Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Heuristics for Network Design Heuristics for Network Design under traffic uncertaintyunder traffic uncertainty

Luca PizziniacoLuca Pizziniaco

Politecnico di MilanoPolitecnico di Milano

AAdvanced dvanced NNetwork etwork TTechnologies echnologies LabLab

Supervised by : Prof. Edoardo Amaldi

OR Gro

up

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

OutlineOutline

The New Approach Optimization model Heuristics Computational Results Economical Impact Conclusions & Future Work

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

The Problem – Classical ApproachThe Problem – Classical Approach

Network Design Allocate Capacity on links Guarantee required Demands Minimize Costs

Assume to know traffic demands Pros: Simpler and “faster” to solve Cons:Traffic is actually well known and

the uncertainty is not accounted

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

The Problem – Stochastic ApproachThe Problem – Stochastic Approach

Network Design Different Scenarios involved

Different traffic matrix in each scenario Probability associated to each scenario

Two phase stochastic programming for modeling the problem Pro: Take into account traffic

characteristic Cons: Huge and very challenging

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Problem OverviewProblem Overview

Given some traffic distributions at the enter of a network

An initial budget for buy capacity over various link in the network

P,DP,D

P,DP,D

P,DP,D

P,DP,D

MaximizeMaximize

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Investor Point of ViewInvestor Point of View

Minimize the costs is not enough Investor points

Minimize costs Guarantee the customers demands Maximize the revenue Analyze investment risk

Trade-Off between earns and risks

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

New ApproachNew Approach

No approach in literature take into account the investor points of view

Our Goals Maximize the Expected ThroughputExpected Throughput Knowing the demands distribution for

each scenario Subject to Limited Budget

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

MIP Probabilistic Model MIP Probabilistic Model

The variable into the model are xijmk : flow on the arc (i,j) for the possible

demand m for the scenario k rij : allocated capacity for arc (i,j)

Φk : throughput for the scenario k

qk : decision variable on the scenario k 1 if scenario k is used for the demensioning 0 otherwise

W is a positive number

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

MIP Probabilistic Model (cont.)MIP Probabilistic Model (cont.)

Expected Throughput

Budget Constraint

Balance Constraint

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

MIP Probabilistic Model (Cont.)MIP Probabilistic Model (Cont.)

Capacity Allocation

Flow Limit Constraint

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Problem SizeProblem Size

Graph G=(V,A) |V| = n (number of nodes) |E| = e (number of edges) s : number of sources t : number of sinks k : number of scenarios m : number of possible demands values

in every scenario MIP Model with size CALCOLAREMIP Model with size CALCOLARE

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Euristhical ApproachEuristhical Approach

Three Heuristics based on EU1 – Scenario Analysis EU2 – Links Analysis EU3 – Paths Analysis

Drastic Reduction in computing time Provide Various information about the

network

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Scenario Based EuristicScenario Based Euristic

Sort in list L all the scenarios on a Metric in accordance with a metric

While L = {} or B > 0 Extract from list Lfirst scenario and allocate

capacity for it Erase dominated scenarios from the list Update the budget B

Score is assigned to each scenario Scenarios are sorted in a List

List maintain only dominating scenario

Complexity : O(KS), where S = Algorithm Complexity for design a scenario

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Links Based EuristicLinks Based Euristic

Allocate capacity to support the scenario with max demands

While dimensioning cost > Budget Foreach link evaluate the probability to use it Decrease by one unit the capacity of the link with

smaller probability Recompute the costs

Supposes that there is no budget for supporting scenario with max demand

Complexity : O((CComplexity : O((Cmaxmax-B)K-N-B)K-N33), where ), where Cmax-B = Maximum cost - Budget

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Path Based EuristicPath Based Euristic

Foreach dst find a path of minimum cost While Budget > Minst{Cst}

Foreach link (i,j) in Pathst such that the probability to be used is maximum Add a unit flow capacity Update the budget

Complexity : O(N2), since Djikstra is used to find paths for every demand dst - CONTROLLARE

VERY

FAS

T HEU

RIST

IC

VERY

FAS

T HEU

RIST

IC

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

A first ResultA first Result

We ese a simple network istance 10 nodes 16 links 1000 scenarios

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

A First Result (cont.)A First Result (cont.)

Exp

ecte

d T

hro

ughput

Budget

Heuristic 2 - 3 give very close results

Heuristic 1 is worst when budget is low

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

CPU Time ComparisonCPU Time Comparison

As Expected: Eu2 is heavyer for low budget value Eu3 is the most efficient

Eu2

Eu1

Eu3

Budget

CPU

Tim

e

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Heuristics QualityHeuristics Quality

How good our heuristics are? Opt

imal

Very Good and Very Good and Pretty Fast!Pretty Fast!

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

The The ScroogeScrooge Investor Investor

Budget constraint is too strong Eu2 assumes the budget is too low

for support all scenario Investor wants to maximize revenue

while paying less Employ all money is always good?

Preserved money could invested elsewhere..

..or used for make creditors happy

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

ImprovementsImprovements

Analyze the derivative Low slopeLow slope

Similar TP values Different B Values

Medium/High slopeMedium/High slope Different TP values Similar B Values

Heuristic 2

Heu

ristic

3

Heur

istics

Mov

es

Heur

istics

Mov

es

Continue to move until derivative reach a certain value

Thro

ughput

Budget

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Some modificationSome modification

Eu2 Price-Quality Approach Continue to iterative delete flow capacity

from the link with small probability Store the Price-Quality ratio (PQR) for each

iteration in time “t” Quality : Expected throughput Price : Dimensioning Cost

Stop when there is no more convenience PQR(t+1) < PQR(t)

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Rusults Rusults

Investor will pay less but he can guarantee the service to the customer

Good service means good revenueGood service means good revenue

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Investments AnalysisInvestments Analysis

RES9RES9

Increase the budget means increase the Increase the budget means increase the revenue and the investments risksrevenue and the investments risks

Low Budget

High Budget

Throughput

Probab

ility

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

ConclusionConclusion

Summary New approach for network design with

economical aspect New heuristics for solving problem in

pretty fast also on huge instances Future Works

Reduce the search space Can this methodology be applied to other

problem of different nature?

Minor Thesis Luca Pizziniaco - Heuristics Under Traffic Uncertainty

Alea Iacta EstAlea Iacta Est

Thanx a Lot… Time for question…if you have !