1. Math443/543 Mathematical Modeling and Optimization

2. A schematic view of modeling/optimization process
assumptions, abstraction,data,simplifications
Real-world
problem
Mathematical
model
makes sense? change the model, assumptions?
optimization algorithm
Solution to
real-world
problem
Solution to
model
interpretation

3. What is a model? Model: A schematic description Mathematical models
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

4. Mathematical models in Optimization
The general form of an optimization model:
min or max f(x1,…,xn) (objective function)
subject to gi(x1,…,xn) ≥ (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.

5. 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)

6. What is 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.

7. Examples of Discrete Optimization Models: Traveling Salesman Problem (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.

8. Examples of Discrete Optimization Models: Job 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 2 6 14 9
time

9. Examples of Discrete Optimization Models: Shortest 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. 3 a d 4 1 1 1 4 7 2 s c t 2 2 1 2 5 b e

10. 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

11. 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

12. Solution Methods for Discrete Optimization Problems
Integer Programming
Network Algorithms
Dynamic Programming
Approximation Algorithms