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

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

Upper Cabin - 6 seats and 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 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

23

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

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 ∑∑ 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

26

Envelope

27

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

28

Combined load limits

29

Combined load limits • Areas ODk – position forward and aft limits – breakpoints of the piecewise linear function

30

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

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 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

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 #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

40

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.

43

More cases A

B

C

D

E

F

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

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)

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 • 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

50

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

52

Without and with incompatibilities

62 s

182 s

53

Thank you http://www.mschyns.be/demonstration/opal_web

54

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

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

Upper Cabin - 6 seats and 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 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

23

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

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 ∑∑ 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

26

Envelope

27

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

28

Combined load limits

29

Combined load limits • Areas ODk – position forward and aft limits – breakpoints of the piecewise linear function

30

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

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 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

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 #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

40

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.

43

More cases A

B

C

D

E

F

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

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)

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 • 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

50

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

52

Without and with incompatibilities

62 s

182 s

53

Thank you http://www.mschyns.be/demonstration/opal_web

54