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Constrained Integer Network Flows. April 25, 2002. 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. Definition

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## Constrained Integer Network Flows

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**Constrained Integer Network Flows**April 25, 2002**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**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 Minimum-Cost Network Flows MCNF**One-to-One (SP)**Find Shortest-Path From s To t b = et - es One-to-All (ASP) Find Shortest-path From s To All Other Vertices b = 1 - |V|es Special Solution Algorithms Label Setting Label Correcting Shortest-Path Problems SP/ ASP**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 Resource-Constrained Shortest Path RCSP**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**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**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 Multicommodity Network Flow MCF**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**ODIMCF1: Node-Arc Formulation**Rows: |V||K| + |E| Variables: |K||E| ODIMCF2: Path-Based Formulation Rows: |K| + |E| Variables: Dependent on Network Structure Origin-Destination Integer MCF ODIMCF2 ODIMCF1**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**ODIMCF: Rail-Car Movement**• Arcs - Catching a Train or Remaining at a Station • Vertex - Station+Train Arrival/Departure Stations Remain at A A a1 a2 a3 a4 Catch a Train B b1 b2 b3 b4 b5 C c1 c2 c3 c4 Time**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**ODIMCF: MPLS Networks**MPLS Switches LSP LSR LSR IP Net IP Net LSR LSR MPLS Network LSR: Label-Switch Router**MCF Specialization**xk Binary l= 0 bk = et - es ODIMCF Variant qk = 1 for all k Binary MCF BMCF**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 RCSP: Current Algorithms RRCSP()**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 RCSP: Proposed Algorithm**Algorithm**Relax Side-Constraint Forming ASP() ASP With sAs 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 RCSP: Proposed Algorithm ASP()**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 RCSP: Proposed Algorithm**RCSP: Proposed Algorithm**Minimum Threat Routing = 0.0 Intermediate Routing Option = 0.44**RCSP: Proposed Algorithm**Minimum Time Routing = 1.0 Accumulated Threat vs Time To Target**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**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**Branch By Forbidding a Set of Arcs For a Commodity**Select Commodity k Find Vertex dAt Which Flow Splits Create 2 Nodes in Tree Each Forbidding ~Half the Arcs at d Has Difficulty Many Commodities |A|/|V| ~2 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**ODIMCF: Proposed Algorithm**• Heuristic Based On Market Prices • Circumstances • Large Sparse Networks • Many Commodities • Arcs Capable of Supporting Multiple Commodities**ODIMCF: Proposed Algorithm**• Arc Costs, cij´ =f(rij, uij, cij, qk)R • Uses Non-Linear Price Curve, p(z, uij) R • Based On • Original Arc Cost, cij • Upper Bound, uij • Current Capacity Usage, rij • Demand of Commodity, qk**ODIMCF: Proposed Algorithm**c´ij = f(rij, uij, cij, qk) As an Area p(z, uij) Demand, qk Current Usage, rij Area = Arc Cost, c´ij Marginal Arc Cost Upper Bound, uij**ODIMCF: Proposed Algorithm**Arc Cost For Increasing rij**ODIMCF: Proposed Algorithm**Total System Cost Total Additional System Cost Additional Cost To Other Commodities Arc Cost To Commodity Current Usage, rij Current System Cost**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 ODIMCF: Proposed Algorithm**Considerations**Form of p(z, uij) Commodity Differentiation Under-Capacitated Net Preferential Routing Selection Strategy Advanced Start Cooling Off of p(z, uij) Step 0 - SP Steps 1… Increasing Enforcement of u ODIMCF: Proposed Algorithm 4 3 2 1 0**ODIMCF: Proposed Algorithm***CPLEX65 Used MIP To Find Integer Solution. All Other Problems Solved As LP Relaxations With No Attempt At Integer Solution.**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**Original Commodities**Demands of 1 Single Origin & Destination SP Aggregations Demands 1 Single Origin Multiple Destinations MCNF BMCF: Proposed Algorithm**Aggregation MCNFs Solved On Modified Network**Each Original Arc Is Replaced With qa Parallel Arcs Parallel Arcs Have Convex Costs Derived From p(z, uij) Upper Bounds of 1 cij i j (0, uij) cij3 (0, 1) cij2 i j (0, 1) cij1 (0, 1) BMCF: Proposed Algorithm**BMCF: Proposed Algorithm**Parallel Arc Costs p(z, uij) Demand, qa = 3 Current Usage, rij cij3 cij2 cij1 Upper Bound, uij One Unit of Flow**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 BMCF: Proposed Algorithm**Expected Contributions**• Will Address Important Problems With Wide Range of Applications • Efficient Algorithms Will Have a Significant Impact in Several Disparate Fields**A - Matrix**x - Vector 0 - Vector of All 0’s 1 - Vector of All 1’s ei - 0 With a 1 at ith Position xi - ith element of x x - Scalar A - Set |A| - Cardinality of A - Empty Set R - Set of Reals B - {0,1}, Binary Set Rmxn - Set of mxn Real Matrices Bm - Set of Binary, m Dimensional Vectors Notation**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 All Networks Are Directed xij Is the Flow Variable for ( i, j) E - Set of Arcs V - Set of Vertices cij , sij i j (lij , uij) Notation: Networks**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 Notation: Problem Abbreviations

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