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
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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
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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
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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.
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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)
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
f1
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
f1lb1
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
f1lb1
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
f1lb1
lb2
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
f1lb1
lb2
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
ideal point
f1lb1
lb2
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Lower bound of a MOIP [Villarreal and Karwan, 1981]
f2
ideal point
f1lb1
lb2
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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.
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Scalarization by ε-constraint
Master Problem
minimize c1x
Ax ≥ b−c2x ≥ −ε
x ≥ 0
Dual
maximize by1 − εy2
Ay1 − c2y2 ≤ c1
y1, y2 ≥ 0
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Approach 1: point-by-point search
f2
f1
ε0
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Approach 1: point-by-point search
f2
f1
ε0
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Approach 1: point-by-point search
f2
f1
ε0
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Approach 1: point-by-point search
f2
f1
ε0
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Approach 1: point-by-point search
f2
f1
ε0
ε1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
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Approach 1: point-by-point search
f2
f1
ε0
ε1
ε2
εk-1
εk
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Approach 2: parallel search 1
f2
f1
ε0
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Approach 2: parallel search 1
f2
f1
ε0
ε1
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Approach 2: parallel search 1
f2
f1
ε0
ε2
ε1
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Approach 2: parallel search 1
f2
f1
ε0
ε2
ε1
ε3
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Approach 2: parallel search 1
f2
f1
ε0
εk
εk-1
ε2
ε1
ε3
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Approach 2: parallel search 1
f2
f1
ε0
εk
εk-1
ε2
ε1
ε3
generate m/kcolumns for RMP
generate m/kcolumns for RMP
generate m/kcolumns for RMP
generate m/kcolumns for RMP
generate m/kcolumns for RMP
generate m/kcolumns for RMP
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Approach 2: parallel search 1
f2
f1
ε0
εk
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Approach 2: parallel search 1
f2
f1
ε0
εk
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Approach 2: parallel search 1
f2
f1
ε0
εk
ε11
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Approach 2: parallel search 1
f2
f1
ε0
εk
ε2
ε11
1
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Approach 2: parallel search 1
f2
f1
ε0
εk
εk-1
ε2
ε1
ε3
1
1
1
1
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Approach 3: parallel search 2
f2
f1
ε0
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Approach 3: parallel search 2
f2
f1
ε0
generate mcolumns for RMP
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Approach 3: parallel search 2
f2
f1
ε0
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Approach 3: parallel search 2
f2
f1
ε0
ε1
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Approach 3: parallel search 2
f2
f1
ε0
ε1
generate mcolumns for RMP
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Approach 3: parallel search 2
f2
f1
ε0
ε1
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Approach 3: parallel search 2
f2
f1
ε0
ε1
ε2
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Approach 3: parallel search 2
f2
f1
ε0
ε1
ε2
generate mcolumns for RMP
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Approach 3: parallel search 2
f2
f1
ε0
ε1
ε2
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Approach 3: parallel search 2
f2
f1
ε0
ε1
ε2
εk
εk-1
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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
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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
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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 )
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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.
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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].
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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
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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.
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