Research Article Optimization Model and Algorithm Design for Airline … · 2019. 7. 31. ·...

Research ArticleOptimization Model and Algorithm Design for Airline FleetPlanning in a Multiairline Competitive Environment

Yu Wang12 Hong Sun2 Jinfu Zhu1 and Bo Zhu1

1College of Civil Aviation Nanjing University of Aeronautics and Astronautics Nanjing Jiangsu 210016 China2School of Air Transportation Management Civil Aviation Flight University of China Guanghan Sichuan 618307 China

Correspondence should be addressed to Yu Wang wangyu2001111163com

Received 1 October 2014 Revised 9 January 2015 Accepted 23 January 2015

Academic Editor Masoud Hajarian

Copyright copy 2015 Yu Wang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a multiobjective mathematical programming model to optimize airline fleet size and structure withconsideration of several critical factors severely affecting the fleet planning processThemain purpose of this paper is to reveal howmultiairline competitive behaviors impact airline fleet size and structure by enhancing the existing route-based fleet planningmodelwith consideration of the interaction between market share and flight frequency and also by applying the concept of equilibriumoptimum to design heuristic algorithm for solving themodelThrough case study and comparison the heuristic algorithm is provedto be effective By using the algorithm presented in this paper the fleet operational profit is significantly increased compared withthe use of the existing route-based model Sensitivity analysis suggests that the fleet size and structure are more sensitive to theincrease of fare price than to the increase of passenger demand

1 Introduction

The fleet planning decision-making process is considered tobe one of the most problematic issues for airline industryAn over-large fleet size would cause an airline unnecessaryexpense since the increasing capital assets account for alarge proportion of the airline operational costs On thecontrary an underestimated fleet size would also result ina great number of passengers overflowing to other marketcompetitors Moreover considering the profit margin of theairline industry around theworld continuously pressured by along-term exposure to a high-cost and low-fare environmentthe irrational fleet composition would necessarily deterioratethe airlinersquos operationTherefore airlinesmay have to developa more practical fleet planning approach to meet passengerdemand with lower costs and more controllable risks at astrategic level

The aim of airline fleet planning is to determine thefleet size and structure for a given operational environmentincluding network characteristics flight schedule and meanfare levels Macro-fleet planning is considered to be oneof the most popular approaches around the world where

network-based passenger demand within a future area isused to estimate the needed number of aircraft of differenttypes for a given candidate aircraft type set However theoversimplifying macro-approach is hardly to reflect theadaptability of a specific type of aircraft flying on route forexample aircraft of common types without modification topropulsion and oxygen system are forbidden to fly on plateauroutes In addition the economic feature is also beyondthe consideration scope of macro-fleet planning approachfor example the passenger-spilling problem in single aisleaircraft with small seating capacity on heavily traveled routesas well as the vacant seat problem in two-aisle aircraft withbig seating capacity flying on less traveled routes

In order to avoid these disadvantages more attentionshave been paid to the application of micro-fleet planningapproaches where passenger demand on a single route orflight leg is accommodated by different types of aircraftThenthe number of different types of aircraft flying on every routeis aggregated to determine the fleet size and structure

Dynamic fleet management is one of the most importantbranches in micro-fleet planning approaches where thefleet assignment technique [1ndash3] is widely used to optimize

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 783917 13 pageshttpdxdoiorg1011552015783917

2 Mathematical Problems in Engineering

the fleet size and structure under the condition that the futuredetailed flight schedule has been already presented In thisaspect Listes and Dekker used time-space network to con-struct a fleet assignment-based model to determine the fleetcomposition They also developed a scenario aggregation-based algorithm to solve the model [4] Wang and Sunutilized simulated annealing algorithm to solve airline fleetplanning problem and discussed a robust airline fleet plan-ning method [5 6] However this kind of approach isbased on a given flight schedule which is hardly simulateddue to the uncertainty of airlinersquos future environment Thisdrawback may result in an unreliable fleet size and structuredeprived from the fleet assignment-based approaches

Therefore recent relevant studies have focused mainlyon route-based fleet planning approaches in which the bestaircraft type or aircraft type mix is assigned to each route tomaximize the fleet operational profit (or minimize the fleet-related costs) In this aspect Schick and Stroup proposed amultiyear fleet planning model with consideration of passen-ger demand constraints and aircraft balance equations aswellas minimum andmaximum flight frequency to minimize thefleet-related-costs [7] Sun et al applied a similar model toanalyze the Chinese market [8] Wang et al presented a newfleet planning approach for those airlines operating in Huband Spoke network in which network effects are highlighted[9] Wei and Hansen discussed the competitive relationshipbetween aircraft size and flight frequency using game theoryIt was concluded that the extra landing fee could reduceflight delay and airport congestion [10 11] Takebayashiconstructed a supply-demand interactionSDI model forHaneda airport He held the view that airlines did not alwaysadopt a downsizing aircraft size strategy in response to airportrunway expansion [12] Tsai et al incorporated the constraintof the European Emissions Trading Scheme into a mixedactivity-based fleet planningmodelThey believed that a self-purchased wide-bodied airplane could benefit from high rev-enue tone kilometers [13] Givoni and Rietveld evaluated theimpact of environmental factors on the choice of aircraft sizeThey thought that environmental improvement could benefitfrom those airlines using large aircraft size [14] Rosskopfet al proposed a multiobjective linear programming modelto study the trade-off between economic-environment goalsThey argued that the environment goal could be achieved by6 improvement at the cost of 3 deviation from economicoptimum [15] Pai analyzed the main factors affecting thechoice of aircraft size and flight frequency [16] Other relevantstudies [17 18] analyzed some external impacting factors onfleet size and structure (eg Brown 1992 Bahram et al1999)

Previous studies on route-based fleet planning approach-es investigated the issue mainly depending on airline itselfFew researches have taken into account the impact of otherairlinesrsquo competitive behaviors on the airlinersquos operationThispaper aims to optimize the fleet size and structure throughnetwork-wide allocating different types of aircraft and flightfrequency under multiairline competitive behaviors Effortsare made to construct a multiobjective model to maximizeeach airline fleet operational profit subject to the availableflight frequency offered to each route and air-crew flying

hours for each aircraft fleet type This study formulates thefleet operational profit as evaluating function including fleetoperational costs penalty cost and passenger spilling costIn addition this paper also develops an effective algorithmto solve the proposed model The validation and benefits ofthe model are shown with a case study Therefore the maincontributions of this paper can be summarized as follows

(i) We develop a new route-based fleet planning modelwhich is capable of reflecting the impact of multiair-line competitive behaviors on fleet size and structure

(ii) We devise a heuristic algorithm for our route-basedmodel and show its validation through Monte Carlosimulation

(iii) Through case study using real airline data we quantifyto analyze the benefits of the model presented in thispaper

(iv) Through sensitivity analysis we find the importantfactors impacting airline fleet size and structure

The remainder of this paper is organized as follows Inthe next section the problem is presented in detail includ-ing mathematical modeling in competitive environment InSection 3 a heuristic algorithm is introduced according toequilibrium optimum theory For a case with real airlinedata the multiobjective function is solved by the proposedalgorithm coding with MATLAB software in Section 4 AndSection 5 is the conclusion of this paper

2 Problem Modeling

In this section the paper illustrates how multiairline com-petitive behaviors affect the choice of aircraft size and flightfrequencyThenwe extend the existing route-based approachlisted in reference [8] to construct a network-wide allocationmodel of the choice of aircraft size and flight frequency withmultiairline competitive behaviors

21 Problem Description Traditional viewpoint generallyholds that an airlinersquos choice of aircraft size and flightfrequency largely depends on the network-wide distributionof the airlinersquos passenger demand However few studies [7ndash9] inversely consider that an airlinersquos choice of aircraft sizeand flight frequency in a competitive environment is alsogreatly affecting the airline capability of capturing passengerdemand 119878 curve presented by Simpson [19] in the last centuryis considered as one of most famous functions around theworld for evaluating airline market share which is defined asa ratio of captured demand for an airline to the total demandin the market It is concluded that airline market share is afunctionwith respect to flight frequency and yields to 119878 shapeRecently Wei and Hansen [20] once again have confirmed119878 curve function widely existing in airline market Based onthese evidences 119878 curve function is used throughout thispaper to analytically express the relationship between airline

Mathematical Problems in Engineering 3

market share and flight frequency The mathematical modelcan be written as

MS119894=

(119873119894)120573

sum119900

119895=1(119873119895)120573 (1)

where MS119894is the market share for airline 119894 on a competitive

route 119873119895is the number of flights that airline 119895 (119895 =

1 2 119894 119900) provides to the competitive route 120573 meansmarket share index Symbol 119900 denotes the total number ofairlines on the route

Previous route-based approachesmust predict an airlinersquosdemand in advance and then use the proper number offlights provided by different types of aircraft to determine theairlinersquos fleet size and structure while satisfying its predicteddemand [7ndash9] This process obviously neglects the impactof the choice of aircraft size and flight frequency on theairline captured passenger demand In order to capture thedesired market share the airline must provide a certain flightfrequency Furthermore the flight frequency that the airlineprovides alone could not wholly determine its market sharewhich is also affected by the flight frequencies provided byother airlines in the same market

In order to facilitate the description of the problem weassume that two airlines (named A and B for simplicity)operate on the same route in which 600 total passengers existThe mean fare level is 1000 Yuan and the market share index120573 = 1 Both of airlines use aircraft type 1 with 50 seats andaircraft type 2 with 100 seats as their candidate aircraft typesSuppose that the unit operational cost of aircraft 1 and aircrafttype 2 is 14000 and 20000 Yuan respectively When airlinesA and B provide three flights with aircraft type 2 respectivelythey all account for half market share and the correspond-ing operational profit reaches 240000 Yuan respectivelyIn this situation when airline A increases the number offlights to six then the market share of airlines A and B ischanged to 67 and 33 respectively The correspondingfleet operational profit of airline A reaches 280000 Yuanbut the operational profit of airline B is reduced to 140000Yuan This evidence shows how one airlinersquos choice of flightfrequency impacts other airlinesrsquo operational profits If weuse four airplanes with type 1 to replace three airplanes withtype 2 for airline A then the fleet operational profit becomes30400YuanThis suggests that an airline can also increase thefleet operational profit through optimizing its fleet structureTherefore the choice of aircraft size and flight frequency notonly affects the airline itself but also significantly influencesother airlinesrsquo decision policies in the same market Usingprevious fleet planning approaches without consideration ofairlinesrsquo competitive behaviors the resulting fleet size andstructure would be unreliable As a result it is necessary forairlines to develop a more practical fleet planning approachto deal with the multiairline competitive behaviors

22 Assumptions and Limitations

(i) Airlines in competitive environment are rationalparticipants and the objectives of them are all tomaximize fleet operational profits

(ii) Airlinemarket share is affected only by the number offlights the airline offered to the market (route)

(iii) All information such as fare price passengerdemand and candidate-aircraft-type-related infor-mation is known during the planning period

(iv) Critical resources such as flight frequency and air-crew flying hours are known and finite during theplanning period Moreover none of them can beexceeded

(1) Sets

119868 = 1 2 119866 set of airlines in a competitive net-work environment indexed by 119894119869119894= 1 2 119867

119894 set of routes in a competitive net-

work environment indexed by 119895119870119894= 1 2 119876

119894 set of candidate aircraft types for

airline 119894 in a competitive network environmentindexed by 119896

(2) Parameters

119901119894119895 the average fare price on route 119895 for airline 119894

119888119894119895119896 the fleet related costs per flight for airline 119894 flying

on route 119895 with aircraft type 119896119904119894119895119896 the available seats offered by airline 119894 flying on

route 119895 with aircraft type 119896119863119895 the passenger demand on route 119895

119887119894119895119896 the average flying hours offered by airline 119894 flying

on route 119895 with aircraft type 119896119865119894119895 the maximum flight frequency offered by airline 119894

on route 119895119879119894119896 the maximum air-crew flying hours in aircraft

fleet type 119896 for airline 119894119906119894119896 the expected aircraft utilization rate of aircraft

type 119896 for airline 119894120573119895 the market share index for route 119895

120576 a torrent term used to reflect the extent to which theplanning manager allows the loss of fleet operationalprofit120575119894119895119896 = 1 if candidate aircraft type 119896 is airworthy on

route119895 for airline 119894 otherwise = 0119872 a sufficient large positive number

(3) Decision Variables

119909119894119895119896 the number of flights on route 119895 flown by aircraft

type 119896 for airline 119894119910119894119895 the spilling number of passengers on route 119895 for

airline 119894

(4) Functions

119911119894 the fleet operational profit of airline 119894 a function

with respect to 119909119894119895119896

and 119910119894119895

MS119894119895 the market share of airline 119894 on route 119895


23 A Unified Evaluating Function In general previousmathematical models [5ndash7] for airline fleet planningapproaches use fleet operational costs as objective functionwhich can hardly reflect the impact of route fare levels onfleet size and structure and finally lead to a conservativeresult Furthermore airworthiness and passenger spilling aretwo important factors that should be sufficiently consideredin evaluating each airlinersquos operation Therefore a unifiedevaluating function (called ldquofleet operational profitrdquo forsimplicity) needs to be defined and it should mainly includethe following components

(i) Fleet operational costs it is the sum of the costitems directly related to the operation on a routewith a given aircraft type including fuel consumptionlanding and take-off fees and gate rental Besidesthat the capital assets related to the depreciation andamortization of different types of aircraft should bealso included in these cost items

(ii) Penalty cost the cost item used to reflect a routewhether a specific aircraft type can fly or not A suffi-cient positive number (119872) will be added to the routeoperational costs when airline uses an unqualifiedaircraft type to fly otherwise the penalty cost equalingzero

(iii) Passenger spilling cost the cost item represents thepassenger revenue loss due to an insufficient aircraftcapacity

(iv) Passenger revenue the revenue item represents thegenerated revenue from an airline capturing passen-gers

24 Mathematical Formulation As is analyzed in Section 21the coupling relationship for an airline itself is not onlyreflected by the interaction between the choice of aircraftsize and flight frequency and the airline capturing passen-ger demand but also reflected by other airlinesrsquo competi-tive behaviors Moreover this coupling relationship wouldbecome more complicated in a network-wide environmentbecause airlinesrsquo competitive behaviors suffer more limita-tions from their critical resources

(i) Maximum flight frequency on each route there existsan upper-bound in airline flight frequency on each routedue to the airspace structure and management model as wellas allocating policy on airport slots over a period of timeThe lack of runway slots especially in congested airports isregarded as a serious bottleneck for airlines to provide betterservices In this context the limited flight frequency tends tobe offered to more profitable markets and the wide-bodiedaircraft types with more seating capacity are used so longas the growth of passenger demand can afford their highoperational costs Therefore the flight frequency is such acritical resource that inevitably affects airline fleet size andstructure The mathematical formulations can be written as

sum

119896isin119870119894

119909119894119895119896le 119865119894119895 forall119894 isin 119868 119895 isin 119869

119894 (2)

Constraints (2) are maximum flight frequency constraintsfor each airline ensuring that the number of flights cannotexceed the maximum flight frequency on each route

(ii) Maximum captured passenger demand on each routean airline captured demand on a route depends on thenumber of flights that both the airline provides and othercompetitive airlines provideMeanwhile an airline could alsouse different mixes of aircraft types to balance its captureddemand and the corresponding supply But in any case airlineenplaned passengers could not exceed the airline captureddemand The mathematical formulations can be written as

sum

119896isin119870119894

119904119894119895119896119909119894119895119896+ 119910119894119895= MS119894119895119863119895 forall119894 isin 119868 119895 isin 119869

119894 (3)

Constraints (3) are passenger flow conservation constraintsfor each airline ensuring the enplaned and spilled passengerson each route equal to the market the airline captures Inaddition MS

119894119895can be mathematically written as

MS119894119895=

(sum119896isin119870119894

119909119894119895119896)120573

sum119894isin119868(sum119896isin119870119894

119909119894119895119896)120573 forall119894 isin 119868 119895 isin 119869

119894 (4)

where term sum119896isin119870119894

119909119894119895119896

is the number of flights that airline 119894provides to route 119895

(iii) Maximum air-crew flying hours of each aircraftfleet type to ensure flight safety the number of air-crewflying hours for each aircraft fleet type is strictly limitedby civil regulations (eg 100 flying hours per month forone pilot) These limitations make some routes incapableof being flown by the desired aircraft types because theair-crew flying hours have been wholly exhausted during aperiod of time (eg one year) This means no qualified pilots(compliance with regulation requirements) are left to drivethe corresponding aircraft to fly these routes This limitationmaybe results in some other substitute aircraft types to flythose routes Therefore the maximum air-crew flying hoursfor each aircraft fleet type necessarily affect the fleet planningprocess and mathematically can be written as

sum

119895isin119869119894


119894 (5)

Constraints (5) are the limitations of available air-crewflying hours ensuring that the number of flying hours for aspecific aircraft fleet type can not exceed the correspondingmaximum number

Constraints (6) are the numeric types and value ranges ofdecision variables

119909119894119895119896ge 0 int 119910

119894119895isin 119877 forall119894 isin 119868 119895 isin 119869

119894 119896 isin 119870

119894 (6)

Note that decision variables 119910119894119895are generally relaxed to

a real numeric type because airline market share typicallybelongs to [0 1]


Finally the multiairline objective function maximizingeach airlinersquos fleet operational profit can be mathematicallywritten as

max 119911119894= sum

119895isin119869119894

sum

119896isin119870119894

(119901119894119895119904119894119895119896minus 119888119894119895119896minus119872120575

119894119895119896) 119909119894119895119896

minus sum

119895isin119869119894

119901119894119895

10038161003816100381610038161003816119910119894119895

10038161003816100381610038161003816 forall119894 isin 119868

(7)

where the first term in right hand means the fleet planningprofit comprised of (1)ndash(4) items described in Section 23The second term also in right hand represents the passengerspilling cost when decision variables 119910

119894119895are positive If not an

offset item is generated to offset the overcalculated value fromthe first term due to the surplus of seating capacity unfilledwith passengers when decision variables 119910

119894119895are negative

3 Algorithm Design

Model (2)ndash(7) is a nonlinear mix integer programmingproblem in which one airlinersquos network-wide distribution ofthe number of flights with different types of aircraft affectsthe fleet operational profit of both the airline and othercompetitors in the same network environment Meanwhileit tends to be accompanied with other airlinesrsquo decisionpolices changed Hence our interests are in the equilibriumoptimal solution [21 22] to the multiobjective function Inthis section we introduce a heuristic algorithm to solve theproposed model

31 Description of Equilibrium Optimum Consider 119899 airlinesin the game of network-wide distribution of aircraft sizeand flight frequency with 119873 and 119894 representing the set ofairlines and a particular airline respectively 119894 isin 119873 Let119883119894be the distributing policy set of airline 119894 and let 120601

119894

be the fleet operational profit of airline 119894 One policy set(1199091 1199092 119909

119899) can be constructed when each airline selects

its own distributing policy 119909119894(119909119894isin 119883119894) We define (1199091015840

119894| 119883) equiv

(1199091 119909

119894minus1 1199091015840

119894 119909119894+1 119909

119899) as a distributing policy set with

airline 119894 adopting policy 1199091015840119894while other competitors remain

distributing policy 119909119895(119895 = 1 119894minus1 119894+1 119899) unchanged

Then the equilibrium optimal solution119883lowast could be achievedwhen each airlinersquos fleet operational profit is all maximized inthe game or mathematically written as

120601119894(119883lowast) = max

(119909119894|119883lowast)isin119883

120601119894(1199091015840

119894| 119883) forall119894 (8)

We introduce formulation (9) as a function so as to easilysolve the equilibrium optimal solution

120593 (119883 1199091015840) = sum

119894isin119868

(120601119894(1199091015840

119894| 119883) minus 120601

119894 (119883)) (9)

Function 120593(119883 1199091015840) is a sum of airlinesrsquo increased profits wheneach airline 119894 changes its distributing policy from 119909

119894to 1199091015840119894 If

policy 119909119894is the best solution to airline 119894 then equation 119909

119894= 1199091015840

119894

holds Otherwise the policy 119909119894is replaced by 1199091015840

119894 Therefore

equilibriumoptimal solution119883lowast will be foundwhen function120593(119883 119909

1015840) equals zero or mathematically written as

119885 (119883lowast) = arg max

1199091015840isin119883

120593 (119883 1199091015840) (10)

32 Optimization Procedure Therefore the model presentedin this paper must be iteratively solved since the solutionto one airline is based on other airlinesrsquo given solutionsaccording to formulations (8)ndash(10) In order to obtain theequilibrium optimum solution each airline is selected inorder of increasing airline number For each airline the mostprofitablemix of aircraft type and route is selected one by oneThen the algorithm determines the corresponding number offlights to ensure that the increased fleet operational profit isalways maximized so long as the remaining number of flightson the route and the remaining air-crew flying hours of theaircraft fleet type are still positive After the algorithm visitingall of feasible mixes the current solution is determinedThen according to a certain principle both current and lastsolutions are used to update the next iterative solution untilthe increased fleet operational profit for each airline cannot be improved any more which suggests the equilibriumoptimum solution is obtained This process can be depictedas the flow chart shown in Figure 1

Algorithm Procedure

Step 1 Initialize the maximum iteration number (119898119886119909119894119905119890119903119886119905119894119900119899 119899119906119898119904) the number of airlines (119886119894119903119897119894119899119890 119899119906119898119904) Setiterative count 119905 = 0 airline number 119894 = 0 and decisionvariables 119909119905

119894119895119896= 0 forall119894 119895 119896

Step 2 If 119905 lt 119898119886119909 119894119905119890119903119886119905119894119900119899 119899119906119898119904 then use formulations(2)ndash(7) to calculate airline fleet operational profit 119901119903119900119891119894119905119905

119894 forall119894

and otherwise go to Step 10

Step 3 Set airline number 119894 = 119894 + 1 If 119894 lt 119886119894119903119897119894119899119890 119899119906119898119904 theninitialize the set of assigningmixes of aircraft types and routesby formulation (11) and otherwise go to Step 9

119905119910119901119890 119903119900119906119905119890 119904119890119905119894

= (119895 119896) | (119901119894119895119904119894119895119896minus 119888119894119895119896) gt 0 119895 isin 119869

119894 119896 isin 119870

119894

(11)

Step 4 Select the mix of aircraft type and route withmaximum fleet operational profit for airline 119894 from the set119905119910119901119890 119903119900119906119905119890 119904119890119905

119894using the following formulation

(119895lowast 119896lowast) = (119895 119896) | max

(119895119896)isin119905119910119901119890 119903119900119906119905119890 119904119890119905119894

119901119894119895119904119894119895119896minus 119888119894119895119896 (12)

Step 5 Calculate current maximum flight frequency119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904

119894119895lowast for airline 119894 using formulation (13) with

aircraft type 119896 flying on route 119895

119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast = min119865

119894119895lowast lfloor

119879119894119896lowast

119887119894119895lowast119896lowast

rfloor (13)


Start

Are they positive

Select the most profitable assigning mix of aircra

Find the minimum number of flight frequencies on the

route according to the remaining flying hours of the aircra fleet type and the remaining number of flight frequencies on the

Find all positive unit assignment

mixes of aircra types and routes

Find the best number of flight frequencies and

provide it to the route for

Calculate the remaining resources for both flight frequency on the route and the flying

Initialize flight frequencyaircra type set and route

set for each airline

All airlines are processed

Yes

Calculate the total fleet operational profits

Yes

Approximates to the last one

Calculate the number of flight frequencies in the next iteration for

each airline

End

Calculate the fleet composition for each airline

Yes

NoNo

Remove the aircra type from aircra type set or remove the route from

No

One of sets is null

No

Yes

No

Reach the maximum iterative count

Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i

profits for airline i from all assigning

airline i

hour of the aircraft fleet type for airline i

route set for airline i

type and route for airline i

Figure 1 Flow chart of optimization procedure

Step 6 If 119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast gt 0 then calculate the rev-

enue loss 119910119905119894119895lowast using formulation (14) and the fleet operational

profit for airline 119894 operating on route 119895lowast using formulation(15) respectively so as to find such 119909

119905

119894119895lowast119896lowast that ensure the

condition 119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905

119894119895lowast119896lowast minus 1) and 119911

119905

119894119895lowast(119909119905

119894119895lowast119896lowast) ge

119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905

119894119895lowast119896lowast isin 0 119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904

119894119895lowast holds

and go to Step 8 Otherwise go to Step 7

119910119905

119894119895lowast = MS119905

119894119895lowast119863119895lowast minus sum

119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905

119894119895lowast119896lowast) = sum

119896isin119870119894

(119901119894119895lowast119904119894119896minus 119888119894119895lowast119896) 119909119905

119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816

MS119905119894119895lowast119863119895lowast minus sum

119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)

where MS119905119894119895lowast = (sum

119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)

Step 7 If lfloor119879119894119896lowast119887119894119895lowast119896lowastrfloor le 0 then set 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896lowast) Otherwise set 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=

calculate the fleet operational profit 119901119903119900119891119894119905119905119894for airline 119894

according to current decision variables 119909119905119894119895119896

(forall119895 119896) and go toStep 3 Otherwise go to Step 4

Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905

119894119895lowast119896lowast 119879119894119896lowast = 119879

119894119896lowast minus 119887119894119895lowast119896lowast119909119905

119894119895lowast119896lowast and

119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890

119904119890119905 = then go to Step 3 Otherwise go to Step 4

Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10

Otherwise update decision variables 119909119905

119894119895119896(forall119894 119895 119896) for all

airlines in competitive environment according to formulation(16) set iterative count 119905 = 119905 + 1 and go to Step 2

119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)

where 120572 is a weighting factor used to reflect the relativeimportance for current decision variables 119909119905

119894119895119896


Table 1 Basic information of candidate aircraft types

Airline number 1 2 3 4 5 6 7 8 9 10 11Candidate aircraft type 1sim7 1sim6 1sim6 1sim6 1sim5 1sim3 1sim2 1sim2 1sim4 1sim2 4 1sim2

Step 10 Calculate the number of different types of aircraft foreach airline by term lceilsum

119895isin119869119894119909119894119895119896119906119894119896rceil (forall119894 isin 119868 119896 isin 119870

119894) with a

given utilization rate of each specific aircraft fleet type andterminate

33 Computation Complexity The algorithm procedurepresents the computation complexity in nature It revealsthat the iterative process is terminated when the maximumiterative count (here denoted by 119872) or the tolerant term issatisfied For each iteration each airline needs to calculatethe unit assignment profits of all mixes of aircraft typesand routes which largely depend on the number of bothcandidate aircraft types and the quasi-operating routes in thecandidate route set According to Step 3 the worst situationwill happen at the time that all unit assignment profits arepositive It suggests that all candidate aircraft types flying oneach route need to be calculated one time or mathematicallythe computation complexity can be written as 119874(119867

119894119876119894) for

airline 119894 and 119874(119868119867119894119876119894) for all airlines Another concern on

computation complexity must be given to the selection of thebest number of flights provided by different types of aircraftfor each airline After selecting the most profitable mix ofaircraft type and route the best number of flights providedby the selected aircraft type on the route must be foundin the interval [0 the current remaining number of flights]according to Step 6Note that the upper-bound of the intervalis determined by the minimum one between the remainingnumber of flights from the maximum flight frequency onthe route and the remaining number of flights convertedfrom the remaining air-crew flying hours of the aircraft fleettype for the given route through using term lfloor119879

119894119895119887119894119895119896rfloor This

also can be mathematically expressed as min119865119894119895 lfloor119879119894119895119887119894119895119896rfloor

Therefore the worst situation will happen at the time that theminimum value reaches the maximum It suggests that therange of the interval is extended to the largest Accordinglythe computation complexity can bemathematically expressedas119874(min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor) for each selected unit profit mix and

119874(119868119867119894119876119894sdotmin119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor) for each airline

Therefore the computation complexity of theproposed algorithm can be mathematically expressed as119874(119872(119868119867

119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot

min119865119894119895 lfloor119879119894119895119887119894119895119896rfloor)) for simplicity

4 Case Study

To demonstrate applications of the model a case study isconducted based on available data from 11 airlines with 20routes and 7 candidate aircraft types in China

41 Basic Information For the sake of simplification 25major airports in China are selected from all the cities beingserved by 11 airlines in 2009 None of the routes are served

Table 2 Basic information of routes and airlines

Route number Demandyear Airline number1 94046 1 4 6 102 26377 1 2 4 63 87934 2 3 4 5 74 103394 1 2 6 75 65260 1 2 5 66 271588 1 4 97 58920 3 88 103490 3 89 602044 1 3 6 7 1010 140488 1 3 6 911 63703 2 4 6 912 57753 2 4 513 223 314 33106 1 4 6 8 1015 55585 1 2 4 5 616 104557 1 3 5 917 28042 2 3 4 6 918 9060 1 3 819 201189 1 3 8 1120 37931 1 2 3 4 5 9

by all eleven airlines On each route there exist four airlineson average including one monopolized market (route) andone route operated by six airlines Seven aircraft types areserved as candidate aircraft types Airline number shownin Table 1 represents the number of airlines participating inthe competitive environment The candidate aircraft typespresent the preliminary choice of aircraft types each airlineprobably selects in the fleet planning process The selectionof candidate aircraft types mainly depends on airlinesrsquo oper-ational data on 20 routes in 2009 More detailed informationis listed in Table 1

The seating capacity of candidate aircraft types 1sim7 is 130162 165 192 198 244 and 298 respectively The operationalscope of each airline is listed in Table 2 in which the secondcolumn represents the statistical data of 20 routes in 2009 Inaddition we suppose that the market share index 120573 = 28 forall routes and the parameters of the proposed algorithm are119898119886119909 119894119905119890119903119886119905119894119900119899 119899119906119898119904 = 100 120572 = 06 and 120576 = 0 respectively

42 Results and Discussions According to the presentedsolution algorithm shown in Section 32 the resulting fleetsize and structure for each airline is listed in Table 3 Eachrow in Table 3 represents the equilibrium optimum numberof different types of aircraft each airline ownsThe last columnin Table 3 shows the resulting fleet operational profit foreach airline Therefore the total fleet operational profits for


Table 3 Fleet planning scheme in the competitive environment

Airline numberFleet planning scheme

Aircraft type Operational profit1 2 3 4 5 6 7

1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7

Figure 2 Proportion of flight frequency on route 3

airlines operating in the competitive environment amount to179168800 Yuan

As can be seen from Table 3 small-scale airlines areeliminated from the competitive environment They areunable to seize any profitablemarkets against the competitorssince few critical resources are given to them Small-scalehere refers to the airline with (relative) few routes to beoperated and few critical resources used to seizemarket shareFigure 2 shows that small-scale airline 7 needs to competewith another four airlines on route 3 to seize themarket shareusing yearly available flight frequencies a proportion of 1 intotal on route 3 from which almost no market share can beobtained according to 119878 curve function It is similar to thecases on route 4 for airline 7 (shown in Figure 3) and routes9 and 1 for airline 10 (shown in Figures 4 and 5)

Note particularly that the proportion for airline 11 onroute 19 shown in Figure 7 reaches 55 but no flight needs tobe offered to the route The reason is that whatever candidateaircraft types are used to operate on route 19 the unitassignment profit is constantly negative This is also the case

22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10


for airline 10 on route 14 (Shown in Figure 6) These resultsshow that small-scale airlines with few critical resourcesavailable to be used can hardly carry out operations onroutes with intensified competition due to their disadvantage


36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10


positions in the aspects of critical resources In order to enterinto these routes one regular approach for these airlines is tolower their fare levels to ensure their unit assignment profitsare changed to the positive values To some extent it is justthe reason why small-scale airlines need to lower their farelevels when entering into a new competitive market

In addition equilibrium optimum solution refers to thefact that one rational participant who changes its decisionpolicy alone can not increase its profit any more so long asother participants keep their decision polices unchanged Inorder to illustrate the effectiveness of the solution algorithmpresented in Section 32 we adopt Monte Carlo simulationto verify the result Detailed procedure of simulation can bedepicted as follows

(i) We select airline as adjustor in order of increasingairline number and randomly generate one decisionpolicy of the selected airline while keeping the deci-sion policies of other unselected airlines unchanged

(ii) Based on (i) we simulate 2000 times for each adjustorin order using Monte Carlo simulation and calculatethe corresponding fleet operational profit for eachtime

6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11


0 200 400 600 800 1000 1200 1400 1600 1800 2000Simulation number

Airline 5 Airline 3Airline 2

Airline 8

Airline 4

Airline 6 Airline 11

Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104

Figure 8 Monte Carlo simulations on absolute error of fleetoperational profit

(iii) We then use corresponding equilibrium optimumvalue to minus each simulated fleet operational profitso as to formulate the absolute error of fleet opera-tional profit for each simulation

(iv) We draw these points in Figure 8 in order of increas-ing absolute error for each adjustor

For each airline Figure 8 shows that all of absolute errorsare all below zero It suggests that there is no one decisionpolicy that could increase one airline fleet operational profitalone when all the decision policies of other airlines remainunchanged and the solution deprived from presented algo-rithm is just the equilibrium optimum

In order to illustrate the benefits of the approach pre-sented in this paper (called ldquocompetitive modelrdquo for simplic-ity) we compare it with the model presented in reference [8](called ldquotraditionalmodelrdquo for simplicity) inwhich the route-based approach optimizing airline fleet size and structure isbased on the assumption that the airlinersquos passenger demandon each route has been already given However an airlinepredicted passenger demand largely depends on the methods


Table 4 Comparisons to the traditional model on related indices

Airline numberCompetitive model Traditional model

Operationalprofitten thousand

Loadfactor

Spillingrate

Operational profitten thousand Load factor Spilling rateH M L H M L H M L

1 654769 090 005 minus6705061 minus7252687 minus8296303 078 059 051 076 072 0712 133094 059 000 minus195280 minus539765 minus885021 037 030 022 031 002 0003 35999 061 000 minus4550185 minus4930516 minus5443666 059 051 040 088 078 0784 233070 076 000 minus2509223 minus3600855 minus3979908 057 029 023 078 061 0585 123559 082 000 minus91541 minus221267 minus415567 055 040 036 057 001 0006 518977 085 005 minus103696 minus393672 minus1315913 072 059 051 048 018 0167 000 000 000 minus44718 minus44718 minus44718 001 001 001 000 000 0008 3234 066 000 minus635043 minus654528 minus932996 090 075 057 084 064 0639 88986 100 000 minus531525 minus617476 minus2231595 088 085 070 077 042 04010 000 000 000 minus5078193 minus5143741 minus5146172 077 044 042 098 096 09611 000 000 000 minus546205 minus546205 minus546205 100 100 100 071 071 071Note H and L represent the highest and lowest values in all simulations respectively M is the average value of all simulating results

and assumptions that the airline uses It means that thedifferent methods and assumptions would present differentpredicting results In order to cover all possible predictedpassenger demand for each airline in a competitive envi-ronment we assume passenger demand predicted by eachairline ranges from 0 to total passenger demand on route andyield to uniformdistributionWe useMonteCarlo simulationto generate passenger demand first for each airline (servedas predicted passenger demand) and obtain the numberof flights of different types of aircraft using the traditionalmodel These resulting values of decision variables are thenput into the competitive model to calculate the actual fleetoperational profit and the values of related indices (all listedin the column of ldquoTraditional modelrdquo) while the decisionpolicies of other airlines remain unchanged during the airlinebeing processed The whole simulating experiment conducts10000 times for each airline and the results are listed in thelast nine columns of Table 4 in which load factor is definedas a ratio of the captured passenger demand to the totalpassenger demand Spilling rate is also a fraction which isreferred to as a ratio of the spilling number of passengerdemand to the total passenger demand As can be seenfrom Table 4 the fleet operational profit for each airline isdramatically decreased by using the traditional model Noneof fleet operational profits resulting from the competitivemodel are lower than the ones from the traditional modelAll the results listed in Table 4 are similar to each other Row1 (airline 1) in Table 4 can be considered as a representativeresult for these comparisons and shows that the resulting fleetoperational profit and load factor deprived from competitivemodel are all much better than the corresponding valuesresulting from the traditional model The spilling rate in row1 also reveals that the competitive model (005) outperformsthe traditional one (071sim076) Note particularly that all fleetoperational profits of airlines 7 10 and 11 become negative incase of any one quasi-route these airlines operate on or anyone candidate aircraft type these airlines are used to operatewith This shows again that the results of airlines 7 10 and

0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3

Increased (demand) ()

Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1

Airline 9 Airline 2Airline 3 Airline 5Airline 8 Airline 7 10 11

Airline 6

Total

times104

minus05

Figure 9 Impact of passenger demand on fleet operational profit

11 deprived from the model presented in this paper are theequilibrium optimum Therefore it could be concluded thatthe relationship between flight frequency andmarket share isa significant factor that must be considered into airline fleetplanning process The competitive model could better reflectthe realistic environment and therefore the results are moreconsistent with realities

43 Sensitivity Analysis Passenger demand and fare price incompetitive environment are the most important parametersthat compel airlines to change their decision policies soas to (relatively) maximize their fleet operational profitsTo analyze these impacts we consider data in case studyas benchmark and increase the proportion of passengerdemand and fare price within interval [0 1] respectivelyThen the solution algorithm shown in Section 32 is usedagain to solve the competitivemodel for each time increase ofpassenger demand and fare price The total fleet operationalprofit shown in Figure 9 is roughly positive proportion to


0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7

Increased (fare price) ()

Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4

Airline 2 Airline 3 Airline 5 Airline 8 Airline 7 10 11 Airline 9

times104

minus1

Figure 10 Impact of fare price on fleet operational profit

0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104

Figure 11 Comparison of passenger demand and fare price on fleetoperational profit

passenger demand For each airline the fleet operationalprofit also shows positive proportion to passenger demandIn particular airlines 7 10 and 11 are still eliminated fromcompetitive environment despite the increasing passengerdemand The reason is that the unit assignment profit of air-craft type and route is unchanged yet In addition Figure 10reveals similar laws in terms of fleet operational profit for theincrease of fare price

However as shown in Figure 11 its increasing rate ismuchfaster than the rate in the increase of passenger demand atthe aggregate level This suggests that the fleet operationalprofit is more sensitive to the increase of fare price than tothe increase of passenger demand for the whole competitiveenvironmentThe reason is that airlines could obtain the extraenhanced revenue without needing to provide any flight inthe increased fare environmentThis suggests that airlines canobtain additional profits without incurring any additional (orfew) costs in maintaining or acquiring more market shareOn the contrary the rapid growth of passenger demand inthe competitive environment cannot make airlines directly

0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand

Figure 12 Comparison of passenger demand and fare price on fleetsize

0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand

Figure 13 Comparison of passenger demand and fare price onaverage seating capacity

acquire more market share The only way to acquire thedesired market share is to provide more flights on the market(route) whichmeansmore related costs incurred and parts offleet operational profits are offset As a result the competitivemodel is more sensitive to fare price than to passengerdemand

Furthermore in order to illustrate the sensitivity of fleetsize and structure we record each of corresponding dataon fleet size and structure in Figures 12 and 13 Fleet sizerefers to the total number of different types of aircraft in thewhole environment On the basis of fleet size fleet structureis defined as the average seating capacity per aircraft

As can be seen from Figures 12 and 13 both fleet size andaverage seating capacity are all roughly positive proportionto passenger demand and fare price at the aggregate levelThey also perform more sensitivity to the increase of fareprice than to the increase of passenger demand This issimilar to the conclusion deprived from Figures 9 10 and


11 The results show that fare price rather than passengerdemand is the major factor contributing to the increaseof fleet size and average seating capacity This seems to becontradiction with the traditional viewpoint in which fleetsize increases with the growth of passenger demand andaverage seating capacity is up-sized only when the availableflight frequency on route cannot meet passenger demandany more Figures 12 and 13 contrarily show that fleet sizeand average seat capacity are all more sensitive to fare priceIt could be explained that the increase of fare price meansmore direct fleet operational profits (without any additionalcosts incurred) airlines could be obtained Based on themairlines could afford more costs incurred from using eitherlarger aircraft types or more aircraft numbers in pursuit ofmaximizing airlinesrsquo fleet operational profits

However the increase of passenger demand just onlymeans the possibility for airlines to capture more passengersTo capture their desired passengers airlines unnecessarilyneed to provide corresponding flights due to the vacant seatswhich may exist on some flights or airlines are actuallyunwilling to provide additional flights to capture these pas-sengers since the costs incurred cannot be offset by the gen-erated revenue These are the reasons that result in the slowgrowth of flight frequency and less sensitivity of fleet size andaverage seating capacity to the increase of passenger demand

5 Conclusions

Motivated by the concern that the assumption of airlinecaptured demand on each route in route network to bepresented first is too simplistic to reflect the true complexityof the relationship betweenflight frequency andmarket sharewe develop a new route-based fleet planning approach inwhich we (i) incorporate 119878 curve function into airline fleetplanning process (ii) construct a multiobjective functionmaximizing each airline fleet operational profit under theconstraints that the number of flights offered to each routeand air-crew flying hours obtained from each aircraft fleettype must not exceed the corresponding maximum numberrespectively and (iii) devise a practical solution procedure tosolve the presented modelThroughMonte Carlo simulationthe solution procedure presented in this paper is proved to beeffective Case studies show that no one airline can increasefleet operational profit by changing its distribution of aircraftsize and flight frequency alone while the distributing policiesof other airlines remain unchanged Sensitivity analysis sug-gests that the fare is the main changing factor to airline fleetsize and structure as opposed to the passenger demand

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work that is described in this paper has been supportedby the National Science Foundation of China (no 61179074)

References

[1] R A Rushmeier and S A Kontogiorgis ldquoAdvances in the opti-mization of airline fleet assignmentrdquo Transportation Sciencevol 31 no 2 pp 159ndash169 1997

[2] C Barnhart A Farahat and M Lohatepanont ldquoAirline fleetassignment with enhanced revenue modelingrdquo OperationsResearch vol 57 no 1 pp 231ndash244 2009

[3] C Barnhart T S Kniker and M Lohatepanont ldquoItinerary-based airline fleet assignmentrdquo Transportation Science vol 36no 2 pp 199ndash217 2002

[4] O Listes and R A Dekker ldquoA scenario aggregation-basedapproach for determining a robust airline fleet composition fordynamic capacity allocationrdquoTransportation Science vol 39 no3 pp 367ndash382 2005

[5] Y Wang and H Sun ldquoHeurist algorithm of fleet planningbased on airline fleet assignmentrdquo System Engineering Theoryamp Practice vol 31 no 2 pp 371ndash378 2011

[6] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo System EngineeringTheory amp Practice vol 33 no 4 pp 963ndash970 2013

[7] G Schick and J Stroup ldquoExperience with a multi-year fleetplanning modelrdquo Omega vol 9 no 4 pp 389ndash396 1981

[8] H Sun P W Zhang and Y Wang ldquoFleet planning approachbased on optimized fleet capacity allocation in airline net-worksrdquo Journal of Southwest Jiaotong University vol 45 no 1pp 111ndash115 2010

[9] Y Wang J F Zhu andW Ge ldquoAirline fleet robust optimizationapproach under stochastic demand with route network effectsrdquoJournal of Southwest Jiaotong University vol 49 pp 727ndash7332014

[10] W Wei ldquoImpact of landing fees on airlinesrsquo choice of aircraftsize and service frequency in duopoly marketsrdquo Journal of AirTransport Management vol 12 no 6 pp 288ndash292 2006

[11] W B Wei and M Hansen ldquoAirlinesrsquo competition in aircraftsize and service frequency in duopoly marketsrdquo TransportationResearch Part E Logistics and Transportation Review vol 43 no4 pp 409ndash424 2007

[12] M Takebayashi ldquoThe runway capacity constraint and airlinesrsquobehavior choice of aircraft size and network designrdquo Trans-portation Research Part E Logistics and Transportation Reviewvol 47 no 3 pp 390ndash400 2011

[13] W-H Tsai K-C Lee J-Y Liu H-L Lin Y-W Chou and S-JLin ldquoA mixed activity-based costing decision model for greenairline fleet planning under the constraints of the EuropeanUnion Emissions Trading Schemerdquo Energy vol 39 no 1 pp218ndash226 2012

[14] M Givoni and P Rietveld ldquoThe environmental implicationsof airlinesrsquo choice of aircraft sizerdquo Journal of Air TransportManagement vol 16 no 3 pp 159ndash167 2010

[15] M Rosskopf S Lehner and V Gollnick ldquoEconomic-environmental trade-offs in long-term airline fleet planningrdquoJournal of Air Transport Management vol 34 pp 109ndash115 2014

[16] V Pai ldquoOn the factors that affect airline flight frequency andaircraft sizerdquo Journal of Air Transport Management vol 16 no4 pp 169ndash177 2010

[17] J H Brown ldquoAirline fleet composition and deregulationrdquoReview of Industrial Organization vol 8 no 4 pp 435ndash4491992

[18] A Bahram C Garland and R Kambiz ldquoThe effects of marketstructure and technology on airline fleet composition after


deregulationrdquo Review of Industrial Organization vol 15 no 1pp 77ndash88 1999

[19] R W Simpson A market share model for US domestic airlinecompetitive markets [PhD thesis] Massachusetts Institute ofTechnology Cambridge Mass USA 1970

[20] W B Wei and M Hansen ldquoImpact of aircraft size and seatavailability on airlinesrsquo demand and market share in duopolymarketsrdquo Transportation Research Part E vol 41 no 4 pp 315ndash327 2005

[21] J Contreras M Klusch and J B Krawczyk ldquoNumericalsolutions to Nash-Cournot Equilibria in coupled constraintelectricity marketsrdquo IEEE Transactions on Power Systems vol19 no 1 pp 195ndash206 2004

[22] J B Krawczyk and S Uryasev ldquoRelaxation algorithms to findNash equilibria with economic applicationsrdquo EnvironmentalModeling amp Assessment vol 5 no 1 pp 63ndash73 2000

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the fleet size and structure under the condition that the futuredetailed flight schedule has been already presented In thisaspect Listes and Dekker used time-space network to con-struct a fleet assignment-based model to determine the fleetcomposition They also developed a scenario aggregation-based algorithm to solve the model [4] Wang and Sunutilized simulated annealing algorithm to solve airline fleetplanning problem and discussed a robust airline fleet plan-ning method [5 6] However this kind of approach isbased on a given flight schedule which is hardly simulateddue to the uncertainty of airlinersquos future environment Thisdrawback may result in an unreliable fleet size and structuredeprived from the fleet assignment-based approaches

Therefore recent relevant studies have focused mainlyon route-based fleet planning approaches in which the bestaircraft type or aircraft type mix is assigned to each route tomaximize the fleet operational profit (or minimize the fleet-related costs) In this aspect Schick and Stroup proposed amultiyear fleet planning model with consideration of passen-ger demand constraints and aircraft balance equations aswellas minimum andmaximum flight frequency to minimize thefleet-related-costs [7] Sun et al applied a similar model toanalyze the Chinese market [8] Wang et al presented a newfleet planning approach for those airlines operating in Huband Spoke network in which network effects are highlighted[9] Wei and Hansen discussed the competitive relationshipbetween aircraft size and flight frequency using game theoryIt was concluded that the extra landing fee could reduceflight delay and airport congestion [10 11] Takebayashiconstructed a supply-demand interactionSDI model forHaneda airport He held the view that airlines did not alwaysadopt a downsizing aircraft size strategy in response to airportrunway expansion [12] Tsai et al incorporated the constraintof the European Emissions Trading Scheme into a mixedactivity-based fleet planningmodelThey believed that a self-purchased wide-bodied airplane could benefit from high rev-enue tone kilometers [13] Givoni and Rietveld evaluated theimpact of environmental factors on the choice of aircraft sizeThey thought that environmental improvement could benefitfrom those airlines using large aircraft size [14] Rosskopfet al proposed a multiobjective linear programming modelto study the trade-off between economic-environment goalsThey argued that the environment goal could be achieved by6 improvement at the cost of 3 deviation from economicoptimum [15] Pai analyzed the main factors affecting thechoice of aircraft size and flight frequency [16] Other relevantstudies [17 18] analyzed some external impacting factors onfleet size and structure (eg Brown 1992 Bahram et al1999)

Previous studies on route-based fleet planning approach-es investigated the issue mainly depending on airline itselfFew researches have taken into account the impact of otherairlinesrsquo competitive behaviors on the airlinersquos operationThispaper aims to optimize the fleet size and structure throughnetwork-wide allocating different types of aircraft and flightfrequency under multiairline competitive behaviors Effortsare made to construct a multiobjective model to maximizeeach airline fleet operational profit subject to the availableflight frequency offered to each route and air-crew flying

hours for each aircraft fleet type This study formulates thefleet operational profit as evaluating function including fleetoperational costs penalty cost and passenger spilling costIn addition this paper also develops an effective algorithmto solve the proposed model The validation and benefits ofthe model are shown with a case study Therefore the maincontributions of this paper can be summarized as follows

(i) We develop a new route-based fleet planning modelwhich is capable of reflecting the impact of multiair-line competitive behaviors on fleet size and structure

(ii) We devise a heuristic algorithm for our route-basedmodel and show its validation through Monte Carlosimulation

(iii) Through case study using real airline data we quantifyto analyze the benefits of the model presented in thispaper

(iv) Through sensitivity analysis we find the importantfactors impacting airline fleet size and structure

The remainder of this paper is organized as follows Inthe next section the problem is presented in detail includ-ing mathematical modeling in competitive environment InSection 3 a heuristic algorithm is introduced according toequilibrium optimum theory For a case with real airlinedata the multiobjective function is solved by the proposedalgorithm coding with MATLAB software in Section 4 AndSection 5 is the conclusion of this paper

2 Problem Modeling

In this section the paper illustrates how multiairline com-petitive behaviors affect the choice of aircraft size and flightfrequencyThenwe extend the existing route-based approachlisted in reference [8] to construct a network-wide allocationmodel of the choice of aircraft size and flight frequency withmultiairline competitive behaviors

21 Problem Description Traditional viewpoint generallyholds that an airlinersquos choice of aircraft size and flightfrequency largely depends on the network-wide distributionof the airlinersquos passenger demand However few studies [7ndash9] inversely consider that an airlinersquos choice of aircraft sizeand flight frequency in a competitive environment is alsogreatly affecting the airline capability of capturing passengerdemand 119878 curve presented by Simpson [19] in the last centuryis considered as one of most famous functions around theworld for evaluating airline market share which is defined asa ratio of captured demand for an airline to the total demandin the market It is concluded that airline market share is afunctionwith respect to flight frequency and yields to 119878 shapeRecently Wei and Hansen [20] once again have confirmed119878 curve function widely existing in airline market Based onthese evidences 119878 curve function is used throughout thispaper to analytically express the relationship between airline



MS119894=

(119873119894)120573

sum119900

119895=1(119873119895)120573 (1)











(1) Sets






(2) Parameters















airline 119894

(4) Functions


with respect to 119909119894119895119896

and 119910119894119895










sum

119896isin119870119894


119894 (2)



sum

119896isin119870119894


119894 (3)



MS119894119895=

(sum119896isin119870119894

119909119894119895119896)120573

sum119894isin119868(sum119896isin119870119894

119909119894119895119896)120573 forall119894 isin 119868 119895 isin 119869

119894 (4)


119909119894119895119896



sum

119895isin119869119894


119894 (5)



119909119894119895119896ge 0 int 119910


119894 119896 isin 119870

119894 (6)





max 119911119894= sum

119895isin119869119894

sum

119896isin119870119894

(119901119894119895119904119894119895119896minus 119888119894119895119896minus119872120575

119894119895119896) 119909119894119895119896

minus sum

119895isin119869119894

119901119894119895

10038161003816100381610038161003816119910119894119895

10038161003816100381610038161003816 forall119894 isin 119868

(7)





3 Algorithm Design



119894




119894| 119883) equiv

(1199091 119909

119894minus1 1199091015840

119894 119909119894+1 119909





120601119894(119883lowast) = max

(119909119894|119883lowast)isin119883

120601119894(1199091015840

119894| 119883) forall119894 (8)


120593 (119883 1199091015840) = sum

119894isin119868

(120601119894(1199091015840

119894| 119883) minus 120601

119894 (119883)) (9)


119894to 1199091015840119894 If


119894= 1199091015840

119894


119894 Therefore



119885 (119883lowast) = arg max

1199091015840isin119883

120593 (119883 1199091015840) (10)


Algorithm Procedure


119894119895119896= 0 forall119894 119895 119896


119894 forall119894



119905119910119901119890 119903119900119906119905119890 119904119890119905119894

= (119895 119896) | (119901119894119895119904119894119895119896minus 119888119894119895119896) gt 0 119895 isin 119869

119894 119896 isin 119870

119894

(11)




(119895119896)isin119905119910119901119890 119903119900119906119905119890 119904119890119905119894

119901119894119895119904119894119895119896minus 119888119894119895119896 (12)




119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast = min119865


119879119894119896lowast

119887119894119895lowast119896lowast

rfloor (13)


Start

Are they positive












Yes


Yes



each airline

End


Yes

NoNo


No

One of sets is null

No

Yes

No


Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i


airline i








119905



119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905


119905

119894119895lowast(119909119905


119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905




119910119905

119894119895lowast = MS119905


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905


119896isin119870119894


119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)


119894=


119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=




Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905




119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890


Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10


119894119895119896(forall119894 119895 119896) for all


119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)


119894119895119896






119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











MS119894=

(119873119894)120573

sum119900

119895=1(119873119895)120573 (1)











(1) Sets






(2) Parameters















airline 119894

(4) Functions


with respect to 119909119894119895119896

and 119910119894119895










sum

119896isin119870119894


119894 (2)



sum

119896isin119870119894


119894 (3)



MS119894119895=

(sum119896isin119870119894

119909119894119895119896)120573

sum119894isin119868(sum119896isin119870119894

119909119894119895119896)120573 forall119894 isin 119868 119895 isin 119869

119894 (4)


119909119894119895119896



sum

119895isin119869119894


119894 (5)



119909119894119895119896ge 0 int 119910


119894 119896 isin 119870

119894 (6)





max 119911119894= sum

119895isin119869119894

sum

119896isin119870119894

(119901119894119895119904119894119895119896minus 119888119894119895119896minus119872120575

119894119895119896) 119909119894119895119896

minus sum

119895isin119869119894

119901119894119895

10038161003816100381610038161003816119910119894119895

10038161003816100381610038161003816 forall119894 isin 119868

(7)





3 Algorithm Design



119894




119894| 119883) equiv

(1199091 119909

119894minus1 1199091015840

119894 119909119894+1 119909





120601119894(119883lowast) = max

(119909119894|119883lowast)isin119883

120601119894(1199091015840

119894| 119883) forall119894 (8)


120593 (119883 1199091015840) = sum

119894isin119868

(120601119894(1199091015840

119894| 119883) minus 120601

119894 (119883)) (9)


119894to 1199091015840119894 If


119894= 1199091015840

119894


119894 Therefore



119885 (119883lowast) = arg max

1199091015840isin119883

120593 (119883 1199091015840) (10)


Algorithm Procedure


119894119895119896= 0 forall119894 119895 119896


119894 forall119894



119905119910119901119890 119903119900119906119905119890 119904119890119905119894

= (119895 119896) | (119901119894119895119904119894119895119896minus 119888119894119895119896) gt 0 119895 isin 119869

119894 119896 isin 119870

119894

(11)




(119895119896)isin119905119910119901119890 119903119900119906119905119890 119904119890119905119894

119901119894119895119904119894119895119896minus 119888119894119895119896 (12)




119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast = min119865


119879119894119896lowast

119887119894119895lowast119896lowast

rfloor (13)


Start

Are they positive












Yes


Yes



each airline

End


Yes

NoNo


No

One of sets is null

No

Yes

No


Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i


airline i








119905



119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905


119905

119894119895lowast(119909119905


119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905




119910119905

119894119895lowast = MS119905


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905


119896isin119870119894


119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)


119894=


119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=




Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905




119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890


Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10


119894119895119896(forall119894 119895 119896) for all


119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)


119894119895119896






119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















sum

119896isin119870119894


119894 (2)



sum

119896isin119870119894


119894 (3)



MS119894119895=

(sum119896isin119870119894

119909119894119895119896)120573

sum119894isin119868(sum119896isin119870119894

119909119894119895119896)120573 forall119894 isin 119868 119895 isin 119869

119894 (4)


119909119894119895119896



sum

119895isin119869119894


119894 (5)



119909119894119895119896ge 0 int 119910


119894 119896 isin 119870

119894 (6)





max 119911119894= sum

119895isin119869119894

sum

119896isin119870119894

(119901119894119895119904119894119895119896minus 119888119894119895119896minus119872120575

119894119895119896) 119909119894119895119896

minus sum

119895isin119869119894

119901119894119895

10038161003816100381610038161003816119910119894119895

10038161003816100381610038161003816 forall119894 isin 119868

(7)





3 Algorithm Design



119894




119894| 119883) equiv

(1199091 119909

119894minus1 1199091015840

119894 119909119894+1 119909





120601119894(119883lowast) = max

(119909119894|119883lowast)isin119883

120601119894(1199091015840

119894| 119883) forall119894 (8)


120593 (119883 1199091015840) = sum

119894isin119868

(120601119894(1199091015840

119894| 119883) minus 120601

119894 (119883)) (9)


119894to 1199091015840119894 If


119894= 1199091015840

119894


119894 Therefore



119885 (119883lowast) = arg max

1199091015840isin119883

120593 (119883 1199091015840) (10)


Algorithm Procedure


119894119895119896= 0 forall119894 119895 119896


119894 forall119894



119905119910119901119890 119903119900119906119905119890 119904119890119905119894

= (119895 119896) | (119901119894119895119904119894119895119896minus 119888119894119895119896) gt 0 119895 isin 119869

119894 119896 isin 119870

119894

(11)




(119895119896)isin119905119910119901119890 119903119900119906119905119890 119904119890119905119894

119901119894119895119904119894119895119896minus 119888119894119895119896 (12)




119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast = min119865


119879119894119896lowast

119887119894119895lowast119896lowast

rfloor (13)


Start

Are they positive












Yes


Yes



each airline

End


Yes

NoNo


No

One of sets is null

No

Yes

No


Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i


airline i








119905



119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905


119905

119894119895lowast(119909119905


119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905




119910119905

119894119895lowast = MS119905


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905


119896isin119870119894


119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)


119894=


119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=




Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905




119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890


Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10


119894119895119896(forall119894 119895 119896) for all


119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)


119894119895119896






119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











max 119911119894= sum

119895isin119869119894

sum

119896isin119870119894

(119901119894119895119904119894119895119896minus 119888119894119895119896minus119872120575

119894119895119896) 119909119894119895119896

minus sum

119895isin119869119894

119901119894119895

10038161003816100381610038161003816119910119894119895

10038161003816100381610038161003816 forall119894 isin 119868

(7)





3 Algorithm Design



119894




119894| 119883) equiv

(1199091 119909

119894minus1 1199091015840

119894 119909119894+1 119909





120601119894(119883lowast) = max

(119909119894|119883lowast)isin119883

120601119894(1199091015840

119894| 119883) forall119894 (8)


120593 (119883 1199091015840) = sum

119894isin119868

(120601119894(1199091015840

119894| 119883) minus 120601

119894 (119883)) (9)


119894to 1199091015840119894 If


119894= 1199091015840

119894


119894 Therefore



119885 (119883lowast) = arg max

1199091015840isin119883

120593 (119883 1199091015840) (10)


Algorithm Procedure


119894119895119896= 0 forall119894 119895 119896


119894 forall119894



119905119910119901119890 119903119900119906119905119890 119904119890119905119894

= (119895 119896) | (119901119894119895119904119894119895119896minus 119888119894119895119896) gt 0 119895 isin 119869

119894 119896 isin 119870

119894

(11)




(119895119896)isin119905119910119901119890 119903119900119906119905119890 119904119890119905119894

119901119894119895119904119894119895119896minus 119888119894119895119896 (12)




119886V119886119894119897 119891119897119894119892ℎ119905 119899119906119898119904119894119895lowast = min119865


119879119894119896lowast

119887119894119895lowast119896lowast

rfloor (13)


Start

Are they positive












Yes


Yes



each airline

End


Yes

NoNo


No

One of sets is null

No

Yes

No


Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i


airline i








119905



119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905


119905

119894119895lowast(119909119905


119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905




119910119905

119894119895lowast = MS119905


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905


119896isin119870119894


119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)


119894=


119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=




Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905




119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890


Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10


119894119895119896(forall119894 119895 119896) for all


119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)


119894119895119896






119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Are they positive












Yes


Yes



each airline

End


Yes

NoNo


No

One of sets is null

No

Yes

No


Yes

i = i + 1

i = i + 1

i = 1

Select airline i

route for airline i


airline i








119905



119894119895lowast(119909119905

119894119895lowast119896lowast) ge 119911

119905

119894119895lowast(119909119905


119905

119894119895lowast(119909119905


119911119905

119894119895lowast(119909119905

119894119895lowast119896lowast + 1) 119909

119905




119910119905

119894119895lowast = MS119905


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896 (14)

119911119905

119894119895lowast (119909119905


119896isin119870119894


119894119895lowast119896

minus 119901119894119895lowast

10038161003816100381610038161003816100381610038161003816100381610038161003816


119896isin119870119894

119904119894119895lowast119896119909119905

119894119895lowast119896

10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


119896isin119870119894119909119905

119894119895lowast119896)120573119895lowast

((sum119896isin119870119894

119909119905

119894119895lowast119896)120573119895lowast

+

sum1198941015840isin119868119894

(sum119896isin1198701198941015840119909(119905minus1)

1198941015840119895lowast119896)120573119895lowast

)


119894=


119894=

119903119900119906119905119890 119905119910119901119890 119904119890119905119894 (119895lowast 119896) | 119896 isin 119870

119894 If 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894=




Step 8 Set 119865119894119895lowast = 119865

119894119895lowast minus 119909119905




119903119900119906119905119890 119905119910119901119890 119904119890119905119894= 119903119900119906119905119890 119905119910119901119890 119904119890119905

119894 (119895lowast 119896lowast) If 119903119900119906119905119890 119905119910119901119890


Step 9 If sum119894isin119868(119901119903119900119891119894119905

119905

119894minus119901119903119900119891119894119905

(119905minus1)

119894) le 120576 then go to Step 10


119894119895119896(forall119894 119895 119896) for all


119909(119905+1)

119894119895119896= lceil120572119909

119905

119894119895119896+ (1 minus 120572) 119909

(119905minus1)

119894119895119896rceil 0 le 120572 le 1 (16)


119894119895119896






119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894) with a



119894119876119894) for



119894119895119887119894119895119896rfloor This




119874(119868119867119894119876119894sdotmin119865



119894119876119894+ 119868119867119894119876119894sdot min119865

119894119895 lfloor119879119894119895119887119894119895119896rfloor)) or 119874(119872(119868119867

119894119876119894sdot


4 Case Study












1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1 0 0 1 5 0 0 0 654769302 0 0 2 0 0 0 0 133093893 0 0 1 1 0 0 0 35998634 0 0 0 1 0 1 0 233070345 0 0 0 2 0 0 0 123558836 0 0 7 0 0 0 0 518977097 0 0 0 0 0 0 0 08 1 0 0 0 0 0 0 3233889 0 0 0 1 0 0 0 889860010 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0Total 1 0 11 10 0 1 0 179168800

12

2

77

71Airline 2

Airline 3

Airline 4

Airline 5Airline 7





22

16

58

4Airline 1

Airline 2

Airline 7

Airline 6


34

34

17

133

Airline 1

Airline 3

Airline 6

Airline 7Airline 10




36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










36

19

40

5Airline 1

Airline 4

Airline 6

Airline 10


11

31

23

17

19

Airline 1

Airline 4

Airline 6

Airline 8

Airline 10






6

28

10

55

Airline 1

Airline 3

Airline 8

Airline 11




Airline 8

Airline 4


Airline 1

Airline 9

Airline 7

Airline 10

0

minus05

minus1

minus15

minus2

minus25

Abso

lute

erro

r

times104










Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Loadfactor

Spillingrate




0 01 02 03 04 05 06 07 08 09 1

0

05

1

15

2

25

3


Flee

t ope

ratio

nal p

rofit

Airline 4Airline 1


Airline 6

Total

times104

minus05





0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 1

0

1

2

3

4

5

6

7


Flee

t ope

ratio

nal p

rofit

Airline 6

Total

Airline 1

Airline 4


times104

minus1


0 01 02 03 04 05 06 07 08 09 11

15

2

25

3

35

4

45

5

55

6

Increased ()

Flee

t ope

ratio

nal p

rofit

Fare price

Passenger demand

times104




0 02 04 06 08 120

25

30

35

40

45

Increased ()

Flee

t siz

e

Fare price

Passenger demand


0 02 04 06 08 1170

175

180

185

190

195

Increased ()

Aver

age s

eatin

g ca

paci

ty Fare price

Passenger demand








5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












5 Conclusions




Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Optimization Model and Algorithm Design for Airline … · 2019. 7. 31. ·...

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Transcript of Research Article Optimization Model and Algorithm Design for Airline … · 2019. 7. 31. ·...