the bi-objective multi-vehicle covering tour problem

The Bi-objective Multi-VehicleCovering Tour Problem (BOMCTP):

formulation and lower-boundcomputation

B.M. SARPONG C. ARTIGUES N. JOZEFOWIEZ

LAAS-CNRS

13/04/2012

Outline

1 Introduction

2 Mathematical formulation of the BOMCTP

3 Column generation for a bi-objective integerproblem

4 Lower bound for the BOMCTP

5 Conclusions and perspectives

1 / 18

The Covering Tour Problem [Gendreau et al., 1997]

Find a minimal-length tour on V ′ ⊆ V such that the nodes of Ware covered by those of V ′.

Vehicle route

May be visited

MUST be visited : T

V

MUST be covered : W

Cover

2 / 18

The Multi-Vehicle CTP [Hachicha et al., 2000]

Find a set of at most m tours on V ′ ⊆ V , having minimum totallength and such that the nodes of W are covered by those of V ′.

The length of each route cannot exceed a preset value p.The number of vertices on each route cannot exceed a presetvalue q.

Vehicle routes

May be visited

MUST be visited : TV

MUST be covered : W

Cover distance

3 / 18

Description of the BOMCTP

ProblemGiven a graph G = (V ∪W ,E ) with T ⊆ V , design a set ofvehicle routes on V ′ ⊆ V .

ObjectivesMinimize the total length of the set of routes.Minimize the cover distance induced by the set of routes.

ConstraintsEach vertex of T must belong to a vehicle route.Each vertex of W must be covered.The length of each route cannot exceed a preset value p.The number of vertices on each route cannot exceed a presetvalue q.

4 / 18

A set-covering model for the BOMCTP

VariablesΩ : set of all feasible routesrk ∈ Ω : feasible route kck : cost of route rk

θk : 1 if route rk is selected in solution and 0 otherwisezij : 1 if vertex vj ∈ V is used to cover vertex wi ∈W and 0otherwiseaik : 1 if rk uses vertex vi ∈ V and 0 otherwiseCovmax : cover distance induced by a set of routes

Objective functions

minimize∑rk∈Ω

ckθk

minimize Covmax5 / 18

A set-covering model for the BOMCTP

Constraints

− zij +∑rk∈Ω

ajkθk ≥ 0 (wi ∈W , vj ∈ V )

∑rk∈Ω

ajkθk ≥ 1 (vj ∈ T )

Covmax − cijzij ≥ 0 (wi ∈W , vj ∈ V )∑vj∈V

zij ≥ 1 (wi ∈W )

Covmax ≥ 0zij ∈ 0, 1 (wi ∈W , vj ∈ V )

θk ∈ 0, 1 (rk ∈ Ω)

6 / 18

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

f2

f1

7 / 18


f2

f1lb1

7 / 18


f2

f1lb1

lb2

7 / 18


f2

ideal point

f1lb1

lb2

7 / 18

Column generation for a bi-objective integer problem

Problem

minimize (c1x , c2x)

Ax ≥ bx ≥ 0 and integer

ProcedureTransform bi-objective problem into a single-objective one bymeans of ε-constraint scalarization.Solve the linear relaxation of the problem obtained fordifferent values of ε by means of column generation.

8 / 18

Scalarization by ε-constraint

Master Problem

minimize c1x

Ax ≥ b−c2x ≥ −ε

x ≥ 0

Dual

maximize by1 − εy2

Ay1 − c2y2 ≤ c1

y1, y2 ≥ 0

9 / 18

Approach 1: point-by-point search

f2

f1

ε0

10 / 18


f2

f1

ε0

ε1

10 / 18


f2

f1

ε0

ε1

ε2

10 / 18


f2

f1

ε0

ε1

ε2

εk-1

10 / 18


f2

f1

ε0

ε1

ε2

εk-1

εk

10 / 18

Approach 2: parallel search 1

f2

f1

ε0

11 / 18


f2

f1

ε0

ε1

11 / 18


f2

f1

ε0

ε2

ε1

11 / 18


f2

f1

ε0

ε2

ε1

ε3

11 / 18


f2

f1

ε0

εk

εk-1

ε2

ε1

ε3

11 / 18


f2

f1

ε0

εk

εk-1

ε2

ε1

ε3

generate m/kcolumns for RMP






11 / 18


f2

f1

ε0

εk

11 / 18


f2

f1

ε0

εk

ε11

11 / 18


f2

f1

ε0

εk

ε2

ε11

1

11 / 18


f2

f1

ε0

εk

εk-1

ε2

ε1

ε3

1

1

1

1

11 / 18


f2

f1

ε0

12 / 18


f2

f1

ε0

generate mcolumns for RMP

12 / 18


f2

f1

ε0

12 / 18


f2

f1

ε0

ε1

12 / 18


f2

f1

ε0

ε1


12 / 18


f2

f1

ε0

ε1

12 / 18


f2

f1

ε0

ε1

ε2

12 / 18


f2

f1

ε0

ε1

ε2


12 / 18


f2

f1

ε0

ε1

ε2

12 / 18


f2

f1

ε0

ε1

ε2

εk

εk-1

12 / 18

The Restricted Master Problem (RMP)

minimize∑

rk∈Ω1

ckθk

Constraints

− zij +∑

rk∈Ω1

ajkθk ≥ 0 (wi ∈W , vj ∈ V )

∑rk∈Ω1

ajkθk ≥ 1 (vj ∈ T )

Covmax − cijzij ≥ 0 (wi ∈W , vj ∈ V )∑vj∈V

zij ≥ 1 (wi ∈W )

−Covmax ≥ −ε

13 / 18

The Restricted Master Problem (RMP)

minimize∑

rk∈Ω1

ckθk

Constraintsdual variables

− zij +∑

rk∈Ω1

ajkθk ≥ 0 (wi ∈W , vj ∈ V ) αij∑rk∈Ω1

ajkθk ≥ 1 (vj ∈ T ) ϕj

Covmax − cijzij ≥ 0 (wi ∈W , vj ∈ V ) γij∑vj∈V

zij ≥ 1 (wi ∈W ) βi

−Covmax ≥ −ε λ

13 / 18

Dual of RMP

maximize − ελ+∑

wi∈Wβi +

∑vj∈T

ϕj

subject to: ∑wi∈Wvj∈V

ajkαij +∑

vj∈Tajkϕj ≤ ck (rk ∈ Ω1)

−λ+∑

wi∈Wvj∈V

γij ≤ 0

− cijγij + βi − αij ≤ 0 (wi ∈W , vj ∈ V )

14 / 18

Definition of sub-problem

Find routes such that ck −∑

wi∈Wvj∈V

ajkαij −∑

vj∈Tajkϕj < 0.

Let α∗hj = αhj if vj ∈ V ,wh ∈W and 0 otherwise.Let ϕ∗j = ϕj if vj ∈ T and 0 otherwise.Let A be the set of arcs formed between two nodes of V .Let xijk = 1 if route rk uses arc (vi , vj) and 0 otherwise.

Note : ck =∑

(vi ,vj )∈Axijkcij and ajk =

∑vi∈V |(vi ,vj )∈A

xijk

So∑

(vi ,vj )∈Acijxijk −

∑(vi ,vj )∈A

∑vh∈W

α∗hjxijk −∑

(vi ,vj )∈Aϕ∗j xijk < 0.

15 / 18

Definition of sub-problem

∑(vi ,vj )∈A

(cij − ϕ∗j −

∑vh∈W

α∗hj

)xijk < 0.

Sub-problemFind elementary paths from the depot to the depot with a negativecost, satisfying the constraints of length and maximum number ofvertices on a path. Costs are set to

cij − ϕ∗j −∑

vh∈Wα∗hj .

An elementary shortest path problem with resource constraintsSolved by the Decremental State Space Relaxation (DSSR)algorithm [Righini and Salani, 2008].

16 / 18

Computational results

Instances:120 random points generated in the [0, 100] x [0, 100] squareSet V taken as first |V | points; W taken as remaining points

RMP coded in C++ and solved with CPLEX 12.2Computer: Intel Core 2 Duo, 2.93 GHz, 2 GB RAM

Table: Averages over 10 random instances for |T | = 1, and q = +∞.

Point-by-point Parallel 1 Parallel 2|V | p time Nb. cols Nb. solved time Nb. cols Nb. solved time Nb. cols Nb. solved

(sec) Gen master (sec) Gen master (sec) Gen master6 5.93 16.7 34.5 46.41 18.7 917.8 4.35 16.8 35.0

40 8 11.76 18.0 34.6 48.61 20.1 925.5 5.49 18.3 35.112 25.01 20.7 34.9 57.21 23.3 947.0 7.12 20.6 35.46 10.94 18.2 38.9 60.87 20.5 1099.7 8.19 22.1 40.6

50 8 22.01 19.6 38.4 64.74 24.0 1115.2 8.99 19.7 38.912 41.74 22.1 37.1 75.85 26.9 1107.8 11.93 22.6 37.56 19.13 19.9 42.7 40.22 20.9 1017.4 9.45 20.4 43.1

60 8 44.56 19.3 42.3 52.78 22.5 1240.1 14.02 21.1 43.412 128.98 23.4 42.6 68.66 26.9 1258.1 22.55 25.8 44.0

17 / 18

Conclusions and perspectives

ConclusionsPossible to have several (and efficient) ways of applyingcolumn generation to bi-objective integer problems.Model for BOMCTP has a weak linear relaxation.

Work in progressInvestigate other intelligent ways of generating columns for abi-objective integer problem.Test developed approaches on different problems (includinganother model for the BOMCTP with a stronger linearrelaxation).Efficiently solve the BOMCTP by a multi-objectivebranch-and-price algorithm.

18 / 18

THANK YOU FOR YOUR ATTENTION.

the bi-objective multi-vehicle covering tour problem

Documents