Maximization of an Airline s Profit

Transcription

1 Maximization of an Airline s Profit Team 8 Wei Jin Bong Liwen Lee Justin Tompkins WIN 15

2 Abstract This project aims to maximize the profit of an airline. Three subsystems will be considered Price and Demand, Fleet Assignment, and Gate Assignment. Detailed description of each subsystem s model, optimization and results is shown. The optimal price of and demand for tickets along each plane route will be found by the Price and Demand subsystem, which will be used by the Fleet Assignment subsystem to estimate the optimal number of planes to be flown on each route in a day. The Gate Assignment subsystem then takes the fleet assignment and calculates the delays at each airport due to the optimal flight schedule. Our problem deals with a schedule for the day, and works to find the optimal solution for a network of five airports. The problem contains both discrete and continuous functions, and optimization tools such as Sequential Quadratic Programming, Neural Networks and Genetic Algorithms are used. Optimal results will be shown, and limitations to these approaches will be discussed. 1

3 Contents FIGURES... 4 INTRODUCTION... 5 SYSTEM LEVEL NOMENCLATURE... 5 SUBSYSTEM DESCRIPTIONS... 5 PRICE AND DEMAND SUBSYSTEM... 5 Problem Statement... 5 Nomenclature... 6 Mathematical Model and Analysis... 6 Objective... 6 Variables... 7 Parameters... 7 Constraint... 8 FLEET ASSIGNMENT SUBSYSTEM Problem Statement Nomenclature Mathematical Model Objective Design Variable(s) Parameters Constraints Impact of Fleet Assignment Optimization Study GATE ASSIGNMENT SUBSYSTEM Problem Statement

4 Nomenclature / Parameters Mathematical Model Objective Design Variable Modifications to Original Model Significance of the Gate Assignment Subsystem Optimization SYSTEM INTEGRATION System Objective Subsystem Integration System Constraints Conflicting Requirements Final System Results REFERENCES APPENDIX A Market Size and Beta Coefficients APPENDIX B Aircraft Specifications APPENDIX C Average route lengths for each origin-destination pair APPENDIX D Average daily demand and fares used for subsystem optimization

5 FIGURES Figure 1: Potential Market Size of Each Route...7 Figure 2: Part-Worth Coefficients. A) Route. B) Number of Connecting Flights...8 Figure 3: Spline Interpolation Results. A) Price Part-Worth Coefficients. B) Time Part-Worth Coefficients...8 Figure 4: Importance of each attribute from the survey results...9 Figure 5: Profit of U 1 with varying competitor prices. A) Price of U 2 is $90 B) Price of U 2 is $ Figure 6: Profit of U 1 with two competitors. Price of U 2 is the same in both figures. A) Price of U 3 is $90 B) Price of U 3 is $ Figure 7: Local maximum profit with different i Figure 8: Local maximum profit with initial values picked using results of the previous run Figure 9: Average Prices of Tickets. A) Average price trend over the day. B) Average price for each route for the entire day Figure 10: Graphical illustration of the fleet assignment subsystem problem Figure 11: The fleet assignment can have significant impacts on the costs and profits of the airline Figure 12: Penalty value for GA does is non-zero because of both difficulties in imposing constraints and the discrete nature of the problem Figure 13: Diagram of the structure of the modified gate assignment and delay subsystem. Locations in red represent locations that have a delay (more than one plane arriving) at this time Figure 14: The cost that is accumulated by the airline over a 24-hour day. Ascending values represent the total amount of cost to airline up to that point and the values on the x-axis represent the time steps with no delays and therefore no added cost Figure 15: The percentage of total penalty cost by airport Figure 16: The percentage of total penalty cost by airport to be compared to Figure Figure 17: Links between subsystems and between subsystems and system formulation Figure 18: Average price for each route. A) Average price for each route B) Lookup table for routes numbers Figure 19: Representation of the Fleet Assignment design variables. The number of planes in the air at each time step is always below Figure 20: Approximation of contribution to fuel costs by each route. The largest contributions come from routes that are longer in length. As with the number of passengers, this approximation is likely to be off due to the use of surrogate functions

6 INTRODUCTION The system of interest for this report is the profit of an airline company. In this maximization problem, three subsystems will be considered: 1) Price and Demand 2) Fleet Assignment 3) Gate Assignment. The Price and Demand subsystem determines the price of tickets and the demand for each route, two important factors in estimating the revenue of the airline over the day; the Fleet Assignment subsystem finds a feasible assignment of planes to each route and provides the fuel costs for the system optimization; the Gate Assignment subsystem manages the gate assignment of aircrafts on the ground to minimize detrimental backups at each airport. SYSTEM LEVEL NOMENCLATURE Symbol R: Fc i : T i : CF f : D i : Cap f : NP f : Meaning The set of routes Number of connecting flights of route i Departure time of route i Cost of fuel per mile of aircraft type f Travel distance of route i Capacity of an aircraft type f Number of aircraft of type f SUBSYSTEM DESCRIPTIONS The system is divided into three subsystems: Price and Demand, Fleet Assignment, and Gate Assignment. In this section, a description of each subsystem is provided. PRICE AND DEMAND SUBSYSTEM Problem Statement This subsystem is to maximize the profit of an airline company over a day. The profit will be determined by number of tickets sold, price of tickets, and the cost. In the beginning, a schedule will be set to decide the departure time of each leg, the number of legs of each route in a day, and the number of connecting flights. Then there will be only one factor, price, needed to be assigned. By assigning a set of price of each route, conjoint analysis will then be applied to estimate the demand of each route. The number of tickets actually sold is bounded by the capacity of aircraft types assigned. Revenue is dependent on the number of tickets sold and the price of that. In this subsystem, cost will be affected only by fuel cost, which is related to the traveling distance of each route, the aircraft type of each leg, and the number of connecting flights. There are some assumptions made to simplify the problem. Consider these two cases are the same route: depart from A and arrive at B and depart from B and arrive at A, so that the combinations of five chosen cities can be reduced from twenty to ten routes. Besides, a value will be assigned as the utilities provided by the competitors when applying conjoint analysis. 5

7 Nomenclature Symbol M: P i : β a : U i : f i : Definition Total market size The ticket price of route i Part-worth Coefficient of attribute a Utility of route i Aircraft type of route i Mathematical Model and Analysis Maximize: ( M eu i S. T. g1: M eu i i e U j j e U j j P i D i CF fi ) Cap fi 0 i R Objective The objective function of the sub-system of demand will be to maximize the profit, which is represented as below: Profit = number of tickets sold price of the ticket cost of fuel The demand of each route is estimated by applying conjoint analysis. First, a survey is created to investigate customer preference on five attributes: route, departure time, price, and number of connecting flights. Conjoint analysis will then be used in the following steps to determine the percentage of market share can be achieved with the given attribute combination. The relation between utility of route i and the four attributes is shown below: U i = β r R i + β fc Fc i + β t T i + β p P i β are the part-worth coefficients of each attributes. R i, Fc i, T i, P i are binary variables represent the choice of route, number of connecting flights, departure time, and price of ticket respectively. Hence the probability of selecting route i among all j alternatives will be: e U i e U j j As mentioned in former section, the utility of alternative choices are set to be 1 in order to simplify the problem in this step. And the estimated demand of route i will be: 6

8 Market Size M eu i e U j j M is the total market size. In the cost model, the cost of a route only depends on fuel consumption, and traveling distance. D i CF fi d is the traveling distance of each route, which is the sum of distance between each connecting flights, and CF is the cost of fuel per mile of aircraft type f. Then the profit model can be represented as the sum of profit of each route: ( M i eu i e U j j P i D i CF fi ) Variables Variable is a set of price of each route. In the progress report, the variables were the five attributes (destination, price, number of connecting flights, departure day, and departure time) which will influence the demand of tickets. To simplify this problem, it is modified to consider only price as variable, and set other attributes as parameters. There will be 372 variables when optimizing the system level problem. Parameters The total market M is estimated by the demand of passengers traveling among the five cities: New York, Chicago, Seattle, Los Angeles, and Atlanta. From the statistics of Airports Council International's year-to-date figures (January to September, 2014), total potential market size over a day is , which is the annual value divided by 365 days, and the demands of each route are shown in Figure 1. These are estimated by the average demand of departure and arrival cities Figure 1: Potential Market Size of Each Route 7

9 Part-worth coefficient Beta Part-worth Coefficient Beta The part-worth coefficients obtained by the conjoint analysis are listed below. The exact figures can be found in Appendix A. 30 A) 60 B) non-stop one two three Figure 2: Part-Worth Coefficients. A) Route. B) Number of Connecting Flights Since departure time and price are continuous, interpolation is needed to obtain the part-worth coefficient of the corresponding attribute values. The spline interpolation results are obtained by MATLAB function spline shown in Figure 3. A) B) Figure 3: Spline Interpolation Results. A) Price Part-Worth Coefficients. B) Time Part-Worth Coefficients Constraint The number of tickets sold for a route should not exceed capacity of aircraft. M eu i e U j j Cap fi 0 i R 8

10 Optimization Study Parametric Study: The importance of each attribute can also be obtained by conjoint analysis. That shows which attribute influence the customer preference more. The importance of attribute i is calculated by max β i min β i 100% j max β j min β j Figure 4 shows the result from the survey. The attribute of departure day is neglected since we do not consider it in the final optimization process. Figure 4: Importance of each attribute from the survey results Given the situation as the table below: R i Fc i T i P i f i U 1 NY-LA Non-stop 6:00 pm $80-$200 Boeing U 2 Atlanta-LA Non-stop 6:00 pm $90/$100 Boeing U 3 NY-LA Non-stop 12:00 pm $90/$100 Boeing Consider U 1 as the utility of route we provide and U 2, U 3 are the utilities provided by competitors. Figure 5 shows the profit of U 1 when U 2 is the only utility provided by competitor. Figure 5A shows the profit of U 1 when the price of U 2 is $90, and Figure 5B is the condition when U 2 is $100. It is obvious that after the price of U 1 goes higher than that of U 2, the revenue will become almost zero. And since the part-worth coefficient beta of R 1 is higher than R 2, preference of route does influence a bit on the result. In Figure 5B the profit of U 1 is expected to decrease after $100 but the threshold is actually around $130. The difference may be the influence of different route. 9

11 A) B) Figure 5: Profit of U1 with varying competitor prices. A) Price of U2 is $90 B) Price of U2 is $100 Figure 6 shows the profit of U 1 when U 2 and U 3 are provided. When U 3 is added, it is expected that adding the same route should influence the profit of U 1. This expectation can be observed in Figure 6B, the threshold of profit dropping is lower than that in Figure 5B. It shows that when the price of U 1 is higher than that of U 3, the revenue decreases. There are not much difference between Figure 5A and Figure 6A. This result matches the expectation that departure time will not influence much on the customer s choice. A) B) Figure 6: Profit of U1 with two competitors. Price of U2 is the same in both figures. A) Price of U3 is $90 B) Price of U3 is $100 Approach: In the beginning, demand of each route are estimated by M eu i e U j j. The M refers to total market size of all the five airports and total utility includes not only competitor but other legs in our schedule. However, this assumption led to a bad optimization result that it took a long time to run and was not able to get a positive profit the by using fmincon. Because of that we made another assumption that the M refers to market size of each route, which is the average of departure and arrival airport, and total utility includes the competitor and legs with the same route in our schedule. Then we iterated different initial points to see the trend of local optimal result. 10

12 Profit ($) Profit ($) First we randomly choose a set of initial values between 10 and 300, then we increase or decrease those values in each iteration: initial values: 260-i, 30+i, 89+i, 60+i, 40+i, 10+i, 280-i, 220-i, 40+i, 130+i, 90+i,. The optimization results are shown in Figure i=50 i=60 i=70 i=80 i=90 i=100 i=110 i=120 Figure 7: Local maximum profit with different i We can find an obvious trend among the results, the local optimal value increases with i increases and the maximum profit happened when i equal to 90. In that iteration, the initial values are around a certain range and the values are close to each other. So the maximum profit may happen when starting with similar initial values. Than we tried another iteration to see the local optimal results starting with different price level: initial values: 20+i, 20+i, 28+i, 20+i, 40+i, 10+i, 20+i, 50+i, 30+i, 30+i, 20+i, 20+i, i=70 i=80 i=90 i=100 i=110 i=120 Figure 8: Local maximum profit with initial values picked using results of the previous run 11

13 Price The maximum happens when i equal to 110, when the initial values fall in the range close to the former iteration. We then choose this value as the optimization result in this subsystem. Discussion of Results: Compare the optimal price that maximize profit with the part-worth coefficient with respect to different attributes, the results are shown in Figure 9. Figure 9A is the average price in each hour. Figure 9A and Figure 3B, which shows the beta value in each hour, follow a similar trend. This implies that when customers are more willing to buy the tickets, the ticket price will be higher. Figure 9B is the average price of each route. Since the utility value of our competitors is set to be one, which is relatively lower than ours, the demand will be almost the same as potential market size shown former in Figure 1. And the price reflects the demand in each route as well. A) B) Figure 9: Average Prices of Tickets. A) Average price trend over the day. B) Average price for each route for the entire day. 12

14 FLEET ASSIGNMENT SUBSYSTEM Problem Statement For this subsystem, the airline schedule is defined as a list of flight legs that the airline plans to fly over the course of the day. Costs and revenue generated are directly affected by fleet assignment decisions made in fulfilling this schedule. Each aircraft that flies incurs costs in the form of fuel costs while revenue is dependent on demand for flights, price of tickets, and the carrying capacities the airline can offer for each flight leg. The objective of this subsystem is to find a fleet assignment that maximizes the profit, defined as the total revenue less costs. The associated fuel costs of the optimal fleet assignment will be returned as an input to the overall system objective. The problem will be tackled by first dividing the day into fifteen minute time blocks. When the day starts (assumed to 0000 hours), a number of flights on various legs with various aircrafts will begin. The number of passengers flown will be subtracted from the average daily demand for that flight route. Respective fuel costs and revenues will be calculated, and the number of planes available will be updated. The number and type of aircrafts that take off will be subtracted from the number of planes available until a length of time equal to the duration of the flights has passed, i.e. after these aircrafts have landed. For example, if a full load Boeing 717 takes off from ATL to ORD at midnight, and since the total number of Boeing 717s owned/leased is 10, the number of Boeing 717s that can still be assigned is 9 until 1.45 am (arrival time rounded up) assuming no other 717 takes off. This procedure is repeated over the entire day for every flight route, subject to constraints. A graphical illustration of the problem is shown below. Figure 10: Graphical illustration of the fleet assignment subsystem problem. In order to have a manageable initial model, only non-stop flights will be considered. Further simplifications include ignoring the costs of storing and retrieving each aircraft types, the assumption that there is no minimum duration that an aircraft needs to be on the ground before it can fly again, and the assumption that all flight legs must be covered. 13

15 A few modifications have been made since the progress report. While we originally wanted to allow the model to generate an optimal flight schedule, this proved too difficult an initial step, and an assumed schedule was used instead to simplify the problem. The number of planes owned or leased by the airline was also modified because using the actual number of aircrafts owned or leased by Delta Airlines resulted in too many design variables to be tractable on our computers. Finally, while the opportunity cost of excess demand beyond carrying capacity was previously considered in the progress report, the revised lower number of aircrafts assumed to be owned or leased by the airline, coupled with the higher than anticipated demand from the Price and Demand subsystem, resulted in much higher excess demand that rendered this an infeasible consideration. For initial demonstration purposes, the focus will be on finding a fleet assignment for a single day. Moreover, the distribution of the aircrafts at the end of the day need not be that of the original distribution at the beginning of the day. If we were to be concerned about the schedule over a longer period, this will be much more important to ensure continuity. Nomenclature Symbol Definition Cij Expected costs associated with assigning aircraft of type i to route j Dj Length of flight route j F Set of aircraft types Gist Number of aircraft of type i at airport s at time t i Index for aircraft type IN Set of flights inbound to airport s J Set of flight legs j Index for flight legs NPi Total number of aircraft of type i OUT Set of flights outbound from airport s Rij Expected revenue associated with assigning aircraft of type i to route j Rangei Flight range of aircraft type i Re Set of all flight legs for aircraft type i crossing a counting line for our time based model s Index for airports S Set of airports t Index for time, departure, and arrival events T Set of all departure and arrival events xij Design variable. Equals 1 if that aircraft of type i is assigned to route j. Equals 0 otherwise. 14

16 Mathematical Model The mathematical model used to analyze this system is based on integer programming formulations for the daily fleet assignment problem described in (Ram Gopalan, 1998) and (C. Barnhart, 2012). The modified model is shown here. (1) Maximize Rij xij Cij xij S.T. i F j J x G NP, i F (2) ij is0 i j Re( i) s S ; ; G G x x 0, ist ist ij ij j IN j OUT i F s S t T (3) xij 1, j J (4) i F x 0;1, i F; j J (5) ij Gist 0, i F; s S; t T (6) x ij D j < Range i i F; j J (7) Objective The objective of this subsystem is to find a fleet assignment that maximizes total profit at the end of the day. This is done by maximizing (1) in the mathematical model. Design Variable(s) The design variables are xij, each of which represent the decision to have aircraft of type i fly route j for each combination of aircraft and route type. Each variable is binary, and takes on the value 1 when the aircraft of type i is assigned to fly flight leg j, and 0 otherwise. Parameters Expected costs, C, will be estimated using the average cost of fuel per gallon and the average fuel consumption for each aircraft type. According to the USDOT s Bureau of Transportation Statistics (BTS) (US Department of Transportation, 2015), the average fuel cost per gallon for 2014 is $2.85. While we originally wanted to use aircrafts that are currently used in actual domestic flights offered by Delta Air Lines between the airports of interest, this resulted in too many design variables to be tractable on our computers. Hence, we assumed that each airport started with two of each aircraft type at the beginning of the day, shown in Appendix B. Flight distances are required to estimate the cost of fuel for each leg. The airports that will be used in this model are: Atlanta International Airport (ATL), John F. Kennedy International Airport (JFK), Los Angeles International Airport (LAX), O Hare International Airport (ORD), and Seattle-Tacoma International Airport (SEA). Average flight route distances were calculated using data from FlightAware s 15

17 IFR route analyzer (FlightAware, 2015). Although multiple routes are usually used between each origin destination pair, the averages of the lengths of these routes are used as a representative distance for each flight leg. A list of average distances between each airport is given in Appendix C. To estimate R, the expected revenue, the daily demand and average fare for each flight leg are needed. These values, in the system level optimization, are obtained from the Price and Demand subsystem. The figures used in the original formulation of this subsystem, before integration of all three subsystems, are shown in Appendix C. Constraints (1) Maximize Rij xij Cij xij S.T. i F j J x G NP, i F (2) ij is0 i j Re( i) s S ; ; G G x x 0, ist ist ij ij j IN j OUT i F s S t T (3) xij 1, j J (4) i F x 0;1, i F; j J (5) ij Gist 0, i F; s S; t T (6) x ij D j < Range i i F; j J (7) This basic model is subjected to a few constraints. First, the number of each type of aircraft in the fleet imposes constraint (2), which states that the total number of aircraft of each type either flying or at an airport cannot exceed the total number of aircraft of that type owned or leased by the company. The inequality, rather than an equality, means that the optimal number of aircrafts could be less than the number of that type of aircraft owned, i.e. some aircrafts can be stored. The costs of storing and retrieving an aircraft will be neglected in this model. To further simplify the model, this check is done only once during each day. Since storing and retrieving an aircraft from storage likely requires time and preparation and hence cannot be performed frequently, this assumption does make some sense in reality. Once the number of aircrafts is set at the beginning (or end) of each day, and assuming that aircrafts are only stored or retrieved at the start of each day, constraint (3) must be satisfied. This constraint implies that at each airport, the difference in number between the incoming and outgoing aircrafts must equal the change in the number of planes on the ground. Another simplifying assumption made in this subsystem model is that there is no minimum duration an aircraft has to be on the ground before it is ready to fly again. Constraint (4) imposes the constraint that every flight leg has to be covered by one, and only one, aircraft. This ensures that every leg in the network is covered, and that there are no redundant duplicate flights. (5) 16

18 is imposed because the design variables are binary either an aircraft type is assigned to a route or it is not. Constraint (6) is imposed because a negative number of aircrafts on the ground makes no sense. Constraint (7) ensures that the route assigned to each aircraft is shorter than the aircraft s maximum range and therefore feasible. Impact of Fleet Assignment To test whether the fleet assignment can indeed significantly affect the profits of the airline, a few initial test cases were ran, each with different initial conditions. In the first run, only Boeing 717s are available, while only Boeing 737s are available in the second. 5 Boeing 717s and 5 Boeing 737s are available in the third and final illustrative run. Since the implementation of the mathematical model includes a random element in the assignment of planes, each initial condition was run five times. As can be seen Figure 11, the different trials resulted in drastically fuel costs, with values ranging from less than $500,000 to over $3 million. Similarly, the estimated profits also showed a large spread with the different initial conditions. This demonstrates that the fleet assignment can indeed have a significant impact on the costs and profits of the airline. Figure 11: The fleet assignment can have significant impacts on the costs and profits of the airline Optimization Study To optimize this subsystem, MATLAB was used. A MATLAB function was first written to implement the mathematical model, and it was used to generate 2000 test points for use in subsequent steps in the optimization. Since this is a discrete problem, we decided to use genetic algorithm (GA) to find an optimal fleet assignment. However, the original MATLAB function did not have the appropriate input-output relation 17

19 to be used in the GA toolbox in MATLAB. We realized that GA was only providing the optimal assignment for the planes leaving at midnight, and not the fleet assignment for the entire day. Hence, a neural network was used to map the fleet assignment for the entire day to the associated profit and cost using the test data generated earlier. The surrogate function for the profit was then used in GA to find an optimal fleet assignment for the entire day. A population of 200 and 54 generations was used when GA was run. The optimal fleet assignment found resulted in a profit of $7.77 million and fuel costs of $3.22 million. Figure 12 shows the convergence plot for the run. Figure 12: Penalty value for GA does is non-zero because of both difficulties in imposing constraints and the discrete nature of the problem According to the MATLAB GA documentation, for the integer GA algorithm, as is in this case, the penalty function is the fitness function for feasible members, and the maximum fitness function among the feasible members of the populations, plus a sum of the constraint violations of the point for infeasible members. This provides an explanation for the negative penalty values seen in the GA runs. As surrogate functions were used to estimate both profit and fuel costs, accurate data at each time step was difficult to obtain. This made it difficult to track with certainty factors such as the number of planes on the ground, and the demand left for each route. Coupled with the fact that approximate surrogate functions were used, this made it difficult to ensure that the original constraints were enforced in the optimization. Furthermore, the genetic algorithm does not perform well in finding true optima of problems, especially in one with as many design variables as there were in this problem. With greater in-depth knowledge on integer programming, and more time to implement the methods, there could be better options and methods to solve this integer binary problem. 18

20 GATE ASSIGNMENT SUBSYSTEM Problem Statement An airline s primary goal is to maximize its profit and to have continued growth over long periods of time. At the center of achieving this goal is to focus on having sound operations with an emphasis on increasing customer satisfaction and completing flights as seamlessly as possible. A major part of these two important aspects involves the arrival and departure of planes as well as successfully getting passengers aboard the aircraft in a timely manner. These issues have been studied under the name the Airport Gate Assignment Problem (AGAP) as way to help airlines improve the efficiency of their daily operations. This area has been studied in terms of both domestic travel as well as in international markets. The model proposed by Ding et al (Ding, 2005) in 2004 aimed to minimize the walking distance for passengers going from the airport entrance to the gate, from the tarmac to their respective transfer gate, and transferring from one gate to another. Their paper explored the over-constrained AGAP, meaning that the number of flights exceeds the number of gates available. The over-constrained version of the problem proposes a better for solution for the AGAP considers circumstances that are arguably more relevant to today s busy airports. In other research, the situation where planes have to wait for a gate to open before deplaning was not considered. As a result the earlier research did not fully examine of the possibilities that a particular airline may encounter. For this project the over-constrained airport gate assignment problem will be addressed for a network of five airports, under the assumption that each airport is considered a gate. The first step in solving the problem of delays was to come up with a schedule for the system of airports. This schedule has twenty routes, and at time t=0, all routes are flown. From t=k, where k can be one or greater that one, the most popular routes, or in other words the routes with the most demand, are flown. The number of planes on the ground and arriving and departing at each airport were recorded. With respect to the gate assignment subsystem, the most important aspect of the information recorded is the number of planes that arrive in the same time step at the same location. When these events occurred there was a cost penalty assigned, which will be discussed in detail in the modifications section of this paper 19

21 Nomenclature / Parameters Symbol Definition N set representing the number of flights, including both flights that are arriving and departing from the airport M set representing the number of gates that are available at an airport n the cardinality of the set N (i.e. the total number of flights) m the cardinality of the set M (i.e. the total number of gates) a i the arrival time for flight i d i the departure time for flight i w k,l the distance a passenger / customer will walk from one gate, k, to another gate, l the amount of passengers who transfer from one flight, i, to another flight, j f i,j Mathematical Model Minimize: n n y i,m+1 + n m+1 m+1 n n f i,j w k,l y i,k y j,l + f i,0 w i,0 f 0,1 w 0,i i=1 i=1 j=1 k=1 l=1 i=1 i=1 Subject to: m+1 y i,k = 1( i, 1 i n) (1) k=1 a i < d i ( i, 1 i n) (2) y i,k y j,k (d j a i )(d i a j ) ( i, j, 1 i, j n, k m + 1) (3) y i,k {0,1}( i, 1 i n, and k, 1 k m + 1) (4) Objective The objective of this subsystem is to minimize the objective function above, and hence to minimize the delay penalty. Design Variable This particular model uses a binary variable y i,k and includes the entrance as gate 0 and the tarmac as gate m+1. The variable y can be equal to 1 or 0, where the values correspond to the assignment of a particular flight. For example y i,k = 1 means indicates that flight i is assigned to gate k, where k is any gate within the airport or the tarmac (m+1). The variable y i,k is equal to 0 otherwise. 20

22 To ensure that two flights cannot be scheduled to a single gate at the same, time the following condition must be met: Constraints (i, j) y i,k = y j,k = 1(k m + 1) a i > d j or a j > d i. m+1 y i,k = 1( i, 1 i n) (1) k=1 a i < d i ( i, 1 i n) (2) y i,k y j,k (d j a i )(d i a j ) ( i, j, 1 i, j n, k m + 1) (3) y i,k {0,1}( i, 1 i n, and k, 1 k m + 1) (4) Each of the above constraints, with the exception of the last constraint, ensures that the model takes in to account the conditions that are present in modern-day air travel. Constraint (1) states that either a flight is assigned to one gate or it is waiting for an open gate on the tarmac. Constraint (2) ensures that the arrival time of each flight is earlier than its departure time. Lastly, constraint (3) ensures that flight schedules will not overlap in the case that two planes are assigned to the same gate. (4) is a result of the binary variable y i,k and the fact that two airplanes cannot use the same gate simultaneously. It is important to note that in the second constraint the terms arrival and departure refer to a plane arriving at a gate with one group of passengers and departing with another group. This is not to be confused with the arrival and departure that applies to the travel plane of a customer. Modifications to Original Model In the original model as proposed by (Xu, 2001), formulation of the problem was in the context of addressing the gate assignment problem at one particular airport. However, to make it easier to integrate the subsystems of our project, the decision was made to focus on five airports: Atlanta (ATL), Chicago (ORD), Los Angeles (LAX), New York (JFK), and Seattle (SEA). As a result there was a need to adapt the model to a network of airports. To account for this difference the assumption was made that each destination would be considered as a gate. This adaptation eliminated the need for the terms that represent the number of transferring passengers and the walking distance between gates. Another major modification came when there was a need to figure out a way to maximize the profit, and therefore assign a cost to the existence of a delay at any of the airports. One of the drawbacks of using a simplified schedule is that there is a lot of information that is lost during each time step. These include the departure city of each arriving plane at each time step and, most importantly, the exact arrival time of each plane. The limitations suggested that there needed to be a simplified penalty system instead of using a method that prorates ticket price with respect to time. The final decision on a penalty system was to assign a 20% penalty to the planes that arrived in the same time step. Meaning that if there is a delay then 21

23 the passengers will receive a twenty percent refund of the average ticket price for the planes being flown in our system. An illustration of this for time t=k is below: Figure 13: Diagram of the structure of the modified gate assignment and delay subsystem. Locations in red represent locations that have a delay (more than one plane arriving) at this time. Significance of the Gate Assignment Subsystem A lack of organization can be a major factor in a customer s decision to fly a particular airline. By minimizing the delays at for an airline, there are economic benefits that come in two forms: increased demand and improved profit. Being viewed as a company that specializes in customer satisfaction will bring more passenger to an airline. In addition minimizing the possible cost associated with delayed flight will directly affect the profit of a particular airline. A lower cost will reduce the amount of money that is subtracted from the overall revenue. Figure 14: The cost that is accumulated by the airline over a 24-hour day. Ascending values represent the total amount of cost to airline up to that point and the values on the x-axis represent the time steps with no delays and therefore no added cost. 22

24 Optimization Because this is a discrete problem, there was a need to optimize the problem using methods that do not require any information about the gradient. Just as there was some information lost in the formation of the system schedule, there is information that is not available about the objective function. In between each point there is no information that can describe the function s gradient over an entire day. When coming up with examples to show the nature of the function s output, the different cases began to show that the delays depended on the location. This aspect of the subsystem has proven to be the most important with respect to the contribution to the total cost. The results of the preliminary example shows that, in general, Atlanta and Seattle have very few delays and Los Angeles or Chicago will have the greatest contribution to the cost to the airline. The two graphs below show the percentage of the total cost from each airport. Figure 15: The percentage of total penalty cost by airport In the first example the output resulted in one delay for both Seattle and Atlanta s airports. The majority of the delays in the system occur in Los Angeles and Chicago, while New York contributes just a moderate amount to the amount of contribution to the total delay cost to the airline of the course of a day. Figure 16 shows the same distribution of cost from the earlier graph. These results fit what would be expected in a real airline with the exception of Atlanta. This airport is generally a popular airport so delays would be expected. As a result of the inability to get certain information, we needed to define a relationship between the schedule and the final output of the subsystem, the delay penalty. This was achieved by using the neural 23

25 network application in MATLAB using 20 neurons and a split. After fitting the data, the surrogate function obtained from neural networks was then put into genetic algorithm in MATLAB. Figure 16: The percentage of total penalty cost by airport to be compared to Figure 15 Genetic algorithm was used will the default number of populations and finished running the optimization of 51 generations. The final value obtained was $526, 243. There were some difficulties in running genetic algorithm for this subsystem. The algorithm used in MATLAB returned an optimal value however, after inspection of the best-fit graph, genetic algorithm did not converge. This result could be due to the lack of information known about the fitness function or due to the initial conditions. When using nongradient search methods there is need to start with a good starting point. A major possibility for the divergence of the algorithm could be that MATLAB s randomly generated initial population may not have been the right point to solve the problem properly. Also there could have been a problem with the function obtained from neural networks. The fit to the data was average given that the R-value for the function was Because of this, genetic algorithm may not have been able to find the solution without a better representation of the data. 24

26 SYSTEM INTEGRATION Figure 17: Links between subsystems and between subsystems and system formulation System Objective The objective of this system level formulation is to maximize profit using the three subsystems described earlier. The profit of the system is the revenue over the day less the fuel costs and delay penalties. Subsystem Integration The system level formulation takes as inputs price and demand from the Price and Demand subsystem, the delay penalties from the Gate Assignment subsystem, and the fuel costs from the Fleet Assignment subsystem. Both the Gate Assignment and Fleet Assignment subsystems use the outputs of the Price and Demand subsystem as inputs, namely price, and price and demand respectively. The Gate Assignment subsystem also takes as input the Fleet Assignment from the Fleet Assignment subsystem. The links between each subsystem, as well as between the subsystems and the system, are depicted in Figure 17. To find the optimal solution for the system, the Price and Demand subsystem was first optimized to determine the optimal price. An optimal fleet assignment was then found using this price and demand, which was then fed into the Gate Assignment subsystem. The optimal fleet assignment also allowed the estimation of the total revenue and total fuel costs. Finally, the Gate Assignment subsystem calculates the delay penalties resulting from this fleet assignment. The overall profit is then calculated by subtracting both the fuel costs and the delay penalties from the total revenue. 25

27 Price ($) System Constraints As all three subsystems are utilized in finding the optimal output for the system level formulation, all constraints on the subsystem level still apply to the problem. No added constraints were thought to be necessary for the system level problem. Conflicting Requirements At the system level, one conflicting requirement was present. Increasing the number of flights per day would increase the revenue of the airline over the entire day since more passengers can be carried. However, doing so would result in higher delays at each airport, increasing the delay penalties assigned. While we were aware of this conflict when optimizing the system level formulation, the assumptions that made on the number of planes that were available limited the number of flights that could be flown and negated the problem posed by this conflict. Final System Results The average price for each route found using the Price and Demand subsystem is shown in Figure A) B) Route # Origin Destination Route Figure 18: Average price for each route. A) Average price for each route B) Lookup table for routes numbers 1 ATL JFK 2 ATL LAX 3 ATL SEA 4 ATL ORD 5 JFK LAX 6 JFK ATL 7 JFK ORD 8 JFK SEA 9 LAX JFK 10 LAX ORD 11 LAX SEA 12 LAX ATL 13 ORD JFK 14 ORD LAX 15 ORD ATL 16 ORD SEA 17 SEA LAX 18 SEA ORD 19 SEA ATL 20 SEA JFK Using these average prices and demands found, the optimal fleet assignment was found by the Fleet Assignment subsystem. Due to the 26

28 Contribution (%) numerous routes and time steps that are involved in the calculation of the fleet assignment, as well as the fact that a black box surrogate function was used in GA, a representation of the fleet assignment is shown here. Figure 19 shows the total number of planes that are in the air (design variables with values 1) at each time step during the day. Figure 19: Representation of the Fleet Assignment design variables. The number of planes in the air at each time step is always below 50. This fleet assignment resulted in approximate fuel costs of $3.22 million. An attempt to breakdown this fuel costs by route is shown in Figure 20. Refer to Figure. 18B for the route lookup table. With this fleet assignment, the approximate number of passengers carried over the entire day was passengers, with the associated revenue being $10.99 million. Given that the delay cost for the day is $526, 243, the profit is: Profit = Revenue - Fuel Cost - Delay Cost = $7.24 million Route Figure 20: Approximation of contribution to fuel costs by each route. The largest contributions come from routes that are longer in length. As with the number of passengers, this approximation is likely to be off due to the use of surrogate functions. These results illustrate a problem with the approach that we have taken to optimize the Fleet Assignment subsystem. With that revenue and number of passengers, the average price of a ticket would be less than $30. However, the average price of tickets from the Price and Demand subsystem was $161. This large 27

29 disparity is likely due to the use of surrogate functions in the optimization process. The use of surrogate functions resulted in difficulties in tracking various aspects in the subsystem, such as the number of planes on the ground at each airport and the demand for each route. The approximation required to obtain the numbers shown above resulted in the difference in ticket prices. 28

30 REFERENCES C. Barnhart, B. S. (2012). Airline Planning and Schedule Development. In L. A. T. L. Jacobs, Quantitative Problem Solving Methods in the Airline Industry (pp ). Springer US. Ding, H. A. (2005). The Over-constrained Airport Gate Assignment Problem. Computers & Operations Research, FlightAware. (2015). IFR Route Analyzer. Retrieved from FlightAware Live Flight Tracking: Ram Gopalan, K. T. (1998). Mathematical models in airline schedule planning: A survey. Annals of Operations Research, US Department of Transportation. (2015, Febuarary 24). Airline Fuel Cost and Consumption. Retrieved from Bureau of Transportation Statistics: Xu, J. a. (2001). The Quadratic Assignment Problem: Evaluation of Exact and Heuristic Algorithms. Proceedings of the 34th Hawaii International Conference on System Sciences. Hawaii. 29

31 APPENDIX A Market Size and Beta Coefficients R LAX-JFK LAX- SEA LAX- ORD LAX- ATL JFK- SEA M JFK- ORD JFK- ATL SEA- ORD SEA- ATL ORD- ATL Route R LAX-JFK LAX- LAX- LAX- JFK- JFK- JFK- SEA- SEA- ORD- SEA ORD ATL SEA ORD ATL ORD ATL ATL β r Number of connecting flights Fc i Nonstop One Two Three β fc Departure time T i 12 am. 3 am. 6 am. 9 am. 12 pm. 3 pm. 6 pm. 9 pm. β t Price of ticket P i β p

32 APPENDIX B Aircraft Specifications Aircraft Type Range (Miles) Capacity (Passengers) Fuel Consumption per Mile (Gallon) + Number of aircrafts Cruise Speed (mph) Boeing Boeing Boeing Boeing McDonnell Douglas MD Approximated by dividing maximum fuel capacity by maximum range. Assume that the same amount of fuel is burned regardless of passenger count. 31

33 APPENDIX C Average route lengths for each origin-destination pair Origin Destination Average Route Length (miles) ATL LAX 1992 ATL JFK 823 ATL ORD 796 ATL SEA 2261 JFK ATL 820 JFK LAX 2556 JFK ORD 799 JFK SEA 2460 LAX ATL 2008 LAX ORD 1848 LAX JFK 2568 LAX SEA 1014 ORD ATL 641 ORD LAX 1776 ORD JFK 752 ORD SEA 1763 SEA ATL 2233 SEA LAX 1790 SEA JFK 2487 SEA ORD

34 APPENDIX D Average daily demand and fares used for subsystem optimization Annual Demand (Passengers) Daily Demand Origin Destination (Passengers) ATL JFK 1000, ATL LAX 900, ATL SEA 500, ATL ORD 500, JFK LAX 1600, JFK ATL 200, JFK ORD 200, JFK SEA 200, LAX JFK 1600, LAX ORD 1200, LAX SEA 960, LAX ATL 900, ORD JFK 1650, ORD LAX 1160, ORD ATL 740, ORD SEA 500, SEA LAX 960, SEA ORD 650, SEA ATL 400, SEA JFK 200, Average Fare ($) 33