 23 23 17 23
Adjustment costs ($) 114 000 114 000 255 000 255 000 255 000 114 000 255 000 Expected penalty values ($) 127 290 127 290 394 387 394 387 394 387 127 290 394 387
Judgement Adjust Adjust Adjust Adjust Adjust Adjust Adjust
TPE–FRA Monthly #ight frequencies (two-way)
Initial Months
proposed
Routes Aircraft Jan. Mar. Jul. Aug. Nov. Dec.
TPE–FRA B747-400 0 0 0 0 0 0 0
TPE–BKK–FRA B747-400 9 8 8 10 10 8 8
Adjustment costs ($) 35 000 35 000 78 000 78 000 35 000 35 000
Expected penalty values ($) 66 587 66 587 276 991 276 991 66 587 66 587
Judgement Adjust Adjust Adjust Adjust Adjust Adjust
TPE–AMS Monthly #ight frequencies (two-way)
Initial Months
proposed
Routes Aircraft Jan. Jul. Aug. Nov. Dec.
TPE–AMS B747-400 0 0 0 0 0 0
TPE–BKK–AMS B747-400 42 36 51 51 36 36
Adjustment costs ($) 206 730 690 768 690 768 206 730 206 730
Expected penalty values ($) 247 966 912 061 912 061 247 966 247 966
Judgement Adjust Adjust Adjust Adjust Adjust
TPE–TYO Monthly #ight frequencies (two-way)
Initial Months
Routes Aircraft proposed Jan. Feb.
TPE–TYO B747-400 133 161 161
A300 0 0 0
Adjustment costs ($) 431 088 431 088
Expected penalty values ($) 1 063 092 1 063 092
Judgement Adjust Adjust
TPE–FRA, TPE–AMS and TPE–TYO during adjustment months were then determined by solving model P2 (Eqs. (16a)–(16f)).
To simplify our case study, we used relevant aircraft op-erating costs reported in Kane [44] to estimate the adjust-ment cost for Boeing 747-400s on the above OD pairs dur-ing adjustment months. We use relevant #ight expenses (in-cluding crew expense, fuel oil taxes, and insurance, etc.) to estimate the adjustment cost per #ight increased during peak-tra?c or abnormal demand-explosion months, and we
used maintenance burdens and #ight depreciation expenses to estimate the adjustment cost per #ight decreased dur-ing valley-tra?c periods. The adjustment costs per Boedur-ing 747-400 #ight on OD pairs TPE–NYC, TPE–FRA, TPE– AMS and TPE–TYO are listed in Table 7. The total
adjust-ment costs for #ights on these OD pairs, CADt
˙r˙s , were then
calculated using Eq. (17), and the penalty values, ˆp(f
˙r˙s(t)),
were calculated using Eq. (18). Table 7 lists the adjusted monthly #ight frequencies, related adjustment costs, ex-pected penalty values, and the results ofadjust/do-nothing
Table 8
Comparisons ofinitial airline network designs with and without adjustments Monthly #ight frequencies (two-way) Initial Months
proposed
Routes Aircraft Jan. Feb. Mar. Jun. Jul. Aug. Oct. Nov. Dec.
TPE–HKG B747-400 26 26 26 26 26 26 26 26 26 26 A300 239 239 239 239 239 239 239 239 239 239 TPE–TYO B747-400 133 161 161 133 133 133 133 133 133 133 A300 0 0 0 0 0 0 0 0 0 0 TPE–BKK B747-400 53 53 53 53 53 53 53 53 53 53 A300 0 0 0 0 0 0 0 0 0 0 TPE–SIN B747-400 0 0 0 0 0 0 0 0 0 0 A300 49 49 49 49 49 49 49 49 49 49 TPE–LAX B747-400 41 41 41 41 41 41 41 41 41 41 TPE–TYO–LAX B747-400 0 0 0 0 0 0 0 0 0 0 TPE–SFO B747-400 21 21 21 21 21 21 21 21 21 21 TPE–TYO–SFO B747-400 0 0 0 0 0 0 0 0 0 0 TPE–NYC B747-400 0 0 0 0 0 0 0 0 0 0 TPE–TYO–NYC B747-400 20 20 17 17 23 23 23 17 20 23 TPE–FRA B747-400 0 0 0 0 0 0 0 0 0 0 TPE–BKK–FRA B747-400 9 8 9 8 9 10 10 9 8 8 TPE–AMS B747-400 0 0 0 0 0 0 0 0 0 0 TPE–BKK–AMS B747-400 42 36 42 42 42 51 51 42 36 36
Initial objective function value +
total expected penalty values ($): 48 249 423 48 062 160 47 065 656 47 266 166 48 455 218 48 455 218 46 999 068 47 186 331 47 580 719
Adjusted objective function value +
total extra adjustment costs ($): 47 285 708 47 588 678 46 699 365 47 228 171 48 443 243 48 443 243 46 680 314 46 372 431 46 728 824
judgements for these unreliable OD pairs. Table 7 shows that the adjustment costs for #ights on these unreliable OD pairs were all less than the expected penalty values, so all were judged to bene8t from adjustment. Table 7 shows that on OD pair TPE–NYC, the #ight frequencies on the route TPE–TYO–NYC will decrease from 20 to 17 #ights in peak-tra?c months, February, March and October, re-spectively, and will increase from 20 to 23 #ights during valley-tra?c months June to August. On OD pair TPE– FRA, #ight frequencies on the route TPE–BKK–FRA will decrease from 9 to 8 #ights in valley-tra?c months, Jan-uary, March, November and December, respectively, and increase from 9 to 10 #ights during peak-tra?c months July to August. And, on the TPE–AMS OD pair, #ight frequen-cies on the route TPE–BKK–AMS will decrease from 42 to 36 #ights in valley-tra?c months, January, November and December, respectively, and increase from 42 to 51 #ights during peak-tra?c months July to August. TPE–TYO #ight frequencies will increase from 133 to 161 #ights in response to the abnormal explosion in demand during January to February.
In addition to the aforementioned judgements, we also compared the total airline network costs after performing adjustments with the total costs ifno adjustments were per-formed. If #ight frequencies on unreliable OD pair ˙r– ˙s in month t (t ∈ t) are not adjusted, the total cost ofthe air-line network should be the sum ofthe initial objective func-tion value ofP1 (i.e., initial estimated total airline operat-ing costs plus the total passenger travel costs) and the total expected losses in airline revenue or increases in passen-ger travel costs in month t. On the other hand, ifthe airline decides to perform adjustments in month t, the total cost ofthe airline network is the sum ofthe objective function value ofP2 in month t and the total extra adjustment costs in month t. Table 8 lists comparisons between the results ifno adjustments are made and ifadjustments are made in each month t ∈ t. In Table 8 we see that the total costs ofthe airline network in every adjustment month after perform-ing adjustments will be less than those ifno adjustments are performed. The #ight frequency adjustments are shown to bene8t the airline and provide #exibility for airline plan-ners to determine responsive #ight frequency plans on OD
pairs with severe #uctuations. Consequently, the reliability evaluation and adjustment procedure presented in this pa-per could provide a post-evaluation and adjustment method for airline network design in response to uncertain demand #uctuations.
5. Conclusions
This study focuses on reliability evaluation of airline net-work designs. The reliability evaluation method proposed in this study evaluates the reliabilities ofproposed monthly #ight frequencies on individual OD pairs on condition that normal/abnormal #uctuations occur on it. We analyze the probabilities ofnormal and abnormal state occurrences, the probabilities oftheir durations, and estimate the reli-abilities for individual OD pairs during the planned year. In responding to such tra?c #uctuations, we also provide a priori adjustment of#ight frequencies by tuning #ight frequencies on only parts of routes with severe tra?c #uctu-ations while still maintaining overall airline network design objectives. To determine whether performing adjustments is justi8ed, we compare the adjustment costs with expected losses in airline revenues and/or increases in passenger travel costs.
Application ofour developed models to 10 selected cities served by the CAL network was performed in a case study. Relatively low reliability value OD pairs were detected by the reliability evaluation when these OD pairs had severe enough monthly tra?c #uctuations to evoke unreliability. Thus, the initially proposed monthly #ight frequencies on these OD pairs were assessed as unreliable. This case study provides a guideline for post-design evaluation of airline net-work designs in response to uncertain #uctuation in passen-ger demand. The results ofthis case study not only show that #ight adjustment costs on unreliable OD pairs are less than expected losses, but also that the total cost ofthe adjusted airline network is less than the total costs ofthe original airline network design when no adjustments are performed. The results also indicate that the #ight frequency adjustment method proposed in this paper may bene8t the airline and its passengers and provide #exibility in decision-making for determining responsive #ight frequency plans on OD pairs with severe #uctuations. The adjusted #ight frequencies can surely match future potential tra?c #uctuations before these frequencies are used for detailed scheduling and are more appropriate for operational planning.
In sum, the reliability evaluation model provides a highly e>ective tool that enables planners to evaluate the perfor-mance ofairline network designs and to assess the impact oftra?c #uctuations on network design performance by taking demand variability, probabilities ofnormal/abnormal state occurrences and their durations into account. This study demonstrates how reliability evaluation might be applied to airline network design problems. Results in
this study shed further light into operational planning and performance-related issues in airline network design. Acknowledgements
The authors would like to thank the National Science Council ofthe Republic ofChina for 8nancially supporting this research under Contract No. NSC 89-2211-E-009-022. The constructive comments ofthe anonymous referees are greatly appreciated.
Appendix A. Greysystematic models for forecasting OD-pair tra(c
The structure ofan OD-pair demand model in this pa-per consists ofsocioeconomic variables that determine the tra?c for an airline OD-pair. GM(1; N) is a polyfac-tor forecasting model for the grey systematic model; herein, GM(1; N) models are developed to predict all OD-pair travel demands on an airline. Formulation of the GM(1; N) model is brie#y described below. Assume an original historic series ofannual tra?c #ows for a given airline’s OD pair r–s (origin r and destination s), F(0)
rs , is Frs(0)= (Frs(0)(1); : : : ; Frs(0)(n)), where n denotes the
number ofyears observed. Accumulated generating op-erations (AGOs), an important feature of grey models, focus largely on reducing data randomness. The AGO
formation of F(0)
rs is Frs(1)= (Frs(1)(1); : : : ; Frs(1)(n)), where
F(1)
rs (k) =kt=1Frs(0)(t), k = 2; 3; : : : ; n and Frs(1)(1) = Frs(0)(1).
Assume that X1rs, X2rs; : : : ; XN−1rs are socioeconomic
vari-ables for polyfactor GM(1; N) models. The original series
ofthese variables, X1rs; X2rs; : : : ; XN−1rs, are, respectively,
X(0) 1rs = (X (0) 1rs(1); X (0) 1rs(2); : : : ; X (0) 1rs(n)), X (0) 2rs = (X (0) 2rs(1); X (0) 2rs (2); : : : ; X2(0)rs(n)); : : : ; X (0) N−1rs = (X (0) N−1rs(1); X (0) N−1rs(2); : : : ; X(0) N−1rs(n)); and X (1) 1rs; X (1) 2rs; : : : ; X (1)
N−1rs are their respective
AGO-series. The GM(1; N) model can be constructed by
formulating a group of di>erential equations for F(1)
rs and X(1) 1rs; X (1) 2rs; : : : ; X (1) N−1rs. That is dF(1) rs dt = −aFrs(1)+ b1X1(1)rs + b2X (1) 2rs + · · · + bN−1X (1) N−1rs; dX(1) 1rs dt = −a1X1(1)rs + u1; dX(1) 2rs dt = −a2X2(1)rs + u2; .. . dX(1) N−1rs dt = −aN−1XN−1(1) rs+ uN−1: (A.1)
In Eq. (A.1), the parameters, a; bi; ai; ui, i=1; 2; : : : ; N −1,
can be determined by applying the least-squares method. The 8rst-order di>erential equation for the AGO-series of each of
socioeconomic variable, X(1) 1rs; X (1) 2rs; : : : ; X (1) N−1rs; is GM(1,1)
