Column Generation for Bi-ObjectiveVehicle Problems with a Min-Max

Objective

B.M. Sarpong1,2 C. Artigues1 N. Jozefowiez1,2

1. LAAS-CNRS, Toulouse, France2. INSA-Toulouse, France

05/09/2013

Outline

Multi-Objective Integer Programming

Bi-Objective VRP with a Min-Max Objective

Application to the BOMCTP

Computational Results

Conclusions

1 / 23

Definition of a Multi-Objective Integer Program

(MOIP) =

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

s.t. x ∈ X

I r ≥ 2 : number of objective functions

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

I X ⊆ Nn : feasible set of solutions

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

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

I y = (y1, y2, . . . , yr ) ∈ Y with yi = fi (x) : vector of objectivefunction values

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Dominance and Pareto Optimality

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

AB

C

D

E

f1

f2

I A Pareto optimal solution is a solution dominated by noother feasible solution.

I The image of a Pareto optimal solution in the objective spaceis said to be nondominated.

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Lower Bound of a MOIP [Villarreal and Karwan, 1981]

ideal

nadir

image of feasible solution

estimate of nondominated region

member of lower bound

f2

f1

Lower BoundA set of points (feasible or not) such that the image of everyfeasible solution is dominated by at least one of the points

Upper BoundA set of mutually nondominated feasible points.

4 / 23

Lower Bound of a MOIP [Villarreal and Karwan, 1981]

ideal

nadir image of feasible solution

estimate of nondominated region

member of lower bound

f2

f1

Lower BoundA set of points (feasible or not) such that the image of everyfeasible solution is dominated by at least one of the points

Upper BoundA set of mutually nondominated feasible points.

4 / 23

Lower Bound of a MOIP [Villarreal and Karwan, 1981]

ideal

nadir image of feasible solution

estimate of nondominated region

member of lower bound

f2

f1

Lower BoundA set of points (feasible or not) such that the image of everyfeasible solution is dominated by at least one of the points

Upper BoundA set of mutually nondominated feasible points.

4 / 23

Lower Bound of a MOIP [Villarreal and Karwan, 1981]

ideal

nadir image of feasible solution

estimate of nondominated region

member of lower bound

f2

f1

Lower BoundA set of points (feasible or not) such that the image of everyfeasible solution is dominated by at least one of the points

Upper BoundA set of mutually nondominated feasible points.

4 / 23

Computing Bound Sets for a BOIP

Lower Bound Set

I Transform the BOIP into single objective by using ascalarization method (weighted sum, ε-constraint, . . . ).

I Solve relaxations of transformed problem for several values ofthe parameters in order to generate a set of points satisfyingthe definition of a lower bound set.

Upper Bound Set

I Generally, obtained through heuristics and metaheuristics.

I The heuristic or metaheuristic need not depend on anyparticular scalarization method.

5 / 23

Column Generation for a BOVRPMMO

Minimize∑

k∈Ω ckθkMinimize Γmax∑

k∈Ω aikθk ≥ bi (i ∈ I)Γmax ≥ σkθk (k ∈ Ω)θk ∈ 0, 1 (k ∈ Ω)

Assumption

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

I Example: σk is an integer.

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Standard ε-Constraint Approach for a BOVRPMMO

Minimize∑

k∈Ω ckθk∑k∈Ω aikθk ≥ bi (i ∈ I)−Γmax ≥ −ε

Γmax ≥ σkθk (k ∈ Ω)θk ∈ 0, 1 (k ∈ Ω)

Advantage

I Each column is valid for all values of ε and so the subproblemalgorithm does not need to keep track of this parameter.

Disadvantage

I Usually need to introduce extra variables and this cannegatively affect the quality of a lower bound set.

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Reformulation of a BOVRPMMO

Minimize∑

k∈Ω ckθkMinimize Γmax∑

k∈Ω aikθk ≥ bi (i ∈ I)Γmax ≥ σkθk (k ∈ Ω)θk ∈ 0, 1 (k ∈ Ω)

Idea Used

I Ω is extended into a new set of columns Ω.

I The feasibility of a column k ∈ Ω depends on σk .

I We define Ωε = k ∈ Ω : σk ≤ ε when we need solutionssatisfying Γmax ≤ ε.

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Reformulation of a BOVRPMMO

Minimize∑

k∈Ω ckθkMinimize Γmax∑

k∈Ω aikθk ≥ bi (i ∈ I)Γmax ≥ σkθk (k ∈ Ω)θk ∈ 0, 1 (k ∈ Ω)

Idea Used

I Ω is extended into a new set of columns Ω.

I The feasibility of a column k ∈ Ω depends on σk .

I We define Ωε = k ∈ Ω : σk ≤ ε when we need solutionssatisfying Γmax ≤ ε.

8 / 23

Reformulation of a BOVRPMMO (Master Problem)

Minimize∑

k∈Ω ckθk∑k∈Ω aikθk ≥ bi (i ∈ I)

θk ∈ 0, 1 (k ∈ Ω)

Advantages

I Both the master and the subproblem are single objectiveproblems.

I No weakening of the linear relaxation for a given value of ε.

Disadvantage

I Need to manage the parameter ε in both the master problemand the subproblem.

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

For any given value of ε, completly solve the linear relaxation ofMP(Ωε) by column generation.

f2

f1

max ε

min ε

ε1

ε2

ε3

ε4

ε5

10 / 23

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

For any given value of ε, completly solve the linear relaxation ofMP(Ωε) by column generation.

f2

f1

max ε

min ε

ε1

ε2

ε3

ε4

ε5

10 / 23

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

For any given value of ε, completly solve the linear relaxation ofMP(Ωε) by column generation.

f2

f1

max ε

min ε

ε1

ε2

ε3

ε4

ε5

10 / 23

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

For any given value of ε, completly solve the linear relaxation ofMP(Ωε) by column generation.

f2

f1

max ε

min ε

ε1

ε2

ε3

ε4

ε5

10 / 23

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

For any given value of ε, completly solve the linear relaxation ofMP(Ωε) by column generation.

f2

f1

max ε

min ε

ε1

ε2

ε3

ε4

ε5

10 / 23

Column Search Strategies: Improved PPS (IPPS)

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

Advantage

I Can cheaply find a large number of columns once the price ofa few columns have been paid.

I Tries to take advantage of the reformulation and the similarsubproblems associated to the different values of ε.

Disadvantage

I No guarantee that a column found by a heuristic will berelevant for other values of ε apart from the current one.

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

Find a set of routes on V ′ ⊆ V with minimum total length andsuch that the nodes of W are covered by those of V ′.

I The number of nodes on each route cannot exceed p.

Vehicle routes

May be visited

DepotV

MUST be covered : W

Cover distance

12 / 23

Some Applications of the MCTP

Design of bi-level transportation networks [Current and Schilling(1994)].

I Aim: Construct a primary route such that all points that arenot on it can easily reach it.

The postbox location problem [Labbe and Laporte (1986)].I Aim: minimize the cost of a collection route through all post

boxes and also ensure that every user is located within areasonable distance from a post box.

Humanitarian logistics. eg. Planning of routes for visiting healthcare teams in developing countries [Current and Schilling, 1994;Hodgson et al. (1998)].

I Medical services can only be delivered to a subset of villages,but all users must be able to reach a visiting medical team.

13 / 23

The Bi-Objective MCTP

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

Objectives

I Minimize the total length of the set of routes.

I Minimize the cover distance induced by the set of routes.

Constraints

I Each route must start and also end at the depot.

I The number of nodes on each route should not exceed p.

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The Cover Distance Induced by a Set of Routes

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The Cover Distance Induced by a Set of Routes

15 / 23

The Cover Distance Induced by a Set of Routes

15 / 23

The Cover Distance Induced by a Set of Routes

15 / 23

The Cover Distance Induced by a Set of Routes

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The Cover Distance Induced by a Set of Routes

15 / 23

A “standard” ε-Constraint Formulation for the BOMCTP

Minimize∑k∈Ω

ckθk (1)

∑vi ∈V \v0

zij ≥ 1 (wj ∈W ) (2)

∑k∈Ω

aikθk − zij ≥ 0 (vi ∈ V \v0,wj ∈W ) (3)

Γmax − dijzij ≥ 0 (vi ∈ V \v0,wj ∈W ) (4)−Γmax ≥ −ε (5)

Γmax ≥ 0 (6)zij ∈ 0, 1 (vi ∈ V \v0,wj ∈W ) (7)θk ∈ 0, 1 (k ∈ Ω) (8)

16 / 23

Reformulation: Master Problem

Minimize∑k∈Ω

ckθk

subject to :∑k∈Ω

ajkθk ≥ 1 (wj ∈W )

θk ∈ 0, 1 (k ∈ Ω)

I A column k ∈ Ω is defined as a route Rk together with asubset Ψk ⊆W of nodes it may cover.

I σk : maxdij : vi ∈ Rk and wj ∈ Ψk.

I ajk : 1 if wj ∈ Ψk , and 0 otherwise.

17 / 23

Reformulation: Subproblem corresponding to ε

S(ε) = mink∈Ω\Ω1

ck −

∑wj ∈W

ajkπj : σk ≤ ε

I S(ε) is associated with a dual vector π, and the value of ε forwhich π was computed.

I An elementary shortest path problem with resourceconstraints.

I Ψk may only contain nodes of the setwj ∈W : ∃vi ∈ Rk with dij ≤ ε.

I Need to modify dominance rule between labels.

18 / 23

IPPS Heuristic for the BOMCTP

19 / 23

IPPS Heuristic for the BOMCTP

19 / 23

IPPS Heuristic for the BOMCTP

19 / 23

IPPS Heuristic for the BOMCTP

19 / 23

IPPS Heuristic for the BOMCTP

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IPPS Heuristic for the BOMCTP

19 / 23

Quality Measures [Ehrgott and Gandibleux (2007)]

d(L,U)

ymax

ymin

Member of L

Member of U

AL

AU

f1

f2

µ1 :=d(L,U)

‖ymax − ymin‖2µ2 :=

AL − AUAL

20 / 23

Experiments

Instances

I |V |+ |W | random points in the [0, 100] x [0, 100] square.I Depot is restricted to lie in the [25, 75] x [25, 75] square.I Set V taken as first |V | points; Set W takes remaining points.

Algorithms and coding

I All codes written in C/C++.I RMP solved with CPLEX 12.4.I Subproblem solved by DSSR algorithm [Boland et al. (2006),

Righini and Salani (2008)].

Computer specifications

I Intel Core 2 Duo, 2.93 GHz, 2 GB RAM.

21 / 23

Comparison of Approaches and Search Strategies

Standard PPS IPPS

p |V | |W | µ1% µ2% µ1% µ2% µ1% µ2%

5 40 80 6.41 23.22 1.07 7.69 1.01 6.095 40 120 4.67 23.35 0.91 11.92 0.92 10.335 50 100 4.74 21.94 0.70 9.48 0.68 7.855 50 150 4.33 26.44 1.20 16.67 1.20 15.41

8 40 80 8.38 25.86 0.32 3.04 0.38 2.468 40 120 6.48 21.70 0.40 4.50 0.40 3.708 50 100 6.13 21.83 0.32 3.83 0.31 3.658 50 150 5.00 18.70 0.31 4.30 0.36 3.66

Table: Quality of Bound Sets

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Comparison of Approaches and Search Strategies

0 200 400 600 800 1000 1200 1400 1600

40 80

40 120

50 100

50 150

40 80

40 120

50 100

50 150

StandardPPSIPPS

Figure: Computational Time (cpu seconds)

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Comparison of Approaches and Search Strategies

0 100 200 300 400 500 600 700

40 80

40 120

50 100

50 150

40 80

40 120

50 100

50 150

StandardPPSIPPS

Figure: Number of DSSR Iterations

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Conclusions

I Methods and models for computing lower bounds are neededin multi-objective optimization.

I Application of column generation to multi-objective problemsseems to have been overlooked.

I Column generation techniques and strategies for singleobjective problems can easily be extended to bi-objectiveproblems.

I Study column generation approach based on otherscalarization methods (eg. weighted sum method)

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Column Generation for Bi-ObjectiveVehicle Problems with a Min-Max

Objective

B.M. Sarpong1,2 C. Artigues1 N. Jozefowiez1,2

1. LAAS-CNRS, Toulouse, France2. INSA-Toulouse, France

05/09/2013

Standard PPS IPPS

p |V | |W | time dssr time dssr time dssr

5 40 80 28 137 50 228 41 1555 40 120 39 163 127 330 94 2015 50 100 69 197 206 390 154 2265 50 150 42 150 393 486 287 247

8 40 80 61 217 113 302 104 2188 40 120 132 299 511 481 504 2938 50 100 281 326 1344 522 1013 3358 50 150 333 380 1525 672 1186 384

Table: Computational Time (cpu seconds)

0 200 400 600 800 1000 1200 1400 1600

40 80

40 120

50 100

50 150

40 80

40 120

50 100

50 150

StandardPPSIPPS

Figure: Computational Time (cpu seconds)

0 100 200 300 400 500 600 700

40 80

40 120

50 100

50 150

40 80

40 120

50 100

50 150

StandardPPSIPPS

Figure: Number of DSSR Iterations