Post on 04-Jul-2015
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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 !