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Constrained Integer Network Flows April 25, 2002

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Constrained Integer Network Flows Traditional Network Problems With Side-Constraints and Integrality Requirements Motivated By Applications in Diverse Fields, Including: –Military Mission-Planning –Logistics –Telecommunications

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Minimum-Cost Network Flows Definition –Minimize Flow Cost –b Represents Demands and Supplies –Special Properties Spanning Tree Basis A Is Totally Unimodular Integer Solutions if b, l, and u Are Integer Row Rank of A Is | V |-1 Special Structure Has Lead To Highly Efficient Algorithms MCNF

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Shortest-Path Problems One-to-One (SP) –Find Shortest-Path From s To t –b = e t - e s One-to-All (ASP) –Find Shortest-path From s To All Other Vertices –b = 1 - |V|e s Special Solution Algorithms –Label Setting –Label Correcting SP/ ASP

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Resource-Constrained Shortest Path Find Shortest Path From s To t Limited By Constraint on a Resource Side-Constraint Destroys Special Structure of MCNF Solutions Non- Integer Unless Integrality Enforced RCSP

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RCSP: Aircraft Routing Time-Critical-Target Available For Certain Time Period Aircraft Need To Be Diverted To Target Planners Wish To Minimize Threats Encountered by Aircraft Multiple Aircraft ( 100s or 1000s ) Considered for Diversion

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RCSP: Aircraft Routing Grid Network Representation Arc Cost: Threat Arc Side-Constraint Value: Time Total Time, Including Decision Making, Is Constrained *Diagonal Arcs Are Included, But Not Shown

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Multicommodity Network Flow Minimize Cost of Flows For All Commodities Total Flow for All Commodities on Arcs Is Restricted Non-Integer Solutions Solution Strategies –Primal Partitioning –Price & Resource Directive Decompositions –Heuristics MCF

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Origin-Destination Integer MCF Specialization of MCF –One Origin & One Destination Per Commodity –Commodity Flow Follows a Single Path Integer-Programming Problem Two Formulations –Node-Arc –Path-Based

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Origin-Destination Integer MCF ODIMCF1: Node-Arc Formulation Rows: | V || K | + | E | Variables: | K || E | ODIMCF2: Path-Based Formulation Rows: | K | + | E | Variables: Dependent on Network Structure ODIMCF2 ODIMCF1

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ODIMCF: Rail-Car Movement Grain-Cars Are “Blocked” for Movement Blocks Move From Origin To Destination through Intermediate Stations Grain-Trains Limited on Total Length and Weight Blocks Need To Reach Destinations ASAP

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ODIMCF: Rail-Car Movement Arcs - Catching a Train or Remaining at a Station Vertex - Station+Train Arrival/Departure Stations B C A Catch a Train Remain at A Time a2a3a4 b2 c2 a1 b5 c3c4c1 b1b3b4

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ODIMCF: MPLS Networks Traffic Is Grouped by Origin-Destination Pair Each Group Moves Across the Network on a Label-Switched Path (LSP) LSPs Need Not Be Shortest-Paths MPLS’s Objective Is Improved Reliability, Lower Congestion, & Meeting Quality-of- Service (QoS) Guarantees

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ODIMCF: MPLS Networks LSR MPLS Network MPLS Switches LSR: Label-Switch Router LSP IP Net

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Binary MCF MCF Specialization –x k Binary – l = 0 –b k = e t - e s ODIMCF Variant –q k = 1 for all k BMCF

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Current & Proposed Algorithmic Approaches

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RCSP: Current Algorithms Lagrangian Relax- ation, RRCSP( ) –Lagrangian 1 Network Reduction Techniques Use Subgradient Optimization To Find Lower Bound Tree Search to Build a Path –Lagrangian 2 Bracket Optimal Solution Changing Finish Off With k- shortest Paths RRCSP( )

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RCSP: Proposed Algorithm Objectives –Solve RCSP For One Origin, s, and Many Destinations, T –Reduce Cumulative Solution Time Compared To Current Strategies Overview –Solves Relaxation (ASP( )) –Relaxation Costs Are Convex Combination of c and s –ASP( ) Solved Predetermined Number of Times

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RCSP: Proposed Algorithm Algorithm –Relax Side- Constraint Forming ASP( ) ASP With s As Origin –Select n Values for 0 1 –Solve ASP( ) For Each Value of –For Each t in T Find Smallest Meeting Side-Constraint For t ASP( )

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RCSP: Proposed Algorithm Aircraft Routing Example –c - Threat on Arcs –s - Time To Traverse Arcs –10 Values for Evaluated –Results Recorded For 2 Points And Target Accumulated Time and Distance For Each Value of

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RCSP: Proposed Algorithm Intermediate Routing Option = 0.44 Minimum Threat Routing = 0.0

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RCSP: Proposed Algorithm Minimum Time Routing = 1.0 Accumulated Threat vs Time To Target

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RCSP: Proposed Algorithm Further Considerations –Normalization of c and s –Reoptimization of ASP( ) –Number of Iterations (Values of ) –Usage As Starting Solution For RCSP –Other Uses

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ODIMCF: Current Algorithms Techniques For Generic Binary IP Branch-and-Price-and-Cut –Designed Specifically For ODIMCF –Incorporates Path-Based Formulation (ODIMCF2) LP Relaxations With Price-Directive Decomposition Branch-and-Bound Cutting Planes

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ODIMCF: Current Algorithms Branch-and-Price- and-Cut (cont.) –Algorithmic Steps Solve LP Relaxation At Each Node Using: –Column-Generation »Pricing Done As SP –Lifted-Cover Inequalities Branch By Forbidding a Set of Arcs For a Commodity –Select Commodity k –Find Vertex d At Which Flow Splits –Create 2 Nodes in Tree Each Forbidding ~Half the Arcs at d –Has Difficulty Many Commodities |A|/|V| ~2

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ODIMCF: Proposed Algorithm Heuristic Based On Market Prices Circumstances –Large Sparse Networks –Many Commodities –Arcs Capable of Supporting Multiple Commodities

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ODIMCF: Proposed Algorithm Arc Costs, c ij ´ = f(r ij, u ij, c ij, q k ) R –Uses Non-Linear Price Curve, p(z, u ij ) R –Based On Original Arc Cost, c ij Upper Bound, u ij Current Capacity Usage, r ij Demand of Commodity, q k

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ODIMCF: Proposed Algorithm Marginal Arc Cost Demand, q k Current Usage, r ij Area = Arc Cost, c´ ij p(z, u ij ) Upper Bound, u ij c´ ij = f(r ij, u ij, c ij, q k ) As an Area

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ODIMCF: Proposed Algorithm Arc Cost For Increasing r ij

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ODIMCF: Proposed Algorithm Total System Cost Current System Cost Total Additional System Cost Additional Cost To Other Commodities Arc Cost To Commodity Current Usage, r ij

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ODIMCF: Proposed Algorithm Basic Algorithm –Initial SP Solutions –Update r –Until Stopping Criteria Met Randomly Choose k Calculate New Arc Costs Solve SP Update r Selection Strategy –Iterative –Randomized Infeasible Inter- mediate Solutions Stopping Criteria –Feasible –Quality –Iteration Limit

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ODIMCF: Proposed Algorithm Considerations –Form of p(z, u ij ) –Commodity Differentiation Under-Capacitated Net Preferential Routing –Selection Strategy –Advanced Start –Cooling Off of p(z, u ij ) Step 0 - SP Steps 1… Increasing Enforcement of u 1 0 2 3 4

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ODIMCF: Proposed Algorithm

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*CPLEX65 Used MIP To Find Integer Solution. All Other Problems Solved As LP Relaxations With No Attempt At Integer Solution.

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BMCF: Proposed Algorithm Modification of Proposed Algorithm For ODIMCF Commodities Are Aggregated By Origin –A is the Set of Aggregations Pure Network Sub-Problems Replace SPs of ODIMCF

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BMCF: Proposed Algorithm Original Commodities –Demands of 1 –Single Origin & Destination –SP Aggregations –Demands 1 –Single Origin –Multiple Destinations –MCNF

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BMCF: Proposed Algorithm Aggregation MCNFs Solved On Modified Network –Each Original Arc Is Replaced With q a Parallel Arcs –Parallel Arcs Have Convex Costs Derived From p(z, u ij ) Upper Bounds of 1 ij (0, u ij ) c ij ij (0, 1) c ij3 (0, 1) c ij2 (0, 1) c ij1

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BMCF: Proposed Algorithm Demand, q a = 3 Current Usage, r ij p(z, u ij ) Upper Bound, u ij Parallel Arc Costs One Unit of Flow c ij1 c ij3 c ij2

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BMCF: Proposed Algorithm Basic Algorithm –Form Aggregates –Solve Initial MCNFs –Update r –Until Stopping Criteria Met Randomly Choose a Create Parallel Arcs Calculate Arc Costs Solve MCNF Update r Considerations –ODIMCF Considerations –ODIMCF vs BMCF –Aggregation Strategy Multiple Aggregations per Vertex Which Commodities To Group

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Expected Contributions Will Address Important Problems With Wide Range of Applications Efficient Algorithms Will Have a Significant Impact in Several Disparate Fields

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Notation A - Matrix x - Vector 0 - Vector of All 0’s 1 - Vector of All 1’s e i - 0 With a 1 at i th Position x i - i th element of x x - Scalar A - Set | A | - Cardinality of A - Empty Set R - Set of Reals B - {0,1}, Binary Set R m x n - Set of m x n Real Matrices B m - Set of Binary, m Dimensional Vectors

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Notation: Networks A - Node-Arc Incidence Matrix x - Arc Flow Variables c - Arc Costs s - Arc Resource Constraint Values u - Arc Upper Bounds l - Arc Lower Bounds b - Demand Vector ij ( l ij, u ij ) c ij, s ij All Networks Are Directed x ij Is the Flow Variable for ( i, j ) E - Set of Arcs V - Set of Vertices

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Notation: Problem Abbreviations MCNF - Minimum- Cost Network Flow SP - Shortest Path ASP - One-To-All Shortest-Path RCSP - Resource Constrained Shortest-Path MCF - Multi- commodity Flow ODIMCF - Origin Destination Integer Multicommodity Network Flow BMCF - Binary Multicommodity Network Flow

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