Automatic Aircraft Cargo Load Planning Sabine Limbourg QuantOM-HEC-University of Liège Michaël Schyns QuantOM-HEC-University of Liège Gilbert Laporte CIRRELT HEC Montréal
Agenda Problem statement Constraints Objective function Mathematical model Experiments OPAL Improvements Topics for next researches 2
Weight and Balance Gross weight maximum allowable L 1 Centre of gravity (CG) L 2 Balance control refers to the location of CG 3
Balance is an issue 4
for aircraft too 5
Improper loading Destruction of valuable equipment Loss of lives Cuts down the efficiency of an aircraft Altitude Maneuverability Rate of climb Speed thus impacting operational costs due to excessive fuel burn 6
Literature on air load planning Chan and Kumar, 2006 Guéret et al., 2003 Heidelberg et al., 1998 Li, Tao and Wang, 2009 Mongeau and Bès, 2003 Nance et al., 2010 Ng, 1992 Sabre, 2007 Souffriau and Vanden Berghe, 2008 Tian et al., 2008, 2009 Tang and Chang, 2010 Yan et al., 2008 Wu, 2010 7
Literature on air load planning Different objectives: optimizing the load inside a container optimizing the load of bulk freight in an aircraft, optimizing passengers aircrafts, minimizing the cost of a flight, optimizing the location of the centre of gravity Constraints taken into account Most of the optimization methods are heuristics Specific cases for specific aircrafts or loads 8
Problem statement Unit Load Device (ULD) Container Pallet and Net 9
Problem statement 10
How do they do it? It is the Load Master s Role to accurately plan the load (loadsheet), complying with all operational and safety requirements. In the past this has been accomplished manually LACHS (Liege Air Cargo Handling Services) 11
12
How do they do it? It is the Load Master s Role to accurately plan the load (loadsheet), complying with all operational and safety requirements. In the past this has been accomplished manually LACHS (Liege Air Cargo Handling Services) and more recently by drag and drop applications TNT (Thomas Nationwide Transport) CHAMP Cargosystems 13
14
15
Main variables and parameters 16
Upper Deck Flight Crew : Captain First Officer First Obsever Second Obsever UpperCabin-6 seatsand 2 bunks 17
Main Deck 18
Lower Deck 19
Lower Deck 20
Main parameters U = the set of ULDs IATA Identification Code The weight of the i th ULD (U i ) is denoted by w i Weight is uniformly distributed inside ULD i 21
Main parameters P = the set of positions Position j is denoted by P j Central arm value of P j : a j = (forward arm + aft arm)/2 List of ULD types that may fit in Laterally, 3 cases: R-L-Covering P L = set of positions on the left side P R = set of positions on the right side 22
Variables x ij = 1 if U i is in P j 0 otherwise Full load Assign each ULD to one position in the aircraft j P x ij = 1 i U 23
Allowable positions Each position accepts only some ULD types x ij = 0 i U, j P U i does not fit in P j One position can accept at most one ULD 1 j P i U x ij Overlaying position x ij x i + ' j' 1 i, i' U, j P, j O J where O J denotes the set of position indices underlying position P j 24
Goals 1. Centre of gravity at the best location Stress on the structure: banana effect 2. Packing Inertia approach 3. Automatic or semi-automatic system 4. Quickly 25
Moment of inertia min i U j P w i ( a j ID) 2 x ij = min I under wi ( a j ID) xij i U j P ε W ε 26
Envelope 27
Lateral balance W [251513,261946[ [261946,266482[ [266482,271018[ [271018,275554[ [275554,280089[ [280089,284625[ [284625,285986[ [285986,287800[ [287800,288031[ D 87115 56768 49032 41286 33548 25802 18065 15745 12648 D wi ( xij i U j P j P R L x ij ) D 28
Combined load limits 29
Combined load limits Areas O D k position forward and aft limits breakpoints of the piecewise linear function 30
Combined load limits Areas O D k position forward and aft limits breakpoints of the piecewise linear function D Ok = maximal weight for area O D k Weight is uniformly distributed inside ULD D = proportion falling in this area o ijk i U j P P j O x D k o D ij ijk φ O D k 31
Cumulative load limits Forward body forward piecewise linear limit function 32
Cumulative load limits Aft body aft piecewise linear limit function 33
Cumulative load limits Forward areas F k - Aft areas T k position forward and aft limits breakpoints of the piecewise linear function F k ( ) = maximal weight from nose (tail) to k limit Weight is uniformly distributed inside ULD f ijk (t ijk ) = proportion falling in this forward (aft) area i U i U k j P P j U c F l c = = 1 φ 1 k j P P U T l= 1 j k c = 1 c φ k x ij x ij f t ijl ijl F T k k 34
Restricted cumulative load limits New limits values: New binary variable: y=0 restricted constraint applied =1 relaxed i U Penalty term: j P P U T l= 1 j R k c = 1 c φ k T k k x ij t ijl Wy R k min w ( a ID)² x + i U j P i j ij 2 L Wy 35
Envelope 36
Mathematical model 37
Case studies Laptop computer Windows XP dualcore 2.5GHz 2.8 GB of RAM CPlex 12 Branch and Cut Cplex algorithm with the default parameters 38
Main case Optimization: 2 s 39
Comparison Automatic Load Master #ULDs % MAC* Time Delta Weight Weight constraints Restricted aft constraint 42 28.001 2 s 4823 kg satisfied no 42 27.601 1200 s 5693 kg satisfied no MAC=Mean Aerodynamic Chord 40
Moment of inertia vs. CG 41 41
Moment of inertia vs. CG 42 42
Moment of inertia vs. CG Inertia Time 5.3E9 0.8 s Min I 1.7E10 13.0 s Min CG The first solution (min I) reduces the stress on the structure but also increases the level of maneuverability. 43
More cases A B C D E F # ULDs 23 26 30 42 42 45 W (kg) 60 418 63 810 59 360 103 975 120 112 107 674 % MAC 27.992 28.007 28.000 27.996 28.001 27.998 % MAC (LM) 26.1 27.5 27.3 28.1 27.601 28 Inertia (min I) 4.4E9 5.3E9 7.3E9 1.8E10 3.1E10 2.5E10 Inertia (min CG) 1.6E10 1.7E10 1.4E10 2.6E10 3.3E10 2.6E10 Time (min I) 1.4 s 0.8 s 1.0 s 1.5 s 2.0 s 2.9 s Time (min CG) 116.6 s 13.0 s 1.9 s 441.9 s 1.2 s 155.7 s Delta Weight (kg) 1 990 580 2 135 1 025 4 823 666 Weight ok ok ok ok ok ok Restricted aft ok ok ok ok NO ok 44
Conclusions Practical problem MIP model Inertia approach packing Large set of realistic constraints Aft restricted weight limits OPAL "Difficult" instances solved in a few seconds Feasible and optimal: CG and constraints Exact solution Interactive software 45
Future work The model has been developed bearing in mind the possibility of future extensions. Multi-destinations How can you load + + in an aircraft? 46
Incompatibilities 47
Lazy constraints Constraints not specified in the constraint matrix of the MIP problem but integrated when violated. Using lazy constraints makes sense when there are a large number of constraints that must be satisfied at a solution, here representing each incompatibility, but are unlikely to be violated if they are left out. 48
Segregation matrix S: the segregation matrix s ik Z + = required segregation distance in inch s ik = 0 if and only if good i can be loaded together with good k without any restrictions s ik > 0 if some segregation conditions between goods i and k are required. s max be the maximum of s ik Note that S is symmetrical and elements of main diagonal are equal to zero. 49
Neighbour positions s max s max P i forward arm of P i aft arm of P i 50
Lazy constraints For each U i to load (i U) For j=i+1 to the number of ULDs to load If s ij >0 then For each position possible P i for U i, For each position possible P j pour U j For each n NL of Pi if (n=j ) x in +x jj 1 51
Without and with incompatibilities 2.4 s 7 s 52
Without and with incompatibilities 62 s 182 s 53
Thank you http://www.mschyns.be/demonstration/opal_web 54