Year over Year Schedule Match Up using LP

Judy Pastor, Continental Airlines

AGIFORS YM 2001

Bangkok, Thailand

Schedule Changes

• One thing is constant in an airline– CHANGE (of Schedule)

• Departure/arrival times

• Frequency

• Equipment – Jet/Express

• Connecting bank structure

• Flight number changes to reflect new throughs

• Seasonal flying

• Good forecasts are necessary for RM• Appropriate flight history is crucial for good

demand forecasts

RM Systems

• RM Systems must be able to construct flight history from past schedules for future flight forecasts

• Heuristics (good guesses) have been implemented in RM systems with varying degrees of success

• Most based on time banding – some primitive methods based on Departure Hour

Time Banding

• Example – AAABBB – 1 Flight/Day– Future schedule shows Flight 123 in market

AAABBB departing at 0935 on Mondays– Past schedule has Flight 999 in market

AAABBB departing at 0830 on Mondays– If the TB parameter for the heuristic is two

hours, then Flight 999 is in same time band as Flight 123

– If the TB parameter is one hour, then Flight 999 is in a new time band

Time Banding

• Heuristics seem to break down when overall structure of a market changes

• In new markets, frequencies may increase if successful entry or decrease if not

• In low frequency markets especially, departure times may shift dramatically– 0600 and 1900 departures to 1100 and 1900

(change may be made to accommodate connections)

Time Banding

• Time Bands can also “drift” over time

• Static Time Band parameters can also be squeezed out over many schedule changes

• More robust way to doing the match up is needed

• One solution: – use the OR practitioners favorite tool:

• The Linear Program (LP)

The Transportation Problem

• Well known problem in OR• Traditional TP concerned with

– distribution of goods from several sources (supply points) to several destinations (demand points)

– above done at minimum total cost

• All goods must be distributed and all demands must be satisfied

• “Dummy Nodes” for possible imbalances

Abstraction

• Use the TP to solve the distribution problem of flights from one schedule to flights in another schedule

• Objective function is to minimize total absolute time difference between flights in schedule 1 (supply points) to flights in schedule 2 (demand points)

• Dummy nodes used to treat changes in frequency from one schedule to another

Model Formulation

• Parameters– aaabbb market for flights– m # flights in aaabbb for Schedule 1– n # flights in aaabbb for Schedule 2

• Data– cmn absdif(depttime) between flight m

and flight n– M big penalty number to encourage

flight matching

Model Formulation

• Decision Variables– Xmn =1, flight m from Schedule 1 matches

with flight n from Schedule 2

– Ym =1 flight m from Schedule 1 is unmatched

– Yn =1 flight n from Schedule 2 is unmatched

Model Formulation

• Objective Function– Minimize SUM(m)SUM(n) (cmn * Xmn) +

SUM(m) (M * Ym) + SUM(n) (M * Yn )

• Minimize the absolute difference between matched flights and penalties for unmatched flights

Model Formulation

• Constraints– Each flight in both schedules must be either

matched or unmatched– Each flight can be matched to at most one other

flight– Correct frequencies in Schedules 1 and 2 must

be preserved– Decision Variables X and Y must be binary (0

or 1)

Model Solution

• Currently implemented to be solved as an LP

• Could be done more efficiently as a transportation problem but requires some programming

• Entire schedule (all markets) done as one big LP though it really consists of many stand alone sub-models

Some Results

OandD Flt Dept Flt Dept

====== ==== ==== ==== ====

DFWIAH 700 600 700 600

DFWIAH 1953 700 702 700

DFWIAH 708 1005 706 900

DFWIAH 710 1125 710 1120

DFWIAH 34 1340 712 1247

DFWIAH 716 1525 716 1540

DFWIAH 718 1704 718 1745

DFWIAH 720 1840

Some Results

OandD Flt Dept Flt Dept

====== ==== ==== ==== ====

EWRSFO 151 740 147 730

EWRSFO 155 915 161 920

EWRSFO 153 1100 149 1045

EWRSFO 114 1410

EWRSFO 37 1540 157 1600

EWRSFO 157 1715 119 1800

EWRSFO 159 1955 159 1945

Some Results

OandD Flt Dept Flt Dept

====== ==== ==== ==== ====

IAHCLL 3462 755

IAHCLL 3098 1045 3518 935

IAHCLL 3524 1310

IAHCLL 3466 1530 3466 1440

IAHCLL 3460 1900 3351 1815

Some Results

OandD Flt Dept Flt Dept

====== ==== ==== ==== ====

IAHDEN 378 700 378 805

IAHDEN 587 923 291 927

IAHDEN 755 1200 1511 1109

IAHDEN 1051 1422 799 1320

IAHDEN 35 1530 1707 1430

IAHDEN 626 1725 1855 1750

IAHDEN 1441 2026 1728 2012

Future Enhancements

• Change from AbsDif function in Obj to AbsDif**2

• Match up of multiple schedules

• Addition of OA schedules – past and present

• Connecting opportunity match up

• Fuzzy logic/Sensitivity Analysis to determine historical quality of information