Math4630/5630Discrete Modeling and

Optimization

A schematic view of modeling/optimization

process

Real-world problem

Mathematical model

Solution to model

Solution toreal-world

problem

assumptions, abstraction,data,simplifications

optimization algorithm

interpretation

makes sense? change the model,

assumptions?

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

of a system, theory, or phenomenon that

accounts for its known or inferred properties

and maybe used for further study of its characteristics.

• Mathematical models– are abstract models– describe the mathematical relationships

among elements in a system

• In this class, mathematical models dealing

with discrete optimization

Mathematical models in Optimization

• The general form of an optimization model:

min or max f(x1,…,xn) (objective function)

subject to gi(x1,…,xn) ≥ 0 (functional constraints)

x1,…,xn S (set constraints)

• x1,…,xn are called decision variables

• In words,

the goal is to find x1,…,xn that

– satisfy the constraints;– achieve min (max) objective function value.

Types of Optimization Models

Stochastic(probabilistic

information on data)

Deterministic(data are certain)

Discrete, Integer(S = Zn)

Continuous(S = Rn)

Linear(f and g are linear)

Nonlinear(f and g are nonlinear)

What is Discrete Optimization?

Discrete Optimization is a field of applied mathematics,

combining techniques from • combinatorics and graph theory, • linear programming, • theory of algorithms,

to solve optimization problems over discrete structures.

Examples of Discrete Optimization Models: Traveling Salesman Problem Traveling Salesman Problem

(TSP)(TSP)

There are n cities. The salesman

starts his tour from City 1,

visits each of the cities exactly once,

and returns to City 1.

For each pair of cities i,j there is a cost cij associated with traveling from City i to City j .

Goal: Find a minimum-cost tour.

Examples of Discrete Optimization Models: Job SchedulingJob Scheduling

There are 4 jobs that should be processed on the same machine. (Can’t be processed simultaneously).

Job k has processing time pk .Here is an example of a possible schedule:

Goal: Find a schedule which minimizes the average completion time of the jobs.

Job 3 Job 1 Job 4 Job 2

time0 2 6 9 14

Examples of Discrete Optimization Models: Shortest Path ProblemShortest Path Problem

In a network, we have distances on arcs ;source node s and sink node t .

Goal: Find a shortest path from the source to the sink.

s

b

a d

e

tc

1 1

1

1

22

2

3

5

4

7

2

4

Problems that can be modeled and solved by discrete

optimization techniques

• Scheduling Problems (production, airline, etc.)

• Network Design Problems

• Facility Location Problems

• Inventory management

• Transportation Problems

Problems that can be modeled and solved by discrete optimization

techniques• Minimum spanning tree problem

• Shortest path problem

• Maximum flow problem

• Min-cost flow problem

• Assignment Problem

Solution Methods for Discrete Optimization Problems

• Integer Programming

• Network Algorithms

• Dynamic Programming

• Approximation Algorithms