Airline Schedule Planning Integrated Flight Schedule Design and Product Line Design

Transcription

1 Diplomarbeit Airline Schedule Planning Integrated Flight Schedule Design and Product Line Design Universität Karlsruhe (TH) Fakultät für Informatik Institut für Algorithmen und Kognitive Systeme Fakultät für Wirtschaftswissenschaft Institut für Wirtschaftstheorie und Operations Research Betreuer (IAKS): Prof. Dr. Jacques Calmet Betreuerin (WIOR): Dr. Cornelia Schoen Autor: Andriniaina Rabetanety Matrikel-Nr.: Sommersemester 2006

2 Hiermit bestätige ich, dass ich diese Arbeit alleine und ohne fremde Hilfe erstellt habe. Ich habe keine anderen als die angegebenen Literaturhilfsmittel verwendet. Karlsruhe, den 22. Juni 2006 Andriniaina Rabetanety

3 Contents 1 Introduction Motivation Outline of the thesis Airline Schedule Design The Five steps of Airline Schedule Design Route Development Schedule Design Fleet Assignment Aircraft Maintenance Routing Crew Paring or Crew Scheduling State-of-the-art of Airline Schedule Design Incremental Airline Schedule Design Integrated models Pricing problem in Airline Schedule Design Solution methodologies Product Line Design Conjoint analysis State-of-the-art of conjoint analysis Preference model Choice rules Integrated model for airline schedule generation and product line design Assumptions Mandatory and Optional flight legs Airline Passenger Demand Airline Supply Fleet Assignment Competitive flights Interaction between passenger demand and airline supply Passenger Mix Model Conjoint analysis at the itinerary attribute level Conjoint analysis at the itinerary level Objective Function Formulations Formulation with utility at itinerary attribute level Formulation with utility at the itinerary level Formulation with utility at the itinerary level without demand segmentation ii

4 5 Solution Approaches Problem class Branch-and-Bound Sequential Quadratic Programming methods Incremental airline schedule model Implementation Framework Numerical Algorithms Group library (eu04cc) Branch-and-Bound algorithm Formal description Improving Branch-and-Bound through search strategy Computational Results Problem with 1 OD-Pair Influence of the consumer preference on the optimum schedule Performance of the first call of the SQP function Performance of the Branch-And-Bound algorithm Truncated Branch-And-Bound algorithm Conclusion Summary of contributions Future Research Directions iii

5 List of Figures 1 Airline Schedule Design subproblems Incremental approach with base and master flight list Time Line Network for an aircraft type Unfeasible schedule based on the Time Line Network figure Feasible schedule based on the Time Line Network figure Flow Chart for incremental solution Formal description of the basic Branch-And-Bound algorithm Branch-and-Bound with Best first search Combination of Best first and Depth first tree search Formal description of the Branch-and-Bound algorithm with combined search strategies Example with 2 OD-Pairs Example with 5 OD-Pairs Time line Network for the example with 5 OD-pairs List of Tables 1 Three itineraries with two attributes Conjoint analysis with two attributes and six level sets Examples of itinerary attributes State-of-the-art of conjoint analysis Numerical values of the parameters for the first example Initial and final point for the first example Parameters for an airline network with 2 OD-Pairs Initial and final point for the second example Influence of the utility to the flight f Initial point with both flight legs included Other convergence point Three outcomes of the SQP function for feasible starting points Size of the constraint matrix Parameters for an example with 5 OD Pairs Approximative evaluation of the performance to find the starting point of the Branch-And-Bound Starting point for the Branch-And-Bound algorithm Output of the Branch-And-Bound algorithm Bad starting point for the Branch-And-Bound algorithm Output of the Branch-And-Bound algorithm with a bad starting point Performance of the complete algorithm with five OD-Pairs

6 1 Introduction The growing gap between airport capacity and airline passenger demand has forced airlines to improve their quality of service. At best, an airline would transport its consumers at their desired time and with their desired level of service. Matching customers expectations enable to capture a large flow of airline passengers but does not ensure the maximization of profit. Regarding to the revenue of an airline company, the airline schedule planning represents the most critical service. This task involves a complex and long lasting decision-making process. Basically marketers choose a set of flights and decide which flights they want to serve and how much they will charge the consumers. Furthermore, they have to take into account a large set of constraints of different types, from the airspace and airport congestion (operational constraint) to the crews working conditions (human resource condition). Not only that the overall size of the problem is enormous, but a bad schedule could terribly affect the revenue and the market share of the airline, causing a loss of millions of euros. Because of the complexity of the problem, the pressure that the planners are undertaken, and the turnover of the airline at stake, the airline industry has been a great concern of the operations research community for the past 30 years. 1.1 Motivation Models of airline schedule design focus nowadays on integrating consecutive steps involved in airline schedule design, that will be presented in section 2.1. And most of them care less of how airline passenger choice might evolve if price of one flight is raised. How many of these passengers will decide to choose another flight? Which other flight will they choose? Will the profit be reduced? In this diploma thesis, we propose a model that answers questions related to the passenger choice, with the maximization of airline revenue as objective function. Unlike previous schedule design efforts, we concentrate on the pricing problem (Product Line Design), namely how to adjust fares to meet the maximum profit, associated with an integrated schedule and fleeting assignment problem (Airline Schedule Design). By this mean, our model can control the capture of airline passenger traffic by adjusting prices of flights. We have based our approach on the integrated model of Lohatepannont M. and Barnhart C. [4] for the airline schedule design and discrete choice models described by Gaul W., Aust E., Baier D. [9] for the preference measurement. 1.2 Outline of the thesis As mentioned earlier, our work is built on two research fields: Airline Schedule Design and Product Line Design. That is the reason why we dedicate two sections of our diploma thesis, section 2 and section 3, to present separately challenges, literature reviews and state-of-the-art in each field. These sections provide a general understanding of fundamental notions, such as the schedule generation and the fleeting assignment model or 2

7 conjoint analysis and discrete LOGIT model, that will be used throughout this thesis. In section 4, we present our approach of the problem and the assumptions that we consider for ease s reason. As for the passenger choice measurement, we differentiate between two approaches: one at itinerary attribute level and the other one at the itinerary level. This section ends up with different formulations of our model and the explanations of the operational constraints. In section 5, we describe common approaches to solve this problem, given the class that this problem belongs to (Mixed Integer Nonlinear Problem): Branch-and-Bound and Sequential Quadratic Programming (SQP) methods. The section 6 deals with our implementation of a solver for this model that realizes a Branch-and-Bound algorithm with a Depth and Best First search strategy, which solves a Sequential Quadratic Programming (SQP) problem at each node. We discuss the programming package used to realize the SQP function and a formal description of the Branch-And-Bound algorithm. In section 7, we discuss the performance of our implementation for a set of examples, in order to estimate whether the optimum of the objective function is found or not and with which percentage the optimum is met. We analyse the influence of the utility and the choice of the initial schedule on the optimal schedule. Finally computational results for bigger problems are compared to estimate the efficiency of our Branch-And-Bound algorithm. Because the problem size grows exponentially, we can not run the Branchand-Bound algorithm until the end. We rather stop the Branch-and-Bound algorithm when a feasible solution close to the optimum is found. In section 7.5, we discuss possible stopping criterium for the truncated Branch-and-Bound. 2 Airline Schedule Design In this section, we introduce first the components of the airline schedule generation problem and the related mathematical models, then the state-of-the-art of Airline Schedule Design by reviewing different integrated models. Here, we give notions that are essential to understand the formulation of our problem. We also position our work in the current research field. 2.1 The Five steps of Airline Schedule Design The airline schedule generation problem takes the airline passenger demand, airport and aircraft characteristics, maintenance and personal requirements as incomes. The outcome is a selection of flight legs (Origin-Destination pair, aircraft s type,arrival/departure time) that maximizes airline profit subject to resource constraints (aircraft and airport capacity, maximal working hour, minimal ground time,...). A flight leg is defined with three attributes: an Origin-Destination pair (an OD pair is a couple of airports), an aircraft s type and an arrival/departure time. Ideally, this large scale problem should be solved all at once. However, due to the enormous amount of parameters and decision variables, it has been divided into five more manageable and sequentially computed 3

8 Figure 1: Airline Schedule Design subproblems. subproblems: Route development, Schedule Design, Fleet Assignment, Aircraft Maintenance Routing and Crew Scheduling. Each subproblem happens at different point in time of the airline schedule design process. Stages are operated in the order, displayed in the figure 1. Each step has particular constraints and purposes, that are the subject of the following subsections Route Development In this stage, planners should decide upon the sets of origin-destination pairs they want to offer. This stage happens 12 months before operation of the airline schedule. In this thesis, we consider that the route development has already been done Schedule Design Decision makers have to determine routes or itineraries for the aircrafts, such that costs are minimized. To choose the set of itineraries is the most determinant part of the process because all the other phases take the generated schedule as input. Schedule can be created for one day and repeated every day. The schedule design involves the two following phases: (1) Frequency Planning. Planners choose the frequency associated to each flight. (2) Timetable Development or route selection. Planners decide when each flight 4

9 will be offered and included in the schedule. The result of the timetable development is a list of flight legs, called base schedule. In this thesis, we focus on the route selection. Since we limit our model to a short period of time, we do not consider the frequency of a flight. However, the frequency planning stage can be operated with the outcome of our model Fleet Assignment Once the schedule is determined, each flight must be assigned to a fleet type so as to minimize assignment cost of flight legs to aircraft types subject to the three following constraints: (1) Assignment constraints. Each flight must be assigned to one fleet. (2) Flow balance or Flow conservation. Considering an airport at a time t, the combined number of aircrafts that land at the airport and aircrafts on the ground immediately before t, compensates the combined number of aircrafts that take off and aircrafts on the ground immediately after t. (3) Plane count. Planners can not use more than the available number of aircrafts. Aircrafts can be on the ground or in the air. The fleet assignment model takes as input the available types of aircraft and a given schedule with fixed departure times. The fleet assignment problem requires a network structure as input to represent the flight legs. In a time-space flight network, a node is an airport, including every activities in this airport (taking off, landing) at a time t. An aircraft movement in space and time is associated with an arc. The network has flight arcs, between departure and arrival airport of the same flight, ground arcs, between two activities in the same airport, and wraparound arcs, between the first and the last node of the time horizon. The related decision variables are the binary fleet assignment variable θ fk and the ground plane count variable z k,a,t. The fleet assignment problem is given by: min( k K c fk θ fk ) (1) f I(k,a,t) θ fk + z k,a,t + θ fk = 1 f Ω (2) k K f O(k,a,t) z k,a,t + a A θ fk z k,a,t = 0 {k,a,t} P (3) f Ω k θ fk N k k K (4) θ fk binary and z k,a,t 0 (5) 5

10 where A : Set of airports, indexed by a. P : Set of nodes of the underlying network. A node is composed by a time, an airport and an aircraft, indexed by {k,a,t}. Ω : Set of flight legs in the flight schedule, indexed by f. K : Set of aircrafts, indexed by k. N k : Number of aircrafts in fleet-type k. Ω k : Set of flight legs that pass the count time when flown by fleet-type k. I(k,a,t) : Set of inbound flight legs to node {k,a,t}. O(k,a,t) : Set of outbound flight legs to node {k,a,t}. c fk : Cost for the assignment of the aircraft k to the flight leg f. δ fm : = 1, when the itinerary m includes the flight leg f, = 0 otherwise. θ fk : = 1, when the flight leg f is flown with the aircraft k, = 0 otherwise. z k,a,t + : The number of aircraft k on the ground at the airport a immediately after time t. z k,a,t : The number of aircraft k on the ground at the airport a immediately before time t. Constraints (2)-(4) express the assignment constraint, the flow conservation and the plane count constraint, described at the beginning of the section Aircraft Maintenance Routing The purpose is to determine feasible aircraft routes, sequences of flight legs flown by an aircraft type, under maintenance requirements. A routing is a set of aircraft routes. Rotations are routings which begin and end at the same airport. Routings and rotations partition flight legs in the schedule. The aircraft maintenance routing problem takes a fleeted schedule and the available number of aircrafts as input. Maintenance requirements are modeled approximatively by aircraft staying overnight at an airport. Each aircraft type needs specific technical checks and a mandatory ground-time. This step ensures that all the flights in the schedule respect maintenance conditions. The aircraft maintenance routing problem has been cast as a network circulation problem with side constraints Crew Paring or Crew Scheduling In the crew scheduling problem, both cabin and cockpit crew are assigned to a flight leg, minimizing the crews costs. An airline crew can only be assigned to an aircraft it is qualified to run. Work schedules must satisfy maximum time-away-from-base (period that flight crews are away from their home station) restriction, and airline crews are not allowed to stay on duty longer than a maximum flying time requirement. The crew paring problem are usually formulated as a set-partitioning problems where each row of the constraint matrix corresponds to a scheduled flight and each column corresponds to a legal crew pairing. 6

11 2.2 State-of-the-art of Airline Schedule Design The decomposition into sequential stages has simplified the model but it has also decreased the accuracy to describe the reality of the problem. Therefore, researchers attempt to integrate two or more consecutive stages of airline schedule design. Another attempt to reduce the size of the problem is the incremental approach, where a schedule is iteratively constructed. In the next section, we review the incremental approach, integrated models and solution methodologies Incremental Airline Schedule Design Actually, planners usually do not build the schedule from scratch but start with an existing schedule. The existing schedule could be the schedule from the last season or last year. Incremental models improve a given schedule by applying modifications to the flight legs. These changes can be: (1) re-scheduling within small time windows, (2) adding subsets of flights to or deleting them from the schedule. Therefore, there are four different approaches to the problem, namely, (1) allowing only flight re-timing, (2) allowing only flight additions and/or deletions, (3) allowing both flight re-timings and flight additions and/or deletions sequentially (4) allowing both flight re-timings and flight additions and/or deletions simultaneously. The process iterates until all combinations of flights have been explored or the optimum objective function has been found. Lohatepanont M., Barnhart C. [4] develop an incremental model allowing only additions and deletions that solves, at each iteration, an integrated model for schedule design and fleet assignment (ISD-FAM) and a passenger mix model (PMM).(for a detailed description 1 see the paper of Barnhart C., Cohn A. [1]). The ISD-FAM gives a feasible good schedule to the PMM, that finds the most profitable flow of passengers over this schedule. The PMM allows to capture the best amount of passengers on each flight leg. The objective function of the passenger mix model is to maximize revenues from the flow of passengers on itineraries. The formulation of the passenger mix problem will be shown in the section 5.3. Lohatepanont M., Barnhart C. underline the decisive interaction between passenger demand and airline supply. The profitability of one itinerary 1 Barnhart C., Cohn A. detail impact, challenges and modelling approach of each steps involved in the schedule planning. Thus they present examples of integrated models and solution approaches such as Branch-and-Price algorithm. 7

12 is indeed correlated with all other itineraries in the schedule, since passengers can be rejected from one flight and take another flight instead. This diploma thesis refers to their incremental approach. However, we integrate a customer choice model to cast the relationship between demand and supply Integrated models Researchers have built integrated models to overcome the simplications introduced by the decomposition in steps. We review the state-of-the-art of integrated models but we restrict our model to the integration of the two first stages: schedule design and fleet assignment. - Schedule Design and Fleet Assignment. Lohatepanont M., Barnhart C. present two integrated models that optimize the selection of legs and the assignment of aircraft types observing the demand and supply interaction: the integrated schedule design and fleet assignment model (ISD-FAM) and the approximate schedule design and fleet assignment model (ASD-FAM). The ASD-FAM holds a constant market share model that uses the recapture mechanism to adjust demand approximately. The ISD-FAM holds a variable market share model that uses demand correction term to adjust demand explicitly. - Fleet Assignment and Aircraft Routing. Barnhart C. et al. [8] develop an integrated model for fleet assignment model and the aircraft routing problem by using strings of flights. Their model contains one core aircraft routing model for each aircraft type under the constraint that each flight leg is assigned with only one aircraft type. Their integrated model achieves a near-optimal fleeting and routing solution. - Aircraft Routing and Crew Scheduling. Klabjan D. [9] proposes to solve the crew pairing problem before the aircraft routing problem. To combine both problems, he adds the so called plane count constraint to the crew pairing problem. These constraint guarantees that forced turns (all pairings that imply a plane turn) can be extended into a feasible rotation if and only if the number of planes on the ground does not exceed the plane count imposed by the fleet assignment model solution Pricing problem in Airline Schedule Design Our model casts the interaction between the consumer choice of itineraries and the optimization of the airline s profit. To date, there is no model of airline schedule design that evaluates the passenger preference with conjoint analysis. But other methods have been used to describe how an airline charges its consumer. Teodorovic D., Kremar-Nozic E. [3] use, for instance, the relationship between market share and flight frequency to define the total expected number of passengers on a route 8

13 as a function of the market share. The decision support system TALLOC determine the passenger behaviour as a pricetime behavioural desirability function, which is calculated for each flight service, as described by Grosche T., Heinzl A. [10]. Soumis, Perland and Rousseau [19] 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 deleting flights. When flights are added or dropped, their heuristic recalculates demand only in markets with significant amount of traffic. Then the passenger selection problem is solved. Their heuristic go through all possible combinations of additions and deletions. By comparison, our model recalculates demands at each deletion or additions of optional flights in every market. Dobson G., Lederer P. [2] propose a model for the competitive choice of flight schedules and route prices in a hub-and-spoke system in order to maximize airline profit. Passengers have preferences that might vary between itineraries. The demand forecasting model is a linear function of the most desired departure time, the duration of the flight and the price. They consider only one class of service, one size of aircraft, and through traffic via hubs. Demand is therefore a weighted logit function. The utility function is supposed to be constant during each slot (the day is divided into 1-hour-slots). The more the real travel time deviates from the ideal one, the more the utility decreases. They develop a two-stage heuristic: First, all candidate flights are assigned to two-hours intervals in the schedule, then non-profitable ones are eliminated. The contributions are measured as the difference between the schedule with all candidate flights and the one without the considered one. In the second stage, schedule feasibility is verified by solving the fleet assignment problem. Instead of a continuous utility function, we have chosen a discrete path-worth utility where each itinerary will be described by a set of attributes. Methods of conjoint analysis are used to get such preference measurements Solution methodologies Airline schedule models are often enormous in terms of number of variables and usually exhibit binary and/or integer variables. They require solution methodologies that decompose and reduce the size of the problem. We give an overview of the most common techniques for large scale problems. Large scale linear problems are often solved by a delayed column generation. In this algorithm, only a subset of columns of the constraint matrix, called the restricted master problem, is solved. Then the column generation subproblem is iteratively solved until we cannot find a column with negative reduced cost, that means an optimal dual function to the original function was found. Erdmann A., Nolte A., Noltemeier A., Schrader R. [6] solve the path-based mixed integer programming formulation with column generation. They cast the aircraft rotation subproblem as the problem of finding shortest paths in 9

14 a constraint network. The purpose is to find the path that generates the cheapest cost. The constrained shortest path algorithm use the same framework as the shortest path algorithm. Lohatepanont M., Barnhart C. [4] construct first a restricted master problem (RMP) of their integrated model and then solve the LP relaxation of the RMP using row and column generation. Branch-and-Cut and Branch-and-Price algorithms use LP relaxations with variable and constraint generation at the subproblems. Constraint generation strengthens the LP relaxation at the subproblems, while variable generation extends their feasible domains. Subproblems are parallely processed. Lagragian relaxation is a common technique for solving large-scale integer problem. Difficult constraints are moved to the objective function with linear penalty. The new problem is then solved by a Lagragian subgradient algorithm. By Benders decomposition, the algorithm solves a mixed integer program with a single continuous variable at every iteration. Branch-and-bound algorithms are developed for problems, in which the decision variables take discrete values from a specified set. A Branch-and-Bound algorithm covers the solution space by dividing the solution space into subregions (branching). For each subregion (node), we calculate an upper bound and a lower bound. The approach bases on the assumption that (for a minimization problem) if the lower bound for a subregion A is greater than the upper bound for another subregion B, then A may be safely discarded from the search. The algorithm is detailed in the section 5.2. The branch-and-bound approach is not a heuristic, but it is an exact procedure. 3 Product Line Design When firms want to introduce products or line of products into a new market with many customer segments, they face the known marketing problem of product line design and pricing. Product line design models take the measurement of customer preference as input. From a sample of consumer data, a global market can thereby be modelled. Product line design models help marketing managers to predict the profitability of a new product in a market. Models adjust prices to maximize either the total consumer welfare or the company profit. Due to the complexity of the problem, most of researchers have focused on the case of single product. Our model integrates a product line design model with multiple products, where products are itineraries. The purpose is to evaluate itinerary prices regarding passenger preferences. Utility measurement is done via conjoint analysis before the schedule generation stage. In the next section, we introduce the stages of a conjoint measurement. 3.1 Conjoint analysis We introduce the idea of conjoint analysis with a basic example, where products are itineraries. We consider three itineraries with prices and durations of the trip as attributes, ranked in table 1. This table does not bring information about the importance of price compared to the duration. The idea of the conjoint analysis is to evaluate both 10

15 Rank Price (euros) Duration (minutes) Table 1: Three itineraries with two attributes Price/Duration Table 2: Conjoint analysis with two attributes and six level sets features conjointly. The ranking of the nine possible combinations of levels for a respondent is displayed in table 2. We can observe that the respondent tends to trade-off duration for price. From this data, we can deduce the preference for each level set. In conjoint analysis, products are described as a set of attributes or features, of which level sets can vary. Respondents rank in terms of preference each combination of attribute level sets, called product profiles. The conjoint analysis allows to determine the relative importance of each product feature by measuring the consumer preference for each product profile. Running a conjoint analysis involves the following steps: - Choose a preference model for measuring the preference function. - Determine attributes for example price, number of stopovers, duration or delay of the trip. - Choose level sets for each attribute. - Define products as a combination of attribute levels. - Choose the form in which the combinations will be presented to the respondents. - Select the technique to be used to analyse the collected data. - Select the measurement scale. - Choose an estimation method. In our model, each itinerary is described by a set of attributes. An itinerary is a trip characterized by an origin-destination pair (a departure airport and an arrival airport). It can contain stopovers and can be flown by different aircrafts. The discrete criteria can be product-oriented as well as schedule-oriented as shown in table 3: 11

16 Product-oriented criteria Price of the flight Reservation/Flight class Sales conditions Quality of Service Schedule-oriented criteria Departure/Arrival airport Duration or delay of the trip Number of stopovers Frequency of the flight Table 3: Examples of itinerary attributes Stage Preference Model Measurement Scale Estimation Procedure Choice Rules Variants of Conjoint Analysis Models Vector Model; Mixed Model; Ideal Point Model; Rating Scale; Paired Comparison; Rank order; Choice-based; Metric and Non-Metric Regression; MONANOVA; PREFMAP;LINMAP; Multiple Regression; LOGIT; PROBIT; Hybrid; Maximum Utility (First Choice); Average Choice; LOGIT; PROBIT; Adaptive CA; bridging CA; Limit CA; Table 4: State-of-the-art of conjoint analysis. 3.2 State-of-the-art of conjoint analysis The table 4 gives the list of common models involved in each step of conjoint analysis. In the next section, we present the three preference models and choice rules for multipleproduct. 3.3 Preference model The first step of conjoint analysis is to define a mathematical model for the preference function. We define N the set of attributes, x jn the j-th level of the n-th attribute, w n the weight for the n-th attribute and u j the preference for the j-th level. The most common models are listed below: - The Ideal Point model establishes an inverse relationship between preferences and the weighted squared distance between the level of the attribute and the ideal point I n for this attribute. The preference is given by: u 2 j = n N w n (x jn I n ) 2 (6) - The Path-Worth model applies a piecewise linear curve f n to each level of the n-th attribute. The preference is given by: u j = n N f n (x jn ) (7) 12

17 - The Vector model is represented by the weighted sum of all levels, where each level of attribute is associated with an individual weight. The preference is given by: u j = n N w n x jn (8) 3.4 Choice rules Choice rules allow to estimate the probability of choice from utility data. The three common models are described below: - First Choice Rule Consumers select the product that maximizes their utility. The probability takes only the value 1 or 0, as shown in equation (9). Dobson G., Kalish S.[12] use the first choice rule in their pricing subproblem to ensure that customers select only the product with the highest utility. { } 1 when uid = max p id = i I(u i d) (9) 0 otherwise where p id : The probability that a customer d chooses the product i. u id : The utility of the d-th customer for the product i. I : The set of all the products, indexed with i. - Bradley-Terry-Luce (BTL) or share of utility rule The probability to select a product corresponds to the relative utility of the product, as shown in equation (10). p id = u id (10) where i I i I u i d > 0, d D. All preference values must be positive. Gaul W., Aust E., Baier D. [13] propose a new approach of the model with a probabilistic consumer behaviour model based on the BTL model, called PROLIN. They extend the formulation of the BTL model by adding a market parameter α that allows a calibration of the model to the market, as shown in equation (11). u i d where p id = i I uα id u α i d (11) α : The market parameter. 13

18 - Multinomial Logit (MNL) Rule The LOGIT model uses an assigned choice probability that is proportional to an increasing monotonic function of the alternative s utility. The probability that a consumer chooses a product is given by dividing the contribution of one product by the contribution of all other products in the same way as the BTL model, as described in equation (12). Negative utilities are possible. p id = exp(u id) (12) exp(u i d) i I Our integrated model will include a Multinomial Logit model as choice rule. For the First-Choice model, market share is the number of trials with highest utility divided by all the trials. For the BTL and MNL model, market share is the arithmetical mean of individual probabilities. There are, among others, two contributions in the conjoint-based product line design literature: Green, Krieger [14] and Kohli, Sukumar [19]. We base our approach on both approaches. The fundamental difference between both formulations is that decisions are made at different level. Green, Krieger decisions are made at the attribute level and Kohli, Sukumar decisions at the product level. By analogy, we will differentiate two formulations of our model as well: one at the itinerary level (product level) and the other at the attribute level. We discuss the formulation of the passenger preference at the itinerary attribute level in section and the one at the attribute level in section Integrated model for airline schedule generation and product line design In this section, we introduce the framework of our work before we formulate our model. We detail the interaction between airline passengers demands and airline supply. Finally we gave three formulations of our model. Notations are given at the end of the thesis. 4.1 Assumptions An airline company seeks to integrate a set of markets, composed by a set of airports. Each OD-pair (couple of source airport and destination airport) defines a market. For example, Paris-Berlin is a market and Berlin-Paris is another market, and both are called opposite markets. Each OD-pair can be flown by a set of itineraries. An itinerary is a sequence of flight legs, that are characterized by an airport-source, an airportdestination, a departure time and an arrival time. Itineraries are products that the airline company wants to offer in a market. For the purpose of conjoint analysis, each product is described by a set of attributes whose level set will be decided by the managers of the marketing s department. Each flight can be direct (without stops) or with one or more stopovers. Flights with stopovers are called via flights. 14

19 Figure 2: Incremental approach with base and master flight list Mandatory and Optional flight legs Planners want to construct a schedule, which is basically a list of itineraries. In the incremental approach, they build it from a base schedule. The base schedule is modified until the best schedule is found. At each iteration, the model takes as input a master flight list, that is composed by mandatory flight legs and optional flight legs. Mandatory flights have to be included in the final schedule while optional flights do not necessary have to. The modifications are the deletion or addition of flight legs. The output is a list of flight legs that maximizes the airline profit. An iteration of the schedule generation process is modelled in figure 2. The final list of flight legs does not have to be a list of rotations (itineraries that start and end at the same airport). We do not restrict our approach to a daily schedule or a schedule that will be repeated. Our model is designed for an airline company, that desires to optimize a subset of their actual schedule by introducing new flights or deleting non-profitable flights. In this case, itineraries are not rotations Airline Passenger Demand Demand for air travel is indeed a derived demand. It is derived from other needs of passengers. People rarely fly from cities to cities for the mere sake of travelling on a plane. Airline passenger demand follows the standard assumptions of a product line design problem, namely, 15

20 (1) Each market is composed of different customer segments of various size. In our thesis, we similarly talk about passenger segment or passenger d. Each segment d of the demand represents a subset of passengers that follows the same consumer behaviour. An example of segment could be students between 20 and 24 years old. In each customer segment, the unconstrained demand is the maximal number of passengers that the airline can capture. (2) Airline passengers choose the flight, that provides the maximum utility. They can not buy two tickets of one flight. Each passenger preference will be determined during the conjoint analysis stage. When the capacity of the aircraft is too small with regard to demand, passenger requests are rejected and the company loses the opportunity to increase its profit. This phenomenon is referred to as spill. But the company can cope with the spill by offering alternative flights to the original offer, a flight with a stopover instead of a direct flight for example. Our model takes the spill-recapture phenomenon into account by controlling demand/passenger choice among available alternatives indirectly through pricing and fare product selection Airline Supply Airlines commonly use a network structure to develop their supply strategy. As exposed in section 2.1.3, the structure underlying our model is a time line or time-space network. The figure 3 displays an example of a time line network for an aircraft type with three airports and four time points. Nodes represent activities of aircraft type, in an airport and at a time t. Arcs represent the movement of aircraft either in the air or on the ground. The network defines the airline company s market. It describes all crucial information to ensure the feasibility of a schedule: input/output flights at each node, ground arcs, number of flights by OD-pairs and number of aircrafts in the air Fleet Assignment The firm owns a finite fleet of different aircrafts or aircraft types. Each aircraft type has special characteristics, such as the number of seats or the minimum ground time. We assume that all aircrafts are available. The airline company can not assign more aircraft capacity to flight legs than the available. Every mandatory flight has to be assigned an aircraft, while some optional flights not. To fulfill operational constraints mentioned in section 2.1.3, the fleet assignment stage takes the time line network as input. Figure 4 displays an unfeasible schedule based on the simple time line network, shown in figure 3. Each colour represents an aircraft. In figure 4, the flow conservation constraint is not fulfilled at the node Paris 15 : 30. The red aircraft disappears after the node Paris 15 : 30. The aircraft has to go either to the node Paris 15 : 55 or to Boston 15 : 55. The aircraft either has to 16

21 Figure 3: Time Line Network for an aircraft type. Figure 4: Unfeasible schedule based on the Time Line Network figure 3. 17

22 Figure 5: Feasible schedule based on the Time Line Network figure 3. stay in the same airport (ground arc) or fly to another airport (normal arc). Figure 5 shows the feasible solution, where the aircraft stays at Paris airport between 15 : 30 and 15 : 55. The assignment cost vary depending on the flight legs flown and the aircraft type used. It represents the main cost in our model Competitive flights A consumer can also be attracted by a competitive flight. Our model takes the competition into account. Customers generally choose the best flight among all possible options: flights offered by the same airline company to another booking class non the same flight or to a low cost flights from other airlines. That is the reason why we model choice behaviour that accounts for competitive flight legs. 4.2 Interaction between passenger demand and airline supply As mentioned in the introduction, we base our approach of airline schedule design on the integrated model of Lohatepannont M. and Barnhart C. [4], where the flow of airline passengers is determined by the passenger mix model. Our model adjusts the price of itinerary considering the consumer preference. In the next subsection, we introduce the passenger mix model and compare it to our model. 18

23 4.2.1 Passenger Mix Model Given a fleeted schedule and the unconstrained itinerary demands, the Passenger Mix Model finds the flow of passengers that maximizes the airlines profit. The formulation is given by: Max fare m x mm (13) m M where variables are: m M m M m M m M δ mf x mm CAP f f Ω (14) x mm b mm D m m M (15) x mm 0 m,m M (16) x mm : the number of passengers who fly on itinerary m that desire to travel on itinerary m. fare m : the average fare of itinerary m. CAP f : the capacity of flight f. b mm : the recovery rate of passengers desiring itinerary m who are offered itinerary m. δ mf : equals 1 if flight f covers itinerary m and 0 otherwise. D m : unconstrained demand on itinerary m. Constraints (14) are the capacity constraints that ensure that the number of passengers on a flight does not exceed the number of seats available. Constraints (15) are the demand constraints ensuring that the total number of passengers that is accommodated or spilled does not exceed the corresponding unconstrained demand. Our approach is to express x mm with consumer preferences and to consider a pricing problem. The number of passengers who fly on one itinerary is now the result of the product between the probability that passengers select this itinerary and the unconstrained demand for the corresponding itinerary. Hence, the demand constraints are not needed anymore because the fraction of passengers on a flight is now always inferior to the unconstrained demand. The probability that passengers choose an itinerary is a function influenced by the price. The price is modelled as a continuous decision variable. With our notations and the assumption of a demand segmentation, we formulate the model as the adjustment of prices that maximizes the profit: Max fare m p dm ω d (17) m M d dem m δ fm p dm ω d CAP f f Ω (18) m M d dem m 0 p dm 1 (19) 19

24 fare m 0 (20) The probability p dm that a passenger decides to fly the itinerary m as a function of price and other fare product attributes will be the subject of sections and We have modelled the preference measurement with the methods of conjoint analysis. We observed two approaches, by comparison to the formulations of Green, Krieger and the one of Kohli, Sukumar [14]: (1) Conjoint analysis with decisions at the itinerary attribute level. (2) Conjoint analysis with decisions at itinerary level. We discuss advantages and disadvantages of each approach in the next two sections Conjoint analysis at the itinerary attribute level Each itinerary is described by a finite set of attributes. Prior to the schedule generation stage, the marketing s department conducts the analysis that gives the utility of the passenger d for the assignment of j-th level to the n-th attribute. The probability that a passenger selects an itinerary for a BTL model is given by equation (21) and for a MNL model by equation (22): p dm = N 1 M + M c m 1 =1 J n n=1 j=1 N 1 ( n=1 j=1 x jnm u djn J n x jnm1 u djn ) (21) where: p dm = M + M c N 1 exp( J n m=1 j=1 N 1 (exp( m 1 =1 n=1 j=1 x jnm u djn ) J n x jnm1 u djn )) (22) N 1 : Set of attributes of an itinerary, indexed by n. J n : Set of level sets for the attribute n, indexed by j. x jnm : = 1, when the n-th attribute is assigned to the j-th level in the itinerary m, = 0 otherwise. These are binary constants for m M c (competitive itineraries) and binary decision variables for m M. u djn : Utility of the passenger d for the assignment of j-th level set to the n-th attribute. Utilities can be negative in a MNL model. The choice of a consumer for an itinerary is considered only if the itinerary is included in the master flight list. Hence, the probability p dm should be equal to zero when the 20

25 itinerary is not served (q m = 0). In equations (21) and (22), we should differentiate the price from the other attributes, because decisions on prices yield revenue. In our model, we consider price as first attribute, indexed with 0 (n = 0). Marketers select the price of an itinerary among a set of possible prices. The choice of charging a passenger with a price level for an itinerary influences the passenger preference for this itinerary. The assignment of the j-th price level to an itinerary has an utility u dj0, given by conjoint analysis. The choice of itinerary prices directly intervenes in the objective function. The price of an itinerary m is shown in equation (23). J 0 x j0m fare jm m M (23) where j=1 J 0 : Set of possible prices, indexed by l. x j0m : =1, when a passenger is charged with the j-th possible price for itinerary m. fare jm : j-th level of the price for the itinerary m. With the contribution of the price, the probability that a consumer selects an itinerary becomes (24) for the BTL model and (25) for the MNL model. p dm = M N J n m 1 =1 n=0 j=1 N J n n=0 j=1 x jnm1 u djn q m1 + x jnm u djn q m M c N J n m 1 = M +1 n=0 j=1 x jnm 1 u djn (24) where p dm = M m 1 =1 N exp( J n n=0 j=1 N exp( J n n=0 j=1 x jnm1 u djn )q m1 + x jnm u djn )q m M c m 1 = M +1 N exp( J n n=0 j=1 x jnm 1 u djn ) (25) N : x jnm1 x jnm 1 q m : Set of attributes of an itinerary, with the price as first attribute and the other attributes are service-oriented. are decision variables. are binary constant. =1 when itinerary m is included in the schedule, = 0 otherwise. In product line design, decision makers must assign exactly one level to each attribute if the product is offered. Furthermore, each product should have all attributes assigned, in order to be introduced in a market. As itineraries are products in our model, every 21

26 itinerary included in the schedule has to fulfill both constraints. Condition (26) ensures that each attribute of an itinerary/fare product is assigned at most one attribute level once. J n j=1 x jnm q m n N, m M (26) To be included in the master flight schedule, each itinerary should have all attributes assigned. Equation (27) guarantees this requirement. J n j=1 x jnm = J n+1 j=1 We cast both requirements in equation (28). J n j=1 x j(n+1)m n N, m M (27) x jnm = q m n N, m M (28) This approach considers the assignment of a level to an attribute (x jnm ) as decision variables, with price as first attribute and other schedule-oriented attributes. The first disadvantage of this approach is the large number of decision variables. The measurement of the utility for each attribute and the determination of decision variables could be computationally challenging. If there are 10 passenger segments, 10 attributes with 10 levels, then there are utilities to measure and M decision variables per itinerary are to be determined. Furthermore the relationship between the choice of itinerary prices and the passenger behaviour is not accurate because of the restriction of the prices to a finite set of possible values. That is why we propose an approach at a higher level where prices are continuous decision variables Conjoint analysis at the itinerary level At the itinerary level, utilities are associated with itineraries, instead of itinerary attributes. Utility as monetary value represents the price that a customer is willing to pay for an itinerary. We assume here that these values are not influenced by the price but are indeed schedule-oriented or related to the quality of service. We suppose that utilities have been determined by conjoint analysis before schedule s operation. Decision makers choose price and select which itineraries as combination of non-price attribute levels to offer. Itineraries as combinations of non-price attributes are defined exogenously prior to the decision making process considered here and can be regarded as input to the optimization model. Prices are continuous variables in this approach. The expression of the probability that a customer chooses an itinerary is given by (29) for a BTL model and (30) for a MNL model: p dm = M m =1 max(u dm fare m,0)q m max(u dm fare m,0)q m + U d0 (29) 22