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

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 generation for a bi-objective integer problem 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 W are covered by those of V ′ . May be visited V MUST be visited : T MUST be covered : W Vehicle route Cover 2 / 18

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 . May be visited V MUST be visited : T MUST be covered : W Vehicle routes Cover distance 3 / 18

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 . 4 / 18

A set-covering model for the BOMCTP Variables Ω : set of all feasible routes r k ∈ Ω : feasible route k c k : cost of route r k θ k : 1 if route r k is selected in solution and 0 otherwise z ij : 1 if vertex v j ∈ V is used to cover vertex w i ∈ W and 0 otherwise a ik : 1 if r k uses vertex v i ∈ V and 0 otherwise Cov max : cover distance induced by a set of routes Objective functions � minimize c k θ k r k ∈ Ω minimize Cov max 5 / 18

A set-covering model for the BOMCTP Constraints � − z ij + a jk θ k ≥ 0 ( w i ∈ W , v j ∈ V ) r k ∈ Ω � a jk θ k ≥ 1 ( v j ∈ T ) r k ∈ Ω Cov max − c ij z ij ≥ 0 ( w i ∈ W , v j ∈ V ) � ≥ 1 ( w i ∈ W ) z ij v j ∈ V Cov max ≥ 0 ∈ { 0 , 1 } ( w i ∈ W , v j ∈ V ) z ij θ k ∈ { 0 , 1 } ( r k ∈ Ω) 6 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 lb 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 lb 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 lb 2 f 1 lb 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 lb 2 f 1 lb 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 ideal point lb 2 f 1 lb 1 7 / 18

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 ideal point lb 2 f 1 lb 1 7 / 18

Column generation for a bi-objective integer problem Problem minimize ( c 1 x , c 2 x ) 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. 8 / 18

Scalarization by ε -constraint Master Problem minimize c 1 x Ax ≥ b − c 2 x ≥ − ε x ≥ 0 Dual maximize by 1 − ε y 2 Ay 1 − c 2 y 2 ≤ c 1 y 1 , y 2 ≥ 0 9 / 18

Approach 1: point-by-point search f2 ε 0 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 f1 10 / 18

Approach 1: point-by-point search f2 ε 0 ε 1 ε 2 ε k-1 ε k f1 10 / 18

Approach 2: parallel search 1 f2 ε 0 f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 ε 2 f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 ε 2 ε 3 f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 ε 2 ε 3 ε k-1 ε k f1 11 / 18

Approach 2: parallel search 1 f2 generate m/k columns for RMP ε 0 generate m/k ε 1 columns for RMP generate m/k ε 2 columns for RMP generate m/k ε 3 columns for RMP generate m/k columns for RMP generate m/k columns for RMP ε k-1 ε k f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε k f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε k f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 1 ε k f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 1 ε 2 1 ε k f1 11 / 18

Approach 2: parallel search 1 f2 ε 0 ε 1 1 ε 2 1 ε 3 1 ε k-1 1 ε k f1 11 / 18

Approach 3: parallel search 2 f2 ε 0 f1 12 / 18

Approach 3: parallel search 2 f2 generate m columns for RMP ε 0 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 generate m columns for RMP ε 1 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 ε 2 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 generate m columns for RMP ε 2 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 ε 2 f1 12 / 18

Approach 3: parallel search 2 f2 ε 0 ε 1 ε 2 ε k-1 ε k f1 12 / 18

The Restricted Master Problem (RMP) � minimize c k θ k r k ∈ Ω 1 Constraints � − z ij + a jk θ k ≥ 0 ( w i ∈ W , v j ∈ V ) r k ∈ Ω 1 � a jk θ k ≥ 1 ( v j ∈ T ) r k ∈ Ω 1 Cov max − c ij z ij ≥ 0 ( w i ∈ W , v j ∈ V ) � ≥ 1 ( w i ∈ W ) z ij v j ∈ V − Cov max ≥ − ε 13 / 18

The Restricted Master Problem (RMP) � minimize c k θ k r k ∈ Ω 1 Constraints dual variables � − z ij + a jk θ k ≥ 0 ( w i ∈ W , v j ∈ V ) α ij r k ∈ Ω 1 � a jk θ k ≥ 1 ( v j ∈ T ) ϕ j r k ∈ Ω 1 Cov max − c ij z ij ≥ 0 ( w i ∈ W , v j ∈ V ) γ ij � z ij ≥ 1 ( w i ∈ W ) β i v j ∈ V − Cov max ≥ − ε λ 13 / 18

Dual of RMP � � maximize − ελ + β i + ϕ j w i ∈ W v j ∈ T subject to: � � a jk α ij + a jk ϕ j ≤ c k ( r k ∈ Ω 1 ) w i ∈ W v j ∈ T v j ∈ V � − λ + ≤ 0 γ ij w i ∈ W v j ∈ V − c ij γ ij + β i − α ij ≤ 0 ( w i ∈ W , v j ∈ V ) 14 / 18

Definition of sub-problem � � Find routes such that c k − a jk α ij − a jk ϕ j < 0 . w i ∈ W v j ∈ T v j ∈ V Let α ∗ hj = α hj if v j ∈ V , w h ∈ W and 0 otherwise. Let ϕ ∗ j = ϕ j if v j ∈ T and 0 otherwise. Let A be the set of arcs formed between two nodes of V . Let x ijk = 1 if route r k uses arc ( v i , v j ) and 0 otherwise. � � Note : c k = and a jk = x ijk c ij x ijk ( v i , v j ) ∈ A { v i ∈ V | ( v i , v j ) ∈ A } � � � α ∗ � ϕ ∗ c ij x ijk − hj x ijk − j x ijk < 0 . So ( v i , v j ) ∈ A ( v i , v j ) ∈ A v h ∈ W ( v i , v j ) ∈ A 15 / 18

Definition of sub-problem � � � c ij − ϕ ∗ � α ∗ j − x ijk < 0 . hj ( v i , v j ) ∈ A v h ∈ W 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 c ij − ϕ ∗ α ∗ � j − hj . v h ∈ W An elementary shortest path problem with resource constraints Solved by the Decremental State Space Relaxation (DSSR) algorithm [Righini and Salani, 2008]. 16 / 18