Optimal Schedule Adjustment for Expected Aircraft Shortage in Multi-Fleet Operations

Transcription

1 Iteratioal Joural of Operatios Research Iteratioal Joural of Operatios Research Vol. 2, No. 1, (2005) Optimal Schedule Adjustmet for Expected Aircraft Shortage i Multi-Fleet Operatios Shagyao Ya *, Chih-Teg Lo, ad Chia-Hug Che Departmet of Civil Egieerig, Natioal Cetral Uiversity, No.300, Jhogda Rd. Chugli, Taiwa R.O.C. Abstract This research develops several etwork models for carriers that both efficietly ad effectively adjust schedule resultig from the expected aircraft shortage for the operatio of multiple fleets as well as o-stop ad oe-stop flights. These models are formulated as pure etwork flow problems or multi-commodity etwork flow problems. The former are solved usig the etwork simplex method while the latter are solved usig a Lagragia relaxatio-based algorithm. A case study regardig the iteratioal operatios of a major Taiwa airlie is preseted. Keywords Expected aircraft shortage, Schedule adjustmet, Time-space etwork, Multi-commodity etwork flow problem, Lagragia relaxatio 1. INTRODUCTION Flight schedule perturbatios may occur i a airlie's regular operatios due to umerous factors. For example, caceled or delayed flights may be caused by cogestio at airports, poor meteorological coditios, equipmet breakdow, late or abset crew members, poorly calculated block times ad service times at certai airports, irregular oboard or package-hadlig processes, late trasfers betwee flights, or sudde war Teodorovic (1988). Besides these factors which ca typically be characterized as real time evets, some expected evets, for example, regular maiteace checks o aircraft or special uses of aircraft i the ear future, may also result i schedule perturbatios. These expected evets, differet from the real time evets, are usually kow some time (a few days or a week) before the evets really happe. Thus, if airlie carriers ca effectively pla their temporary schedules (or hadle schedule perturbatios early) before the expected evets happe, they may reduce their loss of profit ad maitai a better level of service. Aircraft maiteace checks are oe typical expected evet i short-term airlie operatios, as they are ecessary Etschmaier ad Mathaisel (1985). Airlies always adopt maiteace policies that coform with the govermet regulatios. For example, other tha the everyday routie ispectios, may carriers have four other types of ispectios o aircraft. The first major check (a A check) occurs every 65 flight hours or about oce a week. A A check ivolves a visual ispectio of all major systems such as ladig gear, egies, ad cotrol surfaces. B checks are performed every hours ad etail a thorough visual ispectio plus the lubricatio of all movig parts such as horizotal stabilizers ad aileros. C ad D checks are doe about oce every oe to four years respectively, ad require takig the aircraft out of service for up to a moth at a time Wallich (1986). Although carriers have usually cosidered maiteace costraits i their fleet routig ad flight schedulig i short-term operatios Ya ad Youg (1996), the maiteace of aircraft may differ from their schedules, due to the complicated operatio eviromet Teodorovic (1988). Thus, the carrier has to reschedule the dates of the maiteace checks for aircraft i short-term operatios. Usually a A check may ot cause severe schedule perturbatios. Carriers ca do these short checks overight or utilize legthy groud times. However, for a loger check, for example, a B-check, a C-check, or a D-check, the aircraft usually has to be set ito the base plat for a loger time, resultig i a temporary shortage of aircraft ad, therefore, a schedule perturbatio. There is ormally oe maiteace base (usually Taipei) for iteratioal airlie carriers i Taiwa. The available slots that ca be used for maiteace checks (particularly for B, C, ad D-checks) are always limited. Therefore, the maiteace check has to be rescheduled early, so that the aircraft ca be set to the base plat i time for maiteace. Accordig to a major Taiwa airlie, such checks (especially B or C-checks) may occur from time to time i its operatios. They ofte result i perturbatios i flight schedules, ot oly durig the period that the aircraft is i maiteace, but both before ad after. Typically, if the adjustmet to the flight schedule is too early, it may cause uecessary schedule perturbatios ad a decreased level of service. However, if the adjustmet to the flight schedule is too late, it may be ifeasible to sed the aircraft to the maiteace base i time. If the carrier uses the aircraft overtime, the it may cause safety problems. Because of this, whe to start adjustig a flight schedule is a importat issue for airlie carriers, i their short-term operatios. Accordig to a major Taiwa airlie carrier, the available time slots at its base plat are very limited. The plat ad the time slot for a scheduled-check are usually give before * Correspodig author s t320002@cc.cu.edu.tw X Copyright 2005 ORSTW

2 32 a schedule adjustmet is made. The curret process for adjustig the flight schedules of Taiwa carriers is iefficiet ad ieffective from a system perspective, especially for large flight etworks. The process is as follows. Give the aircraft as well as the locatio of the plat ad the time slot for maiteace, the plaers start by choosig a suitable strategy (for example, flight cacellatios, flight delays, ferryig of flights) based o the regular, plaed schedule, projected (with booked) demad o all flights ad the allocatio of the curretly available airplaes. The startig time for adjustig a schedule is always arbitrarily chose. It is typically made to be as early as possible to avoid a ifeasible solutio. The detailed flight/fleet schedule is the adjusted by a trial-ad-error process util a feasible solutio is foud. For example, if a flight caot be served by its scheduled aircraft, the the flight ca be suitably delayed so that a holdig aircraft (or a icomig aircraft) ca be rescheduled to serve this flight. However, if there is o holdig aircraft (or o icomig aircraft), the the flight ca be caceled, or served by a aircraft obtaied from aother statio usig a ferry flight. Such a process geerally ivolves a series of local adjustmets (usually by had) to aircraft routes ad their related flights. As etwork size grows, the process is more iefficiet for fidig the optimal solutio. Usually, a feasible solutio is oly obtaied uder the real time costrait. The draft schedule is the set to the operatig divisio for the applicatio of other costraits (for example crew costraits). The schedule will be executed if it is feasible, otherwise it will be retured to the plaig divisio for revisios. The process is repeated util both the plaig ad operatig divisios are satisfied with the revised schedule. Due to deregulatio, the flight etworks for Taiwa carriers have recetly grow. Thus, it is icreasigly difficult for the traditioal approach to adequately hadle such evets. Perturbatios will geerally affect ot oly the flights ad the scheduled routig for a aircraft, but also may other flights, ad fleet routig as well because of complicated etwork operatios. The poor schedulig of flights, or a etire fleet, may result i a substatial loss of profit, decreased levels of service, or eve safety problems for the airlie carriers. Thus, it would be helpful for carriers to have systematic computerized models to hadle schedule perturbatios efficietly ad effectively, so as to reduce losses resultig from the expected aircraft maiteace. Few optimizatio models exist for solvig maiteace related problems. Feo ad Bard (1989) itroduced a large-scale mixed iteger programmig formulatio for log-term maiteace schedulig problems. The model is used to locate maiteace statios ad to develop flight schedules that better meet the cyclical demads for maiteace. Biro' et al. (1992) itroduced a operative model to adjust the flight schedules i short-term operatios. I particular, they used the ode colorig techique i a graph to adjust the sequece of flights, i order to meet maiteace requiremets. Although, the model was useful for arragig the flight sequeces, may practical schedulig adjustmets, such as flight cacellatios, flight delays, ad the ferryig of idle aircraft, were difficult to use to efficietly adjust flight schedules, especially i a complicated etwork operatio. Cocerig past research o airlie schedule perturbatios e.g. Jedlisky (1967), Etschmaier ad Rothstei (1973), Deckwitz (1984), Teodorovic ad Guberiic (1984), Gershkoff (1987), Teodorovic ad Stojkovic (1990), Jarrah et al. (1993), Mulvey ad Ruszczyski (1995), Ya ad Yag (1996), Du ad Hall (1997), Keyo ad Morto (2003), List et al. (2003) ot much work has bee devoted to the problem of perturbatios ivolved i executig a plaed airlie schedule. I the past, research did ot hadle expected aircraft maiteace. Besides, the aforemetioed models were all focused o the operatios of sigle fleet. Note that airplaes of differet types ca support each other through a ew routig with a temporary flight schedule if the schedule is perturbed temporarily. For example; some idle larger-size aircraft ca serve flights scheduled for small-size aircraft; some flights ca be delayed so that larger-size aircraft ca be rescheduled to serve these flights if it is profitable from the system aspect. Cosequetly, there is ot yet a schedulig model that formulates multi-fleet operatios ad all the practical schedulig rules for hadlig perturbatios that result from the expected aircraft maiteace, i a systematic ad combied framework. I this research, we develop several etwork models to help carriers hadle schedule perturbatios resultig from the expected aircraft maiteace. Other expected evets ca be dealt with i the future. The maiteace types focused o i this research are those that have to be doe i the plat, particularly, the B checks (doe i a day), C checks (doe i a week) or short-term progressive maiteace (doe withi a week). As to A checks, because they ca be doe flexibly either overight or usig a log groud time, it may ot be ecessary to use our models to hadle these. Moreover, the model developed i this research may ot be suitable for D checks, sice D checks are typically carried out i a moth, ad by the the timetable may be chaged. Thus, istead of adjustig a temporary schedule, a ew schedule may be desiged always, icorporatig the D checks. The scope of this research is focused o the operatios of multiple fleets, as well as oe-stop ad o-stop flights. The operatios ivolved i other types of multi-stop flights are left to be addressed i the future. I additio, sice the curret fleet sizes of Taiwa airlie carriers are small ad it is uusual for a carrier to have more tha oe aircraft rescheduled for a B or C check at the same time, we focused o oly oe case of aircraft maiteace for simplificatio. Although the model developed i this research eeds to be modified to hadle cases where more tha oe aircraft is rescheduled for B or C checks, we believe modificatio to be easy, a subject also left for future research. Similar to most fleet routig models e.g. Ya ad

3 33 Yag (1996), Ya ad Youg (1996), the costraits of crew schedulig are excluded i the modelig to facilitate problem solvig. The rest of this paper is orgaized as follows: we first itroduce the modelig approaches icludig the sigle-fleet time-space etwork, the basic model ad several strategic models. These models are the formulated as iteger programs ad their solutios are developed hereafter. Fially, a case study is performed to test the models i practice. 2. MODELING APPROACHES I the modelig, we first itroduce the sigle-fleet time-space etwork, the exted it to the basic model, ad fially develop strategic models based o the basic model. 2.1 The sigle-fleet time-space etwork We suggest usig a time-space etwork (as show i Figure 1) to formulate the sigle-fleet etwork. The etwork is ideed a extesio of the oe i Ya ad Yag (1996), which is used for hadlig schedule perturbatios caused by the breakdow of aircraft. I Figure 1, the horizotal axis represets airport locatios, while the vertical axis represets the time duratio. Each ode represets a specific airport at a specific time. Each arc represets a activity for a airplae. There are four types of arcs; 1) flight arcs, 2) overight arcs, 3) groud arcs ad 4) positio arcs. A flight arc represets a o-stop flight or a oe-stop flight. To reflect additioal charges for cacelig flights, the modificatio of the arc cost of a flight arc is the same as i Ya ad Yag (1996). A groud arc represets the holdig of airplaes at a airport i a time widow. A overight arc represets the holdig of airplaes overight at a airport. A positio arc represets a ferry flight betwee two airports. All of these arcs are made to be the same as those i Ya ad Yag (1996). City 1 City 2 City3 (5) The startig time The maiteace time (6) (1) (2) (3) (4) The recovery time (7) (8) The edig time (1) flight arc (2) groud arc (3) overight arc (4) positio arc (5) iitial supply (6) itermediate supply (7) itermediate demad (8) fial demad Figure 1. The sigle-fleet time-space etwork. The basic etwork cotais a perturbed period, from a startig time to a edig time. The startig time is the earliest time that the system starts the perturbatios. The startig time is desiged as follows. I order to sed the specified aircraft to the maiteace time/space poit i time (the time is whe the aircraft starts its maiteace ad the space is where the maiteace takes place), we add a positio arc, from every other statio to the maiteace time-space poit. To miimize the perturbed period i the schedule, we set the startig time to be the

4 34 latest time that the aircraft's origial route meets the aforemetioed positio arcs. Thus, the aircraft ca use at least oe positio flight to get to the maiteace base i time. Note that if the departure times of the aforemetioed positio arcs are ahead of the startig time, the these arcs should ot be added ito the etwork. It should be metioed that the startig time ca differ if the aircraft is differet, because its origial route also differs, as does the latest time that the route meets the positio arcs. At the startig time, all airplaes located at airports, or i the air, are set as iitial or itermediate ode supplies. To determie the route of a specified aircraft, before fidig the fleet routig, we fid the shortest path from where the specified aircraft is located at the startig time to the maiteace time/space poit. Accordigly, a itermediate ode demad should be placed (i.e. the ode supply should be mius oe) at the time-space poit where the airplae starts its route i the etwork, meaig that the specified airplae is extracted from the system at the startig time. Similar to the case i Ya ad Yag (1996), the specified airplae ca be retured to service after the recovery time (i.e. after fiishig the maiteace), so the edig time that is whe the fleet resumes its ormal operatio is determied to be the same as that i Ya ad Yag (1996). Fial or itermediate ode demads are set accordig to the fleet allocatio at airports or i the air, at the edig time. Sice the specified airplae should be reitroduced ito the system, a itermediate ode supply should be placed at the time/space poit correspodig to the maiteace base ad the recovery time whe the specified airplae is agai ready for service. 2.2 The basic model The basic model is developed based o the sigle-fleet time-space etwork. A example of the basic model is show i Figure 2, assumig that there are three fleets i operatio, each correspodig to a type of aircraft. Sice this research simplifies the case to the expected maiteace of oe aircraft, oly oe etwork (a type B etwork i the example) i the basic model is desiged with positio arcs. Other types of etworks (type A ad type C) are desiged to be similar to type B without positio arcs, because there are o abset aircraft ivolved. Plae A Plae B Plae C City 1 City 2City 3 City 1 City 2City3 City 1City 2 City 3 The startig time a1 The maiteace time c1 b1 c2 a2 The recovery time b2 The edig time a1, a2: plae A (small type) flight arc b1, b2: plae B (media type) flight arc c1, c2: plae C (large type) flight arc Figure 2. The basic model etwork.

5 35 The iteger program formulatig the basic model is show below; Mi St. Ζ= C X + ( C CA ) X + CA (1) M i, j A \ F i, j F i, j F X = Xki b i j O() i k I() i i N, M (2) 0 X U ( i, j) A, M (3) X I ( i, j) A, M (4) where : the umber of aircraft types M : the set of all aircraft types N : the set of all odes i the th type etwork A : the set of all arcs i the th type etwork F : the set of all flight arcs i the th type etwork O(i): the set of head odes for arcs emaatig from ode i I(i): the set of tail odes for arcs poitig ito ode i b : the ode supply/demad of ode i i the th type i etwork. Note that if a specified airplae is take out from the system for maiteace or other reasos at a ode, the its ode supply/demad eed to be mius oe i calculatio. O the cotrary, for a ode if a airplae is retured to the system, the the ode supply/demad eed to be plus oe. C,, X U : arc (, i j ) cost, flow ad upper boud i the th type etwork respectively CA : the cacellatio cost for flight (, i j ) i the th type etwork. The objective of this model is to flow all ode supplies to all ode demads i each etwork at a miimum cost. Sice reveue is formulated as egative cost, the objective fuctio (1) is equivalet to a maximizatio of the system profit. Costrait (2) esures flow coservatio costrait. Costrait (3) esures that all arc flows are withi their upper ad lower bouds, ad costrait (4) esures that all arc flows are itegers. Note that the basic model ca be used to evaluate the cacellatio of flights. 2.3 The strategic models To make the basic model more useful, four practical schedulig rules are icorporated to develop strategic models, i particular; (a) the swap of aircraft types (b) flight delays, (c) the modificatio of oe-stop flights ad (d) the ferryig of idle aircraft. The strategy of the swap of aircraft types is used for evaluatig the swap of aircraft types for some flights for the best use of all aircraft i etwork operatios. The strategy of flight delays is used for the evaluatio of delays for certai flights or all flights i the etwork. he strategy of the modificatio of oe-stop flights is used for evaluatig the cacellatio of oe-stop flight segmets for some or all oe-stop flights whe the aircraft are too few for service, due to a schedule perturbatio. The strategy of the ferryig of idle aircraft is used to evaluate the ferryig of idle aircraft to where ad whe the system eeds them for the best routig. Modificatios for rules (b) ad (d) ca be referred to Ya ad Yag (1996); while modificatios for rule (c) ca be referred to Ya ad Youg (1996). We ote that additioal side costraits should be added to the basic model if rule (b) or (c) is icorporated to form a strategic model Ya ad Yag (1996); Ya ad Youg (1996). However, if oly rule (d) is icorporated ito the basic model, the o additioal side costraits but additioal arcs are added to the basic etwork Ya ad Yag (1996). Rule (a) is suitable for a multi-fleet operatio. A example of the modificatios for this rule is show i Figure 3; assume that there are three types of aircraft (three fleets) i operatio where the capacity of type A is the smallest ad that of type C is the largest. Sice larger aircraft ca serve smaller-type flights, smaller-type flight arcs ca be added ito larger-type fleet etworks. For example, as show i Figure 3, type A flight arcs (e.g. a1 ad a2 ) ca be added ito the type B ad type C etworks. Type B flight arcs (e.g. b1 ad b2 ) ca be added ito the type C etwork. Some of the type B flight arcs (e.g. b1 ad b2 ) may be added ito the type A etwork. If passegers ot able to get o the type A aircraft (because of fewer seats), they ca be reaccommodated o other suitable flights or alterate modes to the same destiatios. Similarly, some type C flight arcs (e.g. c1 ad c2 ) may be added ito the type A or type B etwork. Note that, similar to flight cacellatios, additioal charges may be icurred for reaccommodatig passegers who are uable to get o the smaller-sized aircraft, o suitable flights or alterate modes to the same destiatios. Also ote that, whe usig aother type of aircraft to serve a flight, this type of aircraft should be feasible i terms of the flight mileage ad the associated airport facilities. If flight swaps happe whe usig aother type of aircraft to serve flights, the swap costs (for example, the additioal costs of switchig gates or switchig crew members) should be icluded whe modelig. After all, the smaller-type flight arc cost i the larger-type fleet etwork (e.g. a1 i the type C fleet etwork) equals the larger-type of aircraft's flight expeses, plus the swap cost, mius the

6 36 o-board passeger reveue. Similarly, the larger-type flight arc cost i the smaller-type fleet etwork (e.g. b1 i the type A fleet etwork) equals the smaller-type of aircraft's flight expeses, plus the swap cost, mius the o-board passeger reveue, plus the additioal charges acquired for passegers ot gettig o the smaller-type aircraft. Plae A Plae B Plae C City 1 City 2 City 3 City 1City 2City3 City 1City 2City 3 The startig time a1 a1 a1 The maiteace time c1 c1 c1 c2 b1 b1 b1 c2 c2 a2 a2 a2 The recovery time b2 b2 b2 The edig time a1, a2: plae A (small type) flight arc b1, b2: plae B (media type) flight arc c1, c2: plae C (large type) flight arc Figure 3 Network modificatios for the swap of aircraft types. It should be metioed that although modificatios of the basic model for rules (b), (c) ad (d) i each fleet etwork are referred to Ya ad Yag (1996), ad Ya ad Youg (1996), if rule (a) is applied together with rule (b), the, because at most oe departure time is assiged for a flight, a side costrait should be itroduced for a flight arc ad its alterate flight arcs amog all associated fleet etworks to esure that at most oe flight is served. Similarly, if rule (a) is applied together with rule (c), the the additioal side costrait metioed i Ya ad Youg (1996) should be exteded across all the associated fleet etworks. Besides, to esure that each o-stop flight (which is modified from a oe-stop flight) is served at most oce, a side costrait should be itroduced for each o-stop flight across all the associated fleet etworks. We ote that models cotaiig rule (a) ca be characterized as multi-fleet schedulig models; otherwise they ca be characterized as sigle-fleet schedulig models, because the etworks i ay of them are idepedet of each other. Carriers may create differet models based o the basic model ad the selected schedulig rules, ad the choose the best oe for applicatio. Because the problem size i actual operatios is usually large, we also developed a automatic data process i our research to help users readily apply the models i actual operatios. I particular, with the user's choice of strategies, icludig the swap of possible aircraft types, possible delayed flights, oe-stop flights for possible modificatios, or OD pairs for possible ferryig, this process ca automatically build model iput, develop the strategic model, optimize the model, ad create the fial output, all withi the computer eviromet. We also ote that whe users apply the framework i actual operatios a decisio support system (DSS) icorporatig the automatic data process, the optimizatio algorithm ad a user-friedly iterface would be better. Such a DSS ca be a directio of future research. 3. SOLUTION METHODS The aforemetioed models are formulated as pure

7 37 etwork flow problems or multi-commodity etwork flow problems. I particular, the basic model ad the strategic model usig oly rule (d) are formulated as pure etwork problems. The others are multi-commodity etwork flow problems, which are characterized as NP-hard problems Garey ad Johso (1979). Sice these models are desiged to have optimal solutios, i actual applicatios they do ot geerate ifeasible or ubouded solutios. Referrig to Ya ad Yag (1996), we suggest usig the etwork simplex method to solve the pure etwork flow problems due to its demostrated efficiecy. We modified the Lagragia relaxatio-based algorithm developed by Ya ad Yag (1996) to solve the multi-commodity etwork flow problems. I particular, we used the subgradiet method developed by Camerii et al. (1975), istead of the oe suggested by Fisher (1981), alog with the Lagragia heuristic, the etwork simplex method, ad the Lagragia relaxatio techique to develop the algorithm. After much testig of our problems, we foud that our algorithm performed better tha the oe developed by Ya ad Yag (1996). The solutio steps are summarized as follows: Step 0: Set the iitial Lagragia multipliers. Step 1: Use the techique of Lagragia relaxatio to relax the side costraits with the Lagragia multipliers to form a Lagragia problem. Obviously, the Lagragia problem ca be decomposed ito several idepedet pure etwork flow problems. Optimally solve the Lagragia problem usig the etwork simplex method to get a lower boud. Update the lower boud. Step 2: Apply the Lagragia heuristic Ya ad Yag, (1996) to fid a upper boud ad update the upper boud. Step 3: If the gap betwee the lower boud ad the upper boud is withi a 0.1% error, or the umber of iteratios reaches 500, stop the algorithm. Step 4: Adjust the Lagragia multipliers usig the subgradiet method Camerii et al. (1975). Step 5: Set k = k + 1. Go to Step 1. Sice the solutios obtaied are all arc flows ad are icapable of expressig the route of each airplae, we used the same flow decompositio method Ya ad Yag (1996) to decompose the arc flows ito arc chais, with each arc chai represetig a airplae's route i the perturbed period. Note that the arc chais may ot be uique. The plaers ca choose several solutios ad sed them to other divisios for the applicatio of the other operatig costraits, for example, crew availability. Thus, a satisfactory solutio for all divisios could be relatively easy to fid. 4. CASE STUDY To demostrate our models, we use a case study based o data from a major Taiwa airlie s iteratioal operatios. There are 24 cities ivolved i its operatios. The flight timetable, used for the testig, is rotated oce a week ad icludes 273 flights. About 20 percet of them are oe-stop flights; the others are o-stop flights. There are several types of aircraft ivolved i its operatios, icludig B737 s, AB3 s, AB6 s, MD11 s, B74L s, B744 s ad B747 s. For simplificatio, we had three types of aircraft i this case study; type A idicates B737 s (3 airplaes) with 120 seats, type B icludes AB3 s, AB6 s, MD11 s ad B74L s (17 airplaes) with a average of 269 seats ad type C icludes B744 s ad B747 s (6 airplaes) with a average of 403 seats. For ease i testig, all the cost parameters were set accordig to the airlie's reports ad the Taiwa govermet regulatios, with reasoable simplificatios. Note that accordig to the airlie carrier; the swap cost for differet aircraft types i its operatios is very small compared to the flight cost. For simplificatio, the swap cost betwee differet fleets is assumed to be zero. Besides, sice the airlie has cotracts with other airlies for trasportig passegers uder irregular operatios, without additioal charges, for simplificatio, we assume that the additioal charges, for reaccommodatig passegers ot gettig o the smaller-sized aircraft for suitable flights to the same destiatios, are zero. Note that the cost typically affects the test results. As the case study is oly for demostratio purposes at the curret stage, the evaluatio of the applicatio of this framework to actual operatios is left to future work. For the simplificatio of strategy (b), we oly add a alterate flight arc after each flight arc, each alterate flight deotig a delay of 30 miutes (i other words, we do ot allow a delay of more tha 30 miutes i this test). For strategy (d), we add positio arcs betwee every OD pair every 12 hours from the startig time to the recovery time. We assume that a airplae is rescheduled for a B check at Taipei, startig at 12:00 AM o Wedesday, ad will be ready for service 24 hours later. Cosequetly, the startig time, by defiitio, is 7:00 AM o Wedesday, the maiteace begiig time is 12:00 AM, the recovery time is 12:00 AM o Thursday ad the edig time by defiitio is 15:00 PM o Friday. Note that the startig time (Wedesday 7:00 AM) is the latest time that the schedule has to be adjusted; otherwise, the specified airplae caot be set to Taipei before Wedesday 12:00 AM. Also, the fleet will retur to its ormal operatio after Friday 15:00 PM at the latest. Eightee scearios were doe i this test. Sceario 1 (idicated as Normal ) deotes a ormal operatio. As i Gershkoff (1987), Sceario 2 ( SSP ) applies the successive shortest paths to fidig a series of aircraft routes from the startig time to the edig time. The umber of caceled flights ca thus be calculated after fleet assigmet. To assure that the specified aircraft ca be set to the maiteace time/space poit ad the fleet ca resume its ormal operatios after the edig time, ferry flights could be used. We ote that the SSP method is easy to implemet usig our etworks. I particular, the label correctig algorithm Ahuja (1993) is applicable for solvig

8 38 the fleet assigmet. The SSP method used here is for the prelimiary evaluatio of our models. Scearios 3 through 18 are associated with our strategic models. For example, Sceario B idicates the basic model (flight cacellatios); Ba deotes the combied model for flight cacellatios ad the swap of aircraft types; Bb deotes the combied model for flight cacellatios ad flight delays; Bc deotes the combied model for flight cacellatios ad the modificatio of multi-stop flights; Bd deotes the combied model for flight cacellatios ad the ferryig of idle aircraft; ad Babcd deotes the combied model for flight cacellatios, the swap of aircraft types, flight delays, the modificatio of multi-stop flights ad the ferryig of idle aircraft. Models B ad Bd are pure etwork flow problems ad the other models are multi-commodity etwork flow problems. We used the etwork simplex method to solve the pure etwork flow problems ad applied the Lagragia relaxatio-based algorithm to solve the multi-commodity etwork flow problems. The flow decompositio method Ya ad Yag (996) was used to decompose the lik flows ito arc chais i each fleet etwork, with each arc chai represetig a airplae's route i the perturbed period. Several C programs ad a automatic data process were developed for; (1) the aalysis of raw data (2) the buildig of the basic model (3) the developmet ad solutio of the strategic models ad (4) the output of data. The case study was implemeted o a HP735 workstatio. Sixtee strategic models were tested with problem sizes of up to 4,285 odes ad 14,288 arcs. All of the results idicate that the framework could be useful for actual operatios. The results are summarized i Table 1 ad aalyzed below. strategy (1) computatio time (2) # iteratio (3) objective value (Z U ) (4) Table 1. Results of all scearios coverged gap (5) (Z U -Z L )/Z U # odes (6) # arcs (7) (sec) (NT$) % A B C A B C # side costraits (8) Normal ,529, SSP ,776, ,369 1,369 1,369 2,720 2,799 2, B ,896, ,369 1,369 1,369 2,720 2,799 2, Ba ,584, ,369 1,369 1,369 2,785 2,815 2, Bb ,540, ,369 1,369 1,369 2,728 2,855 2, Bc ,896, ,369 1,403 1,371 2,720 2,918 2, Bd ,314, ,369 1,369 1,369 3,761 4,395 5, Bab ,765, ,369 1,369 1,369 2,851 2,886 2, Bac ,584, ,397 1,403 1,405 2,883 2,934 2, Bad ,601, ,369 1,369 1,369 3,826 4,411 5, Bbc ,540, ,369 1,431 1,373 2,728 3,072 2, Bbd ,540, ,369 1,369 1,369 3,769 4,451 5, Bcd ,314, ,369 1,403 1,371 3,761 4,514 5, Babc ,765, ,419 1,431 1,435 3,026 3,103 3, Babd ,928, ,369 1,369 1,369 3,892 4,482 5, Bacd ,521, ,397 1,403 1,405 3,924 4,530 5, Bbcd ,540, ,369 1,431 1,373 3,769 4,668 5, Babcd ,928, ,419 1,431 1,435 4,067 4,699 5,

9 39 sceario # origial flights (9) # flights served by other fleet (10) Table 1. Results of all scearios (cotiued) # caceled flights (11) # delayed flights (12) # modified multi-stop flights (13) # ferry flights (14) # caceled flights ad flight segmets (15) A B C B/A C/B A B C A B C A B C A B C Total Normal SSP B Ba Bb Bc Bd Bab Bac Bad Bbc Bbd Bcd Babc Babd Bacd Bbcd Babcd Note that the mark ---- i Table 1 idicates that the data is ot available for that sceario. Note that B/A i colum (10) idicates the umber of type B flights served by the type A fleet. Similarly, C/B deotes the umber of type C flights served by the type B fleet. (1) The algorithms performed very well, idicatig they could be useful i practice. I particular, models B ad Bd were optimally solved i about 1 secod of CPU time. The other models coverged withi 0.1% of the error i at most 112 secods of CPU time. Note that the CPU times for etwork geeratio ad data output i each sceario are relatively short compared with model solutios ad ca be eglected. This shows that the etwork simplex algorithm should be efficiet for solvig pure etwork problems, like B ad Bd, ad this research idicates that the Lagragia relaxatio-based algorithm could be efficiet for solvig the multi-commodity etwork flow problems. Compared with the efficiecy of the traditioal approach, the algorithms are superior. (2) All of our models yield a higher profit tha the SSP approach does. The best result (-95,928,199), for Models Babcd ad Babd, is closest to the profit achieved i ormal operatios (-96,529,295). Models Babcd ad Babd cause much less profit loss (NT$ 601,096) tha usig the SSP approach (NT$ 2,752,607). Though the models, B ad Bc, are the worst amog the strategic models, their objective (-93,896,259) is better tha that of the SSP method (-93,776,688). The reaso that our models outperform the SSP method could be that although the SSP method cacels a series of uecoomic (7 flights) flights, it does ot cosider the delayig of flights, the modificatio of multi-stop flights, the ferryig of idle aircraft or a combiatio of these i adjustig the schedule. Through the effective adjustmet of flight schedules or fleet routes, fewer flights are caceled (for example, o flight is caceled i Babcd ). Thus, higher profits are achieved. We ote that if more rules except (c) are icorporated to develop a strategic model, the the model will be more flexible for optimizig a temporary schedule, so that system profits ca be improved more. For example, the result for Babcd is better tha that for Babc, which is better tha Bac. The reader ca see other examples i Table 1. e fid that flight cacellatios, flight delays ad ferry flights are suggested by may of the strategic models, as effective methods for schedule adjustmet. We ote that the strategy of (c) is ot effective i this case. For example, the objectives of B ad Bc, Ba ad Bac, Bb ad Bbc, Bd ad Bcd, Bab ad Babc, or Babc ad Babcd, are same. Note that because there is a covergece error for Bacd, the objective of Bad is

10 40 slightly better tha that of Bacd. Sice the strategy of (c) has bee show to be effective i Ya ad Youg (1996), more cases may eed to perform to verify this, which could be a subject of future work. We also fid that the multi-fleet schedulig models (models cotaiig strategy a ), for example, Ba, Bab, Bac, Bad, Babc, Babd, Bacd, or Babcd yield higher profits (more tha 95,000,000) tha the sigle-fleet schedulig models (models excludig strategy a ) (less tha 95,000,000). From colum (10), all eight multi-fleet schedulig models suggest that a fleet ca temporarily serve other types of flights i schedulig, so as to improve the system profit. For example, four type B flights i Model Ba are served by the type A fleet. The result for Model Ba (-95,584,793) is better tha that for the basic model B (-93,896,259), a improvemet of NT$ 1,688,534 (about 1.77%). The reader ca see other examples i Table 1. This implies that the swap of aircraft types is a importat ad effective strategy i schedule adjustmet i the case. (3) Other tha the profit cosideratios, the degree of schedule perturbatio may be a criteria for carriers to evaluate levels of service for all strategic models. Typically the umber of caceled flights ad delayed flights i the case study may serve as a idex of schedule perturbatio. I this study, sice a flight has at most a delay of 30 miutes, its ifluece o the level of service could be omitted whe compared to a caceled flight. Thus, i terms of the level of service, Models Babcd ad Babd could be the best (o caceled flight) ad SSP the worst (7 caceled flights). Note that the multi-fleet schedulig models caceled less flights (less tha or equal to 1 flight) tha the sigle-fleet schedulig models (more tha 1). Cosequetly, cosiderig both profit ad level of service, the multi-schedulig models perform better tha others. I particular, Models Babcd ad Babd are the best of them all. It should be oted that if a solutio obtaied from a strategic model is associated with a large degree of schedule perturbatio, it might ot be acceptable to carriers uder real operatig costraits (for example, crew availability). The, carriers may choose aother model to fid a solutio with fewer flight schedule perturbatios, or they may make mior modificatios of the perturbed schedule to satisfy the operatig costraits. From this, a DSS might be helpful for the applicatio of these strategic models to real time operatios. 5. CONCLUSIONS We develop several etwork models to help carriers hadle schedule perturbatios resultig from the expected aircraft maiteace, for the operatios of multiple fleets as well as o-stop ad oe-stop flights. These models miimize the schedule-perturbed time so that carriers ca resume their ormal services as soo as possible i order to maitai their levels of service. Besides, these models combie flight cacellatios, the swap of aircraft types, flight delays, the modificatio of multi-stop flights, ad the ferryig of idle aircraft i a combied ad systematic framework to effectively adjust a schedule, so that a carrier ca maitai its profitability. These strategic models are formulated as pure etwork flow problems or multi-commodity etwork flow problems. We used the etwork simplex method to solve the pure etwork flow problems ad developed a Lagragia relaxatio-based algorithm to solve the multi-commodity etwork flow problems. To test the models i practice, a case study regardig the iteratioal operatios of a major Taiwa airlie was performed o a HP735 workstatio. Sixtee strategic models ad the SSP approach were tested, with substatial problem sizes of up to 4,285 odes ad 14,288 arcs. Several C programs ad a automatic data process were developed to apply these models usig a HP735 workstatio. The algorithms performed very well. I particular, models B ad Bd were optimally solved i about oe secod of CPU time; other models coverged to 0.1% of the error i at most 112 secods of CPU time. All our models out performed the SSP approach ad improved the traditioal schedulig process sigificatly. All of these showed that our models could be useful for actual operatios. Sice the case study is oly for demostratio at the curret stage, the evaluatio of impact o the applicatio of this model to actual operatios is left to future work. Although a multi-stop flight is evaluated as to whether to cacel its segmets i our research, how to combie the flight segmets of differet flights to form a multi-stop flight ca be a directio of future research. How to combie several flights of smaller types to form a larger type flight, ad systematic models for hadlig schedule perturbatio caused by other types of expected evets, ca be performed i the future research as well. Fially, a computerized decisio support system for users, useful for applyig these strategic models i practice, could also be aother directio of future research. ACKNOWLEGEMENTS This research was partially supported by a grat from the Natioal Sciece Coucil of Taiwa. We thak the Airlie for providig the test data ad valuable suggestios. REFERENCES 1. Ahuja, R.K., Magati, T.L., ad Orli, J.B. (1993). Network Flows, Theory, Algorithms, ad Applicatios. Pretice Hall, Eglewood Cliffs, New Jersey. 2. Biro', M., Simo, I., ad Ta'czos, C. (1992). Aircraft ad maiteace schedulig support, mathematical isights ad a proposed iteractive system. Joural of Advaced Trasportatio, 26: Camerii, P.K., Fratta, L. ad Maffioli, F. (1975). O improvig relaxatio methods by modified gradiet techiques. Mathematical Programmig Study, 3: Deckwitz, T.A. (1984). Iteractive Dyamic Airplae Schedulig. Flight Trasportatio Laboratory, Report R-84-5, Massachusetts Istitute of Techology.

11 41 5. Du, Y., ad Hall, R., (1997). Fleet sizig ad empty equipmet redistributio for ceter-termial trasportatio etworks. Maagemet Sciece, 43: Etschmaier, M.M. ad Mathaisel, D.F.X. (1985). Airlie schedulig: A overview. Trasportatio Sciece, 19: Etschmaier, M.M. ad Rothstei, M. (1973). Estimatig the Puctuality Rate Iheret i a Airlie Schedule. Techical Report 19, Departmet of Idustrial Egieerig, Uiversity of Pittsburgh. 8. Feo, T.A., ad Bard, J.F. (1989). Flight schedulig ad maiteace base plaig. Maagemet Sciece, 35: Fisher, M.L. (1981). The Lagragia relaxatio method for solvig iteger programmig problems. Maagemet Sciece, 27: Garey, M.R. ad Johso, D.S. (1979). Computers ad Itractability: A Guide to the Theory of NP-completeess.W.H. Freema & Compay, Sa Fracisco. 11. Gershkoff, I. (1987). Aircraft shortage evaluator. preseted at ORSA/TIMS Joit Natioal Meetig, St. Louis, MO. 12. Jarrah, A.I., Yu, G., Krishamurthy, N., ad Rakshit, A. (1993). A decisio support framework for airlie flight cacellatios ad delays. Trasportatio Sciece, 27: Jedlisky, D.C. (1967). The Effect of Iterruptios o Airlie Schedule Cotrol. Master Thesis, Departmet of Aeroautics ad Astroautics, Massachusetts Istitute of Techology. 14. Keyo, A.S., ad Morto, D.P., (2003). Stochastic vehicle routig with radom travel times. Trasportatio Sciece, 37: List, G.F., Wood, B., Nozick, L.K., Turquist, M.A., Joes, D.A., Kjeldgaard, E.A., ad Lawto, C.R., (2003). Robust optimizatio for fleet plaig uder ucertaity. Trasportatio Research E, 39: Mulvey, J.M., ad Ruszczyski, A., (1995). A ew sceario decompositio method for large-scale stochastic optimizatio. Operatios Research, 43: Teodorovic, D. (1988). Airlie Operatios Research. Gordo ad Breach Sciece Publishers, New York. 18. Teodorovic, D. ad Guberiic, S. (1984). Optimal dispatchig strategy o a airlie etwork after a schedule perturbatio. Europea Joural of Operatioal Research, 15: Teodorovic, D. ad Stojkovic, G. (1990). Model for operatioal daily airlie schedulig. Trasportatio Plaig ad Techology, 14: Wallich, P. (1986). Keepig them Flyig. IEEE Spectrum, 23: Ya, S. ad Yag, D. (1996). A decisio support famework for hadlig schedule perturbatio, Trasportatio Research B, 30: Ya, S. ad Youg, H. (1996) A decisio support framework for multi-fleet routig ad multi-stop flight schedulig. Trasportatio Research A, 30: