Column Generation Algorithms for the Capacitated m -Ring-Star Problem - PowerPoint PPT Presentation

Column Generation Algorithms for the Capacitated m -Ring-Star Problem 1 Edna Ayako Hoshino and Cid Carvalho de Souza University of Campinas - UNICAMP Institute of Computing - IC may, 2008 1 Capes/PICDT,CNPq – Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico,FAPESP – Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 1 / 39

Overview Introduction 1 Related Problem Definition of the Problem Examples Motivations 2 Related Work Our Proposal 3 The Techniques Set Covering Model Computational Results 4 Conclusions and Further Works 5 Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 2 / 39

Introduction Related Problem Capacitated Vehicle Routing Problem (CVRP) Figure: CVRP instance with | U | = 15, m = 3 and Q = 6. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 3 / 39

Introduction Related Problem A Solution for the CVRP Figure: A solution for the CVRP instance with m = 3, Q = 6 and graph in Figure 1. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 4 / 39

Introduction Related Problem Capacitated Vehicle Routing Problem (CVRP) Given: a graph G = ( V , E ) where V = { 0 } ∪ U , (depot 0 and a set of customers U ); a fleet of m identical vehicles (each of them having a capacity Q ); costs c e ≥ 0, ∀ e ∈ E ; demands d i ≥ 0 , ∀ i ∈ U . Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 5 / 39

Introduction Related Problem Capacitated Vehicle Routing Problem (CVRP) Given: a graph G = ( V , E ) where V = { 0 } ∪ U , (depot 0 and a set of customers U ); a fleet of m identical vehicles (each of them having a capacity Q ); costs c e ≥ 0, ∀ e ∈ E ; demands d i ≥ 0 , ∀ i ∈ U . The CVRP consists of finding routes for m vehicles such that: each route starts and ends at the depot; each customer is visited by a single vehicle; the total demand of all customers in any route is at most Q , and; the sum of the costs of all routes is minimum. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 5 / 39

Introduction Related Problem Alternative Solution Figure: A solution that allows some customers stay outside of all routes. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 6 / 39

Introduction Definition of the Problem Capacitated m -Ring-Star Problem (CmRSP) Given: a mixed graph G = ( V , E ∪ A ) where V = { 0 } ∪ U ∪ W (depot 0, a set of customers U and a set of Steiner points W ); unitary demands; integer values m and Q ; route costs c e > 0, ∀ e ∈ E = { ( i , j ) : i , j ∈ V } , satisfying the triangular inequalities; connection costs w e > 0, ∀ e ∈ A ⊆ { ij : i ∈ U , j ∈ U ∪ W } . Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 7 / 39

Introduction Definition of the Problem Capacitated m -Ring-Star Problem (CmRSP) Given: a mixed graph G = ( V , E ∪ A ) where V = { 0 } ∪ U ∪ W (depot 0, a set of customers U and a set of Steiner points W ); unitary demands; integer values m and Q ; route costs c e > 0, ∀ e ∈ E = { ( i , j ) : i , j ∈ V } , satisfying the triangular inequalities; connection costs w e > 0, ∀ e ∈ A ⊆ { ij : i ∈ U , j ∈ U ∪ W } . The CmRSP consists of finding m Q -ring-stars with minimum costs and covering all customers. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 7 / 39

Introduction Definition of the Problem Capacitated m -Ring-Star Problem (CmRSP) Given: a mixed graph G = ( V , E ∪ A ) where V = { 0 } ∪ U ∪ W (depot 0, a set of customers U and a set of Steiner points W ); unitary demands; integer values m and Q ; route costs c e > 0, ∀ e ∈ E = { ( i , j ) : i , j ∈ V } , satisfying the triangular inequalities; connection costs w e > 0, ∀ e ∈ A ⊆ { ij : i ∈ U , j ∈ U ∪ W } . The CmRSP consists of finding m Q -ring-stars with minimum costs and covering all customers. a pair ( R , S ) is a Q -ring-star if: R ⊆ E is a cycle passing by the depot 0; ij ∈ S ⊆ A such that i �∈ V [ R ] and j ∈ V [ R ]; | U ∩ ( V [ R ] ∪ V [ S ]) | ≤ Q . Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 7 / 39

Introduction Definition of the Problem Capacitated m -Ring-Star Problem (CmRSP) Given: a mixed graph G = ( V , E ∪ A ) where V = { 0 } ∪ U ∪ W (depot 0, a set of customers U and a set of Steiner points W ); unitary demands; integer values m and Q ; route costs c e > 0, ∀ e ∈ E = { ( i , j ) : i , j ∈ V } , satisfying the triangular inequalities; connection costs w e > 0, ∀ e ∈ A ⊆ { ij : i ∈ U , j ∈ U ∪ W } . The CmRSP consists of finding m Q -ring-stars with minimum costs and covering all customers. a pair ( R , S ) is a Q -ring-star if: R ⊆ E is a cycle passing by the depot 0; ij ∈ S ⊆ A such that i �∈ V [ R ] and j ∈ V [ R ]; | U ∩ ( V [ R ] ∪ V [ S ]) | ≤ Q . We say that a Q -ring-star covers a customer i if i ∈ V [ R ] ∪ V [ S ]. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 7 / 39

Introduction Examples Example of a Q -ring-star l g h c d 3 f 6 2 m 3 i 5 10 0 10 4 e k n 5 5 10 10 3 4 6 r o q p 2 10 j b a Figure: Two 9- ring - star s ( { (0 , g ) , ( g , h ) , ( h , i ) , ( i , j ) , ( j , k ) , ( k , 0) } , { mi , ni , oj , pk , qk } ) and ( { (0 , a ) , ( a , b ) , ( b , c ) , ( c , 0) } , { ( ec , fc } ). Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 8 / 39

Introduction Examples Example of a CmRSP Instance 90 deposito cliente 80 steiner 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 Figure: Instance eil51.tsp with | U | = 25. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 9 / 39

Introduction Examples Example of a Q -ring-star 90 deposito cliente 80 steiner anel estrela 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 Figure: 8-ring-star for the instance in Figure 5. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 10 / 39

Introduction Examples A Solution for the CmRSP instance 90 deposito cliente 80 steiner solucao arcos 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 Figure: A solution for eil51.tsp with m = 3 e Q = 10. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 11 / 39

Motivations Related Work Integer Programming (IP) Formulation Formulation proposed by Baldacci et al and used into a branch-and-cut algorithm. X X (BC) min c e x e + w ij z ij e ∈ E ij ∈ A X subject to x e = 2 m (1) e ∈ δ (0) X x e = 2 y i , ∀ i ∈ V \ { 0 } (2) e ∈ δ ( i ) X z ij + y i = 1 , ∀ i ∈ U (3) ij ∈ A x e ≥ 2 X X X z ij , ∀ S ⊆ V \ { 0 } : S � = {} (4) Q e ∈ δ ( S ) i ∈ U j ∈ S : ij ∈ A y ∈ { 0 , 1 } | V ′ | , z ij ∈ { 0 , 1 } , x ij ∈ { 0 , 1 } . (5) Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 12 / 39

Motivations Related Work Motivations Practical applications arising in telecommunications and logistics; Large fiber optics networks design; Logistics of product distribution; School bus allocation. Just one exact algorithm for CmRSP is reported, to our knowledge; In general, set covering models provide tight relaxations. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 13 / 39

Motivations Related Work Motivations Practical applications arising in telecommunications and logistics; Large fiber optics networks design; Logistics of product distribution; School bus allocation. Just one exact algorithm for CmRSP is reported, to our knowledge; In general, set covering models provide tight relaxations. Our Objective Evaluate the use of a set covering model for the CmRSP together a column generation algorithm to solve it. Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 13 / 39

Our Proposal The Techniques Column Generation (P) min c λ s.a. A λ ≥ 1 , ∀ i ∈ N (6) λ ∈ { 0 , 1 } p (7) Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 14 / 39

Our Proposal The Techniques Column Generation (P) min c λ s.a. A λ ≥ 1 , ∀ i ∈ N (6) λ ∈ { 0 , 1 } p (7) p is exponential in | N | , i.e., the total number of columns is big! Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 14 / 39

Our Proposal The Techniques Column Generation (P) min c λ s.a. A λ ≥ 1 , ∀ i ∈ N (6) λ ∈ { 0 , 1 } p (7) p is exponential in | N | , i.e., the total number of columns is big! Consider just a few number of the columns and construct other columns implicitly by solving the pricing problem . Edna Hoshino, Cid de Souza (IC-UNICAMP) Column Generation for the CmRSP may, 2008 14 / 39