ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING

Transcription

1 ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING Luis Cadarso, Universidad Politécnica de Madrid, Ángel Marín, Universidad Politécnica de Madrid, ABSTRACT In scheduled air transportation, airline profitability is influenced by the airline's ability to construct flight schedules. To produce operational schedules, airlines engage in a complex decision-making process, referred to as airline schedule planning. Because it is impossible to simultaneously solve the entire airline schedule planning problem, the decisions required have historically been separated and optimized in a sequential manner. We propose a multi-objective integrated robust approach for the schedule design phase, considering the passenger behaviour, deciding jointly flight frequencies and timetable. The objectives are passengers' satisfaction and operator costs. We try to fix the timetable ensuring that enough time is available to perform passengers' connections, making the system robust avoiding misconnected passengers. Some test networks are solved in order to demonstrate the achieved robustness and choose an appropriate objective. Keywords: robustness, airline schedule design, multi-objective. INTRODUCTION Commercial aviation operations are supported by what is probably the most complex transportation system and possibly the most complex man-made system in the world (Barnhart and Cohn, 2004). Airports represent the nodes on which the system is built. Aircraft represent the very valuable assets that provide the basic transportation service. Passengers demand transportation between origins and destinations, and request specific travel times. Crews operate the aircraft and provide service to passengers. These entities are coordinated through a flight schedule, comprised of flight legs between airports. In order to produce operational schedules, airlines engage in a complex decision-making process, referred to as airline schedule planning. Most of the time the schedule planning starts from an existing schedule. Then, changes are introduced to the existing schedule to reflect changing demands and environment; this is referred to as schedule development. 1

2 However, we will suppose that a coldstart is needed, that is, we must create the schedule planning from scratch. There are three major components in the schedule development step. The first step, the schedule design, is arguably the most complicated step of all. This step is the one we will treat in this work. The purpose of the second step of schedule development, fleet assignment, is to assign available aircraft types to flight legs such that seating capacity on an assigned aircraft matches closely with flight demand and such that costs are minimized. In base to these flows the network is decomposed into sub networks, each one associated only with a particular fleet type. Given these sub networks, the assignment of individual aircraft to flight legs is done in the aircraft maintenance routing step, the third step. Crew scheduling involves the process of identifying sequences of flight legs and assigning both the cockpit and cabin crews to these sequences. The final goal in airline scheduling is to integrate all phases into a single one. Integrated models would optimize schedules, capacities, pricing and seat inventory. However integrating the planning phases is a big challenge: dynamics and competitive behaviours, organizational coordination... Traditionally the schedule design has been decomposed into two sequential steps. The first, the frequency planning, in which planners determine the appropriate service frequency in a market; and the second one, the timetable development, in which planners place the proposed services throughout the day, subject to network consideration and other constraints. Airline schedule design, including how to determine a network's type, flight frequencies and timetable for each flight leg, is a prerequisite for any airline's operational planning such as fleet assignment, routing and crew assignment. Network design is heavily important since the chosen network type, flight frequency and timetable directly influence the operating effectiveness of the airline and the quality of service provided to passengers. Designing an airline network is an extremely complex task due to the huge number of variables affecting the design, i.e. passenger demand, ground facilities and capacity, the competence, etc. These issues are not always easily modelled and usually result in huge models. The most important issue is probably the demand forecasting. Thus, accurately forecasting the future passenger demand on each market is of priority concern in the planning and design of an airline network. However, accumulating a large number of data with good statistical distribution to develop conventional statistical forecasting models is a challenging task. Besides, the uncertainty in other input data also complicates the design of the airline network. For example, a situation frequently arises in which we cannot know the operational costs for possible new routes in the schedule that have not been performed before. Airlines try to generate the lowest possible operating costs and achieve a higher load factor, while passengers concern about flight frequencies, nonstop flights, and in case of stops minimum stop time. 2

3 In case of intermediate stops in itineraries, passengers must perform a flight connection. In order to accomplish this connection an undetermined time is needed by passengers. Airlines usually design itineraries trying to make big enough connection time. However, this issue makes passengers to be dissatisfied. In order to avoid this situation, a new robustness criterion is introduced. Every intermediate stop will always have a minimum connection time; however, this time will not be enough in some situations. In order to avoid misconnected passengers, a penalty based on statistical data is proposed. In this way, expected misconnected passengers will be penalized, but also accounting for passengers dissatisfaction. State of Art Generally, the approach is to cast these problems as network design models. In the past, there have been efforts to improve the profitability of the schedule. Simpson (1966) presents a computerized schedule construction system that begins by generating demand using a gravity model, then solves the frequency planning problem and, finally, constructs a flight schedule and solves vehicle optimization problem upon that schedule. Chan (1972) provides a framework for designing airline flight schedules covering route generation and route selection. The previous presented work was performed before deregulation of the passenger air transportation industry in Soumis, Ferland and Rousseau (1980) consider the problem of selecting passengers that will fly on their desired itinerary with the objective of minimizing spill costs. No recaptures are considered. Flight schedules are optimized by adding and dropping flights. When flights are added or dropped, their heuristic recalculates demands only in markets with significant amounts of traffic. Then, the passenger selection problem is resolved. Their method can be viewed as an enumeration of all possible combinations of flight additions and deletions. Marín and Salmeron (1996) apply the frequency planning to rail freight transportation. The formulation of the rail freight transportation design model is based on the modelling of the physical network, the services and the demand. They study the problem with the help of nonconvex optimization models which they solved using heuristic methods to obtain the solution for realistic networks. Marín, Barbas and Gallo (1999) propose a model where the timetable was developed from the frequency planning. The objective is in general to minimize the total passenger delay cost. Armacost et al. (2002) describe a new approach for solving the express shipment service network design problem. Under a restricted version of the problem, they transform conventional formulations to a new formulation using what they term composite variables. The formulation relies on two key ideas: first, they capture aircraft routes with a single variable, and second, package flows are implicitly built into the new variables, the composite variables. Lately, researchers have focused on determining incremental changes to flight schedules, producing a new schedule by applying a limited number of changes to the existing schedule. 3

4 Lohatepanont and Barnhart (2004), in their incremental optimization approach select flight legs to include in the flight schedule and simultaneously optimize aircraft assignments to these flight legs. Garcia (2004) extends the previous model and proposes a combination between it and a decision time window model. Lan, Clarke and Barnhart (2006) consider passengers who miss their flight legs due to insufficient connection time. They develop a new approach to minimize passenger misconnections by retiming the departure times of flight legs within a small time window. They present computational results using data from a major U.S. airline and showing that misconnected passengers can be reduced without significantly increasing operational costs. Kim and Barnhart (2007) consider the problem of designing the flight schedule for a charter airline. Exploiting the network structure of the problem, they develop exact and approximate models and solutions, and compare their results using data provided by an airline. They show that the heuristic approach is capable of generating good solutions very quickly. Jiang and Barnhart (2009) propose a dynamic scheduling approach that reoptimizes elements of the flight schedule during the passenger booking process. They recognize that demand forecast quality for a particular departure date improves as it approaches; thus they redesign the flight schedule at regular intervals, using information from both revealed booking data and improved forecasts. Our contributions in this paper include the following: 1. As market demand may be stimulated as a result of changes in the flight schedule, airlines try purposefully to design schedules to capture the largest demands, so we include the demand and supply interaction in the context of airline schedule design. 2. Passengers' itineraries; we have the possibility of representing misconnected passengers due to lack of time to perform intermediate stops. Robustness is introduced avoiding misconnected passengers. 3. Passengers' recapture; as far as we include passengers' itineraries we have the chance of including recapture in a realistic way. 4. Airports capacities for arrivals and departures. We have developed a new integrated robust model to solve the schedule design problem in one unique step. As a proof of the model we have done some computational experience. Synopsis The paper is organized as follows: in section 2, we consider the supply modeling, in which the used time expanded graph is introduced. Section 3 explains the passengers behavior that we have considered, that is, the demand modeling. In section 4, the robustness criterium 4

5 is introduced. In the following section the robust airline scheduling model is introduced. Section 5 proposes the multiobjective optimization. And finally, in section 6 the computational experience is described. Conclusions and references are included in the followings sections. MODELING SUPPLY The objective of schedule design is to develop a schedule, defining an origin, a destination, a departure time, and an arrival time for each service to accommodate passenger demands given available resources. Given the estimated demand for travel, an airline wishes to determine the flight schedule which maximizes its profit while taking into account the satisfaction of its customers. In this system, two components interact: the aircraft flow in the physical network, and passengers using flights to travel. The most important transport supply data include, among others, flight costs, travel time, capacities, etc.; variables include flight frequencies and its timetable. The network is built considering the airports associated to the demand to be met. It is formed by the airports or nodes and all the feasible airway or sections alternatives linking them. The airports are defined by the operations that can be performed within them. The sections are the links between the airports. Each section is characterized by an origin airport and a destination airport. Each section has other technical characteristics as the section time and the capacity of the airplane assigned to it. The time is given by an average speed and the capacity by number of passengers that can be moved in each airplane using the section. In this way, we will consider that the time is discretized by partitioning the planning period, T, into intervals of equal length with starting points 0,1,...,T-1. The intervals' length will be taken as the time unit. As an example, if the period T corresponds to one week, and if the intervals' length is of one hour, then time 0 corresponds to 0:00 a.m. of Monday and time T corresponds to 12:00 p.m. of Sunday. The number of periods is 168. When a section is flown it will be called as flight leg; a flight leg is defined by an origin, destination and a departure time, that is, a flight leg is defined by the pair (s,t), where s is an element of the sections' set, S, and t in {0,1,...,T-1} is the departure time from the origin of S. The set of all possible flight legs is I = S x {0,1,...,T-1}. The proposed supply model is based on the definition of the time graph G= (K,A), where the nodes are: K= {(k,t) / k K is an airport and t {0,1,...,T-1}, and the arcs are: A=A1 A2; with A1= {(k,t),(k',t')} / s S and i I, where i={k,k',st s,t =t'-t} / k,k' K and t,t' {0,1,...,T-1} A2= {(k,t),(k,t+ct)} / t,t+ct {0,1,...,T-1}. 5

6 The arcs in A1 correspond to flight legs, i.e. the possible physical movements of passengers between two airports, st s,t represents that movement's time duration; those in A2 correspond to connection times where CT is the connection time for a flight connection. Note that a section instance or flight leg (s,t) defines in G a path going from the node (k,t) corresponding to the initial airport at time t, to the node (k',t') corresponding to the terminal airport, and (t'-t) is the section time duration. Since the type of flight network that an airline uses has a dominant impact on many of the planning problems, we will mention the common network types. Flight network is an informal name for the geographical network created by the flights operated by an airline timetable. There are three types (Lohatepanont and Barnhart, 2004) of airline networks: linear networks, point-to-point networks and hub-and-spoke networks. The network types described above are the pure definitions. In reality it is seldom the case that an airline has a pure point-to-point or hub-and-spoke network. Most airlines have some sort of hub, housing their main maintenance facilities and crew headquarters. And in most hub-and-spoke networks multiple hubs exist, as well as some direct flights between outlying airports. Supply interacts directly with demand and vice versa. Hub-and-spoke networks illustrate demand and supply interactions. To see this, consider removing a flight leg arriving or departing at a hub airport. The removal of a flight from a hub can have serious effects on passengers in many markets throughout the network. The issue is that the removed flight does not only carry local passengers from the flight's origin city to the flight's destination city, it also carries a significant number of passengers from many other markets that have that flight leg on their itineraries. In order to avoid interactions between flights of the same airline, that is, the competence, a time separation is introduced for departure times of the same sections. Thus, when a section departs, it cannot depart again until some separation time has been spent. In this way, competition for the uptake of demand is avoided. MODELING DEMAND For this work the demand is characterized by the origin, p, and destination, q, airports. Each pair (p,q) is mentioned as the market w. For each pair w the demand of passengers d w is assumed fixed and known datum. This demand will vary in time, that is, it is a dynamic demand, as we can see in Figure 1. For each demand, the passengers from origin to destination are considered in all possible itineraries or routes r, that may be classified by pair w as R w. In schedule design for a given airline, we are interested in its unconstrained market demand, that is, the maximum demand the airline is able to capture. Unconstrained market demand is allocated to itineraries or passenger routes, sequences of connecting flight legs, in each market to determine unconstrained itinerary or route demand. The demand is unconstrained 6

7 because the quantity of interest is measured without taking into account capacity restrictions. In the proposed model, the unconstrained market demand is assigned to passengers' routes taking into account the capacity; in this way we obtain the constrained passengers' route or itinerary demand. Each itinerary or route is defined by a set of sections that connect different airports. It can be composed of one section or more than one including in this last case intermediate stops at different airports. Considering large number of sections for the network, the number of possible passenger itineraries grows exponentially. Different proportions of the same market demand can be routed by different ways in order to use as well as possible the network capacity. Due to the necessity of a symmetric flight schedule, empty flights may appear. In order to avoid them an average demand is required by the airline for flight legs. This requisite could become in disrupted passengers in the real world that we are not taking into account. This is because enforcing a minimum demand, passengers willing to travel at determined time may be obeyed to do it at a different time. In order to avoid this, we also introduce the demandsupply interaction. The demand-supply interaction is represented by the possibility of not attending all the demand. In this way, non-profitable demand will be neglected. Figure 1 Market demand disaggregation To represent passengers' preferences we use market disaggregation, that is, for each market demand we separate it in blocks of demand requiring approximately the same average departure time. These average times can be obtained from passengers' surveys. The market disaggregation is made as in Figure 1. For a specific market, we separate it in groups with the same required average time at w. Passengers' dissatisfaction costs represent the difference between their required departure time and the one assigned by the model, and intermediate stops. When these costs are high or there is not enough capacity, they may be lost to the system or recaptured in other compatible market; this possibility is known as passenger recapture. With partial recapture, only a percentage of passengers will accept travel on an alternative itinerary, and that percentage depends on the desired and the offered alternative itinerary. 7

8 ROBUSTNESS ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING As mentioned before, robustness is introduced through passengers in itineraries with more than one flight, where a connection is mandated. Adding more slack for connection can be good for connecting passengers, but can result in reduced productivity of the fleet; the challenge then is to determine where to add this slack so as to maximize the benefit to passengers without getting worse the network operation (Lan, Clarke and Barnhart (2006)). Every connection is characterized by the minimum time required to perform it. This time varies from airport to airport and it can also vary in the same airport along the day. If a passenger is not able to perform the connection due to lack of time, the passenger will be misconnected. In this way, in itineraries with more than one flight, every passenger is mandated a minimum connection time (MCT) for flight connections. However, this time will not be always enough to perform the connection due to congested airports for example, and passengers will be lost to the system in the real world. We assume that the number of disrupted passengers depends on the available time to perform the flight connection. Once flights' arrival (AT) and departure (DT) times are known, the available connection time (CT) is also known. From airlines historical data, disrupted passengers number variability with connection time might be known and, consequently its number may be calculated for each flight connection. Assigning a statistical distribution to misconnected passengers, the probability of getting misconnected passengers depending on connection time can be calculated. For our test networks the exponential distribution has been chosen; misconnected passengers will decrease exponentially as the available connection time increases. Its probability distribution is as follows:, where depends on the itinerary connection characteristics and is chosen adjusting the probability distribution to historical data; it is supposed that once the connection characteristics are known (airport and time at which it is performed), the assigned gates will be probably known due to historical availability. ECT represents the available excess connection time, that is, the available time exceeding the minimum connection time (ECT=CT-MCT). In this way, given the available excess connection time (ECT), the probability of having misconnected passengers ( ) is: 8

9 Once misconnected passengers are known, they must be removed from the remaining flights of the itinerary, so extra capacity arises in those flights making possible to accommodate other passengers in it in case of disrupted passengers. AIRLINE SCHEDULING MODEL In the proposed model we will assume some issues. We consider the passengers transfer possibility, that is, for every passenger itinerary we will consider the possibility of intermediate stops in the flight. We will consider flights composed of up to two flight legs or one intermediate stop; this issue is mandated for most passengers in hub and spoke networks but not in point to point networks. We will suppose that the unconstrained demand number for each market disaggregation is known; to obtain the actual attended demand we include the demand and supply interaction. In this way, the attended demand will be function of the capacity assigned for each flight leg. We will not enforce the entire demand satisfaction. The neglected demand is penalized in the objective function. However, we do enforce the demand maintaining in its entire flight, that is, in the two flights legs of its itineraries, for example. In supply's side, we include airports capacities, that is, we enforce that the airplanes arrivals and departures must satisfy the airport's runway capacity. Each section has a determined capacity for each condition (for example, weather conditions), so the airplanes number at each period must be limited. In this way, we also include the section time duration dependency on time departure; this is due to the possibility of busy airports, bad weather conditions, etc. In the supply aspect the most important issue is the fleet size; it will determine the flight legs that may be performed in the planning period, and consequently, the attended demand. Finally, we suppose that the schedule will be periodic, that is, the schedule will repeat after the planning period ends. For this purpose, we must take care about airplanes location at the end of the planning period. Its location must be the necessary one to repeat the schedule. In this way, we will enforce for each airport to have the same number of arrivals and departures. As we have said above the schedule design is comprised of two steps: the frequency planning and the timetable development. Historically, this process has been done sequentially, that is, first the frequency planning problem is solved, and then, with frequencies as inputs, the timetable is developed. In this work we define the Robust Airline Scheduling Model (RASM) which solves both at once. The following notation is introduced to explain RASM: Sets: periods' set. 9

10 sections' set. Each section is defined by an origin,, and a destination, : markets' set. We now define the markets by the origin,, destination,, and the average departure time. airports' set. : itineraries' set. itineraries' set composed of more than one section. : itineraries' set attending market. : markets' set attended by itinerary. : itineraries' set containing section as first section. : itineraries' set containing section as second section. : sections' set arriving at airport. : sections' set departing from airport. compatible markets for passenger recapture. : feasible departure time set for the second flight leg in itineraries with more than one flight leg. Parameters: : operating cost in section instance. : passengers' dissatisfaction in market using itinerary at period. : passengers' dissatisfaction due to transhipments times in itinerary with more than one section, being the second one : cost per disrupted passenger from market. recapture rate from market to. Its value depends on markets times. : cost per disrupted passenger in itinerary due to lack of time to perform transhipments. : passenger capacity in each section. : maximum airplane arrival capacity of airport at each time period. : maximum airplane departure capacity of airport at each time period. : maximum airplane capacity in each section and period. : minimum separation time between two consecutive departures of section instances (in periods). : passenger demand for each market. : section instance trip time. We include the section trip time duration dependent on departure time; this is due, i.e., to congested airports or weather conditions which may obey to slow down the airplane. : relative time to the planning period. : 1, if flight leg ( ) is flying at period time. : minimum connection time for each itinerary departing at time period. : minimum average demand required by the airline in section. : fleet size. : likelihood that passengers from ( ) will be misconnected in flight connection with ( ). : it is a real parameter. It represents the time needed to make operative a flight schedule, accounting for network flows. Its value depends on the network size and the fleet diversity 10

11 that will be assigned. Its value grows with both variables, network size and fleet diversity. Its value ranges from 2.25 to Variables: : positive variable. Passengers in itinerary and market departing at period. : positive variable. Passengers using itinerary at period, and the second section, at period. This auxiliary variable represents passengers using a flight leg after a transfer or transhipment. : positive variable. The airline tries to disrupt passengers from market to market. : =1, if section departs at period; 0, otherwise. The RASM is defined by the following objective functions and constraints: Objective Functions Coefficients As we have seen above, we have two different objective functions: one measures the passengers costs and the other one the operator costs. Passengers' Costs Passengers costs are composed of the dissatisfaction. This term measures the difference between the required average departure time and the actual departure time assigned by the model. The more the difference is, the more the penalization is. However, we can suppose that little differences will give little dissatisfaction. In this way, we decide to use a quadratic function for the time penalization. This penalization is as follows: The time dissatisfaction will be null if the assigned departure time is equal to the required average departure time. However, the overall dissatisfaction might not be null, for example, if the route is composed of more than one flight leg. To the previous formulation we add the following term (4), representing the transhipment time: The constants K 1 and K 2 may be calibrated through passengers surveys and transform the time units into costs units. Operator's Costs Disrupted passengers are passengers that the company does not attend due to lack of capacity or high dissatisfaction costs. In this way, we could think that these costs are related 11

12 to passengers because they do not travel. However, we can consider them as spill costs, that is, lost revenue. This costs can be computed as the distance the passengers would go if they were attended by the spill cost. But as we do not have the distance that passengers in each market would go until the model assigns them to routes, we can consider an average distance for each market. In this way, the spill costs may be computed as the product between and the average distance in a market, where is the revenue for available seat-kilometre. However, for disrupted passengers due to lack of time to perform the transhipment, the distance they would have flown is well known and these spill costs may be calculated as the route's distance by. Operating costs are the costs the company incurs due to the operation of flight legs. We include the costs related to each section length,, and those related to the departure or arrival time, for example slot costs. We compute these costs as:, where is the cost for available seat-kilometre, and captures the departure time,, costs modifications. We try to minimize the number of disrupted passengers due to misconnections. In this way we try to introduce the robustness criteria defined above. The expected misconnected passengers ( ) will be as follows: Objective Functions The first objective ( ) function (7) accounts for passengers costs: the first term is the dissatisfaction cost with departure time and, the second one is the dissatisfaction with intermediate stops. The second objective ( function (8) accounts for operator costs, that is, the first term represents operating costs, the second one incurred costs due to disrupted passengers, that is, spill costs, and the last one, costs due to lack of time to perform transhipments. Passengers Constraints 12

13 Constraints (9) ensure the passenger demand allocation through available itineraries in the network; they account for disrupted and recaptured passengers. Group of constraints (10) ensures that passengers in two sections itineraries remain in their trip during the second section; they also consider the average necessary time for performing transhipment between two sections of the same passenger route; note that for the same market demand attended in the same first flight leg, it can be satisfied through different second flight legs. Constraints (11) ensure that each flight leg has an average demand mandated by the airline; misconnected passengers are removed from the flight leg through the term. Constraints (12) ensure that there are enough active sections or flight legs to satisfy the passengers flow; the capacity in these active sections is a very important issue in the model. Depending on this value, the schedule will strongly change. This value must be estimated from demand models, airlines requirements, airports constraints, ect. Once again, misconnected passengers are removed from the flight leg. Flight Legs Constraints Constraints (13) are section capacity constraints; they ensure that the number of aircraft in a section at each period is lower than a maximum number; this capacity may depend on air navigation systems and regulations. We must adequate the aircraft number for every period of time to the allowed one. Group of constraints (14) ensures that the same flight leg does not depart until a specified time has been spent; this time is the separation time between two consecutive flight legs; in this way, competence between flights from the same airline is avoided. Airports Constraints 13

14 Constraints (15)-(16) are airport capacity constraints that try to spare the departures and arrivals at airports at each period; this is mandated by the available slots in the airport to land or take off. Depending on the time, these slots may vary in costs. These costs are included in the operating costs. Fleet and Symmetry Constraints Constraints (17) are the fleet capacity constraints; we must count the necessary aircraft to perform the schedule and compare it to the available ones. Block of constraints (18) ensures that the flight network is symmetric. In this way, the obtained schedule may be repeated periodically. Variable Dominion Constraints (19)-(22) define the variable dominion. As the demand number is an average value, passengers' variables can be defined as positive variables. MULTIOBJECTIVE OPTIMIZATION The ASM has been developed considering the case of multiple design objectives: passengers' and operator's costs. The multiobjective optimization problems (MOP) are generally solved by combining the multiple objectives into one scalar objective, whose solution is a Pareto optimal point for the original MOP. A standard technique in multiobjective optimization is to minimize a positively weighted convex sum of the objectives. It is easy to prove that the minimiser of this combined function is Pareto optimal (Ehrgott, (2005)). But, the problem is up to the user to choose appropriate weights. Until recently, considerations of computational expense forced users to restrict themselves to performing only one such minimization, considering just one set of weights chosen with care. Nowadays, more ambitious approaches aim to minimize convex sums of the objectives for various settings of the convex weights, therefore generating various points in the Pareto set. Though computationally more expensive, this approach gives an idea of the shape of the Pareto surface and provides the user with more information about the trade- 14

15 off among the various objectives. In this, weighted aggregation approach, different objectives are weighted and summed up to one single objective. We can generally define the ASM as follows:, where is a vector containing the problem variables, and is the feasible region of the problem. With multiobjective optimization techniques the problem then becomes as:. A Pareto boundary can be found by assigning varying values to. COMPUTATIONAL RESULTS Introduction Computational tests study the Pareto Optima curves showing different solutions by varying objective coefficients. Then, an appropriate value is chosen for each case in order to compare the achieved robustness in this new approach with a no robust one. All of our computational experience is for tests cases proposed below. Three different networks are studied, all of them Hub and Spoke (HS) networks. In Figure 2 the first network HS1 is shown: it is characterized by one hub and three spokes; each spoke is connected to the hub in both senses. The following network HS2 in Figure 3 has two different hubs and eight outlaying airports. Finally, the third network HS3 is drawn in Figure 4. For each network there are different markets and passenger routes. In network HS1 there are 34 markets and 12 passengers routes; in network HS2 there are 270 markets and 58 passenger routes; and, in network HS3 there are 1260 different markets and 210 passenger routes. Our runs have been performed on a Personal Computer with an Intel Core2 Quad Q9950 CPU at 2.83 GHz and 8 GB of RAM, running under Windows 7 64Bit, and our programs have been implemented in GAMS 23.2/Cplex 12. Figure 2 Air network HS1 Figure 3 Air network HS2 15

16 Figure 4 Air network HS3 The RASM size is shown in Table 1. The RASM number of discrete and continuous variables, constraints and non-zero elements are given for the model in proposed networks. Table 1 RASM size Network Variables Constraints Non-zero Discrete Continuous elements HS HS HS Pareto Curves In order to determine the best value for the multiobjective approach some experiments have been carried out; one group of them for each presented network. Each group of experiments consists of obtaining the Pareto curve for every network by varying the value. However the detailed results are shown, that is, every term in the objective function is drawn in the following figures depending on. In Figure 5 the results for the network HS1 are shown, in Figure 6 for network HS2 and, finally in Figure 7 for network HS3. The blue line represents the objective function values, the red one operating costs, the green line spill costs, and, the purple one dissatisfaction costs. When it is necessary an additional line is drawn in light blue colour, it represents modified spill costs. In the graphics, every marked point represents a solution to the mixed integer model. The value will be chosen accounting for dissatisfaction costs and spill costs. This is due to the fact that operative costs remain almost constant. Thus, the point where dissatisfaction and spill costs are equal will give the optimal value. In order to determine it correctly, disrupted passengers must be accounted for ; at this point, only operator's costs are 16

17 taken into account, so the operator tries to attend all the possible demand without accounting for dissatisfaction costs; thus, passengers not attended at this point will be probably disrupted for every value. So, spill costs curve is moved downward a quantity equal to spill costs at. The intersection between this modified spill costs curve and dissatisfaction curve will give the optimal value ,2 0,4 0,6 0,8 1 Figure 5 Air network HS1 computational results Objective Function Operative costs Spill Costs Dissatisfaction Costs Modified Spill Costs ,2 0,4 0,6 0,8 1 Figure 6 Air network HS2 computational results Objective Function Operating Costs Dissatisfaction Costs Spill Costs ,2 0,4 0,6 0,8 1 Objective Function Operating Costs Dissatisfaction Costs Spill Costs 17

18 Figure 7 Air network HS3 computational results Under the criterion explained above, an objective coefficient value is chosen. This value will be used in the following subsection addressing robustness in the proposed study cases. Although pareto curves have been drawn continuously, we have to note that only the marked points correspond to real obtained solutions, and that there will not probably exist a different solution for every value because the RASM is an integer mixed problem. Robustness As it was explained above, robustness is achieved through passengers that must perform flight connections. In order to demonstrate that a more robust schedule is obtained using the proposed approach, a comparison is made with a no robust Airline Scheduling Model (ASM). The ASM is the same model explained above but removing robustness aspects, that is, the objective function's term in (4) penalizing misconnected passengers, and the terms in constraints (7) and (8) accounting for misconnected passengers in flight leg's minimum demand and capacity, respectively. In Table 2 misconnected passengers are compared for every network. For each network in the first column, the optimal value that has been chosen appears in the second column. In the third column the percentage of misconnected passengers is shown for the ASM; this percentage is calculated with respect to the total number of attended passengers. In the last column the percentage of misconnected passengers is shown for the robust case (RASM). For every network the percentage is sensitively reduced. Table 2 Misconnected passengers Network ASM RASM HS HS HS Robustness is achieved through the reduction in misconnected passengers. However, this reduction is not for free, it has a price: the robustness price. I order to analyse this concept, objective function values are shown in Table 3 for every study case. In the robust case (RASM), objective function's values are greater than the no robust one (ASM). However, this increase may be due to the term of misconnected passengers that is not included in the no robust case (ASM). To clarify this aspect information about some objective function terms' is provided in Tables 4 and 5. In Table 4 operating costs are shown. For HS1 study case, operating costs are greater in the robust approach (RAMR). However for the rest of the study cases these costs are greater for the no robust case (ASM). Consequently, it cannot be said that the price of robustness falls on operating costs. In Table 5 passengers' dissatisfaction costs are written for every study case. One must note that these costs are always greater in the robust case (RASM), that is, in the robust 18

19 approach where misconnected passengers number has been reduced; passengers' dissatisfaction has been increased. This is due to the fact that in the robust approach departure times are chosen accounting not only for passengers satisfaction and capacities but also for misconnected passengers in itineraries with more than one flight leg. Thus, it can be concluded that the price of robustness remains in passengers' satisfaction. Table 3 Objective function Network ASM RASM HS HS HS Table 4 Operating costs Network ASM RASM HS HS HS Table 5 Dissatisfaction costs Network ASM RASM HS HS HS CONCLUSIONS A new robust approach has been proposed to solve the airline scheduling problem, where frequency and timetable problems are jointly solved. In addition, passengers' flows are obtained through the different itineraries in the network. Market demand and supply interaction have been included, making possible to stimulate demand through flight schedule changes. Furthermore, passengers' partial recapture has been included in a realistic way; this due to the fact that market demand is allocated to itineraries. As far as itineraries are composed of more than one flight leg, intermediate stops have been included, accounting for passengers dissatisfaction. Airports' arrival and departure capacities have been included. In the presented approach we suppose that these capacities (slots) are well known, and that they usually are associated to determined gates. It also may be included the purchase of new slots, however, nowadays this issue is a very difficult and time consuming task in some aiports. Robustness has been introduced through itineraries with more than one flight leg. When an intermediate stop must be performed, passengers need some undetermined time to accomplish it. This undetermined time is captured through statistical distribution and it is 19

20 introduced into the model to represent expected misconnected passengers. In this way, the expected costs that the operator would incur due to misconnected passengers are reduced. The model has been tested in three different networks. Computational results show how robustness may be achieved. However, this robustness has a price. The robust approach has been compared with a no robust approach showing the price of the achieved robustness. Future research may integrate the airline scheduling problem and the fleet assignment problem. In this way, the average values used in the proposed approach would be substituted by real values depending on the assigned fleet type. REFERENCES Armacost, A., Barnhart, C. and Ware, K. (2002). Composite Variable Formulations for Express Shipment Service Network Design. Transportation Science, 36, Barnhart, C. and Cohn, A. (2004). Airline Schedule Planning: Accomplishments and Opportunities. Manufacturing & Service Operations Management, 6, Chan, Y. (1972). Route Network Improvement in Air Transportation Schedule Planning. Flight Transportation Laboratory R72-3, Massachusetts Institute of Technology, Cambridge, MA. Garcia, FA. (2004). Integrated Optimization Model for Airline Schedule Design: Profit Maximization and Issues of Access for Small Markets. Department of Civil and Environmental Engineering and the Engineering Systems, Massachusetts Institute of Technology, Cambridge, MA. Ehrgott, M. (2005). Multicriteria Optimization. Springer. Jiang, H. and Barnhart, C. (2009). Dynamic Airline Scheduling. Transportation Science, 43, Kim, D. and Barnhart, C. (2007). Flight Schedule Design for a Charter Airline. Computers & Operations Research, 34, Lan, S., Clarke, J.P. and Barnhart, C. (2006). Planning for Robust Airline Operations: Optimizing Aircraft Routings and Flight Departure Times to Minimize Passenger Disruptions. Transportation Science, 40, Lohatepanont, M. and Barnhart, C. (2004). Airline Schedule Planning: Integrated Models and Algorithms for Schedule Design and Fleet Assignment. Transportation Science, 38, Marín, A. and Salmerón, J. (1996). Tactical Design of Rail Freight Networks. Part I: Exact and Heuristic Methods. European Journal of Operational Research, 90, Marín, A., Barbas, J. and Gallo, G. (1999). Railway Freight Scheduling Using Bender's Decomposition. Working Paper, Universidad Politecnica de Madrid. Simpson, RW. (1966). Computerized Schedule Construction for an Airline Transportation System. Flight Transportation Laboratory, Massachusetts Institute of Technology, Cambridge, MA. Soumis, F., Ferland, JA. and Rousseau, JM. (1980). A Model for Large Scale Aircraft Routing and Scheduling Problems. Transportation Research, 14,