Multi-objective combinatorial optimization: From methods to problems, from the Earth to (almost) the Moon

Transcription

1 Multi-objective combinatorial optimization: From methods to problems, from the Earth to (almost) the Moon Nicolas Jozefowiez Maı tre de confe rences en informatique INSA, LAAS-CNRS, Universite de Toulouse le mardi 03 de cembre 2013

2 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

3 Part I Curriculum vitæ

4 Short CV Ph.D. in Computer Science, Université de Lille 1, France. Title: Modelling and approximate solution of multi-objective vehicle routing problems Temporary assistant professor, Université de Lille 1, France. Fulbright visiting scholar, CU Boulder, CO, USA. Temporary assistant professor, Université de Valenciennes et du Hainaut-Cambrésis, France. Postdoctoral position, ESG-UQAM / CIR- RELT, Montréal, Canada. Assistant professor in Computer Science, INSA / LAAS-CNRS, Toulouse, France. Nicolas Jozefowiez 4 / 51

5 Supervisions: Ph.D Panwadee Tangpattanakul, Multi-objective optimization of Earth observing satellites, French-Thai grant, co-director: P. Lopez. Boadu Mensah Sarpong, Column generation for bi-objective integer programs: Application to vehicle routing problems, Ministry grant, codirector: C. Artigues Leonardo Malta, Transportation problems for door-to-door services, ANR RESPET, codirector: F. Semet Leticia Gloria Vargas Suarez, Multi-objective cumulative vehicle routing problems for humanitarian logistics, Erasmus grant, codirector: S. U. Ngueveu. Nicolas Jozefowiez 5 / 51

6 Supervisions: Postdoct & Master H. Murat Afsar, Optimization of intelligent waste collecting routing, Midi-Pyrénées grant, co-supervisor: P. Lopez. Rodrigo Acuna-Agost, Methods for integrated aircraft-passenger recovery systems, Amadeus S.A., co-supervisor: C. Mancel Oussama Ben Ammar, Bi-objective scheduling on a single machine, Université de Toulouse Myriam Foucras, Multi-modal traveling salesman problem, Université Paul Sabatier. Nicolas Jozefowiez 6 / 51

7 Project participations GEDEON, Projet Région Midi-Pyrénées. Gestion des perturbations dans le transport aérien, Amadeus SA Algo. de PL embarquable pour le rendezvous orbital, CNES Planification mission flexible, R&T CNES. Energy Aware feeding system, ECO- INNOVERA. RESPET, ANR Transports Terrestres Durables. ATHENA, ANR Blanc. Nicolas Jozefowiez 7 / 51

8 Activities 2010 Fulbright jury, French scholar program ROADEF 2010, organizing committee member GdR RO GT2L 7th meeting, organizer ROADEF WG PM2O, co-animator. LAAS laboratory council, member elect. Bureau ROADEF, member (trésorier) ODYSSEUS 2015, organizing committee member. Nicolas Jozefowiez 8 / 51

9 Research area Nicolas Jozefowiez 9 / 51

10 Research area Combinatorial optimization Nicolas Jozefowiez 9 / 51

11 Research area Combinatorial optimization Multi-objective optimization Nicolas Jozefowiez 9 / 51

12 Research area Combinatorial optimization Multi-objective optimization Meta-heuristics Nicolas Jozefowiez 9 / 51

13 Research area Combinatorial optimization Vehicle routing problems Multi-objective optimization Meta-heuristics Nicolas Jozefowiez 9 / 51

14 Research area Combinatorial optimization Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Meta-heuristics Nicolas Jozefowiez 9 / 51

15 Research area Combinatorial optimization Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Meta-heuristics Nicolas Jozefowiez 9 / 51

16 Research area Combinatorial optimization Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Meta-heuristics Nicolas Jozefowiez 9 / 51

17 Research area Combinatorial optimization Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

18 Research area Combinatorial optimization Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

19 Research area Combinatorial optimization Route balancing Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

20 Research area Combinatorial optimization Facultative visits Route balancing Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

21 Research area Combinatorial optimization Labels Facultative visits Route balancing Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

22 Research area Combinatorial optimization Labels Facultative visits Route balancing Scheduling Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

23 Research area Combinatorial optimization Labels Facultative visits Route balancing Scheduling Vehicle routing problems Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

24 Research area Combinatorial optimization Labels Facultative visits Route balancing Scheduling Vehicle routing problems Air transp. and space Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

25 Research area Combinatorial optimization Labels Facultative visits Route balancing Disruption magt. EOS Scheduling Vehicle routing problems Air transp. and space Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

26 Research area Combinatorial optimization Labels Facultative visits Route balancing Disruption magt. EOS Scheduling Vehicle routing problems Air transp. and space Branch-and-cut algorithm Multi-objective optimization Column generation Meta-heuristics Nicolas Jozefowiez 9 / 51

27 Part II Multi-objective optimization

28 Multi-objective optimization problem (MOP) = { minimize F (x) = (f 1 (x), f 2 (x),..., f n (x)) x Ω n 2: number of objectives F : function vector to optimize Ω R m : feasible solution set (solution space) x: a solution Y = F (Ω): objective space y = (y 1, y 2,..., y n ) Y with y i = f i (x): a point in the objective space Nicolas Jozefowiez 11 / 51

29 Pareto dominance x y { f i (x) f i (y) i [1,..., n] f i (x) < f i (y) i [1,..., n] f 2 H Efficient/Pareto-optimal solution Efficient/Pareto-optimal set A E F Non-dominated point Non-dominated set B C G D f 1 Nicolas Jozefowiez 12 / 51

30 Pareto dominance x y { f i (x) f i (y) i [1,..., n] f i (x) < f i (y) i [1,..., n] f 2 H Efficient/Pareto-optimal solution Efficient/Pareto-optimal set A E F Non-dominated point Non-dominated set B C G D f 1 Nicolas Jozefowiez 12 / 51

31 Pareto dominance x y { f i (x) f i (y) i [1,..., n] f i (x) < f i (y) i [1,..., n] f 2 H Efficient/Pareto-optimal solution Efficient/Pareto-optimal set A E F Non-dominated point Non-dominated set B C G D f 1 Nicolas Jozefowiez 12 / 51

32 Pareto dominance x y { f i (x) f i (y) i [1,..., n] f i (x) < f i (y) i [1,..., n] f 2 H Efficient/Pareto-optimal solution Efficient/Pareto-optimal set A E F Non-dominated point Non-dominated set B C G D f 1 Nicolas Jozefowiez 12 / 51

33 Pareto dominance x y { f i (x) f i (y) i [1,..., n] f i (x) < f i (y) i [1,..., n] f 2 H Efficient/Pareto-optimal solution Efficient/Pareto-optimal set A E F Non-dominated point Non-dominated set B C G D f 1 Nicolas Jozefowiez 12 / 51

34 Solution approach A priori approach Consideration of a decision-maker choice set One solution that is optimal (or an approximation) regarding to 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 Nicolas Jozefowiez 13 / 51

35 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 E76-10e E101-08e E151-12c E200-17c E121-07c E101-10c Nicolas Jozefowiez 14 / 51

36 MOP as a decision tool Example: Cumulative Capacitated Vehicle Routing Problem Cumulative length Number of vehicles Nicolas Jozefowiez 15 / 51

37 Scalarization methods Weighted sum method { min (f 1 (x),..., f n (x)) x Ω { min n i=1 λ if i (x) x Ω n λ i = 1 i=1 ɛ-constraint method { min (f 1 (x),..., f n (x)) x Ω min f k (x) x Ω f i (x) ɛ i (i [1, n], i k) Nicolas Jozefowiez 16 / 51

38 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

39 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

40 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

41 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

42 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

43 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

44 Two-phase method [Ulungu & Teghem, 1993] Phase 1 f 2 Dichotomic search Weighted sum objective Only the convex hull Supported solutions Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

45 Two-phase method [Ulungu & Teghem, 1993] Phase 1 Dichotomic search Weighted sum objective Only the convex hull Supported solutions f 2 Phase 2 Enumerative search Bounded by phase 1 solutions Not supported solutions f 1 Nicolas Jozefowiez 17 / 51

46 Iterative ɛ-constraint method f 2 Nicolas Jozefowiez 18 / 51 f 1

47 Iterative ɛ-constraint method f 2 ɛ 0 Nicolas Jozefowiez 18 / 51 f 1

48 Iterative ɛ-constraint method f 2 ɛ 0 Nicolas Jozefowiez 18 / 51 f 1

49 Iterative ɛ-constraint method f 2 ɛ 0 ɛ 1 Nicolas Jozefowiez 18 / 51 f 1

50 Iterative ɛ-constraint method f 2 ɛ 0 ɛ 1 Nicolas Jozefowiez 18 / 51 f 1

51 Iterative ɛ-constraint method f 2 ɛ 0 ɛ 1 ɛ 2 Nicolas Jozefowiez 18 / 51 f 1

52 Iterative ɛ-constraint method f 2 ɛ 0 ɛ 1 ɛ 2 ɛ 3 ɛ 4 ɛ 5 Nicolas Jozefowiez 18 / 51 f 1

53 Intensification / Diversification f2 f2 f2 Intensification Diversification Goals f1 Good intensification f1 Good diversification f1 Nicolas Jozefowiez 19 / 51

54 Intensification / Diversification f2 f2 f2 Intensification Diversification Goals f1 Good intensification f1 Good diversification f1 Usually associated to multi-objective evolutionary algorithms Nicolas Jozefowiez 19 / 51

55 Intensification / Diversification f2 f2 f2 Intensification Diversification Goals f1 Good intensification f1 Good diversification f1 Usually associated to multi-objective evolutionary algorithms What does it mean for methods such as the Two-Phase method? Nicolas Jozefowiez 19 / 51

56 Intensification / Diversification f2 f2 f2 Intensification Diversification Goals f1 Good intensification f1 Good diversification f1 Usually associated to multi-objective evolutionary algorithms What does it mean for methods such as the Two-Phase method? What does it mean for exact methods? Nicolas Jozefowiez 19 / 51

57 Intensification / Diversification f2 f2 f2 Intensification Diversification Goals f1 Good intensification f1 Good diversification f1 Usually associated to multi-objective evolutionary algorithms What does it mean for methods such as the Two-Phase method? What does it mean for exact methods? It should be true all along the search Nicolas Jozefowiez 19 / 51

58 Multi-objective anytime algorithm Nicolas Jozefowiez 20 / 51

59 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 Nicolas Jozefowiez 20 / 51

60 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 Nicolas Jozefowiez 20 / 51

61 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) Nicolas Jozefowiez 20 / 51

62 Representativity f 2 Limit: 3 iterations Non-dominated set Nicolas Jozefowiez 21 / 51 f 1

63 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Nicolas Jozefowiez 21 / 51 f 1

64 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Nicolas Jozefowiez 21 / 51 f 1

65 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Nicolas Jozefowiez 21 / 51 f 1

66 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Nicolas Jozefowiez 21 / 51 f 1

67 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Two-phase method Nicolas Jozefowiez 21 / 51 f 1

68 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Two-phase method Nicolas Jozefowiez 21 / 51 f 1

69 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Two-phase method Nicolas Jozefowiez 21 / 51 f 1

70 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Two-phase method Nicolas Jozefowiez 21 / 51 f 1

71 Representativity f 2 Limit: 3 iterations Non-dominated set ɛ-constraint method Two-phase method Representative set Nicolas Jozefowiez 21 / 51 f 1

72 Uniformity f 2 Nicolas Jozefowiez 22 / 51 f 1

73 Uniformity f 2 Nicolas Jozefowiez 22 / 51 f 1

74 Uniformity f 2 Nicolas Jozefowiez 22 / 51 f 1

75 Uniformity f 2 Nicolas Jozefowiez 22 / 51 f 1

76 Efficiency f 2 S A D C B Two-phase method: iterations / Best search: iterations Nicolas Jozefowiez 23 / 51 f 1

77 Efficiency f 2 S A D C B Two-phase method: iterations / Best search: iterations Nicolas Jozefowiez 23 / 51 f 1

78 Efficiency f 2 S A D C B Two-phase method: iterations / Best search: iterations Nicolas Jozefowiez 23 / 51 f 1

79 Efficiency f 2 S A D C B Two-phase method: iterations / Best search: iterations Nicolas Jozefowiez 23 / 51 f 1

80 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

81 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

82 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

83 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

84 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

85 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: f 1 iterations Nicolas Jozefowiez 23 / 51

86 Efficiency f 2 S A D C B Two-phase method: 20 iterations / Best search: 10 iterations Nicolas Jozefowiez 23 / 51 f 1

87 Part III Multi-objective search tree

88 Upper and lower bounds Upper bound (ub) {x Ω : y ub, y x} Ω Lower bound (lb) [Villareal & Karwan, 1981] {x R n : ( x, y lb, y x) ( y Ω, x lb, x y)} R n Case (1) Case (2) Case (3) Nicolas Jozefowiez 25 / 51

89 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 Nicolas Jozefowiez 26 / 51

90 Example minimize 1.00x x 2 minimize x 3 s.t. 50x x x 1 2x 2 4 x 1 + x 3 2 x 1, x 2 0 and integer x 3 {0, 1, 2} Nicolas Jozefowiez 27 / 51

91 Example Φ = {φ ɛ, ɛ {0, 1, 2}} minimize 1.00x x 2 s.t. 50x x x 1 2x 2 4 x 1 + x 3 2 x 1, x 2 0 x 3 = ɛ Nicolas Jozefowiez 27 / 51

92 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Nicolas Jozefowiez 28 / 51

93 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Nicolas Jozefowiez 28 / 51

94 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Nicolas Jozefowiez 28 / 51

95 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Nicolas Jozefowiez 28 / 51

96 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Nicolas Jozefowiez 28 / 51

97 Search tree ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Unfeasible ɛ = 0 x 1 = 2 x 2 = 4 Unfeasible Unfeasible Unfeasible Unfeasible Unfeasible Number of LP solutions: 15 Nicolas Jozefowiez 28 / 51

98 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Nicolas Jozefowiez 29 / 51

99 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Nicolas Jozefowiez 29 / 51

100 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Nicolas Jozefowiez 29 / 51

101 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Nicolas Jozefowiez 29 / 51

102 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Nicolas Jozefowiez 29 / 51

103 Partial pruning ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 3 ɛ = 1 x 1 = 1 x 2 = 3 Not solved ɛ = 0 x 1 = 1.94 x 2 = 4.92 Unfeasible Not solved ɛ = 0 x 1 = 2 x 2 = 4 Not solved Not solved Unfeasible Not solved Not solved Number of LP solutions: 9 Nicolas Jozefowiez 29 / 51

104 Parallel branching ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 4 ɛ = 1 x 1 = 1 x 2 = 3 Not solved Unfeasible Unfeasible Not solved Nicolas Jozefowiez 30 / 51

105 Parallel branching ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 4 ɛ = 1 x 1 = 1 x 2 = 3 Not solved Unfeasible Unfeasible Not solved Nicolas Jozefowiez 30 / 51

106 Parallel branching ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 4 ɛ = 1 x 1 = 1 x 2 = 3 Not solved Unfeasible Unfeasible Not solved Nicolas Jozefowiez 30 / 51

107 Parallel branching ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 4 ɛ = 1 x 1 = 1 x 2 = 3 Not solved Unfeasible Unfeasible Not solved Nicolas Jozefowiez 30 / 51

108 Parallel branching ɛ = 0 x 1 = 1.94 x 2 = 4.92 ɛ = 1 x 1 = 1 x 2 = 3.5 ɛ = 2 x 1 = 0 x 2 = 2 ɛ = 0 x 1 = 2 x 2 = 4 ɛ = 1 x 1 = 1 x 2 = 3 Not solved Unfeasible Unfeasible Not solved Number of LP solutions: 7 Nicolas Jozefowiez 30 / 51

109 The multilabel traveling salesman problem Nicolas Jozefowiez 31 / 51

110 The multilabel traveling salesman problem G = (V, E) Cost function c on E Nicolas Jozefowiez 31 / 51

111 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Nicolas Jozefowiez 31 / 51

112 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Nicolas Jozefowiez 31 / 51

113 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Minimize the total length Nicolas Jozefowiez 31 / 51

114 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Minimize the total length Minimize the number of labels used Nicolas Jozefowiez 31 / 51

115 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Minimize the total length Minimize the number of labels used IP: Based on [Dantzig et al., 54] + valid inequalities Nicolas Jozefowiez 31 / 51

116 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Minimize the total length Minimize the number of labels used IP: Based on [Dantzig et al., 54] + valid inequalities Lower bound: ɛ-constraint method on the # of labels used (max LP solved L ) Nicolas Jozefowiez 31 / 51

117 The multilabel traveling salesman problem G = (V, E) Cost function c on E A set of labels L = {,,, } Each e E δ e L (data) Minimize the total length Minimize the number of labels used IP: Based on [Dantzig et al., 54] + valid inequalities Lower bound: ɛ-constraint method on the # of labels used (max LP solved L ) Cuts are searched after each LP solution Nicolas Jozefowiez 31 / 51

118 Computational results (I) Comparison with an iterative ɛ-constraint method Same underlying branch-and-cut algorithm MOB&C ɛcm L V #Par #Nodes Seconds Seconds* #Nodes Seconds Nicolas Jozefowiez 32 / 51

119 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 Nicolas Jozefowiez 33 / 51

120 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 Nicolas Jozefowiez 33 / 51

121 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 Nicolas Jozefowiez 33 / 51

122 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 Nicolas Jozefowiez 33 / 51

123 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 Nicolas Jozefowiez 33 / 51

124 Part IV Multi-objective column generation

125 Column generation & bi-objective IP Nicolas Jozefowiez 35 / 51

126 Column generation & bi-objective IP Master problem (MP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j N (j J) Nicolas Jozefowiez 35 / 51

127 Column generation & bi-objective IP Linear relaxation of the MP (LMP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j R + (j J) Nicolas Jozefowiez 35 / 51

128 Column generation & bi-objective IP f2 Linear relaxation of the MP (LMP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j R + (j J) Weighted sum f1 Nicolas Jozefowiez 35 / 51

129 Column generation & bi-objective IP f2 Linear relaxation of the MP (LMP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j R + (j J) Weighted sum f2 ɛ-constraint method f1 f1 Nicolas Jozefowiez 35 / 51

130 Column generation & bi-objective IP f2 Linear relaxation of the MP (LMP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j R + (j J) Weighted sum f2 ɛ-constraint method f1 f1 Restricted master problem (RMP) J J, J << J Nicolas Jozefowiez 35 / 51

131 Column generation & bi-objective IP f2 Linear relaxation of the MP (LMP) minimize j J cr j θ j (r = 1, 2) s.t. j J a ijθ j b i (i I ) θ j R + (j J) Weighted sum f2 ɛ-constraint method f1 f1 Restricted master problem (RMP) J J, J << J Generate columns for the Nicolas Jozefowiez 35 / 51

132 Point-by-point search (PPS) Nicolas Jozefowiez 36 / 51

133 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method Nicolas Jozefowiez 36 / 51

134 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method f1 Nicolas Jozefowiez 36 / 51

135 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration f1 Nicolas Jozefowiez 36 / 51

136 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration f1 Nicolas Jozefowiez 36 / 51

137 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration f1 Nicolas Jozefowiez 36 / 51

138 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration Possible improvements f1 Nicolas Jozefowiez 36 / 51

139 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration Possible improvements Column storage f1 Nicolas Jozefowiez 36 / 51

140 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration Possible improvements Column storage Blind ad-hoc heuristics (Improved PPS)... f1 Nicolas Jozefowiez 36 / 51

141 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration Possible improvements Column storage Blind ad-hoc heuristics (Improved PPS)... f1 Problems: may be caught in a tailing effect, no uniform convergence, no factorization, not good as a heuristic... Nicolas Jozefowiez 36 / 51

142 Point-by-point search (PPS) Scalarization technique = ɛ-constraint method f2 Iterative ɛ-constraint method Full column generation algorithm at each iteration Possible improvements Column storage Blind ad-hoc heuristics (Improved PPS)... f1 Problems: may be caught in a tailing effect, no uniform convergence, no factorization, not good as a heuristic... column search strategies Nicolas Jozefowiez 36 / 51

143 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Nicolas Jozefowiez 37 / 51

144 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem Nicolas Jozefowiez 37 / 51

145 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ Nicolas Jozefowiez 37 / 51

146 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ At each iteration f2 f1 Nicolas Jozefowiez 37 / 51

147 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ At each iteration Select a value ɛ 1 f2 f1 Nicolas Jozefowiez 37 / 51

148 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ f2 At each iteration Select a value ɛ 1 Solve the LRMP for ɛ 1 f1 Nicolas Jozefowiez 37 / 51

149 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ f2 At each iteration Select a value ɛ 1 Solve the LRMP for ɛ 1 Search for a column set J 1 f1 Nicolas Jozefowiez 37 / 51

150 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ f2 At each iteration Select a value ɛ 1 Solve the LRMP for ɛ 1 Search for a column set J 1 For several ɛ k, solve the LRMP π k f1 Nicolas Jozefowiez 37 / 51

151 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ f2 At each iteration Select a value ɛ 1 Solve the LRMP for ɛ 1 Search for a column set J 1 For several ɛ k, solve the LRMP π k Heuristically built columns using J 1 and π k f1 Nicolas Jozefowiez 37 / 51

152 Solve once, generate for all (SOGA) Scalarization technique = ɛ-constraint method Main computational cost: solution of a subproblem The subproblem is similar for several values of ɛ f2 At each iteration Select a value ɛ 1 Solve the LRMP for ɛ 1 Search for a column set J 1 For several ɛ k, solve the LRMP π k Heuristically built columns using J 1 and π k f1 Nicolas Jozefowiez 37 / 51

153 Bi-obj. multi-vehicle covering tour problem Nicolas Jozefowiez 38 / 51

154 Bi-obj. multi-vehicle covering tour problem G = (V W, E, d) Nicolas Jozefowiez 38 / 51

155 Bi-obj. multi-vehicle covering tour problem G = (V W, E, d) depot Nicolas Jozefowiez 38 / 51

156 Bi-obj. multi-vehicle covering tour problem G = (V W, E, d) depot V : nodes that can be visited Nicolas Jozefowiez 38 / 51

157 Bi-obj. multi-vehicle covering tour problem G = (V W, E, d) depot V : nodes that can be visited W : nodes to cover Nicolas Jozefowiez 38 / 51

158 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 Objectives: i) minimize the total length depot V : nodes that can be visited W : nodes to cover Nicolas Jozefowiez 38 / 51

159 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) max wi W min vj V d ij depot V : nodes that can be visited W : nodes to cover Nicolas Jozefowiez 38 / 51

160 A model for the BOMCTP minimize minimize s.t. c k θ k ω k R Γ max ω k R a ik θ k 1 (w i W ) Γ max ρ k θ k (ω k R) θ k {0, 1} (ω k R) Nicolas Jozefowiez 39 / 51

161 A model for the BOMCTP minimize minimize s.t. c k θ k ω k R Γ max ω k R a ik θ k 1 (w i W ) Γ max ρ k θ k (ω k R) θ k {0, 1} (ω k R) ω k R: a tour on V V + W W Nicolas Jozefowiez 39 / 51

162 A model for the BOMCTP minimize minimize s.t. c k θ k ω k R Γ max ω k R a ik θ k 1 (w i W ) Γ max ρ k θ k (ω k R) θ k {0, 1} (ω k R) ω k R: a tour on V V + W W c k : the tour length Nicolas Jozefowiez 39 / 51

163 A model for the BOMCTP minimize minimize s.t. c k θ k ω k R Γ max ω k R a ik θ k 1 (w i W ) Γ max ρ k θ k (ω k R) θ k {0, 1} (ω k R) ω k R: a tour on V V + W W c k : the tour length a ik = 1 if w i W, 0 otherwise. Nicolas Jozefowiez 39 / 51

164 A model for the BOMCTP minimize minimize s.t. c k θ k ω k R Γ max ω k R a ik θ k 1 (w i W ) Γ max ρ k θ k (ω k R) θ k {0, 1} (ω k R) ω k R: a tour on V V + W W c k : the tour length a ik = 1 if w i W, 0 otherwise. ρ k = max wi W min v j V d ij Nicolas Jozefowiez 39 / 51

165 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) Nicolas Jozefowiez 40 / 51

166 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem Nicolas Jozefowiez 40 / 51

167 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Nicolas Jozefowiez 40 / 51

168 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Well-suited for an ɛ-constraint method [Bérubé et al., 2009] Nicolas Jozefowiez 40 / 51

169 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Well-suited for an ɛ-constraint method [Bérubé et al., 2009] R ɛ = {ω k R : ρ k ɛ} Nicolas Jozefowiez 40 / 51

170 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Well-suited for an ɛ-constraint method [Bérubé et al., 2009] R ɛ = {ω k R : ρ k ɛ} No weakening of the linear relaxation for a given ɛ value Nicolas Jozefowiez 40 / 51

171 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Well-suited for an ɛ-constraint method [Bérubé et al., 2009] R ɛ = {ω k R : ρ k ɛ} No weakening of the linear relaxation for a given ɛ value Difference with the mono-objective model: a ik is to be decided Nicolas Jozefowiez 40 / 51

172 Reformulation minimize s.t. c k θ k ω k R ω k R a ik θ k 1 (w i W ) θ k {0, 1} (ω k R) The master problem is a single objective problem The subproblem is a single objective problem Well-suited for an ɛ-constraint method [Bérubé et al., 2009] R ɛ = {ω k R : ρ k ɛ} No weakening of the linear relaxation for a given ɛ value Difference with the mono-objective model: a ik is to be decided Large variety of problems 1 A global objective on the complete solution 2 An objective on the components minimizing the worst case Nicolas Jozefowiez 40 / 51

173 Computational results PPS SOGA p V W Seconds #SP Seconds % #SP % R Nicolas Jozefowiez 41 / 51

174 Part V Multi-objective genetic algorithms

175 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 Nicolas Jozefowiez 43 / 51

176 Set-based optimization [Zitzler et al., 2010] Nicolas Jozefowiez 44 / 51

177 Set-based optimization [Zitzler et al., 2010] population Nicolas Jozefowiez 44 / 51

178 Set-based optimization [Zitzler et al., 2010] f 2 population Standard approach f 1 Nicolas Jozefowiez 44 / 51

179 Set-based optimization [Zitzler et al., 2010] f 2 f 2 population Standard approach f 1 f 1 Set-based approach Nicolas Jozefowiez 44 / 51

180 Set-based optimization [Zitzler et al., 2010] f 2 f 2 population Standard approach f 1 f 1 Set-based approach How to manipulate and define operators? Nicolas Jozefowiez 44 / 51

181 Set-based optimization [Zitzler et al., 2010] f 2 f 2 population Standard approach f 1 f 1 Set-based approach How to manipulate and define operators? Proto-solution Nicolas Jozefowiez 44 / 51

182 Set-based optimization [Zitzler et al., 2010] f 2 f 2 population Standard approach f 1 f 1 Set-based approach How to manipulate and define operators? Proto-solution Multi-objective decoder: a proto-solution several solutions Nicolas Jozefowiez 44 / 51

183 Agile Earth Observation Satellite Captured photograph Method Satellite direction Earth surface Candidate photographs Problem Select and schedule acquisitions Operational constraints, multiple customers max. profit / fairness between customers Biased random-key genetic algorithms [Gonçalves & Resende, 2010] Proto-solution: order to consider the acquisitions Decoder: two different heuristics Strategies to combine the solutions Strict improvement on computational results Nicolas Jozefowiez 45 / 51

184 Vehicle routing problems Nicolas Jozefowiez 46 / 51

185 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint Nicolas Jozefowiez 46 / 51

186 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] Nicolas Jozefowiez 46 / 51

187 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] Nicolas Jozefowiez 46 / 51

188 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] :55 :95 30 :40 :50 :60 :80 : : :120 :90 Nicolas Jozefowiez 46 / 51

189 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] :55 :95 30 :40 :50 :60 :80 : : :120 :90 Nicolas Jozefowiez 46 / 51

190 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] :55 :95 30 :40 :50 :60 :80 : : :120 :90 Nicolas Jozefowiez 46 / 51

191 Vehicle routing problems Proto-solution A giant tour (TSP solution) Example: CVRP ignore the capacity constraint SPLIT operator [Prins, 2004] :55 :95 30 :40 :50 :60 :80 : : :120 :90 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 Nicolas Jozefowiez 46 / 51

192 Part VI Conclusions and perspectives

193 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-objective combinatorial optimization Short term perspectives Heuristics (matheuristics, multi-objective decoder...) Vehicle routing problems (balancing, generalization of problems with facultative visits...) Study of other families of problems to validate the methods Nicolas Jozefowiez 48 / 51

194 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 Nicolas Jozefowiez 49 / 51

195 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 Nicolas Jozefowiez 50 / 51

196 Uncertainty in MOCO Stochastic programming min x X {c1 x + Q 1 (x), c 2 x + Q 2 (x) : Ax = b} Q i (x) = E ζ [v i (h i (ω) T i (ω)x)], v i (s) = min y Y i {q i (ω)y : W i y = 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 Nicolas Jozefowiez 51 / 51