The Bi-objective Multi-Vehicle Covering Tour Problem (BOMCTP): - - PowerPoint PPT Presentation

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The Bi-objective Multi-Vehicle Covering Tour Problem (BOMCTP): - - PowerPoint PPT Presentation

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The Bi-objective Multi-Vehicle Covering Tour Problem (BOMCTP): formulation and lower-bound computation B.M. SARPONG C. ARTIGUES N. JOZEFOWIEZ LAAS-CNRS 13/04/2012 Outline 1 Introduction 2 Mathematical formulation of the BOMCTP 3 Column

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

B.M. SARPONG

  • C. ARTIGUES
  • N. JOZEFOWIEZ

LAAS-CNRS 13/04/2012

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Outline

1 Introduction 2 Mathematical formulation of the BOMCTP 3 Column generation for a bi-objective integer

problem

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 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 tours on V ′ ⊆ V , having minimum total length 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 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 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 preset value q.

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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 vertex vj ∈ V is used to cover vertex wi ∈ W and 0

  • therwise

aik : 1 if rk uses vertex 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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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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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/k columns for RMP generate m/k columns for RMP generate m/k columns for RMP generate m/k columns for RMP generate m/k columns for RMP generate m/k columns 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 ε1

1

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Approach 2: parallel search 1

f2 f1

ε0 εk ε2 ε1

1 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 m columns 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 m columns 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 m columns 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

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 the constraints of length and maximum number of vertices on a path. Costs are set to cij − ϕ∗

j −

  • vh∈W

α∗

hj.

An elementary shortest path problem with resource constraints Solved 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] square Set V taken as first |V | points; W taken as remaining points

RMP coded in C++ and solved with CPLEX 12.2 Computer: 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 master 6 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 35.1 12 25.01 20.7 34.9 57.21 23.3 947.0 7.12 20.6 35.4 6 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 38.9 12 41.74 22.1 37.1 75.85 26.9 1107.8 11.93 22.6 37.5 6 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 43.4 12 128.98 23.4 42.6 68.66 26.9 1258.1 22.55 25.8 44.0

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Conclusions and perspectives 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

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

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THANK YOU FOR YOUR ATTENTION.