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Constrained Integer Network Flows April 25, 2002.

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Presentation on theme: "Constrained Integer Network Flows April 25, 2002."— Presentation transcript:

1 Constrained Integer Network Flows April 25, 2002

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

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

15 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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17 Current & Proposed Algorithmic Approaches

18 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(  )

19 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

20 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( )

21 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

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

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

24 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

25 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

26 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

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

28 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

29 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

30 ODIMCF: Proposed Algorithm Arc Cost For Increasing r ij

31 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

32 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

33 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

34 ODIMCF: Proposed Algorithm

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

38 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

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

40 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

41 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

42 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

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

44 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

45 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

46 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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