17. Upper Deck. Upper Cabin - 6 seats and 2 bunks. Flight Crew : ⢠Captain. ⢠First Officer .... Laptop computer ... Z+ = required segregation distance in inch. ⢠s ik.
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 • L1≤ Centre of gravity (CG) ≤ L2 • 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
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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
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Problem statement
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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)
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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
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14
15
Main variables and parameters
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Upper Deck Flight Crew : • Captain • First Officer • First Obsever • Second Obsever
Upper Cabin - 6 seats and 2 bunks
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Main Deck
18
Lower Deck
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Lower Deck
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Main parameters • U = the set of ULDs • IATA Identification Code
• The weight of the ith ULD (Ui) is denoted by wi • Weight is uniformly distributed inside ULDi 21
Main parameters • P = the set of positions • Position j is denoted by Pj • Central arm value of Pj: aj = (forward arm + aft arm)/2 • List of ULD types that may fit in • Laterally, 3 cases: R-L-Covering – PL= set of positions on the left side – PR= set of positions on the right side 22
Variables • xij = 1 if Ui is in Pj 0 otherwise
Full load • Assign each ULD to one position in the aircraft
∑x j∈P
ij
=1
∀i∈U
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Allowable positions • Each position accepts only some ULD types xij = 0 ∀ i ∈ U, j ∈ P | Ui does not fit in Pj • One position can accept at most one ULD ∑ xij ≤ 1 ∀ j ∈ P i∈U
• Overlaying position
xij + xi ' j ' ≤ 1 ∀ i, i'∈ U, j ∈ P, ∀j’∈ OJ where OJ denotes the set of position indices underlying position Pj
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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 ∑∑ wi (a j − ID ) 2 xij = min I i∈U j∈P
under
∑∑ w (a −ε ≤
i∈U j∈P
i
W
j
− ID) xij ≤ε
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Envelope
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Lateral balance W
D
[251513,261946[
87115
[261946,266482[
56768
[266482,271018[
49032
[271018,275554[
41286
[275554,280089[
33548
[280089,284625[
25802
[284625,285986[
18065
[285986,287800[
15745
[287800,288031[
12648
− D ≤ ∑ wi ( ∑ xij − ∑ xij ) ≤ D i∈U
j∈PR
j∈PL
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Combined load limits
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Combined load limits • Areas ODk – position forward and aft limits – breakpoints of the piecewise linear function
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Combined load limits • Areas ODk – position forward and aft limits – breakpoints of the piecewise linear function D • O k = maximal weight for area ODk
• Weight is uniformly distributed inside ULD D o • ijk = proportion falling in this area
∑
∑x o
D ij ijk i∈U j∈P Pj ∩OkD ≠φ
≤O
D k
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Cumulative load limits Forward body
forward piecewise linear limit function
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Cumulative load limits Aft body
aft piecewise linear limit function
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Cumulative load limits • Forward areas Fk - Aft areas Tk – position forward and aft limits – breakpoints of the piecewise linear function • Fk ( ) = maximal weight from nose (tail) to k limit • Weight is uniformly distributed inside ULD
• fijk (tijk) = proportion falling in this forward (aft) area k
f ijl ≤ F k
∑
∑
∑x
∑
∑
∑x t
i∈U j∈P Pj ∩U kc =1 Fc ≠φ l =1 k
i∈U j∈P Pj ∩U kc=1 Tc ≠φ l =1
ij
ij ijl
≤Tk 34
Restricted cumulative load limits • New limits values: R k ≤T k • New binary variable: y=0 restricted constraint applied =1 relaxed k
∑
∑
∑x t
i∈U j∈P Pj ∩U kc=1 Tc ≠φ l =1
ij ijl
− Wy ≤ R k
• Penalty term:
min ∑∑ wi (a j − ID)² xij + L2Wy i∈U j∈P
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Envelope
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Mathematical model
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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
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Main case
Optimization: 2 s
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Comparison #ULDs % MAC* Time Delta Weight Weight constraints Restricted aft constraint
Automatic
Load Master
42 28.001 2s 4823 kg satisfied no
42 27.601 1200 s 5693 kg satisfied no
MAC=Mean Aerodynamic Chord
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Moment of inertia vs. CG
41 41
Moment of inertia vs. CG
42 42
Moment of inertia vs. CG Min I
Min CG
Inertia
5.3E9
1.7E10
Time
0.8 s
13.0 s
The first solution (min I) reduces the stress on the structure but also increases the level of maneuverability.
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More cases A
B
C
D
E
F
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26
30
42
42
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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
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Inertia (min I)
4.4E9
5.3E9
7.3E9
1.8E10
3.1E10
2.5E10
1.6E10
1.7E10
1.4E10
2.6E10
3.3E10
2.6E10
1.4 s
0.8 s
1.0 s
1.5 s
2.0 s
2.9 s
116.6 s
13.0 s
1.9 s
441.9 s
1.2 s
155.7 s
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
# ULDs
Inertia (min CG) Time (min I) Time (min CG) Delta Weight (kg)
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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
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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
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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 • sik ∈ Z+ = required segregation distance in inch • sik = 0 if and only if good i can be loaded together with good k without any restrictions • sik > 0 if some segregation conditions between goods i and k are required. • smax be the maximum of sik • Note that S is symmetrical and elements of main diagonal are equal to zero. 49
Neighbour positions smax
smax Pi
forward arm of Pi
aft arm of Pi
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Lazy constraints For each Ui to load (i ∈ U) For j=i+1 to the number of ULDs to load If sij>0 then For each position possible Pi’ for Ui, For each position possible Pj’ pour Uj For each n ∈ NL of Pi’ if (n=j’) xin+xjj’≤1 51
Without and with incompatibilities
2.4 s
7s
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Without and with incompatibilities
62 s
182 s
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Thank you http://www.mschyns.be/demonstration/opal_web
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