Constrained Integer Network Flows

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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

MCNF

• Special Structure Has Lead To Highly Efficient
  Algorithms

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Shortest-Path Problems

• 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 - Ves

SP/ ASP

• Special Solution Algorithms
• Label Setting
• Label Correcting

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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

RCSP

• Solutions Non-Integer Unless Integrality Enforced

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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

MCF

• Price Resource Directive Decompositions
• Heuristics

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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
ODIMCF2
ODIMCF1

• ODIMCF2 Path-Based Formulation
• Rows K E
• Variables Dependent on Network Structure

• ODIMCF1 Node-Arc Formulation
• Rows VK E
• Variables KE

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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 - StationTrain Arrival/Departure

Stations
Remain at A
A
a2
a3
a4
a1
Catch a Train
B
b2
b5
b1
b3
b4
C
c2
c3
c4
c1
Time
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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
• MPLSs Objective Is Improved Reliability, Lower
  Congestion, Meeting Quality-of-Service (QoS)
  Guarantees

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

• MCF Specialization
• xk Binary
• l 0
• bk et - es
• ODIMCF Variant
• qk 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

RRCSP(?)

• Lagrangian 2
• Bracket Optimal Solution Changing ?
• Finish Off With k-shortest Paths

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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 ?

ASP(?)

• For Each t in T Find Smallest ? Meeting
  Side-Constraint For t

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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

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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, 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

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ODIMCF Proposed Algorithm
cij f(rij, uij, cij, qk) As an Area
p(z, uij)
Demand, qk
Current Usage, rij
Area Arc Cost, cij
Marginal Arc Cost
Upper Bound, uij
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ODIMCF Proposed Algorithm
Arc Cost For Increasing rij
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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
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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
4
3
2

• Considerations
• Form of p(z, uij)
• Commodity Differentiation
• Under-Capacitated Net
• Preferential Routing
• Selection Strategy
• Advanced Start

1
0

• Cooling Off of p(z, uij)
• Step 0 - SP
• Steps 1 Increasing Enforcement of u

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ODIMCF Proposed Algorithm
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ODIMCF Proposed Algorithm
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ODIMCF Proposed Algorithm
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ODIMCF Proposed Algorithm
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

• Aggregations
• Demands ? 1
• Single Origin
• Multiple Destinations
• MCNF

• Original Commodities
• Demands of 1
• Single Origin Destination
• SP

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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

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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
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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 0s
• 1 - Vector of All 1s
• 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

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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

• All Networks Are Directed
• xij 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