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. 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

4. 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

5. 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

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. 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

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. 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

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 Remain at A A a1 a2 a3 a4 Catch a Train B b1 b2 b3 b4 b5 C c1 c2 c3 c4 Time

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

15. MCF Specialization xk Binary l= 0 bk = et - es ODIMCF Variant qk = 1 for all k Binary MCF BMCF

17. Current & Proposed Algorithmic Approaches

18. 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()

19. 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

20. 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()

21. 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

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

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. 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

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, 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

29. 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

30. ODIMCF: Proposed Algorithm Arc Cost For Increasing rij

31. 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

32. 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

33. 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

34. ODIMCF: Proposed Algorithm

35. ODIMCF: Proposed Algorithm

36. ODIMCF: Proposed Algorithm

37. ODIMCF: Proposed Algorithm *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. Original Commodities Demands of 1 Single Origin & Destination SP Aggregations Demands  1 Single Origin Multiple Destinations MCNF BMCF: Proposed Algorithm

40. 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

41. 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

42. 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

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

44. 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

45. 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

46. 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