Airline flight scheduling for oligopolistic competition with direct flights and a point to point network

JOURNAL OF ADVANCED TRANSPORTATION J Adv Transp 2016; 50:1942 1957 Published online 25 January 2017 in Wiley Online Library (wileyonlinelibrarycom) DOI: 101002/atr1438 Airline flight scheduling for oligopolistic competition with direct flights and a point to point network Ching-Hui Tang 1 * and Yueh-Ling Hsu 2 1 Department of Transportation Science, National Taiwan Ocean University, Keelung, Taiwan 2 Department of Air Transportation, Kainan University, Taoyuan, Taiwan SUMMARY In this research, we consider a flight scheduling problem for oligopolistic competition with direct flights and a point to point network In this type of market situation, passengers are sensitive to the departure time of a flight rather than the transfer time The airline needs to carefully consider the departure times of their competitors when determining their own Therefore, unlike past approaches which have only considered one departure time for a competitorˈs flight, a flight scheduling framework is developed which takes into consideration possible competitor departure times The framework includes two dependent stages which are repeatedly solved during the solution process In addition, an upper bound model is also designed to evaluate the solution quality Numerical tests are performed using data for Taiwanˈs outlying island route which is characterized by the above market situation Satisfactory results are obtained, showing the good performance of the framework Copyright 2017 John Wiley & Sons, Ltd KEY WORDS: flight scheduling; possible departure times; oligopolistic competition; direct flights; point to point network 1 INTRODUCTION Taiwan is an island nation comprised of one main island and several surrounding islands, making air transportation necessary for the increasing numbers of tourists and residents The outlying island routes have become an important and profitable market for Taiwanˈs domestic operations Currently, Taiwanˈs domestic airlines offer direct flights within a point to point network for these outlying island routes because of the short distances between the main island and surrounding islands Tourists and residents are the major passengers for these outlying island routes preferring direct rather than multistop flights In other words, most passengers on these outlying island routes travel direct, passenger desiring connections being rare Given the market situation and passenger characteristics, these direct passengers are more sensitive to a flightˈs departure time rather than the transfer time The flight departure time significantly and directly affects the willingness of a passenger to choose an airline An airline in this sort of market needs to pay serious attention to its flight departure time in order to attract more passengers Currently, each outlying island route in Taiwan is served by at most four airlines of similar size, and competition among them has been continuous over the years, resulting in a market that is characterized by oligopolistic competition In an oligopolistically competitive market like this, where carriers are few and obvious, the timetable drawn up by each specific carrier influences and is influenced by those of the other carriers In particular, each carrier naturally tries to predict each of the othersˈ flight departure times and strives to plan its own in response to its competitorsˈ This mutual guessing and responding leads to variations in each carrierˈs flight departure time and makes prediction difficult *Correspondence to: Ching-Hui Tang, Department of Transportation Science, National Taiwan Ocean University, Keelung 20224, Taiwan E-mail: chtang@mailntouedutw Copyright 2017 John Wiley & Sons, Ltd

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1943 Using one departure time for a competitorˈs flight as input makes the flight scheduling process more difficult and reflective of the actual competitive situation The airline needs to carefully consider possible departure times for all competing flights when determining its own departure times, a planning process which is the focus of this work There have been numerous studies focused on the airline flight scheduling problem including those conducted early on [1 5] More recently, researchers have begun considering passenger-related issues during the flight scheduling process, including passenger preferences and customer satisfaction, passenger choice behaviors, variations in demand, and random market demands [6 11] Obviously, the consideration of passengersˈ issues provides a great improvement in the flight scheduling results because of the systematic and simultaneous consideration of the interrelation between airline supply and passenger demand In addition to the consideration of passenger issues, recently, several other influences and schemes have also been incorporated into the flight scheduling problem, such as public service obligations, subsidy schemes, welfare analysis [12, 13], ticket price decisions [14], and dynamic scheduling [15, 16] These practical considerations and value schemes have enriched the flight scheduling process and made the finding of solutions to these problems of particular ongoing interest in the field A review of past studies shows that early on researchers focused on demand issues and were interested in schedule optimization with consideration of the interrelation between airline supply and passenger demand The consideration of the competitor issue in the flight scheduling problem is an area of developing interest In related studies (for example see [7], [8], [9], [15], and [16]), the competitorsˈ timetables have been considered in the scheduling models However, these models have only used the projected departure time of a competitorˈs flight as an input The incorporation of possible departure times of competitorsˈ flights has not been carefully explored This is the incremental contribution of this research with respect to the literature In this work, we focus on a flight scheduling problem with oligopolistic competition with direct flights and a point to point network in which direct passengers are sensitive to the flight departure times and an airline needs to carefully consider the possible departure times of all their competitorsˈ flights Therefore, unlike in past flight scheduling approaches, the projected departure time of a competitorˈs flight is not utilized as input We develop a flight scheduling framework embodied in two dependent stages In the first stage, we find a departure time for each flight of the target airline with the consideration of its competitorsˈ possible departure times In the second stage, after each flightˈs departure time has been decided upon in the first stage, we further consider other operating constraints in a flight scheduling model to solve for the overall flight schedule, with the objective of maximizing the target airlineˈs profit The two-stage process is repeatedly solved to improve the solution In addition, an upper bound model is designed by incorporating a relaxation of possible flight departure times and a theoretical maximum demand which is used to evaluate the solution quality of the framework The remainder of this paper is organized as follows: In Section 2, the problem is described In Section 3, the flight scheduling framework and the upper bound model are introduced Numerical tests are performed in Section 4 Finally, in Section 5, we provide some conclusions 2 THE PROBLEM DESCRIPTION An oligopolistically competitive market is typically characterized as being controlled by a small number of carriers Specifically, we consider a market in which the airline companies are similar in size The competitors of the target airline are thus finite and obvious, meaning that all other airlines operating the same route are competitors It also means that no single airline dominates the market, and no leader follower relationship exists among these airlines In this study, it is assumed that the participating airlines are optimistic about the market situation and act rationally when making decisions It is also assumed that these airlines operate without alliances or cooperation in terms of flight scheduling among the airlines In fact, there are currently no alliances among Taiwanˈs domestic airlines nor do they cooperate with each other in actual operations or negotiate with each other in the setting of flight schedules It should be mentioned that several elements are included in the timetable, such as the flightˈs departure time, flying time, and arrival time, as well as an airplaneˈs ground handling time, and so on Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1944 C-H TANG AND Y-L HSU Stochastic flight times and ground handling times are not considered In other words, determining the departure time for a flight is equivalent to determining its arrival time and thus is an essential decision for setting a timetable The departure time for a flight is determined based on two sequential methods: 1 The first method is the Nash Equilibrium (NE) which is the central solution concept in game theory The NE is useful for airlines in the same market where no one airline has a better departure time and other airlines do not change their departure times We believe that all airlines in the same market will act rationally to keep the NE The target airline and its competitors have no desire to change their departure times; hence, it is suggested that the target airline also select the NE 2 The second method is used when the NE cannot be found for a flight In this case, the target airline will choose a departure time by eliminating other departure times which the target airline and its competitors will not choose The method will be discussed in more detail in Section 311 3 THE FLIGHT SCHEDULING FRAMEWORK The framework includes two stages which are repeatedly solved during the solution process The structure of the two stages is illustrated in Figure 1 The first stage is designed to find a departure time for a flight for the target airline by taking into consideration the competitorsˈ departure times In particular, we use the passenger choice model proposed by [8] to estimate the passenger demand for the target airline and its competitors In the second stage, a flight scheduling model is employed with the objective of maximizing profit The departure time for a flight for the target airline, as decided upon in the first stage, is treated as an input in the model Other operating constraints, such as aircraft balance, aircraft scale, airport approved quota, and aircraft capacity, are also considered when obtaining the Figure 1 The structure of the two stages Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1945 overall flight schedule and the objective value After this, possible departure times are re-set for each flight, and the process is designed to repeatedly solve the two stages in order to improve the solution In the following sections, we first introduce the two stages and then the repetition process is presented Finally, the upper bound model is proposed 31 Stage 1: Consideration of competitorsˈ departure times We use two sequential methods to find a departure time for a flight We first introduce these two methods in Section 311, and a brief summary is discussed in Section 312 Before introducing them, the symbols used in this section are listed below: d the d th origin destination pair (OD pair) of the target airline f the f th flight of the target airline for an OD pair a the a th flights of the target airline and competitors in a competing combination For ease of modeling, the flight of the target airline is denoted as a = 1 in each competing combination k a the k th a departure time of the a th flight s k a; f a;d the k th a departure time of the a th flight in the f th flightˈs competing combination for the d th OD pair s k 1 ; f the NE for the f th flight of the target airline for the d th OD pair (the f th flight is denoted as a =1 in its competing combination) in the first method p a () the passenger demands for the a th flight of the departure time combination () in the f th flightˈs competing combination of the target airline for the d th OD pair ps k a; f a;d the k th a departure time of the a th flight in the f th flightˈs competing combination for the d th OD pair in the second method ps k 1 ; f the departure time with the largest average passenger demand among all competitorsˈ departure times for the f th flight of the target airline for the d th OD pair in the second method ap k 1; f d the average passenger demand among all competitorsˈ departure times for the k th 1 departure time for the f th flight of the target airline for the d th OD pair in the second method ap k 1 ; f d the largest average passenger demand among all competitorsˈ departure times for the k th 1 : departure time for the f th flight of the target airline for the d th OD pair in the second method pd f d the passenger demand of the f th flight of the target airline for the d th OD pair S f a;d the set of departure times of the a th flights in the f th flightˈs competing combination for the d th OD pair PS f a;d the set of departure times of the a th flights in the f th flightˈs competing combination for the d th OD pair in the second method F d the set of flights of the target airline for the d th OD pair A f d the set of flights in the f th flightˈs competing combination for the d th OD pair D the set of OD pairs of the target airline 311 The two sequential methods Two related elements, the competing combination and the departure time combination, are introduced in advance We establish one competing combination for each flight of the target airline A competing combination is comprised of a set (ie, A f d ) which is composed of the associated flight of the target airline and the competitorsˈ flights Competitorsˈ flights that depart less than one hour before or after the associated flight of the target airline are grouped into the competing combination They are grouped with the target airlineˈs flight with the closest departure time This is designed to avoid multiple departure times for a competitorˈs flight resulting from multiple competing combinations so as to ensure that a flight is only served at one departure time Table I shows the two competing combinations established for F1 and F2 of the target airline The related symbols and sets are listed in the right-hand column It can be seen that the departure time of F3, for competitor 1, is less than 60 min after F1 and before F2, however, because the departure time of F3 is closer to that of F1 than that of F2, it is grouped into a competing combination with F1 For simplicity of modeling, the flight of the target airline is denoted as a = 1 in the associated competing combination (ie, the 1 th element in A f d ) For example, F2 is the 2 nd flight (f = 2) of the target airline and is the 1 th flight (a = 1) in the competing combination Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1946 C-H TANG AND Y-L HSU Table I Examples of competing combinations Departure time Target airline Competitor 1 Competitor 2 Competing combinations 09:00 F5 f = 1 (F1) 09:15 F1 A 1 d ¼ f F1; F3; F5 g 09:45 F3 a = 1 (F1), a = 2 (F3), a = 3 (F5) 10:30 F2 f = 2 (F2) 11:00 F6 A 2 d ¼ f F2; F4; F6 g 11:15 F4 a = 1 (F2), a = 2 (F4), a = 4 (F6) After determination of the competing combination, different departure time combinations are built for each target airline and competitor combination In this study, the gap between the departure times for two flights for the target airline is usually between one and two hours Depending on the density of the target airlineˈs timetable, five possible departure times are set for each flight, so that the number of departure time combinations is 5j Af dj (ie, S f j a;d Af dj ) for each competing combination An example of two flights and the departure time combinations in a competing combination is shown in Table II In this table, p a () represents the number of passengers estimated for the a th flight in the associated depar- ture time combination () For example, p 1 s 1;f ; s1;f 2;d and p 2 s 1;f ; s1;f 2;d represent the number of passengers taking the target airlineˈs flight and the competitorˈs flight in departure time combination s 1;f ; s1;f 2;d Note that, because p a () may vary for different departure time combinations, we use the passenger choice model by [8] to estimate the p a () for each departure time combination Our calculation is similar to that in [8]; the interested reader can refer to their study for a more detailed description Based on the above discussion, the first method, embodied by the NE is represented by Functions! (1) to (3), in which s k 1 ;f ; ; sk a ;f k a;d ; ; s ja f dj ;f indicates the NE of the target airline and its competitors ja f dj;d for the f th flightˈs competing combination for the d th OD pair p a s k 1 ;f ; ; sk a ;f k a;d ; ; s ja f dj ;f ja f dj;d! p a s k 1 ;f k ; ; sk a;f a;d ; ; s ja f dj ;f ja f dj;d! ; k a S f a;d k a ; a A f d ; f F d; d D (1) Departure time of the target airline ¼ s k 1 ;f sk 1 ;f sk 1 ;f ; ; sk a ;f k a;d ; ; s ja f dj ;f ja f dj;d! ; f F d ; d D; (2) pd f d ¼ p 1 s k 1 ;f ; ; sk a ;f k a;d ; ; s ja f dj ;f ja f dj;d! ; f F d ; d D (3) Function (1) is an NE function which indicates that no airline has an advantageous move when its competitors do not change their departure times Constraint (2) indicates that the departure time chosen for each flight of the target airline is s k 1 ;f (ie, the NE for the target airline) Note that the constraint also denotes that each flight can be served once with one departure time Constraint (3) indicates the passenger demand of each flight of the target airline Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1947 Table II An example of different departure time combinations Competitorˈs flight (a = 2) Target airlineˈs flight (a =1) Departure times (s ka;f a;d ) s 1;f s 1;f p 1 s 1;f s 2;f s 3;f 2;d s 2;f 2;d s 3;f 2;d s 4;f 2;d s 5;f 2;d p 2 s 1;f ; s1;f 2;d ; s1;f 2;d p 1 s 1;f p 2 s 1;f p 1 s 3;f ; s3;f 2;d p 2 s 3;f ; s3;f 2;d s 5;f p 1 s 5;f ; s1;f 2;d p 1 s 5;f s 4;f p 2 s 5;f ; s1;f 2;d ; s5;f 2;d ; s5;f 2;d ; s5;f 2;d p 2 s 5;f ; s5;f 2;d Finding the NE is the top priority for a flight Constraint (2) ensures that there is only one departure time per flight, thus our NE is classified as a pure strategy NE 1 However, in some cases, a pure strategy NE may not exist In theory an alternative solution could be found based upon the airlineˈs considerations and preferences, an alliance, or negotiation among players However, in Taiwanˈs domestic operations at present, no alliances exist and airlines do not negotiate with each other in relation to flight scheduling Therefore, we propose solving the problem based on the target airlineˈs considerations and preferences, utilizing the second method detailed in this work After discussion with the target airline, it is assumed that the target airline and its competitors are rational and will not select non-preferred departure times which are worse for themselves than other departure times, regardless of what their competitors might do Thus, non-preferred departure times for the target airline and its competitors can be removed, with the remaining departure times regarded as candidates for a flight Then, for each remaining departure time of the target airline, we calculate the average passenger demand among all competitorsˈ remaining departure times Finally, the remaining departure time with the largest average passenger demand is selected for the target airline Altogether, the second method is formulated as follows: ap k 1;f d ¼ a A f f1gk d a PS f a;d p 1 ps k 1;f ; ; psk a;f a A f fg 1 d PS f a;d k a;d ; ; ps f j A dj j ; f A f dj;d ; k 1 PS f ; f F d; d D; (4) Departure time of the target airline ¼ ps k 1 ;f apk 1 ;f d ¼ Max ap 1;f d ; ; apk 1;f d ; ; ap PSf ;f d ; f F d ; (5) pd f d ¼ apk 1 ;f d ; f F d d D (6) 1 Two kinds of NE can be found: a pure strategy NE and a mixed strategy NE A pure strategy NE is defined as when each player chooses one single strategy with a probability of one and other strategies with a probability of zero A mixed strategy NE is when a player has multiple strategies with non-zero and non-one probabilities Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1948 C-H TANG AND Y-L HSU Constraint (4) is used to estimate the ap k 1;f d s of the target airline Constraint (5) indicates that the departure time chosen for each flight of the target airline is ps k 1 ;f Constraint (6) indicates the passenger demand for each flight of the target arline Based on the second method, the following two remarks are made: Remark 31 In the second method, unlike for the NE, it is not known exactly which departure time the competitors will choose The number of candidate departure times is decreased by removing those that neither the target airline nor its competitors will select, and the expected value criterion is used to evaluate each remaining departure time for the target airline Remark 32 The second method is designed to maximize the expected passenger demand for all competitors departure times after excluding those non-preferred departure times that the airlines will not select If there is no non-preferred departure time for the target airline and it competitors, then the term PS f a;d is replaced by Sf a;d in Functions (4) and (5) From the theoretical point of view, this is a kind of expected value model for a planning problem 312 Brief summary We develop two sequential methods for considering the possible competitor departure times in the process of choosing the departure times for the target airlineˈs flights The first method is used with the NE, and the second method is the application of practical considerations of the target airline The two methods, embodying the theoretical optimal solution produced by the NE and practical solution based upon actual carrier considerations, are combined 12 Stage 2: Finalizing the flight schedule We employ the time-space network technique to develop a mathematical model for finalizing the flight schedule Each time-space network denotes one specific type of airplane movement, as shown in Figure 2 In this stage, the flight departure time obtained in the previous stage, and other operating constraints (such as aircraft balance, aircraft scale, airport approved quota, and aircraft capacity) are considered simultaneously when finalizing the overall flight schedule and the total profit The three types of arcs, flight arcs, ground arcs, and cycle arcs set in the network are described below 1 Flight arc: A flight arc represents a flight connecting two stations, designated as (1) in Figure 2 The departure time for the flight arc is set according to the associated departure time obtained from the first stage The arc flowˈs upper bound is one and the lower bound is zero In addition, because multiple types of aircraft are considered, a side constraint is introduced for multiple flight arcs in different time-space networks to ensure that the sum of the flows is equal to one Thus a flight is served by one suitable type of aircraft 2 Ground arc: A ground arc, designated as (2) in Figure 2, represents the holding of airplanes at an airport within a time window The arc flowˈs upper bound is the apron capacity and the lower bound is zero 3 Cycle arc: A cycle arc represents the continuity between two consecutive planning periods designated as (3) in Figure 2 The arc flowˈs upper bound and lower bound are set the same as those of the ground arcs In particular, the sum of the flows of all cycle arcs denotes the total number of airplanes used during the planning period A constraint needs to be added to ensure that the sum of the flows of all cycle arcs is less than or equal to the available number of airplanes in the associated time-space network r f d c t ij The other symbols used in the model formulation are listed as follows: the ticket fare of the f th flight of the target airline for the d th OD pair the cost of the arc (i,j) in the t th time-space network Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1949 Figure 2 A time-space network (a type of airplane) m t q s b t u t ij the number of available airplanes in the t th time-space network the quota of flights approved for the s th station the airplane capacity in the t th time-space network the upper bound of the arc (i,j) flow in the t th time-space network AA t, AN t the set of all arcs and all nodes in the t th time-space network, respectively E f d the set of flight arcs in all time-space networks of which departure time is the appropriate departure time obtained from the first stage for the f th flight of the target airline for the d th OD pair CA t the set of cycle arcs in the t th time-space network L t s the set of flight arcs connecting the s th airport in the t th time-space network S the set of stations T the set of time-space networks Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1950 C-H TANG AND Y-L HSU THE MODEL FORMULATION (MODEL A) IS WRITTEN BELOW Model A MAX t T d D f F d ij E f d r f d pdf d xt ij c t ij xt ij ; (7) t T ij AA t st x t ij x t li ¼ 0; i AN t ; t T; (8) j AN t l AN t ij CA t x t ij mt ; t T; (9) x t ij qs ; s S; (10) t T ij L t s t T ij E f d x t ij ¼ 1; f F d; d D; (11) t T ij E f d b t pd f ij;d x t ij 0; f F d; d D; (12) 0 x t ij ut ij ; xt ij Integer; ij AAt ; t T: (13) The objective function (7) is to maximize the operating profit Constraint (8) ensures flow conservation at every node (ie, aircraft balance) in each network Constraint (9) ensures that the number of airplanes used does not exceed the available number of airplanes Constraint (10) indicates that the sum of all flights at each airport does not exceed the approved quota Constraint (11) ensures that each flight is served at once with one departure time and airplane type Constraint (12) keeps the passenger demand within the aircraftˈs capacity Constraint (13) holds all the arc flows within their boundsss 33 Repetition process The two stages described above are repeatedly solved to find a better solution The solution process is introduced below and is illustrated in Figure 3 Given the initial timetables of the target airline and its competitors, we set the competing combination and the departure time combination for each flight of the target airline Then, we solve the first stage to find the departure time for a flight (ie, s k 1 ;f or psk 1 ;f ) After that, the second stage is performed by solving Model A to obtain a new timetable and objective value Based on the new timetable, possible departure times are re-set for each flight The competing combination and the departure time combination for each flight of the target airline are also re-set accordingly Then, the passenger demands for the new departure time combinations are re-estimated and the two stages repeated The process is stopped when there is no improvement in the solution after a preset number of iterations That is, the solution process is stopped if a better solution than the Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1951 Figure 3 The process of the framework incumbent one is not found after the number of iterations exceeds a preset limit The best solution obtained during the solution process is used as the final solution 34 The upper bound model The two procedures result in an upper bound model One objective is to relax possible departure times for each flight, and the other is to use the theoretical maximum demand for a departure time as input They are discussed in greater detail below We consider all times available in the time slot of the target airline to be possible departure times for all flights of an OD pair, instead of separately setting five possible departure times for each flight The relaxation process is shown in Figure 4 As can be seen, the union of possible departure times for each flight at each iteration in the framework is just a subset of the relaxation, which forms a condition resulting in an upper bound solution for the maximization problem Accordingly, we consider the flight frequency (ie, the total number of flights) for each OD pair, instead of limiting the service constraint for each flight Thus, x t ij ¼ j F dj; d D; (14) t T ij UE d where Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1952 C-H TANG AND Y-L HSU Figure 4 Relaxation of possible departure times of each flight (OD pair 1 2) UE d the set of flight arcs in the available time slot of the target airline for the d th OD pair in all timespace networks F d the cardinality of the set F d which is the total number of flights of the target airline for the d th OD pair In addition to the above relaxation, we also use the theoretical maximum demand for each departure time in the available time slot as input Instead of using a flight as the target, the departure time is used as the target to produce a competing combination and the associated departure time combinations Because we propose to find the maximum demand for each departure time, other possible departure times are not set for each departure time Five possible departure times are considered for a competitorˈs flight, resulting in five associated departure time combinations for each departure time The passenger demands for the different departure time combinations for each departure time are then estimated to find the maximum one Note that the purpose here is only to find the maximum demand for each departure time It has not yet been decided whether the target airline will provide a flight in the available time slot for each departure time Also note that the maximum demand for each departure time is the theoretically optimal value for the target airline It is assumed that the target airline and its competitors will select the departure time combination with the maximum passenger demand Altogether, with the above relaxation process and the input of the maximum demand, we obtain an upper bound model which can be formulated as a linear integer model that can be optimally solved to find an upper bound solution The upper bound model is formulated as follows: Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1953 Model B MAX r d md ij;d x t ij c t ij xt ij ; (15) t T d D ij UE d t T ij AA t st t T b t md ij;d x t ij 0; ij UE d ; d D; (16) and Constraints (8), (9), (10), (13), (14), where the maximum demand of the flight arc (i,j) of the target airline for the d th OD pair md ij,d 4 NUMERICAL TESTS Numerical tests were performed based on data for airlines serving Taiwanˈs outlying island routes All necessary programs were written in the C++ computer language, and the GUROBI 602 mathematical programming solver was used to solve the problem The tests were performed on an Intel Core i7-3770 CPU with 16-GB of random-access memory using Microsoft Windows 7 41 Data analyses As noted above, real data from airlines operating routes to the outlying island routes during 2014 were used The target airline served four cities (Taipei, Kaohsiung, Kinmen, and Penghu) with eight OD pairs for its outlying island routes Some outlying island routes were served by two airlines, and some by four The operating time was from 06:00 to 21:00 for most OD pairs because of the nighttime curfew in effect at domestic airports in Taiwan During the operating time, the target airline provided about 66 to 72 daily direct flights using two types of aircraft, the ATR 72 and AirBus 320 Because the target airlineˈs and its competitors announce their timetables monthly, we found a timetable for each month of 2014 For ease of testing, the monthly timetables announced by the target airline and its competitors were used as the initial timetables in the framework A total of five possible departure times were set for a flight with a time interval of 30 min before or after the departure time in the announced timetable, with a gap of 15 min In other words, there were 25 (5 2 ) to 625 (5 4 ) departure time combinations generated for a competing combination for each flight of the target airline We also tested different time intervals which are essential to the number of departure time combinations in a competing combination in order to understand the influence of the parameter on the solution The results will be discussed in Section 43 After preliminary tests, the preset number of iterations in the framework was set to be 10, as discussed in Section 44 The parameter MIPGap in GUROBI was set to be 0%, meaning that the second stage (Model A) and the upper bound model (Model B) were solved optimally Other parameters and inputs, such as the ticket price, and the time slot and flight quota at each airport, were set based on actual operating data and Taiwan government regulations 42 Test results Examination of the results in Table III shows that there were 10 months for which the obtained gaps (between the obtained objective values and the upper bounds) were smaller than 3%; only one month had a gap greater than 4% The standard deviation was only 105, showing the stability of the solution quality The solution time is discussed next For seven of the months, more than two hours were needed to solve the problems The minimum, the average, and the maximum solution times were 104, 143, and 218 min, all acceptable for the planning problem As shown in Figure 5, Stage 1 consumed the most Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1954 C-H TANG AND Y-L HSU Table III Exterior results Month Objective value (NT$) Obtained gap (%) Solution time (min) 1 4 323 514 261 104 2 4 838 507 042 161 3 4 525 947 215 125 4 4 474 658 387 156 5 4 370 641 156 132 6 4 885 421 243 148 7 4 902 641 275 218 8 4 600 215 126 101 9 4 487 456 291 161 10 4 435 682 059 118 11 4 320 719 448 179 12 4 339 407 125 119 Average 4 542 067 219 14350 Standard deviation 105 3695 solution time, about 91% to 93% of the total solution times for all cases This result shows the domination of Stage 1 in terms of the computational efficiency of the framework It also shows the complicated characteristics for handling the different departure time combinations of the target airline and its competitors In addition to the above results, we also discuss the variations in the departure time between the timetables obtained with our method and the ones announced by the target airline The percentage variation which is defined as the number of flights for which the departure times obtained with our method are different from the ones in the announced timetable, divided by the number of flights in the associated OD pair, is calculated The maximum, the average, and the minimum percentage variations for all months for each OD pair are listed in Table IV We find that variations in the departure time appeared in all OD pairs because there was no zero value for the minimum percent variation We also found that the percentage variation was higher in the first four OD pairs when four airlines were operating In contrast, the variations in percentage were lower in the OD pairs for operations with two airlines The correlation coefficients of the maximum, the average, and the minimum variation percentages and the number of airlines were 095, 088, and 076, respectively There was a high positive correlation between the percentage variation and the number of airlines More competitors in the market will result in more requirements for improving departure times in the announced timetable of the target airline Figure 5 Percentages of the total solution time taken by the two stages Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1955 Table IV Departure time variation OD pair Percentage variation (%) Max Ave Min Taipei Kinmen 10000 9500 9000 4 Kinmen Taipei 10000 9500 9000 4 Taipei Penghu 7520 5773 4025 4 Penghu Taipei 8050 6038 4025 4 Kaohsiung Penghu 3333 2927 2520 2 Penghu Kaohsiung 3333 2927 2520 2 Kaohsiung Kinmen 4200 3225 2250 2 Kinmen Kaohsiung 4000 3025 2050 2 Coefficient of correlation 095 088 076 Number of airlines 43 Time intervals for possible departure times In previous tests, the possible departure times for a flight were set by adding or subtracting 30 min to or from the time announced in the timetable We tested three other time intervals, 15, 45, and 60 min In the cases of 45 and 60 min, there may be overlap in possible departure times for two flights because of the timetable density The objective values and solution times for different time intervals for each month are shown in Figures 6 and 7, respectively As can be seen in Figure 6, when the time interval was increased to greater than 30 min, there was little difference in the objective values, just a slight increase in the averages of 013% and 014% over 12 months, respectively This was because in the cases of 45 and 60 min, there may be overlap in possible departure times for two flights because of the timetable density, resulting in a slight increase in possible departure times and departure time combinations compared with the case of 30 min These slight increases did not help to significantly improve the solution As shown in Figure 7, in general, when the time intervals were increased, there was an obvious increase in the solution times for most cases Compared with the setting of 30 min, the 45 and 60-min settings produced increases of 2416% and 3857% in the average solution times for all months, respectively Compared with the objective values, the solution time is more greatly affected and sensitive to the time interval, even though there is only a slight increase in possible departure times and departure time combinations 44 Preset numbers of iterations We tested different preset numbers of iterations (from 3 to 20) in the framework As shown in Figure 8, there was an obvious increase in the solution times with an increase in the preset number of iterations After further examination of the number of iterations and the solution time, we found that an increase Figure 6 Objective values for different time intervals for setting departure times Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

1956 C-H TANG AND Y-L HSU Figure 7 Solution times for different time intervals for setting departure times Figure 8 Solution times for different preset numbers of iterations of five iterations would increase the solution time by about 243 times Furthermore, we also observed the number of months for which the gaps obtained were within 3%; see Figure 9 We found that there were 10 months for which obtained gaps were within 3% using 10 and 15 preset iterations For 20 iterations, the number of months for which the gap obtained was less than 3% only increased by one (Month 4), but with a high cost of more than 6 h in solution time All these results indicate that a setting of 10 iterations for the stopping criterion is workable and thus is suggested in our tests Figure 9 Number of months for which the obtained gap was within 3% for different preset numbers of iterations Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr

FLIGHT SCHEDULING FOR DIRECT FLIGHTS UNDER OLIGOPOLISTIC COMPETITION 1957 5 CONCLUSION In this study, we consider a flight scheduling problem for an outlying island route which is characterized by oligopolistic competition with direct flights and a point to point network In an oligopolistic competition scenario, there are few competitors; thus, the timetables drawn up by each carrier influence and are influenced by those of the other carriers In addition, in such a market, with direct flights and point to point transportation, passengers are sensitive to the departure time of a flight, and thus the airline needs to carefully consider the departure times of competitorsˈ flights when determining their own times We develop a framework which includes two stages: consideration of the competitorsˈ departure times and finalizing the schedule The solution is found by repeatedly solving the two stages In addition, an upper bound model is also designed to evaluate the solution quality The framework can be extended to include other types of airline operations, for example, alliances or hub and spoke network In the alliance problem, possible competitor departure times for both allied and non-allied airlines need to be simultaneously considered For the hub and spoke network, not only the flight departure time, but also the transfer time for connecting passengers need to be considered All of these could be directions for future research ACKNOWLEDGEMENTS This research was partially supported by a grant (NSC 102-2410-H-019-014) from the Ministry of Science and Technology of Taiwan We would like to thank the target airline for providing the test data and their valuable opinions REFERENCES 1 Etschmaier MM, Mathaisel DFX Airline scheduling: an overview Transportation Science 1985; 19: 127 138 2 Abara J Applying integer linear programming to the fleet assignment problem Interfaces 1989; 19: 20 28 3 Hane CA, Barnhart C, Johnson EL, Marsten R, Nemhauser GL Sigismondi G The fleet assignment problem: solving a large-scale integer program Mathematical Programming 1995; 70: 211 232 4 Clarke LW, Hane CA, Johnson EL Nemhauser GL Maintenance and crew considerations in fleet assignment Transportation Science 1996; 30: 249 260 5 Desaulniers G, Desrosiers J, Dumas Y, Solomon MM Soumis F Daily aircraft routing and scheduling Management Science 1997; 43: 841 855 6 Yan S, Tseng CH A passenger demand-based model for airline flight scheduling and fleet routing Computers and Operations Research 2002; 29: 1559 1581 7 Lohatepanont M, Barnhart C Airline schedule planning: integrated models and algorithms for schedule design and fleet assignment Transportation Science 2004; 38: 19 32 8 Yan S, Tang CH Lee MC A flight scheduling model for Taiwan airlines under market competitions Omega The International Journal of Management Science 2007; 35: 61 74 9 Yan S, Tang CH Fu TC An airline scheduling model and solution algorithms under stochastic demands European Journal of Operational Research 2008; 190: 22 39 10 Cadarso L, Marín Á Integrated robust airline schedule development Procedia Social and Behavioral Sciences 2011; 20: 1041 1050 11 Cadarso L, Marín Á Robust passenger oriented timetable and fleet assignment integration in airline planning Journal of Air Transport Management 2013; 26(January): 44 49 12 Pita JP, Antunes AP, Barnhart C de Menezes AG Setting public service obligations in low-demand air transportation networks: application to the Azores Transportation Research Part A: Policy and Practice 2013; 54: 35 48 13 Pita JP, Adler N Antunes AP Socially-oriented flight scheduling and fleet assignment model with an application to Norway Transportation Research Part B: Methodological 2014; 61: 17 32 14 Atasoy B, Salani M Bierlaire M An integrated airline scheduling, fleeting, and pricing model for a monopolized market Computer-Aided Civil and Infrastructure Engineering 2014; 29: 76 90 15 Warburg V, Hansen TG, Larsen A, Norman H Andersson E Dynamic airline scheduling: an analysis of the potentials of refleeting and retiming Journal of Air Transport Management 2008; 14: 163 167 16 Jiang H, Barnhart C Robust airline schedule design in a dynamic scheduling environment Computers and Operations Research 2013; 40: 831 840 Copyright 2017 John Wiley & Sons, Ltd J Adv Transp 2016; 50:1942 1957 DOI: 101002/atr