Optimization Process - Discrete Modeling and Optimization - Lecture Slides, Slides of Discrete Structures and Graph Theory

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Math4630/

Discrete Modeling and Optimization

A schematic view of

modeling/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?

Mathematical models in

Optimization

• The general form of an optimization model : min or max f(x 1 ,…,x n ) (objective function) subject to g i (x 1 ,…,x n ) ≥ 0 (functional constraints) x 1 ,…,x n ∈ S (set constraints)
• x 1 ,…,x n are called decision variables
• In words, the goal is to find x 1 ,…,x n 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 = Z n ) Continuous (S = R n ) Linear ( f and g are linear) Nonlinear ( f and g are nonlinear)

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 c ij associated with traveling from City i to City j. Goal: Find a minimum-cost tour.

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 p k . 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 0 2 6 9 14 time

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