Airline Schedule Optimization (Fleet Assignment II) Saba Neyshabouri The Fleet Assignment Model • In order to develop the mathematical optimization model for this problem, some modifications should be made to the underlying time-space network: – Constructing a time –space network for each fleet type – Adding wraparound arcs for each airport to make the possibility of keeping and aircraft at an airport overnight. – Adding the count time to the network to be able to keep track of the total number of aircrafts of a fleet that are assigned. Modified Time-Space Network • By incorporating mentioned modifications our network will change to the following network: Wraparound arc Basic Fleet Assignment Model (FAM) • Based on the assumptions of the model, here is the list of model parameters, data : • Decision Variables: Basic FAM Model • Having defined the parameters and decision variables, the Basic FAM model is as the following: Basic FAM Model (description) • Objective Function – Tries to minimize the (Operational Cost- Generated revenue) of all the assignments of fleet type k to flight leg I • Constraints (assignment constraint): – States that each flight leg must be assigned exactly to one fleet type. Basic FAM Model (description) • Constraints (Fleet balance constraint): – This constraints states that for each node and each fleet type : • All the flights of type k that are arrived at n and are going to stay plus all the flights originating from n that are going to fly out should be equal to the flights on the ground that were waiting until the time at node n plus the number of flights that are going to fly into n. • This constraint is the famous “Flow Balance” constraint in network flow models. • This constraint states that all the flights that are coming to a node should be leaving that node at some point, or to state it differently, total number of aircrafts arriving at an airport (of each type) is equal to the total number of departing aircrafts. • Not satisfying this constraint will cause the model to accumulate all the aircrafts at one node. Basic FAM Model (description) • Constraints (Fleet availability): – This constraints states that for each fleet type : • The Sum of all of flights that has been on the ground during the count time plus all the flights that has been assigned to a flight leg (flying) on that time should be less than or equal to the total number of available aircrafts of that type. • These constraints which are similar to resource availability constraints, will make the optimization model not to assign more than existing aircrafts to flight legs. Basic FAM Model (description) • Constraints (Variable Definition): – This constraints states that for each fleet type and each arc : • Assignment variables are 0-1 variables that shows the decision made about that particular assignment. • Non-negativity constraints for flights on the ground which states that variable can not assume negative values. • Note that for the flights on the ground there is no indication of the variable being integer, while these variables are inherently integer! • In some network problems, The integer constraints can be relaxed thanks to the special structure of the problem, which makes the problem much easier to solve. Basic FAM Weaknesses • Basic FAM is BASIC! • It captures some of the most important constraints of the problem. • It is not covering constraints such as: – – – – Noise restrictions Maintenance requirements Gate restrictions Crew considerations Solving FAM • FAM is an integer, multi-commodity network flow problem with side constraints. • It can be solved (not always easily and not for all the problems) using offthe-shelf optimization software packages such as: – CPLEX – Xpress-MP • Here is an example of a problem size and solution time needed: FAM and Impacts • Here are some examples of the impacts of using Fleet Assignment Models (FAM) : Extending Basic FAM • There are several shortcomings in Basic FAMs: – Spill costs: Revenue lost when the assigned aircraft for that flight cannot accommodate all passenger demand. – Recapture costs: When an airline spills passengers from one flight leg and then books them on other flights in the airline’s network. – Most FAMs consider only aggregate demand and average fare by flight leg or by passenger itinerary which can compromise the accuracy of the estimated spill costs. – Most FAMs assume that demand is static over the schedule period! Example: Problems of Basic FAM • Consider the data for the FAM : (X-Z :connecting through Y) Example: Problems of Basic FAM • Given the data provided in the tables: – Maximum possible revenue is : • Here is the table for each possible fleet assignment combination: – – – Assume fleeting I is selected: Demand for flight 1 is 150 (75+75) and demand for flight 2 is 225 (150+75) by this fleeting, each flight leg will have capacity of 100 (capacity of aircraft A) so 50 passengers on flight 1 and 125 (225-100) passengers on flight 2 will spill. Since the fare for X-Z is less than the sum of fares for X-Y and Y-Z The Revenue maximizing strategy is to spill passengers from X-Z (50) with the cost of 15000 (50*300) and flight 1 will have enough capacity. Because the local fare of Y-Z is less than X-Z, 75 passengers are spilled from Y-Z itinerary (with the cost of 16875) and since 50 are already spilled from X-Z, flight 2 will be at capacity and total spill cost for this fleeting is 15000+16875=31875 Example: Problems of Basic FAM • Following the same line of calculations, the spill cost for each possible fleeting is shown in this table: • Considering the contributions of each fleeting, the optimal solution for this small example is fleeting I. Example: Problems of Basic FAM • If instead of considering network effects, we could calculate spills on a leg-based fashion. – In this case, the objective is to minimize the spill cost for each individual flight leg, independent of the effects on the other flights in the network. – The strategy is to spill passengers greedily in order of increasing fare, until the number of passengers equals the capacity. – In our example, local passengers are always spilled in favor of keeping the connecting passengers with a higher total fare so for the fleeting I: • 50 X-Y passengers are spilled at fare 200 • 125 Y-Z passengers are spilled at fare of 225 Example: Problems of Basic FAM • Calculating the spill costs on a greedy leg-based approach will yield the following table: • Comparing to the network itinerary-based calculation: – The main reason for big differences is that greedy approach does not capture flight leg interdependencies or network effects. Considering Network Effects • For our example, it was easy to enumerate all the possible spill costs for each fleeting, but for problems of real sizes (thousands of flight legs), enumeration can be computationally very expensive if not impossible! • Researchers have developed mathematical models and optimization approaches for large scale problems, and conclude that the benefits of modeling network effects can be significant: – Network-based fleet assignment approach at AA has yielded annual improvements in revenue for 0.54% to 0.77% (Jacobs et al., 1999). – Increased annual contributions from $30 to over $100 million have also been reported as achievable at United Airlines when fleeting decisions are made using networkenhanced FAM instead of a leg-based FAM (Barnhart et al., 2002). Extended Fleet Assignment Problems • To capture network effects and extend Basic FAM to include passenger spill decisions, following inputs to the problem should be considered: – An airline’s flight schedule – Itinerary-based passenger demand – Aircraft operating cost data Extended FAM Modeling • To include the network effects in FAM we need to keep track of number of passengers assigned to each itinerary in airline’s network. Extended FAM Modeling • Data and variables: Extended FAM (IFAM) Model IFAM: Description • Objective function: – The objective is to minimize the sum of the operating cost of flying leg I with aircraft type k for all flight legs and fleet types and the negative of the total revenue. • Constraints: – First 3 constraints are the same as in Basic FAM model IFAM: Description • Constraints: – Passenger flow and capacity constraints: • This constraint makes sure that for each aircraft in a fleet, the number of passengers assigned to that aircraft will not exceed its capacity. – Demand Constraint: • This constraint will limit the total number of passengers traveling on or spilled from itinerary p to the unconstrained demand of p. IFAM: Description • Constraints: – Passenger flow and capacity constraints: • These set of constraints will bound the variables to be positive and also fleet assignment variables should be 0-1. IFAM vs. FAM • Using IFAM will cause the problem size to grow which can cause computation inefficiency or tractability issues, On the other hand it will provide significant economic benefits thanks to considering network effects.