Multi-objective combinatorial optimization: From methods ... · Combinatorial optimization...

Multi-objectivecombinatorial optimization:From methods to problems,

from the Earth to (almost) the Moon

Nicolas Jozefowiez

Maıtre de conferences en informatiqueINSA, LAAS-CNRS, Universite de Toulouse

le mardi 03 decembre 2013

Outline

I. Curriculum Vitæ

II. Multi-objective optimization

III. Multi-objective search tree

IV. Multi-objective column generation

V. Multi-objective genetic algorithms

VI. Conclusions and perspectives

Nicolas Jozefowiez 2 / 51

Part I

Curriculum vitæ

Short CV

2004Ph.D. in Computer Science, Universite deLille 1, France.Title: Modelling and approximate solution ofmulti-objective vehicle routing problems

2005Temporary assistant professor, Universite deLille 1, France.

2006Fulbright visiting scholar, CU Boulder, CO,USA.

2007Temporary assistant professor, Universite deValenciennes et du Hainaut-Cambresis, France.

2008Postdoctoral position, ESG-UQAM / CIR-RELT, Montreal, Canada.

2008 Assistant professor in Computer Science,INSA / LAAS-CNRS, Toulouse, France.

Supervisions: Ph.D.

2013Panwadee Tangpattanakul, Multi-objectiveoptimization of Earth observing satellites,French-Thai grant, co-director: P. Lopez.

2013Boadu Mensah Sarpong, Column generationfor bi-objective integer programs: Applicationto vehicle routing problems, Ministry grant, co-director: C. Artigues.

2012 Leonardo Malta, Transportation problemsfor door-to-door services, ANR RESPET, co-director: F. Semet.

2013 Leticia Gloria Vargas Suarez, Multi-objectivecumulative vehicle routing problems for hu-manitarian logistics, Erasmus grant, co-director: S. U. Ngueveu.

Supervisions: Postdoct & Master

2009H. Murat Afsar, Optimization of intelligentwaste collecting routing, Midi-Pyrenees grant,co-supervisor: P. Lopez.

2010Rodrigo Acuna-Agost, Methods for integratedaircraft-passenger recovery systems, AmadeusS.A., co-supervisor: C. Mancel.

2010• Oussama Ben Ammar, Bi-objective schedul-ing on a single machine, Universite deToulouse 2.

2010• Myriam Foucras, Multi-modal traveling sales-man problem, Universite Paul Sabatier.

Project participations

2009GEDEON, Projet Region Midi-Pyrenees.

2010Gestion des perturbations dans le transportaerien, Amadeus SA.

2010• Algo. de PL embarquable pour le rendez-vous orbital, CNES.

2013Planification mission flexible, R&T CNES.

2015Energy Aware feeding system, ECO-INNOVERA.

2015RESPET, ANR Transports TerrestresDurables.

2017ATHENA, ANR Blanc.

Activities

2010• Fulbright jury, French scholar program.

2010• ROADEF 2010, organizing committeemember.

2011• GdR RO GT2L 7th meeting, organizer.

2013ROADEF WG PM2O, co-animator.

2013LAAS laboratory council, member elect.

2015Bureau ROADEF, member (tresorier).

2015• ODYSSEUS 2015, organizing committeemember.

Research area

Combinatorial optimization

Multi-objectiveoptimization

Meta-heuristics

Vehicle routingproblems

Branch-and-cutalgorithm

Columngeneration

SchedulingAir transp.and space

Route balancingFacultative visitsLabels Disruption magt. EOS

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Route balancing

Facultative visitsLabels Disruption magt. EOS

Research area

Meta-heuristics

Columngeneration

Route balancingFacultative visits

Labels Disruption magt. EOS

Research area

Meta-heuristics

Columngeneration

Route balancingFacultative visitsLabels

Disruption magt. EOS

Research area

Meta-heuristics

Columngeneration

Scheduling

Air transp.and space

Research area

Meta-heuristics

Columngeneration

Scheduling

Air transp.and space

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Research area

Meta-heuristics

Columngeneration

Part II

Multi-objective optimization

Multi-objective optimization problem

(MOP) =

minimize F (x) = (f1(x), f2(x), . . . , fn(x))

x ∈ Ω

• n ≥ 2: number of objectives

• F : function vector to optimize

• Ω ⊆ Rm: feasible solution set (solution space)

• x : a solution

• Y = F (Ω): objective space

• y = (y1, y2, . . . , yn) ∈ Y with yi = fi (x): a point in theobjective space

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Efficient/Pareto-optimal solutionEfficient/Pareto-optimal set

Non-dominated pointNon-dominated set

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Solution approach

A priori approach

• Consideration of a decision-maker choice set

• One solution that is optimal (or an approximation) regardingto this choice set

Interactive approach

• The choice set is updated during the solution

A posteriori approach

• Efficient set (or an approximation)

• The decision-maker chooses among the efficient set

Usefulness

Can every problem be limited to a single objective ? No

Example: fairness between drivers in the CVRP

Taburoute Prins’ GA

Instance Distance Fairness Distance Fairness

E51-05e 524.61 20.07 524.61 20.07E76-10e 835.32 78.10 835.26 91.08E101-08e 826.14 97.88 826.14 97.88E151-12c 1031.17 98.24 1031.63 100.34E200-17c 1311.35 106.70 1300.23 82.31E121-07c 1042.11 146.67 1042.11 146.67E101-10c 819.56 93.43 819.56 93.43

MOP as a decision tool

Example: Cumulative Capacitated Vehicle Routing Problem

Number of vehicles

5 10 15 20 25 30 35 40 45 50| | | | | | | | | |

Scalarization methods

Weighted sum method

min (f1(x), . . . , fn(x))

x ∈ Ω→

∑ni=1 λi fi (x)

x ∈ Ω

n∑i=1

λi = 1

ε-constraint method

min (f1(x), . . . , fn(x))

x ∈ Ω→

min fk(x)

x ∈ Ω

fi (x) ≤ εi (i ∈ [1, n], i 6= k)

Two-phase method [Ulungu & Teghem, 1993]

Phase 1

• Dichotomic search

• Weighted sum objective

• Only the convex hull

• Supported solutions

Phase 2

• Enumerative search

• Bounded by phase 1solutions

• Not supported solutions f1

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• What does it mean for exact methods ?

• It should be true all along the search

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Multi-objective anytime algorithm

Multi-objective evolutionary algorithms

• A population P + mechanisms

• Set-based optimization [Zitzler et al., 2010]

Ψ ∈ P = ψ ⊆ Ω : @x , y ∈ Φ• Multi-objective decoders

Integer programming methods

• A single program, a scalarization method

• Avoid iteration of NP-hard problem solutions

• Search tree, lower bound

Design

• Representativity (Diversity)

• Uniformity (Convergence)

• Factorization (Efficiency)

Design

• Factorization (Efficiency)Nicolas Jozefowiez 20 / 51

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Representativity

Limit: 3 iterations

Non-dominated set

Two-phase method

Representative set

Uniformity

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The search/computational effort should be spread on the completeobjective spaceEach operation should have an impact on the all approximation

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Two-phase method: 20 iterations / Best search: 10 iterations

Part III

Multi-objective search tree

Upper and lower bounds

Upper bound (ub)

x ∈ Ω : @y ∈ ub, y x ⊆ Ω

Lower bound (lb) [Villareal & Karwan, 1981]

x ∈ Rn : (@x , y ∈ lb, y x) ∧ (∀y ∈ Ω,∃x ∈ lb, x y) ⊆ Rn

Case (1) Case (2) Case (3)

Computation of the lower bound

• A single multi-objective integer program

• Lower bound• A set of subproblems Φ• A subproblem φ ∈ Φ = linear relaxation + scalarization

technique

• Computation• Solve a subset Φ ⊆ Φ• Advantage: each φ ∈ Φ is polynomially solvable

• Φ should be kept polynomial or pseudo-polynomial

• Branch-and-cut flowchart is not modified

Example

Φ = φε, ε ∈ 0, 1, 2

minimize −1.00x1 − 0.64x2

minimize x3

s.t. 50x1 + 31x2 ≤ 250

3x1 − 2x2 ≥ −4

x1 + x3 ≤ 2

x1, x2 ≥ 0 and integer

x3 ∈ 0, 1, 2

Example

Φ = φε, ε ∈ 0, 1, 2

minimize −1.00x1 − 0.64x2

s.t. 50x1 + 31x2 ≤ 250

3x1 − 2x2 ≥ −4

x1 + x3 ≤ 2

x3 = ε

x1, x2 ≥ 0

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 1.94 x2 = 4.92UnfeasibleUnfeasible

ε = 0 x1 = 2 x2 = 4UnfeasibleUnfeasible

UnfeasibleUnfeasibleUnfeasible

Number of LP solutions: 15

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

Search tree

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3ε = 2 x1 = 0 x2 = 2

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

ε = 0 x1 = 1.94 x2 = 4.92UnfeasibleNot solved

ε = 0 x1 = 2 x2 = 4Not solvedNot solved

UnfeasibleNot solvedNot solved

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

Partial pruning

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 3ε = 1 x1 = 1 x2 = 3

Not solved

Parallel branching

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 4ε = 1 x1 = 1 x2 = 3

Not solved

UnfeasibleUnfeasibleNot solved

Parallel branching

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 4ε = 1 x1 = 1 x2 = 3

Not solved

Parallel branching

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 4ε = 1 x1 = 1 x2 = 3

Not solved

Parallel branching

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 4ε = 1 x1 = 1 x2 = 3

Not solved

Parallel branching

ε = 0 x1 = 1.94 x2 = 4.92ε = 1 x1 = 1 x2 = 3.5ε = 2 x1 = 0 x2 = 2

ε = 0 x1 = 2 x2 = 4ε = 1 x1 = 1 x2 = 3

Not solved

The multilabel traveling salesman problem

G = (V ,E )

Cost function c on E

A set of labels L = , , ,

Each e ∈ E ← δe ∈ L (data)

Minimize the total length

Minimize the number of labels used

IP: Based on [Dantzig et al., 54] + valid inequalities

Lower bound: ε-constraint method on the # of labels used(max LP solved ≤ |L|)

Cuts are searched after each LP solution

G = (V ,E )

Computational results (I)

Comparison with an iterative ε-constraint method

Same underlying branch-and-cut algorithm

MOB&C εCM

|L| |V | #Par #Nodes Seconds Seconds* #Nodes Seconds

40 20 12.1 606.8 4.2 3.1 1571.0 5.040 30 17.8 1913.0 58.7 42.7 5806.0 67.240 40 21.7 4406.6 503.0 349.8 17462.0 665.840 50 26.6 15360.6 1845.9 1374.5 45306.6 3334.5

50 20 12.4 718.9 4.4 3.4 2296.6 6.850 30 18.8 3248.3 144.0 110.2 12687.6 224.950 40 23.9 8722.7 1374.4 1097.7 36339.4 1636.950 50 27.7 20680.3 4094.0 2902.5 74336.6 5938.4

Computational results (II)

Use of the method as a heuristic

Stop after a percentage of the search tree has been explored

%: percentage of Pareto solutions found

Gap: average over all non efficient solutions of

25% 50% 75%

|L| |V | % Gap % Gap % Gap Seconds

40 20 58.7 1.011 76.0 1.005 87.6 1.002 2.740 30 41.6 1.010 62.9 1.005 83.7 1.002 30.040 40 31.3 1.011 43.8 1.007 80.2 1.002 200.340 50 34.2 1.009 51.9 1.006 71.8 1.003 708.0

50 20 59.7 1.011 69.4 1.009 84.7 1.004 2.950 30 41.0 1.012 63.8 1.005 86.2 1.002 75.450 40 34.3 1.011 51.9 1.005 82.0 1.002 601.850 50 24.5 1.012 40.8 1.007 69.7 1.003 1679.9

25% 50% 75%

40 20 58.7 1.011 76.0 1.005 87.6 1.002 2.740 30 41.6 1.010 62.9 1.005 83.7 1.002 30.040 40 31.3 1.011 43.8 1.007 80.2 1.002 200.340 50 34.2 1.009 51.9 1.006 71.8 1.003 708.0

50 20 59.7 1.011 69.4 1.009 84.7 1.004 2.950 30 41.0 1.012 63.8 1.005 86.2 1.002 75.450 40 34.3 1.011 51.9 1.005 82.0 1.002 601.850 50 24.5 1.012 40.8 1.007 69.7 1.003 1679.9

25% 50% 75%

40 20 58.7 1.011 76.0 1.005 87.6 1.002 2.740 30 41.6 1.010 62.9 1.005 83.7 1.002 30.040 40 31.3 1.011 43.8 1.007 80.2 1.002 200.340 50 34.2 1.009 51.9 1.006 71.8 1.003 708.0

50 20 59.7 1.011 69.4 1.009 84.7 1.004 2.950 30 41.0 1.012 63.8 1.005 86.2 1.002 75.450 40 34.3 1.011 51.9 1.005 82.0 1.002 601.850 50 24.5 1.012 40.8 1.007 69.7 1.003 1679.9

25% 50% 75%

40 20 58.7 1.011 76.0 1.005 87.6 1.002 2.740 30 41.6 1.010 62.9 1.005 83.7 1.002 30.040 40 31.3 1.011 43.8 1.007 80.2 1.002 200.340 50 34.2 1.009 51.9 1.006 71.8 1.003 708.0

50 20 59.7 1.011 69.4 1.009 84.7 1.004 2.950 30 41.0 1.012 63.8 1.005 86.2 1.002 75.450 40 34.3 1.011 51.9 1.005 82.0 1.002 601.850 50 24.5 1.012 40.8 1.007 69.7 1.003 1679.9

25% 50% 75%

40 20 58.7 1.011 76.0 1.005 87.6 1.002 2.740 30 41.6 1.010 62.9 1.005 83.7 1.002 30.040 40 31.3 1.011 43.8 1.007 80.2 1.002 200.340 50 34.2 1.009 51.9 1.006 71.8 1.003 708.0

50 20 59.7 1.011 69.4 1.009 84.7 1.004 2.950 30 41.0 1.012 63.8 1.005 86.2 1.002 75.450 40 34.3 1.011 51.9 1.005 82.0 1.002 601.850 50 24.5 1.012 40.8 1.007 69.7 1.003 1679.9

Part IV

Multi-objective

column generation

Column generation & bi-objective IP

Linear relaxation of the MP (LMP)

minimize∑

j∈Jcrj θj (r = 1, 2)

s.t.∑

j∈Jaijθj ≥ bi (i ∈ I )

θj ∈ R+ (j ∈ J)

Weighted sum ε-constraint method

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Restricted master problem (RMP) J ′ ⊂ J, |J ′| << |J|→ Generate columns for the

Master problem (MP)

minimize∑

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θj ∈ N (j ∈ J)

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minimize∑

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minimize∑

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minimize∑

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Restricted master problem (RMP) J ′ ⊂ J, |J ′| << |J|

→ Generate columns for the

minimize∑

s.t.∑

θj ∈ R+ (j ∈ J)

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Point-by-point search (PPS)

Scalarization technique = ε-constraint method

• Iterative ε-constraintmethod

• Full column generationalgorithm at each iteration

• Possible improvements

• Column storage• Blind ad-hoc heuristics

(Improved PPS)• ...

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Problems: may be caught in a tailing effect, no uniformconvergence, no factorization, not good as a heuristic ...

⇒ column search strategies

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Solve once, generate for all (SOGA)

Main computational cost: solution of a subproblemThe subproblem is similar for several values of ε

• At each iteration

• Select a value ε1

• Solve the LRMP for ε1

• Search for a column set J1

• For several εk , solve theLRMP → π∗k

• Heuristically built columnsusing J1 and π∗k

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Main computational cost: solution of a subproblem

The subproblem is similar for several values of ε

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Bi-obj. multi-vehicle covering tour problem

G = (V ∪W ,E , d), p: max # of nodes in a tour

A solution = a set of tours on V ′ ⊆ V + assignment of W to V ′

Objectives: i) minimize the total length; ii) maxwi∈W minvj∈V ′ dij

V : nodesthat can be visited

W : nodes to cover

G = (V ∪W ,E , d)

, p: max # of nodes in a tour

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

A solution = a set of tours on V ′ ⊆ V

+ assignment of W to V ′

Objectives: i) minimize the total length

; ii) maxwi∈W minvj∈V ′ dij

W : nodes to cover

A model for the BOMCTP

minimize∑ωk∈R

minimize Γmax

s.t.∑ωk∈R

aikθk ≥ 1 (wi ∈W )

Γmax ≥ ρkθk (ωk ∈ R)

θk ∈ 0, 1 (ωk ∈ R)

• ωk ∈ R: a tour on V ′ ⊆ V + W ′ ⊆W• ck : the tour length• aik = 1 if wi ∈W ′, 0 otherwise.• ρk = maxwi∈W ′ minvj∈V ′ dij

minimize∑ωk∈R

minimize Γmax

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• ωk ∈ R: a tour on V ′ ⊆ V + W ′ ⊆W

• ck : the tour length• aik = 1 if wi ∈W ′, 0 otherwise.• ρk = maxwi∈W ′ minvj∈V ′ dij

minimize∑ωk∈R

minimize Γmax

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• ωk ∈ R: a tour on V ′ ⊆ V + W ′ ⊆W• ck : the tour length

• aik = 1 if wi ∈W ′, 0 otherwise.• ρk = maxwi∈W ′ minvj∈V ′ dij

minimize∑ωk∈R

minimize Γmax

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• ωk ∈ R: a tour on V ′ ⊆ V + W ′ ⊆W• ck : the tour length• aik = 1 if wi ∈W ′, 0 otherwise.

• ρk = maxwi∈W ′ minvj∈V ′ dij

minimize∑ωk∈R

minimize Γmax

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• ωk ∈ R: a tour on V ′ ⊆ V + W ′ ⊆W• ck : the tour length• aik = 1 if wi ∈W ′, 0 otherwise.• ρk = maxwi∈W ′ minvj∈V ′ dij

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• The master problem is a single objective problem• The subproblem is a single objective problem• Well-suited for an ε-constraint method [Berube et al., 2009]

• Rε = ωk ∈ R : ρk ≤ ε• No weakening of the linear relaxation for a given ε value• Difference with the mono-objective model: aik is to be decided• Large variety of problems

1 A global objective on the complete solution2 An objective on the components → minimizing the worst case

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• The master problem is a single objective problem

• The subproblem is a single objective problem• Well-suited for an ε-constraint method [Berube et al., 2009]

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• The master problem is a single objective problem• The subproblem is a single objective problem

• Well-suited for an ε-constraint method [Berube et al., 2009]

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• Rε = ωk ∈ R : ρk ≤ ε

• No weakening of the linear relaxation for a given ε value• Difference with the mono-objective model: aik is to be decided• Large variety of problems

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• Rε = ωk ∈ R : ρk ≤ ε• No weakening of the linear relaxation for a given ε value

• Difference with the mono-objective model: aik is to be decided• Large variety of problems

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

• Rε = ωk ∈ R : ρk ≤ ε• No weakening of the linear relaxation for a given ε value• Difference with the mono-objective model: aik is to be decided

• Large variety of problems

Reformulation

minimize∑ωk∈R

s.t.∑ωk∈R

θk ∈ 0, 1 (ωk ∈ R)

Computational results

PPS SOGA

p |V | |W | Seconds #SP Seconds % #SP % |R∗|

5 30 60 18.03 184.6 13.00 .72 112.2 .61 1229.45 30 90 15.38 163.4 12.28 .79 105.6 .64 1041.2

5 40 80 49.50 228.0 36.68 .74 141.4 .62 1609.85 40 120 126.75 330.4 86.39 .68 173.8 .53 2347.6

5 50 100 205.79 390.4 154.51 .75 212.6 .54 2871.25 50 150 392.54 486.4 268.06 .68 223.6 .46 3087.4

8 30 60 49.76 226.8 25.46 .51 136.0 .60 1657.28 30 90 31.26 215.2 18.68 .60 121.8 .56 1284.0

8 40 80 113.41 302.0 86.81 .76 182.8 .60 2252.68 40 120 511.23 480.6 326.63 .63 243.2 .51 3652.4

8 50 100 1343.66 522.0 821.23 .61 289.0 .55 4392.88 50 150 1525.19 672.0 1049.09 .68 305.6 .45 4781.8

Part V

Multi-objective

genetic algorithms

Multi-objective meta-heuristics

Main focus of research on

• Selection

• Mechanisms for diversification

• Mechanisms for intensification

Less focus on

• Operators (crossover), neighborhood

• Encoding

• Usually inspired by a close single objective problem

Set-based optimization [Zitzler et al., 2010]

Standard approach

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Set-based approach

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• How to manipulate and define operators ?

• Proto-solution

• Multi-objective decoder: a proto-solution → several solutions

Standard approach

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Standard approach

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Standard approach

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Standard approach

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Standard approach

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Agile Earth Observation Satellite

Satellite direction

Earth surface

Captured photograph

Candidate photographs

Problem

• Select and scheduleacquisitions

• Operational constraints,multiple customers

• max. profit / fairnessbetween customers

Method

• Biased random-key genetic algorithms [Goncalves & Resende, 2010]

• Proto-solution: order to consider the acquisitions

• Decoder: two different heuristics

• Strategies to combine the solutions

• Strict improvement on computational results

Vehicle routing problems

Proto-solution

• A giant tour (TSP solution)

• Example: CVRP → ignore the capacity constraint

SPLIT operator [Prins, 2004]

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:95:55

Decoder

• Multi-objective Shortest Path Prob. with Resource Constraints

• Dynamic programming [Feillet et al., 2003][Reinhardt & Pisinger, 2011]

• Minimal modification: Label, dominance, extension rules

• Indicator-based evaluation

Proto-solution

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Decoder

Proto-solution

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Decoder

Proto-solution

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Decoder

Proto-solution

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Decoder

Proto-solution

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Decoder

Proto-solution

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Decoder

Proto-solution

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Decoder

Part VI

Conclusions and perspectives

Conclusions and short term perspectives

Research area• Mono- and multi-objective optimization

• Exact algorithms and heuristics

• Land transportation, air transportation, space

Contributions• Problem modeling and studies

• Proposition of new multi-objective meta-heuristics

• Proposition of new multi-objective exact algorithms

• Investigation of lower bound computation for multi-objectivecombinatorial optimization

Short term perspectives• Heuristics (matheuristics, multi-objective decoder ...)

• Vehicle routing problems (balancing, generalization ofproblems with facultative visits ...)

• Study of other families of problems to validate the methodsNicolas Jozefowiez 48 / 51

Multi-objective search tree

• Computation of lower bounds• Additional research for column generation• Multi-objective cutting plane algorithm

• Branching mechanisms

• Pruning mechanisms

• Decision space (variables) / objective space

• Branch-and-price algorithm

• Use of parallelism

• More than two objectives

Collaborative logistics

• Recent trend in logistics

• Supply chain resource pooling between actors

• Flow massification, resource sharing

• A natural ground for multi-objective optimization• Cost• Customer service• Environmental impact• Resource management• Fairness in the consortium ...

• New models / new methods

• ANR RESPET on door-to-door service network

Uncertainty in MOCO

Stochastic programming

minx∈Xc1x + Q1(x), c2x + Q2(x) : Ax = b

Qi (x) = Eζ [vi (hi (ω)− T i (ω)x)], vi (s) = min

y∈Yi

qi (ω)y : W iy = s

• Study of the interaction of the recourse functions

• Adaptation of methods such as Integer L-shaped Method

• Network design

Discrete robust optimization

• Scenarios / regret function

• Regret will not be an objective

• Robust efficient solutions / set

• Study of the interaction objective / scenario

• Adaptation of scenario relaxation methods

Multi-objective combinatorial optimization: From methods ... · Combinatorial optimization...

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