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. Definition

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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
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

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

Side-Constraint Destroys Special Structure of MCNF

Solutions Non-Integer Unless Integrality Enforced

Resource-Constrained Shortest Path

RCSP

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 Flow

MCF

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
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

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

MCF Specialization

xk Binary

l= 0

bk = et - es

ODIMCF Variant

qk = 1 for all k

Binary MCF

BMCF

### Current & Proposed Algorithmic Approaches

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()

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
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()

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
RCSP: Proposed Algorithm

Minimum Threat Routing

 = 0.0

Intermediate Routing Option

 = 0.44

RCSP: Proposed Algorithm

Minimum Time Routing

 = 1.0

Accumulated Threat

vs

Time To Target

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
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
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

Arc Cost To Commodity

Current Usage, rij

Current System Cost

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
Considerations

Form of p(z, uij)

Commodity Differentiation

Under-Capacitated Net

Preferential Routing

Selection Strategy

Cooling Off of p(z, uij)

Step 0 - SP

Steps 1… Increasing Enforcement of u

ODIMCF: Proposed Algorithm

4

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
Original Commodities

Demands of 1

Single Origin & Destination

SP

Aggregations

Demands  1

Single Origin

Multiple Destinations

MCNF

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

cij

i

j

(0, uij)

cij3

(0, 1)

cij2

i

j

(0, 1)

cij1

(0, 1)

BMCF: Proposed Algorithm
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

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
Expected 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

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
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