Constraints Cover constraints Each flight must be assigned to exactly one fleet Balance constraints Number of aircraft of a fleet type arriving at a station must equal the number of aircraft of that fleet type departing Aircraft count constraints Number of aircraft of a fleet type used cannot exceed the number available 2/212021 Barnhart 1.206J/16.77J/ESD. 15J
2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 11 Constraints • Cover Constraints – Each flight must be assigned to exactly one fleet • Balance Constraints – Number of aircraft of a fleet type arriving at a station must equal the number of aircraft of that fleet type departing • Aircraft Count Constraints – Number of aircraft of a fleet type used cannot exceed the number available
Objective Function For each fleet- flight combination: Cost perating cost Spill cost Operating cost associated with assigning a fleet type k to a flight leg i is relatively straightforward to compute Can capture range restrictions, noise restrictions, water restrictions,etc. by assigning "infinite'costs Spill cost for flight leg i and fleet assignment k average revenue per passenger on j MAX(O, unconstrained demand for i-number of seats on k Unclear how to compute revenue for flight legs, given revenue is associated with itineraries 2/212021 Barnhart 1.206J/16.77J/ESD. 15J
2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 12 Objective Function For each fleet - flight combination: Cost Operating cost + Spill cost • Operating cost associated with assigning a fleet type k to a flight leg j is relatively straightforward to compute – Can capture range restrictions, noise restrictions, water restrictions, etc. by assigning “infinite” costs • Spill cost for flight leg j and fleet assignment k = average revenue per passenger on j * MAX(0, unconstrained demand for j – number of seats on k) – Unclear how to compute revenue for flight legs, given revenue is associated with itineraries
Spill cost computation and Underlying assumption Given pill cost for flight leg j and fleet assignment k average revenue per passenger on j* MAX(O, unconstrained demand for i-number of seats on k Implication: a passenger might be spilled from some, but not all, of the flight legs in his/ her itinerary 2/212021 Barnhart 1.206J/16.77J/ESD. 15J
2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 13 Spill Cost Computation and Underlying Assumption Given: – Spill cost for flight leg j and fleet assignment k = average revenue per passenger on j * MAX(0, unconstrained demand for j – number of seats on k) Implication: – A passenger might be spilled from some, but not all, of the flight legs in his/ her itinerary
FAM Spill calculation heuristics ● Fare Allocation Full fare -the full fare is assigned to each leg of the itinerary Partial fare- the fare divided by the number of legs is assigned to each leg of the itinerary Shared fare -the fare divided by the number of capacitated legs is assigned to each capacitated leg in the itinerary Spill cost for each variable Representative fare Aspill fare"is calculated; each passenger spilled results in a loss of revenue equal to the spill fare Integration Sort each itinerary by fare, spill costs are sum of x lowest fare passengers, where x=max 0, demand-capacity) 2/212021 Barnhart 1.206J/16.77J/ESD. 15J
2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 14 FAM Spill Calculation Heuristics • Fare Allocation – Full fare - the full fare is assigned to each leg of the itinerary – Partial fare - the fare divided by the number of legs is assigned to each leg of the itinerary – Shared fare - the fare divided by the number of capacitated legs is assigned to each capacitated leg in the itinerary • Spill Cost for each variable – Representative Fare • A “spill fare” is calculated; each passenger spilled results in a loss of revenue equal to the spill fare – Integration • Sort each itinerary by fare, spill costs are sum of x lowest fare passengers, where x = max{0, demand - capacity}
An illustrative exampl Fleet Type Seats 100 flight 1 flight 2 B 200 Average 0.0 Itinerary Fare Pax X-Y $200 Y-Z $225 150 X-Z $300 Fleet Assign Partial alloc Full alloc Actual Opt FL.1-FL 2 Spill Spill spilled pax $30.000 $38,125 31,87550XZ,75YZ A-B $l1,250 $15,625 12,50025XZ,25XY B-A $22,500 $28,125 28.125 125Y-Z B-B $3,750 $5,625 5.625 25Y-Z 2/21/2021 Barnhart 1.206J/16.77J/ESD. 15J
2/21/2021 Barnhart 1.206J/16.77J/ESD.215J 15 An Illustrative Example X Y Z flight 1 flight 2 Fleet Type Seats A 100 B 200 Market Average Itinerary Fare No. of Pax X-Y 1 $200 75 Y-Z 2 $225 150 X-Z 1-2 $300 75 A-A $30,000 $38,125 50 X-Z, 75 Y-Z A-B $11,250 $15,625 B-A $22,500 $28,125 Fleet Assign. Partial Alloc. Full Alloc. Actual Opt. B-B $3,750 $5,625 Fl. 1- Fl. 2 Spill Spill Spill Spilled Pax 31,875 12,500 28,125 5,625 25 X-Z, 25 X-Y 125 Y-Z 25 Y-Z