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 1 / 24

Definition of a multi-objective optimization problem � min F ( x ) = ( f 1 ( x ) , f 2 ( x ) , . . . , f n ( x )) ( MOP ) = s . t . x ∈ Ω where : n ≥ 2 : number of objective functions F = ( f 1 , f 2 , . . . , f n ) : vector of objective functions Ω ⊆ R m : feasible set of solutions Y = F (Ω) : feasible set in objective space x = ( x 1 , x 2 , . . . , x m ) ∈ Ω : variable vector, variables y = ( y 1 , y 2 , . . . , y n ) ∈ Y with y i = f i ( x ) : vector of objective function values 2 / 24

Dominance and Pareto Optimality A solution x dominates ( � ) another solution y if and only if ∀ i ∈ { 1 , . . . , n } , f i ( x ) ≤ f i ( y ) and ∃ i ∈ { 1 , . . . , n } such that f i ( x ) < f i ( y ) . f2 A B D C E f1 Pareto optimal solution A solution is said to be Pareto optimal if no other feasible solution dominates it. 3 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 lb 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 f 1 lb 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 lb 2 f 1 lb 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 lb 2 f 1 lb 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 ideal point lb 2 f 1 lb 1 4 / 24

Lower bound of a MOIP [Villarreal and Karwan, 1981] f 2 ideal point lb 2 f 1 lb 1 4 / 24

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. 5 / 24

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 6 / 24

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) 7 / 24

Approach 1: standard 8 / 24

Approach 1: standard f2 ε 0 f1 8 / 24

Approach 1: standard f2 ε 0 f1 8 / 24

Approach 1: standard f2 ε 0 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 ε 2 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 ε 2 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 ε 2 f1 8 / 24

Approach 1: standard f2 ε 0 ε 1 ε 2 ε k-1 ε k f1 8 / 24

Approach 2: sequential search (1 iteration of CG at each step) 9 / 24

Approach 2: sequential search (1 iteration of CG at each step) f2 ε 0 f1 9 / 24

Approach 2: sequential search (1 iteration of CG at each step) f2 ε 0 f1 9 / 24

Approach 2: sequential search (1 iteration of CG at each step) f2 ε 0 ε 1 f1 9 / 24

Approach 2: sequential search (1 iteration of CG at each step) f2 ε 0 ε 1 f1 9 / 24

Approach 2: sequential search (1 iteration of CG at each step) f2 ε 0 ε 1 ε 2 ε k-1 ε k f1 9 / 24

Approach 3: parallel search 1 10 / 24

Approach 3: parallel search 1 f2 ε 0 ε 1 0 ε 2 0 ε k-1 0 ε k f1 10 / 24

Approach 3: parallel search 1 f2 generate m/k columns ε 0 generate m/k ε 1 0 columns generate m/k ε 2 0 columns generate m/k columns generate m/k columns ε k-1 0 ε k f1 10 / 24

Approach 3: parallel search 1 f2 ε 0 ε k f1 10 / 24

Approach 3: parallel search 1 f2 ε 0 ε 1 1 ε 2 1 ε k-1 1 ε k f1 10 / 24

Approach 3: parallel search 2 f2 ε 0 ε 1 0 ε 2 0 ε k-1 0 ε k f1 11 / 24

Approach 3: parallel search 2 f2 generate n columns ε 0 ε 1 0 skip ε 2 0 skip generate n columns skip ε k-1 0 ε k f1 11 / 24

Approach 3: parallel search 2 f2 ε 0 ε k f1 11 / 24

Approach 3: parallel search 2 f2 ε 0 ε 1 1 ε 2 1 ε k-1 1 ε k f1 11 / 24

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 ′ . May be visited V MUST be visited : T MUST be covered : W Vehicle route Cover 12 / 24

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 . May be visited V MUST be visited : T MUST be covered : W Vehicle routes Cover distance 13 / 24

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 . 14 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

The cover distance induced by a set of routes 15 / 24

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 node v j ∈ V ∗ ( V \{ v 0 } ) is used to cover node w i ∈ W and 0 otherwise a ik : 1 if r k uses node 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 16 / 24

A set-covering model for the BOMCTP Constraints ( w i ∈ W , v j ∈ V ∗ ) � − z ij + a jk θ k ≥ 0 r k ∈ Ω � ( v j ∈ T ∗ ) a jk θ k ≥ 1 r k ∈ Ω ( w i ∈ W , v j ∈ V ∗ ) Cov max − c ij z ij ≥ 0 � ≥ 1 ( w i ∈ W ) z ij v j ∈ V ∗ Cov max ≥ 0 ( w i ∈ W , v j ∈ V ∗ ) z ij ∈ { 0 , 1 } θ k ∈ { 0 , 1 } ( r k ∈ Ω) 17 / 24

The Restricted Master Problem (RMP) � minimize c k θ k r k ∈ Ω 1 Constraints ( w i ∈ W , v j ∈ V ∗ ) � − z ij + a jk θ k ≥ 0 r k ∈ Ω 1 � ( v j ∈ T ∗ ) a jk θ k ≥ 1 r k ∈ Ω 1 ( w i ∈ W , v j ∈ V ∗ ) Cov max − c ij z ij ≥ 0 � ≥ 1 ( w i ∈ W ) z ij v j ∈ V ∗ − Cov max ≥ − ε 18 / 24

The Restricted Master Problem (RMP) � minimize c k θ k r k ∈ Ω 1 Constraints dual variables � ( w i ∈ W , v j ∈ V ∗ ) − z ij + a jk θ k ≥ 0 α ij r k ∈ Ω 1 ( v j ∈ T ∗ ) � a jk θ k ≥ 1 ϕ j r k ∈ Ω 1 ( w i ∈ W , v j ∈ V ∗ ) Cov max − c ij z ij ≥ 0 γ ij � z ij ≥ 1 ( w i ∈ W ) β i v j ∈ V ∗ − Cov max ≥ − ε λ 18 / 24

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 ∗ ( w i ∈ W , v j ∈ V ∗ ) − c ij γ ij + β i − α ij ≤ 0 19 / 24

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. j = ϕ j if v j ∈ T ∗ and 0 otherwise. Let ϕ ∗ 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 = x ijk c ij and a jk = 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 20 / 24

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 constraints on the length and the maximum number of nodes visited by a path. Costs are set to c ij − ϕ ∗ � α ∗ j − hj v h ∈ W An elementary shortest path problem with resource constraints 21 / 24