column generation for bi-objective integer linear programs ... · based on the principles of lp...

Column Generation for Bi-ObjectiveInteger Linear Programs: Application toBi-Objective Vehicle Routing Problems

Boadu Mensah SARPONG

Directed by :Christian ARTIGUES

Nicolas JOZEFOWIEZ

03 / 12 / 2013

Introduction and Literature ReviewColumn Generation for BOILPs

Application to the BOMCTPComputational Results

Conclusions and Perspectives

Outline

1 Introduction and Literature Review

2 Column Generation for BOILPs

3 Application to the BOMCTP

4 Computational Results

5 Conclusions and Perspectives

Column Generation for BOILPs: Application to BOVRPs 1 / 54




Column GenerationMulti-Objective Integer ProgramsSummary of Contributions

Definitions and Principles of Column Generation

What is column generation ?A method for solving linear programs (LPs) in which there is an exponentialnumber of variables without having to enumerate all the variables a priori

Based on the principles of LP decomposition

Useful in computing dual bounds for integer LPs

Useful as a heuristic in finding feasible solutions of Integer LPs

Where has it been applied ?

Vehicle routing problems

Multi-commodity flow problems

Cutting stock problems

Binary cutting stock problems

Crew rostering

etc






Some Definitions

IP Master Problem (IPM)The original IP having an exponential number of variables (corresponding to thecolumns of the constraint matrix)

LP Master Problem (LPM)The linear relaxation of IPM

Restricted LP Master Problem (RLPM)A copy of LPM in which only a subset of the original columns are present

Sub-problemA problem solved to determine which columns of the constraint matrix of LPM tointroduce into an RLPM






Main Idea and Convergence of Column Generation

RLPM

exponential

LPM

interactions

DRLPM

expo

nent

ial

DLPM

Notes

DLPM : Dual of LPM

LPM has an exponential number of variables (columns)

DLPM has an exponential number of constraints







RLPM

exponential

LPM

interactions DRLPM

expo

nent

ial

DLPM

Notes

The feasible space of RLPM is a subset of the feasible space of LPM

The feasible space of DLPM is a subset of the feasible space of DRLPM

An optimal solution for DRLPM may not be feasible for DLPM







RLPM

exponential

LPM

interactions DRLPM

expo

nent

ial

DLPM

Subproblem(s)

Is the current objective value of RLPM optimal for LPM ?

Are any constraints of DLPM violated in DRLPM (is DRLPM feasible) ?

Depends on the particular problem and the formulation used







RLPM

exponential

LPM

interactions DRLPM

expo

nent

ial

DLPM

Notes

Adding a column to RLPM corresponds to adding a constraint to DRLPM

Feasible space of RLPM enlarges and approaches that of LPM

Feasible space of DRLPM shrinks and approaches that of DLPM






Flow Chart: A Column Generation Algorithm

Start

Choose initialcolumns and

formulate RLPM

Solve RLPMto optimality

Solvesubproblem(s)

Any newcolumn(s)?

Add column(s)to RLPM

Stopyes no






Definition of a Multi-Objective Integer Program

(MOIP) =

min F (x) = (f1(x), f2(x), . . . , fr (x))

s.t. x ∈ X

r ≥ 2 : number of objective functions

F = (f1, f2, . . . , fr ) : vector of objective functions

X ⊆ Nn : feasible set of solutions

Y = F (X ) : feasible set in objective space

x = (x1, x2, . . . , xn) ∈ X : variable vector, variables

y = (y1, y2, . . . , yr ) ∈ Y with yi = fi (x) : vector of objective functionvalues






Pareto Dominance and Optimality

A solution x1 dominates another solution x2 if ∀i ∈ 1, . . . , n,fi (x1) ≤ fi (x2) and ∃i ∈ 1, . . . , n such that fi (x1) < fi (x2)

AB

C

D

E

f1

f2Pareto Optimal Solution : Asolution dominated by no otherfeasible solution

Nondominated Point : The imageof a Pareto optimal solution in theobjective space






Scalarization of a BOILP

Original Problem : P

Minimize (c1)T xMinimize (c2)T x

Ax ≥ bx ≥ 0 and integer

Weighted Sum : P(λ)

Given λ = (λ1, λ2) with λ1, λ2 ≥ 0and λ1 + λ2 = 1

Minimize λ1(c1)T x + λ2(c2)T xAx ≥ b

x ≥ 0 and integer

ε-Constraint : P(ε)

Given ε ∈ R

Minimize (c1)T xAx ≥ b

−(c2)T x ≥ −εx ≥ 0 and integer






Lower and Upper Bounds of a MOIP

ideal

nadir

image of feasible solution

estimate of nondominated regionmember of lower bound

f2

f1

Lower Bound [Villarreal and Karwan, 1981]A set of points (feasible or not) such that the image of every feasible solution isdominated by at least one of the points

Upper BoundA set of mutually nondominated feasible points







ideal

nadir image of feasible solution

estimate of nondominated region

member of lower bound

f2

f1









ideal

nadir image of feasible solution

estimate of nondominated regionmember of lower bound

f2

f1








Constructing Bound Sets [Ehrgott and Gandibleux, 2007]

Idea Used

Apply a weighted sum method

Find convex hull of P(λ) if P(λ)is “easy”

Otherwise, find convex hull of arelaxation of P(λ)

Scalarized Problem : P(λ)

Given λ = (λ1, λ2) with λ1, λ2 ≥ 0and λ1 + λ2 = 1


x ≥ 0 and integer

Nondominated point of P

Convex hull lower bound set of P

Nondominated point of PL

Convex hull lower bound set of PL

f1

f2






Column Generation in Multi-Objective Optimization

Some References

Ehrgott and Tind. Column Generation in Integer Programming withApplications in Multicriteria Optimization. Technical Report of the Facultyof Engineering, University of Auckland, New Zealand, 2007

Khanafer et al. The Min-Conflict Packing Problem. Computers andOperations Research 39, 2012

Peng et al. A new column generation based algorithm for VMAT treatmentplan optimization. Physics in Medicine and Biology, 57(14), 2012

Salari and Unkelbach. A Column-Generation-Based Method forMulti-Criteria Direct Aperture Optimization. Physics in Medicine andBiology, 58, 2013






Thesis Objectives and Contributions

Main Objectives

Design of efficient column generation methods in computing lower boundsof BOILPs (through the study of a bi-objective VRP)

Propose speed-up techniques

Main Contributions

The use of a different scalarization method (ε-constraint) in constructingbound sets

A generalized column generation algorithm for BOILPsBased on either a weighted sum method or an ε-constraint method

Different strategies for implementing the generalized algorithm

Application to a practical problem (the BOMCTP)





Column Search StrategiesColumn Generation for BOIPMMO

Using a Weighted Sum Method or an ε-Constraint Method

LPM(λ)Given λ = (λ1, λ2) with λ1, λ2 ≥ 0and λ1 + λ2 = 1


x ≥ 0

LPM(ε)Given ε ∈ R

Minimize (c1)T xAx ≥ b

−(c2)T x ≥ −εx ≥ 0

Dual of LPM(λ)

Maximize bTπ

ATπ ≤ λ1c1 + λ2c2

π ≥ 0

Dual of LPM(ε)

Maximize bTπ − εϕATπ ≤ c1 + ϕc2

π, ϕ ≥ 0






Similar Subproblem Structure

Subproblem : S(λ)

Find a variable corresponding to acolumn of matrix A and which satisfyan inequality of the form :

λ1c1 + λ2c2 − ATπ < 0

Subproblem : S(ε)

Find a variable corresponding to acolumn of matrix A and which satisfyan inequality of the form :

c1 + ϕc2 − ATπ < 0

Implications and Consequences

Subproblems have similar structure for both scalarization methods

Strategies described for one scalarization method can be adopted for theother

For any of the methods, it is possible to treat more than one subproblem atthe same time when searching for columns






A Generalized Column Generation Algorithm for BOILPs

StartChoose scalarizationmethod and convert

problem

Formulate RLPMfor the chosen

scalarization method

Solve RLPM fordifferent values

of the parameter

Solve subproblem(s)corresponding to

one or more points

Add column(s)to RLPM

Any newcolumn(s)?Stop

yes

no






Column Search Strategies : Point-by-Point Search (PPS)

For any given value of ε, completely solve LPM(ε) by column generation

max ε

min ε

ε1

ε2

ε3

ε4

ε5

f2

f1






Column Search Strategies : k-Step PPS

Temporary skip a value of ε if the objective value of RLPM fails toimprove significantly after k column generation iterations

max ε

min ε

ε1

ε2ε3

ε4

f2

f1








max ε

min ε

ε1

ε2

ε3

ε4

f2

f1








max ε

min ε

ε1

ε2ε3

ε4

f2

f1






Column Search Strategies : Improved PPS (IPPS)

IdeaAt each column generation iteration in PPS, use heuristics (problem dependent)to search for other columns that are relevant for current value of ε and may alsobe relevant for other values

Advantages

Can cheaply find a large number of columns once the price of a few columnshave been paid

Tries to take advantage of the reformulation and the similar subproblemsassociated to the different values of ε

Disadvantages

No guarantee that a column found by a heuristic will be relevant for othervalues of ε apart from the current one






Column Search Strategies : Sequential Search

At each iteration, generate a set of points corresponding to different valuesof ε and then search for a set of columns that are relevant for the points

max ε

min ε

ε1

ε2

ε3

ε1

ε2

ε3

ε4

f2

f1






Column Search Strategies : Solve-Once-Generate-for-All (SOGA)

Solve subproblem corresponding to one point. Combine information fromthe found columns and dual values from the other generated points tosearch for more columns.

max ε

min ε

ε1

ε2

ε3

ε1

ε2

ε3

ε4

f2

f1






BOIP with a Min-Max Objective

Minimize∑

k∈Ω ckθk

Minimize Γmax∑k∈Ω aikθk ≥ bi (i ∈ I)

Γmax ≥ σkθk (k ∈ Ω)

θk ∈ 0, 1 (k ∈ Ω)

Assumptions

There is a finite number of possible values for σk in a range [σmin, σmax]

Example: σk is an integer






Standard ε-Constraint Approach for a BOIPMMO

Minimize∑

k∈Ω ckθk

−Γmax ≥ −ε∑k∈Ω aikθk ≥ bi (i ∈ I)


θk ∈ 0, 1 (k ∈ Ω)

Advantage

Each column is valid for all values of ε and so the subproblem algorithmdoes not need to keep track of this parameter

Disadvantage

Usually need to introduce extra variables and this can negatively affect thequality of a lower bound set






Reformulation of a BOIPMMO

Minimize∑

k∈Ω ckθk

Minimize Γmax∑k∈Ω aikθk ≥ bi (i ∈ I)


θk ∈ 0, 1 (k ∈ Ω)

Idea Used

Ω is extended into a new set of columns Ω

The feasibility of a column k ∈ Ω depends on σk

To obtain a solution satisfying Γmax ≤ ε, “remove” all columns k ∈ Ωhaving σk > ε from the model






Reformulation of a BOIPMMO (IP Master Problem)

Minimize∑

k∈Ω ckθk∑k∈Ω aikθk ≥ bi (i ∈ I)

θk ∈ 0, 1 (k ∈ Ω)

Advantages

Both the master and the subproblem are single objective problems

Possible to use known methods to solve LPM for any fixed value of ε

Disadvantage

Need to manage the parameter ε in both the master problem and thesubproblem





Problem DescriptionFormulation 1 (A “Standard” Formulation)Formulation 2 (Reformulation as a BOIPMMO)

The Multi-Vehicle Covering Tour Problem [Hachicha et al., 2000]

Find a set of routes on V ′ ⊆ V with minimum total length and such thatthe nodes of W are covered by those of V ′

The number of nodes on each route cannot exceed pThe length of each route cannot exceed q

v0

VMay be visited

Must be visited : T

Must be covered : W

Vehicle route

Cover distance (predetermined)






Some Applications of the MTCP

Design of bi-level transportation networks [Current and Schilling,1994]

Construct a primary route such that all points that are not on it can easilyreach it

Location of post boxes [Labbe and Laporte, 1986]

Minimize the cost of a collection route through all post boxes and alsoensure that every user is located within a reasonable distance from a postbox

Humanitarian logistics

Planning of routes to be used by visiting health care teams in developingcountries [Current and Schilling, 1994; Hodgson et al., 1998]






The Bi-Objective MTCP (BOMCTP)

Problem DescriptionGiven a graph G = (V ∪W ,E ), design a set of routes on V ′ ⊆ V . D = (dij) is adistance matrix satisfying the triangle inequality

Objectives

Minimize the total length of the set of routes

Minimize the cover distance induced by the set of routes

Constraints

Each route must start from the depot and also end at the depot

The number of nodes on each route should be at most p

The length of each route should be at most q






The Cover Distance Induced by a Set of Routes

Given a set of routes

Assign each node of W to the closest visted node of V \v0

Take maximum of the assigned distances

v0

VMay be visited

Must be visited : T

Must be covered : W

Vehicle route

Cover distance (induced)






The Cover Distance Induced by a Set of Routes

Given a set of routesAssign each node of W to the closest visted node of V \v0

Take maximum of the assigned distances

v0

VMay be visited

Must be visited : T

Must be covered : W

Vehicle route

Cover distance (induced)






Formulation 1 (A “Standard” Formulation)

Variables Used

Ω : Set of all feasible columns

θk : 1 if column k is selected in solution, 0 otherwise

zij : 1 if vi ∈ V \v0 is used to cover wj ∈W , otherwise

aik : 1 if vi ∈ V is used by column k, 0 otherwise

Γmax : Induced cover distance






Formulation 1 : IP Master Problem and RLPM

Minimize∑

k∈Ω ckθk

Minimize Γmax

Γmax − dijzij ≥ 0 (vi ∈ V \v0,wj ∈W )∑vi∈V\v0 zij ≥ 1 (wj ∈W )∑

k∈Ω aikθk − zij ≥ 0 (vi ∈ V \v0,wj ∈W )∑k∈Ω aikθk ≥ 1 (vi ∈ T\v0)

zij ∈ 0, 1 (vi ∈ V \v0,wj ∈W )

θk ∈ 0, 1 (k ∈ Ω)

RLPM

Relax the integrality constraints in IPM

Restrict Ω to a subset Ω1






Formulation 1 : Dual of LP Master Problem

Maximize∑

wj∈W βj +∑

vi∈T\v0 πi − ελ

subject to :∑vi∈V\v0

wj∈Waikαij +

∑vi∈T\v0 aikπi ≤ ck (k ∈ Ω)

βj − dijγij − αij ≤ 0 (vi ∈ V \v0,wj ∈W )∑vi∈V\v0

wj∈Wγij − λ ≤ 0

SubproblemFind feasible routes k ∈ Ω\Ω1 such that

ck −∑

vi∈V\v0wj∈W

aikαij −∑

vi∈T\v0 aikπi < 0






Formulation 1 : Subproblem

SubproblemAn elementary shortest path problem with resource constraints (ESPPRC)

Minimize∑

(vi ,vj )∈E

(dij − π∗i −

∑wh∈W α∗ih

)xijk

subject to : k ∈ Ω\Ω1

π∗i = πi if vi ∈ T\v0, 0 otherwise

α∗ij = αij if vi ∈ V \v0 and wj ∈W , 0 otherwise

xijk = 1 if route k visits vj immediately after visiting vi , 0 otherwise

Solving the Subproblem (By Dymanic Programming)

Use of the DSSR Algorithm [Boland et al. (2006), Righini and Salani(2008)]






Formulation 1 : Subproblem Algorithm

Definition of a Label: Λ1 = (rc1, p1, q1)

rc1 : Reduced cost up to current node

p1 : Total number of nodes of V \v0 visited up to current node

q1 : Length of route up to current node

Extending a Label Λ1 = (rc1, p1, q1) on vi to vj

We obtain Λ2 = (rc2, p2, q2) where

rc2 = rc1 + reduced cost of arc (vi , vj)

p2 = p1 + 1 and q2 = q1 + dij

Dominance Rule (the “usual” one)

Λ1 dominates Λ2 if Λ1 ≤ Λ2






Formulation 2 (Reformulation as a BOIPMMO)

Variables Used

A column k ∈ Ω is defined as a route Rk together with a subset Ψk ⊆ W ofnodes it may cover

σk : Minimum distance required by Rk\v0 to cover all nodes of Ψk

aik : 1 if vi ∈ Rk , and 0 otherwise

bjk : 1 if wj ∈ Ψk , and 0 otherwise

Γmax : The cover distance induced by a set of routes






Formulation 2 : Reformulation of IPM and Dual of LPM

Reformulated IPM

Minimize∑

k∈Ω ckθk∑k∈Ω aikθk ≥ 1 (vi ∈ T\v0)∑k∈Ω bjkθk ≥ 1 (wj ∈W )

θk ∈ 0, 1 (k ∈ Ω)

Feasibility of a column k ∈ Ω depends on the value of σk

Dual of LPMGiven πi ≥ 0 for vi ∈ T\v0 and βj ≥ 0 for wj ∈W

Maximize∑

vi∈T\v0 πi +∑

wj∈W βj

subject to :∑

vi∈T\v0 aikπi +∑

wj∈W bikβj ≤ ck (k ∈ Ω)






Formulation 2 : Subproblem corresponding to ε

Subproblem : S(ε)

Minimize ck −∑

vi∈T\v0 πi aik −∑

wj∈W βjajk

subject to : k ∈ Ω\Ω1

σk ≤ ε

Notes

S(ε) is associated with dual vectors π and β, and the value of ε for whichthey were computed

Ψk may only contain nodes of the set wj ∈W : ∃vi ∈ Rk with dij ≤ ε

A non-additive elementary shortest path problem with resource constraints

Solved by the DSSR algorithm but need to modify dominance rule betweenlabels






Formulation 2 : Subproblem Algorithm

Extending a Label Λi = (rc1, p1, q1) on vi to vj

Identify any new nodes of W that can be covered by vj and substract their totalprofit (β∗) from the current reduced costWe obtain Λ2 = (rc2, p2, q2) where

rc2 = rc1 + reduced cost of arc (vi , vj)− β∗

p2 = p1 + 1 and q2 = q1 + dij

Dominance Rule (Modified for Non-Additivity)

Λ1 dominates Λ2 if Λ1 5 Λ2 and rc1 ≤ rc2 − F12

F12 represents the sum of the profits associated to nodes of W that arecovered by Λ1 but not yet covered by Λ2

Similar rules proposed by Reinhardt and Pisinger (2011)






IPPS Heuristic for the BOMCTP

Principle

Successively remove a node of W that induces the value of σ

Same vector of dual values used throughout the process

v0

v1

v2

v3

σ

σ

σσ






IPPS Heuristic for the BOMCTP

Principle

Successively remove a node of W that induces the value of σ

Same vector of dual values used throughout the process

v0

v1

v2

v3

σσ

σ

σ






SOGA Heuristic for the BOMCTP

Principle

Reconstruct the set of nodes to cover (Ψk ⊆W )

A different vector of dual values is used for each modification

v0

v1

v2

v3

σ

σ

σ






SOGA Heuristic for the BOMCTP

Principle

Reconstruct the set of nodes to cover (Ψk ⊆W )

A different vector of dual values is used for each modification

v0

v1

v2

v3

σ σ

σ





Weighted Sum Versus ε-ConstraintResults for the BOMCTP : Formulation 1Results for the BOMCTP : Formulation 2

A Comparison of the Weighted Sum and ε-Constraint Methods

The Bi-Objective Set Covering Problem (BOSCP)

Minimize∑n

j=1 c1j xj

Minimize∑n

j=1 c2j xj∑n

j=1 aijxj ≥ 1 i = 1, . . . ,mxj ∈ 0, 1 j = 1, . . . ,m

Notes

No need for column generation on considered instances

Lower bound sets by weighted sum method and ε-constraint methods

Exact nondominated set (obtained from a method by Berube et al., 2009)used as upper bound set

Solved with CPLEX 12.4 on an intel Core 2 Duo, 2.93 GHz, 2 GB RAM






Evaluation of Bound Sets [Ehrgott and Gandibleux (2007)]

d(L,U)

ymax

yminf1

f2

d(L,U)

ymax

ymin

L

U

AL

AU

f1

f2

µ1 :=d(L,U)

‖ymax − ymin‖2µ2 :=

AL − AUAL






Comparison of Lower Bound Sets for the BOSCP

Weighted Sum ε-Constraint

Instance |U∗| time |ext| µ1% µ2% time |L| µ1% µ2%

2scp11A 39 0.00 13 2.9 2.5 0.02 192 2.7 2.12scp11C 10 0.00 21 13.7 34.1 0.01 117 13.9 32.72scp41A 105 0.01 23 1.5 1.3 0.14 1028 1.4 1.22scp41C 24 0.02 17 2.6 4.5 0.04 280 2.7 5.72scp42A 206 0.02 37 0.6 0.5 0.45 1663 0.5 0.42scp42C 87 0.04 51 3.0 3.4 0.44 2245 2.9 3.42scp43A 46 0.04 63 3.6 4.3 0.12 473 3.5 4.12scp43C 12 0.02 38 6.3 11.1 0.04 225 6.3 10.62scp61A 254 0.06 58 1.1 0.8 1.39 2829 1.1 0.82scp61C 28 0.02 17 1.0 3.7 1.02 732 0.9 4.42scp62A 98 0.28 236 3.5 3.6 1.01 1240 3.5 3.72scp62C 6 0.23 180 38.9 46.6 0.00 2 36.6 58.52scp81A 424 0.07 47 0.5 0.4 3.49 5580 0.4 0.32scp81C 14 0.12 77 0.3 0.5 0.60 147 0.4 0.7






Experiments for the BOMCTP

Instances

|V |+ |W | random points in the [0, 100] x [0, 100] square

Depot is restricted to lie in the [25, 75] x [25, 75] square

Set V taken as first |V | points; Set W takes remaining points

Algorithms and Coding

All codes written in C/C++

RLPM solved with CPLEX 12.4

Subproblem solved by DSSR algorithm [Boland et al. (2006), Righini andSalani (2008)]

Computer Specifications

Intel Core 2 Duo, 2.93 GHz, 2 GB RAMColumn Generation for BOILPs: Application to BOVRPs 43 / 54





Formulation 1 : Quality of Bound Sets

PPS k-PPS Sequential

p |T | |V | |W | |L∗| |U| µ1% µ2% |U| µ1% µ2% |U| µ1% µ2%

5 1 30 90 66 29 6.0 27.6 29 6.1 27.5 28 5.9 27.55 1 40 120 67 37 5.1 25.4 29 5.1 25.4 36 5.0 25.25 1 50 150 67 35 4.5 24.8 39 4.6 24.7 36 4.5 24.05 15 30 90 5 6 40.5 55.7 6 40.5 55.7 6 40.4 55.75 20 40 120 7 5 31.2 62.3 6 31.2 62.3 6 31.2 62.35 25 50 150 9 4 28.9 79.1 4 28.9 78.9 5 28.9 79.0

8 1 30 90 63 30 6.5 28.2 29 6.5 28.1 29 6.4 28.08 1 40 120 65 35 5.2 25.7 26 5.3 25.4 36 5.1 25.28 1 50 150 64 36 4.7 24.8 38 4.6 24.5 36 4.7 24.48 15 30 90 2 6 35.4 71.1 6 35.4 71.3 6 35.4 71.08 20 40 120 4 6 9.2 73.8 6 9.1 73.8 6 9.1 73.28 25 50 150 4 5 24.3 87.9 6 24.2 87.3 7 24.2 87.2






Formulation 1 : Computational Times (cpu seconds)

p 5

p 8

0 30 60 90 120 150 180 210 240 27030 140 150 1

30 1540 2050 25

30 140 150 1

30 1540 2050 25

PPSk-PPSSequential






Formulation 1 : Number of Subproblems Solved

PPS k-PPS Sequential

p |T | |V | |W | # Subproblems # Subproblems # Subproblems

5 1 30 90 93 95 735 1 40 120 129 112 905 1 50 150 137 116 1065 15 30 90 153 148 1415 20 40 120 192 174 1585 25 50 150 285 223 206

8 1 30 90 144 141 1328 1 40 120 154 127 1208 1 50 150 199 132 1118 15 30 90 198 177 1698 20 40 120 296 278 2548 25 50 150 334 312 259






Formulation 2 : Quality of Bound Sets

PPS IPPS SOGA

p |T | |V | |W | |L∗| |U| µ1% µ2% |U| µ1% µ2% |U| µ1% µ2%

5 1 30 90 21 21 0.8 2.3 22 0.9 2.0 22 0.9 2.05 1 40 120 27 28 0.4 1.2 29 0.5 1.1 29 0.5 1.25 1 50 150 30 30 0.2 0.5 30 0.2 0.7 31 0.2 0.45 15 30 90 5 6 1.9 15.7 6 1.4 14.5 6 1.8 15.05 20 40 120 5 5 0.8 38.2 5 0.8 46.4 5 0.8 41.35 25 50 150 3 3 3.0 57.7 3 1.4 41.1 3 1.8 35.9

8 1 30 90 22 22 0.7 1.4 23 0.8 1.2 22 0.8 1.18 1 40 120 29 30 0.2 0.7 30 0.3 0.7 30 0.3 0.78 1 50 150 30 30 0.3 0.7 30 0.2 0.5 31 0.3 0.58 15 30 90 5 6 2.8 21.5 6 3.2 23.4 6 3.3 23.58 20 40 120 5 5 3.2 50.2 5 3.4 51.7 5 3.8 50.38 25 50 150 3 3 2.5 64.9 3 2.6 71.5 3 3.0 65.7






Formulation 2 : Computational Times (cpu seconds)

0 250 500 750 1000 1250 1500 1750 2580 2900 322030 140 150 1

30 1540 2050 25

30 140 150 1

30 1540 2050 25

p 5

p 8

PPSIPPSSOGA






Formulation 2 : Number of Subproblems Solved

PPS IPPS SOGA

p |T | |V | |W | # Subproblems # Subproblems # Subproblems

5 1 30 90 163 120 1065 1 40 120 330 201 1745 1 50 150 486 247 2245 15 30 90 141 114 515 20 40 120 236 198 675 25 50 150 185 171 65

8 1 30 90 215 142 1228 1 40 120 481 293 2438 1 50 150 672 384 3068 15 30 90 288 209 1028 20 40 120 564 455 1498 25 50 150 374 342 130






Comparison of Formulations 1 and 2

Formulation 1 Formulation 2

p |T | |V | |W | |U∗| |L| µ1% µ2% |L| µ1% µ2%

5 1 30 90 21 66 5.4 23.0 21 0.8 2.35 1 40 120 28 67 4.2 19.7 27 0.4 1.25 1 50 150 30 67 3.6 21.8 30 0.2 0.55 15 30 90 6 5 9.3 47.5 5 1.9 15.75 20 40 120 5 7 8.5 57.3 5 0.8 38.25 25 50 150 3 9 6.9 62.6 3 3.0 57.7

8 1 30 90 22 63 5.3 21.2 22 0.7 1.48 1 40 120 30 65 5.1 21.1 29 0.2 0.78 1 50 150 30 64 4.5 19.7 30 0.3 0.78 15 30 90 6 2 13.3 56.7 5 2.8 21.58 20 40 120 5 4 8.8 73.1 5 3.2 50.28 25 50 150 3 4 6.1 79.9 3 2.5 64.9






Comparison of Formulations 1 and 2

Instance type: |T | = 1, |V | = 50, |W | = 150, p = 5, q =∞C

over

Dis

tanc

e

Length

Formulation 1 : LBFormulation 2 : LBFormulation 1 : UBFormulation 2 : UB





Conclusions

Conclusions

Methods and models for computing lower bounds are needed inmulti-objective optimization

Application of column generation to multi-objective problems seems to havebeen overlooked

Column generation techniques and strategies for single objective problemscan easily be extended to bi-objective problems

Good lower bounds for bi-objective problems can be obtained by columngeneration in reasonable times if columns are efficiently managed





Perspectives

Short Term

Optimize computer code and continue with tests

Define more general methods for constructing bound sets (eg. for problemswith non-integral objective coefficients or having an exponential number ofsupported nondominated points)

Middle Term

Incorporate strategies in a multi-objective branch and price algorithm

Develop and evaluate different branching rules

Long Term

Extend ideas to problems with more than two objectives

Optimize code execution (eg. parallel versions of algorithms)





List of Publications

International Journals

Sarpong, Artigues, and Jozefowiez. Using Column Generation to ComputeLower Bound Sets for Bi-Objective Combinatorial Optimization Problems.RAIRO - Operations Research. 2013 (Accepted)

International Conferences

Sarpong, Artigues, and Jozefowiez. Column Generation for Bi-ObjectiveVehicle Routing Problems with a Min-Max Objective. In 13th Workshop onAlgorithmic Approaches for Transportation Modelling, Optimization, andSystems (ATMOS). 2013

Sarpong, Artigues, and Jozefowiez. The Bi-Objective Multi-Vehicle CoveringTour Problem: Formulation and Lower Bound. In 5th InternationalWorkshop on Freight Transportation and Logistics, ODYSSEUS. 2012


column generation for bi-objective integer linear programs ... · based on the principles of lp...

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