Multi-objective optimizationand
vehicle routing problems
Nicolas Jozefowiez
INSA, LAAS-CNRS, Universite de Toulouse
le vendredi 28 fevrier 2014
Outline
I. Introduction
II. Classification of objectives
III. Applications
IV. Methods
V. Conclusions and perspectives
Nicolas Jozefowiez 2 / 53
Part I
Introduction
Nicolas Jozefowiez 3 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the
periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problems
Nicolas Jozefowiez 4 / 53
Vehicle routing problems
1
2 3
4
56
A solution: a tour or a collection of tours on a subset of nodes
Constraints: network, resources, reachability, periods ...
From the traveling salesman problem to the periodic two-echelonheterogeneous fleet pick-up and delivery multi-depot distanceconstrained split multi-trip capacitated vehicle routing problemwith multiple soft time windows with multiple profits and beyond
This talk is not about multi-objective shortest path problemsNicolas Jozefowiez 4 / 53
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
Nicolas Jozefowiez 5 / 53
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
f1
f2
A
C
D
B•
•
•
•
E•
F•
G•
H•
Nicolas Jozefowiez 6 / 53
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
f1
f2
A
C
D
B•
•
•
•
E•
F•
G•
H•
Nicolas Jozefowiez 6 / 53
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
f1
f2
A
C
D
B•
•
•
•
E•
F•
G•
H•
Nicolas Jozefowiez 6 / 53
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
f1
f2
A
C
D
B•
•
•
•
E•
F•
G•
H•
Nicolas Jozefowiez 6 / 53
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
f1
f2
A
C
D
B•
•
•
•
E•
F•
G•
H•
Nicolas Jozefowiez 6 / 53
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
Nicolas Jozefowiez 7 / 53
MOP as a decision tool
Example: Cumulative Capacitated Vehicle Routing Problem
Number of vehicles
Cu
mu
lati
vele
ngt
h
1000
1500
2000
2500
3000
3500
4000
4500
—
—
—
—
—
—
—
—
5 10 15 20 25 30 35 40 45 50| | | | | | | | | |
Nicolas Jozefowiez 8 / 53
Survey
1 Y. Park, C. Koelling, ”A solution of vehicle routing problemsin multiple objective environment”, Engineering Costs andProduction Economics, 10, p. 121–132, 1986.
2 J. Current, M. Marsh, ”Multiobjective transportation networkdesign and routing problems: Taxonomy and annotation”,European Journal of Operational Research, 65, p. 4–19, 1993.(4 references)
3 N. J., F. Semet, E-G. Talbi, ”Multi-objective vehicle routingproblems”, European Journal of Operational Research, 189, p.293–309, 2008. (45 references)
4 N. Labadie, C. Prodhon, ”A survey on multicriteria analysis inlogistics: Focus on vehicle routing problems”, Chapter 1 inApplications of Multi-criteria and Game theory approaches,Series in Advanced Manufacturing, Springer, p. 3–29, 2014.(30 references)
Nicolas Jozefowiez 9 / 53
Survey
1 Y. Park, C. Koelling, ”A solution of vehicle routing problemsin multiple objective environment”, Engineering Costs andProduction Economics, 10, p. 121–132, 1986.
2 J. Current, M. Marsh, ”Multiobjective transportation networkdesign and routing problems: Taxonomy and annotation”,European Journal of Operational Research, 65, p. 4–19, 1993.(4 references)
3 N. J., F. Semet, E-G. Talbi, ”Multi-objective vehicle routingproblems”, European Journal of Operational Research, 189, p.293–309, 2008. (45 references)
4 N. Labadie, C. Prodhon, ”A survey on multicriteria analysis inlogistics: Focus on vehicle routing problems”, Chapter 1 inApplications of Multi-criteria and Game theory approaches,Series in Advanced Manufacturing, Springer, p. 3–29, 2014.(30 references)
Nicolas Jozefowiez 9 / 53
Survey
1 Y. Park, C. Koelling, ”A solution of vehicle routing problemsin multiple objective environment”, Engineering Costs andProduction Economics, 10, p. 121–132, 1986.
2 J. Current, M. Marsh, ”Multiobjective transportation networkdesign and routing problems: Taxonomy and annotation”,European Journal of Operational Research, 65, p. 4–19, 1993.(4 references)
3 N. J., F. Semet, E-G. Talbi, ”Multi-objective vehicle routingproblems”, European Journal of Operational Research, 189, p.293–309, 2008. (45 references)
4 N. Labadie, C. Prodhon, ”A survey on multicriteria analysis inlogistics: Focus on vehicle routing problems”, Chapter 1 inApplications of Multi-criteria and Game theory approaches,Series in Advanced Manufacturing, Springer, p. 3–29, 2014.(30 references)
Nicolas Jozefowiez 9 / 53
Survey
1 Y. Park, C. Koelling, ”A solution of vehicle routing problemsin multiple objective environment”, Engineering Costs andProduction Economics, 10, p. 121–132, 1986.
2 J. Current, M. Marsh, ”Multiobjective transportation networkdesign and routing problems: Taxonomy and annotation”,European Journal of Operational Research, 65, p. 4–19, 1993.(4 references)
3 N. J., F. Semet, E-G. Talbi, ”Multi-objective vehicle routingproblems”, European Journal of Operational Research, 189, p.293–309, 2008. (45 references)
4 N. Labadie, C. Prodhon, ”A survey on multicriteria analysis inlogistics: Focus on vehicle routing problems”, Chapter 1 inApplications of Multi-criteria and Game theory approaches,Series in Advanced Manufacturing, Springer, p. 3–29, 2014.(30 references)
Nicolas Jozefowiez 9 / 53
Classification by components
Tour related objectivesmin cost, min makespan, min traveling time, min operative cost,max capacity use, min imbalance/max fairness, min risk, maxprofit ...
Node/arc related objectives
min the individual risk, min disutility, min time window violation(penalties), max customer satisfaction, max access ...
Resource related objectivesmin fleet cost, min number of vehicles, min number of labels ...
Nicolas Jozefowiez 10 / 53
Classification by components
Tour related objectivesmin cost, min makespan, min traveling time, min operative cost,max capacity use, min imbalance/max fairness, min risk, maxprofit ...
Node/arc related objectives
min the individual risk, min disutility, min time window violation(penalties), max customer satisfaction, max access ...
Resource related objectivesmin fleet cost, min number of vehicles, min number of labels ...
Nicolas Jozefowiez 10 / 53
Classification by components
Tour related objectivesmin cost, min makespan, min traveling time, min operative cost,max capacity use, min imbalance/max fairness, min risk, maxprofit ...
Node/arc related objectives
min the individual risk, min disutility, min time window violation(penalties), max customer satisfaction, max access ...
Resource related objectivesmin fleet cost, min number of vehicles, min number of labels ...
Nicolas Jozefowiez 10 / 53
Classification by components
Tour related objectivesmin cost, min makespan, min traveling time, min operative cost,max capacity use, min imbalance/max fairness, min risk, maxprofit ...
Node/arc related objectives
min the individual risk, min disutility, min time window violation(penalties), max customer satisfaction, max access ...
Resource related objectivesmin fleet cost, min number of vehicles, min number of labels ...
Nicolas Jozefowiez 10 / 53
Classification by use
• Extension of classic problems• To enhance the practical aspects of the models• Not only cost driven• CVRP, PVRP, VRPTW ...
• Generalization of classic problems• [Boffey, 1995]• To replace constraints and/or parameters by one or several
objective(s)• VRPTW, TPP, CTP ...
• Real-life problems
Nicolas Jozefowiez 11 / 53
Classification by use
• Extension of classic problems• To enhance the practical aspects of the models• Not only cost driven• CVRP, PVRP, VRPTW ...
• Generalization of classic problems• [Boffey, 1995]• To replace constraints and/or parameters by one or several
objective(s)• VRPTW, TPP, CTP ...
• Real-life problems
Nicolas Jozefowiez 11 / 53
Classification by use
• Extension of classic problems• To enhance the practical aspects of the models• Not only cost driven• CVRP, PVRP, VRPTW ...
• Generalization of classic problems• [Boffey, 1995]• To replace constraints and/or parameters by one or several
objective(s)• VRPTW, TPP, CTP ...
• Real-life problems
Nicolas Jozefowiez 11 / 53
Classification by use
• Extension of classic problems• To enhance the practical aspects of the models• Not only cost driven• CVRP, PVRP, VRPTW ...
• Generalization of classic problems• [Boffey, 1995]• To replace constraints and/or parameters by one or several
objective(s)• VRPTW, TPP, CTP ...
• Real-life problems
Nicolas Jozefowiez 11 / 53
Classification by problems
Traveling salesman problemCapacitated vehicle routing problemCovering tour problemOrienteering problemSelective TSPVehicle routing problem with time windowsDynamic vehicle routing problemTraveling purchaser problemCapacitated arc routing problemMulti-depot VRPLocation routing problemReal life...
Nicolas Jozefowiez 12 / 53
Part II
Classification of objectives
Nicolas Jozefowiez 13 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Classification by attributes
[Vidal et al., 2014]
Assignment
• Single tour
• Optional visits
• Multiple tours
• Multipledepots
• Multipleperiods
Sequence
• Pick-up anddelivery
Evaluation
• Single cost
• Multiple costs
• Labels
• Time windows
Each attribute can be linked to one or several basic multi-objectiveproblems or objectives that can be mixed together
In the following, the first objective will be to minimize the length
Nicolas Jozefowiez 14 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 3
1,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 3
1,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 3
1,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 3
1,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 3
1,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 31,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 31,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 31,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Evaluation attribute
1
4
1
3 31
4
1
3 31,2
4,5
1,2
3,1 3,11,2
4,5
1,2
3,1 3,1
4
3 3
4
3 3
1 1
1 1
1
2 3
4
56
1 Single cost: Bi-objective pollution-routing problem [Demir et
al., 2014]
2 Multiple costs: multi-objective TSP, hazardous material
3 Labels: sum of costs, max. # of labels
4 Time windows: min. # of violated TW, total violation,waiting time...
Nicolas Jozefowiez 15 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - optional visits
1
4
1
3 31
4
1
3 3
1 11
2 3
4
56
1
1
3
3
2
1 Lose profit → Traveling salesman problem with profits
2 Pay a price → Covering tour problem, ring star problem
3 All these problems are the same from a bi-objective point ofview
Nicolas Jozefowiez 16 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)
• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)
• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost
• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours
• Minimize imbalance (cost, # of nodes)• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect
• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect• Minimize makespan
• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect• Minimize makespan• Minimize capacity
• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple tours
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)
• Local objectives → optimize one aspect• Minimize makespan• Minimize capacity• Clustering
Nicolas Jozefowiez 17 / 53
Assignment - Multiple depots
1
4
1
1
4
1
1 11 4
2 3
56
• Location routing problem
• Min. the fixed, set-up or variable costs for depots
Nicolas Jozefowiez 18 / 53
Assignment - Multiple depots
1
4
1
1
4
1
1 11 4
2 3
56
• Location routing problem
• Min. the fixed, set-up or variable costs for depots
Nicolas Jozefowiez 18 / 53
Assignment - Multiple depots
1
4
1
1
4
1
1 11 4
2 3
56
• Location routing problem
• Min. the fixed, set-up or variable costs for depots
Nicolas Jozefowiez 18 / 53
Assignment - Multiple periods
1
4
3
1
4
3
2 21
2
5
3
4
1st period 2nd period
• Balance the work load over the periods
• Marketing → a customer should be served by the same driver
Nicolas Jozefowiez 19 / 53
Assignment - Multiple periods
1
4
3
1
4
3
2 21
2
5
3
4
1st period 2nd period
• Balance the work load over the periods
• Marketing → a customer should be served by the same driver
Nicolas Jozefowiez 19 / 53
Assignment - Multiple periods
1
4
3
1
4
3
2 21
2
5
3
4
1st period 2nd period
• Balance the work load over the periods
• Marketing → a customer should be served by the same driver
Nicolas Jozefowiez 19 / 53
Assignment - Multiple periods
1
4
3
1
4
3
2 21
2
5
3
4
1st period 2nd period
• Balance the work load over the periods
• Marketing → a customer should be served by the same driver
Nicolas Jozefowiez 19 / 53
Assignment - Multiple periods
1
4
3
1
4
3
2 21
2
5
3
4
1st period 2nd period
• Balance the work load over the periods
• Marketing → a customer should be served by the same driver
Nicolas Jozefowiez 19 / 53
Sequence attribute
1
4
1
4
3
2 5 21
2 3
45
2+
2−
3+
3−
1 Pick-up and delivery / Backhaul / Dial-a-ride problems
• Min. the delay between pick-up and delivery, tardiness
Nicolas Jozefowiez 20 / 53
Sequence attribute
1
4
1
4
3
2 5 21
2 3
45
2+
2−
3+
3−
1 Pick-up and delivery / Backhaul / Dial-a-ride problems
• Min. the delay between pick-up and delivery, tardiness
Nicolas Jozefowiez 20 / 53
Sequence attribute
1
4
1
4
3
2 5 21
2 3
45
2+
2−
3+
3−
1 Pick-up and delivery / Backhaul / Dial-a-ride problems• Min. the delay between pick-up and delivery, tardiness
Nicolas Jozefowiez 20 / 53
Sequence attribute
1
4
1
4
3
2 5 21
2 3
45
2+
2−
3+
3−
1 Pick-up and delivery / Backhaul / Dial-a-ride problems• Min. the delay between pick-up and delivery, tardiness
Nicolas Jozefowiez 20 / 53
Sequence attribute
1
4
1
4
3
2 5 21
2 3
45
2+
2−
3+
3−
1 Pick-up and delivery / Backhaul / Dial-a-ride problems• Min. the delay between pick-up and delivery, tardiness
Nicolas Jozefowiez 20 / 53
Useless efficient solutions
Vehicle routing problem with route balancingBest length solution
1
2 3
4
56
78
9 10
Best balance solution
1
2 3
4
56
78
9 10
Nicolas Jozefowiez 21 / 53
Useless efficient solutions
Vehicle routing problem with route balancing
Best length solution
1
2 3
4
56
78
9 10
Best balance solution
1
2 3
4
56
78
9 10
Nicolas Jozefowiez 21 / 53
Useless efficient solutions
Vehicle routing problem with route balancingBest length solution
1
2 3
4
56
78
9 10
Best balance solution
1
2 3
4
56
78
9 10
Nicolas Jozefowiez 21 / 53
Useless efficient solutions
Vehicle routing problem with route balancingBest length solution
1
2 3
4
56
78
9 10
Best balance solution
1
2 3
4
56
78
9 10
Nicolas Jozefowiez 21 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windows
Hierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are few
Nicolas Jozefowiez 22 / 53
Be careful of correlation
Vehicle routing problem with time windowsHierarchical objective [Solomon, 84]:
1 min. the # of vehicles
2 then, min. the length
The two objectives can be conflicting
1
4
1
3 31
4
1
3 36
1
41
2 3
4
56
Empirically (on Solomon’s instances), # of potentially efficientsolutions are fewNicolas Jozefowiez 22 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Is it a real multi-objective problem ?
• Vehicle routing problem with soft time windows
• A first bi-objective vision
1 Minimize the routing cost2 Minimize the violation cost
• Who is paying at the end ?
• From the decision-maker point of view, there is no difference.
• A second bi-objective vision
1 Minimize the routing cost2 Maximize the quality of service
• Two conflicting aspects: company (financial) / customer
Nicolas Jozefowiez 23 / 53
Part III
Examples
Nicolas Jozefowiez 24 / 53
Applications
• Freight transportation
• (Urban/rural) school bus routing
• Hazardous waste transportation
• Waste collection
• Humanitarian logisticsN. J., F. Semet, E-G. Talbi, ”The bi-objective covering tour problem”,
Computers & Operations Research, 34, p. 1929–1942, 2007.
• Green logisticsE. Demir, T. Bektas, G. Laporte, ”The bi-objective Pollution-Routing Problem”,
European Journal of Operational Research, 232, p. 464–478, 2014.
Nicolas Jozefowiez 25 / 53
Mobile healthcare facility routing
M. J. Hodgson, G. Laporte, F. Semet, ”A covering tour model for planning mobile
health care facilities in Suhum district, Ghana”, Journal of Regional Science, 38, p.
621–639, 2011.
Nicolas Jozefowiez 26 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , pOutput: a minimal length set of routes on V ′ ⊆ V s.t.
|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , p
Output: a minimal length set of routes on V ′ ⊆ V s.t.|V ′| ≤ p,∀wi ∈W ,∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
The (multi-vehicle) covering tour problem
Input: a valuated graph G = (V ∪W ,E , d), c , pOutput: a minimal length set of routes on V ′ ⊆ V s.t.
|V ′| ≤ p,∀wi ∈W , ∃vj ∈ V : dij ≤ c
depot
V : nodesthat can be visited
W : nodes to cover
c: cover distance
Nicolas Jozefowiez 27 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
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
depot
V : nodesthat can be visited
W : nodes to cover
Nicolas Jozefowiez 28 / 53
The Suhum district case
Exact algorithm MOEANB Time Ratio GD Time
dry season 48 5577 0.96 0.65 108.8rainy season 19 36 1.00 0.00 5.2
20000
40000
60000
80000
100000
120000
140000
0 100000 200000 300000 400000 500000 600000
couv
ertu
re (
pied
)
longueur (pied)
Solution Pareto optimale
80000
90000
100000
110000
120000
130000
140000
150000
160000
0 50000 100000 150000 200000 250000 300000
couv
ertu
re (
pied
)
longueur (pied)
Solution Pareto optimale
Nicolas Jozefowiez 29 / 53
Two solutions
Best cover / worst length
Dry season
routecover
SuhumVillages that cannot be visited
Villages that can be visited
Rainy season
routecover
SuhumVillages that cannot be visited
Villages that can be visited
Nicolas Jozefowiez 30 / 53
Other studies
• P. C. Nolz, K. F. Doerner, W. J. Gutjahr, R. F. Hartl, ”Abi-objective metaheuristic for disaster relief operationplanning”, Advances in Multi-objective Nature InspiredComputing, p. 167–187, 2010.
• P. C. Nolz, F. Semet, K. F. Doerner, ”Risk approaches fordelivering disaster relief supplies”, OR Spectrum, 33, P.543–569, 2011.
• F. Tricoire, A. Graf, W. J. Gutjahr, ”The bi-objectivestochastic covering tour problem”, Computers & OperationsResearch, 39, p. 1582–1592, 2012.
• S. Rath, W. J. Gutjahr, ”A math-heuristic for the warehouselocation-routing problem in disaster relief”, Computers &Operations Research, 42, p. 25–39, 2014.
Nicolas Jozefowiez 31 / 53
Green logistics
• E. Demir, T. Bektas, G. Laporte, ”A comparative analysis ofseveral vehicle emission models for freight transportation”,Transportation Research Part D: Transport and Environment,6, p. 347–357, 2011.
• 6 fuel consumption models
• T. Bektas, G. Laporte, ”The pollution-routing problem”,Transportation Research Part B, 45, p. 1232–1250, 2011.
• VRP combining distance, speed, vehicle load, and driver wages
• E. Demir, T. Bektas, G. Laporte, ”The bi-objectivePollution-routing problem”, European Journal of OperationalResearch, 232, p. 464–478, 2011.
Nicolas Jozefowiez 32 / 53
Bi-objective PRP
• Assignment attributes• Single cost• Fixed-size fleet
• Evaluation attribute• Time windows
• Decision variables• Arcs in the solution• Speed to travel along an arc
• Objectives• Fuel consumption• Driving time
Nicolas Jozefowiez 33 / 53
Two solutions [Demir et al., 2014]
# of Total Fuel Operational CO2 Fuel Driver Totalroutes distance consumption time emissions cost cost cost
km L h kg £ £ £
Solution A 6 1621.7 321.57 21.16 1008.12 450.20 169.28 619.48Solution B 6 1270.1 233.54 23.21 732.15 326.96 185.68 512.64
Solution A Solution B
Nicolas Jozefowiez 34 / 53
Part IV
Methods
Nicolas Jozefowiez 35 / 53
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
Nicolas Jozefowiez 36 / 53
Scalarization methods
Weighted sum method
min (f1(x), . . . , fn(x))
x ∈ Ω→
min
∑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)
Nicolas Jozefowiez 37 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
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
f2
•
•
•
•
•
•
•
••
•
Nicolas Jozefowiez 38 / 53
Ranking
f1
f2
A•
B•
C•D•
E•F•
G•
H•
1
1
1
1
2
2
2
3
Nicolas Jozefowiez 39 / 53
Ranking
f1
f2
A•
B•
C•D•
E•F•
G•
H•
1
1
1
1
2
2
2
3
Nicolas Jozefowiez 39 / 53
Ranking
f1
f2
A•
B•
C•D•
E•F•
G•
H•
1
1
1
1
2
2
2
3
Nicolas Jozefowiez 39 / 53
Ranking
f1
f2
A•
B•
C•D•
E•F•
G•
H•
1
1
1
1
2
2
2
3
Nicolas Jozefowiez 39 / 53
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 40 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Set-based optimization [Zitzler et al., 2010]
pop
ula
tion
Standard approach
f1
f2
•
•
••
•
Set-based approach
f1
f2
••
•
•••
•
••••
•
•• •
•
• How to manipulate and define operators ?
• Proto-solution
• Multi-objective decoder: a proto-solution → several solutions
Nicolas Jozefowiez 41 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
Vehicle routing problems
Proto-solution
• A giant tour (TSP solution)
• Example: CVRP → ignore the capacity constraint
SPLIT operator [Prins, 2004]
20
10
30 25
15
35
25
30
40
:40 :50 :80 :50
:85
:120
:95:55
:60
: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 42 / 53
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)
Nicolas Jozefowiez 43 / 53
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 44 / 53
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
Nicolas Jozefowiez 45 / 53
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
Nicolas Jozefowiez 45 / 53
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
Nicolas Jozefowiez 46 / 53
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
Nicolas Jozefowiez 46 / 53
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
Nicolas Jozefowiez 46 / 53
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
Nicolas Jozefowiez 46 / 53
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
Nicolas Jozefowiez 46 / 53
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
Nicolas Jozefowiez 46 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 9
Nicolas Jozefowiez 47 / 53
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
Number of LP solutions: 7
Nicolas Jozefowiez 48 / 53
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
Number of LP solutions: 7
Nicolas Jozefowiez 48 / 53
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
Number of LP solutions: 7
Nicolas Jozefowiez 48 / 53
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
Number of LP solutions: 7
Nicolas Jozefowiez 48 / 53
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
Number of LP solutions: 7
Nicolas Jozefowiez 48 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 49 / 53
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
Nicolas Jozefowiez 50 / 53
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
Nicolas Jozefowiez 51 / 53
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
Nicolas Jozefowiez 51 / 53
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
Nicolas Jozefowiez 51 / 53
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
Nicolas Jozefowiez 51 / 53
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
Nicolas Jozefowiez 51 / 53
Part V
Conclusions
Nicolas Jozefowiez 52 / 53
Conclusions
• A need for a better qualification• Not precise, e.g., multi-objective vehicle routing problem• Needed to spread the research• Unified the field
• Standard MOVRP• Define relevant objectives• Define relevant combinations• Define benchmark
• Multi-objective methods• Generic methods or mechanisms• Specific methods or mechanisms for MOVRP• Metaheuristics, branch-and-X algorithms• Matheuristics
Nicolas Jozefowiez 53 / 53
Conclusions
• A need for a better qualification• Not precise, e.g., multi-objective vehicle routing problem• Needed to spread the research• Unified the field
• Standard MOVRP• Define relevant objectives• Define relevant combinations• Define benchmark
• Multi-objective methods• Generic methods or mechanisms• Specific methods or mechanisms for MOVRP• Metaheuristics, branch-and-X algorithms• Matheuristics
Nicolas Jozefowiez 53 / 53
Conclusions
• A need for a better qualification• Not precise, e.g., multi-objective vehicle routing problem• Needed to spread the research• Unified the field
• Standard MOVRP• Define relevant objectives• Define relevant combinations• Define benchmark
• Multi-objective methods• Generic methods or mechanisms• Specific methods or mechanisms for MOVRP• Metaheuristics, branch-and-X algorithms• Matheuristics
Nicolas Jozefowiez 53 / 53
Conclusions
• A need for a better qualification• Not precise, e.g., multi-objective vehicle routing problem• Needed to spread the research• Unified the field
• Standard MOVRP• Define relevant objectives• Define relevant combinations• Define benchmark
• Multi-objective methods• Generic methods or mechanisms• Specific methods or mechanisms for MOVRP• Metaheuristics, branch-and-X algorithms• Matheuristics
Nicolas Jozefowiez 53 / 53