Impact of Capacity and Demand Forecast Modifications on the Revenue: An Example

1. Impact of Capacity and Demand Forecast Modifications on the Revenue:An Example Michael Burkard, Peter Minder Atraxis AG CKCA/Operations Research CH-8058 Zurich Airport Tel. + 41 1 812 56 25 www.atraxis.com

2. Overview • The Bid-Price model for O&D optimization • Description of the constraints in the Bid-Price model • Example with a small virtual network • Interpretation of its solution • Sensitivity of the capacity constraints • Sensitivity of the demand constraints • Conclusion Impact of Capacity and Demand Forecast Modifications on the Revenue

3. The Bid-Price Model for O&D Optimization The idea of the Bid-Price model: ... to maximize the revenue of an entire origin-destination (O&D) network based on the aircrafts’ capacities and the forecasted unconstrained demand per O&D and fare class. • The Bid-Price model is a linear program consisting of • an objective function (maximizing the revenue) • capacity constraints on the legs (aircraft physical capacity) • demand constraints (for each O&D and fare class) Impact of Capacity and Demand Forecast Modifications on the Revenue

4. revenue fare (O&D / fare class) (forecasted) demand number of PAX (O&D / fare class) (remaining) capacity sum over all fare classes sum over all O&Ds The Bid-Price Model: Objective Function Impact of Capacity and Demand Forecast Modifications on the Revenue

5. subject to for all legs l (remaining) capacity of leg l passengers (O&D, fare class) number of PAX on leg l sum over all fare classes sum over all O&Ds i covering leg l The Bid-Price Model: Capacity Constraints Impact of Capacity and Demand Forecast Modifications on the Revenue

6. subject to forecasted demand (O&D, fare class) number of PAX The Bid-Price Model: Demand Constraints for all legs l for all O&Ds i andall fare classes j Impact of Capacity and Demand Forecast Modifications on the Revenue

7. significant O&Ds Example: The Network LON HKG FRA ZRH BKK ROM SIN flown legs Impact of Capacity and Demand Forecast Modifications on the Revenue

8. Example: Input Data (Fares, Capacities and Demand) For reasons of simplicity we use in this example only one fare class. Impact of Capacity and Demand Forecast Modifications on the Revenue

9. 20 / 20 10 / 30 70 / 80 150 / 150 0 / 30 180 / 180 50 / 50 170 / 180 30 / 50 10 / 10 20 / 20 10 / 10 Example: Solution of the Linear Program LON HKG 180/ 200 100 / 100 200/ 200 80 / 80 FRA ZRH BKK 200/ 200 40/ 50 ROM SIN  number of PAX per O&D (solution x) total number of PAX on legs (P) physical capacity on legs (C) forecasted demand (D) Total Revenue: 875.000 Impact of Capacity and Demand Forecast Modifications on the Revenue

10. Sensitivity: Identifying Constraint Status In the Bid-Price model we have • for every leg a capacity constraint • for every O&D two demand constraints (upper and lower bound) We distinguish between 2 types of constraints: • binding (B): if the solution x fulfills a constraint with equality (slack=0) • non-binding (N): otherwise (slack 0) Impact of Capacity and Demand Forecast Modifications on the Revenue

11. Besides the primal solution x of the linear program, we also get a dual solution. For the capacity constraints these values are also called bid prices. Example: Solution (Part 1: Capacity) Impact of Capacity and Demand Forecast Modifications on the Revenue

12.  primal solution x:  new capacity for leg l:  new revenue: Sensitivity: Binding Capacity Constraints If the value of the physical capacity C of a binding constraint is increased by 1, then the optimal value (potential revenue) increases by the value of the corresponding dual variable (=bid price). Example:  l = ZRHSIN, Cl = 200  Cl = Pl  constraint binding corresponding bid price: 1400 Impact of Capacity and Demand Forecast Modifications on the Revenue

13. Example: S O&D fare f BP leg1 BP leg2 accept? fare - BP ROMBKK 1700 0 1200 +500 LONSIN 1600 300 1400 -100 Interpretation of the Bid Prices • The values of the dual solution corresponding to the capacity constraints are also called bid-prices (BP). • They can be interpreted as cut-off values in order to decide whether a new booking for an O&D should be accepted or declined. Impact of Capacity and Demand Forecast Modifications on the Revenue

14. Example: Solution (Part 2: Demand) Impact of Capacity and Demand Forecast Modifications on the Revenue

15.  new demand for O&D i:  new revenue: Sensitivity: Binding Demand Constraints If the value of the forecasted demand D of a binding constraint is increased by 1, then the optimal value (potential revenue) increases by the value of the dual variable. Example: (upper bound constraint: xD)  i =ROMHKG, Di =10  primal solution: xi =10  xi = Di  constraint binding corresponding dual variable: 1800 Impact of Capacity and Demand Forecast Modifications on the Revenue

16. Sensitivity: Feasibility Range of Binding Constraints Where are these modifications valid? • For each binding constraint an interval can be computed within which the values C and D can be modified with the previously described effects on the revenue. • If the new value for C or D lies beyond this interval, then the corresponding constraint may change its status from binding to non-binding and the corresponding dual solution (and therefore the increase in the revenue) changes as well. Impact of Capacity and Demand Forecast Modifications on the Revenue

17. Sensitivity: Non-Binding Constraints • For non-binding constraints an increase of the capacity or demand by 1 has no direct effect on the revenue. • Similarly as for binding constraints, we can compute an interval for each constraint within which C or D can be chosen without influencing the revenue. • When choosing a value of increase exceeding this interval, this may change the status of the constraint from non-binding to binding. Non-binding constraints: Impact of Capacity and Demand Forecast Modifications on the Revenue

18. Sensitivity: Remarks • All previous results shown for an increase of C or D by 1 are also valid (within the corresponding interval) for a decrease by 1, thus decreasing the revenue. • In certain cases, if the linear program is degenerated (i.e. the optimal solution does not have a unique representation), the modifying interval may be reduced to one single point. • In a similar way as for capacity and demand, a sensitivity analysis can also be carried out for fare changes. • In any case the most accurate demand forecast available, should be used! The revenue calculated by the Bid-Price model is based on the assumption of an EXACT demand forecast. General remarks: Impact of Capacity and Demand Forecast Modifications on the Revenue

19. Summary • ... identify the relevant (restricting) demand constraints by determining the binding constraints. • ... calculate the increase/decrease in revenue for these constraints using the dual solution. • ... calculate the intervals within which non-binding constraints can be modified without influencing the revenue. • ... determine, whether after some input data has changed, a re-optimization becomes necessary. • ... give us a better understanding of bid price modifications as a result of capacity changes. The solution of the linear program and its sensitivity analysis allows us to: Impact of Capacity and Demand Forecast Modifications on the Revenue

20. Atraxis – and IT works