The Bi-objective Multi-Vehicle Covering Tour Problem: formulation - - PowerPoint PPT Presentation

The Bi-objective Multi-Vehicle Covering Tour Problem: formulation and lower-bound computation

B.M. Sarpong

• C. Artigues
• N. Jozefowiez

LAAS-CNRS / Universit´ e de Toulouse, France 25/05/2012

Outline

1 Multi-objective optimization problems 2 Column generation for a bi-objective integer

problem

3 The Bi-Objective Multi-Vehicle Covering

Tour Problem

4 Computational results 5 Conclusions

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Definition of a multi-objective optimization problem

(MOP) =

• min F(x) = (f1(x), f2(x), . . . , fn(x))

s.t. x ∈ Ω where : n ≥ 2 : number of objective functions F = (f1, f2, . . . , fn) : vector of objective functions Ω ⊆ Rm : feasible set of solutions Y = F(Ω) : feasible set in objective space x = (x1, x2, . . . , xm) ∈ Ω : variable vector, variables y = (y1, y2, . . . , yn) ∈ Y with yi = fi(x) : vector of objective function values

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Dominance and Pareto Optimality

A solution x dominates () another solution y if and only if ∀i ∈ {1, . . . , n}, fi(x) ≤ fi(y) and ∃i ∈ {1, . . . , n} such that fi(x) < fi(y).

f1 f2 A B C D E

Pareto optimal solution

A solution is said to be Pareto optimal if no other feasible solution dominates it.

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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 f1

lb1

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

f2 f1

lb1

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

f2 f1

lb1 lb2

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

f2 f1

lb1 lb2

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

f2

ideal point

f1

lb1 lb2

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

f2

ideal point

f1

lb1 lb2

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Column generation for a bi-objective integer problem Problem

minimize (c1x, c2x) Ax ≥ b x ≥ 0 and integer

Procedure

Transform bi-objective problem into a single-objective one by means of ε-constraint scalarization. Solve the linear relaxation of the problem obtained for different 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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Approaches to compute the lower bound of a BOIP by CG Approaches

Approach 1: standard Approach 2: sequential search Approach 3:

parallel search 1 parallel search 2 Performance indicator

Execution time (CPU seconds)

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Approach 1: standard

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Approach 1: standard

f2 f1

ε0

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Approach 1: standard

f2 f1

ε0

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Approach 1: standard

f2 f1

ε0

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Approach 1: standard

f2 f1

ε0 ε1

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Approach 1: standard

f2 f1

ε0 ε1

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Approach 1: standard

f2 f1

ε0 ε2 ε1

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Approach 1: standard

f2 f1

ε0 ε2 ε1

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Approach 1: standard

f2 f1

ε0 ε2 ε1

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Approach 1: standard

f2 f1

ε0 εk εk-1 ε2 ε1

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Approach 2: sequential search (1 iteration of CG at each step)

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Approach 2: sequential search (1 iteration of CG at each step)

f2 f1

ε0

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Approach 2: sequential search (1 iteration of CG at each step)

f2 f1

ε0

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Approach 2: sequential search (1 iteration of CG at each step)

f2 f1

ε0 ε1

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Approach 2: sequential search (1 iteration of CG at each step)

f2 f1

ε0 ε1

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Approach 2: sequential search (1 iteration of CG at each step)

f2 f1

ε0 εk εk-1 ε2 ε1

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Approach 3: parallel search 1

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Approach 3: parallel search 1

f2 f1

ε0 εk εk-1 ε2 ε1

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Approach 3: parallel search 1

f2 f1

ε0 εk εk-1 ε2 ε1

generate m/k columns generate m/k columns generate m/k columns generate m/k columns generate m/k columns 10 / 24

Approach 3: parallel search 1

f2 f1

ε0 εk

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Approach 3: parallel search 1

f2 f1

ε0 εk εk-1 ε2 ε1

1 1 1

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Approach 3: parallel search 2

f2 f1

ε0 εk εk-1 ε2 ε1

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Approach 3: parallel search 2

f2 f1

ε0 εk εk-1 ε2 ε1

generate n columns generate n columns skip skip skip 11 / 24

Approach 3: parallel search 2

f2 f1

ε0 εk

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Approach 3: parallel search 2

f2 f1

ε0 εk εk-1 ε2 ε1

1 1 1

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

Find a minimal-length route on V ′ ⊆ V such that the nodes of W are 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 routes on V ′ ⊆ V , having minimum total length and such that the nodes of W are covered by those of V ′. Each node of T ⊆ V ′ must be used by a route. The length of each route cannot exceed a preset value p. The number of nodes on each route cannot exceed a preset value q. Vehicle routes May be visited MUST be visited : T V MUST be covered : W Cover distance

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Description of the BOMCTP Problem

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

Objectives

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

Constraints

Each node of T must belong to a vehicle route. The length of each route cannot exceed a preset value p. The number of nodes on each route cannot exceed a preset value q.

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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

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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

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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

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The cover distance induced by a set of routes

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A set-covering model for the BOMCTP Variables

Ω : set of all feasible routes rk ∈ Ω : feasible route k ck : cost of route rk θk : 1 if route rk is selected in solution and 0 otherwise zij : 1 if node vj ∈ V ∗ (V \{v0}) is used to cover node wi ∈ W and 0 otherwise aik : 1 if rk uses node vi ∈ V ∗ and 0 otherwise Covmax : cover distance induced by a set of routes

Objective functions

minimize

• rk∈Ω

ckθk minimize Covmax

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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 ≥ 0 zij ∈ {0, 1} (wi ∈ W , vj ∈ V ∗) θk ∈ {0, 1} (rk ∈ Ω)

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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

Constraints

dual 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∈W

βi +

• vj∈T ∗

ϕj subject to:

• wi∈W

vj∈V ∗

ajkαij +

• vj∈T ∗

ajkϕj ≤ ck (rk ∈ Ω1) −λ +

• wi∈W

vj∈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∈W

vj∈V ∗

ajkαij −

• vj∈T ∗

ajkϕj < 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)∈A

xijkcij and ajk =

• {vi∈V |(vi,vj)∈A}

xijk So

• (vi,vj)∈A

cijxijk −

• (vi,vj)∈A
• vh∈W

α∗

hjxijk −

• (vi,vj)∈A

ϕ∗

j xijk < 0

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Definition of sub-problem

• (vi,vj)∈A
• cij − ϕ∗

j −

• vh∈W

α∗

hj

• xijk < 0

Sub-problem

Find elementary paths from the depot to the depot with a negative cost, satisfying constraints on the length and the maximum number of nodes visited by a path. Costs are set to cij − ϕ∗

j −

• vh∈W

α∗

hj

An elementary shortest path problem with resource constraints

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Computational results Instances

120 random points generated in the [0, 100] x [0, 100] square Depot is restricted to be in the [25, 75] x [25, 75] square Set V taken as first |V | points; remaining points belong to set W

Algorithms and coding

RMP coded in C++ and solved with CPLEX 12.2 Sub-problem solved by the Decremental State Space Relaxation (DSSR) algorithm [Righini and Salani, 2006]

Computer specifications

Intel Core 2 Duo, 2.93 GHz, 2 GB RAM

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

Nb Cols |V | 40 50 60 generated p 6 8 12 6 8 12 6 8 12 std 3.81 18.06 66.52 7.00 14.29 69.65 16.04 48.69 233.50 2 seq 2.14 3.17 7.57 4.14 8.76 715.86 10.15 23.78 959.14 par 1 5.47 7.39 12.42 30.30 43.78 47.32 35.80 46.38 72.25 par 2 2.74 3.46 4.62 8.08 10.10 14.44 8.83 11.73 16.43 std 3.99 19.67 74.71 7.08 16.08 73.36 16.81 51.19 215.11 5 seq 2.13 3.14 7.13 4.05 7.17 14.94 7.30 13.13 36.97 par 1 4.51 7.45 14.81 9.52 19.89 38.51 24.43 36.31 79.94 par 2 2.45 3.08 4.56 4.76 6.21 7.81 7.77 10.32 14.84 std 3.85 19.10 82.41 7.27 16.76 77.39 16.60 51.10 196.00 10 seq 2.10 3.20 9.83 5.53 6.73 18.19 6.67 11.84 42.24 par 1 5.74 8.26 11.15 11.79 21.52 50.79 21.93 28.59 48.41 par 2 2.34 3.18 4.76 4.48 5.69 11.64 6.82 9.72 14.97 std 4.13 19.82 84.30 7.37 16.89 77.55 17.06 52.03 207.62 20 seq 2.17 3.48 7.19 5.20 8.70 42.90 6.93 12.52 31.39 par 1 6.33 9.04 13.69 13.78 21.45 28.91 19.66 37.38 51.53 par 2 2.36 3.22 5.11 4.67 6.92 9.87 7.25 10.65 17.30

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Conclusions Conclusions

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

Work in progress

Test developed approaches on different problems (including another model for the BOMCTP with a stronger linear relaxation). Investigate other intelligent ways of generating columns for a bi-objective integer problem. Efficiently solve the BOMCTP by a multi-objective branch-and-price algorithm.

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The Bi-objective Multi-Vehicle Covering Tour Problem: formulation and lower-bound computation

B.M. Sarpong

• C. Artigues
• N. Jozefowiez

LAAS-CNRS / Universit´ e de Toulouse, France 25/05/2012