2. Content : 1. Introduction 2. Traffic forecasting and traffic allocation ◦ 2.1. Traffic forecasting ◦ 2.2. Traffic allocation 3. Fleeting problems ◦ 3.1. The daily fleet assignment model  3.1.1. Minimizing number of aircraft  3.1.2. The flight network for the daily fleet assignment model  3.1.3. Integer programming formulation ◦ 3.2. Swapping equipment types in a daily fleet assignment ◦ 3.3. Weekly fleet assignment Mathematical models in airline schedule planning: A survey (Gopalan,1998)

3. 1. Introduction What’s the airline schedule? What’s flight leg? What period of time ? The column labeled frequency indicates the days on which this flight is offered Two different flights may share a flight number. Typically, this is the case for one-stop or two-stop flights. Flt. No.FromToDep.Arr. Frequency 547BOSPIT525p711p12345 1753BOSPHL730p851p1234567..... …..

4. The schedule planning paradigm Schedule developFleet assignmentThrough flight selectionFlight numbering OPERATIONS Crew scheduling recovery from irregular operations Operational difficulties 1- traffic forecasts for the month 2- tactical and strategic initiatives 3- seasonal demand variations 4- … 1- traffic forecasts for the month 2- tactical and strategic initiatives 3- seasonal demand variations 4- …

5. 2. Traffic forecasting and traffic allocation Traffic forecasting: ◦ Airline demand that tends to be seasonal Traffic allocation: ◦ determines how the demand will be allocated across the various available itineraries competing for it. Thus, an allocation model determine the approximate market share of each competing airline based upon the schedule offerings

6. Traffic allocation: the traffic share of an airline: MS i : the market share of airline i FS i : frequency share of airline i in a particular market The factor β is determined from prior history by regression Note: ◦ ignores the fact that passengers have a stronger preference for traveling at certain times of the day

7. Unconstrained demand: ascribe a “desirability” factor to each itinerary. the market share of an airline is the sum of the desirabilities of all itineraries run by the airline divided by the sum of all desirabilities of all itineraries serving the market. we can always estimate the demand for a particular itinerary by multiplying the total market demand by the itinerary’s desirability fraction in practice, heavily traveled itineraries “spill” passengers

8. flight (leg) based spill model: f (x): probability density of demand for a flight (Normal with a given mean µ and variance σ) C: capacity of the aircraft assigned to the flight ES: expected number of spilled passengers ◦ evaluate the expected revenues from assigning different equipment types to a particular flight ◦ the expected opportunity cost of spilling passengers because of insufficient capacity

9. 3. Fleeting problems ( daily fleet assignment) Advantages: widest application in practice Its not too restrictive in practice as most airlines fly the same schedule, at least during weekdays easier to schedule gates, crew, and maintenance More benefits ◦ intangible benefits like ease of scheduling maintenance, crew, and gates. ◦ it is not clear if it is possible to capture more demand by going to a variable fleeting

10. 3. Fleeting problems ( daily fleet assignment) Disadvantages: Airline demand varies by day of the week (it is typically higher on Mondays and Fridays, lower during the middle of the week and lowest on Saturdays) It is a common practice among airlines to solve the daily fleet assignment problem and modify it for weekend flying (i.e. most US domestic carriers)

11. 3.1. The daily fleet assignment (model) : Goal: ◦ matching capacity to demand as much as possible ◦ The objective function used is combination of the operating cost and the opportunity cost of spilling passengers Constraints: ◦ rigid constraints:  the right aircraft should be present at the right place at the right time  to ensure that we do not assign more aircraft of each type than present in our fleet  every flight leg in the schedule has to be assigned exactly one equipment type ◦ constraints relating to crew and maintenance

12. Case study: American, Delta, and USAir have reported routinely solving the problem to near-optimality in a few hours on a workstation, a feat unthinkable just a few years ago

13. A simple schedule of two flights a daily fleet assignment is impossible.

14. 3.1.1. Minimizing number of aircraft Deadheading Forbidden (easy) FIFO Allowed (complicated) repeats daily Min cost/day s.t. n=const Without daily repeats Min n of aircraft polynomial-time algorithms for the problem same aircraft every dayNP-complete

15. 3.1.2. The flight network for the daily fleet assignment model

16. 3.1.3. Integer programming formulation The basic constraint classes ◦ specify that each flight leg gets assigned exactly one equipment type ◦ equipment balance is maintained at every node ◦ number of aircraft used in any equipment type do not exceed the number of aircraft of that equipment in the fleet

17. 3.2. Swapping equipment types in a daily fleet assignment Introduction: changing the equipment type on a specific flight leg from the assigned equipment type to another equipment type. The solution of integer programming method is a set of flight legs of an equipment type that form disconnected components, a situation (called a locked rotation) that may be unacceptable for maintenance routing purposes Use when: ◦ equipment failure ◦ schedule disruptions ◦ unexpected demand ◦ to connect up the components and unlock locked rotations ◦ rebalance a fleeting

18. Same-Day Swap Algorithm The Same-Day Algorithm finds a swap opportunity between two fleets, if one exists, that involves the fewest number of changes to the existing assignment

19. Same-Day Swap Algorithm (steps) : Same-Day Swap Algorithm ◦ Step 1. Remove all the overnight arcs from the flight network. ◦ Step 2. Reverse the direction of all arcs in the remaining flight network that have the equipment type B assigned to them. Call this network the swap network. Let s be the head of f (the flight whose equipment type has to be swapped) and t its tail. ◦ Step 3. Find a path from s to t in the swap network that uses the fewest number of arcs. This can be done by running a shortest-path algorithm from s to t with all arc costs equal to one. The arcs involved in the path represent a same-day swap opportunity that involves the fewest number of changes to the existing assignment.

20. 3.3. Weekly fleet assignment A heuristic solution: ◦ first solve the daily fleet assignment problem for a representative day of the week ◦ solve a two-day fleet assignment problem for only Saturday and Sunday and require the “overnight flow” to match the Friday evening count as suggested by the daily fleet assignment problem Case study: ◦ for most domestic airlines in the US, weekend schedules do differ from weekday schedules. Also, airlines with a strong presence in international flying change their schedules significantly even during the regular week