Model Heuristic Polyhedral Structure B&C Robust Network Design with Several Traffic Scenarios Models and Algorithms Eduardo ´ Alvarez-Miranda 1 Valentina Cacchiani 1 Tim Dorneth 2 unger 2 Frauke Liers 2 Andrea Lodi 1 Tiziano Parriani 1 Michael J¨ Daniel R. Schmidt 2 1 DEIS, Universit` a di Bologna 2 Institut f¨ ur Informatik, Universit¨ at zu K¨ oln Financial support is acknowledged from the Ateneo Italo-Tedesco VIGONI programme and the DFG 1
Model Heuristic Polyhedral Structure B&C Outline 1 A Robust Network Design Model 2 A Large Neighborhood Search Heuristic 3 The Polyhedral Structure 4 A Branch-and-Cut Algorithm 2
Model Heuristic Polyhedral Structure B&C About Robust Network Design Given: undirected graph G = ( V , E ) cost vector c : E → Z ≥ 0 K integer balance vectors b 1 , . . . , b K : V → Z ( “scenarios” ) 3
Model Heuristic Polyhedral Structure B&C About Robust Network Design Given: undirected graph G = ( V , E ) cost vector c : E → Z ≥ 0 K integer balance vectors b 1 , . . . , b K : V → Z ( “scenarios” ) Task: find integer capacities u : E → Z ≥ 0 s.t. there is directed b i -flow f i w.r.t. u for all i = 1 , . . . , K . f i u , v + f i (1) v , u ≤ u u , v for all { u , v } ∈ E ( f i u , v − f i v , u ) = b i � (2) for all v ∈ V v u ∈ δ ( v ) minimize c T u 3
Model Heuristic Polyhedral Structure B&C Flow based IP-formulation � min c u , v u u , v { u , v }∈ E f i u , v + f i v , u ≤ u u , v for all { u , v } ∈ E , i = 1 , . . . , K � ( f i u , v − f i v , u ) = b i for all v ∈ V , i = 1 , . . . , K v u ∈ δ ( v ) u u , v ∈ Z ≥ 0 for all { u , v } ∈ E f i u , v , f i v , u ∈ Z ≥ 0 for all { u , v } ∈ E , i = 1 , . . . , K 4
Model Heuristic Polyhedral Structure B&C Applications streaming networks with changing customer demands planning of communication networks, public transport 5
Model Heuristic Polyhedral Structure B&C Applications streaming networks with changing customer demands planning of communication networks, public transport Related work non-robust, multi-commodity setting [Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007] 5
Model Heuristic Polyhedral Structure B&C Applications streaming networks with changing customer demands planning of communication networks, public transport Related work non-robust, multi-commodity setting [Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007] robust, multi-commodity setting [Koster, Kutschka, Raack 2010, 2011] 5
Model Heuristic Polyhedral Structure B&C Applications streaming networks with changing customer demands planning of communication networks, public transport Related work non-robust, multi-commodity setting [Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007] robust, multi-commodity setting [Koster, Kutschka, Raack 2010, 2011] robust, single commodity, scenarios given by polytope [Ben-Ameur, Kerivin 2005] 5
Model Heuristic Polyhedral Structure B&C Applications streaming networks with changing customer demands planning of communication networks, public transport Related work non-robust, multi-commodity setting [Atamt¨ urk, 2001] [Koster, Orlowski, Raack, Wess¨ aly 2007, 2008] [Avella, Mattia, Sassano 2007] robust, multi-commodity setting [Koster, Kutschka, Raack 2010, 2011] robust, single commodity, scenarios given by polytope [Ben-Ameur, Kerivin 2005] robust, single commodity [Buchheim, Liers, Sanit` a 2008] [Oriolo, Sanit` a] 5
Model Heuristic Polyhedral Structure B&C Complexity polynomial cases single scenario, multiple sources/sinks (minimum cost flow) 6
Model Heuristic Polyhedral Structure B&C Complexity polynomial cases single scenario, multiple sources/sinks (minimum cost flow) Problem is already NP-hard if. . . K = 3 scenarios, binary demands, single-source [Sanit` a 2009] K = 2 scenarios, multiple sources/sinks [Sanit` a 2009] 6
Model Heuristic Polyhedral Structure B&C Complexity polynomial cases single scenario, multiple sources/sinks (minimum cost flow) Problem is already NP-hard if. . . K = 3 scenarios, binary demands, single-source [Sanit` a 2009] K = 2 scenarios, multiple sources/sinks [Sanit` a 2009] open cases K = 2 scenarios, binary demands, single-source 6
Model Heuristic Polyhedral Structure B&C Example (unit edge costs) Scenario 1 2 -2 c ≡ 1 7
Model Heuristic Polyhedral Structure B&C Example (unit edge costs) Scenario 1 Scenario 2 2 -2 1 c ≡ 1 -1 7
Model Heuristic Polyhedral Structure B&C Example (unit edge costs) Scenario 1 Scenario 2 2 2 -2 1 c ≡ 1 1 -1 1 Feasible solution: 2 cost: 4 1 0 1 7
Model Heuristic Polyhedral Structure B&C Example (unit edge costs) Scenario 1 Scenario 2 2 1 2 -2 1 c ≡ 1 1 -1 Feasible solution: 2 cost: 3 0 1 0 7
Model Heuristic Polyhedral Structure B&C Heuristic: Constructive Phase 1 insert auxilliary edge (for all edges e ∈ E ) 0 / 0 capacity that is already installed � c e c e / ∞ additional capacity must be paid 8
Model Heuristic Polyhedral Structure B&C Heuristic: Constructive Phase 1 insert auxilliary edge (for all edges e ∈ E ) 0 / 0 capacity that is already installed � c e c e / ∞ additional capacity must be paid 2 For q = 1 , . . . , K : 1 compute MinCost flow for q -th scenario f e 2 update capacities for subsequent scenarios 0 / ( u e + f e ) c e / ∞ 8
Model Heuristic Polyhedral Structure B&C Heuristic: Constructive Phase 1 insert auxilliary edge (for all edges e ∈ E ) 0 / 0 capacity that is already installed � c e c e / ∞ additional capacity must be paid 2 For q = 1 , . . . , K : Caution: The order matters! 1 compute MinCost flow for q -th scenario f e 2 update capacities for subsequent scenarios 0 / ( u e + f e ) c e / ∞ 8
Model Heuristic Polyhedral Structure B&C Heuristic: Improvement Phase Large Neighborhood Search use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T 9
Model Heuristic Polyhedral Structure B&C Heuristic: Improvement Phase Large Neighborhood Search use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T Definition Let f 1 , . . . , f K be flows from constructive phase Set U e := max { f i e | i = 1 , . . . , K } for all e ∈ E Set E ′ := E \ { e ∈ E | f i e = 0 ∀ i } 9
Model Heuristic Polyhedral Structure B&C Heuristic: Improvement Phase Large Neighborhood Search use constructive phase information to constrain variables in IP find good solution“close”to constructed solution control closeness (neighborhood size) by parameter T Definition Let f 1 , . . . , f K be flows from constructive phase Set U e := max { f i e | i = 1 , . . . , K } for all e ∈ E Set E ′ := E \ { e ∈ E | f i e = 0 ∀ i } Algorithm Limit u e by U e plus some limited tolerance (neighborhood) Solve flow-based IP on ( V , E ′ ). 9
Model Heuristic Polyhedral Structure B&C Improvement phase IP � min c u , v ( U u , v + w u , v ) { u , v }∈ E ′ � ( f i u , v − f i v , u ) = b i for all v ∈ V , i = 1 , . . . , K v u ∈ δ ( v ) f i u , v + f i for all { u , v } ∈ E ′ , i = 1 , . . . , K v , u ≤ U u , v + w u , v � w u , v ≤ T { u , v }∈ E ′ for all { u , v } ∈ E ′ w u , v ≥ 0 for all { u , v } ∈ E ′ u u , v ∈ Z ≥ 0 f i u , v , f i for all { u , v } ∈ E ′ , i = 1 , . . . , K v , u ∈ Z ≥ 0 10
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs set random supply from 1 , . . . , 10 11
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs 2 -2 set random supply from 1 , . . . , 10 11
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs 5 -5 set random supply from 1 , . . . , 10 11
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs -1 1 set random supply from 1 , . . . , 10 11
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs -8 8 set random supply from 1 , . . . , 10 11
Model Heuristic Polyhedral Structure B&C Difficult instances Instances on d -dimensional hypercube opposite nodes on hypercube are terminal pairs -8 8 set random supply from 1 , . . . , 10 if all supplies 1: integrality gap converges to 2 as d → ∞ 11
Model Heuristic Polyhedral Structure B&C Computational Results Cplex 12.3, CS2 code by Goldberg for CP Intel(R) Core(TM) i7 CPU, 64 bit, 1.73 GHz, 6 Gb RAM time limit of 7200 sec. average values 12
Recommend
More recommend
