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Math443/543Mathematical Modeling and

Optimization

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A schematic view ofmodeling/optimization process

Real-world

problem

Mathematical

model

Solution to

model

Solution to

real-world

problem

assumptions,abstraction,data,

simplifications

optimization

algorithm

interpretation

makes sense?

change the model,

assumptions?

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What is a model?What is a model? Model: A schematic description

of a sstem, theor, or phenomenon that

accounts for its known or inferred properties

and mabe used for further stud of its characteristics!

Mathematical models

"are abstract models

"describe the mathematical relationships

among elements in a sstem

#n this class,mathematical models dealing

with discrete optimization

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Mathematical models inOptimization

$he general form of an optimization model:

minor maxf%&',(,&n) %ob*ecti+e function)

subject to gi%&',(,&n) 0 %functional constraints)

&',(,&nS %set constraints)

&',(,&n are calleddecision +ariables

#n words,the goal is to find &',(,&nthat

"satisf the constraints

"achie+e min %ma&) ob*ecti+e function +alue!

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Types ofOptimization Models

Stochastic

%probabilistic

information on data)

Deterministic

%data are certain)

Discrete, Integer

%S Zn)

Continuous

%S Rn)

Linear

%f andgare linear)

Nonlinear

%f andgare nonlinear)

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What is iscrete Optimization?

.iscrete /ptimization

is a field of applied mathematics,

combining techni0ues from combinatorics and graph theor,

linear programming,

theor of algorithms,to sol+e optimization problems

o+er discrete structures!

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1&amples of .iscrete /ptimization

Models: Traveling !alesman "ro#lemTraveling !alesman "ro#lem

$T!"%$T!"%

$here are n cities! $he salesman

starts his tour from 2it ',+isits each of the cities e&actl once,

and returns to 2it '!

3or each pair of cities i,* there is a cost cijassociated with tra+eling from 2it i to 2it * !

4oal:3ind a minimum-cost tour!

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1&amples of .iscrete /ptimization

Models: &o# !ched'ling&o# !ched'ling

$here are 5*obs that should be processed on the samemachine! %Cant be processed simultaneously)!

6ob khas processing time pk!

7ere is an e&ample of a possible schedule:

4oal:3ind a schedule which minimizes

the a+erage completion time of the *obs!

Job 3 Job 1 Job 4 Job 2

time

89

;'5

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1&amples of .iscrete /ptimization

Models: !hortest "ath "ro#lem!hortest "ath "ro#lem

#n a network, we ha+e distances on arcs

source nodesand sink node t.

4oal:3ind a shortest path from the source to the sink!

s

b

a d

e

tc

' '

'

'

9

9

9

9

5

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"ro#lems that can #e modeledand solved #y discrete

optimization techni('es

Scheduling roblems %production, airline, etc!)

@etwork .esign roblems

3acilit ocation roblems

#n+entor management

$ransportation roblems

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"ro#lems that can #e modeledand solved #y discreteoptimization techni('es

Minimum spanning tree problem

Shortest path problem Ma&imum flow problem

Min-cost flow problem

Assignment roblem

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!ol'tion Methods for iscrete

Optimization "ro#lems #nteger rogramming

@etwork Algorithms

.namic rogramming

Appro&imation Algorithms