Incremental Network Design Martin Savelsbergh Georgia Tech Aussois - PowerPoint PPT Presentation

Incremental Network Design Martin Savelsbergh Georgia Tech Aussois 2016

Collaborators • Thomas Kalinowski – University of Newcastle • Dmytro Matsypura – University of Sydney

Outline • Motivation • Incremental Network Design with Maximum Flows – IP formulations – Heuristics – Special case: unit arc capacities • Open questions • Other results • Future research

Motivation Some of my recent research projects: • Optimizing coal export chain – Transporting coal by rail from mines to terminals at a port • Optimizing carbon capture and storage – Transporting CO 2 captured at power plants via a pipeline to subsurface aquifers • Optimizing renewable energy integration – Transporting electricity produced by renewable energy sources to consumers

Motivation Planning Expansions Over Time Network Network Use Operational

Motivation • Where should upgrades/expansions of a network take place? • When should upgrades/expansions of a network take place? • In what sequence should upgrades/expansions of a network take place when budget and/or resources are constrained?

Incremental Network Design • Network optimization problem – Minimum spanning tree, shortest path, maximum flow, minimum cost flow, TSP, … • Multiple periods – Incur the cost / realize the benefit of an optimal solution to the network optimization problem in each period – Minimize the total cost / maximize the total benefit over the planning horizon • Option to invest in capacity expansion – Limited budget to invest in capacity expansion in each period (capacity expansions may decrease cost / increase benefit)

Incremental Network Design SIMPLEST VARIANT: build single edge per period at no cost

Questions of interest • How difficult are incremental network design problems? • If they are difficult, are there constant factor approximation algorithms? • If they are difficult, what size instances can we solve using integer programming techniques? • If they are difficult, are there effective and efficient heuristics?

Incremental Network Design with Maximum Flows • Given: – Graph D=(N, A e U A p ) , • Existing arcs and Potential arcs – Arc capacity c a – Source s – Sink t – Planning horizon T = | A p | – Option to construct one potential arc each period • Decide: – Which arc to construct in each period so as to maximize the total flow over the planning horizon (i.e., sum of the s-t flows in each period)

Incremental Network Design with Maximum Flows Which path to build first? 1-1-6 = 8 1-1-1-3 = 6 0-5-6 = 11 0-0-2-3 = 5 Note: We assume that an arc built in a period can immediately be used in that period.

Incremental Network Design with Maximum Flows COMPLEXITY

Incremental Network Design with Maximum Flows

Incremental Network Design with Maximum Flows if yes-instance 3+6+…+3n+3n+…+3n

Incremental Network Design with Maximum Flows INTEGER PROGRAMMING FORMULATIONS

Integer Programming flow on arc a Maximize flow out of source in period k Flow balance Arc capacity Arc capacity & forcing Budget Consistency arc a built in Initial condition or before period k Concern: Symmetry when multiple arcs need to be build to increase flow Note: Assumes T = |A p |+1 and that an arc built can be used only in subsequent periods

Integer Programming An Alternative Formulation #arcs built to make a flow of f+1 possible #arcs built to make a flow of f+k+1 possible Total flow:

Integer Programming Maximize total flow Flow balance Arc capacity Arc capacity & forcing Consistency Concern: Size when F – f is large

Example 2 2 b a c d 1 1 Solution:

Incremental Network Design with Maximum Flows HEURISTICS

Heuristics STEP 1: Determine the cardinality of a minimum set of potential arcs that have to be build to increase the maximum flow from f+k to at least f+k+1 (MinArcs) capacity of arc a flow on arc a set of potential arcs build arc a or not already built

Heuristics STEP 2: Among the minimal sets of potential arcs that have to be build to increase the maximum flow from f+k to at least f+k+1 choose one that increases the flow the most (MaxVal)

Quickest-increment restricted set of potential arcs A p

Quickest-to-ultimate Notation: r = F - f

Quickest-to-target

Remarks • Heuristics are not polynomial in general • Heuristics are polynomial if the maximum arc capacity of potential arcs is equal to 1 (MinArcs becomes a minimum cost flow problem) • If the maximization of the flow increment is omitted, then quickest-increment can be implemented to run in polynomial time even with general arc capacities

Incremental Network Design with Maximum Flows SPECIAL CASE: All capacities equal to 1

Quickest-to-ultimate path of k potential arcs Quickest-to-ultimate: 0-0-…-0-0 | 1-1-…-1-1 | 2-2 Opt: 0 | 1-1-…-1-1 | 1-1-…-1 | 2

Quickest-improvement Quickest-improvement: 0-0-…-0-0 | 1-1-…-1-1 | 1-1-…-1-1 | 2 Opt: 0-0-…-0-0 | 1-1-…1-1 | 2-2-…-2

Incremental Network Design with Maximum Flows SPECIAL CASE: All capacities equal to 1 and a bipartite graph

Special Case Incremental Network Design with Maximum Cardinality Matchings

Observations • Quickest-improvement does not always produce an optimal solution • Quickest-to-ultimate does not always produce an optimal solution • Quickest-to-ultimate is a 4/3-approximation algorithm

Quickest-increment is not optimal Quickest-increment: (5,8)(7,10) (1,2)(3,4)(5,6)(9,10)(11,12)(13,14) Value: 2*5+6*6+7=53 Optimal: (1,2)(3,4)(5,6) (9,10)(11,12)(13,14) (5,8)(7,10) Value: 3*5+3*6+2*7=54

Quickest-to-ultimate is not optimal Quickest-to-ultimate: (1,2)(3,4)(5,6)(7,8) (9,10)(11,12)(13,14)(15,16) Value: 4*6+4*7+2*8=68 Optimal: (1,2)(9,4) (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) Value: 2*6+7*7+8=69

Incremental Network Design with Maximum Flows OPEN QUESTIONS

Open Questions • Is incremental network design problem with maximum flows and capacities equal to 1 solvable in polynomial time or NP-complete? – Formulation 2 always gives integer solution – Formulation 2 is not TU • Is the incremental network design problem with maximum cardinality matchings solvable in polynomial time or NP-complete? • What are the approximation ratios of quickest-to- ultimate and quickest-improvement for the incremental network design problem with maximum cardinality matchings?

Incremental Network Design with Maximum Flows OTHER RESULTS

Other results • The Incremental Network Design Problem with Shortest Paths is NP-hard • The Incremental Network Design Problem with Minimum Spanning Trees is solvable in O (max{ n 2 , m log m }) time

Incremental Network Design with Maximum Flows FUTURE RESEARCH

• Multi-commodity flow • Investment budgets • Multi-period build times • Uncertainty

QUESTIONS ?

• M. Baxter, T. Elgindy, A.T. Ernst, T. Kalinowski, and M.W.P. Savelsbergh. “Incremental Network Design with Shortest Paths”. EJOR 238, 675-684, 2014. • T. Kalinowski, D. Matsypura, and M.W.P. Savelsbergh. “The Incremental Network Design Problem with Maximum Flows”, EJOR 242, 51-62, 2015. • C. Engel, T. Kalinowski, and M.W.P. Savelsbergh. “The Incremental Network Design Problem with Minimum Spanning Trees”, arXiv:1306.1926 [math.CO].