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Constrained Integer Network FlowsPowerPoint Presentation

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

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 FlowsMCNF

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 ProblemsSP/ 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 PathRCSP

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 FlowMCF

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

Rows: |V||K| + |E|

Variables: |K||E|

ODIMCF2: Path-Based Formulation

Rows: |K| + |E|

Variables: Dependent on Network Structure

Origin-Destination Integer MCFODIMCF2

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

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 AlgorithmsRRCSP()

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 AlgorithmRelax 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 AlgorithmASP()

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 AlgorithmRCSP: 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

- Column-Generation

- Solve LP Relaxation At Each Node Using:

- Algorithmic Steps

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

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 AlgorithmForm 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 Algorithm4

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

Demands of 1

Single Origin & Destination

SP

Aggregations

Demands 1

Single Origin

Multiple Destinations

MCNF

BMCF: Proposed AlgorithmAggregation 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 AlgorithmBMCF: Proposed Algorithm

Parallel Arc Costs

p(z, uij)

Demand, qa = 3

Current Usage, rij

cij3

cij2

cij1

Upper Bound, uij

One Unit of Flow

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 AlgorithmExpected 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

NotationA - 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: NetworksMCNF - 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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