Math443/543 Mathematical Modeling and Optimization

1. Math443/543Mathematical 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 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 maxf(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,…,xnthat • 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? 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 cijassociated with traveling from City i to City j . • Goal: Find a minimum-cost tour.

8. Job 3 Job 1 Job 4 Job 2 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. 2 6 14 0 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