Crew Assignment Problems Dr. AA Trani Associate Professor of Civil

Crew Assignment ProblemsDr. A.A. TraniAssociate Professor of Civil EngineeringCEE 5614- Analysis of Air Transportation Systems1 of 15

General Remarks• Sequentially these problems follow the aircraft assignment problem• Difficult problems because there are substantial constraints associatedwith a crew scheduleHours of operation (< 8 flight hours per day, < 14 hours in a dutycycle, ,< 30 hours per week, < 90 hours per month, and < 1,000hours per year)Vacation/rest days (selected by the pilot/crew member)Exogenous parameters• Very important since crew members represent a sizeable amount of theDOC cost of the airline (senior captains make $240,000 per year at makorU.S. airlines)2 of 15

Crew Scheduling ProblemA small airline uses LP to allocate crew resources tominimize cost. The following map shows the city pairsto be operated.1Denver2Dallas3New YorkMorningAfternoonNightDEN = Denver,DFW = Dallas4MexicoMEX = Mexico City,JFK = New York3 of 15

Crew Scheduling Problem.Flight Number O-D Pair Time of Day100 DEN-DFW Morning200 DFW-DEN Afternoon300 DFW-MEX Afternoon400 MEX-DFW Night500 DFW-JFK Morning600 JFK-DFW Night700 DEN-JFK Afternoon800 JFK-DEN Afternoon4 of 15

Crew Scheduling ProblemDefinition of terms:a) Rotations consists of 1 to 2 flights (to make theproblem simple)b) Rotations cost $2,500 if terminates in the originatingcityc) Rotations cost $3,500 if terminating elsewhereExample of a feasible rotations are (100, 200),(500,800),(500), etc.5 of 15

Crew Scheduling ProblemR iSingleFlightRotationsCost ($) R iTwoflightRotationsCost ($)1 100 3,500 9 100,200 2,5002 200 3,500 10 100,300 3,5003 300 3,500 11 500,800 3,5004 400 3,500 12 500,600 2,5005 500 3,500 13 300,400 2,5006 600 3,500 14 200,100 3,5006 of 15

R iSingleFlightRotationsCost ($) R iTwoflightRotationsCost ($)7 700 3,500 15 600,300 3,5008 800 3,500 16 600,200 3,50017 600,500 3,50018 800,100 3,50019 700,600 3,50020 700,800 3,5007 of 15

Decision VariablesDecision variables:⎛R1i = ⎜⎝ 0if i rotation is usedif i rotation is not used8 of 15

Crew Scheduling ProblemMin Costsubject to: (possible types of constraints)a) each flight belongs to a rotationMinz = 3500 R 1 + 3500 R 2 + 3500 R 3 + 3500 R 4 + 3500+R 5R 6 R 7 R 8 R 9 R 103500 + 3500 + 3500 + 2500 + 35009 of 15

3500 + 2500 + 2500 + 3500+ 3500 R 15R 11 R 12 R 13 R 143500 R 16 + 3500 R 17 + 3500 R 18 + 3500 R 19 + 3500R 20R 1 R 9 R 10 R 14 R 18s.t. (Flt. 100) + + + + = 1(Flt. 200) R 2 + R 9 + R 14 + R 16= 1(Flt. 300) R 3 + R 10 + R 13 + R 15= 1(Flt. 400) R 4 + R 13= 1(Flt. 500) R 5 + R 11 + R 12 + R 17= 110 of 15

(Flt. 600) R 6 + R 12 + R 15 + R 16 + R 17 += 1R 19R 7 R 19 R 20(Flt. 700) + + = 1(Flt. 800) R 8 + R 11 + R 18 + R 20= 111 of 15

Crew Scheduling ProblemProblem statistics:a) 20 decision variables (rotations)b) 8 functional constraints (one for eachflight)c) All constraints have equality signs12 of 15

Crew Scheduling Problem (Matlab)Input Fileminmax=1; % Minimization formulationa=[1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 00 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 00 0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 00 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 0 1 00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 10 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1]b=[1 1 1 1 1 1 1 1]'c=[-3500 -3500 -3500 -3500 -3500 -3500 -3500 -3500 -2500 -3500 -3500 -2500-2500 -3500 -3500 -3500 -3500 -3500 -3500 -3500]13 of 15

as=[1 2 3 4 5 6 7 8]14 of 15

Problem SolutionOptimal Solution (after 8 iterations):bas =[ 9 12 13 4 19 18 20 11]The current basic variable values are :b =1 rotation 9 (100,200) Cost = $2,5001 rotation 12 (500,600) Cost = $2,5001 rotation 13 (300,400) Cost = $2,5000 rotation 4 (400)0 rotation 19 (700,600)0 rotation 18 (800,100)1 rotation 20 (700,800) Cost = $3,5000 rotation 11 (500,800)z = $11,000 dollars to complete all flights (4 crews assigned)15 of 15