multi-objective optimization and vehicle routing problems

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

Vehicle routing problems

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

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

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Efficient/Pareto-optimal solutionEfficient/Pareto-optimal set

Non-dominated pointNon-dominated set

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Pareto dominance

x y ⇔

fi (x) ≤ fi (y) ∀i ∈ [1, . . . , n]

fi (x) < fi (y) ∃i ∈ [1, . . . , n]

Usefulness

Can every problem be limited to a single objective ? No

Example: fairness between drivers in the CVRP

Taburoute Prins’ GA

Instance Distance Fairness Distance Fairness

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

MOP as a decision tool

Example: Cumulative Capacitated Vehicle Routing Problem

Number of vehicles

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

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)

Survey

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 ...

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

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...

Part II

Classification of objectives

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

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Assignment

• Single tour

• Optional visits

• Multiple tours

• Multipledepots

• Multipleperiods

Sequence

Evaluation

• Single cost

• Multiple costs

• Labels

• Time windows

Evaluation attribute

3,1 3,11,2

3,1 3,1

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...

3,1 3,11,2

3,1 3,1

al., 2014]

3,1 3,11,2

3,1 3,1

al., 2014]

3,1 3,11,2

3,1 3,1

al., 2014]

3,1 3,11,2

3,1 3,1

al., 2014]

3 31,2

3,1 3,11,2

3,1 3,1

al., 2014]

3 31,2

3,1 3,11,2

3,1 3,1

al., 2014]

3 31,2

3,1 3,11,2

3,1 3,1

al., 2014]

3 31,2

3,1 3,11,2

3,1 3,1

al., 2014]

Assignment - optional visits

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

Assignment - Multiple tours

• 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

• Global objectives (ex: Q = 3)

• Minimize total cost• Minimize number of tours• Minimize imbalance (cost, # of nodes)

• Global objectives (ex: Q = 3)• Minimize total cost

• Minimize number of tours• Minimize imbalance (cost, # of nodes)

• Global objectives (ex: Q = 3)• Minimize total cost• Minimize number of tours

• Minimize imbalance (cost, # of nodes)• Local objectives → optimize one aspect

• 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

• Local objectives → optimize one aspect• Minimize makespan• Minimize capacity

• Clustering

• Local objectives → optimize one aspect• Minimize makespan• Minimize capacity• Clustering

Assignment - Multiple depots

1 11 4

• Location routing problem

• Min. the fixed, set-up or variable costs for depots

1 11 4

Assignment - Multiple periods

1st period 2nd period

• Balance the work load over the periods

• Marketing → a customer should be served by the same driver

Sequence attribute

2 5 21

1 Pick-up and delivery / Backhaul / Dial-a-ride problems

• Min. the delay between pick-up and delivery, tardiness

Sequence attribute

2 5 21

1 Pick-up and delivery / Backhaul / Dial-a-ride problems

• Min. the delay between pick-up and delivery, tardiness

Sequence attribute

2 5 21

1 Pick-up and delivery / Backhaul / Dial-a-ride problems• Min. the delay between pick-up and delivery, tardiness

Sequence attribute

2 5 21

Sequence attribute

2 5 21

Useless efficient solutions

Vehicle routing problem with route balancingBest length solution

Best balance solution

Vehicle routing problem with route balancing

Best length solution

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

Empirically (on Solomon’s instances), # of potentially efficientsolutions are few

Vehicle routing problem with time windows

Hierarchical objective [Solomon, 84]:

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

Part III

Examples

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.

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.

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

V : nodesthat can be visited

W : nodes to cover

c: cover distance

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

W : nodes to cover

c: cover distance

W : nodes to cover

c: cover distance

W : nodes to cover

c: cover distance

W : nodes to cover

c: cover distance

W : nodes to cover

c: cover distance

W : nodes to cover

c: cover distance

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

W : nodes to cover

c: cover distance

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

W : nodes to cover

G = (V ∪W ,E , d)

, p: max # of nodes in a tour

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

G = (V ∪W ,E , d)

W : nodes to cover

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

+ assignment of W to V ′

Objectives: i) minimize the total length

; ii) maxwi∈W minvj∈V ′ dij

W : nodes to cover

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

100000

120000

140000

0 100000 200000 300000 400000 500000 600000

longueur (pied)

Solution Pareto optimale

100000

110000

120000

130000

140000

150000

160000

0 50000 100000 150000 200000 250000 300000

longueur (pied)

Solution Pareto optimale

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

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.

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.

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

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

Part IV

Methods

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

Scalarization methods

Weighted sum method

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

x ∈ Ω→

∑ni=1 λi fi (x)

x ∈ Ω

n∑i=1

λi = 1

ε-constraint method

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

x ∈ Ω→

min fk(x)

x ∈ Ω

fi (x) ≤ εi (i ∈ [1, n], i 6= k)

Two-phase method [Ulungu & Teghem, 1993]

Phase 1

• Dichotomic search

• Weighted sum objective

• Only the convex hull

• Supported solutions

Phase 2

• Enumerative search

• Bounded by phase 1solutions

• Not supported solutions f1

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Phase 1

Phase 2

••

Ranking

C•D•

E•F•

Ranking

C•D•

E•F•

Ranking

C•D•

E•F•

Ranking

C•D•

E•F•

Multi-objective meta-heuristics

Main focus of research on

• Selection

• Mechanisms for diversification

• Mechanisms for intensification

Less focus on

• Operators (crossover), neighborhood

• Encoding

• Usually inspired by a close single objective problem

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

Standard approach

••

Set-based approach

••

•••

••••

•• •

• How to manipulate and define operators ?

• Proto-solution

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

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Standard approach

••

Set-based approach

••

•••

••••

•• •

• Proto-solution

Proto-solution

• A giant tour (TSP solution)

• Example: CVRP → ignore the capacity constraint

SPLIT operator [Prins, 2004]

:40 :50 :80 :50

:95:55

Decoder

• Multi-objective Shortest Path Prob. with Resource Constraints

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

• Minimal modification: Label, dominance, extension rules

• Indicator-based evaluation

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Proto-solution

:40 :50 :80 :50

:95:55

Decoder

Upper and lower bounds

Upper bound (ub)

x ∈ Ω : @y ∈ ub, y x ⊆ Ω

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

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

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

Computation of the lower bound

• A single multi-objective integer program

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

technique

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

• Φ should be kept polynomial or pseudo-polynomial

• Branch-and-cut flowchart is not modified

Example

Φ = φε, ε ∈ 0, 1, 2

minimize −1.00x1 − 0.64x2

minimize x3

s.t. 50x1 + 31x2 ≤ 250

3x1 − 2x2 ≥ −4

x1 + x3 ≤ 2

x1, x2 ≥ 0 and integer

x3 ∈ 0, 1, 2

Example

Φ = φε, ε ∈ 0, 1, 2

minimize −1.00x1 − 0.64x2

s.t. 50x1 + 31x2 ≤ 250

3x1 − 2x2 ≥ −4

x1 + x3 ≤ 2

x3 = ε

x1, x2 ≥ 0

Search tree

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

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

ε = 0 x1 = 1.94 x2 = 4.92UnfeasibleUnfeasible

ε = 0 x1 = 2 x2 = 4UnfeasibleUnfeasible

UnfeasibleUnfeasibleUnfeasible

Number of LP solutions: 15

Search tree

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

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

Search tree

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

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

Search tree

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

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

Search tree

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

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

Search tree

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

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

Partial pruning

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

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

Not solved

ε = 0 x1 = 1.94 x2 = 4.92UnfeasibleNot solved

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

UnfeasibleNot solvedNot solved

Partial pruning

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

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

Not solved

Partial pruning

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

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

Not solved

Partial pruning

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

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

Not solved

Partial pruning

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

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

Not solved

Partial pruning

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

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

Not solved

Parallel branching

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

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

Not solved

UnfeasibleUnfeasibleNot solved

Parallel branching

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

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

Not solved

Parallel branching

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

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

Not solved

Parallel branching

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

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

Not solved

Parallel branching

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

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

Not solved

The multilabel traveling salesman problem

G = (V ,E )

Cost function c on E

A set of labels L = , , ,

Each e ∈ E ← δe ∈ L (data)

Minimize the total length

Minimize the number of labels used

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

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

Cuts are searched after each LP solution

G = (V ,E )

Computational results (I)

Comparison with an iterative ε-constraint method

Same underlying branch-and-cut algorithm

MOB&C εCM

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

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

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

Computational results (II)

Use of the method as a heuristic

Stop after a percentage of the search tree has been explored

%: percentage of Pareto solutions found

Gap: average over all non efficient solutions of

25% 50% 75%

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

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

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

25% 50% 75%

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

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

25% 50% 75%

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

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

25% 50% 75%

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

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

25% 50% 75%

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

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

Part V

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

Conclusions

multi-objective optimization and vehicle routing problems

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