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Constrained Integer Network Flows - PowerPoint PPT Presentation

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

April 25, 2002

• 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 Flows

MCNF

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

• 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

• 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

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

• 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 MCF

ODIMCF2

ODIMCF1

• 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

• 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

• 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

MPLS Switches

LSP

LSR

LSR

IP Net

IP Net

LSR

LSR

MPLS Network

LSR: Label-Switch Router

xk Binary

l= 0

bk = et - es

ODIMCF Variant

qk = 1 for all k

Binary MCF

BMCF

Current & Proposed Algorithmic Approaches

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()

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

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()

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

Minimum Threat Routing

 = 0.0

Intermediate Routing Option

 = 0.44

Minimum Time Routing

 = 1.0

Accumulated Threat

vs

Time To Target

• Further Considerations

• Normalization of c and s

• Reoptimization of ASP()

• Number of Iterations (Values of )

• Usage As Starting Solution For RCSP

• Other Uses

• 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

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

• Heuristic Based On Market Prices

• Circumstances

• Large Sparse Networks

• Many Commodities

• Arcs Capable of Supporting Multiple Commodities

• 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

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

Arc Cost For Increasing rij

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 Algorithm

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

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

• 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 Algorithm

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

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 Algorithm

• 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

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